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CHAPTER 1
INTRODUCTION Larry W. Mays Department of Civil and Environmental Engineering Arizona State University Tempe, Arizona
1.1 OVERVIEW Since the Egyptian’s and Mesopotamian’s first successful efforts to control the flow of water thousands of years ago, a rich history of hydraulics has evolved. Sec. 1.2 contains a brief description of some ancient hydraulic structures that are found around the world. During the 20th century, many new developments have occurred in both theoretical and applied hydraulics. A number of handbooks and textbooks on hydraulics have been published, as indicated in Fig. 1.1. From the viewpoint of hydraulic design, however, only manuals, reports, monographs, and the like have been published, mostly by government agencies. Unfortunately, many aspects of hydraulic design have never been published as a compendium. This Hydraulic Design Handbook is the first effort devoted to producing a comprehensive handbook for hydraulic design. The book covers many aspects of hydraulic design, with step-by-step procedures outlined and illustrated by sample design problems.
1.2 ANCIENT HYDRAULIC STRUCTURES 1.2.1 A Time Perspective Although humans are newcomers to earth, their achievements have been enormous. It was only during the Holocene epoch (10,000 years ago) that agriculture developed (keep in mind that the earth and the solar system originated 4,600 million years ago). Humans have spent most of their history as hunters and food-gatherers. Only in the past 9,000 to 10,000 years have humans discovered how to raise crops and tame animals. Such changes probably occurred first in the hills to the north of present-day Iraq and Syria. The remains of the prehistoric irrigation works in Mesopotamia and Egypt still exist. Table 1.1 presents a chronology of knowledge about water. Figure 1.2 illustrates the chronology and locations of various civilizations ranging from India to Western Europe. This figure, from O. Neugebaur’s book titled The Exact Sciences in Antiquity, illustrates the Hellenistic period the era of “ancient science,” during which a form of science developed that spread later from Europe to India. This ancient science was dominant until the creation of modern science dominant in Isaac Newton’s time. 1.1 Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
INTRODUCTION
1.2
Chapter One
Abbott’s Computational Hydraulics (1980)
Fischer et al., Mixing in Inland and Coastal Waters (1979)
1980 Freeze and Cherry’s Groundwater (1979)
Graf’s Hydraulics of Sediment Transport (1971)
1970 Streeter and Wylies’ Hydraulic Transients (1967)
U.S. Geological Survey’s Roughness Characteristics of Natural Channels (1967)
Hendersons’ Open-Channel Flow (1966) Leliavsky’s River and Canal Hydraulics (1965) Morris and Wiggert’s Applied Hydraulics in Engineering (1963)
USBR Design of Small Dams (1960)
Daily and Harleman’s Fluid Dynamics (1966)
Linsley and Franzini’s Elements of Hydraulic Engineering (1964)
1960
Chow’s Open-Channel Hydraulics (1959) U.S. Bureau of Reclamation’s Hydraulic Design of Stilling Basin and Energy Dissipators (1958) Stoker’s Water Waves (1957) Parmakiams’ Waterhammer Analysis (1955) King’s Handbook of Hydraulics (1954)
Leliavsky’s An Introduction to Fluvial Hydraulics (1955) Addison’s Treastise on Applied Hydraulics (1954)
U.S. Bureau of Reclamation’s Hydraulic Laboratory Practice (1953) Rich’s Hydraulic Transients (1951) Rouse’s Engineering Hydraulics (1950)
1950
FIGURE 1.1 A selected list of books on hydraulics published between 1900 to 1980.
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INTRODUCTION
Introduction 1.3 1950 Allen’s Scale Models in Hydraulic Engineering (1947)
ASCE’s Hydraulic Models (1942)
Davis and Sorersen’s Handbook of Applied Hydraulics (1942)
Woodward and Posey’s Hydraulics of Steady Flow in Open Channels (1941) 1940 Rouse’s Fluid Mechanics for Hydraulic Engineers (1938) Muskat’s The Flow of Homogeneous Fluids Through Porous Media (1937)
Daugherty’s Hydraulics (1937) Bakhmeteff’s The Mechanics of Turbulent Flow (1936)
Bakhmeteff’s Hydraulics of Open Channels (1932) 1930 Schoder and Dawson’s Hydraulics (1927) Le Conte’s Hydraulics (1926)
1920
Hoyt and Grover’s River Discharge (1916) Hoskins’s A Text–Book on Hydraulics (1911) 1910
Merriman’s Treatise on Hydraulics (1904)
1900 FIGURE 1.1 (Continued)
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INTRODUCTION
1.4
Chapter One TABLE 1.1
Chronology of Knowledge About Water
Prehistorical period 3rd –2nd millennium B.C. 3rd millennium B.C. 3 millennium B.C. Probably very early† 2nd millennium B.C. 8th-6th c. B.C.
6th c. B.C. at the latest
6th c. B.C. at the latest
6th-3rd c. B.C.
Springs Cisterns Dams Wells Reuse of excrement as fertilizer Gravity flow supply pipes or channels and drains, pressure pipes (subsequently forgotten) Long-distance water supply lines with tunnels and bridges, as well as intervention in and harnessing of karst water systems Public as well as private bathing facilities, consisting of: bathtubs or showers, footbaths, washbasins, latrines or toilets, laundry and dishwashing facilities Use of definitely two and probably three qualities of water: potable, subpotable, and nonpotable, including irrigation using storm runoff, probably combined with waste waters Pressure pipes and siphon systems
*Indicates an element discovered, probably forgotten, and rediscovered later. †Indicates an educated guess. Source: Crouch, 1993.
1.2.2 Irrigation Systems 1.2.2.1 Egypt and Mesopotamia. In ancient Egypt, the construction of canals was a major endeavor of the Pharaohs beginning in Scorpio’s time. Among the first duties of provincial governors was the digging and repair of canals, which were used to flood large tracts of land while the Nile was flowing high. The land was checkerboarded with small basins defined by a system of dikes. Problems associated with the uncertainty of the Nile’s flows were recognized. During high flows, the dikes were washed away and villages were flooded, drowning thousands of people. During low flows, the land was dry and no crops could grow. In areas where fields were too high to receive water directly from the canals, water was drawn from the canals or from the Nile by a swape or shaduf (Fig. 1.3), which consisted of a bucket on the end of a cord hung from the long end of a pivoted boom that was counterweighted at the short end (de Camp, 1963). Canals continued to be built in Egypt throughout the centuries. The Sumerians in southern Mesopotamia built city walls and temples and dug canals that were the world’s first engineering works. It also is of interest that these people, fought over water rights from the beginning of recorded history. Irrigation was vital to Mesopotamia, Greek for “the land between the (Tigris and Euphrates) rivers.” An ancient Babylonian curse was, “May your canal be filled with sand” (de Camp, 1963), and even their ancient laws dealt with canals and water rights. The following quotation from approximately the sixth century B.C., illustrates such a law (de Camp, 1963): “The gentleman who opened his wall for irrigation purposes, but did not make his dyke strong and hence caused a flood and inundated a field adjoining his, shall give grain to the owner of the field on the basis of those adjoining.” Because the Tigris and Euphrates carried several times more silt per unit volume of water than the Nile did, flooding problems were more serious in Mesopotamia than in Egypt. As a result the rivers in Mesopotamia rose faster and changed course more often.
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INTRODUCTION
Introduction 1.5
FIGURE 1.2 Chronology and location of different civilizations ranging from India to Western Europe. (Neugebauer, 1993)
The irrigation systems in both Mesopotamia and the Egyptian Delta were of the basin type, opened by digging a gap in the embankment and closed by placing mud back into the gap. (See Fig. 1.4 for a comparison of the irrigation works in Upper Egypt and in Mesopotamia.) Water was hoisted using the swape, Mesopotamian laws required farmers to keep their basins and feeder canals in repair; they also required everyone else to wield hoes and shovels when the rivers flooded or when new canals were required or old ones needed repair (de Camp, 1963). Some canals may have been used for 1,000 years before
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INTRODUCTION
1.6
Chapter One
FIGURE 1.3 Shadufs of the Amarna period, from the tomb of Nefer-Hotep at Thebes. Note irrigation of date palms and other orchard trees and the apparent tank or pool (lower right). The water pattern in the lowest margin suggests lifting out of an irrigation canal. (Davies, 1933, pls. 46 and 47). Figure as presented in Butzer (1976).
they were abandoned and others were built. Even today, 4,000 to 5,000 years later, the embankments of the abandoned canals remain. In fact, these canal systems supported a larger population than lives there today. Over the centuries, Mesopotamian agriculture began to decline because of the salty alluvial soil. In 1258, the Mongols conquered Mesopotamia and destroyed its irrigation systems.
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INTRODUCTION
Introduction 1.7
FIGURE 1.4 Comparative irrigation networks in Upper Egypt and Mesopotamia. A. Example of linear, basin irrigation in Sohag province, ca. AD 1850. B. Example of radial canalization system in the lower Nasharawan region southeast of Baghdad, Abbasid (A.D. 883–1150). Modified from R. M. Adams (1965, (Fig. 9) Same scale as Egyptian counterpart) C. Detail of field canal layout in B. (Simplified from R. M. Adams, 1965, Fig. 10). Figure as presented in Butzer (1976).
The Assyrians also developed extensive pubic works. When Sargon II invaded Armenia in 714 B.C., he discovered the ganãt (Arabic) or kariz (Persian), a system of tunnels used to bring water from an underground source in the hills down to the foothills (Fig. 1.5). Sargon destroyed the system in Armenia but brought the concept back to Assyria. Over the centuries, this method of irrigation spread across the Near East into North Africa and is
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INTRODUCTION
1.8
Chapter One
FIGURE 1.5 Details of the ganãt system. (Biswas, 1970).
still used. Sargon’s son Sennacherib also developed waterworks by damming the Tebitu River and using a canal to bring water to Nineveh, where the water could be used for irrigation without the need for hoisting devices. During high water in the spring, overflows were handled by a municipal canebrake that was built to develop marshes used as game preserves for deer, wild boar, and birds. When this system was outgrown, a new canal 30 mi long was built, with an aqueduct that had a layer of concrete or mortar under the upper layer of stone to prevent leakage. 1.2.2.2 Prehistoric Mexico. During the earliest years of canal irrigation in Mexico, the technology changed little (Fig. 1.6) and the method of flooding tended to be haphazard. The technological achievements were relatively primitive until about 600 or 500 B.C., and few of the early systems remain. Whereas the earlier systems were constructed of loosely piled rocks, the later ones consisted of storage dams constructed of blocks that were mortared together. Some spillways were improved, and floodgates were used in some spillways. (Some dams could be classified as arch dams.) The canals were modified to an extent during this time: Different cross-sectional areas were
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INTRODUCTION
Introduction 1.9
FIGURE 1.6 Regional chronology and dates of developments in various aspects of canal irrigation technology in Mexico. (Doolittle, 1990)
used, some were lined with stone slabs, and the water for irrigation of crops was more carefully controlled. Between 550 and 200 B.C., the irrigation-related features and the entire canal systems were significantly improved. The channelization of stream beds, the excavation of canals, and the construction of dams were probably the most significant improvements. However, the technology stopped improving after 200 B.C., and no significant developments occurred for approximately 500 years. Around 300 A.D., a few new improvements were initiated, but the technology remained essentially the same through the classic period (A.D., 200 – 800/1000) and early postclassical period (A.D. 800/1000–1300). Figure 1.7 is a map of fossilized canals in the Tehuacan Valley in Mexico.
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INTRODUCTION
1.10
Chapter One
FIGURE 1.7 Map of fossilized canals on the Llano de la Taza in the Tehuacan Valley. (Woodbury and Neely, 1972, as presented in Doolittle, 1990)
1.2.2.3 North America. The canal irrigation systems in the Hohokam and Chaco regions stand out as two major prehistoric developments in the American Southwest (Crown and Judge, 1991). The two systems expanded over broad geographic areas of similar size (the Hohokam in Arizona and the Chacoans in New Mexico). Although they were developed at similar times, they apparently functioned independently. Because the two systems evolved in different environments, their infrastructures also differed considerably. The Hohokam Indians inhabited the lower Salt and Gila River valleys near Phoenix, Arizona. Although the Indians of Arizona began limited farming nearly 3000 years ago, construction of the Hohokam irrigation systems probably did not begin until the first few centuries A.D. Who originated the idea of irrigation in Arizona, whether the technology was developed locally or it was introduced from Mexico, is unknown. Figure 1.8 illustrates the extensive system in the Phoenix area, and Fig. 1.9 provides a schematic of the details of its major components. In approximately 1450 A.D., the Hohokam culture declined, possibly for a combination of reasons: flooding in the 1080s, hydrologic degradation in the early 1100s, and the recruitment of laborers by surrounding populations. The major flood in 1358 ultimately destroyed the canal networks, resulting in movement of the people. Among the Pima Indians, who were the successors of the Hohokam Indians, use of canals was either limited or absent. Although the prehistoric people who lived outside the area of Hohokam
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FIGURE 1.8 Canal building in the Salt River Valley with a stone hoe held in the hand without a handle. These were the original engineers, the true pioneers who built, used, and abandoned a canal system when London and Paris were a cluster of wild huts. Turney (1922) (Courtesy of Salt River Project, Phoenix, Arizona)
INTRODUCTION
1.11
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INTRODUCTION
1.12
Chapter One
FIGURE 1.9 Schematic representation of the major components of a Hohokam irrigation system in the Phoenix Basin. (Masse, 1991)
culture also constructed irrigation systems, none approached the grand scale of the Hohokam systems. In the ninth century, the Anasazi people of northwestern New Mexico developed a cultural phenomenon, the remains of which currently consist of more than 2400 archaeological sites and nine towns, each containing hundreds of rooms, along a 9-mi stretch. The Chacoan irrigation system is situated in the San Juan Basin in northwestern New Mexico. The basin has limited surface water, most of it discharge from ephemeral washes and arroyos. Figure 1.10 illustrates the method of collecting and diverting runoff throughout Chaco Canyon. The water collected from the side canyon that drained from the top of the upper mesa was diverted into a canal by either an earthen or a masonry dam near the mouth of the side canyon (Vivian, 1990). These canals averaged 4.5 m in width and 1.4 m in depth; some were lined with stone slabs and others were bordered by masonry walls. The canals ended at a masonry head gate, where water was then diverted to the fields in small ditches or to overflow ponds and small reservoirs.
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INTRODUCTION
Introduction 1.13
FIGURE 1.10 Hypothetical reconstruction of the Rincon–4 North water control system in Chaco Canyon. Similar systems were located at the mouths of all northern side conyons in the lower 15 m of Chaco Canyon. (Adapted by Ron Beckwith from Vivian, 1974, Fig. 9.4)
1.2.3 Dams The Sadd-el-Kafara dam in Egypt, situated on the eastern bank of the Nile near Heluan approximately 30 km south of Cairo, in the Wadi Garawi, has been referred to as the world’s oldest large dam (Garbrecht, 1985). The explorer and geographer George Schweinfurth rediscovered this dam in 1885, and it has been described in a number of publications since that time (see Garbrecht, 1985). It was built between 2950 and 2690 B.C. Although the Jass drinking-water reservoir in Jordon and the diversion dams on the Kasakh River in Russia are probably older, they are much smaller than the Sadd-el-Kafara (Dam of the Pagans). It is unlikely that the Sadd-el-Kafara dam was built to supply water for drinking or irrigation because the dam lies too far from the alabaster quarries situated upstream to have supplied the labor force with drinking water. Furthermore, there is a vast supply of water and fertile land in the nearby Nile valley. The apparent purpose of the dam was to protect installations in the lower wadi and the Nile valley from frequent, sudden floods. The dam was destroyed during construction by a flood; consequently, it was never completed. To date, the dam’s abutments still exist.
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INTRODUCTION
1.14
Chapter One
The dam had an impervious core consisting of rubble, gravel, and weathered material. On both the upstream and downstream sides, the core was bordered by sections of rockfill that supported and protected the core. The diameter of the stones ranged from 0.1 to 0.6 m. One remarkable construction feature is the facing of the section of rockfill where parts of the facing on the upstream side are still well preserved. The dam had an approximate crest length of 348 ft and a base length of 265 ft and was built straight across the wadi at a suitably narrow point, with a maximum height of 32 ft above the valley bed. See Smith (1971) and Upton (1975) for more on dams. Dam building in the Americas began in the pre-Colombian period in the civilizations of Central and South America: the Aztecs in Mexico, the Mayans in Guatemala and Yucatan, and the Incas in Peru. Where as old-world civilizations developed in the valleys of the big rivers, the Nile River, the Euphrates and the Tigris Rivers, the Indus River, and the Yellow River, most of the early civilizations in the New World were not river civilizations. In South America, the civilizations appeared in the semiarid highlands and the arid coastal valleys traversed by small rivers. In Central America, the Mayans, the Aztecs, and the predecessors of the Aztecs were not river civilizations. The Mayans did not practice irrigation; however, they did provide efficient water supplies to several of their large cities. They developed the artificial well (cenote), the underground cistern (chultun), and the large open reservoir (aguado). The Mayans’ failure to develop irrigation may have accelerated their decline. In the Yucatan, the aguados are still found in some places, but the cenote was the major source of water for drinking and bathing.
1.2.4 Urban Water Supply and Drainage Systems Knossos, approximately 5 km from Herakleion, the modern capital of Crete, was among the most ancient and unique cities of the Aegean and Europe. The city was first inhabited shortly after 6000 B.C. and, within 3000 years, it had became the largest Neolithic Settlement in the Aegean (Neolithic age, circa 5700–2800 B.C). During the Bronze age (circa 2800–1100 B.C.), the Minoan civilization developed and reached its culmination as the first Greek cultural miracle of the Aegean world. The Minoan civilization has been subdivided into four periods: the prepalatial period (2800–1900 B.C.), the protopalatial period (1900–1200 B.C.), the neopalatial period (1700–1400 B.C.), and the postpalatial period (1400–1100 B.C.). During the prepalatial period, a settlement at Knossos; was leveled to erect a palace. Little is known about the old palace because it was destroyed in approximately 1700 B.C. A new palace was constructed on leveled fill from the old palace. During the neopalatial period, Knossos was at the height of its splendor. The city covered an area of 75,000 to 125,000 m2 and had a population estimated to be on the order of tens of thousands. The irrigation and drainage systems at Knossos were most interesting. An aqueduct supplied water through tubular conduits from the Kounavoi and Archanes regions and branched out into the city and the palace. Figure 1.11 shows the type of pressure conduits used within the palace for water distribution. The drainage system consisted of two separate conduits: one to collect the sewage and the other to collect rain water (Fig. 1.12). Unfortunately, the Mycenean palace was destroyed by an earthquake and fire in approximately 1450 B.C., as were all the palatial cities of Crete. Anatolia, also called Asia Minor, which is part of the Republic of Turkey, has been the crossroads of many civilizations during the past 10,000 years. During the last 4000 years, going back to the Hittite period (2000–200 B.C.) many remains of ancient urban watersupply systems have been found, including pipes, canals, tunnels, inverted siphons, aqueducts, reservoirs, cisterns, and dams. (see Ozis, 1987 and Ozis and Harmancioglu, 1979).
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INTRODUCTION
Introduction 1.15
FIGURE 1.11 Water distribution pipe in Knossos, Crete. (Photograph by L.W. Mays)
FIGURE 1.12 Urban drainage system in Knossos, Crete. (Photograph by L.W. Mays)
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INTRODUCTION
1.16
Chapter One
An example of one such city is Ephesus, which was founded during the 10th century as an Ionian city out of the Temple of Artemis. In the sixth century B.C., the city settled near the temple, and subsequently was reestablished at its present site, where it developed further during the Roman period. Water was supplied to Ephesus from springs at different sites. Cisterns also supplied well water to the city. Water for the great fountain, built between 4 and 14 A.D., was diverted by a small dam at Marnss and was conveyed to the city by a system 6 km long consisting of one large and two small clay pipe lines. Figure 1.13 shows the type of clay pipes used at Ephesus to distribute water. B.C.
FIGURE 1.13 Water distribution pipe in Ephesus, Turkey. (Photograph by L. W. Mays)
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INTRODUCTION
Introduction 1.17
The latrine, or public toilet shown in Fig. 1.14, was built in the first century A.D. at Ephesus. The toilets were placed side by side with no partitions. In the middle was a square pond, and the floors were paved with mosaics. The Great Theatre at Ephesus, the city’s largest and most impressive building, had a seating capacity for 24,000 people. Built in the Hellenistic period, the theatre was not only a monumental masterpiece but during the early days of Christianity, one major confrontation between Artemis and Christ took place there. Of notable interest from a waterresources viewpoint is the theatre’s intricate drainage system. Figure 1.15 shows a drainage channel in the floor of the theatre. Public baths also were a unique feature in ancient cities: for example, the Skolactica baths in Ephesus had a salon and central heating; a hot bath (caldarium), a warm bath (tepidarium), and a cold bath (frigidarium); and a dressing room (apodyterium). In the second century A.D., the first building had three floors. In the fourth century, a woman named Skolacticia modified the baths, making them accessible to hundreds of people. There were public rooms and private rooms, and people who wished to could stay for many days. Hot water was provided by a furnace and a large boiler. Perge is another ancient city in Anatolia that had a unique urban water infrastructure. The photographs in Fig. 1.16 illustrate the Majestic Fountain (nymphaion), which consisted of a wide basin and a richly decorated architectural facade. Because of its architecture and statues, the fountain was one of Perge’s most magnificent edifices. A water channel ran along the middle, dividing each street and bringing life and coolness to the city. The baths of Perge were magnificent. The first photograph in Fig. 1.17 shows one of the baths of Perge; the second photograph illustrates the storage of water under the floor to keep the water warm. Like the baths in other ancient cities in Anatolia, the baths of Perge had a caldarium, a tepidarium, and a frigidarium.
FIGURE 1.14 A latrine, or public toilet, built at Ephesus, Turkey, in the first century B.C. (Photograph by L. W. Mays)
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INTRODUCTION
1.18
Chapter One
FIGURE 1.15 A drainage channel on the floor of the Great Theater at Ephesus, Turkey. (Photograph by L. W. Mays)
The early Romans devoted much of their time to useful public works projects, including roads, harbor works, aqueducts, temples, forums, town halls, arenas, baths, and sewers. The prosperous early Roman bourgeois typically had a 12–room house, with a square hole in the roof to let rain in and a cistern beneath the roof to store the water. Although the Romans built many aqueducts, they were not the first to do so. King Sennacherio built aqueducts, as did the Phoenicians and the Helenes. The Romans and Helenes needed extensive aqueduct systems for their fountains, baths, and gardens. They also realized that
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INTRODUCTION
Introduction 1.19
FIGURE 1.16 Two views of the Majestic Fountain (nymphaion) in Perge, Anatolia, Turkey. (Photographs by L. W. Mays)
water transported from springs was better for their health than river water and that spring water did not need to be lifted to street level as did river water. Roman aqueducts were built on elevated structures to provide the needed slope for water flow. Knowledge of pipe
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INTRODUCTION
1.20
Chapter One
FIGURE 1.17A View of the baths at Perge, Anatolia, Turkey. (Photographs by L.W. Mays)
making–using bronze, lead, wood, tile, and concrete–was in its infancy, and the difficulty of making strong large pipes was a hinderance. Most Roman piping was made of lead, and even the Romans recognized that water transported by lead pipes was a health hazard. The source of water for a typical Roman water supply system was a spring or a dug well, which usually was equipped with a bucket elevator to raise the water. If the well
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INTRODUCTION
Introduction 1.21
FIGURE 1.17B View of the baths at Perge, Anatolia, Turkey. (Photographs by L.W. Mays)
water was clear and of sufficient quantity, it was conveyed to the city by aqueduct. Also, water from several sources was collected in a reservoir, then conveyed by an aqueduct or a pressure conduit to a distributing reservoir (castellum). Three pipes conveyed the water: one to pools and fountains, one to the public baths for public revenue, and one to private houses for revenue to maintain the aqueducts (Rouse and Ince, 1957). Figures 1.18 and 1.19 illustrate the layout of the major aqueducts of ancient Rome. Figure 1.20 shows the Roman aqueduct in Segovia, Spain, which is probably among the most interesting of Roman remains in the world. This aqueduct, built during the second half of the first century A.D. or the early years of the second century, has a maximum height of 78.9 m. See Van Deman (1934) for more details on Roman aqueducts. Irrigation was not a major concern because of the terrain and the intermittent rivers. However, the Romans did, drain marshes to obtain more farmland and to eliminate the bad air, or “harmful spirits,” rising from the marshes because they believed it caused disease (de Camp, 1963). The disease-carrying mechanism was not the air, (or spirits) but the malaria-carrying mosquito. Empedocles, the leading statesman of Acragas in Sicily during the Persian War in the fifth century B.C., drained the local marshes of Selinus to improve the people’s health (de Camp, 1963). The fall of the Roman Empire extended over a 1000-year period of transition called the Dark Ages during which the concepts of science related to water resources probably retrogressed. After the fall of the Roman Empire, clean water, sanitation, and public health declined in Europe. Historical accounts tell of incredibly unsanitary conditions: polluted water, human and animal wastes in the streets, and water thrown out of windows onto passersby. As a result, various epidemics ravaged Europe. During the same period, the Islamic cultures on the periphery of Europe religiously mandated high levels of personal hygiene, highly developed water supplies, and adequate sanitation systems. For furthen reading see Needham (1959) Payne (1959), Reynolds (1970) Robbins (1946), Sarton (1952-59) and Wittfogel (1956).
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INTRODUCTION
1.22
Chapter One
FIGURE 1.18 Termini of the major aqueducts in ancient Rome. (Evans, 1993)
FIGURE 1.19 The area of Spes Vetus showing the courses of the major aqueducts entering the city above ground. (From R. Lanciani, Forma Urbis Romae), as presented in Evans (1993).
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INTRODUCTION
Introduction 1.23
FIGURE 1.20 Roman aqueduct in Segovia, Spain. (Photograph by L.W. Mays)
1.3 DEVELOPMENT OF HYDRAULICS The historical development of hydraulics as a modern science has been described by Biswas (1970), Rouse (1976), and Rouse and Ince (1963). More recently, the book titled, The Science of Water (Levi, 1995) presents an excellent history of the foundation of modern hydraulics. The reader is referred to these excellent books for details on the development of hydraulics.
1.4 FEDERAL POLICIES AFFECTING HYDRAULIC DESIGN Federal legislation contains policies that can affect the design of various types of hydraulic structures. These policies are listed in Appendix 1.A, where they are categorized into the following sections: environment, health, historic and archeological preservation, and land and water usage. The appendix also lists the abbreviations used in the policies, (adapted from AASHTO, 1991).
1.5 CONVENTIONAL HYDRAULIC DESIGN PROCESS Conventional procedures for hydraulic design are basically iterative trial-and-error procedures. The effectiveness of conventional procedures depends on an engineer’s intuition, experience, skill, and knowledge of hydraulic systems. Therefore, conventional procedures are closely related to the human element, a factor that could lead to inefficient results for the design and analysis of complex systems. Conventional procedures are typically based on
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INTRODUCTION
1.24
Chapter One
using simulation models in a process of trial and error to arrive at an optimal solution. Figure 1.21 presents a depiction of the conventional procedure for design and analysis. For example, determining a least-cost pumping scheme for an aquifer dewatering problem would require one to select the required pump sizes and the site where the aquifer would be dewatered. Using a trial set of pump sizes and sites, a groundwater simulation model is solved to determine whether the water levels are lower than desired. If the pumping scheme (pump size and site) does not satisfy the water levels, then a new pumping scheme is selected and simulated. This iterative process is continued, each time to determine the cost of the scheme. Optimization eliminates the trial-and-error process of changing a design and resimulating it with each new change. Instead, an optimization model automatically changes the design parameters. An optimization procedure has mathematical expressions that describe
Data collection to describe system
Estimate initial design of system
Analyze system design using simulation
Check results of simulation to check performance
Change design
No
Is design satisfactory? Yes
Compute cost or benefits
No
Are costs or benefits ok?
Yes
Stop
FIGURE 1.21 Conventional procedure for hydraulic design and analysis. (Mays and Tung, 1992)
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INTRODUCTION
Introduction 1.25
the system and its response to the system inputs for various design parameters. These mathematical expressions are constraints in the optimization model. In addition, constraints are used to define the limits of the design variables, and the performance of the design is evaluated through an objective function, which could be used to minimize costs. An advantage of the conventional process is that engineers use their experience and intuition to make conceptual changes in the system or to change or add specifications. The conventional procedure can lead to nonoptimal or uneconomical designs and operation policies. Also, the conventional procedure can be extremely time consuming. An optimization procedure requires the engineer to identify the design variables explicitly, the objective function of the measure of performance to be optimized, and the constraints for the system. In contrast to the decision-making process in the conventional procedure, the optimization procedure is more organized because a mathematical approach is used to make decisions. Refer to Mays and Tung (1992) for more detail.
1.6 ROLE OF ECONOMICS IN HYDRAULIC DESIGN 1.6.1 Engineering Economic Analysis Engineering economic analysis is an evaluation process that can be used to compare alternative hydraulic designs and then apply a discounting technique to select the best alternative. To perform this analysis, the engineer must understand several basic concepts, such as equivalence of kind, equivalence of time, and discounting factors. One first step in economic analysis is to find a common unit of value, such as monetary units. Through the use of common value units, alternatives of rather diverse kinds can be evaluated. The monetary evaluation of alternatives generally occurs over a number of years. Each monetary value must be identified by amount and time. Because the time value of money results from the willingness of people to pay interest for the use of money, money at different times cannot be directly combined or compared; first, it must be made equivalent through the use of discount factors, which convert a monetary value at one date to an equivalent value at another date. Discount factors are described using the following notations: i is the annual interest rate, n is the number of years, P is the present amount of money, F is the future amount of money, and A is the annual amount of money. Consider an amount of money P that is to be invested for n years at an interest rate of i percent. The future sum F at the end of n years is determined from the following progression: Period Year 1 Year 2 Year 3 Year n
Amount at beginning of year
Plus interest
iP iP(1+i) iP(1+i)2
iP(1+i)n–1
P (1+i)P (1+i)2P (1+i)n–1P
Amount at end of year (1+i)P (1+i)2P (1+i)3P (1+i)nP
The future sum is then F P(1 i)n
(1.1)
and the single-payment compound amount factor is
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INTRODUCTION
1.26
Chapter One F F (1 i)n , i%, n P P
(1.2)
This factor defines the number of dollars that accumulate after n years for each dollar initially invested at an interest rate of i percent. The single-payment present worth factor (P/F, i%, n) is simply the reciprocal of the single-payment compound amount factor. Table 1.2 summarizes the various discount factors. Uniform annual series factors are used for equivalence between present (P) and annual (A) monetary amounts or between future (F) and annual (A) monetary amounts. Consider the amount of money A that must be invested annually (at the end of each year) to accumulate F at the end of n years. Because the last value of A in the nth year is withdrawn immediately on deposit, it accumulates no interest. The future value F is F A (1 i)A (1 i)2 A (1 i)n–1 A
TABLE 1.2
(1.3)
Summary of Discounting Factors
Type of Discount Factor
Symbol
Given*
Find
Factor
Compound-amount factor
F, i%, n P
P
F
(1 i)n
Present-worth factor
P, i%, n F
F
P
1 n (1 i)
Sinking-fund factor
A, i%, n F
F
A
i (1 i)n 1
Capital-recovery factor
A, i%, n P
P
A
i(1 i)n (1 i)n 1
F, i%, n A
A
F
(1 i)n 1 i
Single-payment factors: P = $1
F
P
F = $1
Uniform annual series factors:
P
(1 i)n 1 i(1 i)n
P,i%,n G
G
P
(1 i)n 1 (1 ni i) i2(1 i)n
Uniform gradient series factors: Uniform gradient series present-worth factor
A
A
A
A
A
P = $1 A
A
A = $1 A
A
F A
P
A
A
A = $1
A
A
A
P
G = $1
A
A
(n-1)G
A
A
3G
P, i%, n A
A
2G
Series present-worth factor
A
G
Series compound-amount factor
F = $1
*The discount factors represent the amount of dollars for the given amounts of $1 for for P, F, A and G. Source: Mays and Tung, 1992.
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INTRODUCTION
Introduction 1.27
Multiply Eq. (1.3) by (1 i); then subtract Eq. (1.3) from the result to obtain the uniform annual series sinking–fund factor: i A A, i%, n (1.4) (1 i)n 1 F F The sinking-fund factor is the number of dollars A that must be invested at the end of each of n years at i percent interest to accumulate $1. The series compound amount factor (F/A) is simply the reciprocal of the sinking-fund factor (Table 1.3), which is the number of accumulated dollars if $1 is invested at the end of each year. The capital-recovery factor can be determined by simply multiplying the sinking fund factor (A/F) by the single-payment compound-amount factor (Table 1.2): A , P
i%, n A F (1.5) F P This factor is the number of dollars that can be withdrawn at the end of each of n years if $1 is invested initially. The reciprocal of the capital-recovery factor is the series presentworth factor (P/A), which is the number of dollars initially invested to withdraw $1 at the end of each year. A uniform gradient series factor is the number of dollars initially invested to withdraw $1 at the end of the first year, $2 at the end of the second year, $3 at the end of the third year, and so on.
1.6.2 Benefit-Cost Analysis Water projects extend over time, incur costs throughout the duration of the project, and yield benefits. Typically, the costs are large during the initial start-up period of construction, followed by operation and maintenance costs only. Benefits typically build up to a maximum over time, as depicted in Fig. 1.22. The present values of benefits (PVB) and costs (PVC) are as follows: b1 b2 bn PVB b0 (1.6) (1 i) (1 i)2 (1 i)n and c1 c2 cn (1.7) PVC c0 (1 i) (1 i)2 (1 i)n
B Benefits (B) and Costs (C)
C
Time FIGURE 1.22 Illustration of how benefits (B) and costs (C) build up over time. (Mays and Tung, 1992)
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INTRODUCTION
1.28
Chapter One
The present value of net benefits is (b1 c1) (b2 c2) (bn cn) PVNB PVB PVC (b0 c0) (1 i) (1 i)2 (1 i)n
(1.8)
To carry out benefit-cost analyses, rules for economic optimization of the project design and procedures for ranking projects are needed. The most important point in planning a project is to consider the broadest range of alternatives. The range of alternatives selected is typically restricted by the responsibility of the water resource agency, the planners, or both. The nature of the problem to be solved also may condition the range of alternatives. Preliminary investigation of alternatives can help to rule out projects because of their technical unfeasibility or costs. Consider the selection of an optimal, single-purpose project design, such as the construction of a flood-control system or a water supply project. The optimum size can be determined by selecting the alternative so that the marginal or incremental current value of costs, ∆PVC, is equal to the marginal or incremental current value of the benefits, ∆PVB, (∆PVB ∆PVC.) The marginal or incremental value of benefits and costs are for a given increase in the size of a project: ∆b1 ∆b2 ∆bn ∆PVB (1 i) (1 i)2 (1 i)n
(1.9)
∆c1 ∆c2 ∆cn ∆PVC (1 i) (1 i)2 (1 i)n
(1.10)
and
When selecting a set of projects, one rule for optimal selection is to maximize the current value of net benefits. Another ranking criterion is to use the benefit-cost ratio (B/C), PVB/PVC: B PVB (1.11) = C PVC This method has the option of subtracting recurrent costs from the annual benefits or including all costs in the present value of cost. Each option will result in a different B/C, ratio, with higher B/C ratios when netting out annual costs, if the ratio is greater than one. The B/C ratio is often used to screen unfeasible alternatives with B/C ratios less than 1 from further consideration. Selection of the optimum alternative is based on the incremental benefit-cost ratios, ∆B/∆C, whereas the B/C ratio is used for ranking alternatives. The incremental benefitcost ratio is PVBAj PVBAk ∆B (1.12) ∆C PVC Aj PVCAk where PVB(Aj) is the present value of benefits for alternative Aj. Figure 1.23 is a flowchart illustrating the benefit-cost method.
1.6.3 Estimated Life Spans of Hydraulic Structures The Internal Revenue Service bulletin gives estimated average lives for many thousands of different types of industrial assets. The lives (in years) given for certain elements of hydraulic projects are listed in Table 1.3. Although such estimates of average lives may be helpful, they are not necessarily the most appropriate figures to use in any given instance. Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
INTRODUCTION
Introduction 1.29 Compute B/C Ratio of Each Alternative
Keep Alternatives With B/C > 1
Rank Alternatives in Order of Increasing Cost
Compare Two Least-Costly Alternatives
Select Next Alternative to Compare
Choose Less Costly Alternative
Select Next Alternative to Compare
Compute Incremental B/C Ratio ∆B/∆C
No
∆B > 1 ∆C
Yes
Choose More Costly Alternative
FIGURE 1.23 Flowchart for a benefit-cost analysis. (Mays and Tung, 1992)
TABLE 1.3
Lives (in years) for Elements of Hydraulic Projects
Barges Booms, log Canals and ditches Coagulating basins Construction equipment Dams: Crib Earthen, concrete, or masonry Loose rock Steel Filters Flumes: Concrete or masonry Steel
12 15 75 50 5 25 150 60 40 50 75 50
Penstocks Pipes: Cast iron 2-4 in. 4-6 in. 8-10 in. 12 in. and over Concrete PVC Steel Under 4 in. Over 4 in. Wood stave 14 in. and larger
50
50 65 75 100 20-30 40 30 40 33
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INTRODUCTION
1.30
Chapter One TABLE 1.3
(Continues)
Wood Fossil-fuel power plants Generators: Above 3000 kva 1000-3000 kva 50 hp-1000 kva Below 50 hp Hydrants Marine construction equipment Meters, water Nuclear power plants
25 28 28 25 17-25 14-17 50 12 30 20
3-12 in. Pumps Reservoirs Standpipes Tanks: Concrete Steel Wood Tunnels Turbines, hydraulic Wells
20 18-25 75 50 50 40 20 100 35 40-50
*Alternating-current generators are rated in kilovolt-amperes (kva). Source: Linsley et al., 1992.
1.7 ROLE OF OPTIMIZATION IN HYDRAULIC DESIGN An optimization problem in water resources can be formulated in a general framework in terms of the decision variables (x), with an objective function to optimize f(x)
(1.13)
g(x) 0
(1.14)
subject to constraints
and bound constraints on the decision variables x x x (1.15) where x is a vector of n decision variables (x1, x2, …, xn), g(x) is a vector of m equations called constraints, and x and x represent the lower and upper bounds, respectively, on the decision variables. Every optimization problem has two essential parts: the objective function and the set of constraints. The objective function describes the performance criteria of the system. Constraints describe the system or process that is being designed or analyzed and can be in two forms: equality constraints and inequality constraints. A feasible solution of the optimization problem is a set of values of the decision variables that simultaneously satisfies the constraints. The feasible region is the region of feasible solutions defined by the constraints. An optimal solution is a set of values of the decision variables that satisfies the constraints and provides an optimal value of the objective function. Depending on the nature of the objective function and the constraints, an optimization problem can be classified as (1) linear vs. nonlinear, (2) deterministic vs. probabilistic, (3) static vs. dynamic, (4) continuous vs. discrete, or (5) lumped parameter vs. distributed parameter. Linear programming problems consist of a linear objective function, and all constraints are linear, whereas nonlinear programming problems are represented by nonlinear equations: that is, part or all of the constraints or the objective functions or both are nonlinear. Deterministic problems consist of coefficients and parameters that can be assigned fixed values, whereas probabilistic problems consist of uncertain parameters that are regarded as random variables.
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INTRODUCTION
Introduction 1.31
Static problems do not explicitly consider the variable time aspect, whereas dynamic problems do consider the variable time. Static problems are referred to as mathematical programming problems, and dynamic problems are often referred to as optimal control problems, which involve difference or differential equations. Continuous problems have variables that can take on continuous values, whereas with discrete problems, the variables must take on discrete values. Typically, discrete problems are posed as integer programming problems in which the variables must be integer values. Lumped problems consider the parameters and variables to be homogeneous throughout the system, whereas distributed problems must account for detailed variations in the behavior of the system from one location to another. The method of optimization used depends up the type of objective function, the type of constraints, and the number of decision variables. Optimization is not covered in this handbook, but it is discussed in detail in Mays and Tung (1992).
1.8 ROLE OF RISK ANALYSIS IN HYDRAULIC DESIGN 1.8.1
Existence of Uncertainties
Uncertainties and the consequent related risks in hydraulic design are unavoidable. Hydraulic structures are always subject to a probability of failure in achieving their intended purposes. For example, a flood control project may not protect an area from extreme floods. A water supply project may not deliver the amount of water demanded. This failure may be caused by failure of the delivery system or may be the result of the lack of supply. A water distribution system may not deliver water that meets quality standards although the source of the water does. The rationale for selecting the design and operation parameters and the design and operation standards are questioned continually. Procedures for the engineering design and operation of water resources do not involve any required assessment and quantification of uncertainties and the resultant evaluation of a risk. Risk is defined as the probability of failure, and failure is defined as an event that causes a system to fail to meet the desired objectives. Reliability is defined as the complement of risk: i.e., the probability of nonfailure. Failures can be grouped into either structural failures or performance failures. Water distribution systems are a good example. A structural failure, such as broken pipe or a failed pump, can result in unmet demand. In addition, an operational aspect of a water distribution system, such as the inability to meet demands at required pressure heads, is a failure despite the lack of a structural failure in any component in the system. Uncertainty can be defined as the occurrence of events that are beyond one’s control. The uncertainty of a hydraulic structure is an indeterministic characteristic and is beyond rigid controls. In the design and operation of these systems, decisions must be made under various kinds of uncertainty. The sources of uncertainties are multifold. First, the ideas of natural uncertainties, model structure uncertainties, model parameter uncertainties, data uncertainties, and operational uncertainties will be discussed. Natural uncertainties are associated with the random temporal and spatial fluctuations that are inherent in natural processes. Model structural uncertainties reflect the inability of a simulation model or design procedure to represent the system’s true physical behavior or process precisely. Model parameter uncertainties reflect variability in the determination of the parameters to be used in the model or design. Data uncertainties include inaccuracies and errors in measurements, inadequacy of the data gauging network, and errors in data handling and
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INTRODUCTION
1.32
Chapter One
transcription. Operational uncertainties are associated with human factors, such as construction, manufacture, deterioration, and maintenance, that are not accounted for in the modeling or design procedure. Uncertainties fall into four major categories: hydrologic uncertainty, hydraulic uncertainty, structural uncertainty, and economic uncertainty. Each category has various component uncertainties. Hydrologic uncertainty can be classified into three types: inherent, parameter, and model uncertainties. Various hydrologic events, such as streamflow or rainfall, are considered to be stochastic processes because of their observable natural, (inherent) randomness. Because perfect hydrologic information about these processes is lacking, informational uncertainties about the processes exist. These uncertainties are referred to as parameter uncertainties and model uncertainties. In many cases, model uncertainties result from the lack of adequate data and knowledge necessary to select the appropriate probability model or from the use of an oversimplified model, such as the rational method for the design of a storm sewer. Hydraulic uncertainty concerns the design of hydraulic structures and the analysis of their performance. It arises mainly from three basic sources: the model, the construction and materials, and the operational conditions of flow. Model uncertainty results from the use of a simplified or an idealized hydraulic model to describe flow conditions, which in turn contributes to uncertainty when determining the design capacity of hydraulic structures. Because simplified relationships, such as Manning’s equation, are typically used to model complex flow processes that cannot be described adequately, resulting in model errors. Structural uncertainty refers to failure caused by structural weakness. Physical failures of hydraulic structures can be caused by saturation and instability of soil, failures caused by erosion or hydraulic soil, wave action, hydraulic overloading, structural collapse, material failure, and so forth. An example is the structural failure of a levee system either in the levee or in the adjacent soil; the failure could be caused by saturation and instability of soil. A flood wave can cause increased saturation of the levee through slumping. Levees also can fail because of hydraulic soil failures and wave action. Economic uncertainty can arise from uncertainties regarding construction costs, damage costs, projected revenue, operation and maintenance costs, inflation, project life, and other intangible cost and benefit items. Construction, damage, and operation or maintenance costs are all subject to uncertainties because of fluctuations in the rate at which construction materials, labor costs, transportation costs, and economic losses, increase and the rate at which costs increase in different geographic regions. Many other economic and social uncertainties are related to inconvenience losses: for example, the failure of a highway crossing caused by flooding, which results in trafficrelated losses. The objective when analyzing uncertainties is to incorporate the uncertainties systematically into the evaluation of loading and resistance. The most commonly used method is the first-order analysis of uncertainties. This method is used to determine the statistics of the random variables loading and resistance, which are typically defined through the use of deterministic models but have uncertain parameter inputs. Chapter 7 provides details of the first-order analysis of uncertainties.
1.8.2 Risk-Reliability Evaluation 1.8.2.1 Load resistance The load for a system can be defined as an external stress to the system, and the resistance can be defined as the capacity of the system to overcome the external load. Although the terms load and resistance have been used in structural
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INTRODUCTION
Introduction 1.33
engineering, they definitely have a place in the types of risk analysis that must be performed for engineering projects involving water resources. If we use the variable R for resistance and the variable L for load, we can define a failure as the event when the load exceeds the resistance and the consequent risk is the probability that the loading will exceed the resistance, P(L R). A simple example of this type of failure would be a dam that fails because of overtopping. The risk would be the probability that the elevation of the water surface in a reservoir exceeds the elevation of the top of the dam. In this case, the resistance is the elevation of the top of the dam, and the loading is the maximum elevation of the water surface of a flood wave entering the reservoir. Because many uncertain variables define both the resistance and loading, both are regarded as random variables. A simple example would be to use the rational equation Q CiA to define the design discharge (loading) for a storm sewer. The loading L Q is a function of three uncertain variables: the runoff coefficient C, the rainfall intensity i, and the drainage area A. Because the three variables cannot be determined with complete certainty, they are considered to be random variables. If the resistance is defined using Manning’s equation, then the resistance is a function of Manning’s roughness factor, the pipe diameter, and the slope (friction slope). The two main contributors to uncertainty in this equation would be the friction slope and the roughness factor i.e., random variables. Thus, the resistance is also is a random variable because it is a function of the other two random variables. It is interesting to note that in the example of the storm sewer, both the loading and the resistance are defined by deterministic equations: the rational equation and Manning’s equation. Both equations are considered to have uncertain design parameters that result in uncertain resistance and loading. Consequently, they are considered to be random variables. In the storm sewer example, as in many types of hydraulic structures, the loading uncertainty is actually the hydrologic uncertainty and the resistance uncertainty is the hydraulic uncertainty. 1.8.2.2 Composite risk The discussion about the hydrologic and hydraulic uncertainties being the resistance and loading uncertainties leads to the idea of a composite risk. The probability of failure defined previously as the risk, P(L R), is actually a composite risk. If only the hydrologic uncertainty, in particular the inherent hydrologic uncertainty, were considered, then this would not be a composite risk. In the conventional design processes of water resources engineering projects, only the inherent hydrologic uncertainties have been considered. Essentially, a large return period is selected and is artificially considered as the safety factor without any regard to accounting systematically for the various uncertainties that actually exist. 1.8.2.3 Safety factor The safety factor is defined as the ratio of the resistance to loading, R/L. Because the safety factor SF R/L is the ratio of two random variables, it also is a random variable. The risk can be written as P(SF 1) and the reliability can be written as P(SF 1). In the example of the storm sewer, both the resistance and the loading are considered to be random variables because both are functions of random variables. Consequently, the safety factor for storm sewer design would also be a random variable. 1.8.2.4 Risk assessment Risk assessment requires several phases or steps, which can vary for different types of water resources engineering projects: (1) identify the risk of hazard, (2) assess load and resistance, (3) perform an analysis of the uncertainties, (4) quantify the composite risk, and (5) develop the composite risk-safety factor relationships. 1.8.2.5 A model for risk-based design The risk-based design of hydraulic structures potentially promises to be the most significant application of uncertainty and risk analy-
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INTRODUCTION
1.34
Chapter One
sis. The risk-based design of hydraulic structures integrates the procedures of economics, uncertainty analysis, and risk analysis in design practice. Such procedures can consider the tradeoffs among risk, economics, and other performance measures in the design of hydraulic structures. When risk-based design is embedded in an optimization framework, the combined procedure is called optimal risk-based design. This approach to design is the ultimate model for the design, analysis, and operation of hydraulic structures and water resource projects that hydraulics engineers need to strive for in the future. Chapter 7 provides detailed discussions on risk-reliability evaluation.
REFERENCES Abbott, M.B., Computational Hydraulics, Pitman, London, 1980. Adams, R.M., Heartland of Cities, Surveys of Ancient Settlement and Land Use on the Central Floodplain of the Euphrates, University of Chicago Press. Addison, H.A., A Treatise on Applied Hydraulics, Chapman and Hall, London, UK, 1954. Akurgal, E., Ancient Civilizations and Ruins of Turkey, 8th ed., Net Turistik Yaylinlar A.S., Istanbul, 1993. Allen, J., Scale Models in Hydraulic Engineering, Longman, Green, London, UK, 1947. American Association of State Highway and Transportation Officials (AASHTO), Model Drainage Manual, AASHTO, Washington, D.C., 1991 American Society of Civil Engineers (ASCE), Hydraulic Models, ASCE Manual 25, ASCE, New York, 1942. Bakhmeteff, B.A., Hydraulics of Open Channels, McGraw-Hill, New York, 1932 Bakhmeteff, B.A., The Mechanics of Turbulent Flow, University Press, Princeton, NJ., 1936. Binnie, G.M., Early Victorian Water Engineers, London: Thomas Telford Ltd., 1981. Biswas, A.K., History of Hydrology, North-Holland Publishing Amsterdam, 1970. Butzer, K.W., Early Hydraulic Civilization in Egypt, University of Chicago Press, Chicago, 1976. Chow, V T., Open-Channel Hydraulics, McGraw-Hill, New York, 1959. Crouch, D.P., Water Management in Ancient Greek Cities, Oxford University Press, New York, 1993. Crown, P.L. and W.J. Judge, eds., Chaco and Hohokam Prehistoric Regional Systems in the American Southwest, School of American Research Press, Sante Fe, NM, 1991. Daily, J.W. and D.R.F. Harleman, Fluid Dynamics, Addison-Wesley Reading, MA, 1966. Dart, A., Prehistoric Irrigation in Arizona: A Context for Canals and Related Cultural Resources, Technical Report 89-7, Center for Desert Archaeology, Tucson, AZ, 1989. Daugherty, R.L., Hydraulics, McGraw-Hill New York, 1937. Davies, N., The Tomb of Nefer-Hotep at Thebes, Vol. 1, Publication 9, Metropolitan Museum of Art Egyption Expedition, New York, 1933. Davis, C.V., and K.E. Sorensen, Handbook of Applied Hydraulics, McGraw-Hill, New York, 1942. de Camp, L.S. , The Ancient Engineers, Dorset Press, New York, 1963. Doolittle, W.E., Canal Irrigation in Prehistoric Mexico, University of Texas Press, Austin, 1990. Evans, H.B., Water Distribution in Ancient Rome, University of Michigan Press, Ann Arbor, 1994. Fischer, H.B., E.J. List, C.Y. Koh, J. Imberger, and N.H. Brocks, Mixing in Inland and Coastal Waters, Academic Press, New York, 1979. FitzSimons, N., Engineering Classics of James Kip Finch, Cedar Press, Kensington, MD, 1978. Freeze, R.A. and J.A. Cherry, Groundwater, Prentice-Hall Inc. Englewood Cliffs, N.J., 1979. Garbrecht, G., Wasserversorgung im Antiken Rom, (Water Supply in Ancient Rome), R. Oldenburg Verlag München, Vienna, 1982.
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INTRODUCTION
Introduction 1.35 Garbrecht, G., “Sadd-el-Kafara: The World’s Oldest Large Dam,” International Water and Power Dam Construction, July 1985. Garraty, J.A. and P. Gay, The Columbia History of the World, Harper & Row, New York, 1972. Graf, W., Hydraulics of Sediment Transport, McGraw-Hill, New York, 1971. Henderson, F.M., Open-Channel Flow, McGraw-Hill, New York, 1976. Hoskins, L.M., A Text-Book on Hydraulics, Henry Holt, New York, 1911. Hoyt, J.C. and N.C. Grover, River Discharge, John Wiley & Sons, New York, 1916. King, H.W., Handbook of Hydraulics, McGraw-Hill, New York, 1954 King, H.W. and C.O. Wisler, Hydraulics, John Wiley and Sons, Inc., N.Y., 1922. Kolupaila, S., Early History of Hydrometry in the United States, Journal of Hydraulic Div, ASCE, 86: 1–52, 1960. Le Conte, J.N., Hydraulics, McGraw-Hill, New York, 1926. Leliavsky, S., An Introduction to Fluvial Hydraulics, Constable, London, UK, 1955. Leliavsky, S., River and Canal Hydraulics, Chapman and Hall, London, UK, 1965. Levi, E., The Science of Water: The Foundation of Modern Hydraulics, ASCE Press, New York 1995. Linsley, R.K., and J.B. Franzini, Elements of Hydraulic Engineering, McGraw-Hill, New York, 1964 Masse, W.B., The Quest for Subsistence Sufficiency and Civilization in the Sonovan Desert, in Chaco and Hohokam Prehistoric Regional Systems in the American Southwest, P.L. Crown and W.J. Judge, editors, pp. 195-223 School of American Research Press, Santa Fe, NM, 1991. Mays, L.W., “Introduction,” in Water Resources Handbook edited by L.W. Mays, ed., pp. 1.3-1.35, McGraw-Hill, New York, 1996. Mays, L.W. and Y.K. Tung, Hydrosystems Engineering and Management, McGraw-Hill, N.Y., N.Y., 1992. Merriman, M., Treatise on Hydraulics, John Wiley & Sons, New York, 1904. Morris, H.M., and J.M. Wiggert, Applied Hydraulics in Engineering, 1st ed, Ronald Press, New York, 1963 Muskat, M., The Flow of Homogeneous Fluids Through Porous Media, McGraw-Hill, New York, 1937. Needham, J., Science and Civilization in China, Vol. I, Cambridge University Press, Cambridge, England, UK, 1954. Neugebauer, O., The Exact Sciences in Antiquity, 2nd ed, Barne & Noble, New York, 1993. Ozis, U., “Ancient Water Works in Anatolia,” Water Resources Development, Buttermorth & Co. Publishers Ltd. 3(1): pp. 55-62 1987. Ozis, U., and N. Harmancioglu, “Some Ancient Water Works in Anatolia,” in Proceedings of the International Seminar on Kaust Hydrogeology, IAHR, Anatalya, Turkey, pp. 380-385 1979. Parmakiams, J., Waterhammer Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1955. Payne, R. The Canal Builders, Macmillan New York, 1959. Reynolds, J., Windmills and Waterwheels, Praeger, New York, 1970. Rich, G.R., Hydraulic Transients, McGraw-Hill, New York., 1951. Robbins, F.W., The Story of Water Supply, Oxford University, London, UK, 1946. Rouse, H., Fluid Mechanics for Hydraulic Engineers, McGraw-Hill, New York, 1938. Rouse, H., ed., Engineering Hydraulics, John Wiley & Sons, New York, 1950 Rouse, H., Hydraulics in the United States, 1776–1976, Iowa Institute of Hydraulic Research, Iowa City, 1976. Rouse, H., and S. Ince, History of Hydraulics, Dover, New York, 1963. Sarton, G., A History of Science, Harvard University Press, Cambridge, 1952–59. Schoder, E.W., and F.M. Dawson, Hydraulics, McGraw-Hill, New York, 1927. Smith, N., A History of Dams, Peter Davies, London, UK, 1971.
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INTRODUCTION
1.36
Chapter One
Stoker, J.J., Water Waves, Interscience, New York, 1957. Streeter, V.L. and E.B. Wylie, Hydraulic Transients, McGraw-Hill, New York, 1967. Turney, O.S., Map of Prehistoric Irrigation Canals, Map. No. 002004, Archaeological Site Records Office, Arizona State Museum, University of Arizona, Tuscon, 1922. Upton, N., An Illustrated History of Civil Engineering, Crane Russak, New York, 1975. U.S. Bureau of Reclamation, Design of Small Dams, U.S. Government Printing Office, Denver, 1960, 1973, 1987. U.S. Bureau of Reclamation, Hydraulic Design of Stilling Basin and Energy Dissipaters, U.S. Government Printing Office, Washington, D.C., 1958, 1963, 1974, and 1978. U.S. Bureau of Reclamation, Hydraulic Laboratory Practice, Monograph 18, Denver, 1953. U.S. Geological Survey, Roughness Characteristics of Natural Channels, Geological Survey WaterSupply Paper No. 1849, Arlington, VA, 1967. Van Deman, E.B., The Building of Roman Aqueducts, Carnegie Institute of Washington, 1934. Vivian, R.G., “Conservation and Diversion: Water-Control Systems in the Anasazi Southwest, in Irrigation Impact on Society, Anthropological papers of the University of Arizona, No. 25, T. Downing and M. Gibson, eds., pp. 95–112, University of Arizona, Tucson, 1974. Vivian, R.G., The Chacoan Prehistory of the San Juan Basin, Academic Press, San Diego, CA, 1990. Wittfogel, K.A., The Hydraulic Civilization: Man’s Role in Changing the Earth, University of Chicago Press, Chicago, 1956. Woodburg, R.B. and J.A. Neely, “Water Control Systems of the Tehuacan Valley,” in The Prehistory of the Tehuacan Valley: Vol. 4, Chronology and Irrigation, R.S. MacNeish and F. Johnson, eds., pp. 81–153, University of Texas Press, Austin, 1972. Woodward, S.M., and C.J. Posey, Hydraulics of Steady Flow in Open Channels, John Wiley & Sons, New York, 1941.
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INTRODUCTION
Introduction 1.37
APPENDIX 1. A
INTRODUCTION A.1 POLICIES BY CATEGORY A.1.1 Environment National Environmental Policy Act: 42 U.S.C. 4321–4347 (P.L. 91–190 and 94–81). Reference - 23 CFR 770–772, 40 CFR 1500–1508, CEQ Regulations, Executive Order 11514 as amended by Executive Order 11991 on NEPA responsibilities. The purpose is to consider environmental factors through a systematic interdisciplinary approach before committing to a course of action. Section 4(f) of the Department of Transportation Act: 23 U.S.C. 138, 49 U.S.C. 303 (P.L. 100–17, 97–449, and 86–670), 23 CFR 771.135. The purpose is to preserve publicly owned public parklands, waterfowl and wildlife refuges, and all historic areas. Economic, Social, and Environmental Effects: 23 U.S.C. 109(h) (P.I. 91–605), 23 U.S.C. 128, 23 CFR 771. The purpose is to assure that possible adverse, economic, social, and environmental effects of proposed highway projects and their locations are fully considered and that final decisions on highway projects are made in the best overall public interest. Public Hearings: 23 U.S.C. 128, 23 CFR 771.111. The purpose is to ensure adequate opportunity for public hearings on the social, economic, and environmental effects of alternative project locations and major design features as well as the consistency of the project with local planning goals and objectives. Surface Transportation and Uniform Relocation Assistance Act of 1987: Section 123(f) Historic Bridges 23 U.S.C. 144(o) (P.L. 100-17). The purpose is to complete an inventory of on-and-off system bridges to determine their historic significance and to encourage the rehabilitation, reuse, and preservation of historic bridges.
A.1.2 Health Safe Drinking Water Act: 42 U.S.C. 300f–300;f-6 (P.L. 93–523 and 99–339), FHPM 6–7–3–3, 23 CFR 650, Subpart E, 40 CFR 141, 149. The purpose is to ensure public health and welfare through safe drinking water. Solid Waste Disposal Act, as amended by the Resource Conservation and Recovery Act of 1976: 42 U.S.C. 6901, et seq., see especially 42 U.S.C. 6961–6964 (P.L. 89–272, 91–512, and 94–580), 23 CFR 751, 40 CFR 256–300. The purpose is to provide for the recovery, recycling, and environmentally safe disposal of solid wastes.
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INTRODUCTION
1.38
A.1.3
Chapter One
Historic and Archeological Preservation
Section 106 of the National Historic Preservation Act, as amended: 16 U.S.C. 470f (P.L. 89–665, 91–243, 93–54, 94–422, 94–458, 96–199, 96–244, and 96–515), Executive Order 11593, 23 CFR 771, 36 CFR 60, 36 CFR 63, 36 CFR 800. The purpose is to protect, rehabilitate, restore, and reuse districts, sites, buildings, structures, and other objects significant in American architecture, archeology, engineering, and culture. Section 110 of the National Historic Preservation Act, as amended: 16 U.S.C. 470h–2 (P.L. 96–515), 36 CFR 65, 36 CFR 78. The purpose is to protect national historic landmarks and record historic properties before demolition. Archeological and Historic Preservation Act: 16 U.S.C. 469–469c (P.L. 93–291) (MossBennett Act), 36 CFR 66 (draft). The purpose is to preserve significant historical and archeological data from loss or destruction. Act for the Preservation of American Antiquities: 16 U.S.C. 431–433 (P.L. 59–209), 36 CFR 251.50–64, 43 CFR 3. Archeological Resources Protection Act: 16 U.S.C. 470aa–11 (P.L. 96–95), 18 CFR 1312, 32 CFR 229, 36 CFR 296, 43 CFR 7. The purpose is to preserve and protect paleontologic resources, historic monuments, memorials, and antiquities from loss or destruction. American Indian Religious Freedom Act: 42 U.S.C. 1996 (P.L. 95–341). The purpose is to protect places of religious importance to American Indians, Eskimos, and Native Hawaiians.
A.1.4 Land and Water Usage Wilderness Act 16 U.S.C. 1131–1136. 36 CFR 251, 293, 43 CFR 19, 8560, 50 CFR 35. The purpose is to preserve and protect wilderness areas in their natural condition for use and enjoyment by present and future generations. Wild and Scenic Rivers Act: 16 U.S.C. 1271–1287, 36 CFR 251, 261, 43 CFR 8350. The purpose is to preserve and protect wild and scenic rivers and immediate environments for the benefit of present and future generations. Land and Water Conservation Fund Act (Section 6(f)): 16 U.S.C. 4601–4 to 1–11 (P.L. 88–578). The purpose is to preserve, develop, and assure the quality and quantity of outdoor recreation resources for present and future generations. Executive Order 11990, Protection of Wetlands, DOT Order 5660. 1A, 23 CFR 777. The purpose is to avoid direct or indirect support of new construction in wetlands whenever a practicable alternative is available. Emergency Wetlands Resources Act of 1986: 16 U.S.C. 3901 note (P.L. 99–645). The purpose is to promote the conservation of wetlands in the U.S. to maintain the public benefits they provide. National Trails Systems Act: 16 U.S.C. 1241–1249, 36 CFR 251, 43 CFR 8350. The purpose is to provide for outdoor recreational needs and encourage outdoor recreation. Rivers and Harbors Act of 1899: 33 U.S.C. 401, et seq., as amended and supplemented, 23 CFR part 650, Subpart H, 33 CFR 114–115. The purpose is to protect navigable waters in the U.S. Federal Water Pollution Control Act (1972), as amended by the Clean Water Act (1977 & 1987): 33 U.S.C. 1251–1376 (P.L. 92–500, 95–217, 100–4), DOT Order 5660.1A, FHWA
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INTRODUCTION
Introduction 1.39
Notices N5000.3 and N5000.4, FHPM 6–7–3–3, 23 CFR 650, Subpart B, E, 771, 33 CFR 209, 40 CFR 120, 122–125, 128–131, 133, 125–136, 148, 230–231. The purpose is to restore and maintain the chemical, physical, and biological integrity of the nation's waters through prevention, reduction, and elimination of pollution. Executive Order 11988, Floodplain Management, as amended by Executive Order 12148, DOT Order 5650.2, FHPM 6–7–3–2, 23 CFR 650, Subpart A, 771. The purpose is to avoid the long and short-term adverse impacts associated with the occupancy and modification of floodplains and to restore and preserve the natural and beneficial values served by floodplains. National Flood Insurance Act: (P.L. 90–448), Flood Disaster Protection Act: (P.L. 93–234) 42 U.S.C. 4001–4128, DOT Order 5650.2, FHPM 6–7–3–2, 23 CFR 650, Subpart A, 771, 44 CFR 59–77. The purpose is to identify flood-prone areas and provide insurance and to require the purchase of insurance for buildings in special flood-hazard areas. Marine Protection Research and Sanctuaries Act of 1972, as amended: 33 U.S.C. 1401–1445 (P.L. 92–532, 93–254, 96–572), 33 CFR 320, 330, 40 CFR 220–225, 227–228, 230–231. The purpose is to regulate the dumping of materials into U.S. ocean waters. Water Bank Act: 16 U.S.C. (P.L. 91–559, 96–182), 7 CFR 752. The purpose is to preserve, restore, and improve wetlands of the U.S. Coastal Zone Management Act of 1972: 16 U.S.C.1 1451–1464 (P.L. 92–583, 94–370, 96–464), 15 CFR 923, 926, 930–931, 23 CFR 771. The purpose is to preserve, protect, develop, and (when possible) restore and enhance the resources of the coastal zone. Coastal Barrier Resource Act, as amended: 16 U.S.C. 3501–3510, 42 U.S.C. 4028 (P.L. 97–348), Great Lakes Coastal Barrier Act of 1988 (P.L. 100–707), 13 CFR 116 Subparts D, E, 44 CFR 71, 205 Subpart N. The purpose is to minimize the loss of human life, wasteful expenditures of federal revenues, and the damage to fish, wildlife, and other natural resources. Farmland Protection Policy Act of 1981: 7 U.S.C. 4201–4209 (P.L. 97–98, 99–198), 7 CFR 658. The purpose is to minimize impacts on farmland and maximize compatibility with state and local farmland programs and policies. Resource Conservation and Recovery Act of 1976 (RCRA), as amended: 42 U.S.C. 690, et seq. (P.L. 94–580, 98–616), 40 CFR 260–271. The purpose is to protect human health and the environment; prohibit open dumping; manage solid wastes; and regulate the treatment, storage transportation, and disposal of hazardous waste. Comprehensive Environmental Response, Compensation, and Liability Act of 1980 (CERCLA), as amended: 42 U.S.C. 9601–9657 (P.L. 96–510), 40 CFR 300, 43 CFR 11. Superfund Amendments and Reauthorization Act of 1986 (SARA) (P.L. 99–499). The purpose is to provide for liability, compensation, cleanup, and emergency response when hazardous substances have been released into the environment and to provide for the cleanup of inactive hazardous waste disposal sites. Endangered Species Act of 1973, as amended: 16 U.S.C. 1531–1543 (P.L. 93–205, 94–359, 95–632, 96–159, 97–304), 7 CFR 355, 50 CFR 17, 23, 25–29, 81, 217, 222, 225–227, 402, 424, 450–453. The purpose is to conserve species of fish, wildlife, and plants facing extinction. Fish and Wildlife Coordination Act: 16 U.S.C. 661–666c (P.L. 85–624, 89–72, 95–616. The purpose is to conserve, maintain, and manage wildlife resources.
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Source: HYDRAULIC DESIGN HANDBOOK
CHAPTER 2
HYDRAULICS OF PRESSURIZED FLOW Bryan W. Karney Department of Civil Engineering University of Toronto, Toronto, Ontario, Canada
2.1 INTRODUCTION The need to provide water to satisfy basic physical and domestic needs; use of maritime and fluvial routes for transportation and travel, crop irrigation, flood protection, development of stream power; all have forced humanity to face water from the beginning of time. It has not been an easy rapport. City dwellers who day after day see water flowing from faucet’s, docile to their needs, have no idea of its idiosyncrasy. They cannot imagine how much patience and cleverness are needed to handle our great friend-enemy; how much insight must be gained in understanding its arrogant nature in order to tame and subjugate it; how water must be “enticed” to agree to our will, respecting its own at the same time. That is why a hydraulician must first be something like a water psychologist, thoroughly knowledgeable of its nature. (Enzo Levi, The Science of Water: The Foundations of Modern Hydraulics, ASCE, 1995, p. xiii.) Understanding the hydraulics of pipeline systems is essential to the rational design, analysis, implementation, and operation of many water resource projects. This chapter considers the physical and computational bases of hydraulic calculations in pressurized pipelines, whether the pipelines are applied to hydroelectric, water supply, or wastewater systems. The term pressurized pipeline means a pipe system in which a free water surface is almost never found within the conduit itself. Making this definition more precise is difficult because even in a pressurized pipe system, free surfaces are present within reservoirs and tanks and sometimes —for short intervals of time during transient (i.e., unsteady) events—can occur within the pipeline itself. However, in a pressurized pipeline system, in contrast to the open-channel systems discussed in Chapter 3, the pressures within the conveyance system are usually well above atmospheric. Of central importance to a pressurized pipeline system is its hydraulic capacity: that is, its ability to pass a design flow. A related issue is the problem of flow control: how design flows are established, modified, or adjusted. To deal adequately with these two topics, this chapter considers head-loss calculations in some detail and introduces the topics of pumping, flow in networks, and unsteady flows. Many of these subjects are treated in greater detail in later chapters, or in reference such as Chaudhry and Yevjevich (1981). Rather than simply providing the key equations and long tabulations of standard values, this chapter seeks to provide a context and a basis for hydraulic design. In addition to the relations discussed, such issues as why certain relations rather than others are used, what various equations assume, and what can go wrong if a relation is used incor2.1 Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
HYDRAULICS OF PRESSURIZED FLOW
2.2
Chapter Two
rectly also are considered. Although derivations are not provided, some emphasis is placed on understanding both the strengths and weaknesses of various approaches. Given the virtually infinite combinations and arrangements of pipe systems, such information is essential for the pipeline professional.
2.2 IMPORTANCE OF PIPELINE SYSTEMS Over the past several decades, pressurized pipeline systems have become remarkably competitive as a means of transporting many materials, including water and wastewater. In fact, pipelines can now be found throughout the world transporting fluids through every conceivable environment and over every possible terrain. There are numerous reasons for this increased use. Advances in construction techniques and manufacturing processes have reduced the cost of pipelines relative to other alternatives. In addition, increases in both population and population density have tended to favor the economies of scale that are often associated with pipeline systems. The need for greater conservation of resources and, in particular, the need to limit losses caused by evaporation and seepage have often made pipelines attractive relative to openchannel conveyance systems. Moreover, an improved understanding of fluid behavior has increased the reliability and enhanced the performance of pipeline systems. For all these reasons, it is now common for long pipelines of large capacity to be built, many of which carry fluid under high pressure. Some of these systems are relatively simple, composed only of series-connected pipes; in others systems, the pipes are joined to form complex networks having thousands of branched and interconnected lines. Pipelines often form vital links in the process chain, and high penalties may be associated with both the direct costs of failure (pipeline repair, cost of lost fluid, damages associated with rupture, and so forth) and the interruption of service. This is especially evident in industrial applications, such as paper mills, mines, and power plants. Yet, even in municipal systems, a pipe failure can cause considerable property damage. In addition, the failure may lead indirectly to other kinds of problems. For example, a mainline break could flood a roadway and cause a traffic accident or might make it difficult to fight a major fire. Although pipelines appear to promise an economical and continuous supply of fluid, they pose critical problems of design, analysis, maintenance, and operation. A successful design requires the cooperation of hydraulic, structural, construction, survey, geotechnical, and mechanical engineers. In addition, designers and planners often must consider the social, environmental, and legal implications of pipeline development. This chapter focuses on the hydraulic considerations, but one should remember that these considerations are not the only, nor necessarily the most critical, issues facing the pipeline engineer. To be successful, a pipeline must be economically and environmentally viable as well as technically sound. Yet, because technical competence is a necessary requirement for any successful pipeline project, this aspect is the primary focus.
2.3 NUMERICAL MODELS: BASIS FOR PIPELINE ANALYSIS The designer of a hydraulic system faces many questions. How big should each pipe be to carry the required flow? How strong must a segment of pipe be to avoid breaking? Are reservoirs, pumps, or other devices required? If so, how big should they be and where should they be situated?
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HYDRAULICS OF PRESSURIZED FLOW
Hydraulics of Pressurized Flow 2.3
There are at least two general ways of resolving this kind of issue. The first way is to build the pipe system on the basis of our “best guess” design and learn about the system’s performance as we go along. Then, if the original system “as built’’ is inadequate, successive adjustments can be made to it until a satisfactory solution is found. Historically, a number of large pipe systems have been built in more or less this way. For example, the Romans built many impressive water supply systems with little formal knowledge of fluid mechanics. Even today, many small pipeline systems are still constructed with little or no analysis. The emphasis in this kind of approach should be to design a system that is both flexible and robust. However, there is a second approach. Rather than constructing and experimenting with the real system, a replacement or model of the system is developed first. This model can take many forms: from a scaled-down version of the original to a set of mathematical equations. In fact, currently the most common approach is to construct an abstract numerical representation of the original that is encoded in a computer. Once this model is “operational,” experiments are conducted on it to predict the behavior of the real or proposed system. If the design is inadequate in any predictable way, the parameters of the model are changed and the system is retested until design conditions are satisfied. Only once the modeller is reasonably satisfied would the construction of the complete system be undertaken. In fact, most modern pipelines systems are modeled quite extensively before they are built. One reason for this is perhaps surprising —experiments performed on a model are sometimes better than those done on the prototype. However, we must be careful here, because better is a relative word. On the plus side, modeling the behavior of a pipeline system has a number of intrinsic advantages: Cost. Constructing and experimenting on the model is often much less expensive than testing the prototype. Time. The response rate of the model pipe system may be more rapid and convenient than the prototype. For example, it may take only a fraction of a second for a computer program to predict the response of a pipe system after decades of projected growth in the demand for water. Safety. Experiments on a real system may be dangerous or risky whereas testing the model generally involves little or no risk. Ease of modification: Improvements, adjustments, or modifications in design or operating rules can be incorporated more easily in a model, usually by simply editing an input file. Aid to communication. Models can facilitate communication between individuals and groups, thereby identifying points of agreement, disagreement, misunderstanding, or issues requiring clarification.Even simple sketches, such as Fig. 2.1, can aid discussion. These advantages are often seen as so overwhelming that the fact that alternative approaches are available is sometimes forgotten. In particular, we must always remember that the model is not reality. In fact, what makes the model useful is precisely its simplicity—it is not as complex or expensive as the original. Stated more forcibly, the model is useful because it is wrong. Clearly, the model must be sufficiently accurate for its intended purpose or its predictions will be useless. However, the fact that predictions are imperfect should be no surprise. As a general rule, systems that are large, expensive, complex, and important justify more complex and expensive models. Similarly, as the sophistication of the pipeline system increases, so do the benefits and advantages of the modeling approach because this
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HYDRAULICS OF PRESSURIZED FLOW
2.4
Chapter Two
EGL 1 ∆H
EGL 2
Valve FIGURE 2.1 Energy relations in a simple pipe system.
strategy allows us to consider the consequences of certain possibilities (decisions, actions, inactions, events, and so on before they occur and to control conditions in ways that may be impossible in practice (e.g., weather characteristics, interest rates, future demands, control system failures). Models often help to improve our understanding of cause and effect and to isolate particular features of interest or concern and are our primary tool of prediction. To be more specific, two kinds of computer models are frequently constructed for pipeline systems planning models and operational model: Planning models. These models are used to assess performance, quantity or economic impacts of proposed pipe systems, changes in operating procedures, role of devices, control valves, storage tanks, and so forth. The emphasis is often on selection, sizing, or modification of devices. Operational models. These models are used to forecast behavior, adjust pressures or flows, modify fluid levels, train operators, and so on over relatively short periods (hours, days, months). The goal is to aid operational decisions. The basis of both kinds of models is discussed in this chapter. However, before you believe the numbers or graphs produced by a computer program, or before you work through the remainder of this chapter, bear in mind that every model is in some sense a fake— it is a replacement, a stand-in, a surrogate, or a deputy for something else. Models are always more or less wrong. Yet it is their simultaneous possession of the characteristics of both simplicity and accuracy that makes them powerful.
2.4 MODELING APPROACH If we accept that we are going to construct computer models to predict the performance of pipeline systems, then how should this be done? What aspects of the prototype can and should be emphasized in the model? What is the basis of the approximations, and what principles constrain the approach? These topics are discussed in this section. Perhaps surprisingly, if we wish to model the behavior of any physical system, a remarkably small number of fundamental relations are available (or required). In essence, we seek to answer three simple questions: where?, what? and how? The following sections provide elaboration.
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HYDRAULICS OF PRESSURIZED FLOW
Hydraulics of pressurized flow 2.5
The first question is resolved most easily. Because flow in a pipe system can almost always be assumed to be one-dimensional, the question of where is resolved by assuming a direction of flow in each link of pipe. This assumed direction gives a unique orientation to the specification of distance, discharge, and velocity. Positive values of these variables indicate flow in the assumed direction, whereas negative values indicate reverse flows. The issues of what and how require more careful development.
2.4.1 Properties of Matter (What?) The question of ‘What?’ directs our attention to the matter within the control system. In the case of a hydraulic system, this is the material that makes up the pipe walls, or fills the interior of a pipe or reservoir, or that flows through a pump. Eventually a modeller must account for all these issues, but we start with the matter that flows, typically consisting mostly of water with various degrees of impurities. In fact, water is so much a part of our lives that we seldom question its role. Yet water possesses a unique combination of chemical, physical, and thermal properties that makes it ideally suited for many purposes. In addition, although important regional shortages may exist, water is found in large quantities on the surface of the earth. For both these reasons, water plays a central role in both human activity and natural processes. One surprising feature of the water molecule is its simplicity, formed as it as from two diatomic gases, hydrogen (H2) and oxygen (O2). Yet the range and variety of water’s properties are remarkable (Table 2.1 provides a partial list). Some property values in the
TABLE 2.1
Selected Properties of Liquid Water
Physical Properties 1. High density—liq < 1 000 kg/m3 2. Density maximum at 4ºC—i.e., above freezing! 3. High viscosity (but a Newtonian fluid)— ≈ 10–3 N · s/m2 4. High surface tension— ≈ 73 N/m 5. High bulk modulus (usually assumed incompressible)—K ≈ 2.07 GPa Thermal Properties 1. Specific Heat—highest except for NH3—c ≈ 4.187 kJ/(kg·ºC) 2. High heat of vaporization—cv ≈ 2.45 MJ/kg 3. High heat of fusion—cf ≈ 0.36 MJ/kg 4. Expands on freezing—in almost all other compounds, solid > liq 5. High boiling point—c.f., H2 (20 K), O2 (90 K) and H2O (373 K) 6 Good conductor of heat relative to other liquids and nonmetal solids. Chemical and Other Properties 1. Slightly ionized—water is a good solvent for electrolytes and nonelectrolytes 2. Transparent to visible light; opaque to near infrared 3. High dielectric constant—responds to microwaves and electromagnetic fields Note: The values are approximate. All the properties listed are functions of temperature, pressure, water purity, and other factors that should be known if more exact values are to be assigned. For example, surface tension is greatly influenced by the presence of soap films, and the boiling point depends on water purity and confining pressure. The values are generally indicative of conditions near 10ºC and one atmosphere of pressure.
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HYDRAULICS OF PRESSURIZED FLOW
2.6
Chapter Two
table —especially density and viscosity values—are used regularly by pipeline engineers. Other properties, such as compressibility and thermal values, are used indirectly, primarily to justify modeling assumptions, such as the flow being isothermal and incompressible. Many properties of water depend on intermolecular forces that create powerful attractions (cohesion) between water molecules. That is, although a water molecule is electrically neutral, the two hydrogen atoms are positioned to create a tetrahedral charge distribution on the water molecule, allowing water molecules to be held strongly together with the aid of electrostatic attractions. These strong internal forces—technically called ‘hydrogen bonds’—arise directly from the non-symmetrical distribution of charge. The chemical behavior of water also is unusual. Water molecules are slightly ionized, making water an excellent solvent for both electrolytes and nonelectrolytes. In fact, water is nearly a universal solvent, able to wear away mountains, transport solutes, and support the biochemistry of life. But the same properties that create so many benefits also create problems, many of which must be faced by the pipeline engineer. Toxic chemicals, disinfection byproducts, aggressive and corrosive compounds, and many other substances can be carried by water in a pipeline, possibly causing damage to the pipe and placing consumers at risk. Other challenges also arise. Water’s almost unique property of expanding on freezing can easily burst pipes. As a result, the pipeline engineer either may have to bury a line or may need to supply expensive heat-tracing systems on lines exposed to freezing weather, particularly if there is a risk that standing water may sometimes occur. Water’s high viscosity is a direct cause of large friction losses and high energy costs whereas its vapor properties can create cavitation problems in pumps, valves, and pipes. Furthermore, the combination of its high density and small compressibility creates potentially dramatic transient conditions. We return to these important issues after considering how pipeline flows respond to various physical constraints and influences in the next section.
2.4.2
Laws of Conservation (How?)
Although the implications of the characteristics of water are enormous, no mere list of its properties will describe a physical problem completely. Whether we are concerned with water quality in a reservoir or with transient conditions in a pipe, natural phenomena also obey a set of physical laws that contributes to the character and nature of a system’s response. If engineers are to make quantitative predictions, they must first understand the physical problem and the mathematical laws that model its behavior. Basic physical laws must be understood and be applied to a wide variety of applications and in a great many different environments: from flow through a pump to transient conditions in a channel or pipeline. The derivations of these equations are not provided, however, because they are widely available and take considerable time and effort to do properly. Instead, the laws are presented, summarized, and discussed in the pipeline context. More precisely, a quantitative description of fluid behavior requires the application of three essential relations: (1) a kinematic relation obtained from the law of mass conservation in a control volume, (2) equations of motion provided by both Newton’s second law and the energy equation, and (3) an equation of state adapted from compressibility considerations, leading to a wavespeed relation in transient flow and justifying the assumption of an incompressible fluid in most steady flow applications. A few key facts about mass conservation and Newton’s second law are reviewed briefly in the next section. Consideration of the energy equation is deferred until steady flow is discussed in more detail, whereas further details about the equation of state are introduced along with considerations of unsteady flow.
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HYDRAULICS OF PRESSURIZED FLOW
Hydraulics of pressurized flow 2.7
2.4.3
Conservation of Mass
One of a pipeline engineers most basic, but also most powerful, tools is introduced. in this section. The central concept is that of conservation of mass a and its key expression is the continuity or mass conservation equation. One remarkable fact about changes in a physical system is that not everything changes. In fact, most physical laws are conservation laws: They are generalized statements about regularities that occur in the midst of change. As Ford (1973) said: A conservation law is a statement of constancy in nature—in particular, constancy during change. If for an isolated system a quantity can be defined that remains precisely constant, regardless of what changes may take place within the system, the quantity is said to be absolutely conserved. A number of physical quantities have been found that are conserved in the sense of Fords quotation. Examples include energy (if mass is accounted for), momentum, charge, and angular momentum. One especially important generalization of the law of mass conservation includes both nuclear and chemical reactions (Hatsopoulos and Keenan, 1965). 2.4.3.1 Law of Conservation of Chemical Species “Molecular species are conserved in the absence of chemical reactions and atomic species are conserved in the absence of nuclear reactions”. In essence, the statement is nothing more a principle of accounting, stating that number of atoms or molecules that existed before a given change is equal to the number that exists after the change. More powerfully, the principle can be transformed into a statement of revenue and expenditure of some commodity over a definite period of time. Because both hydraulics and hydrology are concerned with tracking the distribution and movement of the earth’s water, which is nothing more than a particular molecular species, it is not surprising that formalized statements of this law are used frequently. These formalized statements are often called water budgets, typically if they apply to an area of land, or continuity relations, if they apply in a well-defined region of flow (the region is well–defined; the flow need not be). The principle of a budget or continuity equation is applied every time we balance a checkbook. The account balance at the end of any period is equal to the initial balance plus all the deposits minus all the withdrawals. In equation form, this can be written as follows: (balance)f (balance)i ∑ deposits ∑ withdrawals Before an analogous procedure can be applied to water, the system under consideration must be clearly defined. If we return to the checking-account analogy, this requirement simply says that the deposits and withdrawals included in the equation apply to one account or to a well-defined set of accounts. In hydraulics and hydrology, the equivalent requirement is to define a control volume—a region that is fixed in space, completely surrounded by a “control surface,” through which matter can pass freely. Only when the region has been precisely defined can the inputs (deposits) and outputs (withdrawals) be identified unambiguously. If changes or adjustments in the water balance (∆S) are the concern, the budget concept can be expressed as ∆S Sf Si (balance)f (balance)i Vi Vo
(2.1)
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HYDRAULICS OF PRESSURIZED FLOW
2.8
Chapter Two
where Vi represents the sum of all the water entering an area, and Vo indicates the total volume of water leaving the same region. More commonly, however, a budget relation such as Eq. 2.1 is written as a rate equation. Dividing the “balance’’ equation by ∆t and taking the limit as ∆t goes to zero produces S’ dS I O (2.2) dt where the derivative term S’ is the time rate of change in storage, S is the water stored in the control volume, I is the rate of which water enters the system (inflow), and O is the rate of outflow. This equation can be applied in any consistent volumetric units (e.g., m3/s, ft3/s, L/s, ML/day, etc.) When the concept of conservation of mass is applied to a system with flow, such as a pipeline, it requires that the net amount of fluid flowing into the pipe must be accounted for as fluid storage within the pipe. Any mass imbalance (or, in other words, net mass exchange) will result in large pressure changes in the conduit because of compressibility effects. 2.4.3.2 Steady Flow Assuming, in addition, that the flow is steady, Eq. 2.2 can be reduced further to inflow = outflow or I = O. Since the inflow and outflow may occur at several points, this is sometimes re-written as
冘
Vi Ai
inflow
冘
Vi Ai
(2.3)
outflow
Equation (2.3) states that the rate of flow into a control volume is equal to the rate of outflow. This result is intuitively satisfying since no accumulation of mass or volume should occur in any control volume under steady conditions. If the control volume were taken to be the junction of a number of pipes, this law would take the form of Kirchhoff’s current law—the sum of the mass flow in all pipes entering the junction equals the sum of the mass flow of the fluid leaving the junction. For example, in Fig. 2.2, continuity for the control volume of the junction states that Q1 Q2 Q3 Q4
(2.4)
2.4.4 Newton’s Second Law When mass rates of flow are concerned, the focus is on a single component of chemical species. However, when we introduce a physical law, such as Newton’s law of motion, we obtain something even more profound: a relationship between the apparently unrelated quantities of force and acceleration. More specifically, Newton’s second law relates the changes in motion of a fluid or solid to the forces that cause the change. Thus, the statement that the resultant of all external forces, including body forces, acting on a system is equal to the rate of change of momentum of this system with respect to time. Mathematically, this is expressed as
冘
d(mv) Fext = dt
(2.5)
where t is the time and Fext represents the external forces acting on a body of mass m moving with velocity υ. If the mass of the body is constant, Eq. (2.5) becomes
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HYDRAULICS OF PRESSURIZED FLOW
Hydraulics of pressurized flow 2.9
Q4 CV
Q3
Q1
Q2
FIGURE 2.2 Continuity at a pipe junction Q1 + Q2 = Q3 + Q4
冘
v ma Fext = m d (2.6) dt where a is the acceleration of the system (the time rate of change of velocity). In closed conduits, the primary forces of concern are the result of hydrostatic pressure, fluid weight, and friction. These forces act at each section of the pipe to produce the net acceleration. If these forces and the fluid motion are modeled mathematically, the result is a “dynamic relation” describing the transient response of the pipeline. For a control volume, if flow properties at a given position are unchanging with time, the steady form of the moment equation can be written as
冘 冕
0
Fext =
cs
ρv(v ⴢ n) dA
(2.7)
where the force term is the net external force acting on the control volume and the right hand term gives the net flux of momentum through the control surface. The integral is taken over the entire surface of the control volume, and the integrand is the incremental amount of momentum leaving the control volume. The control surface usually can be oriented to be perpendicular to the flow, and one can assume that the flow is incompressible and uniform. With this assumption, the momentum equation can be simplified further as follows:
冘
Fext = (ρAv)out (ρAv)in ρQ(vout vin)
(2.8)
where Q is the volumetric rate of flow. Example: Forces at an Elbow. One direct application of the momentum relation is shown in Fig. 2.3, which indicates the flows and forces at elbow. The elbow is assumed to be mounted in a horizontal plane so that the weight is balanced by vertical forces (not shown).
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HYDRAULICS OF PRESSURIZED FLOW
2.10
Chapter Two
CV
Fy (PA )1
V1 Fx V2
y
(PA )2
x
FIGURE 2.3 Force and momentum fluxes at an elbow.
The reaction forces shown in the diagram are required for equilibrium if the elbow is to remain stationary. Specifically, the force Fx must resist both the pressure force and must account for the momentum-flux term. That is, taking x as positive to the right, direct application of the momentum equation gives (PA)1 Fx ρQ1
(2.9)
Fx (PA)1 ρQ1
(2.10)
Thus,
In a similar manner, but taking y as positive upward, direct application of the momentum equation gives (PA)2 Fy ρQ( 2)
(2.11)
(here the outflow gives a positive sign, but the velocity is in the negative direction). Thus, Fy (PA)2 + ρQ2
(2.12)
In both cases, the reaction forces are increased above what they would be in the static case because the associated momentum must either be established or be eliminated in the direction shown. Application of this kind of analysis is routine in designing thrust blocks, which are a kind of anchor used at elbows or bends to restrain the movement of pipelines.
2.5 SYSTEM CAPACITY: PROBLEMS IN TIME AND SPACE A water transmission or supply pipeline is not just an enclosed tube— it is an entire system that transports water, either by using gravity or with the aid of pumping, from its source to the general vicinity of the demand. It typically consists of pipes or channels with their associated control works, pumps, valves, and other components. A transmission sys-
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HYDRAULICS OF PRESSURIZED FLOW
Hydraulics of pressurized flow 2.11
tem is usually composed of a single-series line, as opposed to a distribution system that often consists of a complex network of interconnected pipes. As we have mentioned, there are many practical questions facing the designer of such a system. Do the pipes, reservoirs and pumps have a great enough hydraulic capacity? Can the flow be controlled to achieve the desired hydraulic conditions? Can the system be operated economically? Are the pipes and connections strong enough to withstand both unsteady and steady pressures? Interestingly, different classes of models are used to answer them, depending on the nature of the flow and the approximations that are justified. More specifically, issues of hydraulic capacity are usually answered by projecting demands (water requirements) and analyzing the system under steady flow conditions. Here, one uses the best available estimates of future demands to size and select the primary pipes in the system. It is the hydraulic capacity of the system, largely determined by the effective diameter of the pipeline, that links the supply to the demand. Questions about the operation and sizing of pumps and reservoirs are answered by considering the gradual variation of demand over relatively short periods, such as over an average day or a maximum day. In such cases, the acceleration of the fluid is often negligible and analysts use a quasi-steady approach: that is, they calculate forces and energy balances on the basis of steady flow, but the unsteady form is used for the continuity equation so that flows can be accumulated and stored. Finally, the issue of required strength, such as the pressure rating of pipes and fittings, is answered by considering transient conditions. Thus, the strength of a pipeline is determined at least in part by the pressures generated by a rapid transition between flow states. In this stage, short-term and rapid motions must be taken into account, because large forces and dangerous pressures can sometimes be generated. Here, forces are balanced with accelerations, mass flow rates with pressure changes. These transient conditions are discussed in more detail in section 2.8 and in chapter 10. A large number of different flow conditions are encountered in pipeline systems. To facilitate analysis, these conditions are often classified according to several criteria. Flow classification can be based on channel geometry, material properties, dynamic considerations (both kinematic and kinetic), or some other characteristic feature of the flow. For example, on the basis of fluid type and channel geometry, the flow can be classified as open-channel, pressure, or gas flow. Probably the most important distinctions are based on the dynamics of flow (i.e., hydraulics). In this way, flow is classified as steady or unsteady, turbulent or laminar, uniform or nonuniform, compressible or incompressible, or single phase or multiphase. All these distinctions are vitally important to the analyst: collectively, they determine which physical laws and material properties are dominant in any application. Steady flow: A flow is said to be steady if conditions at a point do not change with time. Otherwise a flow is unsteady or transient. By this definition, all turbulent flows, and hence most flows of engineering importance, are technically unsteady. For this reason, a more restrictive definition is usually applied: A flow is considered steady if the temporal mean velocity does not change over brief periods. Although the assumption is not formally required, pipeline flows are usually considered to be steady; thus, transient conditions represent an ‘abnormal’, or nonequilibrium, transition from one steady-state flow to another. Unless otherwise stated, the initial conditions in transient problems are usually assumed to be steady. Steady or equilibrium conditions in a pipe system imply a balance between the physical laws. Equilibrium is typified by steady uniform flow in both open channels and closed conduits. In these applications, the rate of fluid inflow to each segment equals the rate of
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HYDRAULICS OF PRESSURIZED FLOW
2.12
Chapter Two
outflow, the external forces acting on the flow are balanced by the changes in momentum, and the external work is compensated for by losses of mechanical energy. As a result, the fluid generally moves down an energy gradient, often visualized as flow in the direction of decreasing hydraulic grade-line elevations (e.g., Fig. 2.1). Quasi-steady flow. When the flow becomes unsteady, the resulting model that must be used depends on how fast the changes occur. When the rate of change is particularly slow, typically over a period of hours or days, the rate of the fluids acceleration is negligible. However, fluid will accumulate or be depleted at reservoirs, and rates of demand for water may slowly adjust. This allows the use of a quasi-steady or extended-duration simulation model. Compressible and Incompressible. If the density of the fluid is constant—both in time and throughout the flow field—a flow is said to be incompressible. Thus, is not a function of position or time in an incompressible flow. If changes in density are permitted or reguined the flow is compressible. Surge. When the rate of change in flow is moderate, typically occurring over a period of minutes, a surge model is often used. In North America, the term surge indicates an analysis of unsteady flow conditions in pipelines when the following assumptions are made: the fluid is incompressible (thus, its density is constant) and the pipe walls are rigid and do not deform. These two assumptions imply that fluid velocities are not a function of position along a pipe of constant cross-section and the flow is uniform. In other words, no additional fluid is stored in a length of pipe as the pressure changes; because velocities are uniform, the rate at which fluid enters a pipe is always equal to the rate of discharge. However, the acceleration of the fluid and its accumulation and depletion from reservoirs are accounted for in a surge model. Waterhammer. When rapid unsteady flow occurs in a closed conduit system, the transient condition is sometimes marked by a pinging or hammering noise, appropriately called waterhammer. However, it is common to refer to all rapidly changing flow conditions by this term, even if no audible shock waves are produced. In waterhammer models, it is usually assumed that the fluid is slightly compressible, and the pipe walls deform with changes in the internal pressure. Waterhammer waves propagate with a finite speed equal to the velocity of sound in the pipeline. The speed at which a disturbance is assumed to propagate is the primary distinction between a surge and a waterhammer model. Because the wavespeed parameter a is related to fluid storage, the wavespeed is infinite in surge or quasi-steady models. Thus, in effect, disturbances are assumed to propagate instantly throughout the pipeline system. Of course they do no such thing, because the wavespeed is a finite physical property of a pipe system, much like its diameter, wall thickness, or pipeline material. The implication of using the surge or quasi-steady approximation is that the unsteady behavior of the pipe system is controlled or limited by the rate at which the hydraulic boundary conditions (e.g., pumps, valves, reservoirs) at the ends of the pipe respond to the flow and that the time required for the pipeline itself to react is negligible by comparison. Although unsteady or transient analysis is invariably more involved than is steady-state modeling, neglecting these effects in a pipeline can be troublesome for one of two reasons: the pipeline may not perform as expected, possibly causing large remedial expenses, or the line may be overdesigned with respect to transient conditions, possibly causing unnecessarily large capital costs. Thus, it is essential for engineers to have a clear physical grasp of transient behavior and an ability to use the computer’s power to maximum advantage. One interesting point is that as long as one is prepared to assume the flow is compressible, the importance of compressibility does not need to be known a priory. In fact,
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HYDRAULICS OF PRESSURIZED FLOW
Hydraulics of pressurized flow 2.13
all the incompressible, quasi-steady, and steady equations are special cases of the full transient equations. Thus, if the importance of compressibility or acceleration effects is unknown, the simulation can correctly assume compressible flow behavior and allow the analysis to verify or contradict this assumption. Redistribution of water, whatever model or physical devices are used, requires control of the fluid and its forces, and control requires an understanding not only of physical law but also of material properties and their implications. Thus, an attempt to be more specific and quantitative about these matters will be made as this chapter progresses. In steady flow, the fluid generally moves in the direction of decreasing hydraulic grade-line elevations. Specific devices, such as valves and transitions, cause local pressure drops and dissipate mechanical energy; operating pumps do work on the fluid and increase downstream pressures while friction creates head losses more or less uniformly along the pipe length. Be warned, however—in transient applications, this orderly situation rarely exists. Instead, large and sudden variations of both discharge and pressure can occur and propagate in the system, greatly complicating analysis.
2.6 STEADY FLOW The design of steady flow in pipeline systems has two primary objectives. First, the hydraulic objective is to secure the desired pressure and flow rate at specific locations in the system. Second the economic objective is to meet the hydraulic requirements with the minimum expense. When a fluid flows in a closed conduit or open channel, it often experiences a complex interchange of various forms of mechanical energy. In particular, the work that is associated with moving the fluid through pressure differences is releted to changes in both gravitational potential energy and kinetic energy. In addition, the flow may lose mechanical energy as a result of friction, a loss that is usually accounted for by extremely small increases in the temperature of the flowing fluid (that is, the mechanical energy is converted to thermal form). More specifically, these energy exchanges are often accounted for by using an extended version of Bernoulli’s famous relationship. If energy losses resulting from friction are negligible, the Bernoulli equation takes the following form: v22 v21 p1 p2 2g z1 2g z2 γ γ
(2.13)
where p1 and p2 are the pressures at the end points, γ is the specific weight of the fluid, v1 and v2 are the average velocities at the end points, and z1 and z2 are the elevations of the end points with respect to an arbitrary vertical datum. Because of their direct graphical representation, various combinations of terms in this relationship are given special labels, historically called heads because of their association with vertical distances. Thus, Head
Definition
Associated with
Pressure head Elevation head Velocity head Piezometric head Total head
p/γ z v2/2g p/γ z p/γ z v2/2g
Flow work Gravitational potential energy Kinetic energy Pressure elevation head Pressure elevation velocity head
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HYDRAULICS OF PRESSURIZED FLOW
2.14
Chapter Two
A plot of piezometric head along a pipeline forms a line called the hydraulic grade line (HGL). Similarly, a plot of the total head with distance along a pipeline is called the energy grade line (EGL). In the vast majority of municipally related work, velocity heads are negligible and the EGL and HGL essentially become equivalent. If losses occur, the situation becomes a little more complex. The head loss hf is defined to be equal to the difference in total head from the beginning of the pipe to the end over a total distance L. Thus, hf is equal to the product of the slope of the EGL and the pipe length: hf L · Sf . When the flow is uniform, the slope of the EGL is parallel to that of the HGL, the difference in piezometric head between the end points of the pipe. Inclusion of a headloss term into the energy equation gives a useful relationship for describing 1-D pipe flow v22 v21 p1 p2 γ 2g z1 γ 2g z2 hf
(2.14)
In this relation, the flow is assumed to be from Point 1 to Point 2 and hf is assumed to be positive. Using capital H to represent the total head, the equation can be rewritten as H1 H2 hf In essence, a head loss reduces to the total head that would have occurred in the system if the loss were not present (Fig. 2.1). Since the velocity head term is often small, the total head in the above relation is often approximated with the piezometric head. Understanding head loss is important for designing pipe systems so that they can accommodate the design discharge. Moreover, head losses have a direct effect on both the pumping capacity and the power consumption of pumps. Consequently, an understanding of head losses is important for the design of economically viable pipe systems. The occurrence of head loss is explained by considering what happens at the pipe wall, the domain of boundary layer theory. The fundamental assertion of the theory is that when a moving fluid passes over a solid surface, the fluid immediately in contact with the surface attains the velocity of the surface (zero from the perspective of the surface). This “no slip” condition gives rise to a velocity gradient in which fluid further from the surface has a larger (nonzero) velocity relative to the velocity at the surface, thus establishing a shear stress on the fluid. Fluid that is further removed from the solid surface, but is adjacent to slower moving fluid closer to the surface, is itself decelerated because of the fluid’s own internal cohesion, or viscosity. The shear stress across the pipe section is zero at the center of the pipe, where the average velocity is greatest, and it increases linearly to a maximum at the pipe wall. The distribution of the shear stress gives rise to a parabolic distribution of velocity when the flow is laminar. More frequently, the flow in a conduit is turbulent. Because turbulence introduces a complex, random component into the flow, a precise quantitative description of turbulent flow is impossible. Irregularities in the pipe wall lead to the formation of eddy currents that transfer momentum between faster and slower moving fluid, thus dissipating mechanical energy. These random motions of fluid increase as the mean velocity increases. Thus, in addition to the shear stress that exists for laminar flow, an apparent shear stress exists because of the exchange of material during turbulent flow. The flow regime–whether laminar, turbulent, or transitional–is generally classified by referring to the dimensionless Reynold’s number (Re). In pipelines, Re is given as VDρ Re µ
(2.15)
where V is the mean velocity of the fluid, D is the pipe diameter, ρ is the fluid density, and
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HYDRAULICS OF PRESSURIZED FLOW
Hydraulics of pressurized flow 2.15
µ is the dynamic viscosity. Although the exact values taken to limit the range of Re vary with author and application, the different flow regimes are often taken as follows: (1) laminar flow: Re ≤ 2000, (2) transitional flow: 2000 ≤ Re ≤ 4000, and (3) turbulent flow: Re > 4000. These flow regime have a direct influence on the head loss experienced in a pipeline system.
2.6.1 Turbulent Flow Consider an experiment in which a sensitive probe is used to measure flow velocity in a pipeline carrying a flowing fluid. The probe will certainly record the mean or net component of velocity in the axial direction of flow. In addition, if the flow in the pipeline is turbulent, the probe will record many small and abrupt variations in velocity in all three spatial directions. As a result of the turbulent motion, the details of the flow pattern will change randomly and constantly with time. Even in the simplest possible system–an uniform pipe carrying water from a constant-elevation upstream reservoir to a downstream valve–the detailed structure of the velocity field will be unsteady and exceedingly complex. Moreover, the unsteady values of instantaneous velocity will exist even if all external conditions at both the reservoir and valve are not changing with time. Despite this, the mean values of velocity and pressure will be fixed as long as the external conditions do not change. It is in this sense that turbulent flows can be considered to be steady. The vast majority of flows in engineering are turbulent. Thus, unavoidably, engineers must cope with both the desirable and the undesirable characteristics of turbulence. On the positive side, turbulent flows produce an efficient transfer of mass, momentum, and energy within the fluid. In fact, the expression to “stir up the pot” is an image of turbulence; it implies a vigorous mixing that breaks up large-scale order and structure in a fluid. But the rapid mixing also may create problems for the pipeline engineer. This “down side” can include detrimental rates of energy loss, high rates of corrosion, rapid scouring and erosion, and excessive noise and vibration as well as other effects. How does the effective mixing arise within a turbulent fluid? Physically, mixing results from the random and chaotic fluctuations in velocity that exchange fluid between different regions in a flow. The sudden, small-scale changes in the instantaneous velocity tend to cause fast moving “packets” of fluid to change places with those of lower velocity and vice verse. In this way, the flow field is constantly bent, folded, and superimposed on itself. As a result, large-scale order and structure within the flow is quickly broken down and torn apart. But the fluid exchange transports not only momentum but other properties associated with the flow as well. In essence, the rapid and continual interchange of fluid within a turbulent flow creates both the blessing and the curse of efficient mixing. The inherent complexity of turbulent flows introduces many challenges. On one hand, if the velocity variations are ignored by using average or mean values of fluid properties, a degree of uncertainty inevitably arises. Details of the flow process and its variability will be avoided intentionally, thereby requiring empirical predictions of mean flow characteristics (e.g., head-loss coefficients and friction factors). Yet, if the details of the velocity field are analyzed, a hopelessly complex set of equations is produced that must be solved using a small time step. Such models can rarely be solved even on the fastest computers. From the engineering view point, the only practical prescription is to accept the empiricism necessitated by flow turbulence while being fully aware of its difficulties–the averaging process conceals much of what might be important. Ignoring the details of the fluid’s motion can, at times, introduce significant error even to the mean flow calculations. When conditions within a flow change instantaneously both at a point and in the mean, the flow becomes unsteady in the full sense of the word. For example, the downstream valve in a simple pipeline connected to a reservoir might be closed rapidly, creating shock
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HYDRAULICS OF PRESSURIZED FLOW
2.16
Chapter Two
waves that travel up and down the conduit. The unsteadiness in the mean values of the flow properties introduces additional difficulties into a problem that was already complex. Various procedures of averaging, collecting, and analyzing data that were well justified for a steady turbulent flow are often questionable in unsteady applications. The entire situation is dynamic: Rapid fluctuations in the average pressure, velocity, and other properties may break or damage the pipe or other equipment. Even in routine applications, special care is required to control, predict, and operate systems in which unsteady flows commonly occur. The question is one of perspective. The microscopic perspective of turbulence in flows is bewildering in its complexity; thus, only because the macroscopic behavior is relatively predictable can turbulent flows be analyzed. Turbulence both creates the need for approximate empirical laws and determines the uncertainty associated with using them. The great irregularity associated with turbulent flows tends to be smoothed over both by the empirical equations and by a great many texts.
2.6.2 Head Loss Caused by Friction A basic relation used in hydraulic design of a pipeline system is the one describing the dependence of discharge Q (say in m3/s) on head loss hf (m) caused by friction between the flow of fluid and the pipe wall. This section discusses two of the most commonly used head-loss relations: the Darcy-Weisbach and Hazen-Williams equations. The Darcy-Weisbach equation is used to describe the head loss resulting from flow in pipes in a wide variety of applications. It has the advantage of incorporating a dimensionless friction factor that describes the effects of material roughness on the surface of the inside pipe wall and the flow regime on retarding the flow. The Darcy-Weisbach equation can be written as L V2 Q2 hf ,DW f 0.0826 5 Lf D 2g D
(2.16)
where hf ,DW = head loss caused by friction (m), f = dimensionless friction factor, L = pipe length (m), D = pipe diameter (m), V = Q/A = mean flow velocity (m/s), Q = discharge (m3/s), A = cross-sectional area of the pipe (m2), and g = acceleration caused by gravity (m/s2). For noncircular pressure conduits, D is replaced by 4R, where R is the hydraulic radius. The hydraulic radius is defined as the cross-sectional area divided by the wetted perimeter or, R = A/P. Note that the head loss is directly proportional to the length of the conduit and the friction factor. Obviously, the rougher a pipe is and the longer the fluid must travel, the greater the energy loss. The equation also relates the pipe diameter inversely to the head loss. As the pipe diameter increases, the effects of shear stress at the pipe walls are felt by less of the fluid, indicating that wider pipes may be advantageous if excavation and construction costs are not prohibitive. Note in particular that the dependence of the discharge Q on the pipe diameter D is highly nonlinear; this fact has great significance to pipeline designs because head losses can be reduce dramatically by using a large-diameter pipe, whereas an inappropriately small pipe can restrict flow significantly, rather like a partially closed valve. For laminar flow, the friction factor is linearly dependent on the Re with the simple relationship f = 64/Re. For turbulent flow, the friction factor is a function of both the Re and the pipes relative roughness. The relative roughness is the ratio of equivalent uniform sand grain size and the pipe diameter (e/D), as based on the work of Nikuradse (1933), who
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HYDRAULICS OF PRESSURIZED FLOW
Hydraulics of pressurized flow 2.17
experimentally measured the resistance to flow posed by various pipes with uniform sand grains glued onto the inside walls. Although the commercial pipes have some degree of spatial variance in the characteristics of their roughness, they may have the same resistance characteristics as do pipes with a uniform distribution of sand grains of size e. Thus, if the velocity of the fluid is known, and hence Re, and the relative roughness is known, the friction factor f can be determined by using the Moody diagram or the Colebrook-White equation. Jeppson (1976) presented a summary of friction loss equations that can be used instead of the Moody diagram to calculate the friction factor for the Darcy-Weisbach equation. These equations are applicable for Re greater than 4000 and are categorized according to the type of turbulent flow: (1) turbulent smooth, (2) transition between turbulent smooth and wholly rough, and (3) turbulent rough. For turbulent smooth flow, the friction factor is a function of Re: 1 2log (Re兹f苶) (2.17) 兹苶f For the transition between turbulent smooth and wholly rough flow, the friction factor is a function of both Re and the relative roughness e/D. This friction factor relation is often summarized in the Colebrook White equation: e/D 2.51 1 2log + 3 . 7 兹苶f Re兹f苶
(2.18)
When the flow is wholly turbulent (large Re and e/D), the Darcy-Weisbach friction factor becomes independent of Re and is a function only of the relative roughness: 1 1.14 2log (e/D) (2.19) 兹苶f In general, Eq. (2.16) is valid for all turbulent flow regimens in a pipe,, where as Eq. (2.22) is merely an approximation that is valid for the hydraulic rough flow. In a smoothpipe flow, the viscous sublayer completely submerges the effect of e on the flow. In this case, the friction factor f is a function of Re and is independent of the relative roughness e/D. In rough-pipe flow, the viscous sublayer is so thin that flow is dominated by the roughness of the pipe wall and f is a function only of e/D and is independent of Re. In the transition, f is a function of both e/D and Re. The implicit nature of f in Eq. (2.18) is inconvenient in design practice. However, this difficulty can be easily overcome with the help of the Moody diagram or with one of many available explicit approximations. The Moody diagram plots Re on the abscissa, the resistance coefficient on one ordinate and f on the other, with e/D acting as a parameter for a family of curves. If e/D is known, then one can follow the relative roughness isocurve across the graph until it intercepts the correct Re. At the corresponding point on the opposite ordinate, the appropriate friction factor is found; e/D for various commercial pipe materials and diameters is provided by several manufacturers and is determined experimentally. A more popular current alternative to graphical procedures is to use an explicit mathematical form of the friction-factor relation to approximate the implicit Colebrookwhite equation. Bhave (1991) included a nice summary of this topic. The popular network-analysis program EPANET and several other codes use the equation of Swanee and Jain (1976), which has the form f 0.25 (2.20) e 5.74 2 log Re0.9 3.7D
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HYDRAULICS OF PRESSURIZED FLOW
2.18
Chapter Two
To circumvent considerations of roughness estimates and Reynolds number dependencies, more direct relations are often used. Probably the most widely used of these empirical head-loss relation is the Hazen-Williams equation, which can be written as Q Cu CD2.63S0.54
(2.21)
where Cu unit coefficient (Cu 0.314 for English units, 0.278 for metric units), Q discharge in pipes, gallons/s or m3/s, L length of pipe, ft or m, d internal diameter of pipe, inches or mm, C Hazen-Williams roughness coefficient, and S = the slope of the energy line and equals hf /L. The Hazen-Williams coefficient C is assumed constant and independent of the discharge (i.e., R e). Its values range from 140 for smooth straight pipe to 90 or 80 for old, unlined, tuberculated pipe. Values near 100 are typical for average conditions. Values of the unit coefficient for various combinations of units are summarized in Table 2.2. In Standard International (SI) units, the Hazen-Williams relation can be rewritten for head loss as hf ,HW 10.654
1 0.54 Q C
1 4.87 L D
(2.22)
where hf,HW is the Hazen-Williams head loss. In fact, the Hazen-Williams equation is not the only empirical loss relation in common use. Another loss relation, the Manning equation, has found its major application in open channel flow computations. As with the other expressions, it incorporates a parameter to describe the roughness of the conduit known as Manning’s n. Among the most important and surprisingly difficult hydraulic parameter is the diameter of the pipe. As has been mentioned, the exponent of diameter in head-loss equations is large, thus indicating high sensitivity to its numerical value. For this reason, engineers
EGL H1
H3
H2
Q1
Q1
(a)
EGL
Q1 H1
H2
Q2 (b)
FIGURE 2.4 Flow in series and parallel pipes.
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HYDRAULICS OF PRESSURIZED FLOW
Hydraulics of pressurized flow 2.19 TABLE 2.2
Unit Coefficient Cu for the Hazen-Williams Equation
Units of Discharge Q
Units of Diameter D
Unit Coeficient Cu
MGD ft3/s GPM GPD m3/s
ft ft in in m
0.279 0.432 0.285 405 0.278
and analysts must be careful to obtain actual pipe diameters often from manufacturers; the use of nominal diameters is not recommended. Yet another complication may arise, however. The diameter of a pipe often changes with time, typically as a result of chemical depositions on the pipe wall. For old pipes, this reduction in diameter is accounted for indirectly by using an increased value of pipe resistance. Although this approach may be reasonable under some circumstances, it may be a problem under others, especially for unsteady conditions. When ever possible, accurate diameters are recommended for all hydraulic calculations. However, some combinations of pipes (e.g., pipes in series or parallel; Fig. 2.4) can actualy be represented by a single equivalent diamenter of pipe.
2.6.3 Comparison of Loss Relations It is generally claimed that the Darcy-Weisbach equation is superior because it is the oretically based, where as both the Manning equation and the Hazen-Williams expression use empirically determined resistance coefficients. Although it is true that the functional relationship of the Darcy-Weisbach formula reflects logical associations implied by the dimensions of the various terms, determination of the equivalent uniform sand-grain size is essentially experimental. Consequently, the relative roughness parameter used in the Moody diagram or the Colebrook-White equations is not theoretically determined. In this section, the Darcy-Weisbach and Hazen-Williams equations are compared briefly using a simple pipe as an example. In the hydraulic rough range, the increase in ∆hf can be explained easily when the ratio of Eq. (2.16) to Eq. (2.22) is investigated. For hydraulically rough flow, Eq. (2.18) can be simplified by neglecting the second term 2.51 (Re兹f苶) of the logarithmic argument. This ratio then takes the form of hf ,HW e 2 D0.13 1 128.94 1.14 2 log 1.8 52 D Q0.148 C hf ,DW
(2.23)
which shows that in most hydraulic rough cases, for the same discharge Q, a larger head loss hf is predicted using Eq. (2.16) than when using Eq. (2.22). Alternatively, for the same head loss, Eq. (2.22) returns a smaller discharge than does Eq. (2.16). When comparing head-loss relations for the more general case, a great fuss is often made over unimportant issues. For example, it is common to plot various equations on the Moody diagram and comment on their differences. However, such a comparison is of secondary importance. From a hydraulic perspective, the point is this: Different equations should still produce similar similar head discharge behavior. That is, the physical relation between head loss and flow for a physical segment of pipe should be predicted well by
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HYDRAULICS OF PRESSURIZED FLOW
2.20
Chapter Two
any practical loss relation. Said even more simply, the issue is how well the hf versus Q curves compare. To compare the values of hf determined from Eq. (2.16) and those from Eq. (2.22), consider a pipe for which the parameters D, L and C are specified. Using the HazenWilliams relation, it is then possible to calculate hf for a given Q. Then, the DarcyWeisbach f can be obtained, and with the Colebrook formula Eq. (2.18), the equivalent value of roughness e can be found. Finally, the variation of head with discharge can be plotted for a range of flows. This analysis is performed for two galvanized iron pipes with e 0.15 mm. One pipe has a diameter of 0.1 m and a length of 100 m; and the dimensions of the other pipe are D 1.0 m and L = 1000 m, respectively. The Hazen-Williams C for galvanized iron pipe is approximately 130. Different C values for these two pipes to demonstrate the shift and change of the range within which ∆hf is small. The results of the calculated hf – Q relation and the difference ∆hf of the head loss of the two methods for the same discharge are shown in Figs. 2.5 and 2.6. If hf ,DW denotes the head loss determined by using Eq. (2.16) and hf ,HW that using Eq. (2.22), ∆hf (m) can be ∆hf hf ,DW hf ,DW
(2.24)
where by the Darcy-Weisbach head loss hf ,DW is used as a reference for comparison. Figures 2.5 and 2.6 show the existence of three ranges: two ranges, within which hf ,DW hf,DW , and the third one for which hf ,DW hf ,DW. The first range of hf ,DW hf ,DW is at a lower head loss and is small. It seems that the difference the result of ∆hf in this case is the result of the fact that the Hazen-Williams formula is not valid for the hydraulic smooth and the smooth-to-transitional region. Fortunately, this region is seldom important for design purposes. At high head losses, the Hazen-Williams formula tends to produce a discharge that is smaller than the one produced by the Darcy-Weisbach equation. For a considerable part of the curve–primarily the range within which hf ,DW hf ,DW– ∆hf is small compared with the absolute head loss. It can be shown that the range of small ∆hf changes is shifted when different values of Hazen-Williams’s C are used for the calculation. Therefore, selecting the proper value of C, which represents an appropriate 1.2
25
L D e C
1.0
Darcy-Weisbach 20
Hazen-Williams
(m)
0.8 15
= 100 m = 0.1 m = 0.15 mm = 122.806
0.6
0.4
10
0.2 5 0
0
-0.2 0
0.005
0.01
0.015
0.02
Q (cms)
0.025
0.03
0.035
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
Q (cms)
FIGURE 2.5 Comparison of Hazen-Williams and Darcy-Weisbach loss relations (smaller diameter).
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HYDRAULICS OF PRESSURIZED FLOW
Hydraulics of pressurized flow 2.21 9
120
8 Darcy-Weisbach
100
L D e C
7
Hazen-Williams 6 80
= 1000 m = 1.0 m = 0.15 mm = 124.923
(m)
5 4
60
3 40
2 1
20
0 -1
0 0
2
4
6
Q (cms)
8
10
0
2
4
6
8
10
Q (cms)
FIGURE 2.6 Comparison of Hazen-Williams and Darcy-Weisbach loss relations (larger diameter).
point on the head-discharge curve, is essential. If such a C value is used, ∆hf is small, and whether the Hazen-Williams formula or the Darcy-Weisbach equation is used for the design will be of little importance. This example shows both the strengths and the weaknesses of using Eq. (2.22) as an approximation to Eq. (2.16). Despite its difficulties, the Hazen-Williams formula is often justified because of its conservative results and its simplicity of use. However, choosing a proper value of either the Hazen-Williams C or the relative roughness e/D is often difficult. In the literature, a range of C values is given for new pipes made of various materials. Selecting an appropriate C value for an old pipe is even more difficult. However, if an approximate value of C or e is used, the difference between the head-loss equations is likely to be inconsequential. Head loss also is a function of time. As pipes age, they are subject to corrosion, especially if they are made of ferrous materials and develop rust on the inside walls, which increases their relative roughness. Chemical agents, solid particles, or both in the fluid can gradually degrade the smoothness of the pipe wall. Scaling on the inside of pipes can occur if the water is hard. In some instances, biological factors have led to time-dependent head loss. Clams and zebra mussels may grow in some intake pipes and may in some cases drastically reduce discharge capacities.
2.6.4 Local Losses Head loss also occurs for reasons other than wall friction. In fact, local losses occur whenever changes occur in the velocity of the flow: for example, changes in the direction of the conduit, such as at a bend, or changes in the cross-sectional area, such as an aperture, valve or gauge. The basic arrangement of flow and pressure is illustrated for a venturi contraction in Fig. 2.7. The mechanism of head loss in the venturi is typical of many applications involving local losses. As the diagram indicates, there is a section of flow contraction into which the flow accelerates, followed by a section of expansion, into which the flow decelerates. This aspect of the venturi, or a reduced opening at a valve, is nicely described by the continuity equation. However, what happens to the pressure is more interesting and more important. Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
HYDRAULICS OF PRESSURIZED FLOW
2.22
Chapter Two ∆ hf EGL EGL HGL
V2 2g
HGL
Q1 1
2
FIGURE 2.7 Pressure relations in a venturí contraction.
As the flow accelerates, the pressure decreases according to the Bernoulli relation. Everything goes smoothly in this case because the pressure drop and the flow are in the same direction. However, in the expansion section, the pressure increases in the downstream direction. To see why this is significant, consider the fluid distributed over the cross section. In the center of the pipe, the fluid velocity is high; the fluid simply slow down as it moves into the region of greater pressure. But what about the fluid along the wall? Because it has no velocity to draw on, it tends to respond to the increase in pressure in the downstream direction by flowing upstream, counter to the normal direction of flow. That is, the flow tends to separate, which can be prevented only if the faster moving fluid can “pull it along” using viscosity. If the expansion is too abrupt, this process is not sufficient, and the flow will separate, creating a region of recirculating flow within the main channel. Such a region causes high shear stresses, irregular motion, and large energy losses. Thus, from the view point of local losses, nothing about changes in pressure is symmetrical—adverse pressure gradients or regions of recirculating flow are crucially important with regard’s to local losses. Local head losses are often expressed in terms of the velocity head as v2 hl k 2g
(2.25)
where k is a constant derived empirically from testing the head loss of the valve, gauge, and so on, and is generally provided by the manufacturer of the device. Typical forms for this relation are provided in Table 2.3 (Robertson and Crowe, 1993).
2.6.5 Tractive Force Fluid resistance also implies a flux in momentum and generates a tractive force, which raises a number of issues of special significance to the two-phase (liquid-solid) flows found in applications of transport of slurry and formation of sludge. In these situations, Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
HYDRAULICS OF PRESSURIZED FLOW
Hydraulics of pressurized flow 2.23 TABLE 2.3 Description
Local Loss Coefficients at Transitions Sketch
Pipe entrance hL KeV2/2g
Additional Data
K
Source
r/d 0.0 0.1 0.2
Ke 0.50 0.12 0.03
(1)
D2/D1 0.0 0.20 0.40 0.60 0.80 0.90
Kc 0 5 60º 0.08 0.08 0.07 0.06 0.05 0.04
Kc 0 5 180º 0.50 0.49 0.42 0.32 0.18 0.10
D1/D2
KE 0 5 10º
KE 0 5 180º
0.13 0.11 0.06 0.03
1.00 0.92 0.72 0.42 0.16
Contraction
hL KeV22/2g Expansion
0.0 0.20 0.40 0.60 0.80
hL KEV21/2g 90º miter bend 90º miter bend
Threaded pipe fittings
(1)
Without vanes
Kb 1.1
(26)
With vanes
Kb 0.2
(26)
r/d 1 2 4 6 8 10 Globe valve—wide open Angle valve—wide open Gate valve—wide open Gatevalve—half open Return bend Tee 90º elbow 45º elbow
(1)
Kb 0.35 0.19 0.16 0.21 0.28 0.32 Kv 10.0 Kv 5.0 Kv 0.2 Kv 5.6 Kb 2.2 Kt 1.8 Kb 0.9 Kb 0.4
(3) and (13)
(26)
the tractive force has an important influence on design velocities: The velocity cannot be too small or the tractive force will be insufficient to carry suspended sediment and deposition will occur. Similarly, if design velocities are too large, the greater tractive force will increase rates of erosion and corrosion in the channel or pipeline, thus raising maintenance and operational costs. Thus, the general significance of tractive force relates to designing
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HYDRAULICS OF PRESSURIZED FLOW
2.24
Chapter Two
self-cleansing channel and pressure-flow systems and to stable channel design in erodible channels. Moreover, high tractive forces are capable of causing water quality problems in distribution system piping through the mechanism of biofilm sloughing or suspension of corrosion by-products.
2.6.6 Conveyance System Calculations: Steady Uniform Flow A key practical concern in the detailed calculation of pressure flow and the estimation of pressure losses. Because the practice of engineering requires competent execution in a huge number of contexts, the engineer will encounter many different applications in practice. compare, for example Fig. 2.4 to 2.8 In fact, the number of applied topics is so large that comprehensive treatment is impossible. Therefore, this chapter emphasizes a systematic presentation of the principles and procedures of problem-solving to encourage the engineer’s ability to generalize. To illustrate the principles of hydraulic analysis, this section includes an example that demonstrates both the application of the energy equation and the use of the most common head-oss equations. A secondary objective is to justify two common assumptions about pipeline flow: namely, that flow is, to a good approximation, incompressible and isothermal. Problem. A straight pipe is 2500 m long 27 inches in diameter and discharges water at 10ºC into the atmosphere at the rate of 1.80 m3/s. The lower end of the pipe is at an elevation of 100 m, where a pressure gauge reads 3.0 MPa. The pipe is on a 4% slope. 1. Determine the pressure head, elevation head, total head, and piezometric level at both ends of this pipeline. 2. Determine the associated Darcy-Weisbach friction factor f and Hazen-Williams C for this pipeline and flow. 3. Use the known pressure change to estimate the change in density between the upstream and downstream ends of the conduit. Also estimate the associated change
196.7 m 190.0 m
2
B
1 A C 3
D
162.6 m
FIGURE 2.8 Flow in a simple pipe network.
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HYDRAULICS OF PRESSURIZED FLOW
Hydraulics of pressurized flow 2.25
in velocity between the two ends of the pipe, assuming a constant internal diameter of 27 in throughout. What do you conclude from this calculation? 4. Estimate the change in temperature associated with this head loss and flow, assuming that all the friction losses in the pipe are converted to an increase in the temperature of the water. What do you conclude from this calculation? Solution. The initial assumption in this problem is that both the density of the water and its temperature are constant. We confirm at the end of the problem that these are excellent assumptions (a procedure similar to the predictor-corrector approaches often used for numerical methods). We begin with a few preliminary calculations that are common to several parts of the problem. Geometry. If flow is visualized as moving from left to right, then the pipeline is at a 100 m elevation at its left end and terminates at an elevation of 100 0.04 (2500) 200 m at its right edge, thus gaining 100 m of elevation head along its length. The hydraulic grade line—representing the distance above the pipe of the pressure head term P/γ—is high above the pipe at the left edge and falls linearly to meet the pipe at its right edge because the pressure here is atmospheric. Properties. At 10ºC, the density of water ρ 999.7 kg/m3, its bulk modulus K ρ∆ρ/ P/∆ρ 2.26 GPa, and its specific heat C 4187 J/(kg · ºC). The weight density is γ ρg 9.81 kN/m3. Based on an internal diameter of 27 in, or 0.686 m, the cross-sectional area of the pipe is π π A D2 (0.686)2 0.370 m2 4 4 Based on a discharge Q 1.80 m3/s, the average velocity is Q 1.80 m3/s V 2 4.87 m/s A 0.370 m Such a velocity value is higher than is typically allowed in most municipal work. 1. The velocity head is given by v2 hv 1.21 m 2g Thus, the following table can be completed: Variable Pressure (MPa)
Expression P
Upstream
Downstream
3.0
0.0
Pressure head (m)
P /γ
305.9
0.0
Elevation head (m)
z
100.0
200.0
405.9
200.0
407.1
201.2
Piezometric head (m) Total head (m)
P /γ z P /γ z v /2g 2
2. The head loss caused by friction is equal to the net decrease in total head over the length of the line. That is, hf 407.1 201.2 205.9 m. Note that because this pipe is of uniform diameter, this value also could have been obtained from the piezometric head terms.
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HYDRAULICS OF PRESSURIZED FLOW
2.26
Chapter Two
From the Darcy-Weisbach equation, we can obtain the following expression for the dimensionless f: hf D (205.9)(0.686) f 0.047 (2500)(1.21) v2 L 2g Alternatively, from the Hazen-Williams equation that Q 0.278 C D2.63 (hf /L)0.54, we obtain the following for the dimensional C: Q 1.8 C = 67.2 0.278 D2.63(hf/L)0.54 0.278(0.686)2.63(205.9/2500)0.54 These values would indicate a pipe in poor condition, probably in much need of repair or replacement. 3. In most problems involving steady flow, we assume that the compressibility of the water is negligible. This assumption is easily verified since the density change associated with the pressure change is easily computed. In the current problem, the pressure change is 3.0 MPa and the bulk modulus is 2200 MPa. Thus, by definition of the bulk modulus K, ∆ρ ∆P 3 0.0014 ρ K 22 00 Clearly, even in this problem, with its unusually extreme pressure changes, the relative change in density is less than 0.2 percent. The density at the higher pressure (upstream) end of the pipe is ρ1 ρ2 ∆ρ 999.7 (1 0.0014) 1001.1 kg/m3. Using the mass continuity equation, we have ρ(AV)1 (ρAV)2 In this case, we assume that the pipe is completely rigid and that the change in pressure results in a change in density only (in most applications, these terms are likely to be almost equally important). In addition, we assume that the velocity we’ve already calculated applies at the downstream end (i.e., at Location 2). Thus, the continuity equation requires ρ 999.7 V1 V2 2 4.87 4.86 m/s 1001.1 ρ1 Obviously, even in this case, the velocity and density changes are both negligible and the assumption of incompressible flow is an extremely good one. 4. Assuming that the flow is incompressible, the energy dissipated, Pd, can be computed using work done in moving the fluid through a change in piezometric flow (in fact, the head loss is nothing more than the energy dissipating per unit rate of weight of fluid transferred). Thus, Pd γQhf Strictly speaking, this energy is not lost but is transferred to less available forms: typically, heat. Since energy is associated with the increase in temperature of the fluid, we
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HYDRAULICS OF PRESSURIZED FLOW
Hydraulics of pressurized flow 2.27
can easily estimate the increase in temperature of the fluid that would be associated with the dissipation of energy, assuming that all the heat is retained in the fluid. That is, Pf ρQ c∆T ρgqhf . Solving for the temperature increase gives (9.91m/s2)(205.9 m) gh ∆T f 4187 J/(kg ·ºC) 0.48ºC. c We conclude that the assumption of isothermal flow also is an excellent one.
2.6.7 Pumps: Adding Energy to the Flow Although water is the most abundant substance found on the surface of the earth, its natural distribution seldom satisfies an engineer’s partisan requirements. As a result, pumping both water and wastewater is often necessary to achieve the desired distribution of flow. In essence, a pump controls the flow by working on the flowing fluid, primarily by discharging water to a higher head at its discharge flange than is found at the pump inlet. The increased head is subsequently dissipated as frictional losses within the conduit or is delivered further downstream. This section provides a brief introduction to how pumps interact with pipe systems. Further details are found in Chap. 10. How exactly is the role of a pump quantified? The key definition is the total dynamic head (TDH) of the pump. This term describes the difference between the total energy on the discharge side compared with that on the suction side. In effect, the TDH HP is the difference between the absolute total head at the discharge and suction nozzle of the pump: that is, V2 V2 HP hp hp 2g d 2g s
(2.26)
where hp hydraulic grade line elevation (i.e., pressure-plus-elevation head with respect to a fixed datum), and subscripts d and s refer to delivery and suction flanges, respectively. Typically, the concern is how the TDH head varies with the discharge Q; for a pump, this HQ relation is called the characteristic curve. What the TDH definition accomplishes can be appreciated better if we consider a typical pump system, such as the one shown in Fig. 2.9. In this relation, the Bernoulli equation relates what happens between Points 1 and 2 and between Points 3 and 4, but technically it cannot be applied between 2 and 3 because energy is added to the flow. However, the TDH definition spans this gap. To see this more clearly, the energy relation is written between Points 1 and 2 as HS HPS hfs
(2.27)
where Hs is the head of the suction reservoir, HPS is the total head at the suction flange of the pump, and hfs is the friction loss in the suction line. Similarly, the energy relation is written between Points 3 and 4 as HPD HD hfd
(2.28)
where Hd is the head of the discharge reservoir, HPD is the head at the discharge flange of the pump, and hfd is the friction loss in the discharge line. If Eq. (2.27) is then added to Eq. (2.28), the result can be rearranged as Hpd Hps Hd Hs hfd hfs
(2.29)
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HYDRAULICS OF PRESSURIZED FLOW
2.28
Chapter Two HPD HGL HD
4 HST
HS
HP
HGL 3
HPS 1 2
Pump
FIGURE 2.9 Definition sketch for pump system relations.
which can be rewritten using Eq. (2.26) as Hp Hst hf
(2.30)
where Hst is the total static lift and hf is the total friction loss. The total work done by the pump is equal to the energy required to lift the water from the lower reservoir to the higher reservoir plus the energy required to overcome friction losses in both the suction and discharge pipes.
2.6.8 Sample Application Including Pumps Problem. Two identical pumps are connected in parallel and are used to force water into the transmission/distribution pipeline system shown in Fig. 2.10. The elevations of the demand locations and the lengths of C 120 pipe also are indicated. Local losses are negligible in this system and can be ignored. The demands are as follows: D1 1.2 m3/s, D2 1.6 m3/s, and D3 = 2.2 m3/s. The head-discharge curve for a single pump is approximated by the equation H 90 6Q1.70 1. What is the minimum diameter of commercially available pipe required for the 4.2 km length if a pressure head is to be maintained at a minimum of 40 m everywhere in the system? What is the total dynamic head of the pump and the total water horsepower supplied for this flow situation? 2. For the system designed in the previous questions the demand can shift as follows under certain emergency situations: D1 0.8 m3/s, D2 1.2 m3/s, and D3 4.2 m3/s. For this new demand distribution, can the system maintain a residual pressure head of 20 m in the system?
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HYDRAULICS OF PRESSURIZED FLOW
Hydraulics of pressurized flow 2.29
Solution. Total flow is Qt D1 D2 D3 1.2 1.6 2.2 5.0 m3/s and, each pump will carry half of this flow: i.e., Qpump Qt /2 2.5 m3/s. The total dynamic head of the pump Hpump is Hpump 90 6(2.5)1.7 61.51 m which allows the total water power to be computed as Power 2 (Qpump Hpumpγ) Qt Hpumpγ Thus, numerically, N m3 Power 5.0 (61.51m) 9810 3 3017 kW m s
which is a huge value. The diameter d1 of the pipe that is 4.2 km long, the head loss ∆hi caused by friction for each pipe can be determined using the Hazen-Williams formula since the flow can be assumed to be in the hydraulic rough range. Because d1 is unknown, ∆h2, ∆h3, and ∆h1 are calculated first. The site where the lowest pressure head occurs can be shown to be at Node 2 (i.e., the highest node in the system) as follows: 1
1
Q3 0.54 0.54 2.2 800 3.80 m ∆h3 L3 2.63 2.63 0.278Cd 0.278(120)(1.067 )
Because the head loss ∆h3 is less than the gain in elevation of 10 m, downstream pressures increase; thus, Node 2 (at D2) will be critical in the sense of having the lowest pressure. Thus, if the pressure head at that node is greater than 40 m, a minimum pressure head of 40 m will certainly be maintained throughout the pipeline. Continuing with the calculations, 1
1
0.54 0.54 3.8 Q2 ∆h2 L2 1000 2.30 m 2.63 2.63 0.278Cd 0.278(120)(1.524 )
D2 D1
D3
0m
0m 1000''φ 6
80
250 m
255 m
245 m
240 m
00
m
42
Pump
FIGURE 2.10 Example pipe and pump system.
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HYDRAULICS OF PRESSURIZED FLOW
2.30
Chapter Two
Now, the pressure head at Node 2 is hp2 zR Hpump ∆h1 ∆h2 z2 40 m which implies that ∆h1 (zR z2) Hpump ∆h2 40 (240 255) 61.51 2.30 40 4.21 m where z is the elevation and the subscripts R and 2 denote reservoir and Node 2, respectively. Thus, the minimum diameter d1 is
冢
1 2.63
冣
Q d1 0.278CS0.54
冢
5.0 0.54 4.21 0.278(120) 4200
冣
1 2.63
= 2.006 m
Finally, the minimum diameter (d1 2.134 m) of the commercially available pipe is therefore 84 in. Under emergency conditions (e.g., with a fire flow), the total flow is Qt D1 D2 D3 0.8 1.2 4.2 6.2 (m3/s). Note that with an increase in flow, the head lost resulting from friction increases while the head supplied by the pump decreases. Both these facts tend to make it difficult to meet pressure requirements while supplying large flows. More specifically, Hpump 90 6(3.1)1.7 48.90 m and
冢
4.2 ∆h3 800 0.278(120)(1.067)2.63
1 0.54
冣
12.6 m
Because this loss now exceeds the elevation change, Node 3 (at D3) now becomes critical in the system; minimum pressures now occur at the downstream end of the system. Other losses are
冢
5.4 ∆h2 1000 0.278(120)(1.524)2.63
1 0.54
冣
4.4 m
and
冢
6.2 ∆h1 4200 0.278(120)(2.134)2.63
1 0.54
冣
4.6 m
Thus, the pressure head at Node 3 is hp3 (zR z3) Hpump ∆h1 ∆h2 ∆h3 5 48.9 12.6 4.4 4.6 22.3 m Clearly, a residual pressure head of 20 m is still available in the system under emergency situations, and the pressure requirement is still met, though with little to spare!
2.6.9 Networks—Linking Demand and Supply In water supply and distribution applications, the pipes, pumps, and valves that make up the system are frequently connected into complex arrangements or networks. This topological complexity provides many advantages to the designer (e.g., flexibility, reliability, water quality), but it presents the analyst with a number of challenges. The essential problems associated with “linked” calculations in networks are discussed in Chap. 9. Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
HYDRAULICS OF PRESSURIZED FLOW
Hydraulics of pressurized flow 2.31
2.7 QUASI-STEADY FLOW: SYSTEM OPERATION The hydraulics of pressurized flow is modified and adjusted according to the presence, location, size, and operation of storage reservoirs and pumping stations in the system. This section discusses the criteria for and the approach to these components, introducing the equations and methods that will be developed in later chapters. A common application of quasi-steady flow arises in reservoir engineering. In this case, the key step is to relate the rate of outflow O to the amount of water in the reservoir (i.e., its total volume or its depth). Although the inflow is usually a known function of time, Eq. (2.2) must be treated as a general first-order differential equation. However, the solution usually can be approximated efficiently by standard numerical techniques, such as the Runge-Kutta or Adams-type methods. This application is especially important when setting operating policy for spillways, dams, turbines, and reservoirs. One simple case is illustrated by the example below. Usually, reservoir routing problems are solved numerically, a fact necessitated by the arbitrary form of the input function to the storage system and the sometimes complex nature of the storage-outflow relation. However, there are occasions when the application is sufficiently simple to allow analytical solutions. Problem. A large water-filled reservoir has a constant free surface elevation of 100 m relative to a common datum. This reservoir is connected by a pipe (L 50 m, D 6 cm, and f constant 0.02, hf = fLV2/2gD) to the bottom of a nearby vertical cylindrical tank that is 3 m in diameter. Both the reservoir and the tank are open to the atmosphere, and gravity-driven flow between them is established by opening a valve in the connecting pipeline. Neglecting all minor losses, determine the time T (in hours) required to raise the elevation of the water in the cylindrical tank from 75 m to 80 m. Solution. If we neglect minor losses and the velocity head term, the energy equation can be written between the supply reservoir and the finite area tank. Letting the level of the upstream reservoir be hr, the variable level of the downstream reservoir above datum being h and the friction losses being hf , the energy equation takes on the following simple form: hr h hf This energy relation is called quasi-steady because it does not directly account for any transient terms (i.e., terms that explicitly depend on time). A more useful expression is obtained if we use the Darcy-Weisbach equation to relate energy losses to the discharge Q VA: fL v2 fL Q2 8fL hf 2 25 Q2 gπ D D 2g D 2gA What is significant about this expression, however, is that all the terms involved in the last fraction are known and can be treated as a single constant. Thus, we can solve for Q and rewrite it as Q C兹h 苶苶 h where C2 gπ2D5/8fL r 苶苶, Thus far, we have a single equation involving two unknowns: the head h in the receiving tank and the discharge Q between them. A second relation is required and is given by the continuity equation. Because the flow can be treated as incompressible, the discharge in the tank (i.e., the tanks area At times its velocity of dh/dt) must equal the discharge in the pipe Q. in symbols,
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HYDRAULICS OF PRESSURIZED FLOW
2.32
Chapter Two
dh At Q dt Thus, using the energy equation, we have, C dh h A兹h苶苶 r 苶苶 dt t Separating variables and integrating gives
冕
h2
h1
dh 兹h苶苶 h r 苶苶
冕
t
0
C A dt t
and performing the integration and using appropriate limits gives
冢
冣
C 2 兹h苶苶 h苶1 兹h 苶苶 h苶2 t r 苶苶 r 苶苶 At Finally, solving for t gives the final required expression for quasi-steady flow connecting a finite-area tank to a constant head reservoir:
冢
冣
2A t = t 兹h苶苶 h苶1 兹h 苶苶 h苶2 r 苶苶 r 苶苶 C
The numerical aspects are now straightforward: gπ2(0.06)5 gπ2 (0.06)5 0.5 C = or C m5/2/s 3.068(10)3 m5/2/s 8·1 8 1 If hr 100 m, h1 75 m, h2 80 m, than we have 2 π4 (3 m)2 苶5苶苶 m 兹2 苶0苶苶) m 2432.6 s t = 3.068(10)3m5/2/s (兹2
冢
冣
冢
冪莦莦莦莦莦莦莦
冣
Converting to minutes, this gives a time of about 40.5 minutes (0.676 hr). In problems involving a slow change of the controlling variables, it is often simple to check the calculations. In the current case, a good approximation can be obtained by using the average driving head of 22.5 m (associated with an average tank depth of 77.5 m). This average head, in turn, determines the associated average velocity in the pipeline. Using this “equivalent” steady velocity allows one to estimate how much time is required to fill the tank by the required 5 m. The interested reader is urged to try this and to verify that this approximate time is actually relatively accurate in the current problem, being within 6 s of the “exact” calculation.
2.8 UNSTEADY FLOW: INTRODUCTION OF FLUID TRANSIENTS Hydraulic conditions in water distribution systems are in an almost continual state of change. Industrial and domestic users often alter their flow requirements while supply conditions undergo adjustment as water levels in reservoirs and storage tanks change or as pumps are turned off and on. Given this dynamic condition, it is perhaps surprising that steady state considerations have so dominated water and wastewater engineering. The following sections provide an introduction to unsteady flow in pipe systems—a topic that is neglected too often in pipeline work. The purpose is not too create a fluid transients expert but to set the stage for Chap. 12, which considers these matters in greater detail.
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HYDRAULICS OF PRESSURIZED FLOW
Hydraulics of pressurized flow 2.33
2.8.1 Importance of Water Hammer Pressure pipe systems are subjected to a wide range of physical loads and operational requirements. For example, underground piping systems must withstand mechanical forces caused by fluid pressure, differential settlement, and concentrated loads. The pipe must tolerate a certain amount of abuse during construction, such as welding stresses and shock loads. In addition, the pipe must resist corrosion and various kinds of chemical attack. The internal pressure requirement is of special importance, not only because it directly influences the required wall thickness (and hence cost) of large pipes, but also because pipe manufacturers often characterize the mechanical strength of a pipeline by its pressure rating. The total force acting within a conduit is obtained by summing the steady state and waterhammer (transient) pressures in the line. Transient pressures are most important when the rate of flow is changed rapidly, such as by closing a valve or stopping a pump. Such disturbances, whether caused intentionally or by accident, create traveling pressure and velocity waves that may be of large magnitude. These transient pressures are superimposed on steady-state values to produce the total pressure load on a pipe. Most people have some experience with waterhammer effects. A common example is the banging or hammering noise sometimes heard when a water faucet is closed quickly. In fact, the mechanism in this simple example typifies all pipeline transients. The kinetic energy carried by the fluid is rapidly converted into strain energy in the pipe walls and fluid. The result is a pulse wave of abnormal pressure that travels along the pipe. The hammering sound indicates that a portion of the original kinetic energy is converted into acoustic form. This and other energy-transformation losses (such as fluid friction) cause the pressure wave to decay gradually until normal (steady) pressures and velocities are once again restored. It turns out that waterhammer phenomena are the direct means of achieving all changes in fluid velocity, gradual or sudden. The difference is that slow adjustments in velocity or pressures produce such small disturbances that the flow appears to change smoothly from one value to another. Yet, even in these cases of near equilibrium, it is traveling pressure waves that satisfy the conservation equations. To illustrate why this must be so, consider the steady continuity equation for the entire pipe. This law requires that the rate at which fluid leaves one end of a conduit must be equal to the rate at which it enters the other end. The coordination between what happens at the two ends of the pipeline is not achieved by chance or conspiracy. It is brought about by the same physical laws and material properties that cause disturbances to propagate in the transient case. If waterhammer waves were always small, the study of transient conditions would be of little interest to the pipeline engineer. This is not the case. Waterhammer waves are capable of breaking pipes and damaging equipment and have caused some spectacular pipeline failures. Rational design, especially of large pipelines, requires reliable transient analysis. There are several reasons why transient conditions are of special concern for large conduits. Not only is the cost of large pipes greater, but the required wall thickness is more sensitive to the pipe’s pressure rating as well. Thus, poor design—whether it results in pipeline failure or the hidden costs of overdesign—can be extremely expensive for large pipes. Despite their intrinsic importance, transient considerations are frequently relegated to a secondary role when pipeline systems are designed or constructed. That is, only after the pipelines profile, diameter, and design discharge have been chosen is any thought given to transient conditions. This practice is troublesome. First, the pipeline may not perform as expected, possibly causing large remedial expenses. Second, the
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HYDRAULICS OF PRESSURIZED FLOW
2.34
Chapter Two
line may be overdesigned and thus unnecessarily expensive. This tendency to design for steady state conditions alone has been particularly common in the water supply industry. In addition, there has been a widely held misconception that complex arrangements of pipelines reflect or dampen waterhammer waves. Although wave reflections in pipe networks do occur, attenuation depends on many factors and cannot be guaranteed. Networks are not intrinsically better behaved than simple pipelines are, and some complex systems may respond even more severely to transient conditions (Karney and McInnis, 1990). The remainder of this chapter introduces, in a gentle and nonmathematical way, several important concepts relating to transient conditions. Although rigorous derivations and details are avoided, the discussion is physical and accurate. The goal is to answer two key questions: How do transients arise and propagate in a pipeline? and under what circumstances are transient conditions most severe? Transient conditions in pressure pipelines are modeled using either a “lumped” or “distributed” approach. In distributed systems, the fluid is assumed to be compressible, and the transient phenomena occur in the form of traveling waves propagating with a finite speed a. Such transients often occur in water supply pipes, power plant conduits, and industrial process lines. In a lumped system, by contrast, the flow is considered to be incompressible and the pipe walls are considered to be inelastic. Thus, the fluid behaves as a rigid body in that changes in pressure or velocity at one point in the flow system are assumed to change the flow elsewhere instantaneously. The lumped system approximation can be obtained either directly or in the limit as the wavespeed a becomes unbounded in the distributed model. The slow oscillating water level in a surge tank attached to a short conduit typifies a system in which the effects of compressibility are negligible. Although the problem of predicting transient conditions in a pipeline system is of considerable practical importance, many challenges face the would-be analyst. The governing partial differential equations describing the flow are nonlinear, the behavior of even commonly found hydraulic devices is complex, and data on the performance of systems are invariably difficult or expensive to obtain. The often-surprising character of pulse wave propagation in a pipeline only makes matters worse. Even the basic question of deciding whether conditions warrant transient analysis is often difficult to answer. For all these reasons, it is essential to have a clear physical grasp of transient behavior.
2.8.2 Cause of Transients In general, any change in mean flow conditions in a pipeline system initiates a sequence of transient waves. In practice, we are generally concerned with changes or actions that affect hydraulic devices or boundary conditions attached to the conduit. The majority of these devices are used to provide power to the system or to control the flow in some way. The following list illustrates how some transient conditions can originate, although not all of the them are discussed further here: 1. Changes in valve settings (accidental or planned; manual or automatic) 2. Starting or stopping of either supply or booster pumps 3. Changes in the demand conditions, including starting or arresting a fire flow 4. Changes in reservoir level (e.g., waves on a water surface or the slow accumulation of depth with time) 5. Unstable device characteristics, including unstable pump characteristics, valve instabilities, the hunting of a turbine, and so on Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
HYDRAULICS OF PRESSURIZED FLOW
Hydraulics of pressurized flow 2.35
6. Changes in transmission conditions, such as when a pipe breaks or buckles 7. Changes in thermal conditions (e.g., if the fluid freezes or if changes in properties are caused by temperature fluctuations) 8. Air release, accumulation, entrainment, or expulsion causing dramatic disturbances (e.g., a sudden release of air from a relief valve at a high point in the profile triggered by a passing vehicle); pressure changes in air chambers; rapid expulsion of air during filling operations 9. Transitions from open channel to pressure flow, such as during filling operations in pressure conduits or during storm events in sewers. 10. Additional transient events initiated by changes in turbine power loads in hydroelectric projects, draft-tube instabilities caused by vortexing, the action of reciprocating pumps, and the vibration of impellers or guide vanes in pumps, fans, or turbines
2.8.3 Physical Nature of Transient Flow In pipeline work, many approximations and simplifications are required to understand the response of a pipe system following an initialization of a transient event. In essence, this is because the flow is both unsteady in the mean as well as turbulent. Many of these assumptions have been confirmed experimentally to the extent that the resulting models have provided adequate approximations of real flow behavior. Yet, it is wise to be skeptical about any assumption and be cautious about mathematical models. As we have stressed, any model only approximates reality to a greater or lesser extent. Still, even in cases where models perform poorly, they may be the best way of pinpointing sources of uncertainty and quantifying what is not understood. An air of mystery often surrounds the development, role, and significance of transient phenomena in closed conduits. Indeed, the complexity of the governing differential equations and the dynamic nature of a system’s response can be intimidating to the novice. However, a considerable understanding of transient behavior can be obtained with only the barest knowledge about the properties of fluid and a few simple laws of conservation. When water flows or is contained in a closed conduit so that no free surface is present—for example, in a typical water supply line—the properties of the flowing fluid have some direct implications to the role and significance of transient conditions. For a water pipeline, two properties are especially significant: water’s high density and its large bulk modulus (i.e., water is heavy and difficult to compress). Surprisingly, these two facts largely explain why transient conditions in a pipeline can be so dramatic (see also, Karney and McInnis, 1990): 2.8.3.1 Implication 1. Water has a high density. Because water has a high density (⬇ 1000 kg/m3) and because pipelines tend to be long, typical lines carry huge amounts of mass, momentum, and kinetic energy. To illustrate, assume that a pipeline with area A 1.0 m2 and length L 1000 m is carrying fluid with a velocity v 2.0 m/s. The kinetic energy contained in this pipe is then KE 12 mv2 12 ρLAv2 ⬇ 2,000,000 J Now this is a relatively ordinary situation: the discharge is moderate and the pipe is not long. Yet the pipe still contains energy equivalent to, say, 10,000 fast balls or to a pickup truck falling from a 30-story office tower. Clearly, large work interactions are required to change the flow velocity in a pipeline from one value to another. Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
HYDRAULICS OF PRESSURIZED FLOW
2.36
Chapter Two
In addition to kinetic energy, a pipeline for liquid typically transports large amounts of mass and momentum as well. For example, the above pipeline contains 2(106) kg m/s of momentum. Such large values of momentum imply that correspondingly large forces are required to change flow conditions (Further details con be found in Karney. 1190). 2.8.3.2 Implication 2. Water is only slightly compressible. Because water is only slightly compressible, large head changes occur if even small amounts of fluid are forced into a pipeline. To explain the influence of compressibility in a simple way, consider Fig. 2.11, which depicts a piston at one end of a uniform pipe. If this piston is moved slowly, the volume containing the water will be altered and the confining pressure will change gradually as a result. Just how much the pressure will change depends on how the pipe itself responds to the increasing pressure. For example, the bulk modulus of water is defined as ∆P K ⬇ 2,070 MPa (2.31) ∆ρ/ρ Thus, if the density of the fluid is increased by as little as one-tenth of 1 percent, which is equivalent to moving the imagined piston a meter in a rigid pipe, the pressure will increase by about 200 m of head (i.e., 2 MPa). If the pipe is not rigid, pressure increases are shared between the pipe walls and the fluid, producing a smaller head change for a given motion of the piston. Typical values are shown in the plot in Fig. 2.11. For example, curve 2 indicates typical values for a steel pipe in which the elasticity of the pipe wall and the compressibility of the fluid are nearly equal; in this case, the head change for a given mass imbalance (piston motion) is about half its previous value. Note that it is important for the conduit to be full of fluid. For this reason, many options for accommodating changes in flow conditions are not available in pipelines that can be used in channels. Specifically, no work can be done to raise the fluid mass against gravity. Also note that any movement of the piston, no matter how slowly it is accomplished, must be accommodated by changes in the density of the fluid, the dimension of the conduit, or both. For a confined fluid, Cauchy and Mach numbers (relating speed of change to speed of disturbance propagation) are poor indexes of the importance of compressibility effects. 2.8.3.3 Implication 3. Local action and control. Suppose a valve or other device is placed at the downstream end of a series-connected pipe system carrying fluid at some steady-state velocity V0. If the setting of the valve is changed—suddenly say, for simplicity, the valve is instantly closed—the implications discussed above are combined in the pipeline to produce the transient response. We can reason as follows: The downstream valve can only act locally, providing a relationship between flow through the valve and the head loss across the valve. In the case of sudden closure, the discharge and velocity at the valve becomes zero the instant the valve is shut. However, for the fluid mass as a whole to be stopped, a decelerating force sufficient to eliminate the substantial momentum of the fluid must be applied. But how is such a force generated? We have already mentioned that gravity cannot help us because the fluid has no place to go. In fact, there is only one way to provide the required decelerating force—the fluid must be compressed sufficiently to generate an increase in pressure large enough to arrest the flow. Because water is heavy, the required force is large; however, since water is only slightly compressible, the wave or disturbance will travel quickly. In a system like the one shown in Fig. 2.11, a pressure wave of nearly 100 m would propagate up the pipeline at approximately 1000 m/s. In many ways, the response of the system we have described is typical. For closed conduit systems, the only available mechanism for controlling fluid flows is the propagations of shock waves resulting from the elasticity of the fluid and the pipeline. In essence, tranDownloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
HYDRAULICS OF PRESSURIZED FLOW
Hydraulics of pressurized flow 2.37 L
pipe
Pressure head (m)
piston
←
200
100
0.5
1.0
x
Piston movement (m) FIGURE 2.11 Relation between piston motion (‘mass imbalance’) and head change in a closed conduit
sient considerations cause us to look at the flow of fluid in a pipeline in a new way: For any flow, we consider not only its present significance but also how this condition was achieved and when it will change because, when change occurs, pressure pulses of high magnitude may be created that can burst or damage pipelines. Although this qualitative development is useful, more complicated systems and devices require sophisticated quantitative analysis. The next section briefly summarizes how more general relations can be obtained. (Greater detail is provided in Chap. 12.)
2.8.4 Equation of State-Wavespeed Relations In pipeline work, an equation of state is obtained by relating fluid pressure to density through compressibility relations. Specifically, the stresses in the wall of the pipe need to be related to the pressure and density of the fluid. The result is a relationship between the fluid and the properties of the pipe material and the speed at which shock waves are propagated (wavespeed or celerity). The most basic relation describing the wavespeed in an infinitely elasticly fluid is usually written as follows:
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HYDRAULICS OF PRESSURIZED FLOW
2.38
Chapter Two
aγ
冪莦Kρ
(2.32)
where a is the wavespeed, γ is the ratio of the specific heats for the fluid, K is the bulk modulus of the fluid, and ρ is the fluid density. If a fluid is contained in a rigid conduit, all changes in density will occur in the fluid and this relation still applies. The following comments relate to Eq. (2.32): 1. As fluid becomes more rigid, K increases and, hence, a increases. If the medium is assumed to be incompressible, the wavespeed becomes infinite and disturbances are transmitted instantaneously from one location to another. This is not, strictly speaking, possible, but at times it is a useful approximation when the speed of propagation is much greater than the speed at which boundary conditions respond. 2. For liquids that undergo little expansion on heating, γ is nearly 1. For example, water at 10ºC has a specific heat ratio (γ) of 1.001. 3. Certain changes in fluid conditions can have a drastic effect on celerity (or wavespeed) values. For example, small quantities of air in water (e.g., 1 part in 10,000 by volume) greatly reduce K, because gases are so much more compressible than liquids are at normal temperatures. However, density values (ρ) are affected only slightly by the presence of a small quantity of gas. Thus, wavespeed values for gas-liquid mixtures are often much lower than the wavespeed of either component taken alone. Example: Elastic Pipe The sonic velocity (a) of a wave traveling through an elastic pipe represents a convenient method of describing a number of physical properties relating to the fluid, the pipe material, and the method of pipe anchoring. A more general expression for the wavespeed is a
K/ρ 冪莦 1 c KD/Ee w
(2.33)
1
where K is the bulk modulus of the fluid, ρw is the density of the fluid, E is the elastic modulus of the pipe material, and D and e are the pipe’s, diameter and wall thickness, respectively. The constant c1 accounts for the type of support provided for the pipeline. Typically, three cases are recognized, with c1 defined for each as follows (µ is the Poison’s ratio for the pipe material): Case a. The pipeline is anchored only at the upstream end: µ c1 1 2
(2.34)
Case b. The pipeline is anchored against longitudinal movement. c1 1 µ2
(2.35)
Case c. The pipeline has expansion joints throughout.
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HYDRAULICS OF PRESSURIZED FLOW
Hydraulics of pressurized flow 2.39
c1 1
(2.36)
Note that for pipes that are extremely rigid, thick-walled, or both, c1KD/Ee → 0 and Eq. 2.33 can be simplified to a 兹K 苶/ρ 苶苶 w that which recovers the expression for the acoustic wavespeed in an infinite fluid (assuming γ 1). For the majority of transient applications, the wavespeed can be regarded as constant. Even in cases where some uncertainty exists regarding the wavespeed, the solutions of the governing equations, with respect to peak pressures, are relatively insensitive to changes in this parameter. It is not unusual to vary the wave celerity deliberately by as much as 15 percent to maintain a constant time step for solution by standard numerical techniques (Wylie and Streeter, 1993). (Again, further details are found in Chap. 12.) Wavespeeds are sensitive to a wide range of environmental and material conditions. For example, special linings or confinement conditions (e.g., tunnels); variations in material properties with time, temperature, or composition; and the magnitude and sign of the pressure wave can all influence the wavespeed in a pipeline. (For additional details, see Wylie and Streeter, 1993.Chaudhry, 1987; or Hodgson, 1983).
2.8.5 Increment of Head-Change Relation Three physical relations—Newton’s second law, conservation of mass and the wavespeed relation—can be combined to produce the governing equations for transient flow in a pipeline. The general result is a set of differential equations for which no analytical solution exists. It is these relations that are solved numerically in a numerical waterhammer program. In some applications, a simplified equation is sometimes used to obtain a first approximation of the transient response of a pipe system. This simple relation is derived with the assumption that head losses caused by friction are negligible and that no interaction takes place between pressure waves and boundary conditions found at the end of pipe lengths. The resulting head rise equation is called the Joukowsky relation: a ∆H ∆V g
(2.37)
where ∆H is the head rise, ∆H is the change in velocity in the pipe, a is the wavespeed, and g is the acceleration caused by gravity. The negative sign in this equation is applicable for a disturbance propagating upstream and the positive sign is for one moving downstream. Because typical values of a/g are large, often 100 s or more, this relation predicts large values of head rise. For example, a head rise of 100 m occurs in a pipeline if a/g 100 s and if an initial velocity of 1 m/s is suddenly arrested at the downstream end. Unfortunately, the Joukowsky relation is misleading in a number of respects. If the equation is studied, it seems to imply that the following relations are true: 1. The greater the initial velocity (hence, the larger the maximum possible ∆V), the greater the transient pressure response. 2. The greater the wavespeed a, the more dramatic the head change. 3. Anything that might lower the static heads in the system (such as low reservoir levels or large head losses caused by friction) will tend to lower the total head (static plus dynamic) a pipe system is subject to. Although these implications are true when suitable restrictions on their application are enforced, all of them can be false or misleading in more complicated hydraulic systems.
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HYDRAULICS OF PRESSURIZED FLOW
2.40
Chapter Two
It is important to be skeptical about simple rules for identifying “worst case” scenarios in transient applications. Karney and McInnis (1990) provide further elaboration of this point. However, before considering even a part of this complexity, one must clarify the most basic ideas in simple systems.
2.8.6 Transient Conditions in Valves Many special devices have been developed to control and manage flows in pipeline systems. This is not surprising because the inability to control the passage of water in a pipeline can have many consequences, ranging from the minor inconvenience of restrictive operating rules to the major economic loss associated with pipeline failure or rupture. The severity of the problem clearly depends on the timing and magnitude of the failure. In essence, control valves function by introducing a specified and predictable relationship between discharge and pressure drop in a line. When the setting of a valve (or, for that matter, the speed of a pump) is altered, either automatically or by manual action, it is the head-discharge relationship that is controlled to give the desired flow characteristics. The result of the change may be to increase or reduce the pressure or discharge, maintain a preset pressure or flow, or respond to an emergency or unusual condition in the system. It is a valve control function that creates most difficulties encountered by pipeline designers and system operators. Valves control the rate of flow by restricting the passage of the flow, thereby inducing the fluid to accelerate to a high velocity as it passes through the valve even under steady conditions. The large velocities combine with the no-slip condition at the solid boundaries to create steep velocity gradients and associated high shear stresses in the fluid; in turn, these shear stresses, promote the rapid conversion of mechanical energy into heat through the action of turbulence of the fluid in the valve. The net result is a large pressure drop across the valve for a given discharge through it; it is this ∆h-Q relationship for a given opening that makes flow control possible. However, the same high velocities also are responsible for the cavitation, noise, energy loss, wear, and maintenance problems often associated with valves even under steady conditions. This section presents an overview of control valve hydraulics and considers the basic roles that control valves play in a pipeline. Valves are often classified by both their function and their construction. Valves can be used for on/off control or for pressure or flow control, and the physical detail of the valve’s construction varies significantly depending on the application. The kind of valves used can range from traditional gate and globe valves to highly sophisticated slow-closing air valves and surge-anticipating valves. The actuator that generates the valve’s motion also varies from valve to valve, depending on whether automatic or manual flow control is desired. Many kinds of valves can be used in a single pipeline, creating challenging interactions for the transient analyst to sort out. The most basic of these interactions is discussed in more detail in the following section. 2.8.6.1 Gate discharge equation. Among the most important causes of transient conditions in many pipelines is the closure of regulating and flow control valves. The details of how these valves are modeled can be influential in determining the maximum pressure experienced on the lines. For this reason, and because some knowledge of valve behavior is required to interpret the output from a simulation program, it is worthwhile to briefly review valve theory. Consider a simple experiment in which a reservoir, such as the one shown in Fig. 2.12, has a valve directly attached to it. If we initially assume the valve is fully open, the discharge through the valve Q0 can be predicted with the usual orifice equation: Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
HYDRAULICS OF PRESSURIZED FLOW
Hydraulics of pressurized flow 2.41
Q0 (Cd A v )0 兹2苶g苶∆ 苶H 苶0苶
(2.38)
where Cd is the discharge coefficient, Av is the orifice area, g is the acceleration caused by gravity, ∆H0 is the head difference across the valve, and the subscript 0 indicates that the valve is fully open. If the valve could completely convert the head difference across it into velocity, the discharge coefficient Cd would be equal to 1. Since full conversion is not possible, Cd values are inevitably less than 1, with values between 0.7 and 0.9 being common for a fully open valve. The product of the orifice area Av and the discharge coefficient Cd is often called the “effective area” of the valve. The effective area, as determined by details of a valve’s internal construction, controls the discharge through the valve. Equation (2.38) is valid for a wide range of heads and discharges: For example, the solid curve in the plot above depicts this relation for a fully open valve. Yet, clearly the equation must be altered if the setting (position) of the valve is altered because both the discharge coefficient and the orifice valve area would change. Describing a complete set of a valve’s characteristics would appear to require a large set of tabulated Cd Av values. Fortunately, a more efficient description is possible. Suppose we take a valve at another position and model its discharge in a way that is analogous to the one shown in Eq. (2.38). That is: Q (Cd Av) 兹2 苶g苶∆ 苶H 苶
(2.39)
where both Cd and Av will, in general, have changed from their previous values. If Eq. (2.39) is divided by Eq. (2.38), the result can be written as Q Es τ 兹∆ 苶H 苶
(2.40)
In Eq. (2.40), Es is a new valve constant representing the ratio of the fully open discharge to the root of the fully open head difference: Q0 Es 兹∆ 苶苶 H苶0 In essence, Es “scales” the head losses across a fully open valve for its size, construc tion, and geometry. In addition, τ represents the nondimensional effective gate opening: C Av τ d (Cd Av)0 Using τ-values to represent gate openings is convenient, because the effective range is from 0.0 (valve fully closed) to 1.0 (valve fully open). The precise way the τ-value changes as a valve is closed varies from valve to valve. The details of this “closure curve” determine the head-discharge relationship of the valve and thus often have a marked influence on transient conditions in a pipeline. 2.8.6.2 Alternate valve representation. In the literature relating to valves, and as was introduced earlier in this chapter, it is common to model local losses as a multiplier of the velocity head: v2 ∆H ψ 2g
(2.41)
where v is the average velocity in the pipeline upstream of the valve and ψ is the alternative valve constant. This apparently trivial change has a detrimental effect on numerical
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HYDRAULICS OF PRESSURIZED FLOW
Chapter Two
∆H
∆H ∆H0
τ = 0.0
τ = 0.5
τ = 1.0
∆H0 Closed
2.42
Valve
Q0
Q
FIGURE 2.12 Relation between head and discharge in a valve.
calculations: ψ now varies from some minimum value for a fully open valve to infinity for a closed valve. Such a range of values can cause numerical instabilities in a transient program. For this reason, the reciprocal relationship involving τ-values is almost always preferred in transient applications. 2.8.6.3 Pressure Regulating Valves. In many applications, the valve closure relations are even more complicated than is the case in the situation just described. Pressure-regulating valves are often installed to maintain a preset pressure on their downstream side; they accomplish this function by partially closing, thus inducing a greater pressure drop across the valve. However, if a power failure or other transient condition were now to occur in the line, any “active” pressure-regulating valve would start from an already partially closed position. Depending on its initial setting, a pressure-regulating valve may close in a time that is much less than its design or theoretical value. The influence of the initial valve position is most severe for regulating valves breaking the largest pressures, which are often associated with relatively low head losses in the remainder of the line. Thus, when a pressure-regulating valve is used, the most severe transient conditions can occur in a system transmitting small flows.
2.8.7 Conclusion Transient fluid flow, variously called waterhammer, oil hammer, and so on, is the means of achieving a change in steady-state flow and pressure. When conditions in a pipeline are changed, such as by closing a valve or starting a pump, a series of waves are generated. These disturbances propagate with the velocity of sound within the medium until they are dissipated down to the level of the new steady state by the action of some form of damping or friction. In the case of flow in a pipeline, these fluid transients are the direct means of achieving all changes in fluid velocity, gradual or sudden. When sudden changes occur, however, the results can be dramatic because pressure waves of considerable magnitude can occur and are capable of destroying the pipe. Only if the flow is regulated extremely slowly is it possible to go smoothly from one steady state to another without large fluctuations in pressure head or pipe velocity. Clearly, flow control actions can be extremely important, and they have implications not only for the design of the hydraulic system but also for other aspects of system design and operation. Such problems as selecting the pipe layout and profile, locating control ele-
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HYDRAULICS OF PRESSURIZED FLOW
Hydraulics of pressurized flow 2.43
ments within the system, and selecting device operating rules as well as handling the ongoing challenges of system management are influenced by the details of the control system. A rational and economic operation requires accurate data, carefully calibrated models, ongoing predictions of future demands and the response of the system to transient loadings, and correct selection of both individual components and remedial strategies. These design decisions cannot be regarded as an afterthought to be appended to a nearly complete design. Transient analysis is a fundamental and challenging part of rational pipeline design.
REFERENCES Bhave, P.R. 1991. Analysis of Flow in Water Distribution Networks, Technomic Publishing Inc., Lancaster, PA, 1991. Chaudhry, H. M., Applied Hydraulic Transients, Van Nostrand Reinhold, New York, NY, 1987. Chaudhry, M. H., and V. Yevjevich, Closed Conduit Flow, Water Resources Publications, Littleton, CO, 1981. Ford, K. W. Classical and Modern Physics, Vol. 1, Xerox College Publishing, Lexington, Ma, 1973. Hatsopoulos, N, and J. H. Keenan, Principles of General Thermodynamics, John Wylie and Sons, New York, 1965. Hodgson, J., Pipeline Celerities Master’s of Engineering thesis, University of Alberta, Edmonton, Alberta, Canada. 1983. Jeppson, R.W. Analysis of Flow in Pipe Networks. Ann Arbor Science Publishers, Stoneham, MA, 1976. Karney, B. W., “Energy Relations in Transient Closed Conduit Flow.,” Journal of Hydraulic Engineering, 116: 1180—1196, 1990. Karney, B. W., and D. M. McInnis, “Transient Analysis of Water Distribution Systems,” Journal of the American Water Works Association, 82(7): 62—70, 1990. Nikuradse, “Stonm ungs gesetze in rauhen Rohre.” Forsch-Arb, Ing.-Wes. Itett 361, 1933. Roberson, J. A., and C. T. Crowe, Engineering Fluid Mechanics, Houghton Mifflin, Boston, MA, 1993. Swamee, P. K, and A. K. Jain, “Explicit Equations for Pipe Flow Problems,” Journal of Hydraulic Engineering, 102: 657—664, 1997. Wylie, B. E., and V. L. Streeter, Fluid Transients in Systems, Prentice-Hall, Englewood Cliffs, NJ, 1993.
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Source: HYDRAULIC DESIGN HANDBOOK
CHAPTER 3
HYDRAULICS OF OPEN CHANNEL FLOW Richard H. French Desert Research Institute, University and Community College System of Nevada Reno, Nevada
3.1 INTRODUCTION By definition, an open channel is a flow conduit having a free surface: that is, a boundary exposed to the atmosphere. The free surface is essentially an interface between two fluids of different density. Open-channel flows are almost always turbulent, unaffected by surface tension, and the pressure distribution within the fluid is hydrostatic. Open channels include flows ranging from rivulets flowing across a field to gutters along residential streets and highways to partially filled closed conduits conveying waste water to irrigation and water supply canals to vital rivers. In this chapter, the basic principles of open channel hydraulics are presented as an introduction to subsequent chapters dealing with design. By necessity, the material presented in this chapter is abbreviated—an abstract of the fundamental concepts and approaches—for a more detailed treatment, the reader is referred to any standard references or texts dealing with the subject: for example, Chow (1959), French (1985), Henderson (1966), or Chaudhry (1993) As with any other endeavor, it is important that a common vocabulary be established and used: Critical slope (Sc): A longitudinal slope such that uniform flow occurs in a critical state. Flow area (A): The flow area is the cross-sectional area of the flow taken normal to the direction of flow (Table 3.1). Froude number (Fr): The Froude number is the dimensionless ratio of the inertial and gravitational forces or V Fr gD
(3.1)
where V average velocity of flow, g gravitational acceleration, and D hydraulic depth. When Fr 1, the flow is in a critical state with the inertial and gravitational forces in equilibrium; when Fr 1, the flow is in a subcritical state and the gravitational forces are dominant; and when Fr 1, the flow is in a supercritical state and the inertial forces are dominant. From a practical perspective, sub – and supercritical flow can be differentiated simply by throwing a rock or other object into the flow. If ripples from the rock 3.1 Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
CircularFigure T3.1-6
Triangle with unequal side slopes
Triangle with equal side slopes
Trapezoid with unequal side slopes
Trapezoid with equal side slopes
Rectangle
(1)
1 (θ sinθ )d2o 8
0.5y2 (z1 z2 )
zy2
0.5θdo
y (1 z21 1 z22)
2y1 z 2
b y (1 z2 1 z) 2
b 2y1 z2
b 2y
Wetted Perimeter P (3)
1 by 0.5y2(z1 z2) 2
(b zy)y
by
Area A (2)
Channel Section Geometric Properties
Channel Definition
TABLE 3.1
b 2zy
b
Top Width T (5)
2y( d y) o
y (z1 z2)
0.5y2 ( z1 z2 ) y (1 z21 1 z2) 2
sinθ d 0.25 1 θ o
2zy
zy z2 21
by 0.5y2(z1 z2) b y (z1 + z2) b y(1 z21 1 z2) 2
(b zy)y z2 b 2y1
by b 2y
Hydraulic Radius R (4)
1 θ sinθ 8 sin(0.5θ)
0.5y
0.5y
by 0.5y2(z1 z2) b y(z1 z2)
(b zy)y b 2zy
y
Hydraulic Depth D (6)
HYDRAULICS OF OPEN CHANNEL FLOW
3.2
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HYDRAULICS OF OPEN CHANNEL FLOW
Hydraulics of Open–Channel Flow 3.3
progress upstream of the point of impact, the flow is subcritical; however, if ripples from the rock do not progress upstream but are swept downstream, the flow is supercritical. Hydraulic depth (D). The hydraulic depth is the ratio of the flow area (A) to the top width (T) or D A/T (Table 3.1). Hydraulic radius (R). The hydraulic radius is the ratio of the flow area (A) to the wetted perimeter (P) or R A/P (Table 3.1). Kinetic energy correction factor (α). Since no real open-channel flow is one-dimensional, the true kinetic energy at a cross section is not necessarily equal to the spatially averaged energy. To account for this, the kinetic energy correction factor is introduced, or
3 3 α γV A ∫∫γv dA 2g 2g
and solving for α, ∫∫v3dA α 3 VA
(3.2)
When the flow is uniform, α 1 and values α of for various situations are summarized in Table 3.2. Momentum correction coefficient (β): Analogous to the kinetic energy correction factor, the momentum correction factor is given by βρQV ∫∫ρv2dA ∫∫v2dA β 2 VA
(3.3)
When the flow is uniform, β 1 and values of β for various situations are summarized in Table 3.2 Prismatic channel. A prismatic channel has both a constant cross-sectional shape and bottom slope (So). Channels not meeting these criteria are termed nonprismatic. Specific energy (E). The specific energy of an open-channel flow is V2 E y α 2g
(3.4)
where y depth of flow and the units of specific energy are length in meters or feet.
TABLE 3.2
Typical Values of α and β for Various Situations Situation Min.
Regular channels, flumes, spillways Natural streams and torrents Rivers under ice cover River valleys, overflooded
Value of α Avg. Max.
Min.
Value of β Avg.
Max.
1.10
1.15
1.20
1.03
1.05
1.07
1.15 1.20 1.50
1.30 1.50 1.75
1.50 2.00 2.00
1.05 1.07 1.17
1.10 1.17 1.25
1.17 1.33 1.33
Source: After Chow (1959).
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HYDRAULICS OF OPEN CHANNEL FLOW
3.4
Chapter Three
Specific momentum (M). By definition, the specific momentum of an open-channel flow is Q2 M + zA gA
(3.5)
Stage: The stage of a flow is the elevation of the water surface relative to a datum. If the lowest point of a channel section is taken as the datum, then the stage and depth of flow (y) are equal if the longitudinal slope (So) is not steep or cos (θ) ≈ 1, where θ is the longitudinal slope angle. If θ 10o or So 0.18, where So is the longitudinal slope of the channel, then the slope of the channel can be assumed to be small. Steady. The depth (y) and velocity of flow (v) at a location do not vary with time; that is, (∂y/∂t 0) and (∂v/∂t 0). In unsteady flow, the depth and velocity of flow at a location vary with time: that is, (∂y/∂t ≠ 0) and (∂v/∂t ≠ 0). Top width (T). The top width of a channel is the width of the channel section at the water surface (Table 3.1). Uniform flow. The depth (y). flow area (A), and velocity (V) at every cross section are constant, and the energy grade line (Sf), water surface, and channel bottom slopes (So) are all parallel. Superelevation (∆y). The rise in the elevation of the water surface at the outer channel boundary above the mean depth of flow in an equivalent straight channel, because of centrifugal force in a curving channel. Wetted perimeter (P). The wetted perimeter is the length of the line that is the interface between the fluid and the channel boundary (Table 3.1).
3.2 ENERGY PRINCIPLE 3.2.1 Definition of Specific Energy Central to any treatment of open-channel flow is that of conservation of energy. The total energy of a particle of water traveling on a streamline is given by the Bernoulli equation or p V2 H z α γ 2g where H total energy, z elevation of the streamline above a datum, p pressure, γ fluid specific weight, (p/γ) pressure head, V2/2g velocity head, and g acceleration of gravity. H defines the elevation of the energy grade line, and the sum [z (p/γ)] defines the elevation of the hydraulic grade line. In most uniform and gradually varied flows, the pressure distribution is hydrostatic (divergence and curvature of the streamlines is negligible) and the sum [z + (p/γ)] is constant and equal to the depth of flow y if the datum is taken at the bottom of the channel. The specific energy of an open-channel flow relative to the channel bottom is V2 Q2 E y α y α 2 2g 2gA
(3.6)
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HYDRAULICS OF OPEN CHANNEL FLOW
Hydraulics of Open–Channel Flow 3.5
where the average velocity of flow is given by Q V A
(3.7)
where Q flow rate and A flow area. The assumption inherent in Eq. (3.6) is that the slope of the channel is small, or cos(θ) 1. If θ 10° or So 0.18, where So is the longitudinal slope of the channel, Eq. (3.6) is valid. If θ is not small, then the pressure distribution is not hydrostatic since the vertical depth of flow is different from the depth measured perpendicular to the bed of the channel.
3.2.2 Critical Depth If y in Eq. (3.6) is plotted as a function of E for a specified flow rate Q, a curve with two branches results. One branch represents negative values of both E and y and has no physical meaning; but the other branch has meaning (Fig. 3.1). With regard to Fig. 3.1, the following observations are pertinent: 1) the portion designated AB approaches the line y E asymptotically, 2) the portion AC approaches the E axis asymptotically, 3) the curve has a minimum at point A, and 4) there are two possible depths of flow—the alternate depths—for all points on the E axis to the right of point A. The location of point A, the minimum depth of flow for a specified flow rate, can be found by taking the first derivative of Eq. (3.6) and setting the result equal to zero, or dE Q2 dA 1 3 0 dy gA dy
(3.8)
yc
y
It can be shown that dA (T dy) or (dA/dy T) (French, 1985). Substituting this result, using the definition of hydraulic depth and rearranging, Eq. (3.8) becomes
Specific Momentum
FIGURE 3.1 Specific energy and momentum as a function of depth when the channel geometry and flow rate are specified.
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HYDRAULICS OF OPEN CHANNEL FLOW
3.6
Chapter Three
Q2 dA Q2 T V2 1 3 1 2 1 0 gA dy gA A gD or V2 D 2g 2
(3.9)
V Fr 1 gD
(3.10)
and
which is the definition of critical flow. Therefore, minimum specific energy occurs at the critical hydraulic depth and is the minimum energy required to pass the flow Q. With this information, the portion of the curve AC in Fig. 3.1 is interpreted as representing supercritical flows, where as AB represents subcritical flows. With regard to Fig. 3.1 and Eq. (3.6), the following observations are pertinent. First, for channels with a steep slope and α ≠ 1, it can be shown that Fr
V gD cos(θ) α
(3.11)
Second, E – y curves for flow rates greater than Q lie to the right of the plotted curve, and curves for flow rates less than Q lie to the left of the plotted curve. Third, in a rectangular channel of width b, y D and the flow per unit width is given by Q q b
(3.12)
q V y
(3.13)
and
Then, where the subscript c indicates variable values at the critical point,
(3.14)
Vc2 y c 2 2g
(3.15)
q2 yc g
1/3
and (3.16) yc 2 Ec 3 In nonrectangular channels when the dimensions of the channel and flow rate are specified, critical depth is calculated either by the trial and error solution of Eqs. (3.8), (3.9), and (3.10) or by use of the semiempirical equations in Table 3.3.
3.2.3 Variation of Depth with Distance At any cross section, the total energy is V2 H y z 2g
(3.17)
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HYDRAULICS OF OPEN CHANNEL FLOW
Hydraulics of Open–Channel Flow 3.7 TABLE 3.3
Semiempirical Equations for the Estimation of yc
Channel Definition (1)
Equation for yc in terms of Ψ α Q2/g (2)
Rectangle Figure T3.1-1
Ψ 2 b
TrapezoidFig ure T-3.1-2
Ψ 0.27 b 0.81 0.75 b1.25 z 30z
0.33
TriangleFigure T3.1-4 2Ψ 0.20 2 z
CircleFigure T3.1-6 1.01 0.2 6 Ψ do
0.25
Source: From Straub (1982).
where y depth of flow, z elevation of the channel bottom above a datum, and it is assumed that and cos(θ) are both equal to 1. Differentiating Eq. (3.17) with respect to longitudinal distance, V2 d 2 g dH dy dz (3.18) dx dx dx dx
dH dz where d the channel botx the change of energy with longitudinal distance (Sf), d x tom slope (So), and, for a specified flow rate,
V2 d 2g Q2 dA dy Q2T dy dy 3 3 (Fr)2 dx gA dy dx gA dx dx Substituting these results in Eq. (3.18) and rearranging, dy So Sf dx 1 Fr2
(3.19)
which describes the variation of the depth of flow with longitudinal distance in a channel of arbitrary shape.
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HYDRAULICS OF OPEN CHANNEL FLOW
3.8
Chapter Three
FIGURE 3.2 Channel with a compound section.
3.2.4 Compound Section Channels In channels of compound section (Fig. 3.2), the specific energy correction factor α is not equal to 1 and can be estimated by N K3i A2i i1
α
K3 2 A
(3.20)
where Ki and Ai as follows the conveyance and area of the ith channel subsection, respectively, K and A are conveyance are as follows:
Ki
Ai
N
K
i1
and
N
A
i1
N number of subsections, and conveyance (K) is defined by Eq. (3.48) in Sec. 3.4. Equation (3.20) is based on two assumptions: (1) the channel can be divided into subsections by appropriately placed vertical lines (Fig. 3.2) that are lines of zero shear and do not contribute to the wetted perimeter of the subsection, and (2) the contribution of the nonuniformity of the velocity within each subsection is negligible in comparison with the variation in the average velocity among the subsections.
3.3 MOMENTUM 3.3.1 Definition of Specific Momentum The one-dimensional momentum equation in an open channel of arbitrary shape and a control volume located between Sections 1 and 2 is
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HYDRAULICS OF OPEN CHANNEL FLOW
Hydraulics of Open–Channel Flow 3.9
γ γz1A1 γ z2A2 Pf Q (V2 V1) g
(3.21)
where γ specific weight of water, Ai flow area at sections 1 and 2; Vi average velocity of flow at sections 1 and 2, Pf horizontal component of unknown force acting between Sections 1 and 2 and wzi distances to the centroids of the flow areas 1 and 2 from the free surface. Substitution of the flow rate divided by the area for the velocities and rearrangement of Eq. (3.21) yields
Pf Q2 Q2 gA1 z1A1 gA2 z2A2 γ or
where
P f M1 M2 γ
(3.22)
Q2 Mi ⴙ zi Ai gAi
(3.23)
and M is known as the specific momentum or force function. In Fig. 3.1, specific momentum is plotted with specific energy for a specified flow rate and channel section as a function of the depth of flow. Note that the point of minimum specific momentum corresponds to the critical depth of the flow. The classic application of Eq. (3.22) occurs when Pf 0 and the application of the resulting equation to the estimation of the sequent depths of a hydraulic jump. Hydraulic jumps result when there is a conflict between the upstream and downstream controls that influence the same reach of channel. For example, if the upstream control causes supercritical flow while the downstream control dictates subcritical flow, there is a contradiction that can be resolved only if there is some means to pass the flow from one flow regime to the other. When hydraulic structures, such as weirs, chute blocks, dentated or solid sills, baffle piers, and the like, are used to force or control a hydraulic jump, Pf in Eq. (3.22) is not equal to zero. Finally, the hydraulic jump occurs at the point where Eq. (3.22) is satisfied (French, 1985).
3.3.2 Hydraulic Jumps in Rectangular Channels In the case of a rectangular channel of width b and Pf 0, it can be shown (French, 1985) that y2 [1 8(F r1 )2 1] (3.24) 0.5 y1 or y1 8 (F r2 )2 1] (3.25) 0.5 [1 y2 y1 2(Fr2)2 4(Fr2)4 16(Fr2)6 ... y2 Equations (3.24) and (3.25) each contain three independent variables, and two must be known before the third can be found. It must be emphasized that the downstream depth of flow (y2) is not the result of upstream conditions but is the result of a downstream control—that is, if the downstream control produces the depth y2 then a hydraulic jump will form. The second form of Eq. (3.25) should be used when (Fr2)2 0.05 (French, 1985).
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HYDRAULICS OF OPEN CHANNEL FLOW
3.10
Chapter Three
3.3.3 Hydraulic Jumps in Nonrectangular Channels In analyzing the occurrence of hydraulic jumps in nonrectangular but prismatic channels, we see that no equations are analogous to Eqs. (3.24) and (3.25). In such cases, Eq. (3.22) could be solved by trial and error or by use of semiempirical equations. For example, in circular sections, Straub (1978) noted that the upstream Froude number (Fr1) can be approximated by y 1.93 (3.26) Fr1 c y1 and the sequent depth can be approximated by y2 (3.27 Fr1 1.7y2 c y1 y1.8c (3.28) Fr1 1.7y2 0.73 y1 For horizontal triangular and parabolic prismatic channel sections, Silvester (1964, 1965) presented the following equations. For triangular channels: y2 2.5 y1
y 2 1 1.5 (Fr1)2 1 1 y2
(3.29)
y2 y1
For parabolic channels with the perimeter defined by y aT2/2, where a is a coefficient: y2 2.5 y1 1.5 1 1.67 (Fr1)2 1 (3.30) y1 y2
FIGURE 3.3 Analytic curves for estimating sequent depths in a trapezoidal channel (From Silvester, 1964)
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HYDRAULICS OF OPEN CHANNEL FLOW
Hydraulics of Open-Channel Flow 3.11
In the case of trapezoidal channels, Silvester (1964) presented a method for graphical solution in terms of the parameter b k (3.31) zy1 In Fig. 3.3, the ratio of (y2/y1) is plotted as a function of Fr1 and k.
3.4 UNIFORM FLOW 3.4.1 Manning and Chezy Equations For computational purposes, the average velocity of a uniform flow can be estimated by any one of a number of semiempirical equations that have the general form V CR x S
y
(3.32)
where C a resistance coefficient, R hydraulic radius, S channel longitudinal slope, and x and y are exponents. At some point in the period 1768–1775 (Levi, 1995), Antoine Chezy, designing an improvement for the water system in Paris, France, derived an equation relating the uniform velocity of flow to the hydraulic radius and the longitudinal slope of the channel, or S V C R
(3.33)
where C is the Chezy resistance coefficient. It can be easily shown that Eq (3.33) is similar in form to the Darcy pipe flow equation. In 1889, Robert Manning, a professor at the Royal College of Dublin (Levi, 1995) proposed what has become known as Manning’s equation, or φ V = R2/3S n
(3.34)
where n is Manning’s resistance coefficient and φ 1 if SI units are used and φ 1.49 if English units are used. The relationship among C, n, and the Darcy-Weisbach friction factor (f) is
8g φ C R1/6 n f
(3.35)
At this point, it is pertinent to observe that n is a function of not only boundary roughness and the Reynolds number but also the hydraulic radius, an observation that was made by Professor Manning (Levi, 1995).
3.4.2 Estimation of Manning’s Resistance Coefficient Of the two equations for estimating the velocity of a uniform flow, Manning’s equation is the more popular one. A number of approaches to estimating the value of n for a channel are discussed in French (1985) and in other standard references, such as Barnes (1967), Urquhart (1975), and Arcement and Schneider (1989). Appendix 3.A lists typical values of n for many types of common channel linings.
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HYDRAULICS OF OPEN CHANNEL FLOW
3.12
Chapter Three
In an unvegetated alluvial channel, the total roughness consists of two parts: grain or skin roughness resulting from the size of the sediment particles and form roughness because of the existence of bed forms. The total coefficient n can be expressed as n n’ n”
(3.36)
where n’ portion of Manning’s coefficient caused by grain roughness and n” portion of Manning’s coefficient caused by form roughness. The value of n’ is proportional to the diameter of the sediment particles to the sixth power. For example, Lane and Carlson (1953) from field experiments in canals paved with cobbles with d75 in inches, developed n’ 0.026d751/6
(3.37)
and Meyer-Peter and Muller (1948) for mixtures of bed material with a significant proportion of coarse-grained sizes with d90 in meters developed n’ 0.038d901/6
(3.38)
In both equations, dxx is the sediment size such that xx percent of the material is smaller by weight. Although there is no reliable method of estimating n”, an example of the variation of f for the 0.19 mm sand data collected by Guy et al. (1966) is shown in Fig. 3.4. The n values commonly found for different bed forms are summarized in Table 3.4. The inability to estimate or determine the variation of form roughness poses a major problem in the study of alluvial hydraulics (Yang, 1996). Use of Manning’s equation to estimate the velocity of flow in channels where the primary component of resistance is from drag rather than bed roughness has been questioned (Fischenich, 1996). However, the use of Manning’s equation has persisted among engineers because of its familiarity and the lack of a practical alternative. Jarrett (1984) recognized that
FIGURE 3.4 Variation of the Darcy-Weisbach friction factor as a function of unit stream power.
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HYDRAULICS OF OPEN CHANNEL FLOW
Hydraulics of Open-Channel Flow 3.13
guidelines for estimating resistance coefficients for high-gradient streams with stable beds composed of large cobbles and boulders and minimally vegetated banks (So 0.002) were based on limited data. Jarrett (1984) examined 21 high-gradient streams in the Rocky Mountains and developed the following empirical equation relating n to So and R (in feet): 0.39S00.38 n R0.16
(3.39)
Jarrett (1984) stated the following limitations on the use of Eq. (3.39): First, the equation is applicable to natural main channels with stable bed and bank materials (gravels, cobbles, boulders) with no backwater. Second, the equation can be used for 0.002 So 0.04 and 0.15 R 2.1 m (0.5 R 7.0 ft). Results of the regression analysis indicated that for R 2.1 m ( 7.0 ft), n did vary significantly with depth; therefore, as long as the bed and bank material remain stable, extrapolation to larger flows should not result in significant error. Third, the hydraulic radius does not include the wetted perimeter of the bed particles. Fourth, the streams used in the analysis had relatively small amounts of suspended sediment. Vegetated channels present unique challenges from the viewpoint of estimating roughness. In grass-lined channels, the traditional approach assumed that n was a function of vegetal retardance and VR (Coyle, 1975). However, there are approaches more firmly based on the principles of fluid mechanics and the mechanics of materials (Kouwen, 1988; Kouwen and Li, 1980.) Data also exist that suggest that in such channels flow duration is not a factor as long as the vegetal elements are not destroyed or removed. Further, inundation times, and/or hydraulic stresses, or both that are sufficient to damage vegetation have been found, as might be expected, to reduce the resistance to flow (Temple, 1991). Petryk and Bosmajian (1975) presented a relation for Manning’s n in vegetated channels based on a balance of the drag and gravitational forces, or Cd (Veg)d 1/2 n R2/3 2g
(3.40)
where Cd a coefficient accounting for the drag characteristics of the vegetation and (Veg)d the vegetation density. Flippin-Dudley (1997) has developed a rapid and objective procedure using a horizontal point frame to measure (Veg)d . Equation (3.40) is limited because there is limited information regarding Cd for vegetation (Flippin-Dudley et al., 1997).
3.4.3 Equivalent Roughness Parameter k In some cases, an equivalent roughness parameter k is used to estimate n. Equivalent roughness, sometimes called “roughness height,” is a measure of the linear dimension of roughness elements but is not necessarily equal to the actual or even the average height of these elements. The advantage of using k instead of Manning’s n is that k accounts for changes in the friction factor due to stage, whereas the Manning’s n does not. The relationship between n and k for hydraulically rough channels is
R1/6 n R log10 12.2 k
(3.41)
where Γ 32.6 for English units and 18.0 for SI units.
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HYDRAULICS OF OPEN CHANNEL FLOW
3.14
Chapter Three
TABLE 3.4
Equivalent Roughness Values of Various Bed Materials
Material
k (ft) (2)
(1) Brass, copper, lead, glass Wrought iron, steel Asphalted cast iron Galvanized iron Cast iron Wood stave Cement Concrete Untreated gunite Drain tile Riveted steel Rubble masonry Straight, uniform earth channels Natural streambed
k (m) (3)
0.0001–0.0030 0.0002–0.0080 0.0004–0.0070 0.0005–0.0150 0.0008–0.0180 0.0006–0.0030 0.0013–0.0040 0.0015–0.0100 0.01–0.033 0.0020–0.0100 0.0030–0.0300 0.02 0.01
0.00003048–0.0009 0.0001–0.0024 0.0001–0.0021 0.0002–0.0046 0.0002–0.0055 0.0002–0.0009 0.0004–0.0012 0.0005–0.0030 0.0030–0.0101 0.0006–0.0030 0.0009–0.0091 0.0061 0.0030
0.1000-3.0000
0.0305-0.9144
Sources: From Ackers C (1958), Chow (1959), and Zegzhda (1938).
With regard to Eq. (3.41), it is pertinent to observe that as R increases (equivalent to an increase in the depth of flow), n increases. Approximate values of k for selected materials are summarized in Table 3.4. For sand-bed channels, the following sediment sizes have been suggested by various investigators for estimating the value of k: k d65 (Einstein, 1950), k d90 (Meyer-Peter and Muller, 1948), and k d85 (Simons and Richardson, 1966).
3.4.4 Resistance in Compound Channels In many designed channels and most natural channels, roughness varies along the perimeter of the channel, and it is necessary to estimate an equivalent value of n for the entire perimeter. In such cases, the channel is divided into N parts, each with an associated wetted perimeter (Pi), hydraulic radius (Ri), and roughness coefficient (ni), and the equivalent roughness coefficient (ne) is estimated by one of the following methods. Note that the wetted perimeter does not include the imaginary boundaries between the subsections. 1. Horton (1933) and Einstein and Banks (1950) developed methods of estimating ne assuming that the average velocity in each of the subdivisions is the same as the average velocity of the total section. Then
ne
N
Pini3/2
i1
P
2/3
(3.42)
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HYDRAULICS OF OPEN CHANNEL FLOW
Hydraulics of Open-Channel Flow 3.15
2. Assuming that the total force resisting motion is equal to the sum of the subsection resisting forces,
ne
N
1/2
(Pin i2)
i1
P
(3.43)
3. Assuming that the total discharge of the section is equal to the sum of the subsection discharges, PR5/3 (3.44) ne N PiRi5/3 ni i1
4. Weighting of resistance by area (Cox, 1973),
N
niAi
i1
ne
(3.45)
A
5. The Colebatch method (Cox, 1973).
N
ne
Aini3/2
i1
A
2/3
(3.46)
3.4.5 Solution of Manning’s Equation The uniform flow rate is the product of the velocity of flow and the flow area, or
Q VA AR2/3S (3.47) n In Eq. (3.47), AR2/3 is termed the section factor and, by definition, the conveyance of the channel is
(3.48) K AR2/3 n Before the advent of computers, the solution of Eq. (3.34) or Eq. (3.47) to estimate the depth of flow for specified values of V (or Q), n, and S was accomplished in one of two ways: by trial and error or by the use of a graph of AR2/3 versus y. In the age of the desktop computer, software is used to solve the equations of uniform flow. Trial and error and graphical approaches to the solution of the uniform flow equations can be found in any standard reference or text (e.g., French, 1985).
3.4.6 Special Cases of Uniform Flow 3.4.6.1 Normal and critical slopes. If Q, n, and yN (normal depth of flow) and the channel section are defined, then Eq. (3.47) can be solved for the slope that allows the flow to occur as specified; by definition, this is a normal slope. If the slope is varied while the discharge and roughness are held constant, then a value of the slope such that normal flow Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
HYDRAULICS OF OPEN CHANNEL FLOW
3.16
Chapter Three
occurs in a critical state can be found: that is, a slope such that normal flow occurs with Fr 1. The slope obtained is the critical slope, but it also is a normal slope. The smallest critical slope, for a specified channel shape, roughness, and discharge is termed the limiting critical slope. The critical slope for a given normal depth is gn2DN Sc (3.49)
2RN4/3 where the subscript N indicates the normal depth value of a variable and, for a wide channel, gn2 (3.50) Sc
2yc1/3 3.4.6.2 Sheetflow. A special but noteworthy uniform flow condition is that of sheetflow. From the viewpoint of hydraulic engineering, a necessary condition for sheetflow is that the flow width must be sufficiently wide so that the hydraulic radius approaches the depth of flow. With this stipulation, the Manning’s equation, Eq. (3.48), for a rectangular channel becomes
Q TyN5/3 S (3.51) n where T sheetflow width and yN normal depth of flow. Then, for a specified flow rate and sheetflow width, Eq. (3.51) can be solved for the depth of flow, or nQ 3/5 (3.52) yN T S The condition that the value of the hydraulic radius approaches the depth of flow is not a sufficient condition. That is, this condition specifies no limit on the depth of flow, and there is general agreement that sheetflow has a shallow depth of flow. Appendix 3.A summarizes Manning’s n values for overland and sheetflow.
3.4.6.3 Superelevation. When a body of water moves along a curved path at constant velocity, it is acted for a force directed toward the center of the curvature of the path. When the radius of the curve is much larger than the top width of the water surface, it can be shown that the rise in the water surface at the outer channel boundary above the mean depth of flow in a straight channel (or superelevation) is V2T ∆y (3.53) 2gr where r the radius of the curve (Linsley and Franzini, 1979). It is pertinent to note that if the effects of the velocity distribution and variations in curvature across the channel are considered, the superelevation may be as much as 20 percent more than that estimated by Eq. (3.53) (Linsley and Franzini, 1979). Additional information regarding superelevation is available in Nagami et al., (1982) and U.S. Army Corps of Engineers (USACE, 1970).
3.5 GRADUALLY AND SPATIALLY VARIED FLOW 3.5.1 Introduction The gradual variation in the depth of flow with longitudinal distance in an open channel is given by Eq. (3.19), or
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HYDRAULICS OF OPEN CHANNEL FLOW
Hydraulics of Open-Channel Flow 3.17
So Sf dy 2 1 Fr dx and two cases warrant discussion. In the first case, because the distance over which the change in depth is short it is appropriate to assume that boundary friction losses are small, or Sf 0. When this is the case, important design questions involve abrupt steps in the bottom of the channel (Fig. 3.5) and rapid expansions or contractions of the channel (Fig. 3.6). The second case occurs when Sf ≠ 0. 3.5.2 Gradually Varied Flow with Sf ⴝ 0 When Sf 0 and the channel is rectangular in shape and has a constant width, Eq. (3.19) reduces to dy dz (1 Fr2) 0 (3.54) dx dx and the following observations are pertinent (the observations also apply to channels of arbitrary shape): 1. If dz/dx 0 (upward step) and Fr 1, then dy/dx must be less than zero—depth of flow decreases as x increases. 2. If dz/dx 0 (upward step) and Fr 1, then dy/dx must be greater than zero—depth of flow increases as x increases.
FIGURE 3.5 Definition of variables for gradually varied flow over positive and negative steps.
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HYDRAULICS OF OPEN CHANNEL FLOW
3.18
Chapter Three
FIGURE 3.6 Definition of variables for gradually varied flow through contracting and expanding channel sections.
3. If dz/dx 0 (downward step) and Fr 1, then dy/dx must be greater than zero— depth of flow increases as x increases. 4. If dz/dx 0 (downward step) and Fr 1, then dy/dx must be less than zero—depth of flow decreases as x increases. In the case of a channel of constant width with a positive or negative step, the relation between the specific energy upstream of the step and the specific energy downstream of the step is E1 = E2 + ∆z
(3.55)
In the case dz/dx 0, if the channel is rectangular in shape but the width of the channel changes, it can be shown (French, 1985) that the governing equation is y dT dy (1 Fr2) Fr2 0 b dx dx
(3.56)
The following observations also apply to channels of arbitrary shape: 1. If db/dx 0 (width increases) and Fr 1, then dy/dx must be greater than zero–depth of flow increases as x increases. 2. If db/dx 0 (width increases) and Fr 1, then dy/dx must be less than zero—depth of flow decreases as x increases.
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HYDRAULICS OF OPEN CHANNEL FLOW
Hydraulics of Open-Channel Flow 3.19
3. If db/dx 0 (width decreases) and Fr 1, then dy/dx must be less than zero— depth of flow decreases as x increases. 4. If db/dx 0 (width decreases) and Fr 1, then dy/dx must be greater than zero— depth of flow increases as x increases. In this case, the relation between the specific energy upstream of the contraction (expansion) and the specific energy downstream of the step contraction (expansion) is E1 E2
(3.57)
It is pertinent to note that in the case of supercritical flow, channel expansions and contractions may result in the formation of waves. Additional information regarding steps, expansions, and contractions can be found in any standard reference or text on open-channel hydraulics (e.g., French, 1985). 3.5.3 Gradually Varied Flow with Sf ⴝ 0 In the case where Sf cannot be neglected, the water surface profile must estimated. For a channel of arbitrary shape, Eq. (3.19) becomes So Sf n2 Q2 P4/3 dy S 2 o QT Q2 T dx 13 1 gA gA3
(3.60)
For a specified value of Q, Fr and Sf are functions of the depth of flow y. For illustrative purposes, assume a wide channel; in such a channel, Fr and Sf will vary in much the same way with y since P T and both Sf and Fr have a strong inverse dependence on the flow area. In addition, as y increases, both Sf and Fr decrease. By definition, Sf So when y yN. Given the foregoing, the following set of inequalities must apply:
and
Sf So
for
y yN
Fr 1
for
y yc
Sf So
for
y yN
Fr 1
for
y yc
These inequalities divide the channel into three zones in the vertical dimension. By convention, these zones are labeled 1 to 3 starting at the top. Gradually varied flow profiles are labeled according to the scheme defined in Table 3.5. For a channel of arbitrary shape, the standard step methodology of calculating the gradually varied flow profile is commonly used: for example, HEC-2 (USACE, 1990) or HECRAS (USACE, 1997). The use of this methodology is subject to the following assumptions: (1) steady flow, (2) gradually varied flow, (3) one-dimensional flow with correction for the horizontal velocity distribution, (4) small channel slope, (5) friction slope (averaged) constant between two adjacent cross sections, and (6) rigid boundary conditions. The application of the energy equation between the two stations shown in Fig. 3.7 yields
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Zone 1
(2)
M1
C1
S1
(1)
Mild 0 So Sc
Critical So Sc 0
Steep So Sc 0 S2
C2
M2
(3)
Zone 2
Profile Designation
Classifications of Gradually Varied Flow Profiles
Channel Slope
TABLE 3.5
C3
M3
(4)
Zone 3
Type of Curve (6) Backwater (dy/dx 0) Drawdown (dy/dx < 0) Backwater (dy/dx 0) Backwater (dy/dx 0) Parallel to channel bottom (dy/dx 0) Backwater (dy/dx 0) Backwater (dy/dx 0) Drawdown (dy/dx 0)
Relation of y to yN and yc (5) y yN yc yN y yc yN yc y y yc yN y yN yc
yc yN y y yc yN yc y yN
Supercritical
Subcritical
Supercritical
Uniform critical
Subcritical
Supercritical
Subcritical
Subcritical
Type of Flow (7) HYDRAULICS OF OPEN CHANNEL FLOW
3.20
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None
Horizontal
So 0
Adverse
None
(2)
(1)
So 0
Zone 1
Channel Slope
TABLE 3.5: (Continued)
A2
H2
(3)
Zone 2
A3
H3
Drawdown (dy/dx 0) Backwater (dy/dx 0)
yN yc y
Backwater (dy/dx 0)
yN yc y
yN y yc
Drawdown (dy/dx 0)
yN y yc
Backwater (dy/dx 0)
yc yN y
S3
(4)
Type of Curve (6)
Relation of y to yN and yc (5)
Zone 3
Supercritical
Subcritical
Supercritical
Subcritical
Supercritical
Type of Flow (7)
HYDRAULICS OF OPEN CHANNEL FLOW
3.21
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HYDRAULICS OF OPEN CHANNEL FLOW
3.22
Chapter Three
V2 α1 1 2g
En
erg
yG
rad
eL
y1
z1
Wa ter
Cha
nne
ine
(S )
hf x
f
Sur
face
l Bo
ttom
V2 α2 2 2g
(S ) w
(S ) o
y2 z2
xL FIGURE 3.7 Energy relationship between two channel sections.
V2 V2 z1 α1 1 z2 α2 2 hf + he 2g 2g
(3.61)
where z1 and z2 elevation of the water surface above a datum at Stations 1 and 2, respectively, he eddy and other losses incurred in the reach, and hf reach friction loss. The friction loss can be obtained by multiplying a representative friction slope, Sf , by the length of the reach, L. Four equations can be used to approximate the friction loss between two cross sections: Q1 Q2 Sf K1 K2
2
(average conveyance)
(3.62)
Sf1 Sf2 Sf (average friction slope) 2
(3.63)
2 S Sf2 Sf f1 (harmonic mean friction slope) Sf1 Sf2
(3.64)
Sf S S f1 f2 (geometric mean friction slope)
(3.65)
and
The selection of a method to estimate the friction slope in a reach is an important decision and has been discussed in the literature. Laurenson (1986) suggested that the “true” friction slope for an irregular cross section can be approximated by a third-degree poly-
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HYDRAULICS OF OPEN CHANNEL FLOW
Hydraulics of Open-Channel Flow 3.23
nomial. He concluded that the average friction slope method produces the smallest maximum error, but not always the smallest error, and recommended its general use along with the systematic location of cross sections. Another investigation based on the analysis of 98 sets of natural channel data showed that there could be significant differences in the results when different methods of estimating the friction slope were used (USACE, 1986). This study also showed that spacing cross sections 150m (500 ft) a part eliminated the differences. The eddy loss takes into account cross section contractions and expansions by multiplying the absolute difference in velocity heads between the two sections by a contraction or expansion coefficient, or
V2 V2 he Cx α1 2 α2 2 2g 2g
(3.66)
There is little generalized information regarding the value of the expansion (Ce) or the contraction coefficient (Cc). When the change in the channel cross section is small, the coefficients Ce and Cc are typically on the order of 0.3 and 0.1, respectively (USACE, 1990). However, when the change in the channel cross section is abrupt, such as at bridges, Ce and Cc may be as high as 1.0 and 0.6, respectively (USACE, 1990). With these comments in mind, V2 H1 z1 α1 1 (3.67) 2g and V2 H2 z2 α2 2 (3.68) 2g With these definitions, Eq. (3.61) becomes H1 H2 hf he
(3.69)
Eq. (3.69) is solved by trial and error: that is, assuming H2 is known and given a longitudinal distance, a water surface elevation at Station 1 is assumed, which allows the computation of H1 by Eq. (3.67). Then, hf and he are computed and H1 is estimated by Eq. (3.67). If the two values of H1 agree, then the assumed water surface elevation at Station 1 is correct. Gradually varied water surface profiles are often used in conjunction with the peak flood flows to delineate areas of inundation. The underlying assumption of using a steady flow approach in an unsteady situation is that flood waves rise and fall gradually. This assumption is of course not valid in areas subject to flash flooding such as the arid and semiarid Southwestern United States (French, 1987). In summary, the following principles regarding gradually varied flow profiles can be stated: 1. The sign of dy/dx can be determined from Table 3.6. 2. When the water surface profile approaches normal depth, it does so asymptotically. 3. When the water surface profile approaches critical depth, it crosses this depth at a large but finite angle. 4. If the flow is subcritical upstream but passes through critical depth, then the feature that produces critical depth determines and locates the complete water surface profile. If the upstream flow is supercritical, then the control cannot come from the downstream.
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HYDRAULICS OF OPEN CHANNEL FLOW
3.24
Chapter Three
5. Every gradually varied flow profile exemplifies the principle that subcritical flows are controlled from the downstream while supercritical flows are controlled from upstream. Gradually varied flow profiles would not exist if it were not for the upstream and downstream controls. 6. In channels with horizontal and adverse slopes, the term “normal depth of flow” has no meaning because the normal depth of flow is either negative or imaginary. However, in these cases, the numerator of Eq. (3.60) is negative and the shape of the profile can be deduced. Any method of solving a gradually varied flow situation requires that cross sections be defined. Hoggan (1989) provided the following guidelines regarding the location of cross sections: 1. They are needed where there is a significant change in flow area, roughness, or longitudinal slope. 2. They should be located normal to the flow. 3. They should be located in detail—upstream, within the structure, and downstreamat structures such as bridges and culverts. They are needed at all control structures. 4. They are needed at the beginning and end of reaches with levees. 5. They should be located immediately below a confluence on a main stem and immediately above the confluence on a tributary. 6. More cross sections are needed to define energy losses in urban areas, channels with steep slopes, and small streams than needed in other situations. 7. In the case of HEC-2, reach lengths should be limited to a maximum distance of 0.5 mi for wide floodplains and for slopes less than 38,550 m (1800 ft) for slopes equal to or less than 0.00057, and 370 m (1200 ft) for slopes greater than 0.00057 (Beaseley, 1973).
3.6
GRADUALLY AND RAPIDLY VARIED UNSTEADY FLOW
3.6.1 Gradually Varied Unsteady Flow Many important open-channel flow phenomena involve flows that are unsteady. Although a limited number of gradually varied unsteady flow problems can be solved analytically, most problems in this category require a numerical solution of the governing equations. Examples of gradually varied unsteady flows include flood waves, tidal flows, and waves generated by the slow operation of control structures, such as sluice gates and navigational locks. The mathematical models available to treat gradually varied unsteady flow problems are generally divided into two categories: models that solve the complete Saint Venant equations and models that solve various approximations of the Saint Venant equations. Among the simplified models of unsteady flow are the kinematic wave, and the diffusion analogy. The complete solution of the Saint Venant equations requires that the equations be solved by either finite difference or finite element approximations. The one dimensional Saint Venant equations consist of the equation of continuity ∂y ∂v ∂y y u 0 ∂t ∂x ∂x
(3.70a)
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HYDRAULICS OF OPEN CHANNEL FLOW
Hydraulics of Open-Channel Flow 3.25
and the conservation of momentum equation ∂v ∂v ∂y v g g(So Sf) 0 ∂t ∂x ∂x
(3.71a)
An alternate form of the continuity and momentum equations is ∂y ∂(Au) T 0 ∂t ∂x
(3.70b)
1 ∂v ∂y v ∂v Sf So 0 g ∂t ∂x g ∂x
(3.71b)
and
By rearranging terms, Eq. (3.71b) can be written to indicate the significance of each term for a particular type of flow, or 1 ∂v ∂y v ∂v Sf So steady steady, nonuniform unsteady, nonuniform ∂x g ∂x g ∂t
(3.72)
Equations (3.70) and (3.71) compose a group of gradually varied unsteady flow models that are termed complete dynamic models. Being complete, this group of models can provide accurate results; however, in many applications, simplifying assumptions regarding the relative importance of various terms in the conservation of momentum equation (Eq. 3.71) leads to other equations, such the kinematic and diffusive wave models (Ponce, 1989). The governing equation for the kinematic wave model is ∂Q ∂Q ( V) 0 ∂t ∂x
(3.73)
where a coefficient whose value depends on the frictional resistance equation used ( = 5/3 when Manning’s equation is used). The kinematic wave model is based on the equation of continuity and results in a wave being translated downstream. The kinematic wave approximation is valid when tRSoV 85 y
(3.74)
where tR time of rise of the inflow hydrograph (Ponce, 1989). The governing equation for the diffusive wave model is ∂Q ∂Q Q ∂2Q ( V ) ∂t ∂x 2TSo ∂x2
(3.75)
where the left side of the equation is the kinematic wave model and the right side accounts for the physical diffusion in a natural channel. The diffusion wave approximation is valid when (Ponce, 1989), g 0.5 tRSo 15 y
(3.76)
If the foregoing dimensionless inequalities ( Eq. 3.74 and 3.76) are not satisfied, then the complete dynamic wave model must be used. A number of numerical methods can be used to solve these equations (Chaudhry, 1987; French; 1985, Henderson, 1966; Ponce, 1989).
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HYDRAULICS OF OPEN CHANNEL FLOW
3.26
Chapter Three
3.6.2 Rapidly Varied Unsteady Flow The terminology “rapidly varied unsteady flow” refers to flows in which the curvature of the wave profile is large, the change of the depth of flow with time is rapid, the vertical acceleration of the water particles is significant relative to the total acceleration, and the effect of boundary friction can be ignored. Examples of rapidly varied unsteady flow include the catastrophic failure of dams, tidal bores, and surges that result from the quick operation of control structures such as sluice gates. A surge producing an increase in depth is termed a positive surge, and one that causes a decrease in depth is termed a negative surge. Furthermore, surges can go either upstream or downstream, thus giving rise to four basic types (Fig. 3.8). Positive surges generally have steep fronts, often with rollers, and are stable. In contrast, negative surges are unstable, and their form changes with the advance of the wave. Consider the case of a positive surge (or wave) traveling at a constant velocity (wave celerity) c up a horizontal channel of arbitrary shape (Fig. 3.8b). Such a situation can result from the rapid closure of a downstream sluice gate. This unsteady situation is converted to a steady situation by applying a velocity c to all sections; that is, the coordinate system is moving at the velocity of the wave. Applying the continuity equation between Sections 1 and 2 (V1 c)A1 (V2 c)A2
(3.77)
Since there are unknown losses associated with the wave, the momentum equation rather than the energy equation is applied between Sections 1 and 2 or γ γA1z1 γA2z2 y1(V1 c)(V2 c V1 c) (3.78) g
y1 y2
y2 y1
FIGURE 3.8 Definition of variables for simple surges moving in an open channel.
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HYDRAULICS OF OPEN CHANNEL FLOW
Hydraulics of Open-Channel Flow 3.27
y1 y2
y2 y1
FIGURE 3.8 Definition of variables for simple surges moving in an open channel.
where boundary friction has been ignored. Eliminating V2 in Eq. (3.78) by manipulation of Eq. (3.77) yields A2 0.5 g A1 (A1z1 A2z2) V1 c (3.79) A1 A2
In the case of a rectangular channel, Eq. (3.79) reduces to y2 y2 0.5 V1 c g y1 1 2 y y1 1
(3.80)
When the slope of a channel becomes very steep, the resulting supercritical flow at normal depth may develop into a series of shallow water waves known as roll waves. As these waves progress downstream, they eventually break and form hydraulic bores or shock waves. When this type of flow occurs, the increased depth of flow requires increased freeboard, and the concentrated mass of the wavefronts may require additional structural factors of safety. Escoffier (1950) and Escoffier and Boyd (1962) considered the theoretical conditions under which a uniform flow must be considered unstable. Whether roll waves form or not is a function of the Vedernikov number (Ve), the Montuori number (Mo), and the concentration of sediment in the flow. When the Manning equation is used, the Ve is Ve 2 Fr 3
(3.81a)
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HYDRAULICS OF OPEN CHANNEL FLOW
3.28
Chapter Three
TABLE 3.6
Shape Factor for Common Channel Sections
Channel Definition (1)
(2)
Rectangle
y
b b 2y
b
Trapezoid Trapezoid with unequal side slopes
R( 1 z21 1 z22 ) 1 T
Circle
θ sin(θ) 1 θ[1 cos(θ)]
and if the Chezy equation is used Ve 1 Fr (3.81b) 2 Fr should be computed using Eq. (3.11) and a channel shape factor (Table 3.6) or dP 1 R dA
(3.82)
When Mo 1, flow instabilities should expected. The Montuori number is given by gSfL Mo V2
(3.83)
It is appropriate to note that in some publications (e.g., Aisenbrey et al., 1978) Mo is the inverse of Eq. (3.83). Figure 3.9 provides a basis for deciding whether roll waves will form in a given situation. In the figure, data from Niepelt and Locher (1989) for a slurry flow are also plotted. The Niepelt and Locher data suggest that flow stability also is a function of the concentration of sediment.
3.7 CONCLUSION The foregoing sections provide the basic principles on which the following chapters on design are based. Two observations are pertinent. First, open-channel hydraulics is incrementally progressing. That is, over the past several decades, there have been incremental advances that primarily have added details, often important details, but no major new advances. Second, open-channel hydraulics remains a one-dimensional analytic approach. However, the assumption of a one-dimensional approach may not be valid in many situations: for example, nonprismatic channels, flow downstream of a partially breached dam, or lateral flow over a spillway. In some of these cases, the one-dimensional approach may provide an approximation that is suitable for design. In other cases, however, a two– or three– dimensional approach should be used. Additional information regarding two– and three– dimensional approaches can be found in Chaudhry (1993). Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
HYDRAULICS OF OPEN CHANNEL FLOW
FIGURE 3.9 Flow stability as a function of the Vedernikov and Montuori numbers for clear water and slurry flow. (Based on data from Montuori, 1963; Niepelt and Locher, 1989
gSfL V2
Hydraulics of Open-Channel Flow 3.29
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HYDRAULICS OF OPEN CHANNEL FLOW
3.30
Chapter Three
REFERENCES Ackers, P., “Resistance to Fluids flowing in Channels and Pipes,” Hydraulic Research Paper No. 1, Her Majety’s Stationery Office, London, 1958. Aisenbrey, A. J., Jr., R. B., Hayes, H. J., Warren, D. L., Winsett, and R. B. Young, Design of Small Canal Structures, U.S. Department of Interior, Bureau of Reclamation, Washington, DC 1978. Arcement, G. J., and V. R. Schneider, “Guide for Selecting Manning’s Roughness Coefficients for Natural Channels and Flood Plains,” Water Supply Paper 2339, U.S. Geological Survey, Washington, DC, 1989. Barnes, H. H., “Roughness Characteristics of Natural Channels,” U.S. Geological Survey Water Supply Paper No. 1849, U.S. Geological Survey, Washington, DC. 1967. Beasley, J. G., An Investigation of the Data Requirements of Ohio for the HEC-2 Water Surface Profiles Model, Master’s thesis, Ohio State University, Columbus, 1973. Chaudhry, M. H., Open-Channel Flow, Prentice-Hall, New York 1993. Chaudhry, M. H., Applied Hydraulic Transients, Van Nostrand Reinhold, New York, 1987. Chow, V. T., Open-Channel Hydraulics, McGraw-Hill, New York, 1959. Cox, R. G., “Effective Hydraulic Roughness for Channels Having Bed Roughness Different from Bank Roughness,” Miscellaneous Paper H-73-2, U.S. Army Engineers Waterways Experiment Station, Vicksburg, MS, 1973. Coyle, J. J. “Grassed Waterways and Outlets,” Engineering Field Manual, U.S. Soil Conservation Service, Washington, DC, April, 1975, pp. 7-1–7-43. Einstein, H. A., “The Bed Load Function for Sediment Transport in Open Channel Flows. Technical Bulletin No. 1026, U.S. Department of Agriculture, Washington, DC, 1950. Einstein, H. A., and R. B. Banks, “Fluid Resistance of Composite Roughness,” Transactions of the American Geophysical Union, 31(4): 603–610, 1950. Escoffier, F. F., “A Graphical Method for Investigating the Stability of Flow in Open Channels or in Closed Conduits Flowing Full,” Transactions of the American Geophysical Union, 31(4), 1950. Escoffier, F. F., and M. B. Boyd, “Stability Aspects of Flow in Open Channels,” Journal of the Hydraulics Division, American Society of Civil Engineers, 88(HY6): 145–166, 1962. Fischenich, J. C., “Hydraulic Impacts of Riparian Vegetation: Computation of Resistance,” EIRP Technical Report EL-96-XX, U.S. Army Engineers Waterways Experiment Station, Vicksburg, MS, August 1996. Flippin-Dudley, S. J., “Vegetation Measurements for Estimating Flow Resistance,” Doctoral dissertation, Colorado State University, Fort Collins, 1997. Flippin-Dudley, S. J., S. R. Abt, C. D. Bonham, C. C. Watson, and J. C. Fischenich, “A Point Quadrant Method of Vegetation Measurement for Estimating Flow Resistance,” Technical Report No. EL-97-XX, U.S. Army Engineers Waterways Experiment Station, Vicksburg, MS, 1997. French, R. H., Hydraulic Processes on Alluvial Fans. Elsevier, Amsterdam, 1987. French, R. H., Open-Channel Hydraulics, McGraw-Hill, New York, 1985. Guy, H. P., D. B. Simons, and E. V. Richardson, “Summary of Alluvial Channel Data from Flume Experiments, 1956-61,” Professional Paper No. 462-1, U.S. Geological Survey, Washington, DC, 1966. Henderson, F. M., Open Channel Flow, Macmillan, New York, 1966. Hoggan, D. H., Computer-Assisted Floodplain Hydrology & Hydraulics, McGraw-Hill, New York, 1989. Horton, R. E., “Separate Roughness Coefficients for Channel Bottom and Sides,” Engineering News Record, 3(22): 652–653, 1933.
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HYDRAULICS OF OPEN CHANNEL FLOW
Hydraulics of Open-Channel Flow 3.31 Jarrett, R. D., “Hydraulics of High-Gradient Streams,” Journal of Hydraulic Engineering, American Society of Civil Engineers, 110(11): 1519–1539, 1984. Kouwen, N., “Field Estimation of the Biomechanical Properties of Grass,” Journal of Hydraulic Research, International Association and Hydraulic Research, 26(5): 559–568, 1988. Kouwen, N., and R. Li, “Biomechanics of Vegetative Channel Linings,” Journal of the Hydraulics Division, American Society of Civil Engineers, 106(HY6): 1085–1103, 1980. Lane, E. W., and E. J. Carlson, “Some Factors Affecting the Stability of Canals Constructed in Coarse Granular Materials,” Proceedings of the Minnesota International Hydraulics Convention, September 1953. Laurenson, E. M., “Friction Slope Averaging in Backwater Calculations,” Journal of Hydraulic Engineering, American Society of Civil Engineers 112(12),1151–1163 1986. Levi, E., The Science of Water: The Foundation of Modern Hydraulics, Translated from the Spanish by D. E. Medina, ASCE Press, New York, 1995. Linsley, R. K. and J. B. Franzini, Water Resources Engineering, 3rd ed., Mc-Graw-Hill, New York, 1979. Meyer-Peter, P. E., and R. Muller, “Formulas for Bed Load Transport,” Proceedings of the 3rd International Association for Hydraulic Research, Stockholm, 1948, pp. 39–64. Montuori, C., Discussion of “Stability Aspects of Flow in Open Channels,” Journal of the Hydraulics Division, American Society of Civil Engineers 89(HY4): 264–273, 1963. Nagami, M., R. Scavarda, G. Pederson, G. Drogin, D. Chenoweth, C. Chow, and M. Villa, Design Manual: Hydraulic, Design Division, Los Angeles County Flood Control District, Los Angeles, CA, 1982. Niepelt, W. A., and F. A. Locher, “Instability in High Velocity Slurry Flows, Mining Engineering, Society for Mining, Metallurgy and Exploration, 1989, pp. 1204–1209. Petryk, S., and G. Bosmajian, “Analysis of Flow Through Vegetation,” Journal of the Hydraulics Division, American Society of Civil Engineers, 101(HY7): 871–884, 1975. Ponce, V.M., Engineering Hydrology: Principles and Practices, Prentice–Hall, Englewood Cliffs, NJ, 1989. Richardson, E. V., D. B.Simons, and P. Y. Julien, Highways in the River Environment, U.S. Department of Transportation, Federal Highway Administration, Washington, DC, 1987. Silvester, R., “Theory and Experiment on the Hydraulic Jump,” Proceedings of the 2nd Australasian Conference on Hydraulics and Fluid Mechanics, 1965, pp. A25–A39. Silvester, R., “Hydraulic Jump in All Shapes of Horizontal Channels,” Journal of the Hydraulics Division, American Society of Civil Engineers, 90(HY1): 23–55, 1964. Simons, D. B., and E. V. Richardson, “Resistance to Flow in Alluvial Chºannels,” Professional Paper 422-J, U.S. Geological Survey, Washington, DC, 1966. Simons, Li & Associates, SLA Engineering Analysis of Fluvial Systems, Fort Collins, CO, 1982. Straub, W. O. “A Quick and Easy Way to Calculate Critical and Conjugate Depths in Circular Open Channels,” Civil Engineering, 70–71, December 1978. Straub, W. O., Personal Communication, Civil Engineering Associate, Department of Water and Power, City of Los Angeles, January 13, 1982. Temple, D. M., “Changes in Vegetal Flow Resistance During Long-Duration Flows,” Transactions of the ASAE, 34: 1769–1774, 1991. Urquhart, W. J. “Hydraulics,” in Engineering Field Manual, U.S. Department of Agriculture, Soil Conservation Service, Washington, DC, 1975. U.S Army Corps Engineers, HEC-RAS River Analysis System, User’s Manual, U.S. Army Corps of Engineers Hydrologic Engineering Center, Davis, CA, 1997. U.S Army Corps Engineers, “HEC-2, Water Surface Profiles, Userís Manual,” U.S. Army Corps of Engineers Hydrologic Engineering Center, Davis, CA, 1990.
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HYDRAULICS OF OPEN CHANNEL FLOW
3.32
Chapter Three
U.S Army Corps Engineers, “Accuracy of Computed Water Surface Profiles,” U.S. Army Corps of Engineers Hydrologic Engineering Center, Davis, CA, 1986. U.S Army Corps Engineers, “Hydraulic Design of Flood Control Channels,” EM 1110-2-1601. U.S. Army Corps of Engineers, Washington, DC, 1970. Yang, C. T., Sediment Transport: Theory and Practice, McGraw-Hill, NewYork, 1996. Zegzhda, A.P., Theroiia Podobija Metodika Rascheta Gidrotekhnickeskikh Modele (Theory of Similarity and Methods of Design of Models for Hydraulic Engineering), Gosstroiizdat, Leningrad, 1938.
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HYDRAULICS OF OPEN CHANNEL FLOW
Hydraulics of Open-Channel Flow 3.33
APPENDIX 3.A
VALUES OF THE ROUGHNESS COEFFICIENT n Values of the Roughness Coefficient n* Type of Channel
Minimum
A Closed Conduits flowing partly full A–1 Metal a. Brass, smooth 0.009 b. Steel 1. Lockbar and welded 0.010 2. Riveted and spiral 0.013 c. Cast iron 1. Coated 0.010 2. Uncoated 0.011 d. Wrought iron 1. Black 0.012 2. Galvanized 0.013 e. Corrugated metal 1. Subdrain 0.017 2. Storm drain 0.021 A–2 Non-metal a. Lucite 0.008 b. Glass 0.009 c. Cement 1. Neat, surface 0.010 2. Mortar 0.011 d. Concrete 1.Culvert, straight and free of debris 0.010 2. Culvert, with bends, connections, and some debris 0.011 3. Finished 0.011 4. Sewer and manholes, inlet, etc, straight 0.013 5. Unfinished, steel form 0.012 6. Unfinished, smooth wood form 0.012 7. Unfinished, rough wood form 0.015 e. Wood 1. Stave 0.010 2. Laminated, treated 0.015
Normal
Maximum
0.010
0.013
0.012 0.016
0.014 0.017
0.013 0.014
0.014 0.016
0.014 0.016
0.015 0.017
0.019 0.024
0.030 0.030
0.009 0.010
0.010 0.013
0.011 0.013
0.013 0.015
0.011
0.013
0.013 0.012
0.014 0.014
0.015 0.013 0.014
0.017 0.014 0.016
0.017
0.020
0.012 0.017
0.014 0.020
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HYDRAULICS OF OPEN CHANNEL FLOW
3.34
Chapter Three
Type of Channel
f. Clay 1. Common drainage tile 2. Vitrified sewer 3. Vitrified sewer with manholes, inlet, etc. 4. Vitrified subdrain with open joint g. Brickwork 1. Glazed 2. Lined with cement mortar h. Sanitary sewers coated with sewage slimes with bends and connections i. Paved invert, sewer, smooth bottom j. Rubble masonry, cemented k.Polyethylene pipe l. Polyvinyl chloride B. Lined or Built–up Channels B–1 Metal a. Smooth steel surface 1. Unpainted 2. Painted b. Corrugated B–2 Nonmetal a. Cement 1. Neat, surface 2. Mortar b. Wood 1. Planed, untreated 2. Planed, creosoted 3. Unplaned 4. Plank with battens 5. Lined with roofing paper c. Concrete 1. Trowel finish 2. Float finish 3. Finished, with gravel on bottom 4. Unfinished 5. Gunite, good section 6. Gunite, wavy section 7. On good excavated rock 8. On irregular excavated rock
Minimum
Normal
Maximum
0.011 0.011
0.013 0.014
0.017 0.017
0.013
0.015
0.017
0.014
0.016
0.018
0.011 0.012
0.013 0.015
0.015 0.017
0.012
0.013
0.016
0.016 0.018 0.009 0.010
0.019 0.025 — —
0.020 0.030 — —
0.011 0.012 0.021
0.012 0.013 0.025
0.014 0.017 0.030
0.010 0.011
0.011 0.013
0.013 0.015
0.010 0.011 0.011 0.012 0.010
0.012 0.012 0.013 0.015 0.014
0.014 0.014 0.015 0.018 0.017
0.011 0.013
0.013 0.015
0.015 0.016
0.015 0.014 0.016 0.018 0.017 0.022
0.017 0.017 0.019 0.022 0.020 0.027
0.020 0.020 0.023 0.025 — —
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HYDRAULICS OF OPEN CHANNEL FLOW
Hydraulics of Open-Channel Flow 3.35
Type of Channel
Minimum
Normal
Maximum
0.015 0.017
0.017 0.020
0.020 0.024
0.016 0.020 0.020
0.020 0.025 0.030
0.024 0.030 0.035
0.017 0.020 0.023
0.020 0.023 0.033
0.025 0.026 0.036
0.011 0.012
0.013 0.015
0.015 0.018
0.017 0.023 0.013
0.025 0.032 0.015
0.030 0.035 0.017
0.013 0.016 0.030
0.013 0.016 —
— — 0.500
0.016 0.018 0.022 0.022
0.018 0.022 0.025 0.027
0.020 0.025 0.030 0.033
0.023 0.025
0.025 0.030
0.030 0.033
0.030
0.035
0.040
0.028
0.030
0.035
0.025
0.035
0.040
0.030
0.040
0.050
0.025 0.035
0.028 0.050
0.033 0.060
d. Concrete bottom float with sides of 1. Dressed stone in mortar 2. Random stone in mortar 3. Cement, rubble masonry, plastered 4. Cement rubble masonry 5. Dry rubble or riprap e. Gravel bottom with sides of 1. Formed concrete 2. Random stone in mortar 3. Dry rubble or riprap f. Brick 1. Glazed 2. In cement mortar g. Masonry 1. Cemented rubble 2. Dry rubble h. Dressed ashlar i. Asphalt 1. Smooth 2. Rough J. Vegetal cover C–1 Excavated or Dredged C.1 General a. Earth, straight and uniform 1. Clean and recently completed 2. Clean, after weathering 3. Gravel, uniform section, clean 4. With short grass, few weeds b. Earth, winding and sluggish 1. No vegetation 2. Grass, some weeds 3. Dense weeds or aquatic plants in deep channels 4. Earth bottom and rubble sides 5. Stony bottom and weedy banks 6. Cobble bottom and clean sides c. Dragline-excavated or dredged 1. No vegetation 2. Light brush on banks
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HYDRAULICS OF OPEN CHANNEL FLOW
3.36
Chapter Three
Type of Channel
Minimum
d. Rock cuts 1. Smooth and uniform 0.025 2. Jagged and irregular 0.035 e. Channels not maintained, weeds and brush uncut 1. Dense weeds, high as flow depth 0.050 2. Clean bottom, brush on sides 0.040 3. Same, highest stage of flow 0.045 4. Dense brush, high stage 0.080 C.2 Channels with maintained vegetation and velocities of 2 and 6 ft/s a. Depth of flow up to 0.7 ft 1. Bermuda grass, Kentucky bluegrass, buffalo grass Mowed to 2 in 0.07 Length 4 to 6 in 0.09 2. Good stand, any grass Length approx. 12 in 0.18 Length approx. 24 in 0.30 3. Fair stand, any grass Length approx. 12 in 0.014 Length approx. 24 in 0.25 b. Depth of flow up to 0.7–1.5 ft 1. Bermuda grass, Kentucky bluegrass, buffalo grass Mowed to 2 in 0.05 Length 4–6 in 0.06 2. Good stand, any grass Length approx. 12 in 0.12 3. Length approx. 24 in 0.20 Fair stand, any grass Length approx. 12 in 0.10 Length approx. 24 in 0.17 D Natural streams D–1 Minor streams (top width at flood stage < 100 ft) a. Streams on plain 1. Clean, straight, full stage no rifts or deep pools 0.025 2. Same as above, but with more stones and weeds 0.030 3. Clean, winding, some pools and shoals 0.033 4. Same as above, but with some weeds and stones 0.035
Normal
Maximum
0.035 0.040
0.040 0.050
0.080 0.050 0.070 0.100
0.120 0.080 0.110 0.14
0.045 0.05 0.09 0.15 0.08 0.13
0.035 0.04 0.07 0.10 0.16 0.09
0.030
0.033
0.035
0.040
0.040
0.045
0.045
0.050
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HYDRAULICS OF OPEN CHANNEL FLOW
Hydraulics of Open-Channel Flow 3.37
Type of Channel
Minimum
Normal
5. Same as above, lower stages more ineffective slopes and sections 0.040 6. Same as no. 4, more stones 0.045 7. Sluggish reaches, weedy, deep pools 0.050 8. Very weedy, reaches, deep pools or floodways with heavy stand of timber and underbrush 0.075 b. Mountain streams, no vegetation in channel, banks usually steep, trees and brush along banks submerged at high stages 1. Bottom: gravels, cobbles and few boulders 0.030 2. Bottom: cobbles with large boulders 0.040 D–2 Floodplains a. Pasture, no brush 1. Short grass 0.025 2. High grass 0.030 b. Cultivated areas 1. No crop 0.020 2. Mature row crops 0.025 3. Mature field crops 0.030 c. Brush 1. Scattered brush, heavy weeds 0.035 2. Light brush and trees in winter 0.035 3. Light brush and trees in summer 0.040 4. Medium to dense brush in winter 0.045 5. Medium to dense brush in summer 0.070 d. Trees 1. Dense willows, summer, straight 0.110 2. Cleared land with tree stumps, no sprouts 0.030 3. Same as above but with a heavy growth of sprouts 0.050 4. Heavy stand of timber, a few down trees, little undergrowth, flood stage below branches 0.080 5. Same as above, but with flood stage reaching branches 0.100
Maximum
0.048 0.050
0.055 0.060
0.070
0.080
0.100
0.150
0.040
0.050
0.050
0.070
0.030 0.035
0.035 0.050
0.030 0.035 0.040
0.040 0.045 0.050
0.050
0.070
0.050
0.060
0.070
0.110
0.070
0.110
0.100
0.160
0.150
0.200
0.040
0.050
0.060
0.080
0.100
0.120
0.120
0.160
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HYDRAULICS OF OPEN CHANNEL FLOW
3.38
Chapter Three
Type of Channel
Minimum
Normal
Maximum
D–3 Major streams (top width at flood stage > 100 ft); the n value is less that for minor streams of similar description because banks offer less effective resistance a. Regular section with no boulders or brush 0.025 b. Irregular and rough section 0.035
— —
0.060 0.100
— — — —
0.020 0.030 0.040 0.025
— —
0.015 0.020
0.17 0.17 0.20 0.20 0.10 0.30 0.05
— — — — — — —
0.80 0.48 0.40 0.30 0.20 0.40 0.13
0.09 0.05
— —
0.34 0.25
0.008 0.06 0.06 0.30 0.04 0.07 0.17
— — — — — — —
0.012 0.22 0.16 0.50 0.10 0.17 0.47
0.10
—
0.20
0.10 0.08 0.04 0.02
— — — —
0.15 0.12 0.10 0.05
D–4 Alluvial sandbed channels (no vegetation and data is limited to sand channels with D50 < 1.0 mm a. Tranquil flow, Fr < 1 1. Plane bed 0.014 2. Ripples 0.018 3. Dunes 0.020 4. Washed out dunes or transition 0.014 b. Rapid flow, Fr > 1 1. Standing waves 0.010 2. Antidunes 0.012 E. Overland Flow (Sheetflow) E–1 Vegetated areas a. Dense turf b. Bermuda and dense grass c. Average grass cover d. Poor grass cover on rough surface e. Short prairie grass f. Shrubs and forest litter, pasture g. Sparse vegetation h. Sparse rangeland with debris 1. 0% cover 2. 20% cover E–2 Plowed or tilled fields a. Fallow—no residue b. Conventional tillage c. Chisel plow d. Fall disking e. No till—no residue f. No till (20–40% residue cover) g. No till (100% residue cover) E–3 Other surfaces a. Open ground with debris b. Shallow flow on asphalt or concrete c. Fallow fields d. Open ground, no debris f. Asphalt or concrete
Source: From Chow (1959), Richardson et al. (1987), Simons, Li, & Associates (SLA), 1982, and others. * The values in bold are recommended for design
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Source: HYDRAULIC DESIGN HANDBOOK
CHAPTER 4
SUBSURFACE FLOW AND TRANSPORT Mariush W. Kemblowki and Gilberto E. Urroz Utah Water Research Laboratory Utah State University Logan, Utah
4.1
INTRODUCTION
This chapter begins with the mathematical description of the constitutive relationships for flow and transport in porous media. Following this, simple analytical solutions are presented for a variety of subsurface flow and transport problems. The principles of flow and transport are outlined, and solutions are provided for practical problems of flow and transport in both the saturated and the unsaturated zones. The latter includes problems of transport in the vapor phase. The major focus is on the processes that are relevant to subsurface mitigation.
4.2
CONSTITUTIVE RELATIONSHIPS
This section presents the basic concepts and laws used to describe flow and transport in the subsurface. In particular, the constitutive relationships defining the fluid flow in fully and partially saturated media are given as well as the relationships that describe diffusive and dispersive mass fluxes in porous media. Finally, we show the relations used to describe partitioning of chemicals in the subsurface environment.
4.2.1
Darcy’s law
Consider the flow of a fluid through a pipe filled with a granular material, as shown in Fig. 4.1. In the figure, z1 and z2 represent the elevations of the pipe centerline above a reference level at Sections 1 and 2, respectively, whereas p1/γ and p2/γ represent the water pressure head at Sections 1 and 2, respectively. We define the piezometric head at any location in the porous media as h z p/γ
(4.1)
where γ specific weight (weight per unit volume) of water, typically, γ 9810 N/m3 or 62.4 lb/ft3. Let q be the average water velocity in the cross section of the pipe: i.e., q Q/A
(4.2)
4.1
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SUBSURFACE FLOW AND TRANSPORT
4.2
Chapter Four
∆h p1
γw
p2
1
γw
h1
L
2
h2
z1 z2 Arbitrary datum FIGURE 4.1 Porous media flow.
where Q volumetric discharge (volume per unit time) and A total cross-sectional area of the pipe (including the soil matrix). French hydrologist Henry Darcy discovered that the average flow velocity could be estimated from q K (h1 h2)/L
(4.3)
where L is the distance, measured along the pipe, between cross sections 1 and 2, and K is a parameter that depends on the nature of the porous media as well as on the properties of the transported fluid. For water, K is known as the hydraulic conductivity or the coefficient of permeability. Typical values of K are given in many references (e.g., Bureau of Reclamation, 1985) see tables 4.1 ard 4.2. Eq. (4.3) is known as Darcy's law and is commonly used to model the flow of fluids in porous media. Notice that the velocity V is not the fluid velocity in the soil pores, it is an average velocity calculated over the entire area of the flow cross section. The average pore velocity is calculated as q v (4.4) θW where θW is the volumetric water moisture content. Note that for saturated flow, θW n, where n is porosity. Darcy’s law (i.e., Eq. 4.3) also can be written more concisely as q KI
(4.5)
where I is the hydraulic gradient defined as (h1 h2) 4.6) I L Hydraulic conductivity K is a function of aquifer and fluid properties –specifically of the intrinsic soil permeability k, fluid viscosity µ, and fluid density ρ–and is given by K k ρg/µ
(4.7)
For saturated flow of a constant density fluid in isotopic porous media, the Darcy law can be written as
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SUBSURFACE FLOW AND TRANSPORT
Subsurface Flow and Transport 4.3 ∂h qi K ∂x
j
(4.8)
For anisotropic media, the Darcy law is written as qi Kij ∂h ∂xj
(4.9)
where Kij is the conductivity tensor. For saturated flow of a fluid of variable density in anisotropic porous media, we have k il,ij ∂p ∂z qi so ρfluid g ∂xi µfluid ∂xi
(4.10)
where ksoil,ij is the intrinsic permeability tenser. Finally, for unsaturated flow of variabledensity fluid in anisotropic porous media, we have ∂z kr(θ)ksoil,ij ∂p qi ρfluid g ∂xj µfluid ∂xj
(4.11)
where kr(θ) relative permeability of the porous media. Relative permeability is a function of soil saturation, which in turn is a function of the capillary pressure. These relationships for partially saturated flow are discussed in the next section.
4.2.2 Unsaturated Flow–Constitutive Relationship In unsaturated flow, the concern is water movement in the zone above the water table. In this case, the water saturation Sw is a function of the difference between air and water pressures because the water is resulting from held by capillary forces resulting from surface tension. This difference is known as the capillary pressure and is defined as haw ha hw
(4.12)
Typically, in unsaturated flow theory we assume negligible resistance to the gas-phase flow in porous media; as a result, we also can assume that the gas-phase pressure is uniform and equal to the atmospheric pressure. Hence haw hw. The aqueous pressure in the unsaturated zone is lower than the atmospheric pressure; thus, the capillary pressure is positive. The negative pressure head hw also is known as the soil matrix suction Ψ. Thus, the total head in the aqueous phase is h Ψ + z. To remove water from the pore space i.e., to reduce the water content we have to apply more negative pressure to the aqueous phase, i.e., increase the capillary pressure haw. This relationship is typically called the soil-water retention curve, and can be expressed by the following commonly used parametric models: the Brooks-Corey (BC) model and the van Genuchten (VG) model. The BC model is λ θθ θe r Ψ n θr hb
(4.13)
for Ψ hb and is otherwise (capillary fringe zone), θe 1
(4.14)
where n porosity, θ volumetric moisture content (equal to n Sw), θe effective volumetric moisture content, θr residual (irreducible) moisture content, λ Brooks-Corey parameter, and hb capillary fringe height. The van Genuchten model is Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
SUBSURFACE FLOW AND TRANSPORT
4.4
Chapter Four
θ θe θ r θr [1 (αΨ)n]m (4.15) n where: n curve fitting parameter that depends on the type of soil, α soil property index, α ≈ 1/hb, and m 1 1/n. Note that parameter n depends on pore-size distribution. For a well-graded soil (wide pore-size distribution, which results in a flatter moisture content curve θ(Ψ)), n is small, whereas for poorly graded soils (narrow pore-size distribution, which results in a steeper moisture content curve θ(Ψ), n is large: typically values of n higher than 2.5. Also note that although the BC and VG models are the most commonly used models in analysis, there is no restriction on using different mathematical representations to describe the characteristics of soil-water retention. For example, a simple exponential model, such as θe exp( β Ψ), is in some cases, sufficient to describe the physics of the retention. As the moisture content in partially saturated media decreases, so does the volume of pores available to fluid flow. Thus, hydraulic conductivity for the partially saturated media depends on the water content and, in turn, on the metric suction. To describe this relationship, we modify the value of intrinsic permeability k by the factor of kr(θ) or kr(Ψ), called relative permeability. Several models for kr are shown below: BC model of relative permeability: 2 3λ
kr θe λ
(4.16)
kr θe0.5(1 (1 θe1/m)m)2
(4.17)
VG model of relative permeability:
Mualem model of relative permeability: kr exp [ α Ψ]
(4.18)
4.2.3 Diffussive and Dispersive fluxes 4.2.3.1 Molecular diffusion. Molecular diffusion describes the process by which a contaminant species dissolved in an environmental fluid moves from regions of higher concentration to regions of lower concentration. When the only mechanism affecting the diffusion of the contaminant species is the random motion of its molecules, the process is referred to as molecular diffusion. The mass flux of a solute along a single direction, in a liquid or gaseous body, is described by Fick's law:
冢 冣
dC q D dx
(4.19)
In this equation, q mass flux of solute per unit area per unit time [ML2 T1], D diffusion coefficient (L2 T 1), C solute concentration mass of solute/volume of solution (M/L3), dC/dx concentration gradient along the x direction. The minus sign in Eq. (4.19) indicates that the solute flux will go from regions of larger concentration to those of lower concentration. Values of the diffusion coefficient, D, depend on the type of solute and the type of environmental fluid. For major cations and amnions dissolved in water, values of D range from 1 109 to 2 109 m2/s (Fetter, 1994).
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SUBSURFACE FLOW AND TRANSPORT
Subsurface Flow and Transport 4.5
4.2.3.2 Molecular diffusion in porous media Molecular diffusion in a porous medium is affected by the nature of the medium. For example, the paths that diffusing molecules follow in a porous medium are, in general, more complicated than if they were diffusing in water. Although Eq. (4.19) can still be used to describe the diffusion in a porous medium, the diffusion coefficient must be modified, and Eq. (4.19) is rewritten in terms of an effective diffusion coefficient. One widely accepted expression for the effective diffusion coefficient in porous media is the Millington-Quirk equation θ3.33 solution D Do n2
(4.20)
where Do is the molecular diffusion coefficient of the compound in pure solution fluid, θsolution is the solution fluid-filled porosity of the soil, and n is the total porosity of the soil. 4.2.3.3 Mechanical dispersion and macro-dispersion. Mechanical dispersion refers to the component of dispersion caused by differences in velocity at the pore level that are a consequence of the pore geometry. Water will move at different rates as a result of differences in pore sizes and tortuosity. A contaminant dissolved in the water flowing through a porous medium will be dispersed in both the longitudinal and transverse directions because of the fluctuations in the water velocity field. A way to incorporate the influence of pore geometry in the dispersion process is to define longitudinal and transverse dispersivities αL and αT. Longitudinal and transverse mechanical dispersion coefficients can thus be defined in terms of the dispersivities and the average pore velocity. For example, the longitudinal mechanical dispersion coefficient (DL)mech will be given by (DL)mech αL v. This dispersion coefficient is not treated separately from the effective diffusion coefficient defined in (8); instead, they are both combined in a coefficient of hydrodynamic dispersion. Thus, Fick’s Law, which describes molecular diffusion in a fluid, can be used to describe longitudinal and transverse dispersion in a porous medium if the diffusion coefficient D in Eq. (4.19) is replaced by a coefficient of longitudinal (or transverse) hydrodynamic dispersion, DL or DT, given by DL αL v D*
(4.21)
DT αT v D*
(4.22)
or
where αL, αT longitudinal and transverse dispersivities, respectively, and D* effective porous-media diffusion coefficient. In addition to the pore-scale dispersion, we also have formation-scale dispersion or, more accurately, spreading, which is a result of the variability in transport velocity caused by the heterogeneity of the hydraulic conductivity field. In terms of magnitude, this microdispersive flux is significantly larger than the one related to mechanical dispersion. Mathematically, macrodispersive flux is the flux equal to the expected value of the product of Darcian velocity (q’) and of the contaminant concentration (C’) fluctuations: q macrodispersive q’ C’
(4.23)
In otherwords, this flux can be described using an expression similar to the Fickian diffusion equation (Eq. 4.19) with a macrodispersivity coefficient. In the most general (threedimensional) case, the equation defining the macrodispersive flux in the flowing fluid is
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SUBSURFACE FLOW AND TRANSPORT
4.6
Chapter Four
∂苶苶 C qi ( Aij苶 v ) ∂xj
(4.24)
where Aij represents the macrodispersivity tensor and the bar indicates averaged quantities. The fluctuations q’ and c’ result from the heterogeneous nature of the aquifer. This is expressed in the way the macrodispersivity tensor is estimated. For example, the longitudinal macrodispersivity is estimated using σ 2 λ1 A11 f γ2
(4.25)
where σf2 variance of log-conductivity (f Ln[K]), λ1 correlation scale in the direction of flow, and γ is given by σf2 q γ exp KgJ1 6
(4.26)
In summary, the longitudinal and transverse components of the total diffusive and dispersive mass flux (per unit bulk area) in heterogeneous geologic formations is estimated as follows: ∂C qL (θFLUID AL v FLUID D) ∂x
(4.27)
∂C qT,HOR (θFLUIDAT,HORvFLUID D) ∂y
(4.28)
∂C qT,vert (θFLUIDAT,vertvFLUID D) ∂y
(4.29)
where D Millington-Quirk effective dispersion coefficient. 4.2.4
Partitioning
Equilibrium partitioning and sorption are the most common chemical processes that affect reactive transport. These processes are dealt with by equating the total concentration to the sum of the concentrations in each phase multiplied by their respective volumes. Furthermore, by equating the concentration in each phase to the concentration in a common phase —say, the concentration in water— the total concentration can be expressed in terms of the common phases concentration and a retardation coefficient R CT θWATERCWATER(1
冘
I WATER
K θ1 ) θWATERCWATERR 1 θWATER
(4.30)
where: KI partitioning coefficient between the ith phase and the common phase KI CI / CWATER, θI the volumetric content of the ith phase, and θWATER the volumetric content of the common phase. This type of equilibrium partitioning is frequently used to describe the relationships between concentrations in the following scenarios: (1) vapor and aqueous phases, (2) soil and vapor phases, (3) soil and aqueous phases, (4) partitioning of a tracer between aqueous and NAPL or DNAPL phases, and (5) partitioning of a tracer between vapor and NAPL or DNAPL phases.
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SUBSURFACE FLOW AND TRANSPORT
Subsurface Flow and Transport 4.7
For partitioning of compounds present in a NAPL or DNAPL mixture and aqueous phase, we use CI,WATER xISI
(4.31)
where SI is the solubility of compound I in water and xI is its mole fraction in the mixture. Finally, for partitioning of compounds present in a NAPL or DNAPL mixture and vapor phase, use Pvi[atm]* Mw,i[g/mole] Vi[mg/L] 103[mg/g]xi R 0.0821[L atm/mole °K]T[°K]
(4.32)
where VI is the vapor concentration of compound I, Pv,I is the vapor pressure of compound I, MW,I is its molecular weight, R gas constant, and T temperature in °K. 4.2.5 Degradation In addition to partitioning, degradation of compounds also may affect the fate and transport of reactive compounds significantly. Typically, degradation is modeled using either power-order decay models or growth-process-based models. In environmental subsurface hydrology, three basic power-order models are used: (1) zero-order decay, (2) first-order decay, and (3) a combination of the first two models. According to these models, the total decrease of mass in unit bulk volume caused by degradation is expressed by ∂CT ∂t
冘
θIK0,I
I PHASES
冘
θIK1,ICI
(4.33)
I PHASES
where K0,I zero-order degradation rate of the compound in phase I, K1,I first-order degradation rate of the compound in phase I, and CI = mass/volume concentration of the compound in phase I. In addition to the zero-order and first-order degradation processes, the Monad kinetics is frequently used to describe oxygen limited aerobic degradation of organic compounds in the aqueous phase. According to the Monod model, the degradation rate in terms of the total concentration is expressed by the following system of equations: ∂CT CWATER OWATER θWATERMtτ ∂t KC CWATER KO OWATER
(4.34)
∂OT CWATER OWATER θWATERχMtτ ∂t KC CWATER KO OWATER
(4.35)
and
where CWATER is the aqueous concentration of the contaminant, OWATER is the dissolved oxygen concentration, Mt is the total concentration of the active microbial biomass, τ is the maximum rate of organic solute utilization, KC is the concentration of the organic solute at which the utilization rate is half the maximum, KO is the electron acceptor (oxygen) concentration at which the utilization rate is half the maximum, and χ is the substrate utilization ratio.
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SUBSURFACE FLOW AND TRANSPORT
4.8
4.3
Chapter Four
FLOW AND TRANSPORT IN SATURATED ZONES
In this section, we present some solutions to saturated flow and transport problems that are encountered in the practice of subsurface hydrology. We begin with well hydraulics, an understanding of which is important to the design of pump-and-treat systems, and discuss several models of transport of soluble plumes.
4.3.1 Flow to a Single Well Darcy's law describes the average water flux through a porous medium when the local hydraulic gradient is known. To determine discharges we use the law of conservation of mass for the water, also known as the continuity equation. For the analysis of wells, it is assumed that the flow toward the well caused by the well pumping is radially symmetric. In this situation, it is convenient to use the equation of continuity in radial coordinates (r, θ). For steady-state state flow toward a single well in a confined aquifer without recharge, the equation reads dh Q 2πKB (4.36) dr where K and B aquifer conductivity and thickness, respectively. The equation of continuity for well flow in an unconfined aquifer is similar to Eq. (4.19): namely, dh Q 2πrKh (4.37) dr with B replaced by the variable flow depth h. Implicit in Eq. (4.36) and (4.37) is the Dupuit-Forchheimer assumption, the implication of which is that the flow in the aquifer can be assumed to be practically horizontal. Equations (4.36) and (4.37) are used to obtain solutions for the steady-state discharge to a well in confined and unconfined aquifers, respectively, if the piezometric heads (confined aquifers) or water-table elevations (unconfined aquifers) h1 and h2 are known at two radial distances r1 and r2, respectively:
冢 冣
冢 冣
Q 2πKB(h2 h1)/ln(r2/r1) 2πT(h2 h1)/ln(r2/r1)
(4.38)
where T KB aquifer transmissivity, and Q πKD(h22 h12)/ln(r2/r1)
(4.39)
The solutions given in Eqs. (4.38) and (4.39) assume that the well penetrates to the impermeable bottom of the aquifer and that there is no recharge into the aquifer. They also assume an infinitely large aquifer with no interaction with surface streams or impermeable boundaries. Well solutions are often given in terms of the drawdown s as a function of the radial distance r. The drawdown is defined as sHh
(4.40)
where H is the elevation of the original piezometric surface before pumping at the well starts. The distance R for which h H and s 0 is called the radius of influence of the well. Using the concepts of drawdown and radius of influence, the steady-state flow equation in a confined aquifer can be rewritten as Q s LnR 2πT r
(4.41)
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SUBSURFACE FLOW AND TRANSPORT
Subsurface Flow and Transport 4.9
By combining the solutions for the steady state flow in confined and unconfined aquifers, one can derive a relationship between the drawdown calculated from confined conditions (assuming constant in space and time aquifer thickness equal to H), and that estimated for unconfined conditions (when the change in aquifer thickness caused by pumping is taken into account): sUNC H 兹H 苶2苶 苶苶s 2苶CON 苶苶 FH
(4.42)
Using this formula, known as Jacob’s correction, one can initially assume constant aquifer thickness B H in calculations and use “confined” aquifer solutions to calculate drawdown, then correct the drawdown using Jacob’s correction. This approach is particularly useful when dealing with transient flow. For transient flow conditions in an aquifer with constant flow thickness, the transient drawdown is given by Q ∞ exp[x] Q s (r,t) 兰 dx W [u(r,t)] 4πT u 4πT x
(4.43)
r2S u(r,t) 4Tt
(4.44)
where
and S storativity (for confined aquifers) or porosity (for unconfined aquifers) and W(u) is known in subsurface hydrology as well function and in mathematics as exponential integral. This function is tabulated in almost every groundwater hydrology textbook. It also is available in many engineering mathematics software packages, such as Mathematica©, as a library function.
4.3.2 Superposition and Convolution For a time-variable pumping rate, the principle of convolution can be used to estimate the transient drawdown. This approach is strictly valid for linear systems: i.e., systems in which the response (drawdown) is a linear function of the excitement (pumping rate). The linearity assumption is strictly valid for confined aquifers only; however, as long as the drawdowns do not exceed 20% of the initial aquifer thickness, it also may be used for unconfined aquifers. Using the convolution approach, the transient drawdown for a pumping rate changing in a step-wise fashion is given by s(r,t) 1 4πTk
冘 n
1
(QK QK 1)W(u(r,∆tK 1))
(4.45)
where the drawdown is estimated at time t, tn t tn 1, QK is the pumping rate for tK 1 t t K,tO 0, Q0 0.0, and ∆tK 1 t – tK 1. When several wells are present, the superposition approach is used to estimate the cumulative drawdown by adding the drawdown contributions from all the wells: s(t) 1 4πT L
冘 m
QLW(u(rL,t))
(4.46)
1
where rL is the distance between the point of interest (where the drawdown is estimated) and well L. When several wells are pumping at variable rates, the superposition and convolution approaches are used simultaneously. The superposition principle also can be used to superimpose the drawdown on the natural (ambient) flow conditions. Using this principle leads to
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SUBSURFACE FLOW AND TRANSPORT
4.10
Chapter Four
h(x, y, t) H(x, y, t) s(x, y, t)
(4.47)
where h transient potentiometric surface that combines ambient conditions and well impact, H potentiometric surface under natural (ambient) conditions, and s transient drawdown.
4.3.3
Interception Wells
With respect to contaminant transport in the subsurface, interception wells are used to trap the contaminant plume within the well flow field. It is assumed in this analysis that there is an ambient steady-state uniform flow through the aquifer. The combination of wellrelated flow and ambient uniform flow satisfies the conditions of two-dimensional potential flow in a horizontal plane, where the discharge described by stream function y is related to potential φ, or the piezometric head h. The extent of the aquifer through which water travels to the well and is captured by it is called the capture zone. The derivation of the analytical solution for steady-state flow capture-zone uses the following assumptions: (1) ahomogeneous, isotropic, infinitely large aquifer, (2) uniform flow, (3) no leakage, (4) aquifer storativity or specific yield neglected, (not relevant for steady-state analysis), (5) hydrodynamic dispersion neglected, (6) the Dupuit assumption applies, and (7) the well is fully penetrating and pumping at a constant rate. Three important parameters are used in delineating the capture zone: namely, the stagnation point, the upgradient maximum width of the capture zone, and the equation for the capture zone boundary. For a confined aquifer, the distance from the well to the stagnation point (measured in the direction of the uniform flow) is Qw xSTAG (4.48) 2πTI where Qw well discharge, T aquifer transmissivity KB (K aquifer permeability, B aquifer depth), and I natural hydraulic gradient: i.e., the gradient responsible for the ambient steady-state uniform flow in the aquifer. The upgradient divide, defined by the maximum width of the capture zone far upgradient of the well, for the confined aquifer is given by Q wDIV w TI
(4.49)
and the equation of the dividing streamline is y x 2πTIy tan π Qw
冢
冣
(4.50)
The procedure for delineating the capture zone consists of the following steps: (1) estimate the location of the stagnation point (xSTAG, 0), (2) estimate the maximum width of the capture zone wDIV, and (3) vary y between zero and wDIV/2 and use the capture zone boundary to estimate the boundary location (x,y).
4.3.4 Partially Penetrating Wells Performance of wells that penetrate only partially through the bearing strata is discussed in this section. The simplest case consists of a well that is barely penetrating into an semi-
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SUBSURFACE FLOW AND TRANSPORT
Subsurface Flow and Transport 4.11
infinite porous medium so that the aquifer flow is three-dimensional and spherically symmetric. In this case, the following relationship applies between the flow into the partially penetrating well Qp and the flow to a fully penetrating one Q: R Qp rw (4.51) ln rw Q B where B is the aquifer thickness, rw is the well radius, and R is the radius of influence of the partially penetrating well. Because, in general, rw B, then the equation above indicates that the spherical flow to a partially penetrating well is highly inefficient compared with simple radial flow: i.e., for the same drawdown in the well, it results in a significantly smaller pumping rate. In the general case of partial penetration, one may consider the total drawdown sT, which consists of the drawdown equivalent to that of a fully penetrating well s and additional head loss because of the partial penetration of the well ∆s: sT s ∆s
(4.51)
Additional head loss for a well penetrating from the top (or the bottom) of the aquifer is estimated as follows: Q(1 p) (1 p)h ∆s ln s 2πTp rw
(4.53)
where p penetration factor; p hs/B; hs penetration depth; and B aquifer thickness (Fig. 4.2). For the well centrally positioned in the aquifer, the following formula is used: Q(1 p) (1 p)h ∆s ln s (4.54) 2πTp 2rw Thus, when the pumping rate is defined for a well, we calculate the drawdown correction ∆s and add it to the full penetration drawdown s. However, when the drawdown is given for a well, we have to recalculate the pumping rate. In this case, the true pumping
hs B
B/2 hs hs/2
2rw FIGURE 4.2 Partially-penetrating well.
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SUBSURFACE FLOW AND TRANSPORT
4.12
Chapter Four
rate is given by S ∆s T Qs Qs Q S s ∆s T
(4.55)
where s drawdown defined at the well, Q pumping rate estimated using s, and Qp actual pumping rate 4.3.5
Well Duplets
Well duplets, each of which consists of one pumping and one recharge well, are frequently used as a means of injecting and removing aquifer mitigation solutes, such as co-solvents, surfactants, or both. Typically, the recharge well is positioned directly upgradient from the discharge well, and the magnitudes of pumping and injection rates are the same. In this case, the two wells form a flow circulation cell: i.e., all the injected water is pumped out by the discharge well. In the case of co-solvent flushing, it is important to understand what region of the aquifer is subject to the mitigation: i.e., what the boundary is of the circulation cell. This boundary is defined by the upgradient and downgradient stagnation points (xSTAG, 0) and (xSTAG, 0) and the cell boundary equation. For the x-axis parallel to the direction of ambient flow and the origin of the coordinate system located at the mid-point between the two wells, the two stagnation points, (xSTAG, 0) and (xSTAG, 0) are given as the roots of the quadratic equation qoBd 1 x/d x/d 2 2 0 2π (x/d 1) (x/d 1) Qw
(4.56)
where qo ambient flux, 2d distance between the wells, and Qw pumping/injection rate. The boundary of the circulation cell is defined by
qoBy 1 y/d y/d 1 tan1 tan1 2π Qw 2 (x/d 1) (x/d 1)
(4.57)
The circulation cell is symmetric with respect to the y axis. The cell delineation procedure consists of estimating the locations of the stagnation points and varying x between zero and xSTAG and using the cell-boundary equation to solve for y. This implicit equation can be solved by any calculation software, such as Mathematica© or MS Excel.
4.3.6 Transport Equations The following general form of mass transport equation in the saturated zone is derived assuming one-dimensional advective and three-dimensional diffusive-dispersive transport in the aqueous phase, linear partitioning of a compound between the three phases (water-soil(D)NAPL), and first-order degradation in the aqueous phase. For this conditions, we have ∂CW ∂ R ∂xi
冢冢α v θ 冣∂x冣 v∂x λC D
∂CW
∂CW
i
W
W
i
(4.58)
1
where CW aqueous phase concentration, v pore-water velocity, and θW volumetric moisture content, and αi longitudinal, transverse horizontal, and transverse vertical
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SUBSURFACE FLOW AND TRANSPORT
Subsurface Flow and Transport 4.13
macrodispersivities, and D Millington-Quirk dispersion coefficient, and λ first-order degradation rate in the aqueous phase, and R retardation factor, kSOILγβ kNAPLθNAPL R1 θW θW
(4.59)
where kSOIL water-soil partitioning coefficient, kSOIL S/CW, S weight/weight concentration of absorbed compound in soil, kNAPL NAPL – water partitioning coefficient, kNAPL CNAPL/CW, CNAPL concentration of compound in the NAPL phase, γB bulk density of dry soil, and θNAPL volumetric NAPL content. Assuming that the diffusive fluxes are negligible compared the macrodispersive fluxes, the transport equation can be simplified to yield ∂CW ∂ ∂CW ∂CW αivR vR λRCW ∂xi ∂xi ∂t ∂xi
(4.60)
where vR v/R and λR λ/R. 4.3.7 Selected Analytical Solutions Closed-form solutions are available for a variety of flow, boundary, and initial conditions. Van Genuchten and Alves (1982) presented a good summary of these solutions. Some of the most useful solutions are presented below. 4.3.7.1 One-dimensional transport with step change in concentration–no degradation. This simple case has the initial condition C(x,0) 0 for x 0, and it is subject to the following boundary conditions: C(0,t) Co, t 0 and C(∞,t) 0, t 0. The solution of the transport equation for these conditions is given by
冢
冣
x vRt x x vRt CO C(x,t) Erfc expErfc 1/2 1/2 2 2(αxvRt) αx 2(αxvRt)
(4.61)
4.3.7.2 One-dimensional transport with step change in concentration and first-order degradation. The initial and boundary conditions are the same as in Sec. 4.7.1. The solution is given by
x 4λRαx 1/2 CO C(x,t) exp1 1 vR 2 2αx
Erfc
冤
冢
冣
4λ α 1/2 x vR 1 Rx t vR 1/2 2(αxvRt)
冥
(4.62)
where Erfc complementary error function. 4.3.7.3 Continuous point injection, 2-D dispersive transport, no retardation, and no degradation. A tracer is continuously injected at a rate Q (per unit depth of the aquifer)
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SUBSURFACE FLOW AND TRANSPORT
4.14
Chapter Four
with a concentration Co into a uniform flow field from a point (x 0, y 0). Let the uniform velocity be vx. The asymtotic solution, i.e., for t →∞, is given by
vx Co C(x,y) exp x Ko 2π兹D 2DLDT 苶苶 LDr
v x y 莦莦 冪莦 D莦莦 4D D莦莦莦 2 x
2 L L
T
2
(4.63)
where Ko modified Bessel function of the second kind and of 0th order (Bear, 1972). The time-dependent solution is vx CoQ expx Ko[W(0,β)] W(t,β) C(x,y) 2π兹D 2DL 苶苶 D 苶 L T
(4.64)
where β
y 莦 冪莦4v莦D莦莦冢莦Dx莦莦莦莦 D 冣 2 x
L
2
2
L
T
(4.65)
W (t, b) leaky well function (see, for example, Hunt, 1983, p. 100). 4.3.7.4 Point slug injection into a uniform flow field—3-D transport and retardation. In this case, a slug of contaminant of the mass M CoV is injected at point (0,0,0). The transient distribution of concentration is described by VoCo C(x, y, z, t) 8(πvRt)3/2(αxαyαz)1/2 y2 z2 (x vRt)2 exp 4αyvRt 4αzvRt 4αxvRt
(4.66)
4.3.7.5 Continuous injection from a finite-sized source with retardation and degradation. In this case, consider transport from a rectangular source that is perpendicular to the direction of flow. The source width is Y, and its depth below the water table is Z. The transient concentration distribution in the presence of retardation and degradation is given by x 1 4λRαx 1/2 C C(x, y, z, t) o exp1 vR 8 2αx αx 1/2 x vRt(1 4λRv) Erfc 2(αxvRt)1/2 y Y/2 Erfc 2(αyx)1/2
y Y/2 Erfc 1/2 2(αyx)
zZ Erfc 1/2 2(αzx)
z Z Erfc 1/2 2(αzx)
(4.67)
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SUBSURFACE FLOW AND TRANSPORT
Subsurface Flow and Transport 4.15
4.4 FLOW AND TRANSPORT IN UNSATURATED ZONE — AQUEOUS PHASE In this section, we briefly discuss the flow continuity equations and present some simple solutions to selected flow problems. That discussion is followed by a presentation of mass transport in the water phase of unsaturated zone.
4.4.1 Flow in an Unsaturated Zone The continuity equation for an unsaturated flow system can be written as ∂θ ∇.q ∂t
(4.68)
∂qx ∂qy ∂qz ∂θ ∂t ∂y ∂z ∂x
(4.69)
or
Combining Darcy's law with the mass continuity equation, we can write the final flow equations is ∂θ ∇(K(Ψ)∇h) ∂t
(4.70)
The flow equations can be simplified for horizontal and vertical flow conditions. a. One-dimensional horizontal flow: ∂θ ∂Ψ ∂ K(Ψ) ∂t ∂x ∂x
(4.71)
where θ volumetric water content. In this, the contribution of the elevation head, z, vanishes, since ∂z/∂x 0. b. One-dimensional vertical flow: ∂Ψ ∂θ ∂ K(Ψ) 1 ∂t ∂z ∂z
(4.72)
Note that the flow equations are characterized by the presence of two dependent, albeit related, variables: namely, θ and Ψ. To simplify this situation, we describe the relationship between θ and Ψ by a term called soil diffusivity D(θ) as K(θ) D(θ) (4.73) C(θ) where C(θ) is called specific moisture capacity and is defined as ∂θ C(θ) ∂Ψ Using these definitions, the flow equation can be written as follows: c. One-dimensional horizontal flow: ∂θ ∂ ∂θ Dx(θ) ∂x ∂t ∂x
(4.74)
(4.75)
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SUBSURFACE FLOW AND TRANSPORT
4.16
Chapter Four
d. One-dimensional 1-D vertical flow: ∂θ ∂θ ∂ (Dz(θ) Kz(θ)) ∂z ∂z ∂z
(4.76)
As can be seen, we now have only one dependent variable: namely, θ. The only limitation of this formulation is that specific moisture capacity C(θ) becomes zero in the capillary fringe zone, thus making the solution impossible. Therefore, this formulation is valid only in the partially saturated zone (water content less than saturated value), not in the capillary fringe. Another way to solve this problem, which does not have the limitation discussed above, is to formulate the flow equations in terms of soil suction ψ. For the vertical flow, we obtain
冢
冣
∂Ψ ∂Ψ 1) C(θ) ∂ Kz(Ψ)( ∂z ∂z ∂z
(4.74)
Exercise. Consider steady-state vertical infiltration from the soil surface to the water table at depth L. The relative hydraulic conductivity of the soil is described by the following exponential law: kr(Ψ) exp[ α Ψ]
(4.78)
Derive an expression for the vertical distribution of h. After the first integration of the flow equation, we obtain ∂Ψ Kz(Ψ) 1 q (4.79) ∂z where q infiltration rate and K is a function of capillary pressure y. Kz(Ψ) Ksatexp[αΨ]
(4.80)
where h z ψ. Substitution of these relationships into the flow equation leads to q expα(h z)∂h KSAT ∂z
(4.81)
q exp[αh]dh exp[αx]dx KSAT
(4.82)
or
After the second integration, we obtain q 1 (4.83) exp[αh] exp[αz] C1 α αKSAT Substituting the boundary condition h(0) 0 and solving for C1 yields the following expression for the total head h: q (4.84) h(z) 1 ln (exp[αz] 1) 1 α KSAT 4.4.2 Transport in an Unsaturated Zone The mass continuity equation for an unsaturated flow system with advection and diffusion/dispersion in the aquoeus phase, diffusion in the vapor phase, partitioning between four phases (soil, water, vapor, and (D)NAPL), and first-order degradation in the aqueous phase can be written as
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SUBSURFACE FLOW AND TRANSPORT
Subsurface Flow and Transport 4.17
∂ ∂CW D kAIR ∂CW ∂CW v λCW R αiv D ∂xi θW ∂t θW ∂xi ∂x1 v
(4.85)
where CW aqueous phase concentration; v pore-water velocity; θW volumetric moisture content; αi longitudinal, transverse horizontal, and transverse vertical macrodispersivities; D Millington-Quirk dispersion coefficient; Dv Millington-Quirk dispersion coefficient in the vapor phase; λ first-order degradation rate in the aquoeus phase; and R retardation factor; kSOILγB kNAPLθNAPL kAIRθAIR R1 θW θW θW
(4.86)
where kSOIL water-soil partitioning coefficient, kSOIL S/CW, S weight/weight concentration of absorbed compound in soil, kNAPL NAPL-water partitioning coefficient, kNAPL CNAPL/CW, CNAPL concentration of compound in the NAPL phase, kAIR vaporwater partitioning coefficient, kAIR V/CW, V concentration of compound in the vapor phase, γB bulk density of dry soil, θAIR volumetric vapor content, and θNAPL volumetric NAPL content. Assuming again that the diffusive fluxes are negligible compared with the macrodispersive and advective fluxes, the transport equation can be simplified to yield ∂CW ∂CW ∂CW ∂ αivR vR λRCW ∂xi ∂t ∂xi ∂xi
(4.87)
where vR v/R and λR λ/R. Note that the form of this equation is the same as the form of the one for transport in the saturated zone; therefore, the analytical solutions presented for the saturated transport are valid for the unsaturated conditions. This is particularly true for the one-dimensional (vertical) transport equations, which are of primary interest in the case of unsaturated fate and transport of compounds.
4.5
FLOW AND TRANSPORT IN VAPOR PHASE
This section, presents the soil vapor flow equations. This is followed by selected solutions to vapor flow problems. Finally, we discuss diffusive transport in the vapor phase.
4.5.1 Soil Vapor Flow The flow of gases in a porous medium can be described by combining the following equations: a. The equation of continuity is ∂ρ ∇(ρv) n ∂t
(4.88)
with ρ vapor density, v Darcy’s flux vector, θAIR air-filled porosity, and t time. b. The perfect gas law is pM ρ m v RT
(4.89)
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SUBSURFACE FLOW AND TRANSPORT
4.18
Chapter Four
with m gas mass, V gas volume, p absolute pressure, M molar mass, T absolute temperature ºK, and R universal gas constant. c. Darcy’s law for vapor flow is v k ∇p (4.90) µ where k soil intrinsic permeability and µ gas viscosity. The resulting governing equation is 1953 2nµ ∂p ∇2p2 (4.91) k ∂t (Bruce et al., 1953). d. The molar flux [moles/unit area-time] is given by k q p∇p µRT
(4.92)
For one-dimensional flow, the governing equation reduces to ∂2p2 2θAIR µ ∂p k ∂x2 ∂t
(4.93)
and for radial flow, the governing equation is ∂2p2 1 ∂p2 2θAIR µ ∂p . r ∂r k ∂r2 ∂t
(4.94)
For steady state, exact analytical solutions for gas flow are obtained by using the following transformation (Cho, 1991): K ( p2 pr2 ) m (4.95) 2 where m is referred to as the discharge potential. The governing equation now becomes Laplace's equation: ∇2m 0 (4.96) For example, the exact solution for the point source in three-dimensional space is given by Q m (4.97) 4πr where Q source strength and r distance from source point. To estimate the time required to achieve steady-state vapor flow, Johnson et al. (1990) presented a method based on the solution of radial flow of vapor to a well. Their results are summarized in Fig. 2 of Johnson et al. (1990 b) for values of k corresponding to sandy soil. They also presented a method to estimate vapor flow rates, pressure distributions, and vapor velocities in unsaturated soils based on the steady-state solution to the governing equation of vapor radial flow: namely, 2nµ ∂p ∇2p2 k ∂t
(4.98)
The pressure p can be expressed in terms of the ambient pressure pAtm and a deviation p’ from this pressure: p’ is equivalent to the vacuum that would be measured in the soil. If this substitution is used in the flow equation and if we neglect the product p'2 relative to the product pAtm p’ (linearization), then the resulting equation for radial flow is
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SUBSURFACE FLOW AND TRANSPORT
Subsurface Flow and Transport 4.19
θAIRµ ∂p’ 1 ∂ ∂p’ r kpATM ∂t r ∂r ∂r
(4.99)
The solution to this equation for the following boundary conditions is p’ 0, for r → ∞
冢 冣
Q ∂p’ limr→0 r 2πBk/µ ∂r
(4.100)
as given by Q p’ 4πB(k/µ)
∞
兰
Q exp[ x]dx W(u) 4πB(k/µ)
r2θAIRµ u 4kp ATM
(4.101)
where Q is the volumetric flow rate to the vapor well. The well function W(u) is tabulated in almost all groundwater textbooks. The behavior of the integral is such that for (r2θAIRµ/4kpAtmt) 0.001, its value is close to the asymptotic steady-state limit. Exercise. Given the following parameters—hydraulic conductivity K 10–2 cm/sec, air viscosity µAIR 0.018 cp, volumetric air content (equal to porosity) θ AIR 0.3, and local pressure gradient dp/dx 0.01 atm/cm—estimate the pore-vapor velocity. From the conversion table (Domenico and Schwartz, 1990), we have k[darcy] K[m/s]*1.04*105
(4.102)
Thus, k 10.4 darcy. The vapor flux is given by ∂z kr(θ)kfluid ∂p qi ρg ∂xi µfluid ∂xi
(4.103)
Assuming that kr 1.0, we obtain kr*k[darcy] ∂p atm 10.4* 0.01 q[cm/sec] 5.78[cm/sec] µ[centipoise] ∂x cm 0.018
(4.104)
Finally, we estimate the pore-vapor velocity v q/θAIR 19.26 cm/s. 4.5.2 Transport in Vapor Phase It is usual to assume that whenever advective vapor flow is present, it dominates the transport process and the diffusive/dispersive processes can be neglected. In this case, the fate and transport equation for a compound that partitions between the four phases (soil, water, vapor, and (DNAPL) and is subject to first-order degradation in the aqueous phase is ∂V v ∂V λ θW k V R AW AIR ∂t ∂x1 θAIR
(4.105)
where: V vapor phase concentration, vAIR pore-air velocity, θW volumetric moisture content, θAIR volumetric vapor content, λ first-order degradation rate in the aqueous phase, kA W air-water partitioning coefficient, kAW CW/V, and R retardation factor,
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SUBSURFACE FLOW AND TRANSPORT
4.20
Chapter Four
kA WkSOILγB kA WkNAPLθNAPL kA WθW R1 θAIR θAIR θAIR
(4.106)
where: kSOIL water-soil partitioning coefficient, kSOIL S/CW, S weight/weight concentration of absorbed compound in soil, kNAPL NAPL-water partitioning coefficient, kNAPL CNAPL/CW, CNAPL concentration of compound in the NAPL phase, kAIR vaporwater partitioning coefficient, kAIR V/CW, V concentration of compound in the vapor phase, γB bulk density of dry soil, θAIR volumetric vapor content, and θNAPL volumetric NAPL content. Division of the transport equation by the retardation factor yields ∂V ∂V vAIR,R λ*RV ∂x1 ∂t
(4.107)
where vAir,R vAIR/R and λR kA WλθW/(θAIR R). Note that the form of this equation is the same as the form of the one for transport in the saturated zone, except for the absence of the dispersive term. Therefore, the analytical solutions presented for the saturated transport are valid for the vapor transport conditions. When there is no advective transport in the vapor phase, the transport equation must include diffusive fluxes in vapor and aqueous phases to yield θW ∂V ∂ Dv ∂V Dk ∂V R A W λ kA WV ∂t ∂xi ∂xi ∂xi θAIR
(4.108)
where D Millington-Quirk dispersion coefficient for the aqueous phase, and Dv Millington-Quirk dispersion coefficient in the vapor phase. Assuming that the dispersion coefficients do not vary in space leads to the following form of the transport equation: ∂V ∂2V ˆ ˆ D λV ∂t ∂x 2i
(4.109)
Dv DkA W ˆ D R
(4.110)
θW ˆλ λ θAIRR
(4.111)
where
and
Again, we note the similarity of this fate and transport equation to the one presented for saturated transport and conclude that all the analytical solutions presented in Sec. 4.2 can, in principle, be used to analyze vapor phase transport. Exercise. Given that water saturation Sw 0.20, porosity n 0.4, compound concentration in soil vapor at depth L 2 m Co 100 mg/L, compound concentration at the soil surface Cs 0.01, and molecular diffusion coefficient of the compound Do 0.087cm2/sec, estimate the compound mass flux in the vapor phase at the soil surface. The effective vapor-phase diffusion coefficient is given by θ 3.33 D Do A 0.0026 cm3/s n2
(4.20)
and the mass flux of compound A is estimated as follows:
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SUBSURFACE FLOW AND TRANSPORT
Subsurface Flow and Transport 4.21 3 3 ∂C Co Cs 0.0026[cm2/s] 100*10 [mg/cm ] 1.3[mg/(cm2s)] (4.112) q D D ∂x 200[cm] L
Exercise. Consider advective transport of a compound in vapor phase. The compound partitions between vapor and aqueous phases according to the relationship VKC
(4.113)
where V concentration in vapor phase, C concentration in aqueous phase, and K partitioning coefficient. Given vapor flux q, porosity n, and water saturation Sw, estimate the apparent (retarded) velocity of the compound. From the advective transport equation, we have ∂CT ∂V q ∂t ∂x
(4.114)
SW CT C SW n V(1 SW)n Vn (1 SW) 1 K(1 SW
(4.115)
where
Thus, for nonretarded tracers, we have ∂V ∂V V ∂x
(4.116)
q v n(1 Sw)
(4.117)
where pore-vapor velocity is
whereas for retarded compounds, we have ∂V ∂V vR ∂t ∂x
(4.118)
where the retarded velocity is given by v v 1 vR SW R K(1 SW)
(4.119)
Exercise. Consider diffusive vertical transport of a compound in vapor phase. The compound is subject to first-order degradation in the aqueous phase at rate λ and to partitions between vapor and aqueous phases according to the following relationship: VKC
(4.113)
where V concentration in vapor phase, C concentration in aqueous phase, and K partitioning coefficient. At the depth of 100 cm below the ground surface, the vapor concentration of the compound was measured to be Vo, whereas at the ground surface the concentration was Vs. Given the compound’s diffusion coefficient in vapor phase Do, porosity n, and water saturation Sw, estimate the diffusive flux of the compound at the soil surface. The relevant mass transport equation is given by
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SUBSURFACE FLOW AND TRANSPORT
4.22
Chapter Four
∂ V λnS C 0 D w ∂x2 2
(4.120)
where θ 3.33 D Do A n2 Substituting C V/K into the mass transport equation leads to ∂2V 2 λ*2V 0 ∂x
(4.19)
(4.121)
where λnS 2 λ* w KD We solve the modified mass transport equation to obtain V(x) C1exp[ λ∗x] C2exp[λ∗x]
(4.122)
(4.123)
where constants C1 and C2 are obtained from the boundary conditions V(0) Vo, and V(100) Vs
(4.124)
The compound's mass flux at the soil surface is estimated from ∂V(x) q D ∂x
@x 100
(4.125)
REFERENCES Bear, J., and Y. Bachmat, Introduction to Modeling of Transport Phenomena in Porous Media, Kluwer Academic, Dordrecht, The Netherlands, 1990. Bear, J., Dynamics of Fluids in Porous Media, Dover Publications, New York,k 1972. Bear, J., Hydraulics of Groundwater, McGraw Hill, New York, 1979. Bouwer H., Groundwater Hydrology, McGraw Hill, New York, 1978. Bruce, G. H., D. W. Peaceman, and H. H. Rachford, Jr., 1953. “Calculations of Unsteady-State Gas Flow Through Porous Media,” Petroleum Transactions, AIME, 198: 1953. Bureau of Reclamation, Ground Water Manual (Reprint), U.S. Department of the Interior Washington, DC, 1995. Cedergren H. R., Seepage, Drainage, and Flow Nets, 3rd ed., John Wiley & Sons, Inc., New York, 1989. Charbeneau, R. J., “Kinematic Models for Soil Moisture and Solute Transport,” Water Resources Research, 20: 699–706, June, 1984. Chirlin, G. R., “A Critique of the Hvorslev Method for Slug Test Analysis: The Fully Penetrating Well,” Ground Water Monitoring Review, 130–138, 1989. Cho, J. S., 1991. Forced Air Ventilation for Remediation of Unsaturated Soils Contaminated by VOC., Publication No. EPA/600/S2–91/016, U.S. Environmental Protection Agency, Washington, D.C.
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SUBSURFACE FLOW AND TRANSPORT
Subsurface Flow and Transport 4.23 Dagam, G., “Solute transport in heterogeneous porous formations,” Journal of Fluid Mechanics, 145: 151–177, 1984. De Josselin Jong, G. “Singularity Distribution for the Analysis of Multiple-Fluid Flow Through Porous Media,” Journal of Geophysical Research, 65: 3739–3758, 1960. De Marsily G., Quantitative Hydrogeology—Groundwater Hydrology for Engineers, Academic Press, San Diego, CA, De Smedt F., and P. J. Wirenga, “Solute Transport Through Soil With Nonuniform Water Content,” Soil Science Society of America Journal, 42. (1): 1978. Domenico, P. A., and F. W. Schwartz, Physical and Chemical Hydrogeology, John Wiley & Sons, New York, 1990. Dullien F. A. L., Porous Media: Fluid Transport and Pore Structure, 2nd ed., Academic Press, San Diego, CA, 1992. Edelman J. H., Groundwater Hydraulics of Extensive Aquifers, 2nd., International Institute for Land Reclamation and Improvement, Bulletin No. 13, The Netherland, 1983. Fetter, C. W., Contaminant Hydrogeology, Macmillan, New York, 1993. Fetter, C. W., Applied Hydrogeology, Simon & Schuster, Company Englewood, NJ, 1994. Freeze, R. A., and J. A. Cherry, Groundwater, Prentice-Hall, Englewood Cliffs, NJ, 1979. Gelhar, L. W. and C. L. Axness. “Three-Dimensional Stochastic Analysis of Macrodispersion in Aquifers”, Water Resources Research, 19 (1): 161–180, 1983. Germann, P. F., M. S. Smith, and G. W. Thomas, “Kinematic Wave Approximation to the Transport of Escherichia coli in the Vadose Zone,” Water Resources Research. 23 (7), 1281–1287, 1987. Girinsky, N. K. Determination of the Coefficient of Permeability, Gosgeolizdat, 1950. Grubb, S. “Analytical Model for Estimation of Steady-state Capture Zones of Pumping Wells in Confined and Unconfined Aquifers,” Ground Water, 31(1) 27–32, 1993. Hantush, M. S., Hydraulics of Wells, “in Advances in Hydroscience,” V. T. Chow, ed., Academic Press, New York, 1964. Hantush, M. S. “Growth and Decay of Groundwater-mounds in Response to Uniform Percolation.” Water Resources Research, 3 (1): 227–234, 1967. Harr, M. E., Groundwater and Seepage, Dover Publications, New York, 1990. Haverkmp, R., M. Vauclin, J. Touma, P. J. Wierenga, and G. Vachaud, A “Comparison of Numerical Simulation Models for One-Dimensional Infiltration,” Soil Science Society America Journal, 41: 285–294, 1977. Hinchee, R. E. ed., Air Sparging for Site Remediation, Lewis Publishers, Boca Raton, FL, 1994. Huisman, L., Groundwater Recovery, Winchester Press, New York, 1972. Hunt, B. “Seepage to Collection Gallery Near Seacoast,” Water Resources Research, 21: 311–316, 1985. Hunt, B., Mathematical Analysis of Groundwater Resources, Butterworths, London, UK, 1983. Jaffe, P. R., and R. A. Ferrara, “Desorption Kinetics in Modeling of Toxic Chemicals,” Journal of Environmental Engineering, American Society of Civil Engineers,109: 859–867, 1983. Javandel, I., and C. F. Tsang, “Capture-zone Type Curves: A Tool for Aquifer Cleanup.” Ground Water, 24: 616–625, 1985.
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4.24
Chapter Four
I., C. Doughty, and C. F. Tsang, Groundwater Transport: Handbook of Mathematical Models, American Geophysical Union, Washington, DC, 1987. Johnson, P. C., M. W. Kemblowski, and J. D. Colthart, “Quantitative Analysis for the Cleanup of Hydrocarbon-Contaminated Soils by in-situ Soil Venting,” Ground Water, 1990. Johnson, P. C., C. C. Stanley, M. W. Kemblowski, D. L. Byers, and J. D. Colthart. A “Practical Approach to the Design, Operation, and Monitoring of in situ Soil-Venting Systems.” Ground Water Monitoring Review, Spring, 1990. Jury, W. A., “Chemical Transport Modeling: Current Approaches and Unresolved Problems,” Chemical Mobility and Reactivity in Soil Systems, 1983, pp. 49–64. Jury, W. A., R. Grover, W. F. Spencer, and W. J. Farmer, “Modeling Vapor Losses of Soil Incorporated Triallate,” Soil Science Society of America Journal, 44: 445–450, 1980. Jury. W. A., W. F. Spencer, and W. J. Farmer, “Use of Models for Assessing Relative Volatility, Mobility, and Persistence of Pesticides and other Trace Organics in Soil Systems.” Hazard Assessment of Chemicals: Current Developments, Vol. 2, 1983. Keely, J. F. and C. F. Tsang, “Velocity Plots and Capture Zones of Pumping Centers for GroundWater Investigations.” Ground Water, 21: 701–714, 1983. Kishi, Y. and Y. Fukuo, “Studies on Salinization of Groundwater,” I. Journal of Hydrology, 35: 1–29, 1977. Kool, J. B., J. C. Parker, and M. T. van Genchten, “Parameter Estimation for Unsaturated Flow and Transport Modles—A Review.” Journal of Hydrology, 91: 255–293, 1987. Kozeny, J., Thorie und Berchnung der Brunnen. Wasserkraft und Wasserwirtschaft, Nos. 8–10, 1933. Marino, M. A., “Artificial Groundwater Recharge: I. Circular Recharging Area,” Journal of Hydrology, 25: 201–208, 1975. Marshall, T. J., J. W. Holmes, and C. W. Rose, Soil Physics, 3rd ed., Cambridge University Press, Cambridge, UK, 1996. McElwee, C., and M. Kemblowski, “Theory and Application of an Approximate Model of Saltwater Upconing in Aquifers,” Journal of Hydrology, 115: pp 139–163, 1990. McWhorter, D. B. Steady and Unsteady Flow of Fresh Water in Saline Aquifers, Water Management Technical Report No. 20, Council of U.S. Universities for Soil and Water Development in Arid and Sub-Humid Areas, 1972. Musa, M. and M. W. Kemblowski. “Effective Capture Zone for a Single Well,” Submitted to Ground Water July 1994. Newsom, J. M., and J. L. Wilson, “Flow of Ground Water to a Well Near a Stream - Effect of Ambient Ground-water Flow Direction.” Ground Water, 25: 703–711, 1988. Oberlander, P. L. and R. W. Nelson, “An Idealized Ground-Water Flow and Chemical Transport Model (S-PATHS),” Ground Water, 22: 441–449, 1984. Ostendorf, D. W., R. R. Noss, and D. O. Lederer, “Landfill Leachate Migration through Shallow Unconfined Aquifers,” Water Resources Research, 20: 291–296, 1984. Palmer, C. M., Principles of Contaminant Hydrogeology, Lewis Publishing, Chelsea, MI, 1992. Pankow, J. F., R. L. Johnson, and J. A. Cherry, “Air Sparging in Gate Wells in Cutoff Walls and Trenches for Control of Plumes of Volatile Organic Compounds (VOCs),” Ground Water, 31: 654–663, 1993.
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SUBSURFACE FLOW AND TRANSPORT
Subsurface Flow and Transport 4.25 Parker, J. C., and M. T. van Genuchten, Determining Transport Parameters from Laboratory ad Field Tracer Experiments, Virginia Agricultural Experiment Station Bulletin No. 84–3, Virginia Polytechnic Institute and State University, Blacksburg, 1984. Parker, J. C., and M. T. van Genuchten, “Flux-averaged and Volume-Averaged Concentrations in Coninuum Approaches to Solute Transport,” Water Resources Research, 20: 886–872, 1984. Parker, J. C., K. Unlu, and M. W. Kemblowski, “A Monte Carlo Model to Assess Effects of Land Disposed E & P Waste on Groundwater,” SPE Annual Technical Conference & Exhibition, 1993. Philip, J. R. “The theory of infiltration: 1. The infiltration equation and its solution.” Soil Science. 83: 345–357, 1957. Raudkivi, A. J., and R. A. Callander, Analysis of Groundwater Flow, Edward Arnold, London, UK, 1976. Rosenshein, J., and G. D. Bennet, eds., Groundwater Hydraulics, American Geophysical Union, Washington, DC, 1984. Rubin, H., and G. F. Pinder., “Approximate Analysis of Upconing.” Advances in Water Resources, 1 (2): 97–101, 1977. Sallam, A., W. A. Jury, and J. Letey, “Measurement of Gas Diffusion Coefficient Under Relatively Low Air-Filled Porosity,” Soil Science Society of America Journal 48:3–6, 1983. Schiegg, H. O., “Considerations on Water,. Oil and Air in Porous Media,” Water Science Technology, 17: 467–476. 1984. Shafer, J. M., “Reverse Pathline Calculation of Time-Related Capture Zones in Nonuniform Flow,” Ground Water 25: 283–289, 1987. Sikkema, P. C. and J. C. Van Dam, “Analytical Formulae for the Shape of the Interface in a SemiConfined Aquifer,” Journal of Hydrology, 56: 201–220, 1982. Sposito, G., and W. A. Jury, “Inspectional Analysis in the Theory of Water Flow Through Unsaturated Soil,” Soil Science Society of America Journal, 42 (1): 1985. Sposito, G., “Chemical Models of Inorganic Pollutants in Soils,” CRC Critical Reviews in Environmental Control, 15 (1): 1–24, undated. Strausberg, S. I. “Estimating Distances to Hydrologic Boundaries from Discharging Well Data,” 19th Annual Meeting of the Rocky Mountain Section of the Geological Society of America, Las Vegas, NV, 1966. Thornton, J. S., and W. L. Wootan, Jr., “Venting for the Removal of Hydrocarbon Vapors from Gasoline Contaminated Soil,” Journal of Environmental Science and Health, A17 (1), 31–44, 1982. Todd, D. K., Groundwater Hydrology, 2th John Wiley & Sons, New York, 1980. Todd, D. K., “Salt-Water Intrusion and Its Control.” Journal of the American Water Works Associations, 180–187, 1973. U. S. Department of Agriculture Agricultural Research Service, Analytical Solutions of the OneDimensional Convective-Dispersive Solute Transport Equation, Technical Bulletin No. 1661, Unlu, K., M. W. Kemblowski, J. C. Parker, D. Stevens, P. K. Chong, and I. Kamil, “A Screening Model for Effects of Land-Disposed Wastes on Groundwater Quality” Journal of Contaminant Hydrology, 11: 27–49, 1992. Washington, DC, 1982 Van Genuchten M. T. and W. J. Alves, “Analytic Solution of the One–Dimensional Convective Solute Transport Equation, Technical Bulletin. 1661, U.S. Departament of Agriculture, Washington D.C., 1982. Ward, C. H., M. B. Tomson, P. B. Bedient, and M. D. Lee, “Transport and Fate Processes in the Subsurface,” Water Resources Symposium, Vol 13, WARSAG, 1987.
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SUBSURFACE FLOW AND TRANSPORT
4.26
Chapter Four
Warrick, A. W. , J. W. Biggar, and D. R. Nielsen, “Simultaneous Solute and Water Transfer for an Unsaturated Soil,” Water Resources Research. 7: 1216–1225, 1971. Watson, K. K., and M. J. Jones, “Algebraic Equations for Solute Movement During Absorption,” Water Resources Research, 20: 1131–1136, 1984. Wilson, J. L., and L. W. Gelhar, “Analysis of Longitudinal Dispersion in Unsaturated Flow 1: The Analytical Method,” Water Resources Research, 17 (1): 122-130, 1984. Wilson, J. L., and P. J. Miller, “Two-Dimensional Plume in Uniform Ground-Water Flow Discussion,” Journal of the Hydraulics Division, American Society of Civil Engineers, 103 (HY12): 1567–1570, 1979. Wilson, J. L., and P. J. Miller., “Two-Dimensional Plume in Uniform Ground-Water Flow,” Journal of the Hydraulics Division, American Society of Civil Engineers, 104 (HY4): 503–514, 1978. Wirojanagud, P., and R. Charbeneau., “Saltwater Upconing in Unconfined Aquifers,” Journal of Hydraulic Engineering, American Society of Civil Engineers, 111: 417–434, 1985. Yeh, G. T., Analytical Transient One-, Two-, and Three-Dimensional Simulation of Waste Transport in the Aquifer System, Environmental Sciences Division Publication No. 1439, Oak Ridge National Laboratory, Oak Ridge, TN, 1981.
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Source: HYDRAULIC DESIGN HANDBOOK
CHAPTER 5
ENVIRONMENTAL HYDRAULICS Richard H. French Water Resources Center Desert Research Institute University and Community College System of Nevada Reno, Nevada
Steven C. McCutcheon Ecosystems Research Division National Exposure Research Laboratory U.S. Environmental Protection Agency Athens, Georgia
James L. Martin AScI Corporation Athens, Georgia
5.1 INTRODUCTION The thermal, chemical, and biologic quality of water in rivers, lakes, reservoirs, and near coastal areas is inseparable from a consideration of hydraulic engineering principles; therefore, the term environmental hydraulics. In this chapter we discuss the basic principles of water and thermal budgets as well as mixing and dispersion.
5.2 WATER AND THERMAL BUDGETS 5.2.1 Water Budget A water budget is a statement of the law of conservation of mass or (change in storage) (input) (output)
(5.1)
and the expressions of the water budget can range from simple to very complex. For example, consider the lake or reservoir shown in Figure 5.1. For this situation, a generic water budget could be written as follows: dS s (Ic Io Ig Pr Rr) (Ev Tr Gs Oc W) (5.2) dt 5.1 Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
ENVIRONMENTAL HYDRAULICS
5.2
Chapter Five
FIGURE 5.1 A hypothetical lake illustrating the variables in the water budget.
where Ic channel inflow rate, Io overland inflow rate, Ig groundwater inflow rate, Pr precipitation rate, Rr return flow rate, Ev evaporation rate, Tr transpiration rate, Gs groundwater seepage rate, Oc channel outflow rate, W consumptive withdrawal, and Ss lake/reservoir storage rate at time t (volume). The solution of Eq (5.2) quantifies the terms, and, in many cases, the goal of the modeling effort is to estimate the value of a single term or group of terms: for example, evapotranspiration (Ev Tr). The reliability of using a water budget is directly related to the accuracy of the prediction techniques used, the availability and quality of gauged data, and the time period involved. Among the methods of evaluating the individual terms in Eq. (5.2) are the following: • Channel inflow and outflow ( Ic and Oc )—gauging, statistical simulation. • Overland inflow (Io)—gauging, rainfall-runoff relationships. • Groundwater inflow and seepage rate (Ig and Gs)—seepage equations, gauging. • Precipitation (Pr)—gauging, statistical simulation (Smith, 1993). • Evaporation and transpiration (E and T)—gauging, evaporation/transpiration prediction relationships (Bowie et al. 1985; Shuttleworth, 1993). • Return flow and withdrawal (Rr and W)–gauging.
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ENVIRONMENTAL HYDRAULICS
Environmental Hydraulics 5.3
5.2.2 Thermal Budget The total thermal budget for a body of water includes atmospheric heat exchange at the air water interface (usually the dominant process), the effects of inflows (tributaries, wastewater, and cooling water discharges), heat resulting from chemical-biological reactions, and heat exchange with the stream bed. In the following sections, the primary components of the air-water interface heat budget will be briefly discussed; for further details the reader is referred to Bowie et al., (1985), McCutcheon (1989), or Shuttleworth (1993). Atmospheric heat exchange at the air-water interface is given by H Qs Qsr Qa Qar Qbr Qe Qc
(5.3)
where H net surface heat flux, Qs shortwave radiation incident to the water surface [3–300 (kcal/m2)/h], Qsr reflected shortwave radiation [5–25 (kcalⴢm2)/h], Qa incoming longwave radiation from the atmosphere (225–360 kcal/m2/h), Qar reflected longwave radiation [5–15 (kcalⴢm2)/hr], Qbr longwave back radiation emitted by the water body [220–345 (kcalⴢm2)/h], Qe energy utilized by evaporation [25–900 (kcalⴢm2)/h], and Qc energy convected to or from the body of water (35–50 kcalⴢm2 /hr). Note that the ranges given are typical for the middle latitudes of the United States (Bowie et al., 1985). The equations for estimating the terms of the thermal budgets use a mixed set of units, and appropriate conversions among the different units used are provided in Table 5.1. 5.2.2.1 Net atmospheric shortwave radiation (Qs Qsr) The net shortwave radiation (Qsn) is that portion of the incident shortwave radiation captured at the ground, taking into account losses caused by reflection. Although solar radiation can be measured with specialized meteorological stations equipped with radiometers, these instruments require painstaking calibration and maintenance. In most cases, measured values of solar radiation are not available at the location of interest and must be estimated from equations. Among the formulations for estimating net shortwave solar radiation is 2
Qsn Qs Qsr 0.94Qsc(1 0.65C c)
(5.4)
where Qsc clear sky solar radiation [kcalⴢm2)/h) and Cc fraction of sky covered by clouds (Anderson, 1954; Ryan and Harleman, 1973). It is pertinent to note that Eq. (5.4)
TABLE 5.1
Useful Energy Conversions for Energy Budget Calculations
1BtuⲐft2/day
=
0.131 W/m2
=
0.271 Ly/day
=
0.113 (kcalⴢm2)/h
1 watt/m2
=
7.61 Btuⴢft2)/day
=
2.07 Ly/day
=
0.86 (kcalⴢm2)/h
1 Ly/day
=
0.483 W/m2
=
3.69 (Btu/ⴢft2)/day =
0.42 (kcalⴢm2)/h
1 (kcalⴢm2)/hr =
1.16 W/m2
=
2.40 Ly/day
=
8.85 (Btuⴢft2)/day
1 kpa
=
10 mb
=
7.69 mm Hg
=
0.303 in (Hg)
1 mb
=
0.1 kpa
=
0.769 mm Hg
=
0.03 in (Hg)
1 mm Hg
=
1.3 mb
=
0.13 kpa
=
0.039 in (Hg)
1 in Hg
=
33.0 mb
=
25.4 mm Hg
=
3.3 kpa
Abbreviations Ly Langleys; mb millibar; and Btu British Thermal Unit
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ENVIRONMENTAL HYDRAULICS
5.4
Chapter Five
assumes average reflectance at the Water’s surface and uses clear sky solar radiation. In some situations, the effects of atmospheric attenuation are much greater than normal and more complex equations are required (e.g., 1972). Clear sky radiation (Qsc) can be estimated as a function of calendar month and latitude from Fig. 5.2. Shortwave solar radiation is absorbed at the water’s surface and penetrates the water column, depending on the wavelength of the radiation, the properties of the water, and the matter suspended in the water. The degree of penetration of shortwave solar radiation (sunlight) into the water column has a significant effect not only on water temperature but also on the rate of photosynthesis by aquatic plants and the general clarity, color and aesthetic quality of the water. The penetration of shortwave solar radiation is described by I Ioexp (k e y)
(5.5)
where I light intensity at depth y, Ke extinction coefficient, and Io light intensity at the surface (y 0). Values of the extinction coefficient can be estimated by several methods. For example, measurement of total light penetration into a water column can be made by using a pyreheliometer positioned at the surface that measures the total incoming solar radiation. Simultaneously, an underwater photometer is lowered and the radiation is recorded at each of a series of depths throughout the water column. Then, a value of Ke can be estimated by linear least–squares regression. An alternative but traditional, simpler, and less accurate method to estimate Ke is to lower a target into the water column until, by eye, the target just disappears. A standardized target (Secchi disk) is commonly used, and a number of investigators (Beeton, 1958; French et al., 1982; Sverdrup et al, 1942;) have developed empirical relationships between, the Secchi disk depth (ys) and the extinction coefficient of the form. (1.2 to 1.9) Ke (5.6) ys Finally, the depth (ye) at which 1 percent of the surface radiation still remains (the euphotic depth) is given from Eq. (5.5) as 4.61 ye Ke
(5.7)
5.2.2.2 Net atmospheric long-wave radiation (Qa – Qar) Atmospheric radiation is characterized by much longer wavelengths than solar radiation because the major emitting elements are water vapor, carbon dioxide, and ozone. The approach generally used to estimate this flux involves the empirical estimation of an overall atmospheric emissivity and the use of the Stephan-Boltzman law (Ryan and Harleman, 1973). Swinbank (1963) developed the following equation, which has been used in many water quality models: 2
Qan Qa Qar 1.16 1013(1 0.17C c)(Ta 460)6
(5.8)
where Qan net long–wave atmospheric radiation (Btuⴢft2/day), Cc fraction of sky covered by clouds, and Ta dry bulb air temperature (ºF). 5.2.2.3 Long-wave back radiation (Qbr) The long-wave back radiation from a water surface in most cases is the largest of all the fluxes in the heat budget (Ryan and Harleman, 1973). The emissivity of a water surface is well known; therefore, this flux can be estimated with a high degree of accuracy as a function of the water surface temperature:
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ENVIRONMENTAL HYDRAULICS
FIGURE 5.2 Clear sky solar radiation. (From Hamon et al. 1954)
Environmental Hydraulics 5.5
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ENVIRONMENTAL HYDRAULICS
5.6
Chapter Five
Qbr 0.97σT4s
(5.9)
where Qbr long–wave back radiation (cal/m2/s), Ts surface water temperature (0K), and σ Stefan-Boltzman constant (1.357 10–8 calⴢm2/s/K4) 5.2.2.4 Evaporative heat flux (Qe) Evaporative heat loss (kcal/m2/s) occurs as a result of the change of state of water from a liquid form to vapor and is estimated by Qa ρLwEv
(5.10)
where Lw latent heat of vaporization ( 597 0.57Ts, kcal/kg), Ts surface water temperature (ºC), Ev evaporation rate (m/s), and ρ water density (kg/m3). A standard expression for evaporation from a natural water surface is Ev (a bW)(es ea)
(5.11)
where Ev evaporation rate (m/s), a and b empirical coefficients, W wind speed at some specified distance above the water surface (m/s), es saturation vapor pressure at the temperature of the water surface (mb), and ea vapor pressure of the overlying atmosphere (mb). In many cases, the empirical coefficient a has been taken as zero with 1 10–9 b 5 10–9 (Bowie et al., 1985). The saturated vapor pressure can be estimated (Thackston, 1974) by 9501 es exp 17.62 T s 460
(5.12)
where es is in inches of Hg, and Ts water surface temperature (ºF). There are a number of ways of estimating ea, depending on the available data. For example, if the relative humidity (RH) is known, then e RH ea (5.13) s and then if the wet bulb temperature and atmospheric pressure are known (Brown and Barnwell, 1987) Twb 32 ea es 0.000367Pa(Ta Twb)1 1571
(5.14)
where all pressures are in (in Hg), all temperatures are in (ºF), Pa atmospheric pressure, and Twb wet bulb temperature. The relationship among the air and wet bulb temperatures (ºF) and relative humidity (Thackston, 1974) is Twb (0.655 0.36RH)Ta
(5.15)
There are many equations for estimating the rate of evaporation. For example, Jobson (1980) developed a modified formula that was used in the temperature modeling of the San Diego Aqueduct and subsequently was modified for use on the Chattahoochee River in Georgia (Jobson and Keefer, 1979). McCutcheon (1982) noted that, in many models, the wind speed function is a catchall term that compensates for many factors, such as (1) numerical dispersion in some models, (2) the effects of wind direction, fetch, channel width, sinuosity, bank, and tree height, (3) the effects of depth, turbulence, and lateral velocity distribution; and (4) the stability of air moving over the stream. (Fetch is the distance over which the wind blows or causes shear over the water’s surface.) Finally, it is
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ENVIRONMENTAL HYDRAULICS
Environmental Hydraulics 5.7
important to note that evaporation estimators that work well for lakes or reservoirs will not necessarily provide the same level of performance when used in streams, rivers, or constructed open channels. 5.2.2.5 Convective heat flux (Qc). Convective heat is transferred between air and water by conduction and is transported to or from the air-water interface by convection. The convective heat flux is related to the evaporative heat flux (Qe) by the Bowen ratio (Bowie et al., 1985), or
Q Ts Ta RB c (6.19 10–4)Pa Qe es ea
(5.16)
where all temperatures are in (ºC), all pressures are in (mb), and RB Bowen ratio. 5.2.2.6 Conclusion. The foregoing is a brief summary of the approaches used most frequently to estimate surface heat exchange in numerical models. The reader is referred to other publications for a more detailed discussion of the approaches (Bowie et al., 1985) and meteorological data requirements (Shanahan, 1984). Note that each situation should be considered carefully from the viewpoint of specific factors that must be taken into account. For example, in most lakes, estuaries, and deep rivers, the thermal flux through the bottom is not significant. However, in water bodies with depths less than 3 m (10 ft), bed conduction of heat can be significant in determining the diurnal variation of temperatures within the body of water (Jobson, 1980, Jobson and Keefer, 1979).
5.3 EFFECTS AND CAUSES OF STRATIFICATION 5.3.1 Effects The density of water is strongly affected by temperature and the concentrations of dissolved and suspended solids. Regardless of the cause of differences in water density, water with the greatest density is found at the bottom, whereas water with the least density resides at the surface. When density gradients are strong, vertical mixing is inhibited. Stratification is the establishment of distinct layers of water of different densities (Mills et al., 1982). Stratification is enhanced by quiescent conditions and is destroyed by in a body of waterphenomenasc that encourage mixing (wind stress, turbulence caused by large inflows, and destabilizing changes in water temperature). In many bodies of water (rivers, lakes, and reservoirs), stratification is the single most important phenomena affecting water quality. When stratification is absent, the water column is mixed vertically and dissolved oxygen (DO) is present in the vertical water column from the top to the bottom: that is, fully mixed water columns do not have DO deficit problems. For example, when stratification occurs, in reservoirs and lakes mixing is limited to the epliminion or surface layer. Since stratification inhibits, vertical mixing is inhibited by stratification, and reaeration of the bottom layer (the hypoliminion) is inhibited if not eliminated. The thermocline (the layer of steep thermal gradient between the epiliminion and hypoliminion) limits not only mixing but also photosynthetic activity as well. The hypolimnion has a base oxygen demand and benthic matter and the settling of particulate matter, from the epiliminion only adds to this demand. Therefore, while the demands of DO in the hypoliminion increase during the period of stratification, inhibition of mixing between the epiliminion and the hypolimnion and the lack of photosynthetic activity deplete the DO concentrations in the
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ENVIRONMENTAL HYDRAULICS
5.8
Chapter Five
hypolimnion. Finally, a rule of thumb suggests that when water temperature is the predominant cause of differences in water density a temperature gradient of at least 1ºC/m is required to define the thermocline (Mills et al., 1982). The density of water can be estimated by ρ ρT ∆ρs
(5.17)
where ρ water density (kg/m3), ρT water density as a function of temperature, and ∆ρs increments in density caused by solids. 5.3.2 Water Density as a Function of Temperature A number of formulations have been proposed to estimate ρT and among these are ρT 999.8452594 6.793952 10 2 Te 9.095290 103 Te2 1.001685 104 Te3 1.120083 10
6
(5.18)
Te 6.536332 10 Te 9
4
5
where Te water temperature in ºC (Gill, 1982). 5.3.3 Water Density as a Function of Dissolved Solids or Salinity and Suspended Solids In most cases, data for dissolved solids are in the form of total dissolved solids (TDS); however, in some cases, salinity may be specified. The density increment for dissolved solids can be estimated by ∆ρTDS CTDS(8.221 104 3.87 106 Te 4.99 108 Te2)
(5.19)
(Ford and Johnson, 1983), where CTDS concentration of TDS (g/m3 or mg/L). If the concentration of TDS is specified in terms of salinity (Gill, 1982). ∆ρSL CSL(0.824493 4.0899 103 Te 7.6438 105 Te2 8.2467 107 Te 5.3875 109 Te ) 3
4
3
1.5 (5.72466 10 C SL 6
1.6546 10
4
1.0277 10 Te
2
4
Te ) 4.8314 10
2
CSL
(5.20)
where CSL concentration of salinity (kg/m3). The density increment for suspended solids is 1 ∆ρss Css1. 10 3 SG
(5.21)
where SG specific gravity of the suspended sediment. (Ford and Johnson, 1983). The total density increment caused by solids is then ∆ρs (∆ρTDS or ∆ρSL) ∆ρSS
(5.22)
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ENVIRONMENTAL HYDRAULICS
Environmental Hydraulics 5.9
5.4 MIXING AND DISPERSION IN OPEN CHANNELS Turbulent diffusion (mixing) refers to the random scattering of particles in a flow by turbulent motions, whereas dispersion is the scattering of particles by the combined effects of shear and transverse turbulent diffusion. Shear is the advection of a fluid at different velocities at different positions within the flow. When a tracer is injected into a homogeneous channel flow, the advective transport process can be viewed as composed of three stages. In the first stage, the tracer is diluted by the flow in the channel because of its initial momentum. In the second stage, the tracer is mixed throughout the cross section by turbulent transport processes. In the third stage, longitudinal dispersion tends to erase longitudinal variations in the tracer concentration. In some cases, the second stage is eliminated because the tracer discharge has a significant amount of initial momentum associated with it; however, in many cases, the tracer flow is small and the momentum associated with it is insignificant. In the latter case, the first transport stage is eliminated. In this treatment, only the second and third transport stages will be treated, with the implied assumption that if there is a first stage, it can be treated separately. The reader is cautioned that, in this chapter, y is the vertical coordinate direction and z is the transverse coordinate direction.
5.4.1 Vertical Turbulent Diffusion To develop a quantitative expression for the vertical turbulent diffusion coefficient, consider a relatively shallow flow in a wide rectangular channel. It can be shown that the vertical transport of momentum in such a flow is given by dv τ vρ (5.23) dy where τ shear stress at a distance y above the bottom boundary, ρ fluid density, v vertical turbulent diffusion coefficient, and v longitudinal velocity (French, 1985). Because the one-dimensional vertical velocity profile and shear distribution are known, it can be shown that y y v kv*yd 1 (5.24) y y d d where k von Karman’s turbulence constant (0.41), yd depth of flow, v* shear velocity ( 兹g苶苶S yd苶)苶, and S longitudinal channel slope (French, 1985). The depth–averaged value of v is
苶v苶 0.067ydv*
(5.25)
When the fluid is stably stratified, mixing in the vertical direction is inhibited, and one often quoted formula expressing the relationship between the unstratified and stratified vertical mixing coefficient was provided by Munk and Anderson (1948):
v vs 1 3.33 Ri)1.5
(5.26)
where vs the stratified vertical mixing coefficient.
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ENVIRONMENTAL HYDRAULICS
5.10
Chapter Five
5.4.2 Transverse Turbulent Diffusion In the infinitely wide channel hypothesized to derive Eq. (5.24), there is no transverse velocity profile; therefore, a quantitative expression for t , the transverse turbulent diffusion coefficient, cannot be derived from theory. The following equations to estimate t derived from experiments by Fischer et al., (1979), and Lau and Krishnappen (1977). In straight rectangular channels, an approximate average of the results available is
t 0.15ydv* 50%
(5.27)
where the ±50 percent indicates the error incurred in estimating t. In natural channels, t is significantly greater than the value estimated by Eq. (5.27). For channels that can be classified as slowly meandering with only moderate boundary irregularities
t 0.60ydv* 50%
(5.28)
If the channel has curves of small radii, rapid changes in channel geometry, or severe bank irregularities, then the value of t will be larger than that estimated by Eq. (5.28). For example, in the case of meanders, Fischer (1969) estimated that V 2y3 t 252d * R cv
(5.29)
where a slowly meandering channel is one in which TV (5.30) 2 Rcv* and Rc radius of the curve. As stated above, the complete advective transport process in a two-dimensional flow can be conveniently viewed as composed of three stages. In the second stage, the primary transport mechanism is turbulent diffusion, and a comparison of Eqs. (5.25) and (5.27) shows that the rate of transverse mixing is roughly 10 times greater than the rate of vertical mixing. Thus, the rate at which a plume of tracer spreads laterally is an order of magnitude larger than is the rate of spread in the vertical direction. However, most channels are much wider than they are deep. In a typical case, it will take approximately 90 times as long for a plume to spread completely across the channel as it will take to mix in the vertical dimension. Therefore, in most applications, it is appropriate to begin by assuming that the tracer is uniformly distributed over the vertical. In a diffusional process in which the tracer is added at a constant mass flow rate (M*) at the center line of a bounded channel (∂C/∂z 0 at z 0 and ∂C/∂z 0 at z T), the downstream concentration of tracer is given approximately by C 1 C ’ 兹4苶苶 πx' 苶n
冘 ∞
∞
(z' 2n z ')2 o exp 4x'
冤
(z' 2n z ')2 o exp 4x'
冥
(5.31)
where M* C' VT yd
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ENVIRONMENTAL HYDRAULICS
Environmental Hydraulics 5.11
xt x' VT 2 and z z' T A reasonable criterion for the distance required for “complete mixing” (where the concentration is within 5 percent of its mean value everywhere in the cross section) from a center-line discharge is 0.1V T2 L t
(5.32)
If the pollutant is discharged at the side of the channel, the width over which the mixing must take place is twice that for center-line injection, but the boundary conditions are otherwise identical and Eq. (5.32) applies if T is replaced with 2T.
5.4.3 Longitudinal Dispersion After a tracer becomes mixed across the cross section, the final stage in the mixing process is the reduction of longitudinal gradients by dispersion. If a conservative tracer is discharged at a constant rate into a channel, the flow rate of which also is constant, there is no need to be concerned about dispersion; however, in the case of an accidental release (spill) of a tracer into a channel or the release is cyclic, dispersion is important. The onedimensional equation governing longitudinal dispersion is ∂C 苶 ∂苶 C ∂2 苶 C V K S ∂t ∂x ∂2x
(5.33)
where K the longitudinal dispersion coefficient and S sources or sinks of materials. The initial work in dispersion, beginning with Taylor (1954), assumed a prismatic channel. However, natural streams have bends, sandbars, side pools, in-channel pools, bridge piers, and other natural and anthropogenic changes, and every irregularity in the channel contributes to longitudinal dispersion. Some channels may be so irregular that no reasonable approximation of dispersion is possible: for example, a mountain stream consisting of pools and riffles. Fischer et al. (1979) presented a number of methods of approximating K in a natural open channel. Of these, the most practical is 0.011V2 T2 K ydv*
(5.34)
Equation (5.33) depends on a crude estimate of t and does not reflect the existence of “dead zones” in natural channels. However, it does have the advantage of relying only on the usually available estimates of depth, velocity, width, and surface slope. With regard to the solution of the dispersion equation, the following observations are pertinent: 1. The longitudinal dispersion analysis is not valid until the end of the initial period, when 0.4V T2 x t
(5.35)
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ENVIRONMENTAL HYDRAULICS
5.12
Chapter Five
2. In the case of a slug of dispersing material (mass M), the longitudinal length of the cloud after the initial period can be estimated approximately by 0.5 2K T2 xt 0.07 L 4 2 t VT
(5.36)
and the peak concentration within the dispersing cloud is M Cmax 4 Kx A V
冪莦莦莦莦
(5.37)
Note that the observed value of the peak concentration will generally be less than this estimate because some of the material is trapped in dead zones and some of the typical tracers (Martin and Mc Cutcheon, 1999) sorb onto sediment particles.
5.5 MIXING DISPERSION IN LAKES AND RESERVOIRS Important factors in the hydraulic design, operation, and analysis of spills in reservoirs and lakes include (1) determining vertical stratification to guide lake monitoring and the design withdrawal structures, (2) locating the plunge point or separation point to determine how inflows mix, (3) computing the dilution and mixing of inflows and the time required to travel through a reservoir or lake, and (4) determining the quality of withdrawals or outflows and effects on the quality of reservoir water. The elevation and flow through withdrawal structures at dams are selected to control flooding and achieve certain water-quality targets or standards. The stratification, mixing, and travel of inflows are determined to design water-intake structures at dams or other locations in lakes, to forecast the habitat and fisheries that a proposed reservoir may support, and to track chemical spills or flood waters through reservoirs. This section is based on Chaps. 8 and 9 in Martin and McCutcheon (1999), which provide a number of sample calculations. Many lakes and reservoirs stratify for part of the year into an epilimnion, thermocline, and hypolimnion illustrated in Fig. 5.3. The depth and thickness of the thermocline or metalimnion vary with location and time of the year and even time of the day to a limited extent. The thermocline represents the interface between a well-mixed surface layer, or epilimnion, and the cooler, deeper hypolimnion. In freshwater lakes, the thermocline is defined by a minimum temperature gradient of 1ºC/m. When a distinct interface does not exist, the thermocline, epilimnion, and hypolimnion may not be defined. Mixing processes also are different in riverine, transition, and lacustrine zones (Fig. 5.3). Mixing in the riverine zone is dominated by advection and bottom shear, and turbulence is generally dissipated under the same conditions. Seiche, wind mixing, boundary shear, boundary intrusion, withdrawal shear, internal waves, and dissipation of turbulence generated elsewhere cause mixing in the lacustrine zone. Buoyancy resulting from stable stratification stabilizes or prevents mixing. In the transition zone, ending at the plunge point or separation point, buoyancy begins to balance the advective force of the inflow. There are three sources of energy for mixing: (1) inflows from tributaries, overland runoff, and discharges, (2) withdrawal at dams, discharges at control structures, and natural outflows, and (3) wind shear, solar heating and cooling, heat conduction and evaporation, and other meteorological forces.
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ENVIRONMENTAL HYDRAULICS
FIGURE 5.3 Mixing processes in zones of lakes and reservoirs. (Modified from Fischer, et al.1979)
Environmental Hydraulics 5.13
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ENVIRONMENTAL HYDRAULICS
5.14
Chapter Five
Shallow lakes and reservoirs that do not stratify are normally analyzed in the same fashion as rivers or as a completely mixed body of water. For a completely mixed system, the residence time (T in seconds or more typically years) or time for an inflow to travel through the body of water is simply tr φ/Q, where φ is the volume of the lake (m3) and Q is the sum of the inflows or the average reservoir discharge (m3/s). Freshwater lakes tend to stratify when the mean depth exceeds 10 m and the residence time exceeds 20 days (Ford and Johnson, 1986). The densimetric or internal Froude number Frd (Norton et al., 1968) provides a better indication of the stratification potential of a reservoir where LLQ Vo Frd Frp 冷∆ρ冷 冷∆ρ冷 yavgφ yo gρ gρ
冪莦莦莦 冪莦莦莦
(5.38)
LL the length of the reservoir (m), yavg the its mean depth (m), g gravitational acceleration (m/s2), ∆ρ the difference in density over the depth for the internal Fr or between the inflow and surface waters of the lake or reservoir at the plunge point or separation point (kg /m3), r average density of the lake for the internal Fr or density of the inflow (Turner, 1973) at plunge or separation points (kg/m3), Vo the average velocity of the inflow (m/s), and yo the hydraulic depth or cross-sectional area divided by the top width of the inflow (m). The Fr at the plunge point Frp, also defined in Eq. (5.38), will be used in the next section. For design projections, the dimensionless density gradient ∆ρ/(ρyavg) normally is taken to be 10-6 m–1 (Norton et al., 1968). If Fr >> 1/π, the reservoir is expected to be well mixed. If Fr 0.007, ξ is on the order of 1.18 and the density current depth is the crit-
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ENVIRONMENTAL HYDRAULICS
5.18
Chapter Five
ical depth (Akiyama and Stefan, 1984). However, the entrained fraction ξ is highly variable. The dilution of concentrations or temperatures resulting from mixing in plunging flows follows from a simple mass or heat balance ξCa C Cp 1ξ
(5.47)
where C is the inflow concentration (g/m3 or mg/L3) or temperature (ºC), Ca is the ambient concentration (g/m3 or mg/L3) or temperature (ºC) of the lake, and Cp is the concentration (g/m3 or mg/L3) or temperature (ºC) of the plunging flow after initial mixing. The mixing after plunging results from bottom shear as well as shear at the interface of the underflow with ambient lake water. For a triangular cross section, the entrainment coefficient is (Imberger and Patterson, 1981). 1 2 E CkC3/2 (5.48) D Fr b 2 where laboratory experiments indicate that Ck is approximately 3.2 (Hebbert et al. 1979), CD the dimensionless bottom drag coefficient defined following Eq (5.42), Frb the internal fronde number ub Frb (5.49) 兹苶h b苶苶b where ub underflow velocity, hb underflow depth, and b relative density difference. The entrainment coefficient E is a constant for a specific body of water. yuf (6/5)Ex yO The depth or thickness of the underflow (m) is a linear function of the entrainment coefficient (Hebbert et al. 1979; Imberger and, 1981), where x is the distance downstream from the plunge point (m) and yo is the initial thickness of the underflow (m) that is approximately equal to the depth at the plunge point. If entrainment is limited, the depth of the underflow remains approximately constant as long as the bottom slope remains constant. The increase in flow rate because of entrainment for an underflow in a triangular cross section is solved iteratively as y 5/3 Q(x) Q1 u 1 y 1 (5.50) where Q1 the discharge (m3/s) and y1 the depth (m) from the previous calculation step. For the initial iteration, Q1 the discharge at the plunge point Qp (m3/s) and y1 the plunge point depth yo (m). Because of more significant differences in density and less internal mixing contrasted with the epilimnion, underflows tend to remain more coherent than overflows. Sediment– laden underflows, especially, tend to travel to the lake outlet or dam. 5.5.5 Interflow Mixing After experiencing approximately 15 percent entrainment at the plunge point (for mild slopes) and mixing as an underflow, an interflow intrudes into a lake at the depth at which neutral buoyancy is achieved. The turbulence generated by bottom shear is dissipated quickly, and entrainment into the interflow is dominated by interfacial shear with ambient lake water above and below the intrusion layer.
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Environmental Hydraulics 5.19
When the momentum of inflow is small, an interflow is analogous to a withdrawal from a dam discussed in Sec. (5.5.6). Interflows are governed chiefly by three conditions based on the dimensionless number R FriGr1/3 where Fri is the internal Fr defined in Eq. (5.51) and Gr is the Grashof number (Gr), both of which are computed at the depth of intrusion. The internal Froude Number computed at the intrusion depth is qI QI Fri (5.51) NL 2I BINL2I where qI the interflow rate per unit width following entrainment at the intrusion point ( m2/s), LI the length of the reservoir at the level of intrusion (m), QI the interflow rate (m2/s), BI the intrusion width (m), and N the buoyancy frequency (s-1) expressed as N
冪 莦gρ∆莦yρ莦莦 I
I I
(5.52)
where ∆ρI density difference between the layers into which the flow is intruding (kg/m3), ρI density of the intrusion (kg/m3), and yI the thickness of the depth of the intrusion (m). The dimensionless Grashof number Gr is the square of the ratio of the dissipation time to the internal wave period or N2L 4 Gr 2I v
(5.53)
where v the vertically averaged diffusivity (m2/s). Generally, if Gr 1, then an internal wave field will decay slowly, but if Gr 1 then viscous dissipation damps waves quickly (Fischer et al. 1979). Imberger and Patterson (1981) also introduced a dimensionless time variable tN t* Gr1/6 where t time(s), which, along with the Prandtl number Pr v /t, where t is vertically averaged diffusivity of heat (m2/s), is used to define three interflow conditions: 1. If R 1, the intrusion is governed by a balance of the inertial and buoyancy forces so that the actual intrusion length Li is proportional to time, as given by (Ford and Johnson, 1983; Imberger et al., 1976). Li 0.44Li 兹R 苶苶 t* 0.44 兹q 苶苶t IN苶
(5.54)
If the speed of the intrusion is constant or uniform, the velocity vI is Li/t, so that g∆ρIyI 1/2 苶苶 vI 0.44 兹q IN 0.194 ρ I where ρm the density of the intrusion. The difference in density in the computation of the buoyancy frequency is that occurring over the thickness of the intrusion hm, which, along with the relationship um qm/hm, can be substituted into the above equation to yield an alternative formulation for the speed of intrusion. The thickness of the interflow can be solved by assuming uniform flow (Ford and Johnson, 1983). q2m 1/3 hm 2.99 ∆ m gρ ρm
(5.55)
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ENVIRONMENTAL HYDRAULICS
5.20
Chapter Five
where hm is generally distributed equally above and below the center line of the intrusion. 2. If R t*R P2/3 r , then the flow regime is dominated by the balance between viscous and buoyancy forces and the intrusion length becomes Li 0.57 L R2/3 t*5/6
(5.56)
The thickness of the interflow is hm 5.5LmGr–1/6
(5.57)
In this regime, the flow is generally distributed so that 64 percent lies above the center line of the intrusion (Imberger, 1980); thus, the half-thickness (hma) of the interflow above the center line is given by hma 3.5LmGr–1/6
(5.58)
and the half-thickness below the center line is given by hmb 2.0Lm Gr–1/6
(5.59)
3. If P t* R then the flow regime is dominated by viscosity and diffusion and the intrusion length becomes 2/3 r
-1
Li = C LR3/4 t*3/4
(5.60)
where C is a coefficient, that generally is unknown (Fischer et al. 1979). Ford and Johnson (1986) indicated that unless dissolved solids dominate the density profile (i.e., Pr is high), intrusions into most reservoirs have R 1, where inertia and buoyancy dominate. Because the difference in density varies with the location of the limits of the interflow zone above and below its center line, the solution proceeds by estimating a value of hm and then by computing the difference in density, which is then used to compute a revised estimate of hm. This process is repeated until convergence occurs. The equations for intrusion require information on both the morphometry of the reservoir and the temperature distribution. The widths used in the formulations should represent the conveyance width (Ford and Johnson, 1983). Because the time for the intrusion to pass through a lake can be relatively long, the flow rates used in the calculations should represent an average value over the period of intrusion. To estimate the time scale in their analysis of intrusions in DeGray Lake in Arkansas, Ford and Johnson (1983) used the length of the lake and ∆ρm and hm across the thermocline. For DeGray Lake, the intrusion time scale ranged from 4 to 6 days. Changes in outflow during the period of the intrusion also can affect the movement through the lake. Interflows may stall and collapse if the inflow or outflow ends. Interflows also may be diverted or mixed because of changes in meteorological conditions that influence epilimnion mixing and thermocline depth. The temperature or density of the interflow will remain constant. However, the interflow will spread laterally and the thickness will increase caused by entrainment of ambient water. The resulting concentrations can be computed from a mass balance vinCBhn constant, where vin is the velocity of the interflow; C the concentration or temperature; B the reservoir width, which may vary with distance from the separation or detachment point; and hn the thickness of the interflow.
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FIGURE 5.4 Reservoir withdrawal. (Adapted from Martin and McCutcheon, 1998)
Environmental Hydraulics 5.21
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ENVIRONMENTAL HYDRAULICS
5.22
Chapter Five
5.5.6 Outflow Mixing The withdrawal velocity profile is used in models CE-QUAL-R1 (Environmental Laboratory, 1985) and CE-QUAL-W2 (Cole and Buchak, 1993) and in calculations to predict the effects of withdrawals on reservoir and tail race water quality. The extent of a withdrawal zone (Fig. 5.4) strongly depends on the ambient lake stratification and release rate, location of the withdrawal, and reservoir bathymetry. For a given outflow rate and location, the withdrawal zone thins as the density gradient increases. Depending on the degree of stratification, withdrawal rate and location, and other factors related to the design of the dam and the bathymetry of the reservoir, the withdrawal zone may be thin or may extend to the reservoir bottom or water surface. Within the withdrawal zone, the velocity distribution will vary from a maximum velocity to zero at the limits of the zone, depending on the shape of the density profile. The maximum velocity is not necessarily centered on the withdrawal port. A number of methods predict the extent of withdrawal zones and the resulting velocity distributions. Fischer et al. (1979) described methods of computing withdrawal patterns similar to those used in the analysis of interflows in the previous section. The Box Exchange Transport, Temperature, and Ecology of Reservois (BETTER) model and the SELECT model based on the original work of Bohan and Grace (1973) are the more practical approaches. The BETTER model, applied to a number of Tennessee Valley Authority reservoirs, computes the thickness of the withdrawal zone above and below the outlet elevation from ∆y cw Qout, where Qout the total outflow rate and cw is a thickness coefficient. The model assumes a triangular or Gaussian flow distribution to distribute flows within the withdrawal zone (Bender et al. 1990). The SELECT model (Davis et al. 1985) computes the in-pool vertical distribution of outflow and concentrations of water quality constituents, the outlet configuration and depth, and the discharge rate (Stefan et al. 1989). The SELECT code also is applied as subroutines in generalized reservoir models, such as CE-QUAL-R1 (Environmental Laboratory, 1985). The model is based on the following equations. The theoretical limits of withdrawal (Bohan and Grace, 1973) were modified by Smith et al. (1985) to include the withdrawal angle as Q ut θ (5.61) 3o ZN π where Z distance from the port center line to the upper or lower withdrawal limit; θ the withdrawal angle (radians); and N the buoyancy frequency [g∆ρ/(ρZ)]1/2, in which ∆ρ the difference in density between that at the upper or lower withdrawal limit and at the port centerline; and ρ the density (kg/m3) at the port center line. The convention is that ∆ρ is positive for stably stratified flows such that ∆ρ ρ (upper limit) ρ (withdrawal port) or ∆ρ ρ (withdrawal port) ρ (lower limit). The elevation of the water surface, the bottom, of the reservoir, and the withdrawal port and the density profile must be known. The equation must be solved iteratively since both the distance from the port center line Z and the density as a function of Z are unknown. A typical solution procedure where the upper and lower withdrawal zones can form freely within the reservoir without interference at the surface or bottom is as follows: 1. Rearrange the equation as Qout Z3Nθ/π 0. 2. Check to see if interference exists by, first, using Z equal to the distance from the port,s center line to the surface. Estimate the density at the center line of the withdrawal port and the water surface and substitute the values into the rearranged equation. If the solution is not-zero and is positive, surface interference exists.
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FIGURE 5.5 Definition of withdrawal characteristics. (From Martin and McCutcheon, 1998)
Environmental Hydraulics 5.23
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ENVIRONMENTAL HYDRAULICS
5.24
Chapter Five
Similarly, substitute the distance from the port center line to the bottom, along with the density at the bottom of the reservoir, and determine if a bottom interference exists. 3. If both of the evaluations from Step 2 are negative, the withdrawal zone forms freely in the reservoir. The limit of the surface withdrawal zone above the port can be determined by using iterative estimates of values for Z and the density at the height above the center line until the equation approaches zero to within some tolerance. The lower limit of withdrawal below the port center line can be determined in a similar manner. 4. If surface or bottom interference exists, a theoretical withdrawal limit can be determined using values of Z computed using elevations above the water’s surface for surface interference or below the reservoir’s bottom for bottom interference. However, this solution requires an estimate of density for regions outside the limits of the reservoir. Davis et al. (1985) estimated these densities by linear interpolation using the density at the port center line and the density at the surface or bottom of the reservoir. For the case where one withdrawal limit intersects a boundary and the other does not, the freely forming withdrawal limit cannot be estimated precisely using the rearranged equation. Smith et al. (1985) proposed an extension to estimate the limit of the freely forming layer similar to that described above D d D d Q 0.125(D d)3θ 1 out π 1 π sin π , N D D
(5.62)
where d the distance from the port center line to the boundary of interference (m) and D the distance between the free withdrawal limit and the boundary of interference (m) shown in Fig. 5.5. The length scale in the buoyancy frequency N is D in place of Z, and ∆ρ is the difference in the density between that at the surface for withdrawals that extend to the surface and between the lower free limit or density at the bottom for withdrawals that extend to the bottom and upper free limit. For consistency with the definition of stable stratification as positive, the convention is that ∆ρ ρ(surface layer) ρ(free limit) or ∆ρ ρ(upper free limit) ρ(bottom layer). Once the limits of withdrawal are established, the distribution of withdrawal velocity is estimated by dividing the reservoir into layers, the density of which is determined at the center line of each layer. The computation of the vertical velocity distribution is based on the location of the maximum velocity, which can be estimated from (Bohan and Grace, 1973). 2 Z YL Hsin1.57 L H
(5.63)
where YL the distance from the lower limit to the elevation of maximum velocity (m) shown in Fig. 5.4, H the vertical distance between the upper and lower withdrawal limits (m), and ZL the vertical distance between the outlet center line and the lower withdrawal limit (m). If the withdrawal intersects a physical boundary, the theoretical withdrawal limit is used, which may be above the water’s surface or below the reservoir’s bottom. Once the location of the maximum velocity Vmax (m/s) is determined, the normalized velocity VN(I) V(I)/Vmax in each layer I is estimated for withdrawal zones that intersect a boundary as (Bohan and Grace, 1973).
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ENVIRONMENTAL HYDRAULICS
Environmental Hydraulics 5.25
y(I) ∆ρ(I) 2 VN(I) 1 YL ∆ρmax or for a withdrawal that does not intersect a boundary
(5.64)
y(I) ∆ρ(I) 2 VN(I) 1 − (5.65) YL ∆ρMAX where V(I) the velocity in layer I (m/s), y(I) the vertical distance from the elevation of maximum velocity to the center line of layer I (m), YL the vertical distance from the elevation of maximum velocity to the upper or lower withdrawal limit (m) determined by whether the centerline of layer I is above or below the point of maximum velocity, ∆ρ(I) the density difference between the elevation of maximum velocity and the center line of layer I, and ∆ρmax the difference in density between the point of maximum velocity and the upper or lower withdrawal limit. If the withdrawal intersects the surface or the bottom, velocities are calculated for locations either above the water’s surface or below the reservoir’s bottom and the distribution is truncated at the reservoir’s boundaries to produce the final velocity distribution. The flow rate in each layer I is VN(I) q(I) Qout m
冘
(5.66)
VN(I)
I=1
where Qout the total release rate and m the number of layers. The quality of the release can be determined from a simple flow-weighted average or mass balance as q(I) C(I) C RV N
冘
(5.67)
q(I)
I1
where CR the concentration or temperature of waterquality constituent C in the release and C(I) the concentration or temperature in each layer. For discharge over a weir, the withdrawal limit Z and average velocity in the withdrawal zone Vweir is derived from the densimetric Froude number [Eq. 5.38] as (Grace, 1971, Martin and McCutcheon, 1998), C1g∆ρ(Z Hw) C2∆ρ(Z Hw) 0 Vweir ρ H wρ 2
(5.68)
where ∆ρ the difference in density between the weir crest and the lower withdrawal limit, ρ the density at the weir crest elevation, Hw head above the weir crest elevation, Z distance between the crest elevation and the lower withdrawal limit, and C1 and C2 are constants, which have values of ZH C1 0.54 and C2 0 for w 2.0 Hw and
(5.69) ZH C1 0.78 and C2 0.70 for w 2.0 Hw
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ENVIRONMENTAL HYDRAULICS
5.26
Chapter Five
5.5.7 Mixing Caused by Meteorological Forces Wind–generated waves and convective cooling cause significant mixing at the water surface. Wind shear causes waves at the surface and at each density interface within a lake or reservoir, such as the thermocline, and larger scale surface mixing by Langmuir circulation results from sustained wind. Wind setup, seiche, and upwelling are caused by meteorological events that generate mixing over much larger areas. Internal waves are caused by shearing currents set up by both wind and other currents and, although not as obvious as surface waves, these can be larger and more effective in causing mixing. The intensity of wave mixing and turbulence is a direct result of wind energy or the energy in other shearing currents. The basic characteristics of waves are amplitude or height between trough and crest and the length between crests. The wave period is the time required for successive waves to pass a given point. Progressive waves move with respect to a fixed point, whereas standing waves remain stationary while water and air currents move past. The height and period of wind waves are related to wind speed, duration, and fetch. Fetch is the distance over which the wind blows or causes shear over the water’s surface. As fetch increases, the wavelength increases; long wavelengths are only produced in the presence of a long fetch. The shortest wavelengths require only limited contact between wind and water. Waves with a wavelength less than 2π cm (6.28 cm) are capillary waves, which are not important in the modeling of lakes and reservoirs. The more important gravity waves have wavelengths longer than 2π cm. The two types of gravity waves are short waves and long waves, distinguished by the interaction with the benthic boundary. The wavelength of short waves seen by eye on lakes and reservoirs is much less than the water’s depth, and they are not affected by bottom shear. Long waves, such as lake seiche, are influenced by bottom friction. Seiches are periodic oscillations of the water’s surface and density interfaces resulting from a displacement. Shortwave motion is circular in a vertical plane, making a complete revolution as each successive wave passes. The orbital motion mixes surface layers or layers at an interface. With no net advection of water, the overall effect is dispersive. Thus, the mixing terms in transport and water quality models are generally increased to account for wave mixing, especially in the epilimnion. In a few cases, specific mixinglength formulas Kent and Pritchard, 1957, Rossby and Montgomery, 1935; were derived for wave mixing, but these formulas have not been applied in current models of water quality. No appreciable orbital motion occurs below a depth of approximately one-half the wavelength in unstratified flow, a depth referred to as the wind mixed depth. The wind mixed depth increases with fetch because the wave height and wavelength increase with increasing fetch. This is illustrated by a simple relationship discovered by Lerman (1978) relating fetch to the depth of the summer thermocline for a wide variety of lakes of different sizes and shapes. As wavelength becomes longer in relation to the depth, or as water becomes shallower, wave orbits become increasingly flatter or elliptical. As the orbits flatten, the motion of the water essentially becomes horizontal oscillation (Smith, 1975) so that the motion of the water caused by, long waves is more advective rather than dispersive. For long waves, the wave speed or celerity is c (gY)0.5. As short waves enter shallow water, the bottom affects orbital motion. From this point inland to the line where wave breaking occurs, the depth is less than one–half the wave period. In this shore zone, wave velocity decreases with the square root of the depth, which results in a corresponding increase in wave height. Waves distort as water at the crest moves faster than the wave, creating an instability. These unstable waves may eventually collapse, forming breakers or whitecaps, depending on the wave steepness of the waves, the wind speed and direction, the direction of the waves, and the shape and rough-
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ENVIRONMENTAL HYDRAULICS
Environmental Hydraulics 5.27
ness of the bottom. A spilling breaker tends to form over a gradually shoaling bottom and tends to break over long distances, with the wave collapsing downward in front of the wave. Plunging breakers occur when the bottom shoals rapidly or when the direction of the wind opposes the wave. The plunging breaker begins to curl and then collapses before the curl is complete. A plunging or surging breaker does actually not break or collapse but forms a steep peak as the wave moves up the beach. The type of breaking wave and the associated energy controls beach erosion, aquatic plant growth, surf-zone mixing, and the exchange of contaminants between surface and ground waters. After breaking, waves continue to move up a gradually sloping beach until the force of gravity forces the water back. The extent to which the water runs up the beach is called the swash zone. The movement of the swash up the beach may result in the deposition of particles and debris, causing swash marks at the highest point of the zone. Wave run–up in the swash zone also sets up an imbalance of momentum along the porous beach face that pumps contaminants into and out of the beach (McCutcheon, 1989). In large lakes and reservoirs with an extremely long fetch, parallel pairs of large vertical vortices or circulatory cells known as Langmuir circulation develop at an angle of 15º clockwise with the general direction of a sustained wind, when wave and current conditions are favorable. The depth of the vortices depends on stratification and may interact with internal waves formed on the thermocline, deepening over the troughs of internal waves. Where the counterrotating Langmuir cells converge, visible streaks or bands form on the surface that tend to accumulate floating debris. In the convergence zone, downward velocities of 2–6 cm/s carry surface waters toward the thermocline. These downward currents move in a circular fashion and turn upward into a divergence zone midway between the Langmuirstreaks. Water near the thermocline moves to a zone near the surface at a velocity of about 1 to 2 cm/s over a larger area. As first proposed by Langmuir (1938), this type of large-scale circulation also contributes to the vertical mixing of the epilimnion. Like smaller-scale orbital wave mixing, the effect of Langmuir circulation is lumped into values selected for the eddy viscosities and eddy diffusivities of the epilimnion. Because of the smaller differences in density across density interfaces within a body of water, internal waves travel more slowly than do surface waves, but they achieve greater wave heights. Internal waves include standing waves, such as seiches (Mortimer, 1974) and internal hydraulic jumps (French, 1985), but most are progressive waves that radiate energy from the point at which the waves were generated (Ford and Johnson, 1986). Wind shear, water withdrawals, hydropower releases, and thermal discharges as well as local disturbances produce internal waves. The most significant mixing between stratified layers occurs when internal waves break (Turner, 1973). Before breaking, internal waves mix the water adjacent to the interface and sharpen the density interface to increase the likelihood of breaking. When wave breaking does occur, the entrained water is mixed through the adjacent layer. Among the most important internal waves is the seiche. As defined above, seiches are periodic oscillations of the water surface and density interfaces resulting from a displacement. Displacements are typically caused by large scale wind events or large withdrawals. Sustained wind across a lake surface increases the elevation the water’s surface at the downwind boundary of the lake, causing wind setup. As the wind subsides, the water’s surface tilt or displacement results in a sloshing motion, or seiche, of the lake surface and in thermocline if the lake is stratified. If hydropower operations or reservoir releases change the net flow toward the dam, the water piles up at the dam and forms a seiche, often resulting in noticeable differences in thermocline depths between periods of operation and nonoperation, such as between weekdays and weekends. More rarely, a seiche may result from earthquakes or other geologic events. During the rocking or sloshing, potential energy is converted to kinetic energy and is dissipated by bottom friction.
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ENVIRONMENTAL HYDRAULICS
5.28
Chapter Five
Wind setup in Lake Erie may exceed 2 m during severe storms (Wetzel, 1975), but for a moderate storm blowing over the long axis of Green Bay, Wisconsin the wind setup has reached approximately 12 cm (Martin and McCutcheon, 1998). An estimate of wind setup can be obtained from the onedimensional equation of motion assuming constant depth, negligible bottom stress, and steadystate conditions in an unstratified lake, or ∂ξ ρaCDu2w v2* (5.70) ∂x gρy gy where ξ the deviation of the water’s surface (m), x the horizontal distance (m), ρa the density of air (kg/m3), ρ the density of water (kg/m3), CD the dimensionless drag coefficient, uw the wind speed (m/s), y the water depth (m), and v* the friction velocity in water (m/s) or (τs/ρ)0.5, in which τs is the surface shear stress (kg/m– s2). The term ∂ξ/∂x is positive in the direction of the wind. The divergence between wind and shear force is negligible in shallow lakes and reservoirs but not in deep oceans.
5.6 PLUME AND JET HYDRAULICS A jet is the discharge of a fluid from an opening into a large body of the same or similar fluid that is driven by momentum. A plume is a flow that, while resembling a jet, is the result of an energy source providing the fluid with positive or negative buoyancy rather than momentum relative to its surroundings. Many discharges into the environment are discussed in terms of negatively or positively buoyant jets, implying that they derive from sources that provide both momentum and buoyancy. In such cases, the initial flow is driven primarily by the momentum of the fluid exiting the opening; however, if the exiting fluid is less or more dense than the surrounding fluid, it is subsequently acted on by buoyancy forces. Jets and plumes can be classified as either laminar or turbulent, with the difference between the two being described by a Reynolds number, as with pipe flow. Near the source of the flow, the flow of a jet or plume is controlled entirely by the primary initial conditions that include the mean velocity of the jet’s exit, the geometry of the exit, and the initial difference in density between the discharge and the surrounding, or ambient, fluid. Secondary initial conditions include the intensity of the exiting turbulence and the distribution of the velocity. Following Fischer et al. (1979), the factors of prime importance to jet dynamics can be defined as follows: 1. Mass flux —the mass of fluid passing a jet cross section per unit time
冕 (ρu)dA a
mass flux =
A
(5.71)
where A is the cross-sectional area of the jet and u the time-averaged velocity of the jet in the axial direction. 2. Momentum flux —amount of momentum passing a jet cross section per unit time
冕 (ρu )dA a
momentum flux =
2
A
(5.72)
3. Buoyancy flux —buoyant or submerged weight of the fluid passing a jet cross section per unit of time
冕 (g∆ρu)dA a
buoyancy flux =
A
(5.73)
where ∆ρ the difference in density between the surrounding fluid and the fluid in the jet. It is convenient to define g(∆ρ)/g g' as the effective gravitational acceleration. Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
ENVIRONMENTAL HYDRAULICS
Environmental Hydraulics 5.29
5.6.1 Simple Jets The two dimensional or plane jet issuing from a slot and the round jet issuing from a nozzle into a quiescent ambient fluid are among the simplest cases of jets that can be considered. These jets have been studied extensively, and there is a reasonable understanding of how they behave. The boundary between the ambient and jet fluids is sharp at any instant, and if a tracer were present in the jet fluid, time-averaged measurements would show a Gaussian distribution of tracer concentration (C) across the jet or 2 C exp–k y j x Cm
(5.74)
where the subscript m the value of C on the jet axis, x the distance along the jet axis, kj experimental coefficients, and y the transverse (or radial) distance from the jet axis. The Gaussian distribution also is valid for the time-averaged velocity profile across the jet provided that the measurement is taken downstream of the zone of established flow. In the case of a circular jet, the length of the zone of established flow is approximately 10 orifice diameters downstream. Downstream of the zone of established flow, the jet continues to expand and the mean velocity and tracer concentrations decrease. Within the zone of established flow, the velocity and concentration profiles are self-similar and can be described in terms of a maximum value (measured at the jet’s center line) and a measure of the width or, in the case of the velocity, distribution: y v (5.75) f vm bw where vm the value of v on the jet’s center line, y a coordinate transverse to the jet’s axis, and bw the value of x at which v is some specified fraction of vm (often taken as either 0.5 or 0.37; Fischer et al. 1979). The functional form of f in Eq, (5.75) is most often taken as Gaussian. Almost all the properties of turbulent jets that are important to engineers can be deduced from dimensional analysis combined with empirical data (Fischer et al. 1979). These results are summarized in Table 5.2.
5.6.2 Simple Plumes Because the simple plume has no initial volume or momentum flux (e.g., smoke rising from a fire), all variables must be a function of only the buoyancy flux (B), the vertical distance from the origin (y), and the viscosity of the fluid where ∆ρ B g o Q g'oQ ρ
(5.76)
and ∆ρo difference in density between the plume fluid and the ambient fluid and go’ apparent gravitational acceleration. Results similar to those for jets are summarized in Table 5.3, and the numerical constants given are from Chen and Rodi (1976).
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ENVIRONMENTAL HYDRAULICS
5.30
Chapter Five
TABLE 5.2
Summary of the Properties of Turbulent Jets
Parameter
Round Jet
Plane Jet
πD V0 4
b0 y0 V0
πD2V20 4
b0y0V20
Characteristic length scale lQ
Q0 兹M苶0苶
Q2 0 M0
Maximum time-averaged velocity Vm
l Q vm (7.0 0.1)Q M y
Maximum time-averaged tracer concentration Cm
l Cm Q (5.6 0.1) C0 y
Mean dilution
y Q (0.25 0.01) Q0 lQ
2
Initial volume flow rate Qo Initial momentum flux Mo
Ratio Cm/Cav
1.4 0.1
l Q vm (2.41 0.04)Q M y l Cm Q (2.38 0.04) C0 y y Q\Q0 (0.50 0.02) lQ 1.2 0.1
Source: After Fischer et al. 1979.
TABLE 5.3
Summary of Plume Properties
Parameter
Round Plume
Plane Plume
Maximum time-averaged velocity vm
(4.7 0.2)B1/3 y-1/3
1.66 B1/3
Maximum time-average tracer concentration Cm
(9.1 0.5)M B-1/3 y-5/3
2.38M B-1/3 B-1
Volume flux Q
(0.15 0.015)B1/3 y5/3
0.34 B1/3y
Ratio Cm/Cavg
1.4 0.2
0.81 0.1
Source: After Fischer et al. 1979.
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ENVIRONMENTAL HYDRAULICS
Environmental Hydraulics 5.31
REFERENCES Akiyama, J. and H. Stefan, “Onset of Underflow in Slightly Diverging Channels,” Journal of the Hydraulics Division, American Society of Civil Engineers113(HY7), 1987, pp. 825–844. Akiyama, J., and H. Stefan, “Plunging Flow into a Reservoir, Theory,” Journal of the Hydraulics Division, American Society of Civil Enginners, 110(HY4): 484–499, 1984. Anderson, E. R. “Energy Budget Studies, Part of Water Loss Investigations — Lake Hefner Studies.”, U.S. Geological Survey, Professional Paper No. 269, Washington, DC, 1954. Beeton, A. M., “Relationship Between Secchi Disk Readings and Light Penetration in Lake Huron,” American Fisheries Society Transactions, 87:73–79, 1958. Bender, M. D., G. E. Hauser, M. C. Shiao and W. D. Proctor, “BETTER: A Two-Dimensional Reservoir Water Quality Model, Technical Reference Manual and User’s Guide,” Report No. WR28-2-590-152, Tennessee Valley Authority, Engineering Laboratory, Norris, TN, 1990. Bohan, J. P., and J. L. Grace, Jr., “Selective Withdrawal from Man-Made Lakes: Hydraulics Laboratory Investigation,” Technical Report No. H-73-4, U.S. Army Waterways Experiment Station, Vicksburg, MS, 1973. Bowie, G. L., W. B. Mills, D. B. Porcella, C. L. Campbell, J. R. Pagenkopf, G. L. Rupp, K. L. Johnson, W. H. Chan, S. A. Gherini, and C. E. Chamberlin, “Rates, Constants, and Kinetics Formulations in Surface Water Quality Modeling”, 2nd ed. EPA/600/3–85/040, U.S. Environmental Protection Agency, Environmental Research Laboratory, Athens, GA 1985. Brown, L. C. and T. O. Barnwel, The Enhanced Stream Water Quality Models QUAL2E and QUAL2E-UNCAS: Documentation and User Manual,. EPA/600/3-87/007, U.S. Environmental Protection Agency, 1987. Chapra, S. C., and K. H. Reckhow, Engineering Approaches for Lake Management, Butterworth, Boston, 1983. Chen, C. J., and W. Rodi, A Review of Experimental Data of Vertical Turbulent Buoyant Jets, Hydraulic Research Report No. 193, Iowa Institute of Hydraulics, Iowa City, IA, 1976. Cole, T. M., and E. M. Buchak, CE-QUAL-W2: A Two Dimensional, Laterally Averaged, Hydrodynamic and Water Quality Model, Version 2.0, User Manual, Instruction Report No. EL–95–1, U.S. Army Engineer Waterways Experiment Station, Vicksburg, MS, 1995. Davis, J. E., J. P. Holland, M. L.Schneider, and S. C. Wilhelms, SELECT, A Numerical, One Dimensional Model for Selective Withdrawal, Technical Report, U.S. Army Engineer Waterways Experiment Station, Vicksburg, MS, 1985. Environmental Laboratory, CE-QUAL-R1: A Numerical One-Dimensional Model of Reservoir Water Quality, User’s Manual, Instruction Report No. E-82-1, U.S. Army Engineer Waterways Experiment Station, Vicksburg, MS, 1985. Fischer, H. B., “The Effects of Bends on Dispersion in Streams,” Water Resources Research, 5 496–506, 1969. Fischer, H. B., E. J. List, R. C. Y. Koh, J. Imberger, and N. H. Brooks, Mixing in Inland and Coastal Waters, Academic Press, New York, 1979. Ford, D. E., and M. C. Johnson, An Assessment of Reservoir Mixing Processes, Technical Report No E-86-7, U.S. Army Engineer Waterways Experiment Station, Vicksburg, MS, 1986. Ford, D. E., and M. C. Johnson, An Assessment of Reservoir Density Currents and Inflow Processes, Technical Report No. E-83-7, U.S. Army Engineer Waterways Experiment Station, Vicksburg, MS, 1983. Ford, D. E., and M. C. Johnson, “Field Observations of Density Currents in Impoundments,” in H. G. Stefan, ed. Proceedings Symposium on Surface Water Impoundments, ASCE, New York, 1981. French, R. H., Open-Channel Hydraulics, McGraw-Hill, New York, 1985. French, R. H., J. J. Cooper, and S. Vigg, “Secchi Disc Relationships,” AWRA, Water Resources Bulletin, 18 (1):121–123, 1982. Gill, A. E., “Appendix 3, Properties of Seawater,” in Atmosphere-Ocean Dynamics, Academic Press, New York, 1982, pp. 599–600.
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5.32
Chapter Five
Gu, R., S. C. McCutcheon, and P. F. Wang, “Modeling Reservoir Density Underflow and Interflow from a Chemical Spill in a River,” Water Resources Research, 32:695–705, 1996. Grace, J. L., Jr., "Selective Withdrawal Characteristics of Weirs: Hydraulics Laboratory Investigation," Technical Report H-71-4, U.S. Army Engineer Waterways Experiment Station, Vicksburg, MS, 1971. Hamon, R. W., L. L. Weiss, and W. T. Wilson, “Insulation as an Empirical Function of Daily Sunshine Duration,” Monthly Weather Review, 82(6), 1954. Hebbert, J., J. Imberger, I. Loh, and J. Patterson, “Collie River Flow into Wellington Reservoir,”, Journal of the Hydraulics Division, American Society of Civil Engineers 105(HY(5): 533–545, 1979. Hutchinson, G. E., A Treatise on Limnology: Vol. 1. Geography, Physics and Chemistry, John Wiley Sons, New York, 1957. Imberger, J., “Selective Withdrawal: A Review,” in T, Carstens and T. McClimans, eds., Proceedings of the 2nd International Symposium on Stratified Flow, International Association for Hydraulic Research, Tapir, Trondheim, Norway, 1980. Imberger, J. and J. C. Patterson, “Dynamic Reservoir Simulation Model — DYRESM:5,” in H.B. Fischer, ed., Transport Models for Inland and Coastal Waters, Academic Press, Orlando, FL, 1981, pp. 310–561. Imberger, J., R. T. Thompson, and C. Fandry, “Selective Withdrawal from a Finite Rectangular Tank,” Journal of Fluid Mechanics, 78:489–512, 1976. Jain, S. C., “Plunging Phenomena in Reservoirs,” in H. G. Stefan, ed., Proceedings of the Symposium on Surface Water Impoundments, American Society of Civil Engineers, new York, 1981. Jobson, H. E., “Thermal Modeling of Flow in the San Diego Aqueduct, California, and Its Relation to Evaporation,” Professional Paper No. 1122, U.S. Geological Survey, Washington, DC, 1980. Jobson, H. E., and T. N. Keefer, “Modeling Hight Transient Flow, Mass, and Heat Transport in the Chattahoochee River Near Atlanta, Georgia,” Professional Paper No. 1136, U.S. Geological Survey, Washington, DC, 1979. Kent, R. E. and D. W. Pritchard, “A Test of Mixing Length Theories in a Coastal Plain Estuary,” Journal of Marine Research, 1:456–466, 1957. Kao, T. W., “Principal State of Wake Collapse in a Stratified Fluid, Two-Dimensional Theory,” Physics of Fluids, 19:1071–1074, 1976. Langmuir, I., “Surface Motion of Water Induced by Wind,” Science, 87:119–123, 1938. Lau, Y. L., and B. G. Krishnappen, “Transverse Dispersion in Rectangular Channels,” ASCE, Journal of the Hydraulics Division, American Society of Civil Engineers 103(HY10):1173–1189, 1977. Lerman, A., ed, Lakes: Chemistry, Geology, Physics, Springer-Verlag, New York, 1978. Martin, J. L. and S. C.McCutcheon, Hydrodynamics and Transport for Water Quality Modeling, CRC Press–Boca Raton, FL, 1999. McCutcheon, S. C., Water Quality Modeling: Vol. 1. Transport and Surface Exchange in Rivers, CRC Press, Boca Raton, FL, 1989. McCutcheon, S. C., “Discussion with Harvey Jobson on Windspeed Coefficients for Stream Temperature Modeling,” Memorandum, U.S. Geological Survey, NSTL Station, MS, March 29, 1982. Mills, W. B., J. D. Dean, D. B. Porcella, S. A. Gherini, R. J. M. Hudson, W. E. Frick, G. L. Rupp, and G. L. Bowie, “Water Quality Assessment: A Screening Procedure for Toxic and Conventional Pollutants,” EPA–600/6–82–004b, U.S. Environmental Protection Agency, Athens, GA, 1982. Mortimer, C. H., "Lake Hydrodynamics," International Association of Applied Limnology Mitteilungen, 20:124-197, 1974. Munk, W., and E. R. Anderson, “Notes on a Theory of the Thermocline,” Journal of Marine Research, 7:276–295, 1948.
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Environmental Hydraulics 5.33 Norton, W. R., L. A. Roesner, and G. T. Orlob, Mathematical Models for Predicting Thermal Changes in Impoundments, EPA Water Pollution Control Research Series, U.S. Environmental Protection Agency, Washington, DC, 1968. Rossby, G. G. and R. Montgomery, “The Layers of Frictional Influence in Wind and Ocean Currents,” Papers Phy. Oc. Meth., III(3), 1935. Ryan, P. J. and D. R. F. Harleman, An Analytical and Experimental Study of Transient Cooling Pond Behavior. Technical Report No. 161, R.M. Parsons Laboratory, Massachusetts Dsitributate of Technology, 1973. Safaie, B. “Mixing of Buoyant Surface Jet over Sloping Bottom,” ASCE, Journal Waterway, Port, Coastal and Ocean Engineering DIvision, 105(WW4): 357–373, 1979. Savage, S. B. and J. Brimberg, “Analysis of Plunging Phenomenon of Density Currents in Reservoirs,” IAHR, 00, 13(2): 187–204, 1973. Shanahan, P., “Water Temperature Modeling: A Practical Guide,” in Proceedings of the U.S. Environmental Protection Agency Stormwater and Water Quality Users Group Meeting, LAMR, jounal, Hydraulic, Research, , April 1984. Shuttleworth, W. J., “Evaporation,” in D. R. Maidment, ed., Handbook of Hydrology, McGrawHill, New York, 1993. Singh, B., and C. R. Shah, “Plunging Phenomenon of Density Currents in Reservoirs,” La Houille Blanche, 26(1): 59-64, 1971. Smith, I. R., Turbulence in Lakes and Rivers, Scientific Publication No. 29, Freshwater Biological Association, Ambleside, Cumbria, UK, 1975. Smith, J. A., “Precipitation,” in D. R. Maidment, ed., Handbook of Hydrology, McGraw-Hill, New York, 1993. Smith, D. R., S. C. Wilhelms, J. P. Holland, M. S. Dortch, and J. E. Davis, Improved Description of Selective Withdrawal Through Point Sinks, Technical Report No, E–87–2, U.S. Army Engineer Waterways Experiment Station, Vicksburg, MS, 1985. Stefan, H. G., R. B. Ambrose, Jr., and M. S. Dortch, "Formulation of Water Quality Models for Streams, Lakes and Reservoirs: Modeler’s Perspective," Miscellaneous Paper E-89-1, .S. Army Engineer Waterways Experiment Station, Vicksburg, MS., 1989. Sundaram, T. R., "A Theoretical Model for Seasonal Thermocline Cycle of Deep Temperate Lakes," Proceedings of the 16th Conference of Great Lakes Research, 1973, pp. 1009–1025. Sverdrup, H. U., M. W. Johnson, and R. H. Fleming, The Oceans, Prentice–Hall, Englewood Cliffs, NJ, 1942. Swinbank, W. C., “Longwave Radiation from Clear Skies.” Quarterly Journal of the Royal Meteorological Society, 89: 339-348, 1963. Taylor, G. I., " The Dispersion of Matter in a Turbulent Flow Through a Pipe," Proceedings of the Royal Society of London, Series A, 223: 446-468, 1954. Thackston, E. L., Effect of Geographical Variation on Performance of Recirculating Cooling Ponds, EPA-660/2-74-085, U.S. Environmental Protection Agency, Corvallis, OR, 1974. Tennessee Valley Authority,(TVA) “Heat and Mass Transfer Between a Water Surface and the Atmosphere.” TN Report No. 14, TVA Water Resources Research Engineering Laboratory, Norris, TN, 1972. Turner, J. S., Buoyancy Effects in Fluids, Cambridge University Press, Cambridge, UK, 1973. Wetzel, R. G., Limnology, Saunders College Publishing, Philadelphia, PA, 1975, and 1983. Wunderlich, W. O. and R. A. Elder, “Mechanics of Flow Through Man-Made Lakes,” in Man-Made Lakes: Their Problems and Environmental Effects, W. C. Ackerman, G. F. White, and E. B. Worthington, eds., American Geophysical Union, Washington, DC, 1973.
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Source: HYDRAULIC DESIGN HANDBOOK
CHAPTER 6
SEDIMENTATION AND EROSION HYDRAULICS Marcelo H. García Department of Civil and Environmental Engineering University of Illinois at Urbana-Champaign Urbana, IL
6.1 INTRODUCTION Since the beginning of mankind, sedimentation processes have affected water supplies, irrigation, agricultural practices, flood control, river migration, hydroelectric projects, navigation, fisheries, and aquatic habitat. In the last few years, sediment also has been found to play an important role in the transport and fate of pollutants; thus, sedimentation control has become an important issue in water quality management. Toxic chemicals can become attached to, or adsorbed by, sediment particles and then be transported to and deposited in other areas. By studying the quantity, quality, and characteristics of sediment in rivers and streams, scientists and engineers can determine the sources of the sediment and evaluate the impact of pollutants on the aquatic environment. In the United States, sedimentation control is a multibillion-dollar issue. For example, approximately $500 million are spent every year to dredge waterways and harbors for navigation purposes. Most of the dredged sediment is the result of substantial soil erosion in watersheds. Estimates by the U.S. Department of Agriculture indicate that annual offside costs of sediment derived from copland erosion are on the order of $2 billion to $6 billion, with an additional $1 billion arising from loss in compared productivity. The sediment cycle starts with the process of erosion, where by particles or fragments are weathered from rock material. Action by water, wind, glaciers, and plant and animal activities all contribute to the erosion of the earth’s surface. Fluvial sediment is the term used to describe the case where water is the key agent for erosion. Natural, or geologic, erosion takes place slowly, over centuries or millennia. Erosion that occurs as a result of human activity may take place much faster. It is important to understand the role of each cause when studying sediment transport. Any material that can be dislodged is ready to be transported. The transportation process is initiated on the land surface when raindrops result in sheet erosion. Rills, gullies, streams, and rivers then act as conduits for the movement of sediment. The greater the discharge, or rate of flow, the higher the capacity for sediment transport. The final process in the cycle is deposition. When there is not enough energy to transport the sediment, it comes to rest. Sinks, or depositional areas, can be visible as newly deposited material on a floodplain, on bars and islands in a channel, and on deltas. Considerable deposition occurs that may not be apparent, as on lake and river beds. A knowledge of sediment dynamics is an integral part of understanding the aquatic ecosystem. This chapter presents fundamental aspects of the erosion, transport, and deposition of sediment in the environment. The emphasis is on the hydraulics of bedload and suspend6.1 Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
SEDIMENTATION AND EROSION HYDRAULICS
6.2
Chapter Six
ed load transport in rivers, with the goal of establishing the background needed for sedimentation engineering. Because of their relevance, the hydraulics of both reservoir sedimentation and turbidity currents also is considered. Emphasis is placed on noncohesive sediment transport, where the material involved can be silt, sand, or gravel. When possible, the behavior of both uniform-sized material and sediment mixtures is analyzed. Although such topics as cohesive sediment transport, debris and mud flows, alluvial fans, river meandering, and sediment transport by wave action are not discussed here, it is hoped that the material covered in this chapter will provide a firm foundation to tackle problems in those. For more information on sediment transport and sedimentation engineering, readers are referred to Allen (1985), Ashworth et al. (1996), Bogardi (1974), Bouvard (1992), Carling and Dawson (1996), Chang (1988), Coussot (1997), Fredsøe and Deigaard (1992), Garde and Ranga Raju (1985), Graf (1971), Jansen et al. (1979), Julien (1992), Mehta (1986), Mehta et al. (1989a, 1989b), Morris and Fan (1998), Nakato and Ettema (1996), National Research Council (1996), Nielsen (1992), National Research council (1996), Parker and Ikeda (1989), Raudkivi (1990, 1993), Renard et al. (1997), Sieben (1997), Simons and Senturk (1992), Sloff (1997), van Rijn (1997), Yalin (1972, 1992), Yang (1996), and Wan and Wang (1994).
6.2 HYDRAULICS FOR SEDIMENT TRANSPORT 6.2.1 Flow Velocity Distribution Consider a steady, turbulent, uniform, open-channel flow having a mean depth H and a mean flow velocity U (Fig. 6.1). The channel is extremely wide and its bottom has a mean slope S and a surface roughness that can be characterized by an effective height ks (Brownlie, 1981b). When the bottom of the channel is covered with sediment having a mean size or diameter D, the roughness height ks will be proportional to that diameter. Because of the weight of the water, the flow exerts on the bottom a tangential force per unit bed area known as the bed shear stress τb, which can be expressed as: τb ρgHS
(6.1)
where ρ is the water density and g is the gravitational acceleration. With the help of the boundary shear stress, it is possible to define the shear velocity u* as
FIGURE 6.1 Definition diagram for open-channel flow over an erodible bed.
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SEDIMENTATION AND EROSION HYDRAULICS
Sedimentation and Erosion Hydraulics 6.3
u* τ/ρ b
(6.2)
The shear velocity, and thus the boundary shear stress, provides a direct measure of the intensity of flow and its ability to entrain and transport sediment particles. The size of the sediment particles on the bottom determines the surface roughness, which in turn affects the flow velocity distribution and its sediment transport capacity. Since flow resistance and sediment transport rates are interrelated, the ability to determine the role played by the bottom roughness is important. Research has shown (Schlichting, 1979) that the flow velocity distribution is well represented by: 1 (6.3) uu lnz const. κ * where u is the time-averaged flow velocity at distance z above the bed and κ is known as Von Karman’s constant and is equal to 0.4. For obvious reasons, the above law is known as the logarithmic law of the wall. It strictly applies only in a thin layer near the bed. It is empirically found to apply as a reasonable approximation throughout most of the flow in many rivers. If the bottom boundary is sufficiently smooth (a condition rarely satisfied in rivers), turbulence will be drastically suppressed in an extremely thin layer near the bed. In this region, a linear velocity profile will hold: u*z uu v *
(6.4)
where ν is the kinematic viscosity of water. This law merges with the logarithmic law near z δv, where (6.5) δv 11.6 ν u* denotes the height of the viscous sublayer. In the logarithmic region, the constant of integration introduced above has been evaluated from data to yield 1 u z uu ln * 5.5 κ ν *
(6.6)
Most boundaries in river flow are rough. Let ks denote an effective roughness height. If ks/δv 1, then no viscous sublayer will exist. The corresponding logarithmic velocity profile is given by 1 ln z 8.5 1 ln 30 z uu κ κ ks ks *
(6.7)
As noted above, this relation often holds as a first approximation throughout the flow in a river. It is by no means exact. The conditions ks/δν » 1 for rough turbulent flow and ks/δν « 1 for smooth turbulent flow can be rewritten to indicate that u*ks/ν should be much larger than 11.6 for turbulent rough flow and much smaller than 11.6 for turbulent smooth flow. A composite form that represents both ranges, as well as the transitional range between them, can be written as 1 ln z B uu κ s ks *
(6.8)
with Bs as a function of Re* u*ks/ν, which can be estimated with
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SEDIMENTATION AND EROSION HYDRAULICS
6.4
Chapter Six
Bs 8.5 [2.5 ln(Re*) 3]e0.127[ln(Re*)]2
(6.9)
as proposed by Yalin (1992).
6.2.2 Relations for Channel Resistance Most river flows are indeed hydraulically rough. Equation (6.7) can be used to obtain an approximate expression for depth-averaged velocity U that is reasonably accurate for many flows. Using the following integral: H
U 1 udz H 0
(6.10)
but changing the lower limit slightly to avoid the fact that the logarithmic law is singular at z 0, the following result is obtained: H
U 1 1 z ln 8.5 dz u* H ks κ ks
(6.11)
U 1 H 1 H ln 6 ln 11 u* κ ks κ ks
(6.12)
or, performing the integration
This relation is known as Keulegan's resistance relation for rough flow. An approximation to Keulegan's relation is the Manning-Strickler power form H 1/6 U 8 u* ks
(6.13)
Between Eqs. (6.2) and (6.12), a resistance relation can be found for bed shear stress: τb ρCf U 2
(6.14)
where the friction coefficient Cf is given by –2 Cf 1 ln 11H κ ks
(6.15)
If Eq. (6.13) is used instead of Eq. (6.12), the friction coefficient takes the form 1/6 –2 Cf 8 H ks
(6.16)
It is useful to show the relationship between the friction coefficient Cf and the roughness parameters in open-channel flow relations commonly used in practice. Between Eqs. (6.1) and (6.14), the following form of Chezy's law can be derived: U CcH1/2S1/2
(6.17)
where the Chezy coefficient Cc is given by the relation
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Sedimentation and Erosion Hydraulics 6.5 g 1/2 Cc Cf
(6.18)
A specific evaluation of Chezy's coefficient can be obtained by substituting Eq. (6.15) into Eq. (6.18). It is seen that the coefficient is not constant but varies as the logarithm of H/ks. A logarithmic dependence is typically a weak one, partially justifying the common assumption that Chezy's coefficient in Eq. (6.17) is a constant. Substituting Eq. (6.16) into Eqs. (6.17) and (6.18), Manning's law is obtained: U = 1 H2/3S1/2 n
(6.19)
k 1/6 n = s 8g1/2
(6.20)
where Manning's n is given by
The above relation is often called the Manning-Strickler form of Manning's n.
6.2.3 Fixed-Bed and Movable-Bed Roughness It is clear that to use the above relations for channel flow resistance, a criterion for evaluating ks is necessary. Nikuradse (1933) proposed the following criterion: Suppose a rough surface is subjected to a flow. The equivalent roughness ks of that surface is equal to the diameter of sand grains that, when glued uniformly to a completely smooth wall and then subjected to the same external conditions, yields the same velocity profile. Nikuradse used sand glued to the inside of pipes to conduct this evaluation. Extending Nikuradse's concept of equivalent grain roughness to the case of rivers and streams, ks can be assumed to be proportional to a representative sediment size Dx, ks = αsDx
(6.21)
Suggested values of αs, which have appeared in the literature, are listed in Table 6.1 (Yen, 1992). Different sizes of sediment have been suggested for Dx in Eq. (6.21). Statistically, D50 (the grain size for which 50% of the bed material is finer) is most readily available and meaningful. Physically, a representative size larger than D50 is more meaningful to estimate
TABLE 6.1
Ratio of Nikuradse Equivalent Roughness Size and Sediment Size for Rivers.
Investigator
Measure of Sediment Size, Dx
αs = ks /Dx
Ackers and White (1973) Strickler (1923) Keulegan (1938) Meyer-Peter and Muller (1948) Thompson and Campbell (1979) Hammond et al. (1984) Einstein and Barbarossa (1952) Irmay (1949) Engelund and Hansen (1967)
D35 D50 D50 D50 D50 D50 D65 D65 D65
1.23 3.3 1 1 2.0 6.6 1 1.5 2.0
Lane and Carlson (1953)
D75
3.2
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6.6
Chapter Six
TABLE 6.1.
(Continued)
Investigator
Measure of Sediment Size, Dx
Gladki (1979) Leopold et al. (1964) Limerinos (1970) Mahmood (1971) Hey (1979), Bray (1979) Ikeda (1983) Colosimo et al. (1986) Whiting and Dietrich (1990) Simons and Richardson (1966) Kamphuis (1974) van Rijn (1982)
D80 D84 D84 D84 D84 D84 D84 D84 D85 D90 D90
αs = ks /Dx 2.5 3.9 2.8 5.1 3.5 1.5 36 2.95 1 2.0 3.0
SOURCE: Adapted from Yen (1992)
flow resistance because of the dominant effect by large sediment particles. In flow over a geometrically smooth, fixed boundary, the apparent roughness of the bed ks can be computed using Nikuradse's approach. However, once the transport of bed material has been instigated, the characteristic grain diameter and the thickness of the viscous sublayer no longer provide the relevant length scales. The characteristic length scale in this situation is the thickness of the layer where the sediment particles are being transported by the flow, usually referred to as the bedload layer. Once the bed shear stress τb exceeds the critical shear stress for particle motion τc, the apparent bed roughness ka can be estimated as follows (Smith and McLean, 1997): (τb τc) ka α0 ks (ρs ρ)g
(6.22)
where α0 26.3, ks is Nikuradse's fixed-bed roughness, and ρs is the bed sediment density. This approach is particularly suitable for sand bed rivers. Under intense sediment transport conditions, bedforms, such as dunes, can develop. In this situation, the apparent roughness also will be influenced by the form drag caused by the presence of bedforms. Nikuradse's approach is valid only for grain-induced roughness. Methods for flow resistance in the presence of both bedforms and grain roughness are presented later.
6.3 SEDIMENT PROPERTIES 6.3.1 Rock Types The solid phase of the problem embodied in sediment transport can be any granular substance. In engineering applications, however, the granular substance in question typically consists of fragments ultimately derived from rocks–hence the name sediment transport. The properties of these rock-derived fragments, taken singly or in groups of many particles, all play a role in determining the transportability of the grains under fluid action. The
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Sedimentation and Erosion Hydraulics 6.7
important properties of groups of particles include porosity and size distribution. The most common rock type one is likely to encounter in the river or coastal environment is quartz. Quartz is a highly resistant rock and can travel long distances or remain in place for long periods without losing its integrity. Another highly resistant rock type that is often found together with quartz is feldspar. Other common rock types include limestone, basalt, granite, and more esoteric types, such as magnetite. Limestone is not a resistant rock; it tends to abrade to silt rather easily. Silt-sized limestone particles are susceptible to solution unless the water is buffered sufficiently. As a result, limestone typically is not a major component of sediments at locations distant from its source. On the other hand, it often can be the dominant rock type in mountain environments. Basaltic rocks tend to be heavier than most rocks composing the earth’s crust and typically are brought to the surface by volcanic activity. Basaltic gravels are relatively common in rivers that derive their sediment supply from areas subjected to vulcanism in recent geologic history. Basaltic sands are much less common. Regions of weathered granite often provide copious supplies of sediment. Although the particles produced by weathering are often in the granule size, they often break down quickly to sand size. Sediments in the fluvial or coastal environment in the size range of silt, or coarser, are generally produced by mechanical means, including fracture or abrasion. The clay minerals, on the other hand, are produced by chemical action. As a result, they are fundamentally different from other sediments in many ways. Their ability to absorb water means that the porosity of clay deposits can vary greatly over time. Clays also display cohesivity, which renders them more resistant to erosion.
6.3.2 Specific Gravity The specific gravity of sediment is defined as the ratio between the sediment density ρs and the density of water ρ. Some typical specific gravities for various natural and artificial sediments are listed in Table 6.2.
6.3.3 Size Herein, the notation D is used to denote sediment size, the typical units of which are millimeters (mm) for sand and coarser material or microns (µ) for clay and silt. Another standard way of classifying grain sizes is the sedimentological Φ scale, according to which TABLE 6.2 Specific Gravity of Rock Types and Artificial Material Rock type or material quartz limestone basalt magnetite plastic coal walnut shells
Specific gravity ρs /ρ 2.60 2.70 2.60 2.80 2.70 2.90 3.20 3.50 1.00 1.50 1.30 1.50 1.30 1.40
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6.8
Chapter Six
D 2Φ
(6.23)
Taking the logarithm of both sides, it is seen that 1n(D) Φ log2(D) 1n(2)
(6.24)
Note that the size Φ 0 corresponds to D 1 mm. The usefulness of the Φ scale will become apparent upon a consideration of grain size distributions. The minus sign has been inserted in Eq. (6.24) simply as a matter of convenience to sedimentologists, who are more accustomed to working with material finer than 1 mm than they are with coarser material. The reader should always recall that larger Φ implies finer material. The Φ scale provides a simple way of classifying grain sizes into the following size ranges in descending order: boulders, cobbles, gravel, sand, silt, and clay. (Table 6.3). Note that the definition of clay according to size (D 2) does not always correspond to the definition of clay according to mineral. That is, some clay-mineral particles can be coarser than this limit, and some silt-sized particles produced by grinding can be finer than that. In general, however, the effect of viscosity makes it difficult to grind up particles in water to sizes finer than 2. In practical terms, there are several ways to determine grain size. The most popular way for grains ranging from Φ 4 to Φ 4 (0.0625 to 16 mm) is with the use of sieves. Each sieve has a square mesh, the gap size of which corresponds to the diameter of the largest sphere that would fit through it. Thus, the grain size D so measured corresponds exactly to the diameter only in the case of a sphere. In general, the sieve size D corresponds to the smallest sieve gap size through which a given grain can be fitted. For coarser grain sizes, it is customary to approximate the grain as an ellipsoid. Three lengths can be defined. The length along the major (longest) axis is denoted as a, the length along the intermediate axis is denoted as b, and the length along the minor (smallest) axis is denoted as c. These lengths are typically measured with a caliper. The value b is then equated to grain size D. For grains in the silt and clay sizes, many methods (hydrometer, sedigraph, and so forth) are based on the concept of equivalent fall diameter. That is, the terminal fall velocity vs of a grain in water at a standard temperature is measured. The equivalent fall diameter D is the diameter of the sphere having exactly the same fall velocity under the same conditions. Sediment fall velocity is discussed in more detail below. A variety of other more recent methods for sizing fine particles rely on blockage of light beams. The blocked area can be used to determine the diameter of the equivalent circle: i.e., the projection of the equivalent sphere. It can be seen that all the above methods can be expected to operate consistently as long as grains shape does not deviate too greatly from a sphere. In general, this turns out to be the case. There are some important exceptions, however. At the fine end of the spectrum, mica particles tend to be platelike; the same is true of shale grains at the coarser end. Comparison with a sphere is not necessarily an especially useful way to characterize grain size for such materials.
6.3.4 Size Distribution Any sample of sediment normally contains a range of sizes. An appropriate way to characterize these samples is by grain size distribution. Consider a large bulk sample of sediment of given weight. Let pf(D)—or pf(Φ)—denote the fraction by weight of material in the sample of material finer than size D(Φ). The customary engineering representation of
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9 8 8 7 7 6 6 5 5 4 4 3 3 2 2 1 1 0 01 12 23 34 45 56 67 78 89
4,096 2,048 2,048 1,024 1,024 512 512 256 256 128 128 64 64 32 32 16 16 8 84 42 2.000 1.000 1.000 0.500 0.500 0.250 0.250 0.125 0.125 0.062 0.062 0.031 0.031 0.016 0.016 0.008 0.008 0.004 0.004 0.0020 0.0020 0.0010 0.0010 0.0005 0.0005 0.00024
SOURCE: Adapted from Vanoni, 1975.
Very large boulders Large boulders Medium boulders Small boulders Large cobbles Small cobbles Very coarse gravel Coarse gravel Medium gravel Fine gravel Very fine gravel Very coarse sand Coarse sand Medium sand Fine sand Very fine sand Coarse silt Medium silt Fine silt Very fine silt Coarse clay Medium clay Fine clay Very fine clay
Φ
Size Range
Millimeters
Sediment Grade Scale
Class Name
TABLE 6.3
2,000 1,000 1,000 500 500 250 250 125 125 62 62 31 31 16 16 8 84 42 21 1 0.5 0.5 0.24
Microns 160 80 80 40 40 20 20 10 10 5 5 2.5 2.5 1.3 1.3 0.6 0.6 0.3 0.3 0.16 0.16 0.08
Inches
2 1/2 5 9 16 32 60 115 250
5 10 18 35 60 120 230
Approximate Sieve Mesh Openings per Inch Tyler U.S. standard
SEDIMENTATION AND EROSION HYDRAULICS
Sedimentation and Erosion Hydraulics 6.9
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6.10
Chapter Six
the grain size distribution consists of a plot of pf 100 (percentage finer) versus log10(D): that is, a semilogarithmic plot is used. The same size distribution plotted in sedimentological form would involve plotting pf100 versus Φ on a linear plot. The size distribution pf(Φ) and size density p(Φ) by weight can be used to extract useful statistics concerning the sediment in question. Let x denote some percentage, say 50%; the grain size Φx denotes the size such that x percent of the weight of the sample is composed of finer grains. That is, Φx is defined such that pf (x) x (6.25) 100 It follows that the corresponding grain size of equivalent diameter is given by Dx, where Dx 2 Φx
(6.26)
The most commonly used grain sizes of this type are the median size D50 and the size D90: i.e., 90% of the sample by weight consists of finer grains. The latter size is especially useful for characterizing bed roughness. The density p(Φ) can be used to extract statistical moments. Of these, the most useful are the mean size Φm and the standard deviation σ. These are given by the relations.
Φm Φp(Φ)dΦ;
σ2 (Φ Φm)2p(Φ)DΦ
(6.27a, b)
The corresponding geometric mean diameter Dg and geometric standard deviation σg are given as Dg 2Φm;
σg 2σ
(6,28a,b)
Note that for a perfectly uniform material, σ 0 and σg 1. As a practical matter, a sediment mixture with a value of σg less than 1.3 is often termed well sorted and can be treated as a uniform material. When the geometric standard deviation exceeds 1.6, the material can be said to be poorly sorted (Diplas and Sutherland, 1988). In fact, one never has the continuous function p(Φ) with which to compute the moments of Eqs. (6.27a, and b). Instead, one must rely on a discretization. To this end, the size range covered by a given sample of sediment is discretized using n intervals bounded by n 1 grain sizes Φ1, Φ2,…, Φn 1 in ascending order of Φ. The following definitions are made from i 1 to n: Φ i 1(Φi Φi1) 2
(6.29a)
pi pf(Φi) pf (Φi1)
(6.29b)
Eqs. (6.27a and b) now discretize to
n
Φm
i1
n
i pi Φ
σ2
(Φ i Φm)2pi
(6.30)
i1
In some cases, especially when the material in question is sand, the size distribution can be approximated as gaussian on the Φ scale (i.e., log-normal in D). For a perfectly Gaussian distribution, the mean and median sizes coincide: Φm Φ50 1(Φ84 Φ16) 2
(6.31)
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Furthermore, it can be demonstrated from a standard table of the Gauss distribution that the size Φ displaced one standard deviation larger that Φm is accurately given by Φ84; by symmetry, the corresponding size that is one standard deviation smaller than Φm is Φ16. The following relations thus hold: 1(Φ84 Φ16) 2
(6.32a)
Φm 1(Φ84 Φ16) 2
(6.32b)
Rearranging the above relations with the aid of Eqs. (6.28a and b) and Eqs. (6.31 and 6.32a), D84 1/2 σg (6.33a) D16
Dg (D84D16)1/2
(6.33b)
It must be emphasized that the above relations are exact only for a gaussian distribution in Φ. This is not often the case in nature. As a result, it is strongly recommended that Dg and σg be computed from the full size distribution via Eqs. (6.30a and b) and (6.28a and b) rather than the approximate form embodied in the above relations.
6.3.5 Porosity The porosity p quantifies the fraction of a given volume of sediment that is composed of void space. That is, volume of voids
p volume of total space If a given mass of sediment of known density is deposited, the volume of the deposit must be computed, assuming that at least part of it will consist of voids. In the case of well-sorted sand, the porosity often can take values between 0.3 and 0.4. Gravels tend to be more poorly sorted. In this case, finer particles can occupy the spaces between coarser particles, thus reducing the void ratio to as low as 0.2. Because so-called open-work gravels are essentially devoid of sand and finer material in their interstices, they may have porosities similar to sand. Freshly deposited clays are notorious for having high porosities. As time passes, the clay deposit tends to consolidate under its own weight so that porosity slowly decreases. The issue of porosity becomes of practical importance with regard to salmon spawning grounds in gravel-bed rivers, for example (Diplas and Parker, 1985). The percentage of sand and silt contained in the sediment is often referred to as the percentage of fines in the gravel deposit. When this fraction rises above 20 or 26 percent by weight, the deposit is often rendered unsuitable for spawning. Salmon bury their eggs within the gravel, and a high fines content implies a low porosity and thus reduced permeability. The flow of groundwater necessary to carry oxygen to the eggs and remove metabolic waste products is impeded. In addition, newly hatched fry may encounter difficulty in finding enough pore space through which to emerge to the surface. All the above factors dictate lowered survival rates. Chief causes of elevated fines in gravel rivers include road building and clear-cutting of timber in the basin.
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6.12
Chapter Six
6.3.6 Shape Grain shape can be classified in a number of ways. One of these, the Zingg classification scheme, is illustrated here (Vanoni, 1975). According to the definitions introduced earlier, a simple way to characterize the shape of an irregular clast (stone) is by lengths a, b, and c of the major, intermediate, and minor axes, respectively. If the three lengths are equal, the grain can be said to be close to a sphere in shape. If a and b are equal but c is much larger, the grain should be rodlike. Finally, if c is much smaller than b, which in turn, is much larger than a, the resulting shape should be bladelike.
6.3.7 Fall Velocity A fundamental property of sediment particles is their fall velocity. The relation for terminal fall velocity in quiescent fluid vs can be presented as
1 Rf 4 3 CD(Rp)
1/2
(6.34)
where s Rf v R gD
(6.35a)
vD Rp s v
(6.35b)
and the functional relation CD CD(Rp) denotes the drag curve for spheres. This relation is not particularly useful because it is not explicit in vs; one must compute fall velocity by trial and error. One can use the equation for CD given below CD 24 (1 0.152Rp1/2 0.0151Rp) Rp
(6.36)
and the definition R gD D Rep (6.37) to obtain an explicit relation for fall velocity in the form of Rf versus Rep. In Fig. 6.2, the ranges for silt, sand, and gravel are plotted for 0.01 cm2/s (clear water at 20ºC) and R 1.65 (quartz). A good summary of relations for terminal fall velocity for the case of nonspherical (natural) particles can be found in Dietrich (1982), who also proposed the following useful fit: Rf exp{b1 b2ln(Rep) b3[ln(Rep)]2 b4[ln(Rep)]3 b5[ln(Rep)]4}
(6.38)
where b1 2.891394, b2 0.95296, b3 0.056835, b4 0.002892, and b5 0.000245 6.3.8 Relation Between Size Distribution and Stream Morphology The study of sediment properties and, in particular, size distribution is most relevant to the context of stream morphology. The following discussion points out some of the more interesting issues.
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FIGURE 6.2 Sediment fall velocitydiagram
Sedimentation and Erosion Hydraulics 6.13
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6.14
Chapter Six
In Fig. 6.3, several size distributions from the sand-bed Kankakee River in Illinois, are shown (Bhowmik et al., 1980). The characteristic S shape suggests that these distributions might be approximated by a gaussian curve. The median size D50 falls near 0.3 to 0.4 mm. The distributions are tight, with a near absence of either gravel or silt. For practical purposes, the material can be approximated as uniform. In Fig. 6.4, several size distributions pertaining to the gravel-bed Oak Creek in Oregon, are shown (Milhous, 1973). In gravel-bed streams, the surface layer (“armor” or “pavement”) tends to be coarser than the substrate (identified as “subpavement” in the figure). Whether the surface or substrate is considered, it is apparent that the distribution ranges over a much wider range of grain sizes than is the case in Fig. 6.3. More specifically, in
FIGURE 6.3 Particle size distribution of bed materials in Kankakee River, Illinois. (Bhowmik et al., 1980)
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FIGURE 6.4 Size distribution of bed material samples in Oak Creek. Oregon. Source: (Milhous, 1973)
Sedimentation and Erosion Hydraulics 6.15
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6.16
Chapter Six
the distributions of the sand-bed Kankakee River, Φ varies from about 0 to about 3, whereas in Oak Creek, Φ varies from about 8 to about 3. In addition, the distribution of Fig. 6.4 is upward-concave almost everywhere and thus deviates strongly from the gaussian distribution. These two examples provide a window toward generalization. A river can be loosely classified as sand-bed or gravel-bed according to whether the median size D50 of the surface material or substrate is less than or greater than 2 mm. The size distributions of sandbed streams tend to be relatively narrow and also tend to be S shaped. The size distributions of gravel-bed streams tend to be much broader and to display an upward-concave shape. Of course, there are many exceptions to this behavior, but it is sufficiently general to warrant emphasis. More evidence for this behavior is provided in Fig. 6.5. Here, the grain size distributions for a variety of stream reaches have been normalized using the median size D50. Four sand-bed reaches are included with three gravel-bed reaches. All the sand-bed distributions are S shaped, and all have a lower spread than the gravel-bed distributions. The standard deviation is seen to increase systematically with increasing D50(White et al., 1973). The three gravel-bed size distributions differ systematically from the sand-bed distributions in a fashion that accurately reflects Oak Creek (Fig. 6.4). The standard deviation in all cases is markedly larger than any of the sand-bed distributions, and the distributions
FIGURE 6.5 Dimensionless grain-size distribution for different rivers (White et al., 1973)
are upward-concave except perhaps near the coarsest sizes.
6.4 THRESHOLD CONDITION FOR SEDIMENT MOVEMENT
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Sedimentation and Erosion Hydraulics 6.17
When a granular bed is subjected to a turbulent flow, virtually no motion of the grains is observed at some flows, but the bed is mobilized noticeably at other flows. Factors that affect the mobility of grains subjected to a flow are summarized below:
grain placement
randomness
turbulence
forces on grain
fluid lift
mean & turbulent drag
gravity
In the presence of turbulent flow, random fluctuations typically prevent the clear definition of a critical, or threshold condition for motion: The probability for the movement of a grain is never precisely zero (Lavelle and Mofjeld, 1987). Nevertheless, it is possible to define a condition below which movement can be neglected for many practical purposes.
6.4.1 Granular Sediment on a Stream Bed Figure 6.6 is a diagram showing the forces acting on a grain in a bed of other grains. When critical conditions exist and the grain is on the verge of moving, the moment caused by the critical shear stress τc about the point of support is just equal to that of the weight of the grain. Equating these moments gives (Vanoni, 1975):
FIGURE 6.6 Forces acting on a sediment particle on an inclined bed
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6.18
Chapter Six
ca τc 11 (γs γ) Dcos φ(tan θ tanφ) c2a2
(6.39)
in which γs specific weight of sediment grains, γ specific weight of water, D diameter of grains, is the slope angle of the stream, the angle of repose of the sediment, c1 and c2 are dimensionless constants, and a1 and a2 are lengths shown in Fig. 6.6. Any consistent set of units can be used in Eq. (6.39). For a horizontal bed, Eq. (6.39) reduces to ca τc 11 (γs γ)D tan θ (6.40) c2a2 For an adverse slope (i.e., 0), ca τc 11 (γs γ)D cos φ(tan θ tan φ) c2a2
(6.41)
Equations (6.39), (6.40), and (6.41) cannot be used to give τc because the factors c1, c2, a1, and a2 are not known. Therefore, the relation between the pertinent quantities is expressed by dimensional analysis, and the actual relation is determined from experimental data. Figure 6.7 is such a relation, first presented by Shields (1936) and carries his name. The curve is expressed by dimensionless combinations of critical shear stress τc, sediment and water specific weights γs and γ, sediment size D, critical shear velocity u*c τ/ρ c and kinematic viscosity of water ν. These quantities can be expressed in any consistent set of units. Dimensional analysis yields, u D τc τc* f *c (γs γ)D ν
(6.42)
The Shields values of τc* are commonly used to denote conditions under which bed sediments are stable but on the verge of being entrained. Not all workers agree with the results given by the Shields curve. For example, some workers give τc* 0.047 for the dimensionless critical shear stress for values of R* u*D/ν in excess of 500 instead of 0.06, as shown in Fig. 6.7. Taylor and Vanoni (1972) reported that small but finite amounts of sediment were transported in flows with values of τc* given by the Shields curve. The value of τc to be used in design depends on the particular case at hand. If the situation is such that grains that are moved can be replaced by others moving from upstream, some motion can be tolerated, and the Shields values can be used. On the other hand, if grains removed cannot be replaced, as on a stream bank, the Shields value of τc are too large and should be reduced. The Shields diagram is not especially useful in the form of Fig. 6.7 because to find τc, * one must know u* τ/ρ c. The relation can be cast in explicit form by plotting τc versus Rep, noting the internal relation gD D R u*D u* (τ*)1/2Rep R gD
(6.43)
ρs ρ where R is the submerged specific gravity of the sediment. A useful fit is given ρ by Brownlie (1981a): τ*c 0.22Re0.6 0.06 exp(17.77Re0.06 ) p p
(6.44)
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SEDIMENTATION AND EROSION HYDRAULICS
FIGURE 6.7 Shields diagram for initiation of motion. Source Vanoni (1975)
Sedimentation and Erosion Hydraulics 6.19
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SEDIMENTATION AND EROSION HYDRAULICS
6.20
Chapter Six
With this relation, the value of τc* can be computed readily when the properties of the water and the sediment are given. The value of bed-shear stress τb for a wide rectangular channel is given by τb γHS, as shown earlier. The average bed-shear stress for any channel is given by τb γRhS, in which Rh the hydraulic radius of the channel cross section. 6.4.2 Granular Sediment on a Bank A sediment grain on a bank is less stable than one on the bed because the gravity force tends to move it downward (Ikeda, 1982). The ratio of the critical shear stress τwc for a particle on a bank to that for the same particle on the bed τc is (Lane, 1955) τwc = cos φ1 τc
tan φ 2 1 1 tan φ
(6.45)
where φ1 is the slope of the bank and θ is the angle of repose for the sediment. Values of θ are
FIGURE 6.8 Angle of repose of granular material. (Lane, 1955)
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Sedimentation and Erosion Hydraulics 6.21
given in Fig. 6.8 after Lane (1955) and also can be found in Simons and Senturk (1976).
6.4.3 Granular Sediment on a Sloping Bed Equation (6.39) shows that τc diminishes as the slope angle φ increases. For extremely small φ’s, τc is given by Eq. (6.40). Taking the ratio between Eqs. (6.39) and (6.40) yields tan φ τcφ cos φ1 tan θ τcο
(6.46)
τcφ is the critical shear stress for sediment on a bed with a slope angle φ, and τco is the critical shear stress for a bed with an extremely small slope. The value of τco can be found from the Shields diagram or with Eq. (6.44). Equation (6.46) is for positive φ, which is positive for downward sloping beds. For beds with adverse slope, φ is negative and the term tan φ/tan θ in Eq. (6.46) is positive. 6.4.4 Sediment Mixtures Several authors have offered empirical or quasi-theoretical extensions of the above relations to the case of mixtures (e.g., Wilcock, 1988). Let Di denote the characteristic grain size of the ith size range in a mixture. Furthermore, let Dsg denote the geometric mean size of the surface (exchange, active) layer. Most of the generalizations can be written in the following form (Parker, 1990): D β τ*ci τ*cgi (6.47) Dsg Here τ ci τ*ci b (6.48a) ρRgDi and τ sg τ*cg bc (6.48b) ρRgDsg where τbci and τbcsg denote the values of the dimensioned critical shear stress required to move sediment of sizes Di and Dsg in the mixture, respectively, and β is an exponent taking a value given below; β 0.9
(6.49)
Figure 6.9 shows the similarity between four different published expressions having the general form given by Eq. (6.47), which is of interest because it includes the effect of hiding. For uniform material, the critical Shields stress is defined by Eq. (6.44). Consider two flumes, one with uniform size Da and the other with uniform size Db. For sufficiently coarse material (u*D/ν » 1 or Rep » 1), the critical Shields stress must be the same for both sizes (Fig. 6.7). It follows from Eq. (6.42) that where τbca and τbcb denote the dimensioned boundary shear stresses for cases a and b respectively, D τbcb τbcab Da
(6.50)
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6.22
Chapter Six
FIGURE 6.9 Critical shear stress for sediment mixture (Source: Misri et al., 1983)
For the case of mixtures, on the other hand, it is seen from Eqs. (6.47) and (6.48) that D 1β D 0.1 τbci τbcsgi
τbcsg i Dsg Dsg
(6.51)
Comparing Eqs. (6.50) and (6.51), it is seen that a finer particle (Db Da, or alternatively, Di Dsg) is more mobile than a coarser particle. For example, suppose that one grain size is four times coarser than another. If two uniform sediments are being compared, it follows from Eq. (6.50) that the critical shear stress for the coarser material is four times that of the finer material. In the case of a mixture, however, the critical shear stress for the coarser material is only about 40.1, or 1.15 times that for the finer material. A finer particle in a mixture is thus seen to be only a little more mobile than its coarser-sized brethren, where uniform beds of fine material are much more mobile than are uniform beds of coarser material. The reason is that finer particles in a mixture are relatively less exposed to the flow; they tend to hide in the lee of coarser particles. By the same token, a particle is relatively more exposed to the flow when most of its neighbors are finer. A method to calculate the critical shear stress for motion of uniform and heterogeneous sediments was proposed by Wiberg and Smith (1987) on the basis of the fluid mechanics of initiation of motion, which takes into account both roughness and hiding effects.
6.5 SEDIMENT TRANSPORT
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Sedimentation and Erosion Hydraulics 6.23
6.5.1
Sediment Transport Modes
The most common modes of sediment transport in rivers are bedload and suspended load. In the case of bedload, the particles roll, slide, or saltate over each other, never deviating too far above the bed. In the case of suspended load, the fluid turbulence comes into play carrying the particles well up into the water column. In both cases, the driving force for sediment transport is the action of gravity on the fluid phase; this force is transmitted to the particles via drag. The same phenomena of bedload and suspended load transport occur in a variety of other geophysical contexts. Sediment transport is accomplished in the near-shore lake and oceanic environment by wave action. Turbidity currents carry sediment into lakes, reservoirs, and the deep sea. The phenomenon of sediment transport can sometimes be disguised in rather esoteric phenomena. When water is supercooled, large quantities of particulate frazil ice can form. As this water moves under a frozen ice cover, the phenomenon of sediment transport in rivers is stood on its head. The frazil ice particles float rather than sink and thus tend to accumulate on the bottom side of the ice cover rather than on the river bed. Turbulence tends to suspend the particles downward rather than upward. In the case of a powder snow avalanche, the fluid phase is air and the solid phase consists of snow particles. The dominant mode of transport is suspension. These flows are close analogies of turbidity currents, insofar as the driving force for the flow is the action of gravity on the solid phase rather than the fluid phase. That is, if all the particles drop out of suspension, the flow ceases. In the case of sediment transport in rivers, it is accurate to say that the fluid phase drags the solid phase along. In the case of turbidity currents and powder snow avalanches, the solid phase drags the fluid phase along. Desert sand dunes provide an example for which the fluid phase is air, but the dominant mode of transport is saltation rather than suspension. Because air is so much lighter than water, quartz sand particles saltate in long, high trajectories, relatively unaffected by the direct action of turbulent fluctuations. The dunes themselves are created by the effect of the fluid phase acting on the solid phase. They, in turn, affect the fluid phase by changing the resistance. Among the most interesting sediment–transport phenomena are debris flows, slurries, and hyperconcentrated flows. In all these cases, the solid and fluid phases are present in similar quantities. A debris flow typically carries a heterogeneous mixture of grain sizes ranging from boulders to clay. Slurries and hyperconcentrated flows are generally restricted to finer grain sizes. In most cases, it is useful to think of such flows as consisting of a single phase, the mechanics of which are highly non-Newtonian. The study of the movement of grains under the influence of fluid drag and gravity becomes even more interesting when one considers the link between sediment transport and morphology. In the laboratory, the phenomenon can be studied in the context of a variety of containers, such as channel and wave tanks, specified by the experimentalist. In the field, however, the fluid-sediment mixture constructs its own container. This new degree of freedom opens up a variety of intriguing possibilities. Consider the river. Depending on the existence or lack of a viscous sublayer and the relative importance of bedload versus suspended load, a variety of rhythmic structures can form on the river bed. These include ripples, dunes, antidunes, and alternate bars. The first three of these can have a profound effect on the resistance to flow offered by the river bed. Thus, they act to control river depth. River banks themselves also can be considered to be a self-formed morphological feature, thus specifying the entire container. The container itself can deform in plan. Alternate bars cause rivers to erode their banks in a rhythmic pattern, thus allowing for the onset of meandering. Fully developed river meandering implies an intricate balance between sediment erosion and deposition. If a
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SEDIMENTATION AND EROSION HYDRAULICS
6.24
Chapter Six
stream is sufficiently wide, it will braid rather than meander, dividing into several intertwining channels. Rivers create morphological structures at much larger scales as well. These include canyons, alluvial fans, and deltas. Turbidity currents create similar structures in the oceanic environment. In the coastal environment, the beach profile itself is created by the interaction of water and sediment. On a larger scale, offshore bars, spits, and capes constitute rhythmic features created by wave-current-sediment interaction. The boulder levees often created by debris flows provide another example of a morphologic structure created by a sediment-bearing flow. The floodplains of most sand-bed rivers often contain copious amounts of silt and clay finer than approximately 50 µ. This material is often called wash load because it moves through the river system without being present in the bed in significant quantities. Increased wash load does not cause deposition on the bed, and reduced wash load does not cause erosion because it is transported well below capacity. This is not meant to imply that the wash load does not interact with the river system. Wash load in the water column exchanges with the banks and the floodplain rather than the bed. Greatly increased wash load, for example, can lead to thickened floodplain deposits, with a consequent increase in bankfull channel depth. The emphasis here is the understanding of bedload and suspended load transport in rivers, with the goal of providing the knowledge needed to do sound sedimentation engineering, particularly with problems involving stream restoration and naturalization.
6.5.2 Shields Regime Diagram In the context of rivers, it is useful to have a way to determine what kind of sedimenttransport phenomena can be expected for different flow conditions and different characteristics of sediment particles. In Fig. 6.10, the ordinates correspond to bed shear stresses written in the dimensionless form proposed by Shields τb HS τ* ρgRD RD
(6.52)
and the particle Rep, defined by Eq. (6.37) is used for the abscissa values. There are three curves in the diagram which make it possible to know, for different values of (τ*, Rep), if the given bed sediment will go into motion, and if this is the case whether or not the prevailing mode of transport will be in suspension or as bedload. The diagram also can be used to predict what kind of bedforms can be expected. For example, ripples will develop in the presence of a viscous sublayer and fine-grained sediment. If the viscous sublayer is disrupted by coarse sediment particles, then dunes will be the most common type of bedform. The Shields regime diagram also shows a clear distinction between the conditions observed in sand-bed rivers and gravel-bed rivers at bankfull stage. If one wanted to model in the laboratory sediment transport in rivers, the experimental conditions would be different, depending on the river system in question. As could be expected, the diagram also shows that in gravel-bed rivers, sediment is transported as bedload. In sand-bed rivers, on the other hand, suspended load and bedload transport coexist most of the time. The regime diagram is valid for steady, uniform, turbulent flow conditions, where the bed shear stress τb can be estimated with Eq. (6.1). The ranges for silt, sand, and gravel also are included. In the diagram, the critical Shields stress for motion was plotted with the help of Eq. (6.44). The critical condition for suspension is given by the following ratio: u v* 1 (6.53) s
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FIGURE 6.10 Shields regime diagram. (Source: Gary Parker)
Sedimentation and Erosion Hydraulics 6.25
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SEDIMENTATION AND EROSION HYDRAULICS
6.26
Chapter Six
where u* is the shear velocity and vs is the sediment fall velocity. Equation (6.53) can be transformed into τ∗s 12 Rf
(6.54)
where 2 * (6.55) τ∗s u gRD and Rf is given by Eq. (6.35a) and can be computed for different values of Rep with the help of Eq. (6.38). Finally, the critical condition for viscous effects (ripples) was obtained with the help of Eq. (6.5) as follows: 11.6 *ν 1 (6.56) uD which in dimensionless form can be written as
11.6 2 τ*v Rep
(6.57)
Relations (6.44), (6.54), and (6.57) are the ones plotted in Fig. 6.10. The Shields regime diagram should be useful for studies concerning stream restoration and naturalization because it provides the range of dimensionless shear stresses corresponding to bankfull flow conditions for both gravel- and sand-bed streams.
6.6 BEDLOAD TRANSPORT 6.6.1
The Bed Load Transport Function
Bedload particles roll, slide, or saltate along the bed. The transport thus occurs tangential to the bed. In a case where all the transport is directed in the streamwise, or s direction, the volume bedload-transport rate per unit width (n direction) is given by q; the units are length3/length/per time, or length2/time. In general, q is a function of boundary shear stress τb and other parameters; that is, q q(τb, other parameters)
(6.58)
In general, bedload transport is vectorial, with components qs and qn in the s and n directions, respectively.
6.6.2 Erosion Into and Deposition from Suspension The volume rate of erosion of bed material into suspension per unit time per unit bed area is denoted as E. The units of E are length3/length2/time, or velocity. A dimensionless sediment entrainment rate Es can thus be defined with the sediment fall velocity vs: E vsEs
(6.59)
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Sedimentation and Erosion Hydraulics 6.27
In general, Es can be expected to be a function of boundary shear stress τb and other parameters. Erosion into suspension can be taken to be directed upward normal: i.e., in the positive z direction. Let c denote the volume concentration of suspended sediment (m3 of sediment/m3 of sediment-water mixture), averaged over turbulence. The streamwise volume transport rate of suspended sediment per unit width is given by H
qs c udz
(6.60)
0
In a two-dimensional case, two components, qSs and qSn, result, where H
qSs c udz
(6.61a)
0
H
qSn c vdz
(6.61b)
0
Deposition onto the bed is by means of settling. The rate at which material is fluxed vertically downward onto the bed (volume/area/time) is given by vscb, where cb is a nearbed value of c. The deposition rate D realized at the bed is obtained by computing the component of this flux that is actually directed normal to the bed: D vscb
(6.62)
6.6.3 The Exner Equation of Sediment Mass Conservation for Uniform Material Consider a portion of river bottom, where the bed material is taken to have a (constant) porosity λp. Mass balance of sediment requires the following equation to be satisfied: ∂ [mass of bed material] net mass bedload inflow rate ∂t net mass rate of deposition from suspension. A datum of constant elevation is located well below the bed level, and the elevation of the bed with respect to such datum is given by η. Then, bed level changes as a result of bedload transport, sediment entrainment into suspension, and sediment deposition onto the bed can be predicted with the help of ∂η ∂q ∂q (1 λp) s n vs (cb Es) ∂t ∂s ∂n
(6.63)
To solve the Exner equation, it is necessary to have relations to compute bedload transport (i.e., qs and qn), near-bed suspended sediment concentration cb, and sediment entrainment into suspension Es. The basic form of Eq. (6.63) was first proposed by Exner (1925). 6.6.4
Bedload Transport Relations
A large number of bedload relations can be expressed in the general form
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SEDIMENTATION AND EROSION HYDRAULICS
6.28
Chapter Six
q* q*(τ*, Rep, R)
(6.64)
Here, q* is a dimensionless bedload transport rate known as the Einstein number, first introduced by H. A. Einstein in 1950 and given by q q* RgD D
(6.65)
The following relations are of interest. In 1972, Ashida and Michiue introduced q* 17(τ* τ*c) [(τ*)1/2 (τ∗c)1/2]
(6.66)
and recommend a value of τc* of 0.05. It has been verified with uniform material ranging in size from 0.3 mm to 7 mm. Meyer-Peter and Muller (1948) introduced the following: q* 8(τ* τ*c)3/2
(6.67)
where τ*c 0.047. This formula is empirical in nature and has been verified with data for uniform gravel. Engelund and Fredsøe (1976) proposed, q* 18.74(τ* τ*c) [(t*)1/2 0.7(τ∗c)1/2]
(6.68)
where τ 0.05. This formula resembles that of Ashida and Michiue because the derivation is almost identical. Fernandez Luque and van Beek (1976) developed the following, * c
q* 5.7(τ* τ*c)3/2
(6.69)
where τ∗c varies from 0.05 for 0.9 mm material to 0.058 for 3.3. mm material. The relation is empirical in nature. Wilson (1966): q* 12(τ* τ∗c)3/2
(6.70)
where τ∗c was determined from the Shields diagram. This relation is empirical in nature; most of the data used to fit it pertain to very high rates of bedload transport. Einstein (1950): q* q*(τ*)
(6.71)
where the functionality is implicitly defined by the relation 1 1
(0.143/τ*)2
43.5q* et2dt 1 43.5q* (0.413/τ*)2
(6.72)
Note that this relation contains no critical stress. It has been used for uniform sand and gravel. Yalin (1963): 1n(1 a2s) q* 0.635s(τ*)1/2 1 (6.73) a2s where τ* τ* a2 2.45(R 1)0.4 (τ∗c)1/2; s *c (6.74) τc
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SEDIMENTATION AND EROSION HYDRAULICS
Sedimentation and Erosion Hydraulics 6.29
and τ∗c is evaluated from a standard Shields curve. Two constants in this formula have been evaluated with the aid of data quoted by Einstein (1950), pertaining to 0.8 mm and 28.6 mm material. Parker (1978): (τ* 0.03)4.5 q* 11.2*3 (6.75) τ developed with data sets pertaining to rough mobile-bed flow over gravel. Several of these relations are plotted in Fig. 6.11. They tend to be rather similar in nature. Scores of similar relations could be quoted. To date, only few research groups have attempted complete derivations of the bedload function in water. They are Wiberg and Smith (1989), Sekine and Kikkawa (1992), García and Niño (1992), Niño and García, (1994, 1998), and Niño et al., (1994).
FIGURE 6.11 Bedload transport relations. (Parker, 1990)
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6.30
6.6.5
Chapter Six
Bedload Transport Relation for Mixtures.
Relatively few bedload relations have been developed specifically in the context of mixtures (e.g., Bridge and Bennett, 1992). One of these is presented below as an example. The relationship of Parker (1990) applies to gravel-bed streams. The data used to fit the relation are solely from two natural gravel-bed streams: Oak Creek in Oregon and the Elbow River in Alberta, Canada. The relation is surface-based; load is specified per unit of fractional content in the surface layer. The surface layer is divided into N size ranges, each with a fractional content Fi by volume, and a mean phi size φi; Di 2φi. The arithmetic mean of the surface size on the phi scale φ and the corresponding arithmetic standard deviation σφ are given by φ ΣFiφi;
σ2φ ΣFi(φi φ)2
(6.76a, b)
The corresponding geometric mean size Dsg and the geometric standard deviation σsg of the surface layer are given by Dsg 2φ
σsg 2σφ
(6.77a, b)
In the Parker relation, the volume bedload transport per unit width of gravel in the ith size range is given by the product qiFi (no summation), where qi denotes the transport per unit fraction in the surface layer. The total volume bedload transport rate of gravel per unit width is qT, where qT qiFi
(6.78)
The relation does not apply to sand. Thus, before using the relation for a given surface distribution, the sand content of the grain-size distribution must be removed and Fi must be renormalized so that it sums to unity over all sizes in excess of 2 mm. If pi denotes the fraction volume content of material in the ith size range in the bedload, it follows that q Fi pi i qiFi
(6.79)
The parameter qi is made dimensionless as follows: Rgqi W*si (τb/ρ)3/2Fi
(6.80)
A dimensionless Shields stress based on the surface geometric mean size is defined as follows: τb τ*sg (6.81) ρRgDsg Let φsgo denote a normalized value of this Shields stress, given by τ* go φsgo * s τ rsgo
(6.82)
τ*rsgo 0.0386
(6.83)
where
corresponds to a “near-critical” value of Shields stress. The Parker relation can then be Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
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Sedimentation and Erosion Hydraulics 6.31
expressed in the form W*si 0.0218 G [ωφsgogo(δi)]
(6.84a)
In the above relationship, go denotes a hiding function given by D δi i Dsg
go(δi) δ0.0951 ; i
(6.84b)
The parameter ω is given by the relationship σφ ω 1 (ω 1) σφo o
(6.84c)
where σφo and ωo are specified as functions of φsgo in Fig. 6.12. The function G is specified as
5474(1 0.853/φ) G[φ] exp[14.2(φ 1) 9.28(φ 1) ] φ 4.5
2
Mo
φ 1.65 1 φ 1.65 φ1
(6.85)
and is shown in Fig. 6.13. Here, Mo 14.2 and φ is a dummy variable for the argument in Eq. (6.84) and is not to be confused with the φ grain-size scale. An application of Eq. (6.84) to uniform material with size D results in the relation τ* q* 0.0218(τ*)3/2G 0.0386
where q q* ; gR D D
τb τ* ρgRD
(6.86)
(6.87)
and q denotes the volumetric sediment transport per unit width. In Fig. 6.11, Eq. (6.86) is compared to several other relations and selected laboratory data for uniform material. The figure is adapted from Figs. 6b and 7 in Wiberg and Smith (1989), where reference to the data and equations can be found. The data pertain to 0.5 mm sand and 28.6 mm gravel. Equation (6.86) shows a reasonable correspondence with the data and with several other relations for uniform material. The Parker relationship (Eq. 6.84) can be used to predict mobile or static armor in gravel streams. Note that there is no formal critical stress in the formulation; instead for φ 1, the transport rates become extremely small. For the computation of bedload transport in poorly sorted gravel-bed rivers, the above formulation has been used to implement a series of programs named “ACRONYM” (Parker, 1990). The program “ACRONYM1” provides an implementation of the surface-based bedload transport equation presented in Parker (1990). It computes the magnitude and size distribution of bedload transport over a bed surface of given size distribution, on which a given boundary shear stress is imposed. The program “ACRONYM2” inverts the same bedload transport equation, allowing for calculation of the size distribution at a given boundary shear stress. The program was used to compute mobile and static armor size distributions in Parker (1990) and Parker and Sutherland (1990).
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Chapter Six
FIGURE 6.12 Plots of ω0 and σφ0 versus φsg0, the asymptotes are noted on the plot. (Parker, 1990)
6.32
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SEDIMENTATION AND EROSION HYDRAULICS
FIGURE 6.13 Plot of G and GT versus φ50. (Parker, 1990)
Sedimentation and Erosion Hydraulics 6.33
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SEDIMENTATION AND EROSION HYDRAULICS
6.34
Chapter Six
The program “ACRONYM3” allows for the computation of aggradation or degradation to a specified active or static equilibrium final state. To this end, Parker’s method (1990) is combined with a resistance relation of the Keulegan type. In the program, both constant width and water discharge are assumed. The program “ACRONYM4” is directed toward the wavelike aggravation of self-similar form discussed in Parker (1991a, 1991b). It uses Parker’s method and a resistance relation of the Manning-Strickler type to compute downstream fining and slope concavity caused by selective sorting and abrasion.
6.7 BEDFORMS The formation and behavior of sediment waves produced by moving water are, in equal measure, intellectually intriguing and of great engineering importance. Because of the central role they play in river hydraulics, fluvial ripples, dunes, and bars have received extensive attention from engineers for at least the past two centuries, and even more intensive descriptive study from geologists. Such studies can be divided into three categories according to the approach followed: analytical, empirical, or statistical. Analytical models for bedforms have been proposed since 1925 (Anderson, 1953; Blondeaux et al., 1985; Colombini et al., 1987; Engelund, 1970; Exner, 1925; Fredsoe, 1974, 1982; Gill, 1971; Haque and Mahmood, 1985; Hayashi, 1970; Kennedy, 1963, 1969; Parker, 1975; Raudkivi and Witte, 1990; Richards, 1980; Smith, 1970; Tubino and Seminara, 1990). Empirical methods include the following works (Coleman and Melville, 1994; Colombini et al., 1990; García and Niño, 1993; Garde and Albertson, 1959; Ikeda, 1984; Jaeggi, 1984; Kinoshita and Miwa, 1974; Menduni and Paris, 1986: Ranga Raju and Soni, 1976; Raudkivi, 1963; van Rijn, 1984; Yalin, 1964; Yalin and Karahan, 1979). Statistical models for bedforms have been advanced by the following authors Annambhotla et al.,1972; Hino, 1968; Jain and Kennedy, 1974; Nakagawa and Tsujimoto, 1984; Nordin and Algert, (1966). Despite all the research that has been done, there is presently no completely reliable predictor for the conditions of occurrence and characteristics of the different bed configurations (ripples, dunes, flat bed, antidunes).
6.7.1 Dunes, Antidunes, Ripples, and Bars The ripples, dunes, and antidunes illustrated in Fig. 6.14 are the classic bedforms of erodible-bed open-channel flow. On the one hand, they are a product of the flow and sediment transport; on the other hand, they profoundly influence the flow and sediment transport. In fact, all the bedload formulas quoted previously are strictly invalid in the presence of bedforms. The adjustments necessary to render them valid are discussed later. Ripples, dunes, and antidunes are undular (wavelike) features that have wavelengths Λ and wave heights ∆ that scale no larger than on the order of the flow depth H, as defined below. 6.7.1.1 Dunes. Well-developed dunes tend to have wave heights D scaling up to about one-sixth of the depth: i.e.,
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Sedimentation and Erosion Hydraulics 6.35
FIGURE 6.14 Schematic of different bedforms. (Vanoni, 1975)
∆ 1 H 6
(6.88)
Dune wavelengths can vary considerably. A fairly typical range can be quantified as dimensionless wave number k, where 2πH k Λ
(6.89)
0.25 k 4.0
(6.90)
This range is
Dunes invariably migrate downstream. Typically, they are approximately triangular in shape and usually (but not always) possess a slip face, beyond which the flow is separated for a certain length. A dune progresses forward as bedload accretes on the slip face. Generally, little bedload is able to pass beyond the face without depositing on it, whereas most of the suspended load is not directly affected by it. Let c denote the wave speed of the dune. The bedload transport rate can be estimated as the volume of material transported forward per unit bed area per unit time by a migrating dune. If the dune is approximated as triangular in shape, the following approximation holds: q 1 ∆ c(1 λp) (6.91) 2 Dunes are characteristic of subcritical flow in the Froude sense. In a shallow-water (long wave) model, the Froude criterion (Fr) dividing subcritical and supercritical flow is
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6.36
Chapter Six
Fr 1
(6.92)
U Fr gH
(6.93)
where
Dunes, however, do not qualify as long waves because their wavelength is of the order of the depth. A detailed potential flow analysis over a wavy bed yields the following (wavenumber dependent) criterion for critical flow over a bedform (Kennedy, 1963). (6.94) Fr2 1 tanh(k) k Note that as k → 0(Λ → ∞) tanh(k) → k, and condition (6.92) is recovered in the longwave limit. For dunes to occur, then, the following condition must be satisfied: F2 1 tanh (k) k
(6.95)
Both dunes and antidunes cause the water surface as well as the bed to undulate. In the case of dunes, the undulation of the water surface is usually of much smaller amplitude than that of the bed; the two are nearly 180o out of phase. Dunes also can occur in the case of wind-blown sand. Barchan dunes are commonly observed in the desert. In addition, they can be found in the fluvial environment in the case of sand (in supply insufficient to cover the bed completely) migrating over an immobile gravel bed. 6.7.1.2 Antidunes. Antidunes are distinguished from dunes by the fact that the undulations of the water surface are nearly in phase with those of the bed. They are associated with supercritical flow in the sense that F2 1 tanh (k) (6.96) k Antidunes may migrate either upstream or downstream. Upstream-migrating antidunes are usually rather symmetrical in shape and lack a slip face. Downstream-migrating antidunes are rarer; they have a well-defined slip face and look rather like dunes. The distinguishing feature is the water surface undulations, which are pronounced in the case of antidunes. The potential-flow criterion dividing upstream-migrating antidunes from downstreammigrating antidunes is 1 F2 (6.97) k tanh (k) Values lower than the value in Eq. 6.97 are associated with upstream-migrating antidunes. 6.7.1.3 Ripples. Ripples are dunelike features that occur only in the presence of a viscous sublayer. They look much like dunes because they migrate downstream and have a pronounced slip face. They generally are much more three-dimensional in structure than are dunes, however, and have little effect on the water surface. A criterion for the existence of ripples is the existence of a viscous sublayer. Recalling that the thickness of the viscous sublayer is given by δv 11.6v/u*, it follows that ripples form when
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Sedimentation and Erosion Hydraulics 6.37
uD (6.98) * 11.6 ν 6.7.1.4 Bars. Bars are bedforms in rivers that scale the channel width. They include alternate bars in straight streams, point bars in meandering streams, and pool bars in braided streams. In straight streams, the minimum channel slope S necessary for alternate-bar formation is given by 0.15 exp1.07B M D g S (6.99) B 12.9 Dg (Jaeggi, 1984), where B is the channel width, Dg is the geometric mean size of the bed sediment, as given by Eq. (6.82a), and M is a parameter that varies from 0.34 for uniformsized bed material to 0.7 for poorly sorted material. Scour depth (Sd) caused by alternate bar formation can be estimated with B Sd 0.76∆AB , B 0.15 6 Dg
(6.100)
where ∆AB is the total height of the alternate bar. 6.7.1.5 Progression of bedforms. Various bedforms are associated with various flow regimes. In the case of a sand-bed stream with a characteristic size less than about 0.5 mm, a clear progression is evident as flow velocity increases. This is illustrated in Fig. 6.14. The bed is assumed to be initially flat. At low imposed velocity U, the bed remains flat because no sediment is moved. As the velocity exceeds the critical value, ripples are formed first. At higher values, dunes form and coexist with ripples. For even higher velocities, well-developed dunes form in the absences of ripples. At some point, the velocity reaches a value near the critical value in the Froude sense: Table 6.4 Summary of Bedform Effects on Flow Configuration Bed Form or Configuration (1)
Dimensions (2)
Shape (3)
Ripples
Wavelength less than Roughly triangular in approx 1 ft; height less profile, with gentle, than approx 0.1 ft. slightly convex upstream slopes and downstream slopes nearly equal to the angle of repose. Generally short-crested and three-dimensional.
Bars
Lengths comparable to the channel width. Height comparable to mean flow depth.
Profile similar to ripples. Plan form variable.
Behavior and Occurrence (4) Move downstream with velocity much less than that of the flow. Generally do not occur in sediments coarser than about 0.6 mm.
Four types of bars are distinguished: (1) point, (2) alternating, (3) transverse, and (4) tributary. Ripples may occur on upstream slopes.
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6.38
Chapter Six
Table 6.4 (Continued) Bed Form or Configuration (1)
Dimensions (2)
Shape (3)
Behavior and Occurence (4)
Dunes
Wavelength and height Similar to ripples. greater than ripples but less than bars.
Upstream slopes of dunes may be covered with ripples. Dunes migrate downstream in manner similar to ripples.
Transition
Vary widely
Vary widely.
A configuration consisting of a heterogeneous array of bed forms, primarily low amplitude ripples and dunes interspersed with flat regions.
Flat bed
—
—
A bed surface devoid of bed forms. May not occur for some ranges of depth and sand size.
Antidunes
Wave length 2πV2/g (approx)* Height depends on depth and velocity of flow.
Nearly sinusoidal in profile. Crest length comparable to wavelength.
In phase with and strongly interact with gravity watersurface waves. May move upstream, downstream, or remain stationary, depending on properties of flow and sediment.
*Reported by Kennedy (1969). Source: Vanoni (1975).
i.e., Eq. (6.94). Near this point, the dunes often are suddenly and dramatically washed out. This results in a flat bed known as an upper-regime (supercritical) flat bed. Further increases in velocity lead to the formation of antidunes and, finally, to the chute and pool pattern. The last of these is characterized by a series of hydraulic jumps. In the case of a bed coarser than 0.5 mm, the ripple regime is replaced by a zone characterized by a lower-regime (subcritical) flat bed. Above this lies the ranges for dunes, the upper-regime flat bed, and antidunes. The effect of bedforms on flow resistance is summarized in Table 6.4. As noted earlier for equilibrium flows in wide straight channels, the relation for bed resistance can be expressed in the form τb ρCfU2
(6.101)
where Cf denotes a bed-friction coefficient. If the bed were rigid and the flow were rough, Cf would vary only weakly with the flow, according to the logarithmic law embodied in Eq. (6.12). As a result, the relation between τb and U is approximately parabolic.
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Sedimentation and Erosion Hydraulics 6.39
FIGURE 6.15 Variations of bed shear stress τb and Darcy-Weisbach friction factor f with mean velocity U in flow over a fine sand bed. (Raudkivi, 1990)
The effect of bedforms is to increase the bed shear stress to values often well above that associated with the skin friction of a rough bed alone. In Fig. 6.15, a plot of τb versus U is given for the case of an erodible bed. At extremely low values of U, the parabolic law is followed. As ripples, then dunes are formed, the bed shear stress rises to a maximum value. At this maximum value, the value of Cf is seen to be as much as five times the value without dunes. It is clear that dunes play an important role regarding bed resistance. The increased resistance results from form drag in the lee of the dune. As the flow velocity increases further, dune wavelength gradually increases and dune height diminishes, leading to a gradual reduction in resistance. At some point, the dunes are washed out, and the parabolic law is again satisfied. At even higher velocities, the form drag associated with antidunes appears; it is usually not as pronounced as that of dunes.
6.7.2 Dimensionless Characterization of Bedform Regime Based on the above arguments, it is possible to identify at least three parameters governing bedforms at equilibrium flow. These are Shields stress τ*, shear Reynolds number Re u*D/ν, and Fr. A characteristic feature of sediment transport is the proliferation of dimensionless parameters. This feature notwithstanding, Parker and Anderson (1977) showed that equilibrium relations of sediment transport for uniform material in a straight channel can be expressed with just two dimensionless hydraulic parameters, along with a particle Re (e.g., Rep or Re) and a measure of the denstiy difference (e.g., R).
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6.40
Chapter Six
FIGURE 6.16 Bedform predictor proposed by Simons and Richardson (1966).
In the case of bedforms, then, the following classification can be proposed: bedform type function (π1, π2; Rep, R)
(6.102)
Here, any independent pair of hydraulic variables π1, π2 applicable to the problem can be specified because any one pair can be transformed into any other independent pair. For example, the pair τ* and Fr might be used or, alternatively, S and H/D. One popular discriminator of bedform type is not expressed in dimensionless form at all. It is the diagram proposed by Simons and Richardson (1966), (Fig. 6.16). In the diagram, regimes for ripples and dunes, transition to the upper-regime plane bed, and upperregime plane bed and antidunes are shown. The two hydraulic parameters are abbreviated to a single one, stream power τbU, and the particle Re is replaced by grain size D. The diagram is applicable only for sand-bed streams of relatively small scale. Liu’s discriminator (1957), shown in Fig. 6.17, uses one dimensionless hydraulic para-
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Sedimentation and Erosion Hydraulics 6.41
FIGURE 6.17 Criteria for bedforms proposed by Liu (1957)
FIGURE 6.18 Bedform classification. (after Chabert and Chauvin, 1963)
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6.42
Chapter Six
meter u*/vs (a surrogate for τ*) and the particle Rep. The diagram is of interest because it covers sizes much coarser than those of Simons and Richardson. It is seen that the various regimes become compressed as grain size increases. In the case of extremely coarse material, the flow must be supercritical for any motion to occur. As a result, neither ripples or dunes are expected. In fact, dunes can occur over a limited range in the case of coarse material. This is illustrated in Fig. 6.18. The diagram shows that Re must be less than approximately 10(δv D) for ripples to form. Recalling that uD R g DD Re * (τ*)1/2 ν ν
(6.103)
and using a critical value of τ* of approximately 0.03, it is seen from Eq. (6.101) and the conditions R 1.65, ν 0.01 cm2/s that the condition Re 10 corresponds to a value of D of approximately 0.6 mm. For coarser grain sizes, the dune regime is preceded by a fairly wide range consisting of a lower-regime flat bed. Many gravel-bed rivers never leave this lower-regime flat bed
FIGURE 6.19a Bed-form chart for Rg = 4.5–10 (D50 = 0.12 mm–0.200 mm)
FIGURE 6.19b Bed-form chart for Rg = 4.5–10 (D50 = 0.12 mm–0.200 mm)
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Sedimentation and Erosion Hydraulics 6.43
FIGURE 6.19c Bed-form chart for Rg = 4.5–10 (D50 = 0.15 mm–0.32 mm)
FIGURE 6.19d Bed-form chart for Rg = 16–26 (D50 = 0.228 mm–0.45 mm)
FIGURE 6.19e Bed-form chart for Rg = 24–48 (D50 = 0.4 mm–0.57 mm)
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6.44
Chapter Six
FIGURE 6.19f Bedform chart. A, B, C, D, E, F (after Vanoni, 1974)
region, even at bankfull flow. The diagram in Fig. 6.18 is not suited to the description of upper-regime flow. A complete set of diagrams for the case of sand is shown in Fig. 6.19a to f, (Vanoni, 1974). The two hydraulic parameters are Fr and H/D; the particle Re used in the plot is equal to Rep/R, and constant R is set at 1.65. Note how the transition to upper regime
FIGURE 6.20 Bedform classification (after van Rijin, 1984)
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Sedimentation and Erosion Hydraulics 6.45
occurs at progressively lower values of Fr for relatively deeper flow (in the sense that H/D becomes large). A bedform classification scheme that includes both the lower and the upper regime was proposed by Van Rijn (1984). The scheme is based on a dimensionless particle diameter D* and the transport-stage parameter T defined, respectively, as Rg 1/3 2/3 D* D50 R ep ν2
(6.104)
τ*s τ*c T τ*c
(6.105)
and
where τs* is the bed shear stress caused by skin or grain friction, and τc* is the critical shear stress for motion from the Shields diagram. Van Rijn (1984) suggested that ripples form when both D* 10 and T 3, as shown in Fig. 6.20. Dunes are present elsewhere when T 15, dunes wash out when 15 T 25, and upper flow regime starts when T 25. In the lower regime, the geometry of bedforms refers to representative dune height ∆ and wavelength ∆ as a function of the average flow depth H, median bed particle diameter D50, and other flow parameters such as the transport-stage parameter T, and the grain shear Reynolds number Re. The bedform height and steepness predictors proposed by van Rijn (1984) are
FIGURE 6.21a,b Bedform height and steepness (after van Rijn, 1984)
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6.46
Chapter Six D50 0.3 ∆ 0.11 (1 e 0.5T)(25 T) H H
(6.106)
D50 0.3 ∆ 0.015 (1 e 0.5T) (25 T) Λ H
(6.107)
and
The bedform length obtained from dividing these two equations, Λ 7.3H, is close to the theoretical value Λ 2πH, derived by Yalin (1964). The agreement with laboratory data is good, as shown in Fig. 6.21a and b, but both curves tend to underestimate the bedform height and steepness of field data (Julien, 1995; Julien and Klaasen, 1995). For instance, lower-regime bedforms are observed in the Mississippi River at values of T well beyond 25. Large dunes on alluvial rivers often display small dunes moving along their stoss face (Amsler and García, 1997; Klaasen et al., 1986), resulting in additional form drag that is not accounted for in relations derived from laboratory observations. What is needed is a predictor for bedforms in large alluvial rivers based on field observations,
6.7.3 Effect of Bedforms on River Stage The presence or absence of bedforms on the bed of a river can lead to some curious effects on a river’s stage. According to a standard Manning-type relation for an nonerodible bed, the following should hold: U 1 H2/3S1/2 n
(6.108)
FIGURE 6.22 Flow velocity versus hydraulic radius for the Rio Grande (after Nordin, 1964)
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Sedimentation and Erosion Hydraulics 6.47
Here, the channel is assumed to be wide enough to allow the hydraulic radius to be replaced with the depth H. According to Eq. (6.108), if the energy slope remains relatively constant, depth should increase monotonically with increasing velocity. This would indeed be the case for a rigid bed. In a sand-bed stream, however, resistance decreases as U increases over a wide range of conditions. At equilibrium, τb ρCfU2 ρgHS
(6.109)
This decrease in resistance implies that depth does not increase as rapidly in U as it would for a rigid-bed open channel. In fact, as transition to upper regime is approached, the bedforms can be wiped out suddenly, resulting in a dramatic decrease in resistance. The result can be an actual decrease in depth as velocity increases (Fig. 6.22). It is often found that the discharge at which the dunes are obliterated is a little below bankfull in sand-bed streams. As a result, flooding is not as severe as it would be otherwise. The precise point of transition is generally different, depending on whether the discharge is increasing or decreasing. This can lead to double-valued stage-discharge relations, (Fig. 6.22).
6.8 EFFECT OF BEDFORMS ON FLOW AND SEDIMENT TRANSPORT 6.8.1 Form Drag and Skin Friction As was seen in Sec. 6.7.3, bedforms can have a profound influence on the flow resistance and thus on the sediment transport in an alluvial channel. To characterize the importance of bedforms in this regard, it is of value to consider the forces that contribute to the drag force on the bed. Consider, for example, the case of normal flow in a wide rectangular channel. In the presence of bedforms, Eq. (6.1) must be amended to τb ρgHS
(6.110)
where τb is an effective boundary shear stress, where the overbear denotes averaging over the bedforms and can be defined as the streamwise drag force per unit area, where H now represents the depth averaged over the bedforms. In most cases of interest, the two major sources of the effective boundary shear stress τb are skin friction, which is associated with the shear stresses, and the form drag, which is associated with the pressure. That is, τb τbs τbf
(6.111)
where τbs is the shear stress caused by skin friction and τbf is the shear stress caused by form drag. The important thing to realize is that form drag results from a net pressure distribution over an entire bedform. At any given point along the surface of the bedform, the pressure force acts normal to the body. For this reason, form drag is ineffective in either moving bedload sediment or entraining sediment into suspension. In the case of dunes in rivers, because the flow usually separates in the lee of the crest, the form drag is often substantial. The part of the effective shear stress that governs sediment transport is thus seen to
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SEDIMENTATION AND EROSION HYDRAULICS
6.48
Chapter Six
be the skin friction. To render any of the bedload formulas presented in Sec. 6.6.4 valid in the presence of bedforms, it is necessary to replace the Shields stress τ* by the Shields stress τ*s associated with skin friction only: τs (6.112) τ*s b ρRgD The fact that the form drag needs to be excluded to compute sediment transport does not by any means imply that it is unimportant. It is often the dominant source of boundary resistance and thus plays a crucial role in determining the depth of flow. This will be considered in more detail below.
6.8.2
Shear Stress Partitions
6.8.2.1 Einstein partition. Einstein (1950) was among the first to recognize the necessity to distinguish between skin friction and form drag. He proposed the following simple scheme to partition the two. Equation (6.101) is amended to represent an effective boundary shear stress averaged over bedforms: τb ρCfU2
(6.113)
where Cf now represents a resistance coefficient that includes both skin friction and form drag. For a given flow velocity U, Einstein computed the skin friction as follows: τbs ρCfsU2
(6.l14)
where Cfs is the frictional resistance coefficient that would result if bedforms were absent. For example, in the case of rough turbulent flow, Eq. (6.15) may be used: 1 H 2 Cfs 1n11 s ks
(6.115)
(In fact, Einstein presented a slightly different formula, which allows for turbulent smooth and transitional flow as well.) The parameter Hs denotes the depth that would result in the absence of bedforms (but with U held constant). This depth is per force less than H because the resistance is less in the absence of bedforms. The remaining problem is how to calculate Hs. Einstein restricted his arguments to the case of normal flow. In this case, Eq. (6.15) holds: that is, τb ρCfU2 ρgHS
(6.116a)
τbs ρCfsU2 ρgHsS
(6.116b)
and
Now, between Eqs. (6.113) and (6.116b), the following relation is obtained for Hs: U2 1 H 2 Hs 1n(11 s ) gS ks
(6.117)
For given values of U, ks, and S (averaged over bedforms), Eq. (6.117) is easily solved
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Sedimentation and Erosion Hydraulics 6.49
iteratively for Hs. Once Hs is known, it is not difficult to complete the partition. From Eq. (6.109), it follows that τbf τb τbs.
(6.118)
In analogy to Eqs. (6.111), (6.112), and (6.114), the following definitions are made: τbf ρCffU2 ρgHfS
(6.119)
Cf Cfs Cff
(6.120a)
H Hs Hf
(6.120b)
from which it follows that
and
Here, Cff denotes the resistance coefficient associated with form drag and Hs denotes the extra depth (compared to the case of skin friction alone) that results from form drag. Up to this point, it is assumed the U, S, and ks are given. If, for example, H also is known, τb can be calculated from Eq. (6.110). After Hs, Cfs, and τbs are computed from Eqs. (6.113) to (6.115), it is possible to compute τbf, Hf, and Cff from Eqs. (6.116) and (6.118). 6.8.2.2 Example of the Einstein partition. Consider a sand-bed stream at a given cross section with a slope of 0.0004, a mean depth of 2.9 m, a value of median bed sediment size of 0.35 mm, and a discharge per unit width of 4.4 m2/s. Assume that the flow is at near-normal conditions. Compute values of τbs, τbf, Cfs, Cff, Hs, and Hf . Solution: U 4.4/2.9 1.52 m/s. An appropriate estimate of ks for a sand-bed stream is ks 2.5D50
(6.121)
Solving Eq. (6.115) by successive approximation, it is found the Hs 1.047 m. The following values then hold: τbs 4.11 newton’s/m2
(τ*s
0.725)
τbf 7.27 newtons/m
* f
(τ
1.283)
τb 11.38 newtons/m2
(τ*
2.008)
2
Cfs 0.00178 Cff 0.00315 Cf 0.00493
(Cf–1/2 14.5)
Hs 1.047 m Hf 1.842 m H 2.9 m In the above relations, τ (6.122) τ*f bf ρRgD denotes a form Shield stress. In the above case, only some 30% of the total Shields stress (skin form) contributes to moving sediment. Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
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6.50
Chapter Six
The Einstein method provides a way of partitioning the boundary shear stress if the flow is known. It does not provide a direct means of computing form drag. A method proposed by Nelson and Smith (1989) overcomes this difficulty. 6.8.2.3 Nelson-Smith partition. Nelson and Smith considered flow over a dune; the flow is taken to separate in the lee of the dune. On the basis of experimental observations, they use the following relation for form drag: Df B 1 ρcD∆U2r (6.123) 2 Here, Df denotes that portion of the streamwise drag force Dfs that is caused by form drag, B is the channel width, and Ur denotes a reference velocity to be defined below. They evaluate the drag coefficient cD as cD 0.21
(6.124)
It follows that Df τbf 1 ρcD U2r (6.125) 2 B· The reference velocity Ur is defined to be the mean velocity that would prevail between z ks and z ∆ if the bedforms were not there. From the logarithmic profile represented by Eq. (6.7), this is found to be given by Ur ∆ 1 [ln(30) 1] κ k τbρ s s/
(6.126)
It is now assumed that a rough logarithmic law with roughness ks prevails from z ks to z D, and a different rough logarithmic law with roughness kc prevails from z D to z H. Here kc represents a composite roughness length, including the effects of both skin friction and form drag. The two laws are thus u(z) z 1 ln 30 , κ k τbρ / s s
ks z ∆
(6.127a)
and u(z) z 1 (τ bs τ bfρ )/ κ ln 30 kc ,
∆zH
(6.127b)
Nelson and Smith (1989) matched the above two laws at the level z ∆. After some manipulation, it is found that τbs τbf ln(30 ∆/ks τ = l n(30 ∆/kc) bs
2
(6.128)
The partition requires a prior knowledge of total boundary shear stress τb τbs τbf as well as roughness height ks, dune height ∆, and dune wavelength Λ. Between Eqs. (6.123) and (6.124), ∆ ∆ τbs/f τb τbs 1cD 2 ln 30 1 2 Λκ ks
τ 2
bs
(6.129)
This equation can be solved for τbs, and thus τbf. The value of kc is then obtained from Eq. (6.128). 6.8.2.4 Example of the Nelson-Smith Partition. The example is chosen to be rather Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
SEDIMENTATION AND EROSION HYDRAULICS
Sedimentation and Erosion Hydraulics 6.51
similar to the previous one: H 2.9 m, S 0.0004, ks 2.5 D50, D50 0.35 mm, ∆ 0.4 m, and Λ 15 m. The technique, which requires no iteration, yields the following results: τbs τbf τb kc Cfs Cff Cf Hs Hf H
4.45 newtons/m2 6.93 newtons/m2 11.38 newtons/m2 0.0311 m 0.00130 0.00203 0.00333 1.134 m 1.766 m 2.9 m
(τ*s (τ*f (τ*
(C1/2 f
0.785) 1.223) 2.008)
17.3)
In computing friction coefficients, the following relationship was used for the depth-averaged flow velocity: U 1 1n 11 H (6.130) k (τ τ /ρ ) c bs bf The Nelson-Smith method does not require the assumption of quasi-normal flow.
6.8.3 Empirical Formulas for Stage-Discharge Relations To use either the Einstein or Nelson-Smith partitions, it is necessary to know in advance the total effective boundary shear stress τb. In general, this is not known. As a result, the relations in themselves cannot be used to predict the boundary shear stress (as well as the contributions from skin friction and form drag), and thus depth H, for a flow of, say, given slope S and discharge per unit width qw. A number of empirical techniques have been proposed to accomplish this. Only three are presented here; they are known to perform well for sand-bed streams with dune resistance. 6.8.3.1 Einstein-Barbarossa Method. The method of Einstein and Barbarossa (1952) is applicable for the case of dune resistance in a sand-bed stream. It assumes an empirical relation of the following form:
Cff fnτ*s35
(6.131)
τs τ*s35 b ρRgD35
(6.132)
Here,
The Einstein-Barbarossa plot is shown in Fig. 6.23. Note that it implies that Cff declines for increasing τ*s35. That is, the relation applies in the range for which increased intensity of flow causes a decrease in form drag.
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SEDIMENTATION AND EROSION HYDRAULICS
6.52
Chapter Six
FIGURE 6.23 Flow resistance due to bedforms. [after Einstein et al. (1952).]
In the Einstein-Barbarossa method, Cfs is computed from a relation similar to Eq. (6.113). That relation is used here to illustrate the method, which uses the Einstein partition for skin friction and form drag. 6.8.3.2 Application of the Einstein-Barbarossa Method. The Einstein-Barbarossa method is now used to synthesize a depth-discharge relation: that is, a relation between H and water discharge Q is obtained. It is assumed that the river slope S and the sizes D50 and D35 are known. The river is taken to be wide enough so that the hydraulic radius Rh ≅ H; otherwise, Rh should be used in place of H. In addition, the cross-sectional shape is known, allowing for specification of the following geometric relation: B B(H)
(6.133)
(It also is assumed that auxiliary relations for area A, wetted perimeter P and Rh as functions of H are known.) A range of values of Hs is arbitrarily assumed, ranging from an extremely shallow depth to nearly bankfull depth (recall that Hs H). For each value of Hs, the calculation proceeds as follows: Hs → Cfs
Eq. (6.115)
Cfs, Hs → U Hs → τbs → τ τ*s35 → Cff Cff, U → Hf
Eq. (6.116b) * s35
Eq. (6.116b), (6.132) Eq. (6.131); use the diagram Eq. (6.119)
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SEDIMENTATION AND EROSION HYDRAULICS
Sedimentation and Erosion Hydraulics 6.53 H Hs Hf
Eq. (6.120b)
Q UH B(H)
Eq. (6.133)
The result can be plotted in terms of H versus Q. The analysis can be continued for bedload transport rates. That is, the parameter τbs can be computed from τs τ*s b (6.134) ρRgD50 and this parameter can be substituted into an appropriate bedload transport equation to obtain q. The volume bedload transport rate Qb is then computed as Qb q·B
(6.135)
6.8.3.3 Engelund-Hansen Method. The method of Engelund and Hansen (1967) also applies specifically for sand-bed streams. It is generally more accurate than the method of Einstein and Barbarossa, to which it is closely allied. The method assumes quasi-uniform material; it is necessary to know only a single grain size D. Roughness height ks is computed from Eq. (6.121). The method uses the Einstein partition. Skin friction is computed using Eq. (6.112). Form drag is computed from the following empirical relation: τ*s f(τ*) where
τb ; τ* ρRgD
τs τ*s b ρRgD
(6.136)
(6.137a, b)
Equation (6.134) is shown graphically in Fig. 6.24. It has two branches, each corresponding to lower-regime and upper-regime flows. The two do not meet smoothly, implying the possibility of a sudden transition. The point of transition is not specified, which suggests the possibility of double-valued rating curves. The lower-regime branch of Eq. (6.136) is given by τ*s 0.06 0.4·(τ*)2
(6.138)
The upper branch satisfies the relation τ∗s τ*
(6.139)
over a range; this implies an upper-regime plane bed. For higher values of Shields stress, τ* again exceeds τ*s implying antidune resistance. 6.8.3.4 Application of the Engelund-Hansen Method. The procedure parallels that of Einstein-Barbarossa relatively closely. It is assumed that the values of S and D as well as the cross-sectional geometry are known. Values of Hs are selected, ranging from a low value to near bankfull. The calculation then proceeds as follows: Hs → Cfs → U
Eq. (6.115) and (6.116b)
Hs → τbs → τ
Eq. (6,116b), (6.137b)
* s
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SEDIMENTATION AND EROSION HYDRAULICS
Chapter Six
FIGURE 6.24 Relation between grain shear stress and total shear stress (after Engelund and Hansen, 1976)
6.54
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SEDIMENTATION AND EROSION HYDRAULICS
Sedimentation and Erosion Hydraulics 6.55 τ*s → τ*
Eq. (6.134); use Equation (6.136) or plot
τ* → τb → H
Eq. (6.137b) and (6.116a)
Q = UH B(H)
Eq. (6.133)
The value of τ*s can then be used to calculate bedload transport rates in a fashion that is completely analogous to the procedure outlined for the Einstein-Barbarossa method. 6.8.3.5 Brownlie method. There are almost as many empirical resistance predictors for rivers as there are sediment transport relations. A fairly comprehensive summary of the older methods can be found in ASCE Manual No. 54 (Vanoni, 1975). A recent empirical method offered by Brownlie (1981a) has proved to be relatively accurate. It does not involve a decomposition of bed shear stress; instead it gives a direct predictor of depthdischarge relations. The complete method can be found in Brownlie (1981a), where the relation is presented for the case of lower-regime dune resistance in a sand-bed stream. It takes the form HS 0.3724(˜qS)0.6539 S0.09188 σ0.1050 g D50
(6.140)
where σg denotes the geometric standard deviation of the bed material, and q˜ denotes a dimensionless water discharge per unit width, given by qw q˜ R gD 50 D50
(6.141)
For known S, D50, and σg, qw, and thus Q qwB is computed directly as a function of depth H.
6.9 SUSPENDED LOAD 6.9.1
Mass Conservation of Suspended Sediment
Suspended sediment differs from bedload sediment in that it can be diffused throughout the vertical column of fluid via turbulence. Here, the local mean volume concentration of suspended sediment is denoted as c. As long as the suspended sediment under consideration is coarse enough not to undergo Brownian motion (i.e., silt or coarser), molecular effects can be neglected. Suspended particles are transported solely by convective fluxes. For an arbitrary volume of sediment-water mixture in the water column, the equation of mass balance of suspended sediment can be written in words as ∂ (6.142) [mass in volume] [net mass inflow rate] ∂t Insofar as the choice of volume V is entirely arbitrary, the following sediment conservation equation, averaged over turbulence-induced fluctuations about the mean, can be obtained:
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SEDIMENTATION AND EROSION HYDRAULICS
6.56
Chapter Six
´c´ ∂c u ∂c v ∂c (w v ) ∂c ∂u´c´ ∂v´c´ ∂w ∂s ∂n s ∂s ∂n ∂z ∂t ∂z
(6.143)
where u, v, and w are the mean flow velocities in the s, n, and z directions, respectively, and the terms u´c´, v´c´, and w ´c´ are sediment fluxes caused by turbulence, also known as Reynolds fluxes. The simplest closure assumption for these terms is
and
c u´c´ Dd ∂ ∂s
(6.144a)
c v´c´ Dd ∂ ∂n
(6.144b)
c w´c´ Dd ∂ ∂z
(6.144c)
where the kinematic eddy diffusivity Dd is assumed to be a scalar quantity. To solve Eq. (6.143), boundary conditions are needed.
6.9.2 Boundary Conditions Equation (6.143), when closed with a Fickian assumption, such as Eq. (6.144a, b, and c), represents an advection-diffusion equation for suspended sediment. The condition of vanishing flux of suspended sediment across (normal to) the water surface defines the upper boundary condition. If uniform steady flow over a flat (when averaged over bedforms) bed is considered, the surface boundary condition for the net vertical flux of sediment reduces to F szz H 0
(6.145)
w´c´ Fsz vsc
(6.146)
where
is the net vertical flux of sediment. The boundary condition at the bed differs from the one at the water surface because it must account for entrainment of sediment into the flow from the bed and for deposition from the flow onto the bed. The mean flux of suspended sediment onto the bed is given by D, where D vscb
(6.147)
denotes the volume rate of deposition of suspended sediment per unit time per unit bed area. Here cb denotes a near-bed value of c. The component of the Reynolds flux of suspended sediment near the bed that is directed upward normal to the bed may be termed the rate of erosion, or more accurately, entrainment of bed sediment into suspension per unit bed area per unit time. The entrainment rate E is thus given by E w´c´
(6.148)
where w´ and c´ denote turbulent fluctuations around both the mean vertical fluid velocity and the mean sediment concentration, respectively. The “overbar” denotes averaging over turbulence. The term near bed used to avoid possible singular behavior at the bed
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SEDIMENTATION AND EROSION HYDRAULICS
Sedimentation and Erosion Hydraulics 6.57
(located at z 0). It is seen from the above equations that the net upward normal flux of suspended sediment at (or rather just above) the bed is given by Fsznear bed vs(Es cb)
(6.149a)
Es ≡ E vs
(6.149b)
where
denotes a dimensionless rate of entrainment of bed sediment into suspension. The required bed boundary condition, then, is a specification of Es. Typically, a relation of the following form is assumed: Es Es(τbs, other parameters)
(6.150a)
where τbs denotes the boundary shear stress caused by skin friction. Furthermore, it is assumed that an equilibrium steady, uniform suspension has been achieved. It follows that there should be neither net deposition on (F sz 0) nor erosion from (F sz 0) the bed. That is, Fsz 0, yielding Es cb
(6.150b)
This relation simply states that the entrainment rate equals the deposition rate; thus, there is no net normal flux of suspended sediment at the bed.
6.9.3 Equilibrium Suspension in a Wide Rectangular Channel Consider normal flow in a wide, rectangular open channel. The bed is assumed to be
FIGURE 6.25 Definition diagram for sediment entrainment and deposition
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SEDIMENTATION AND EROSION HYDRAULICS
6.58
Chapter Six
erodible and has no curvature when averaged over bedforms. The z-coordinate is quasivertical, implying low channel slope S. Similarly, the suspension is assumed to be in equilibrium. That is, c is a function of z alone (Fig. 6.25). The flow and suspension are uniform in s and n and steady in time; thus, Eq. (6.143) reduces to w´c´ vsc 0
(6.151)
It is appropriate to close this equation with the assumption of an eddy diffusivity, as in Eq. (6.144c); thus, Eq. (6.151) becomes c v c 0 Dd d (6.152) s dz Equation (6.152) has a simple physical interpretation. The term vsc represents the rate of sedimentation of suspended sediment under the influence of gravity; it is always directed downward. If all the sediment is not to settle out, there must be an upward flux that balances this term. The upward flux is provided by the effect of turbulence, acting to yield a Reynolds flux. According to Eq. (6.144c), this flux will be directed upward as long as dc/dz 0. It follows that the equilibrium suspended-sediment concentration decreases for increasing z: therefore, turbulence diffuses sediment from zones of high concentration (near the bed) to zones of low concentration (near the water surface).
6.9.4 Eddy Diffusivity Further progress requires an assumption for the kinematic eddy diffusivity Dd. The simple approach taken here is that of Rouse (1957), which involves the use of the Prandtl analogy. The argument is as follows: Fluid mass, heat, momentum, and so on should all diffuse at the same kinematic rate because of turbulence and thus have the same kinematic eddy diffusivity because each is a property of the fluid particles, and the fluid particles are what is being transported by Reynolds fluxes. Although the Prandtl analogy is by no means exact, it has proved to be a reasonable approximation for many turbulent flows. Its application to sediment is more of a problem. Inertial effects might cause the sediment particles to lag behind the fluid, resulting in a lower eddy diffusivity for sediment than for the fluid. Furthermore, the mean fall velocity of sediment grains should reduce their time of residence in any given eddy, again reducing the diffusive effect. If the particles are not too large, however, it may be possible to equate the vertical diffusivity of the sediment with the vertical eddy viscosity (eddy diffusivity of momentum) of the fluid as the first approximation. This is done here. The velocity profile is approximated as logarithmic throughout the depth. To account for the possible existence of bedforms, the turbulent rough law embodied in Eq. (6.127b) is used: u(z) 1 z ln 30 u* kc
(6.153)
Here kc is a composite roughness chosen to include the effect of bedforms, as outlined in Sec. 6.8.2.3. Furthermore, according to Eq. (6.2), the bed shear stress is given by ρu2* ≡ τb
(6.154)
where b is chosen to be close to the bed: i.e.,
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SEDIMENTATION AND EROSION HYDRAULICS
Sedimentation and Erosion Hydraulics 6.59
b « 1 H Now the kinematic eddy viscosity Dd is defined as u τ ρ Dd d dz
(6.155)
(6.156)
where the distribution of fluid shear stresses τ is given by τ τb 1 z H
(6.157)
From the above equations, it is quickly found that Dd κu*z1 z H
(6.158)
where κ 0.4 is Von Karman’s constant. The above relation is the Rousean relation for the vertical kinematic eddy viscosity. The form predicted is parabolic in shape. Although strictly applying to the turbulent diffusion of fluid momentum, it is equated to the eddy diffusivity of suspended sediment mass below. If Dd is averaged in the vertical, the following result is obtained: (6.159) Dd κ u*H 0.0667 u*H 6
FIGURE 6.26 Vertical suspended sediment distribution (after Vanoni, 1961).
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SEDIMENTATION AND EROSION HYDRAULICS
6.60
Chapter Six
This relation is useful to estimate the longitudinal dispersion of fine-grained sediment in rivers and streams.
6.9.5 Rousean Distribution of Suspended Sediment The nominal “near bed” elevation in applying the bottom boundary condition is taken to be z b, where b is a distance taken to be extremely close to the bed: i.e., satisfying condition Eq. (6.155). In the Rousean analysis, this value cannot be taken as z 0 because Eq. (6.153) is singular there. Equation (6.158) is now substituted into Eq. (6.152), which is then integrated from the nominal bed level to distance z above the bed in z. The resulting form can be cast as z
z
Z z
Hdz ln H z dcc Z z b z(H z) b
(6.160)
b
where Z denotes the Rouse number, a dimensionless number given by vs Z κu* Further reduction yields the following profile: Z (H z)/z c cb (H b)/b
(6.161)
(6.162)
Some sample profiles of suspended sediment plotted in Rousean form are provided in Fig. 6.26. Note that from Eq. (6.150b), cb is equal to the dimensionless sediment entrainment rate Es in the case of the present equilibrium suspension. This provides an empirical means to evaluate Es as a function of τbs and other parameters, as will be shown. 6.9.6
Vertically Averaged Concentrations: Suspended Load
Assuming that a value of near-bed elevation b is chosen approximately, Eq. (6.162) can be used to evaluate a depth-averaged volume suspended-sediment concentration C, defined by H (6.163) C 1 c(z)dz Hb Using Eq. (6.162), then C cbI1 (Z,ζb)
(6.164a)
where
z Z (1 ζ)/ ζ Il dζ; ζb (1 ζb)/ζb
ζb b H
(6.164b, c)
In the above relation, ζ z/H; the integral is evaluated easily by means of numerical techniques. Einstein (1950) represented I1 in the form Il (0.216)1ζbIl
(6.165)
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SEDIMENTATION AND EROSION HYDRAULICS
Sedimentation and Erosion Hydraulics 6.61
FIGURE 6.27a Function I1 in terms of ξb = b/H for values of Z:
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SEDIMENTATION AND EROSION HYDRAULICS
6.62
Chapter Six
FIGURE 6.27b Function –I2 in terms of ξb = b/H for values of Z:
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SEDIMENTATION AND EROSION HYDRAULICS
Sedimentation and Erosion Hydraulics 6.63
where I1 is given in tabular form in the attached Fig. 6.27a. The streamwise suspended load qs was seen in Eq. (6.61a) to be given by the relation H
qs c(z)u (z)dz
(6.166)
b
Reducing with the aid of Eqs. (6.153) and (6.162), we find that 1 qs cbu*H Il·ln 30 H I2 kc
(6.167)
Here,
1n(ζ)dζ.
1 (1 ζ)/ζ I2(Z, ζb) ( 1 ζb)/ζb ζb
Z
(6.168)
The integral I2 is again evaluated easily numerically: Einstein provides the relation I2 (0.216)1ζbI2
(6.169)
where I2 is given in tabular form in Fig. 6.27b. Brooks (1963) also proposed an interesting way to calculate suspended load discharge from velocity and concentration parameters. It is apparent that further progress is predicated on a method for evaluating the “reference concentration” cb, or equivalently (for the case of equilibrium suspensions) the sediment entrainment rate Es Such a relation is necessary to model transport of suspended sediment (e.g., Celik and Rodi, 1988).
6.9.7 Relation for Sediment Entrainment A number of relations are available in the literature for estimating the entrainment rate of sediment into suspension Es (and thus the reference concentration cb for the case of equilibrium). Table 6.5 summarizes all the relations that are available. García and Parker (1991) performed a detailed comparison of eight such relations against data. The relations were checked against a carefully selected set of data pertaining to equilibrium suspensions of uniform sand. In this case, it is possible to measure cb directly at some near-bed elevation z b, and to equate the result to Es according to Eq. (6.150b) The data consisted of some 64 sets from 10 different sources, all pertaining to laboratory suspensions of uniform sand with a submerged specific gravity R near 1.65. Information about the bedforms was typically not sufficient to allow for a partition of boundary shear stress in accordance with Nelson and Smith (1989). As a result, the shear stress caused by skin friction alone τbs and the associated shear velocity caused by skin friction u*s, given by τbs ρu2*s
(6.170)
were computed using Eq. (6.114) and the following relation for ks, ks 2 D
(6.171)
or a similar method.
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0
0.65γoT cb 1γT
0.65 cb 3 (1 λb1)
Zyserman and Fredsoe (1994)
García and Parker (1991)
Akiyama and Fukushima (1986)
Celik and Rodi (1984)
Van Rijn (1984)
0.331(θ' 0.045)1.75 cb 0.331 (θ' –0.045)1.75 1 0.46
AZu5 Es 1 A Zu5 0.3
Z Es 3 1012Z10 1c ; Zc Z Zm Z Es 0.3; Z Zm
Es 0; Z Zc
k Cm cb 0 I
*
D T1.5 cb 0.015 s 0 b D .3
c k1 k2 u* Ω 1 Itakura and Kishi b vs τ* (1980)
Smith and McLean (1977)
Engelund and Fredsoe (1976; 1982)
q* cb 0.5 23.2 τ*s
Formula
Ao
exp(A2o) ∞ exp(– ξ2)dξ
2 *
m
1
b
s
0.05
A 1.310-7
u g0.5 12R Zu *s Rnp; u*s Um; C' 18log b ; n 0.6; vs C' 3Ds
u Z * R0.5 ; Zc 5; Zm 13.2 vs p
η z/H; ηb 0.05; ko 1.13
k Cm 0.034 1 s H
(u s)2 θ' * RgDs
b
dη;
; β 1.0
vs/0.4u*
0.25
4
η gRuH Uv; I = 1ηη 1η 0.06
; ∆b is the mean dune height
1/3
gR D* Ds v2
βπ 6 τ*s 0.06
k 1; Ao = *3 k4; τ
1+
k1 0.008; k2 0.14; k3 0.143 ; k4 2.0
* Ω τ k4 + k3
;p
0.5
τ*s τc* T ; γo 2.4·103 τc*
λb
βpπ τ*s 0.06 6 0.027(R 1)τ*s
Parameters
Existing formulas to estimate sediment entrainment or near-bed concentration under equilibrium conditions.
b 2Ds
b 0.05H
b 0.05H
b 0.05H
∆ b b if ∆b known 2 else b ks* bmin 0.01H
b 0.05H
b αoτ*s τ∗cDs ks αo = 26.3
b 2Ds
b 2Ds
Reference Height
6.64
Einstein (1950)
Author
TABLE 6.5
SEDIMENTATION AND EROSION HYDRAULICS
Chapter Six
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SEDIMENTATION AND EROSION HYDRAULICS
Sedimentation and Erosion Hydraulics 6.65
Es
Zu FIGURE 6.28 Sediment entrainment function (after García and Parker, 1991).
The data covered the following ranges: 0.0002
0.06
0.70
7.50
H/D:
240
2400
Rep
3.50
37.00
Es: u*s/vs:
The range of values of Rep corresponds to a grain size ranging from 0.09 mm to 0.44 mm. Except for the relatively small values of H/D, the values cover a range that includes typical field sand-bed streams. Three of the relations for Es performed particularly well and are presented here. The first is the relation of García and Parker (1991). The reference level is taken to be 5 percent of the depth: that is, b ζb 0.05. H
(6.172)
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SEDIMENTATION AND EROSION HYDRAULICS
6.66
Chapter Six
FIGURE 6.29 Comparison of predicted and observed near-bed concentration for García-Parker function
The good performance of this relation is not overly surprising because the relation was fitted to the data. The relation takes the form AZ5u Es (6.173a) 1 A Z5u 0.3 where
A 1.3 107
(6.173b)
u Zu *s Rep0.6 vs
(6.173c)
and
Equation (6.173a) is compared against the data in Fig. 6.28. Predicted values of Es are compared with observed values in Fig. 6.29. A second relation that performed well is that of Van Rijn (1984), which takes the form Es 0.015 D (τ*s /τ*c 1)1.5Rep0.2 (6.174) b where τ*s denotes the Shields stress caused by skin friction, given by Eq. (6.112). For the purposes of the present comparison, b was again set equal to 5 percent of the depth: i.e., Eq. (6.172) was used. Van Rijn computed τbs from relations that are similar to Eqs. (6.115) and (6.116b). Van Rijn's relations are 2 1 Cfs ln 12 H (6.175a) ks
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SEDIMENTATION AND EROSION HYDRAULICS
Sedimentation and Erosion Hydraulics 6.67
where, for uniform material, ks 3 D
(6.175b)
Note that in Eq. (6.175), the total depth H is used, in contrast to Eq. (6.115) where Hs is used. In performing the comparison, García and Parker (1991) estimated τ*s from a fit to the
FIGURE 6.30 Comparison of predicted and observed near-bed concentration for van Rijn function
FIGURE 6.31 Comparison of predicted and observed near-bed concentration for Smith-McLean function.
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SEDIMENTATION AND EROSION HYDRAULICS
6.68
Chapter Six
Shields curve due to Brownlie (1981a). This fit is given by Eq. (6.44). Predicted and observed values of Es are presented in Fig. 6.30. A third relation that performs well is that of Smith and McLean (1977) which can be expressed as γo(τ*s / τ*c 1) Es 0.65 (6.176a) 1 γo(τ*s / τ*c 1) where γo 0.0024
(6.176b)
The value b at which E is to be evaluated is given by the following relation: b 26.3(τ*s / τ*c 1)D ks
(6.176c)
where ks denotes the equivalent roughness height for a fixed bed. For the purpose of comparison, García and Parker used Eq. (6.171) to evaluate ks and used Eq. (6.115) to evaluate τbs. Critical Shields stress was evaluated with Eq. (6.44). Predicted and observed values of Es are shown in Fig. 6.31. 6.9.8 Entrainment Relation for Sediment Mixtures García and Parker (1991) provided a generalized treatment for the entrainment rate in the case of mixtures. Let the grain-size range of bed material be divided into N subranges, each with mean size φj on the phi scale and geometric mean diameter Dj 2φj where j 1...N. Let Fj denote the volume fraction of material in the surface layer of the bed in the jth grain range. In analogy to Eq. (6.148), it is assumed that Ej vsj Fj E(Zuj)
(6.177a)
where Ej denotes the volume entrainment rate for the jth subrange and the functional relation between Es and Zuj is given by Eq. (6.173a). The parameter Zuj is specified as
u D Zuj λm *s Repj0.6 j vsj D50
0.2
(6.177b)
In the above relations, vsj denotes the fall velocity of grain size Dj in quiescent water, D50 denotes the median size of the surface material in the bed gD j Dj R Repj ν
(6.177c)
m 1 0.288σφ
(6.177d)
and the parameter λm is given by
Here, σφ denotes the arithmetic standard deviation of the bed surface material on the phi scale, given by Eq. (6.30). The García-Parker relation for mixtures reduces smoothly to the relation for uniform material in the limit as σφ → 0. It was developed and tested with three sets of data from two rivers: the Rio Grande and the Niobrara River. Recently, the García-Parker formulation also has been used to interpret observations of sediment entrainment into suspension by bottom density currents (García and Parker, 1993).
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SEDIMENTATION AND EROSION HYDRAULICS
Sedimentation and Erosion Hydraulics 6.69
6.9.9 Example of Computation of Sediment Load and Rating Curve. Consider the example of a stream in Sec. 6.8.2. For this stream, S 0.0004 and D 0.35 mm (uniform material). At bankfull flow, the stream width is 75 m. For flows below bankfull, the following relation holds: Q 0.1 B Bbf Qbf
where the subscript bf denotes bankfull. Assume that the stream is wide enough to equate the hydraulic radius Rh with the cross-sectionally averaged depth H. Compute the depth-discharge relations for flows up to bankfull (lower regime only) using the Engelund-Hansen method. Plot H versus Q. Use the results of the EngelundHansen method to compute values of τs* as well. Use the values of τ*s to compute the bedload discharge Qb qb B using the AshidaMichiue formulation. For each value of H and U, back-calculate the composite roughness kc. Then compute the suspended load Qs qs B from the Einstein formulation and the relation for Es by García and Parker. Plot Qb, Qs, and QT Qb Qs as functions of water discharge Q. Solution: In this example, the flow depth, bedload discharge, and suspended load discharge are computed as a function of flow discharge for a stream with the following properties: S
= 0.0004
Ds = 0.35 mm = 3.5 10 -4 m R = 1.65 B = 75 m at bankfull H = 2.9 m at bankfull
The calculations are performed for flows up to bankfull. For flows below bankfull, the following relation is used to calculate the stream width:
B Q 0.1 UHB 0.1 Bbf Qbf Qbf where the subscript bf indicates bankfull values. Solving for the stream width B,
UH B Bbf Qbf
0.1 1/0.9
(i)
(ii)
The methods used to determine Q, Qb, Qs, and Qbf are described below. A computer program can be written, or a spreadsheet can be used, to perform the necessary calculations. All computations and results are summarized in Table 6.6. 6.9.9.1 Depth-discharge calculations. The depth-discharge relation is computed using the Engelund-Hansen method. The calculations are performed by assuming a value for Hs (the flow depth that would be expected in the absence of bedforms), then calculating the actual flow depth (H) and the flow discharge (Q). Hs is varied between 0.22 m and the bankfull value of 2.9 m. The first step in calculating the depth-discharge relation is to compute the resistance coefficient caused by skin drag (Cfs) from Hs:
H 1 Cfs 1n 11 s ks
2
(iii)
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0.00314
0.00261
0.00236
0.00220
0.00209
0.00201
0.00194
0.00188
0.00184
0.00180
0.00176
0.00173
0.00170
0.00167
0.00165
0.00163
0.00161
0.00159
0.00157
0.00156
0.00154
0.00153
0.00152
0.00150
0.00150
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
1.10
1.20
1.30
1.40
1.50
1.60
1.70
1.80
1.90
2.00
2.10
2.20
2.30
2.40
2.416
Cfs
0.10
Hs
TABLE 6.6
2.512
2.502
2.440
2.376
2.311
2.244
2.176
2.107
2.036
1.963
1.888
1.811
1.732
1.651
1.566
1.478
1.387
1.291
1.190
1.083
0.969
0.844
0.706
0.548
0.353
U (m/s)
1.673
1.662
1.593
1.524
1.455
1.385
1.316
1.247
1.177
1.108
1.039
0.970
0.900
0.831
0.762
0.693
0.623
0.554
0.485
0.416
0.346
0.277
0.208
0.139
0.069
*s
2.008
2.001
1.958
1.913
1.867
1.820
1.772
1.722
1.671
1.619
1.564
1.508
1.450
1.388
1.325
1.258
1.187
1.111
1.031
0.943
0.846
0.737
0.608
0.443
0.152
*
Width
2.90
2.89
2.83
2.76
2.70
2.63
2.56
2.49
2.41
2.34
2.26
2.18
2.09
2.00
1.91
1.82
1.71
1.60
1.49
1.36
1.22
1.06
0.88
0.64
0.22
75.00
74.94
74.54
74.13
73.70
73.26
72.79
72.30
71.78
71.24
70.67
70.05
69.40
68.70
67.94
67.12
66.22
65.22
64.09
62.80
61.28
59.43
57.04
53.54
45.28
Depth H(m) B (m)
Flow
Computation of Total Sediment Load.
546.35
541.87
514.00
486.40
459.07
432.01
405.24
378.78
352.64
326.83
301.37
276.29
251.59
227.32
203.50
180.15
157.34
135.09
113.48
92.58
72.50
53.36
35.36
18.78
3.52
Discharge (m3/s) Q
29.53002
29.21083
27.24386
25.32585
23.45795
21.64136
19.87737
18.16739
16.51293
14.91565
13.37737
11.90008
10.48599
9.13758
7.85765
6.64937
5.51640
4.46304
3.49443
2.61681
1.83808
1.16862
0.62296
0.22362
0.01296
q*b
0.000778
0.00077
0.000718
0.000667
0.000618
0.00057
0.000524
0.000479
0.000435
0.000393
0.000352
0.000313
0.000276
0.000241
0.000207
0.000175
0.000145
0.000118
9.21E-05
6.89E-05
4.84E-05
3.08E-05
1.64E-05
5.89E-06
3.41E-07
qb
0.0583429
0.0576648
0.0534987
0.0494585
0.0455465
0.0417649
0.0381161
0.0346026
0.0312273
0.0279932
0.0249034
0.0219616
0.0191715
0.0165376
0.0140645
0.0117577
0.0096231
0.0076678
0.0058999
0.0043292
0.0029674
0.0018297
0.0009360
0.0003154
0.0000155
Qb
0.002583
0.002629
0.00294
0.003298
0.003711
0.00419
0.004746
0.005398
0.006163
0.007069
0.008147
0.009439
0.010998
0.012897
0.015228
0.01812
0.021746
0.026344
0.032245
0.039902
0.049901
0.062825
0.078267
0.088314
0.019602
kc
12.3864
12.3453
12.0854
11.8197
11.548
11.2697
10.9843
10.6913
10.3901
10.0799
9.75981
9.42887
9.08589
8.72944
8.3578
7.96885
7.55992
7.12756
6.66722
6.17265
5.63483
5.03995
4.36472
3.56378
2.51997
Zu
0.033651
0.033158
0.030147
0.027265
0.024516
0.021906
0.019441
0.017123
0.014957
0.012944
0.011087
0.009385
0.007839
0.006448
0.00521
0.00412
0.003176
0.002372
0.001703
0.00116
0.000737
0.000422
0.000206
7.47E-05
1.32E-05
Es
Rouse No.Z 2.792324511
2.02073406
2.165604409
1.706170815
1.513554374
1.54381954
1.460859923
1.343850856
0.10667 1.311551538
0.10648 1.313813452
0.10531 1.328407943
0.1041
0.10285 1.360235314
0.10155 1.377669964
0.10019 1.396282541
0.09878 1.416224513
0.09731 1.437677192
0.9577
0.09414 1.486041241
0.9243
0.09062
0.08869 1.577374615
0.08663 1.614926151
0.08441 1.657421368
0.082
0.07935 1.763054519
0.07641 1.830896085
0.07309 1.914196574
0.06923
0.0646
0.05868 2.384016902
0.0501
0.02936 4.764548652
u*
0.28
0.28
0.27
0.27
0.26
0.265
0.26
0.25
0.24
0.245
0.235
0.23
0.225
0.215
0.208
0.205
0.195
0.188
0.175
0.145
0.15
0.14
0.13
0.105
0.048
I1
0.59
0.59
0.59
0.58
0.57
0.57
0.56
0.545
0.535
0.53
0.525
0.51
0.49
0.48
0.47
0.45
0.42
0.41
0.405
0.5
0.39
0.38
0.32
0.26
0.13
2I2
0.014027
0.013716
0.011335
0.009779
0.007974
0.006896
0.005651
0.004475
0.003484
0.002891
0.002177
0.001665
0.001253
0.000887
0.000621
0.000435
0.000283
0.000175
9.64E-05
4.04E-05
2.16E-05
8.26E-06
2.68E-06
4.23E-07
7.36#-09
qs
1.052007
1.027782
0.844959
0.724892
0.587721
0.505155
0.411357
0.323569
0.250092
0.20598
0.153872
0.116671
0.86947
0.60905
0.042174
0.029231
0.018721
0.011424
0.006181
0.002538
0.001325
0.000491
0.000153
2.26E-05
3.33E-07
Qs
1.1103496
1.0854472
0.8984579
0.7743510
0.6332673
0.5469200
0.4494731
0.3581711
0.2813193
0.2339729
0.1787758
0.1386321
0.1061188
0.0774422
0.0562386
0.0409882
0.0283438
0.0190916
0.0120810
0.0068674
0.0042927
0.0023207
0.0010887
0.0003380
0.0000158
Qt
SEDIMENTATION AND EROSION HYDRAULICS
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SEDIMENTATION AND EROSION HYDRAULICS
Sedimentation and Erosion Hydraulics 6.71
where is the von Karman constant (0.4) and ks is given by ks 2.5 Ds 2.5(3.5 104) 8.75 104m
(iv)
The depth-averaged flow velocity (U) can be found from Cfs and Hs: U
g CH S s
(v)
fs
The Shields stress caused by skin friction (τs*) is given by τ HS τ*s bs s ρgRDs RDs
(vi)
According to Engelund-Hansen, the total Shields stress for the lower regime can be found from the following relation: τ*
0.06 τ 0 .4 ∗
s
(vii)
The flow depth can be calculated from the Shields stress as follows: τ*RD H s S
(viii)
Finally, the discharge can be calculated from the results of Eqs. (v) and (viii): Q UHB
(ix)
where B must be adjusted according to Eq. (ii) for flows less than bankfull. A plot of the depth-discharge relation is shown in Fig. 6.32. 6.9.9.2 Bedload discharge calculations. The dimensionless bedload transport rate (q*) is found from the Ashida-Michiue formulation: q* 17(τ*s τ*c)[(τ∗s)0.5 (τ*c)0.5]
(x)
where τs* is calculated in Eq. (vi) and τc* is taken to be 0.05. The bedload transport rate per unit width (qb) is given by qb q* gR D D s s
(xi)
Therefore, the bedload transport rate (in m3/s) is given by Qb qbB
(xii)
Again, B must be adjusted according to Eq. (ii) for flows less than bankfull. 6.9.9.3 Sediment load discharge calculations. The Einstein formulation is used to compute the suspended load transport rate per unit width (qs):
qs 1cbu*H I1ln 30 H I2 κ kc
(xiii)
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SEDIMENTATION AND EROSION HYDRAULICS
Chapter Six
FIGURE 6.32 Example of flow discharge rating curve
6.72
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SEDIMENTATION AND EROSION HYDRAULICS
Sedimentation and Erosion Hydraulics 6.73
where H S u* g
(xiv)
If the suspension is assumed to be at equilibrium, cb Es. The dimensionless rate of entrainment (Es) is calculated with the relation of García and Parker (1991): AZ5u Es 1 A Z5u 0.3
(xv)
where A is equal to 1.3 10-7 and
and
u Zu *s Rep0.6 vs
(xvi)
u*s g H sS
(xvii)
R gD s Ds Rep ν
(xviii)
Notice that for the entrainment formulation, the shear velocity associated with skin friction u*s must be used. The temperature is assumed to be about 20°C; therefore, the kinematic viscosity is about 106 m2/s. An iterative method, or Eq. (6.38), is used to calculate the terminal fall velocity of the sediment particles vs, which is found to be 5.596 102 m/s. The composite roughness (kc) is calculated according to the following relation: κU kc 11 H exp u*
(xix)
The parameters I1 and I2 are found by numerical integration of the following equations: 1ζ
ζ
I1 b 1 ζb
Z
1
Z
dζ
(xx)
ln(ζ)dζ
(xxi)
ζb
and ζb I2 1 ζb
1ζ
ζ
Z
1
Z
ζb
where ςb is taken to be 0.05 and vs Z (xxii) κu* The numerical integrations can be performed with Numerical Recipes subroutines (Press et al., 1986) or can be obtained from Figs. 6.27a and b. The suspended load transport rate per unit width calculated according to Eq. (xiii) is used to compute the suspended load transport rate (in m3/s): Qs qsB
(xxiii)
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SEDIMENTATION AND EROSION HYDRAULICS
6.74
Chapter Six
For flows less than bankfull, B must be adjusted according to Eq. (ii). 6.9.9.4 Determination of bankfull flow discharge (Qbf). The flow discharge at bankfull (Qbf) is determined by assuming that up to bankfull flow, lower regime conditions exist. The bankfull flow depth for this stream is assumed to be 2.9 m. Then, for bankfull flow, the total shear stress τ* is HS 2.9 · .0004 2.01 τ* 1.65 · 3.5 104 RDs
(xxiv)
From Engelund-Hansen, τ*s 0.06 0.4(τ*)2 0.06 0.4 · (2.01)2 1.67
(xxv)
τ*sRDs 1.67 1.65 (3.5 104) Hs 2.42 m 0.0004 S
(xxvi)
H 1 Cfs ln 11s ks κ
U
2
1 2.42 ln 11 0.4 8.75 104
2
1.5 103
g CH S 9.8 1 1 ·. 52 .42 1· 00 .0 0 0 4 2.51 m/s s
fs
3
(xxvii)
(xxviii)
and Qb UHB 2.51 2.9 75 546.35 m3/s
(xxix)
A plot of Qb, Qs, and QT Qb Qs as functions of water discharge are shown in Fig. 6.33. For flows up to 100 m3/s, the bedload discharge is larger than the suspended load discharge. As the flow discharge increases, the suspended load is much larger than the bedload all the way up to bankfull flow conditions. Also notice that the composite roughness kc increases first with flow discharge for low flows but, from then on, decreases monotonically as the bedforms begin to be washed out by the flow. For bankfull conditions, the bedforms have a small effect on flow resistance in this particular problem.
6.10 DIMENSIONLESS RELATIONS FOR TOTAL BED MATERIAL LOAD IN SAND-BED STREAMS 6.10.1 Form of the Relations In the analysis presented in previous sections, the guiding principle has been the development of mechanistically accurate models of the bedload and suspended load components
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SEDIMENTATION AND EROSION HYDRAULICS
FIGURE 6.33 Example sediment discharge rating curves for bedload, suspended load, and total load
Sedimentation and Erosion Hydraulics 6.75
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SEDIMENTATION AND EROSION HYDRAULICS
6.76
Chapter Six
of bed material load. The total bed material load is then computed as the sum of the two. That is, where q denotes the volume bedload transport rate per unit width and qS denotes the volume suspended load transport rate per unit width (bed material only), the total volume transport rate of bed material per unit width is given by qt q qs
(6.178)
Another simpler approach is to ignore the details of the physics of the problem and instead use empirical techniques, such as regression analysis, to correlate dimensionless parameters involving qt to dimensionless flow parameters inferred to be important for sediment transport. This can be implemented in the strict sense only for equilibrium or quasi-equilibrium flows: i.e., for near-normal conditions. The resulting relations are no better than the choice of dimensionless parameters to be correlated. They also are less versatile than physically based relations because their application to nonsteady, nonuniform flow fields is not obvious. On the other hand, they have the advantage of being relatively simple to use and of having been calibrated to sets of both laboratory and field data often deemed to be trustworthy. Here, four such relations are presented, those of Engelund and Hansen (1967), Brownlie (1981a), Yang (1973), and Ackers and White (1973). They apply only to sandbed streams with relatively uniform bed sediment. The first two relations are the most complete because each is presented as a pair of relations for total load and hydraulic resistance. The latter two are presented as relations for total load only. In most cases, it will be necessary for the user to specify a relation for hydraulic resistance as well to perform actual calculations; the latter relations for load give no guidelines for this. The importance of using transport and hydraulic resistance relations as pairs cannot be overemphasized. Consider, for example, the simplest generalization beyond the assumption of normal flow: i.e., the case of quasi-steady, gradually varied, one-dimensional flow. The governing equations for a wide rectangular channel can be written as d V2 H S Sf ds 2g
(6.179a)
UH qw
(6.179b)
τb U2 Sf Cf gH ρgH
(6.180)
where the friction slope Sf is given as
A slightly more general form for nonrectangular channels is d 1 Q2 2 ξb S Sf ds 2 gA
(6.181a)
UA Q
(6.181b)
where A is the channel cross-sectional area and the friction slope Sf is given as τb U2 Cf Sf ρgRh gRh
(6.182)
In the above equations, Rh denotes the hydraulic radius and ξb denotes the water surface elevation above the deepest point in the channel. Note that in the case of normal flow, the momentum equations reduce to Sf S, or τb Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
SEDIMENTATION AND EROSION HYDRAULICS
Sedimentation and Erosion Hydraulics 6.77
ρgHS for the wide rectangular case and τb ρgRhS for the nonrectangular case. However, in the case of gradually varied flow, Sf ≠ S; in this case, the bed slope S cannot be used as a basis for calculating sediment transport. The appropriate choice is Sf, so that from Eq. (6.182), for example, τb ρgRhSf
(6.183)
For the case of gradually varying flow, then, it should be apparent that the friction slope necessary to perform sediment transport calculations must be obtained from a predictor of hydraulic resistance. A few parameters are introduced here. Let Q denote the total water discharge and Qst denote the total volume bed material sediment discharge. Furthermore, let Ba denote the “active” width of the river over which bed material is free to move. In general, Ba is usually less than water-surface width B as a result of the common tendency for the banks to be cohesive, vegetated, or both. Thus, it follows that Q Bqw
(6.184a)
Qst Baqt
(6.184b)
and
One dimensionless form for dimensionless total bed material transport is qt*: qt q*t (6.185) R gD D where D is a grain size usually equated to D50. Another commonly used measure is concentration by weight in parts per million, here called Cs, which can be given as ρ Qst Cs 106 s ρQ ρsQst
(6.186)
6.10.2 Engelund-Hansen Relations 6.10.2.1 Sediment transport. This relation is among the simplest to use for sediment transport and also among the most accurate. It was determined for a relatively small set of laboratory data, but it also performs well as a field predictor. It takes the form Cf qt* 0.05 (τ*)5/2
(6.187)
where Cf is the total resistance coefficient (skin friction plus form drag) and τ* denotes the total (skin friction plus form drag) Shields stress based on the size D50. 6.10.2.2 Hydraulic resistance. The hydraulic resistance relation of Engelund and Hansen (1967) has already been introduced; it must be written in several parts. The key relation for skin friction is
Rhs U C1/2 fs 2.5 · 1n 11 ks g RS hs
(6.188a)
where ks (2 2.5) · D50. Here, Rhs denotes the hydraulic radius caused by skin friction, which often can be approximated by Hs. The relation for form drag can be written in the following form: Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
SEDIMENTATION AND EROSION HYDRAULICS
6.78
Chapter Six
τ*s f(τ*)
(6.188b)
τ*s 0.06 0.4 · (τ*)2
(6.188c)
where for lower regime,
and for upper regime, τ* 1
τ*
τ * s
[0.298 0.702 · (τ )
τ* 1
* 1.8 (1/1.8)
]
(6.188d)
An approximate condition for the transition between lower and upper regime is τ*s 0.55
(6.188e)
Computational procedure for normal flow. The water discharge Q, slope S, and grain size D50 must be known. In addition, channel geometry must be known so that B, Ba, A, H, P, and Rh are all known functions of stage (water-surface elevation) ξ. The procedure is best outlined assuming that Rhs is known and that Q is to be calculated, rather than vice verse. For any given value of Rhs (or Hs), U can be computed from Eq. (6.188a). Noting that τs* RhsS/(RD50) and τ* RhS/(RD50), τ*, and thus Rh can be computed from Eq. (6.188b-e). The plot of Rh versus ξ is used to determine ξ, which is then used to determine B, Ba, H, A, P, and so on. Discharge Q is then given by Q UBH. In actual implementation, this process is reversed (Q is given and Rhs and so forth are computed). This requires an iterative technique; Newton-Raphson is not difficult to implement. Once the calculation of hydraulic resistance is complete, it is possible to proceed to the computation of total bed material load Qst. The friction coefficient Cf is given by (gRhS)/U2. Putting the known values of Cf and τ* into (6.187), qt*, and thus qt can be computed. It follows that Qst qtBa. Computational procedure for gradually varied flow. To implement the method for gradually varied flow, it is necessary to recast the above formulation into an algorithm for friction slope Sf, which replaces S everywhere in the formulation of Eqs. (6.188a-e). The formulation is then solved in conjunction with Eqs. (6.179a and b) or Eqs. (6.181a and b) to determine the appropriate backwater curve. Once Cf and τb are known everywhere, the sediment transport rate can be calculated from Eq. (6.187).
6.10.3 Brownlie Relations 6.10.3.1 Sediment transport. The Brownlie relations are based on regressions of more than 1000 data points pertaining to experimental and field data. For normal or quasi-normal flow, the transport relation takes the form
Rh Cs 7115cf (Fg Fgo)1.978 S 0.6601 D50
0.3301
(6.189a)
where Fg U R gD 50
(6.189b)
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Sedimentation and Erosion Hydraulics 6.79
Fgo 4.596(τ*c)0.5293 S0.1045 σ0.1606 g
(6.189c)
τ∗c 0.22Y 0.06 · 107.7Y
(6.189d)
Y Re0.6 p
(6.189e)
and
In Eq. (6.189a), cf 1 for laboratory flumes and 1.268 for field channels. The parameters τ∗c and Rep are the ones previously introduced in this chapter. 6.10.3.2 Hydraulic resistance. The Brownlie relations for hydraulic resistance were determined by regression from the same set of data used to determine the relation for sediment transport. The relation for lower regime flow is Rh S 0.3724(˜qS)0.6539 S 0.09188 σg0.1050 D50
(6.190a)
The corresponding relation for upper regime flow is Rh S 0.2836(˜qS)0.6248 S 0.08750 σ0.08013 g D50
(6.190b)
In the above relations, qw q˜ (6.190c) gD 50 D50 The distinction between lower and upper regime is made as follows. For S 0.006, the flow is always assumed to be in upper regime. For S 0.006, the largest value of Fg at which lower regime can be maintained is taken to be Fg 0.8F´g
(6.190d)
and the smallest value of Fg for which upper regime can be maintained is taken to be Fg 1.25Fg´
(6.190e)
Fg´ 1.74S1/3
(6.190f)
In the above relations,
6.10.3.3 Computational procedure for normal flow. It is necessary to know Q, S, D50, σg, and cross-sectional geometry as a function of stage. The computation is explicit, although trial and error may be required to determine the flow regime. Hydraulic radius is computed from Eq. (6.190a) or Eq. (6.190b), and the result can be substituted into Eq. (6.189a) to determine the concentration Cs in parts per million by weight. The transport rate Qst is then computed from Eq. (6.186). 6.10.3.4 Computational procedure for gradually varied flow The Brownlie relation is not presented in a form which obviously allows for extension to gradually varied flow. The most unambiguous procedure, however, is to replace S with Sf in the resistance relation, and couple it with a backwater calculation on order to determine Sf . The friction slope is then substituted into Eq. (6.189a) in place of the bed slope in order to determine the sediment transport rate.
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6.80
Chapter Six
6.10.4 The Ackers–White Relation Like the Brownlie relation for sediment transport, this relation is based on a massive regression. Several years after is was presented, a corresponding relation for hydraulic resistance was also presented. The relation for hydraulic resistance, however, does not appear to be among the best predictors. As a result, only the load equation is presented here. It takes the form
A 1
c s D50 U Cs 106 Rh u* where
Fgr
n
(6.191a)
m
aw
n U U* U’*1n Fgr ; U’* R gD 50 Rh 32 log 10 D50
(6.191b,c)
The parameters n, and Aaw are determined as a functions of Dgr, where Dgr Rep2/3
(6.191d)
in the following fashion. If Dgr 60, then n 0; m 1.5
(6.191e and f)
Aaw 0.17; c 0.025
(6.191g and h)
and If 1 Dgr 60, then
9.66 1.34 m Dgr
n 1 0.56log(Dgr);
(6.191i and j)
0.23 Aaw 0.14 D gr
(6.191k)
log(c) 2.86log(Dgr) [log(Dgr)]2 3.53
(6.191)
Note that all logarithms here are base 10, and u* retains its previously introduced meaning as shear velocity.
6.10.5 Yang Relation This relation also was determined by regression. Its form is
US US log(Cs) a1 a2 log c vs vs
(6.192a)
where
vsD50 u* a1 5.435 0.286 log 0.457 log vs
(6.192b)
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FIGURE 6.34a Ratio of concentration calculated by the Engelund and Hansen (1967) technique to observed concentration as a function of observed concentration, for field data.
Sedimentation and Erosion Hydraulics 6.81
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Chapter Six
FIGURE 6.34b Ratio of concentration calculated by the Engelund and Hansen (1967) technique to observed concentration as a function of observed concentration, for laboratory data.
6.82
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SEDIMENTATION AND EROSION HYDRAULICS
FIGURE 6.34c Ratio of concentration calculated by the Brownlie (1981b) technique to observed concentration as a function of observed concentration, for laboratory data.
Sedimentation and Erosion Hydraulics 6.83
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Chapter Six
FIGURE 6.34d Ratio of concentration calculated from Brownlie (1981b) technique to observed concentration as a function of observed concentration, for field data.
6.84
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SEDIMENTATION AND EROSION HYDRAULICS
FIGURE 6.34e Ratio of concentration calculated by the Ackers and White (1973) technique to observed concentration as a function of observed concentration, for laboratory data.
Sedimentation and Erosion Hydraulics 6.85
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Chapter Six
FIGURE 6.34f Ratio of concentration calculated by the Ackers and White (1973) technique to observed concentration as a function of observed concentration, for field data.
6.86
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FIGURE 6.34g Ratio of concentration calculated by the Yang (1973) technique to observed concentration as a function of observed concentration, for laboratory data.
Sedimentation and Erosion Hydraulics 6.87
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Chapter Six
FIGURE 6.34h Ratio of concentration calculated by the Yang (1973) technique to observed concentration as a function of observed concentration, for field data.
6.88
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Sedimentation and Erosion Hydraulics 6.89
vsD50 u* a1 1.799 0.409 log 0.314l og vs
(6.192c)
and Uc denotes a critical flow velocity given by
2.05 if u D 70 2.5 if 1.2 u D 70. U v logu D 0.06 *
*
c
s
50
*
50
(6.192d)
50
Not that the logarithms are all base 10 and that vS retains its previous meaning as fall velocity.
6.10.6 Comparison of the Relations Against Data In the following eight diagrams (Fig. 6.34a-h) taken from Brownlie (1981b) all four relations are compared against first laboratory, then field data. The plots are in terms of the ratio of calculated versus observed concentration as a function of observed concentration Cs in parts per million by weight. In the case of a perfect fit, all the data would fall on the line corresponding to a ratio of unity. The middle dotted line on each diagram shows the median value of this ratio; the upper and lower dotted lines correspond to the 84th percentile and the 16th percentile. The closer the median value is to unity and the smaller the spread is between the two dotted lines, the better is the predictor. The Engelund-Hansen relation is seen to be a good predictor of both laboratory and field data despite its simplicity. The Brownlie relation gives the best fit of both the laboratory and field data shown. This is partly to be expected because the relation was determined by regressing against the data shown in the figures. The Ackers-White relation predicts the laboratory data essentially as well as the Brownlie relation does, but its predictions of field data are relatively low. The Yang equation does a good job with the laboratory data but a rather poor job with the field data.
6.11 HYDRAULICS OF RESERVOIR SEDIMENTATION 6.11.1 Introduction The construction of reservoirs allows for the controlled storage of water.To develop a successful reservoir, the characteristics of the sites sediment transport must be considered. As a matter of course, water backed up behind a dam will experience a marked decrease in sediment-carrying capacity. As a result, if site characteristics are correct, large quantities of sediment will be deposited within the reservoir basin. Over time, the reservoir will, in effect, fill with sediment, greatly decreasing its storage capacity. In 1988, Morris and Fan published an excellent handbook on reservoir sedimentation. When designing a reservoir, it is important to predict the progress of sedimentation. In practice, these predictions are often carried out using empirical and semiempirical methods that have been developed through observation and measurements of operating reservoirs. Although these methods do provide helpful design information, the drawback is that Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
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6.90
Chapter Six
they are not firmly rooted in the physics of sediment transport. Instead, they provide a prediction based on a synthesis of past observations. As a result, an engineer conducting these calculations easily loses touch with the basic mechanisms governing reservoir sedimentation. Sedimentation is treated as a bulk process, and the relative role of bed-load versus suspended-load transport is not always fully understood. The following exercise presents a view of reservoir sedimentation based on theoretical relations. A gorgelike reservoir is considered to allow for a 1-D model (Hotchkiss and Parker, 1991). The following conditions are given: the flow per unit width qw 1.427 m2/s is taken as constant. The stream has an initial slope S 0.0003. The sediments mean diameter and fall velocity are Ds 0.3 mm and vs 4.25 cm/s respectively. Suppose a reservoir is placed at some point on the river so that the water surface is raised and held at an elevation equal to 10 m above the elevation of the initial bed at the dam site. Obviously, the dam will generate a backwater effect, which in turn will reduce the flows ability to transport sediment through the reservoir. The quasi-steady state approximation will be used to develop a model of reservoir sedimentation based on the governing equations of conservation of momentum, bedload and suspended-load relations, and the Exner equation. The model will be used to predict the level of reservoir sedimentation and delta progression for time intervals of 2, 5, 10, 20, and 30 years. First, the model will be run considering bed-load transport only; second, suspended load also will be included to help identify the relative roles these two forms of transport play. The flow discharge per unit width qw used herein is equivalent to the “dominant” water discharge which, if continued constant for an entire year, would yield the mean annual sediment discharge. Of course, it is impractical to assume that a model as simple as the one presented here could replace the empirical methods of predicting reservoir sedimentation. After all, a steady flow, 1-D, constant reservoir-elevation model seriously limits the model’s application, and transport relations are not easily transposed from site to site. Still, the following provides an understanding of the physical mechanisms causing reservoir sedimentation. An ideal reservoir-sedimentation model would be based in sediment transport physics while respecting (and matching) the vast quantity of empirical observations available.
6.11.2 Theoretical Considerations As in any sediment transport study, it is first necessary to identify the appropriate resistance and bedload transport relations that hold for the site under consideration. For this model, the following relations have been chosen: τ* q*b 11.2τ*1,5 1 *c τ
(6.193)
U u*
(6.194)
4.5
and
Cf1/2 8.1 h ks
1/6
where τ* is the Shields Stress; τc* is the critical Shields stress, which is taken to have a value of 0.03; h stands for the flow depth; and ks is the roughness height. Cf is the resistance coefficient, and qb* is the Einstein dimensionless bedload transport defined below: qb q*b gR D D
(6.195)
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Sedimentation and Erosion Hydraulics 6.91
where qb is the volumetric bedload transport per unit width having the dimensions of m2/s, R is the submerged specific gravity (taken as 1.65 for quartz), and D is the mean diameter of the sediment particles. The following conservation relation can be used for suspended-load sediment routing: dUhC dC qw vs(Es roC) dx dx
(6.196)
where x is the coordinate in the streamwise direction, qw Uh, C is the average volumetric suspended sediment concentration, and roC cb near-bed sediment concentration. The shape factor ro is given by the approximate relationship (Parker et al., 1987):
ro 1 31.5 u* vs
1.46
(6.197)
Therefore, the suspended load transport (volume per unit width per unit time) through a section can be evaluated as the product of the average sediment concentration and the flow discharge per unit width: qs qwC
(6.198)
All that remains is to evaluate Es, the sediment-entrainment coefficient. This is accomplished with the García-Parker relation: AZ5u Es 1 AZ u5 0.3
(6.199a)
where A 1.3 107
(6.199b)
and (6.199c) Zu u* Re0.6 vs p With the above equations and the assumption of a rectangular cross section (qw Uh), one can calculate the normal flow and equilibrium transport conditions for the river. These calculations, shown next, will serve as the initial conditions for the sedimentation study.
6.11.3 Computation of Normal Flow Conditions From the Manning-Strickler relation (Eq. 6.194),
qw h U 8.1 h ks
1/6
(ghS)1/2
where ks 2.5 Ds and qw (2.5Ds)1/6 h 8.1gS Now, qb can be computed:
0.6
1.427m2/s · (2.5 · 0.0003)1/6 0.987m 8.19.8 ·.0 0003
0.987m · 0.0003 0.5982 hS τ* RDs 1.65 · 0.0003
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6.92
Chapter Six
Then τ*c q*b 11.2 τ*1.5 1 τ*
4.5
0.03 qb 11.2(0.5982)1.5 1 0.5982
4.5
4.108
q*b q*bR gD s Ds 4.1081 .6 5·.8 9m /s 2·0.0 003 · 0.0003 qb 8.58 105 m2/s Estimation of C, Es and ro: u*s τ*sg RD 0 .5 98·.6 15·.8 9m /s 2·0 .0 003m 0.0538 m/s Rep R gD D/υ
0.0003m 9 .8 1 m /s · 1 .6 5 · 0.0 0 0 3 m 1 10 m /s 2
6
2
Rep 20.98 u 0.0538m/s(20.89)0.6 Zu *s Re0.6 7.841 p vs 4.25 102m/s
u ro 1 31.6 * vs
1.46
0.0538m/s 1 31.5 4.25 102m/s
1.46
ro 23.33. Then 1.3 107 (7.841)5 Es 3.8 · 103 1.3 107 (7.841)5 1 0.3 For equilibrium conditions, entrainment and deposition rates are the same thus, with the help of Eq. (6.196), E 3.8 103 C s 1.63 105 ro 23.33 and, finally, qs can be computed as qs UCH qw C 1.427m2/s 1.63 105 qs 2.33 104m2/s
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Sedimentation and Erosion Hydraulics 6.93
Next, it is necessary to identify the governing equations. Raising the water surface through a control structure results in the development of backwater effects. The backwater profile can be calculated using the standard 1-D St. Venant equation expressed in terms above parameters, with U being the flow velocity in x, the streamwise direction. The symbol η stands for the elevation of the bed above the datum: ∂(η h) ∂U ∂U U2 U g Cf . ∂t ∂x h ∂x
(6.200)
The backwater change in the water depth will cause a change in the transport of sediment. This phenomenon can be captured using the Exner equation with λp being the porosity of the bed sediment (taken at 0.3): 1 ∂ ∂η (q qs) (1 λp) ∂ x b ∂t
(6.201)
Notice that a time differential appears in both of the above equations; this reflects the fact that both hydraulic and transport conditions change continuously in time. The two equations are coupled through η, the bed elevation. Of course, a simultaneous solution of both equations, including the time derivative, is difficult. To simplify the model and expedite a solution, the quasi-steady-state approximation can be used. Not surprisingly, analysis has shown, that the time scale for sedimentological changes is much larger than that for changes in flow condition. Simply put, if the time changes of hydraulic conditions are driven by changes in sediment transport, they will occur slowly. Within an appropriate time step, the flow conditions can be considered to be steady. In this way, it is possible to drop the time differential in the St. Venant equation: dU g d(η h) C U2 U f dx dx h
(6.202)
Equations (6.201) and (6.202), in conjunction with continuity (qw Uh), provide the theoretical basis for the following analysis. In the quasi-steady-state analysis, the backwater curve resulting from a forced raise in water elevation is calculated first, (Eq. 6.202). The new water depths for the time step are used to calculate a new bed position (Eq. 6.201), and these values are used in the next time step to determine a new backwater profile. The procedure repeats for each time step.
6.11.5 Discussion of Method As discussed in a previous section, initial “normal” flow conditions can be calculated through consideration of the resistance and transport relations. Far away from the dam, where backwater effects are negligible, normal flow and equilibrium transport conditions will exist. A numeric scheme and computational grid must be chosen to evaluate the quasisteady-state governing equations as they relate to reservoir sedimentation. First, it is necessary to develop a spatial computational grid. The grid used in this numerical experiment begins 40 km upstream of the front near the dam face. The length is divided into reaches of 200 m, resulting in 201 nodes to be evaluated. This length allows for initial backwater computation to very nearly reach the normal depth at the upstream end. Using this grid, it is possible to develop a numerical scheme for solving the governing equations. To begin the simulation, a backwater calculation starting at the downstream end of the grid must be conducted. Combining the momentum equation (Eq. 6.202) and water
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6.94
Chapter Six
continuity (qw Uh) yields: dH Sf dx
(6.203)
where q2w H hη 2gh2 and the friction slope Sf is given by C qw2 Sf f gh3
(6.204)
(6.205)
If the value of h (and thus H) is known at node i 1, its value at node i (upstream) can be calculated using the following finite difference scheme: Hi Hi1 1(Sf,i 1 Sf,i)∆x 2
(6.206)
The above expression can be expanded and written as a function of hi: qw2 qw2 hi ηi Hi 1 1Sf,i 1 x 1 xCf 0 D(hi) 2 2 2 2ghi ghi3
(6.207)
Now, a Newton-Raphson method can be used to evaluate hi. In this method, an arbitrary guess at hi can be refined by hi using the expression D(hi ) ∆hi D´(hi )
(6.208)
FIGURE 6.35 Water surface elevation before and after reservoir construction
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Sedimentation and Erosion Hydraulics 6.95
FIGURE 6.36 Development of delta for bedload only.
FIGURE 6.37 Development of delta for both bedload and suspended load
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6.96
Chapter Six
where dD 1 F 2 3 x Sf,i D′(hi) r,i dhi 2 hi
(6.209)
The computation begins at the downstream end of the problem, where the initial bed elevation must be specified according to the normal flow conditions that existed before raising the water surface by 10 m. The first jump is ∆x 200 m, and each subsequent jump is from node, to node with ∆x 200 m. Once the computation has progressed to the final node, the hydraulic conditions for the initial bed condition are known. Variables such as U, qb , and qs can be calculated easily for each node once the water depth is known. Knowing the hydraulic conditions, the next necessary step is to evaluate the corresponding change in bed elevation. This is accomplished with the Exner equation written in the backward finite difference form ∆t ηi,j1 ηi,j [(qb,i1 qs,i1) (qb,i qs,i)] ∆x(1 λp)
(6.210)
where j is the current time step, j 1 is the next time step, and i 1 is the node immediately upstream of the node being calculated. The calculation proceeds in the downstream direction. For the first node, the same technique is used, and the initial “normal” bedload and suspended-load transport “feed rate” are used for the upstream values. It is crucial to choose a time step that upholds the assumptions inherent to the quasi-steadystate approximation. Here, for bedload transport only, a time step of 0.01 year (3.65 days) is used, and for runs with both suspended and bedload transport, a time step of 0.001 year (365 days) is used. These time steps are small enough to maintain theoretical integrity and numeric stability. The calculated elevations are fed into the next time step for the adjustment of hydraulic conditions. The above procedure is continued for each time step. To complete the numerous computations necessary for this procedure, a computer pro-
FIGURE 6.38a Delta location and height after 30 years; (A) bedload only.
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Sedimentation and Erosion Hydraulics 6.97
FIGURE 6.38b Delta location and height after 30 years.
gram must be written to facilitate the numerical computations.
6.11.6 Results The initial conditions for the water surface profile and bed elevation are shown in Fig. 6.35. Changes in river profile with time under the consideration of bedload transport only are shown in Fig. 6.36. The initial condition and the conditions after 2, 5, 10, 20, and 30 years are plotted. Figure 6.37 presents the same data, taking into account both bed and suspended sediment transport. Not surprisingly, the delta formation is accelerated considerably when total load (bed and suspended) is considered. Both the heights and lengths of deltas are greater for total load calculations for all time steps. Of course, varying delta formations result in a variation in backwater effects. When both bedload and suspended-load are considered, the backwater effects are more dramatic. The delta formed after 30 years are shown in Fig. 6.38A and B. After 30 years, the total load condition produces a delta reaching a length of approximately 36 km. Considering only
FIGURE 6.39 Turbidity current flowing into a laboratory reservoir. (Bell, 1942)
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6.98
Chapter Six
bedload results in a prediction of a delta only 28.2 km long. Holding the downstream elevation constant results in a considerable backwater effect driven by sedimentation. Although this elevation assumption is not wholly realistic, the results serve to illustrate the threat of flooding associated with reservoir sedimentation. As the reservoir “silts” in, it will be unable to hold the same amount of water without producing a commensurate increase in water stage. If one is interested in estimating the amount of time necessary to fill a reservoir, it is possible to simply divide the reservoirs “filling” volume per unit width by the normal sediment inflow at the upstream end. To determine the “filling” volume, it is necessary to consider the initial bed condition and the full bed conditions. The filling volume per unit width is estimated at 360,000 m3/m. Assuming bedload only and dividing the filling volume by the normal bedload inflow results in an approximate filling time of approximately 130 years. If total load is considered, an approximate filling time of 35 years is determined. This agrees well with the results of the model; Fig. 6.38b shows that the total-load model predicts that the reservoir will be approximately “full” around 40 years. Neither the above calculation nor the developed computer model considers the effect of sediment compaction, which may play an important role in increasing the time required to fill a reservoir. In general, the above results clearly indicate that suspended load plays a major role in reservoir sedimentation. Not considering suspended-load results in a considerable underestimation of the progress and effects of reservoir sedimentation. If the suspended load of the incoming flows is high, plunging may occur and turbidity currents will develop. Turbidity flows can transport fine-grained sediment for long distances, hence having a profound effect on reservoir sedimentation and water quality.
6.12 HYDRAULICS OF TURBIDITY CURRENTS 6.12.1 Introduction Turbidity currents are currents of water laden with sediment that move downslope in otherwise still bodies of water. Consider the situation illustration in Fig. 6.39. After plunging, a turbidity current moves along the bed of a laboratory reservoir (Bell, 1942). It is seen that when the flow goes from the sloping portion onto the flat portion, there is a two-fold increase in current thickness, indicating a change in flow regime through a hydraulic jump. There are a number of field situations where a similar slope-induced hydraulic jump can take place (García, 1993, García, and Parker, 1989). An important engineering aspect of turbidity currents concerns the impact these flows have on the water quality and sedimentation in lakes and reservoirs. Turbidity flows were observed in lakes and man-made reservoirs long before their occurrence in the ocean became apparent. This situation usually occurs during flood periods, when rivers carry a large amount of sediment in suspension. In China, where the suspended load in most rivers is extremely large, the venting of turbidity currents through bottom outlets to reduce the siltation of reservoirs has become common practice. Even though the bed slopes of lakes and reservoirs are orders of magnitude smaller than those in the ocean, turbidity currents are still capable of traveling long distances without losing their identities: e.g., more than 100 km in Lake Mead. An excellent account of numerical methods to model turbidity currents in reservoirs can be found in Sloff (1997). The ability of turbidity currents to transport sediment also has been put to use for the disposal of mining tailings (Normark and Dickson, 1976) and ash from power station boilers. Environmental concern has reduced waste disposal into lakes, but in the ocean, the
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SEDIMENTATION AND EROSION HYDRAULICS
FIGURE 6.40 Turbidity current flowing downslope through a quiescent body of water. (after García, 1994)
Sedimentation and Erosion Hydraulics 6.99
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dumping of mining tailing continues (Hay, 1987a, 1987b).
6.12.2 Governing Equations A detailed derivation of the governing equations for two-dimensional turbidity currents can be found in Parker et al. (1986). Here, the equations of motion are presented in layeraveraged form. The situation described in Fig. 6.40 is considered. A steady, continuous turbidity current is flowing downslope through a quiescent body of water, which is assumed to be infinitely deep and unstratified except for the turbidity current itself. The cross section is taken to be rectangular, with a width many times longer than the underflow thickness; therefore, variation in the lateral direction can be neglected. The bed has a constant small slope S and is covered with uniform sediment of geometric mean diameter Dsg and fall velocity vs; the x coordinate is directed downslope tangential to the bed, and the z coordinate is directed upward normal to the bed. The submerged specific gravity of the sediment is denoted by R (ρs/ρ 1), where ρs is the density of the sediment and ρ is the density of the clear water. Local mean downstream-flow velocity and volumetric sediment concentration are denoted as u and c, respectively. The suspension is dilute, hence c « 1 and Rc « 1 are assumed to hold everywhere. The parameters u and c are assumed to maintain similar profiles as the current develops in the downslope direction. The layer-averaged current velocity U and volumetric concentration C and the layer thickness h are defined via a set of moments (Parker et al., 1986): ∞
Uh udz
(6.211a)
0 ∞
U2h u2dz
(6.211b)
0 ∞
UCh ucdz
(6.211c)
0
The equation of fluid mass balance integrates in the upward normal direction to yield dUh (6.212) ewU dx where ew is the coefficient of entrainment of water from the quiescent water above the current. The equation of sediment conservation takes the layer-averaged form dUCH (6.213) vs(Es cb) dx where cb is the near-bed concentration of suspended sediment evaluated at z 0.05h and Es is a dimensionless coefficient of bed sediment entrainment into suspension. The integral momentum balance equation takes the form dU2h 1 d gRChS gR (Ch2) u2* dx 2 dx
(6.214)
where u* denotes the bed-shear velocity. The equations of sediment mass, fluid mass, and flow momentum balance must be closed appropriately with algebraic laws for ew, u*, Es, and cb. The water entrainment coefficient ew is known to be a function of the bulk Richardson number (Ri), which can be defined as gRCh Ri U2
(6.215)
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SEDIMENTATION AND EROSION HYDRAULICS
Sedimentation and Erosion Hydraulics 6.101
FIGURE 6.41 Water entrainment coefficient as a function of Richardson number (After Parker et al., 1987)
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Chapter Six
FIGURE 6.42 Plot of bed friction coefficient cD versus Reynolds number. (after Parker et al., 1987)
6.102
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SEDIMENTATION AND EROSION HYDRAULICS
Sedimentation and Erosion Hydraulics 6.103
FIGURE 6.43 Plot of shape factor ro versus µ u*/Vs. (after Parker et al., 1987)
FIGURE 6.44 Plot of the sediment entrainment coefficient Es for both open-channel suspensions and density currents. (after García and Parker, 1993)
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and is equal to one over the square of the densimetric Fr U/(gRCh)1/2, often used in stratified flow studies. A useful equation for the water entrainment coefficient plotted in Fig. 6.41 is the following (Parker et al., 1987): 0.075 ew (1 718Ri2.4)0.5
(6.216)
It is customary to take the bed shear stress to be proportional to the square of the flow velocity so that u*2 CDU2, where CD is a bed friction coefficient. Values of CD for turbidity currents have been found to vary between 0.002 and 0.05, as shown in Fig. 6.42 (Parker et al., 1987). The near-bed concentration cb can be related to the layer-averaged concentration C by a shape factor ro cb/C, which is approximately equal to 2 for sediment-laden underflows, as shown in Fig. 6.43 (Parker et al., 1987). The sediment entrainment coefficient Es is known to be a function of bed shear stress and sediment characteristics (García and Parker, 1991). The formulation of García and Parker (Eq. 6.173a) is plotted in Fig. 6.44, where data on sediment entrainment by sediment-laden density currents also are included (García and Parker, 1993).
6.12.3 Plunging Flow The necessary conditions for plunging to occur in a reservoir may vary as a function of the physical parameters that produce flow stratification. These parameters are sometimes known in advance from measurements in the field. Akiyama and Stefan (1984) generalized several expressions that were derived from laboratory experiments, field measure-
FIGURE 6.45 Plot of plunging flow depth versus (q2/g)1/3, including field and laboratory data García (1996)
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Sedimentation and Erosion Hydraulics 6.105
ments, or theoretical analysis as a function of the parameters involved in plunging:
q2w hp 12 Frp gRC
1/3
(6.217)
where hp flow depth at plunging, qw flow discharge per unit width, and Frp densimetric Fr at plunging defined by Frp U (gRChp)1/2
(6.218)
The value of Frp has been found to range from 0.2 to 0.8 (Morris and Fan, 1998). If there is not enough suspended sediment, plunging will not occur and a turbidity current will not develop. Figure 6.45 shows field and laboratory data for the flow depth at plunging as a function of the inflow parameters (García, 1996).
6.12.4 Internal Hydraulic Jump The bulk Ri, given by Eq. (6.215), is an important parameter governing the behavior of stratified slender flows, such as turbidity currents (Turner, 1973). This parameter has a critical value Ric near unity so that the range Ri Ric corresponds to a high-velocity supercritical turbid flow regime, and the range Ri Ric corresponds to a low-velocity subcritical turbid flow regime. The change from supercritical flow to subcritical flow is accomplished via an internal hydraulic jump, as illustrated in Fig. 6.39. Therein, a turbidity current undergoes a hydraulic jump induced by a change in bed slope in the proximity of a laboratory reservoir. Conservation of momentum gives the following relation (García, 1993): h2 1 R 8 i11 1 1 h1 2
(6.219)
which is analogous to Belanger’s equation for open-channel flow hydraulic jumps. For a known prejump Ri1, Eq. (6.219) gives the ratio of the sequent current thickness h2 to the initial current thickness h1. The subcritical flow, forced by some type of control acting farther downstream, will influence the location of the jump and thus the length of the waterentraining supercritical flow upstream of the jump. In laboratory experiments, the downstream boundary conditions are usually imposed by the experimenter (e.g., weir, sluice gate, outfall) because of the finite length of experimental facilities. In the ocean or lakes, where a current may travel several hundred kilometers without losing its identity, the control of the flow will operate through deposition of sediment and bed friction.
6.12.5 Application: Turbidity Current in Lake Superior As an example, the case of turbidity currents produced by the discharge of taconite tailings by the Reserve Mining Company into Lake Superior at Silver Bay, Minnesota, is considered. Over a period of 20 years, the man-made turbidity currents formed a delta with a steep front followed by a depositional fan. Normark and Dickson (1976) used field observations to infer that the transition from the delta slope to the fan slope took place through a hydraulic jump, whereas Akiyama and Stefan (1985) used numerical modeling to show a clear tendency by the flow to become supercritical shortly after reaching the fan region. However, the lack of knowledge about the role played by the hydraulic jump has made
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Chapter Six
FIGURE 6.46 Simulation of turbidity current undergoing a hydraulic jump in Lake Superior, Minnesota. (after García, 1993)
6.106
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SEDIMENTATION AND EROSION HYDRAULICS
Sedimentation and Erosion Hydraulics 6.107
flow computations along the subcritical region practically impossible. Such computations can now be simplified through the knowledge gained in laboratory experiments (García, 1994). This is illustrated by the following numerical experiment. The lake bed topography at Silver Bay in Lake Superior is modeled in a one-dimensional configuration, as illustrated in Fig. 6.46. The delta slop angle is 17º, and the fan slope is 1.5º. The delta-fan slope transition takes place between 600 m to 900 m from the shore. The equations of motion (6.212), (6.213), and (6.214) are solved using a simple standard step method (García and Parker, 1986). The water entrainment ew and sediment entrainment Es coefficients are estimated with relationships proposed by Parker et al. (1987) and García and Parker (1991; 1993), respectively. A constant bed friction coefficient CD 0.02 and a shape factor ro 2 are used. Initial flow conditions at the tailings discharge point similar to those used by Akiyama and Fukushima (1986) are used: i.e., Uo 0.6 m/s, ho 1 m, Φo 0.1 m2/s, and Rio 0.5. The tailings have a mean particle size Dsg 40 mm and a submerged specific gravity R 2.1. Particle fall velocity is estimated to be vs 0.14 cm/s. Lateral spreading of the flows is ignored. The computations march downslope starting at the head of the delta, and at approximately 0.6 km from the tailings’ discharge point, the flow starts to slow down because of the slope transition. If the current depth at the end of the fan region could be known, the jump location could be determined with a simple “backwater” computation. Because this information is not available, the hydraulic jump is assumed to take place at 0.9 km from the inlet. According to the laboratory observations, water entrainment from above, as well as bed sediment entrainment into suspension, can be neglected after the jump. Under these assumptions, Eq. (6.213) can be integrated with the help of Eq. (6.212), and an expression for the spatial variation of the volumetric layer-averaged sediment concentration C is obtained, v r x´ C Cje q s o
w
(6.220)
where C is the value of Cj at the hydraulic jump and x′ is distance measured from the jump’s location. Since the flow discharge per unit width qw is constant in the subcritical flow region (ew 0), Eq. (6.220) can be used to compute the volumetric sediment transport rate per unit width CUh, at any location after the jump. The variation in current thickness between the jump’s location and a point located 2.2 km from the inlet is shown in Fig. 6.46. The profile is obtained by first computing the value of C at 2.2 km with the help of Eq. (6.220), then by doing a “backwater” computation in an iterative manner until the computed current thickness at 0.9 km coincides with the current thickness obtained with the supercritical flow computation and the hydraulic jump Eq. (6.219). The flow discharge per unit width computed at the jump’s location is qw 48 m2/s. For such flow discharge, Eq. (6.220) predicts that the turbidity current, after experiencing a hydraulic jump, will travel approximately 80 km before dying out as a result of deposition of sediment.
REFERENCES Akiyama and Fukushima, 1986, “Entrainment of noncohesive sediment into suspension.” 3rd Int.Symp. on River Sedimentation, S. Y. Wang, H. W. Shen, and L. Z. Ding, eds., Univ. of Mississipi, 804–813. Akiyama and Stefan, 1984, “ Plunging Flow into a Reservior: Theory,” (American Society of Civil Engineer. Journal of Hydraulic Engineering) 110(4), 484–499 Akiyama and Stefan, 1985, “Turbidity Current with Erosion and Deposition,” (American Society of Civil Engineering), Journal of Hydraulic Engineering. 111(12), 1473–1496.
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Ackers, P., and W. R. White, “Sediment Transport: New Approach and Analysis,” Journal of Hydraulic Engineering, American Society of Civil Engineer 99:2041–2060, 1973. Allen, J. R. L., Physical Processes of Sedimentation, Elsevier, New York, 1970. Amsler, M. L., and M. H. García, Discussion of “Sand-Dune Geometry of Large Rivers During Floods,” Journal of Hydraulic Engineering, American Society of Civil Engineer 123:582–584, 1997. Anderson, A. G., “The Characteristics of Sediment Waves Formed by Flow in Open Channels,” Proceedings of the 3rd Midwestern Conference on Fluid Mechanics, University. of Minnesota, Minneapolis, March 1953. Annambhotla, V. S., W. W. Sayre, and R. H. Livesey, “Statistical Properties of Missouri River Bed Forms,” Journal Waterways, Harbors, & Coastal Engineering Division,American Society of Civil Engineer 98(WW4):489–510, 1972. Ashida, K., and M. Michiue, “Study on Hydraulic Resistance and Bed-load Transport Rate in Alluvial Streams.” Transactions of the Japan Society of Civil Engineering 206:59–69, 1972. Ashworth, P .J., S. J. Bennett, J. L. Best, and S. J. Mclelland, eds., Coherent Flow Structures in Open Channels, John Wiley & Sons, Chichester, England, 1996. Bell, H. S., Stratified Flow in Reservoirs and Its Use in Prevention of Silting, Miscellaneous Publication No. 491, U.S. Department of Agriculture, 1942. Bhowmik, N. G., A. P. Bonini, W. C. Bogner, and R. P. Byrne, Hydraulics of Flow and Sediment Transport in the Kankakee River in Illinois, Report of Investigation 98, Illinois State Water Survey, Champaign, IL, 1980. Blondeaux, P., G. Seminara, and M. Tubino, “A Unified Bar-Bend Theory of River Meanders,” Journal of Fluid Mechanics, 157:449–470, 1985. Bogardi, J., Sediment Transport in Alluvial Streams, Akademiai Kiado, Budapest, Hungary, 1974. Bouvard, M., Mobile Barrages and Intakes on Sediment Transporting Rivers, International Association for Hydraulic Research Monograph Series, A. A. Balkema, Rotterdam, 1992. Bridge, J. S., and S. J. Bennett, “A Model for the Entrainment and Transport of Sediment Grains of Mixed Sizes, Shapes, and Densities,” Water Resources Research, 28:337–363, 1992. Brooks, N. H., “Calculation of Suspended Load Discharge from Velocity and Concentration Parameters,” Proceedings of the Federal Interagency Sedimentation Conference, Miscellaneous Publication No. 970, Agricultural Research Service, 1963, pp. 229–237. Brownlie, W. R., “Prediction of Flow Depth and Sediment Discharge in Open Channels,” Report No. KH–R–43A, Keck Laboratory of Hydraulics and Water Resources, California Institute of Technology, Pasadena, CA.1981a. Brownlie, W. R., “Re-examination of Nikuradse Roughness Data,” Journal of the Hydraulics Division, American Society of Civil Engineer 107(HY1):115–119, 1981b. Brownlie, W. R., Errata, “Re-examination of Kikuradse Roughness Data,” Journal of the Hydraulics Division, American Society of Civil Engineer 108(HY8):971, 1982. Carling, P. A., and M. R. Dawson, Advances in Fluvial Dynamics and Stratigraphy, John Wiley & Sons, Chichester, England, 1996. Celik, I., and W. Rodi, “Modeling Suspended Sediment Transport in Nonequilibrium Situations,” Journal of Hydraulic Engineering, American Society of Civil Engineer 114:1157–1191, 1988. Chabert, J., and J. L. Chauvin, “Formation de Dunes et de Rides dans les Modèles Fluviaux,” Bull. Cen. Rech. ess. Chatou, No. 4, 1963. Chang, H. H., Fluvial Processes in River Engineering, John & Sons Wiley, New York, 1988. Coleman, S. E., and B. W. Melville, “Bed-Form Development,” Journal of Hydraulics Engineering, American Society of Civil Engineer 120:554–560, 1994. Colombini, M., G. Seminara, and M. Tubino, “Finite-Amplitude Bars,” Journal of Fluid Mechanics, 181:213–232, 1987. Colombini, M., M. Tubino, and P. Whiting, “Topographic Expression of Bars in Meandering Channels,” Proceedings of the 3rd International Workshop on Gravel-Bed Rivers, Florence, Italy, 1990. Coussot, P., Mudflow Rheology and Dynamics, International Association for Hydraulic Research
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Sedimentation and Erosion Hydraulics 6.109 Monograph Series, A.A. Balkema, Rotterdam, Netherlands, 1997. Dietrich, W. E., “Settling Velocities of Natural Particles,” Water Resources Research, 18:1615–1626, 1982. Diplas, P., and G. Parker, Pollution of Gravel Spawning Grounds Due to Fine Sediment, Report No. 240, EPA/R-808683-01-1, St. Anthony Falls Hydraulics Laboratory, Univ. of Minnesota, Minneapolis, Minnesota. 1985. Diplas, P., and A. Sutherland, “Sampling Techniques for Gravel Sized Sediments,” American Society of Civil Engineer Journal of Hydraulic Engineering, 114:481–501, 1988. Einstein, H. A., The Bedload Function for Bedload Transportation in Open Channel Flows, Technical Bulletin No. 1026, U.S.D.A., Soil Conservation Service, 1950. Einstein, H. A., and N. L. Barbarossa, “River Channel Roughness,” Transactions of the American Society of Civil Engineers, 117:1121–1146, 1952. Engelund, F., “Instability of Erodible Beds,’ Journal of Fluid Mechanics, 42:225–244, 1970. Engelund, F., and J. Fredsoe, “A Sediment Transport Model for Straight Alluvial Channels,” Nordic Hydrology, 7:293–306, 1976. Engelund, F., and E. Hansen. A Monograph on Sediment Transport in Alluvial Streams, Teknisk Vorlag, Copenhagen, Denmark, 1967. Exner, F. M., “Über die Wechselwirkung Zwischen Wasser und Geschiebe in Flüssen,” Sitzenberichte der Academie der Wissenschaften, Wien, Austria, Sec. IIA, 134:199, 1925. Fernandez Luque, R., and R. van Beek, “Erosion and Transport of Bed Sediment.” Journal of. Hydraulics Research, 14:127–144, 1976. Fredsoe, J., “On the Development of Dunes in Erodible Channels,” Journal of Fluid Mechanics, 64(1):1–16, 1974. Fredsoe, J., “Shape and Dimensions of Stationary Dunes in Rivers,” Journal of the Hydraulics Division, American Society of Civil Engineer 108(HY8):932–947, 1982. Fredsoe, J., and R. Deigaard, Mechanics of Coastal Sediment Transport, Advances Series on Ocean Engineering, Vol. 3, World Scientific Publishing, Singapore, 1992. García, M. H., “Hydraulic Jumps in Sediment-Laden Bottom Currents,” Journal of Hydraulic Engineering, American Society of Civil Engineer 199:1094-1117, 1993. García, M. H., “Depositional Turbidity Currents Laden with Poorly-Sorted Sediment,” Journal of Hydraulic Engineering, American Society of Civil Engineer 120:1240–1263, 1994. García, M. H., Environmental Hydrodynamics, Universidad Nacional del Litoral Publication Center, Santa Fe, Argentina (in Spanish), 1996. García, M. H., and Y. Niño “Lagrangian Description of Bedload Transport by Saltating Particles,” Proceedings of the 6th International Symposium Stochastic Hydraulics, Taipei, Taiwan, 259-266, 1992. García, M. H., and Y. Niño, “Dynamics of Sediment Bars in Straight and Meandering Channels: Experiments on the Resonance Phenomenon,” IAHR Journal of Hydraulic Research, 31:739–761, 1993. García, M. H., and G. Parker, “On the Numerical Prediction of Turbidity Currents,” Proc. 3rd Int’l Symp. River Sedimentation, The University of Mississippi, University, Mississippi, 1556-1565, 1986. García, M. H. and G. Parker, “Experiments on Hydraulic Jumps in Turbidity Currents Near a Canyon-Fan Transition,” Science, 117:393–396, 1989. García, M., and G. Parker, “Entrainment of Bed Sediment into Suspension,” Journal of Hydraulic Engineering, American Society of Civil Engineer 117:414–435, 1991. García, M., and G. Parker, “Experiments on the Entrainment of Sediment into Suspension by a Dense Bottom Current,” Journal of Geophysical Research, 98(C3):4793–4807, 1993. Garde, R. J., and Albertson, 1959. “Sand Waves and Regimes of Flow in Alluvial Channels” Proceedings of the 8th International Association for Hydraulic Research Congress, Montreal, Canada, Vol. IV, pa.per 28 Garde, R. J., and K. G. Ranga Raju, Mechanics of Sediment Transportation and Alluvial Stream
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Problems, 2nd ed., Wiley Easter Limited, New Delhi, India, 1985. Gill, M. A., “Height of Sand Dunes in Open Channel Flows,” Journal of the Hydraulics Division, American Society of Civil Engineer 97(HY12):2067–2074, 1971. Graf, W. H., Hydraulics of Sediment Transport, McGraw-Hill, New York, 1971. Haque, M. I., and K. Mahmoud, “Geometry of Ripples and Dunes,” Journal of Hydraulic Engineering, American Society of Civil Engineer 111(1):48–63, 1985. Hay, A. E., “Turbidity Currents and Submarine Channel Formation in Rupert Inlet, British Columbia, 1: Surge Observations,” Journal of Geophysical Research, 92(C3):2875–2881, 1987a. Hay, A. E., “Turbidity Currents and Submarine Channel Formation in Rupert Inlet, British Columbia, 2: The Roles of Continuous and Sure-Type Flow,” Journal of Geophysical Research, 92(C3):2883–2900, 1987b. Hayashi, T., “Formation of Dunes and Antidunes in Open Channels,” Journal of the Hydraulics Division, American Society of Civil Engineer 96(HY2):357–367, 1970. Hino, M., “Equilibrium-Range Spectra of Sand Waves Formed by Flowing Water,” Journal of Fluid Mechanics, 34(3):565–573, 1968. Hotchkiss, R. H., and G. Parker, “Shockfitting of Aggradational Profiles Due to Backwater,”Journal of Hydraulic Engineering, American Society of Civil Engineer 117(9):1129–1144, 1991. Ikeda, S., “Incipient Motion of Sand Particles on Side Slopes,”Journal of the Hydraulics Division, American Society of Civil Engineer 108(HY1):95–114, 1982. Ikeda, S., “Prediction of Alternate Bar Wavelength and Height,” Journal of the Hydraulics Division, American Society of Civil Engineer 110(4):371–386, 1984. Ikeda, S., and G. Parker, eds., “River Meandering,” Water Resources Monograph No. 12, American Geophysical Union, Washington D.C 1989. Jaeggi, M. N. R., “Formation and Effects of Alternate Bars,” Journal of Hydraulic Engineering, American Society of Civil Engineer 10:142–156, 1984. Jain, S. C., and J. F. Kennedy, “The Spectral Evolution of Sedimentary Bed Forms,” Journal of Fluid Mechanics, 63:301–314, 1974. Jansen, P. P., L. van Bendegom, J. van den Berg, M. de Vries, and A. Zanen, Principles of River Engineering: The Non-Tidal Alluvial River, Pitman, London, UK, 1979. Julien, P.Y., Erosion and Sedimentation, Cambridge University Press, 1995. Julien, P. Y., and G. J. Klaassen, “Sand-Dune Geometry of Large Rivers during Floods,” Journal of Hydraulic Engineering, American Society of Civil Engineer 121:657–663, 1995. Kennedy, J. F., “The Mechanics of Dunes and Antidunes in Erodible-Bed Channels,” Journal of Fluid Mechanics, 16:521-544, 1963. Kennedy, J. F., 1969. “The Formation of Sediment Ripples, Dunes, Antidunes,” Annual R2 views of Fluid Mechanics Vol. 1, pp. 147–167, 1969. Kinoshita, R., and H. Miwa, “River Channel Formation Which Prevents Downstream Translation of Transverse Bars” (in Japanese), Shinsabo, 94:12–17 1974. Klaassen, G. J., H. J. M. Ogink, and L. C. van Rijn, “DHL-Research on Bed Forms, Resistance to Flow and Sediment Transport,” Procedings of the 3rd International Symposium River Sedimentation, Jackson, MI, March-April 1986, PP. 58–820 Lane, E.W., “Design of Stable Channels,” Transactions of the American Society of Civil Engineer, 81(745):1–17, 1955. Lavelle, J. W., and H. O. Mofjeld, “Do Critical Stresses for Incipient Motion and Erosion Really Exit?” ASCE Journal of Hydraulic Engineering, 113:370–385, 1987. Liu, H. K., “Mechanics of Sediment-Ripple Formation,” American Society of Civil Engineer Journal of the Hydraulics Division, 83(HY2):1–23, 1957. Menduni, G., and E. Paris, AOn the Prediction of Geometric Characteristics of Dunes,” Proceedings of the International Symposium on River Sedimentation, Jackson, MI, March-April 1986, pp. 665–674. Mehta, A. J., E. J. Hayter, W. R. Parker, R. B. Krone, and A. M. Teeter, “Cohesive Sediment Transport I: Process Description,” Journal of Hydraulic Engineering, American Society of Civil Engineer 115:1076–1093, 1989a. Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
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Sedimentation and Erosion Hydraulics 6.111 Mehta, A. J., E. J. Hayter, W. R. Parker, R. B. Krone, and A. M. Teeter, “Cohesive Sediment Transport II: Application,” Journal of Hydraulic Engineering, American Society of Civil Engineer 115:1094–1113, 1989b. Meyer-Peter, E., and R. Muller, “Formulas for Bedload Transport.” Proceedings of the 2nd Congress of the International Association for Hydraulic Research, Stockholm, 1948, pp. 39–64, Milhous, R. T., “Sediment Transport in a Gravel Bottomed Stream,” doctoral dissertation, Oregon State University, Corvallis, 1973. Misri, R. I., Garde, R. J., and Ranga Raju, K. G., 1983, “Experiments on Bedload Transport of Nonuniform sands and Gravels,” in Proceedings of 2nd International Symposium on River Sedimentation, Water Resources and Electric power Press, China, 450-450. Morris, G. L., and J. Fan, Reservoir Sedimentation Handbook, Mc Graw–Hill Book Co, N.Y., 1988 Nakagawa, H., and T. Tsujimoto, “Special Analysis of San Bed Instability,” Journal of Hydraulic Engineering, American Society of Civil Engineer 110:457–483, 1984. Nakato, T., and R. Ettema, eds., Issues and Directions in Hydraulics, Proceedings of the Iowa Hydraulics College, Iowa City, May 1995, A. A. Balkema, Rotterdam, 1996. National Research Council, “Alluvial Fan Flooding,” Water Science and Technology Board, National Academy Press, Washington, DC, 1996. Nelson, J. M., and J. D. Smith, “Flow in Meandering Channels with Natural Topography,” in S. Ikeda and G. Parker, eds., River Meandering, Water Resources Monograph No. 12, American Geophysical Union, 1989, pp. 69–102. Nielsen, P., Coastal Bottom Boundary Layers and Sediment Transport, Advances Series on Ocean Engineering, Vol. 4, World Scientific Publishing Singapore, 1992. Nikuradse, J., “Laws of Flow in Rough Pipes” (in German), National Advisory Committee for Aeronautics Tech Memo No. 1292, Washington, DC, 1950. Niño, Y., and M. H. García, “Gravel Saltation II: Modeling,” Water Resources Research, 30(6):1915–1924, 1994. Niño, Y., and M. H. García, “Using Langrangian Particle Saltation Observations for Bedload Sediment Transport Modelling,” Hydrological Processes, 12:1197–1218, 1998. Niño, Y., M. H. García, and L. Ayala, “Gravel Saltation I: Experiments,” Water Resources Research, AGU 30:1907–1914, 1994. Nordin, C. F. “Aspects of flow resistance and sediment transport: Rio Grande ner Bernalillo,” Water supply paper No. 1498-H, U.S. Geological Survey, Washington, D C 1963. Nordin, C. F. Jr., and J. H. Algert, “Spectral Analysis of Sand Waves,” Journal of the Hydraulics Division, American Society of Civil Engineer 92(HY5):95–114, 1966. Normark and Dickson, 1976, “Man–mode Turbidity Currents in Lake Superior,” Sedimentology, Vol. 23, pp 815–831. Parker, G., “Sediment Inertia as a Cause of River Antidunes,” Journal of the Hydraulics Division, American Society of Civil Engineer 101(HY2):211–221, 1975. Parker, G., “Self-Formed Straight Rivers with Equilibrium Banks and Mobile Bed, Part 2. The Gravel River,” Journal of Fluid Mechanics, 89(1):127–146, 1978. Parker, G., “Surface-Based Bedload Transport Relation for Gravel Rivers,” Journal of Hydraulic Research, 28(4):417–436, 1990. Parker, G., “Selective Sorting and Abrasion of River Gravel: Theory,” Journal of Hydraulic Engineering, American Society of Civil Engineer 117(HY2):131–149, 1991a. Parker, G., “Selective Sorting and Abrasion of River Gravel: Applications,” Journal of Hydraulic Engineering, American Society of Civil Engineer 117:(HY2):150–171, 1991b. Parker, G., and A. G. Anderson, “Basic Principles of River Hydraulics,” Engineer Journal of the Hydraulics Division, American Society of Civil Engineer 103(HY9):1077–1087, 1977. Parker, G., and E. D. Andrews, “Sorting of Bedload Sediment by Flow in Meander Bends,” Water Resources Research, 24:1361–1373, 1985. Parker and Ikeda, 1989, “ River Meandering,” Water Resources Mouograph No 12, American Geophysical Union, Washington D.C. 485p.
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6.112
Chapter Six
Parker, G., and A. J. Sutherland, “Fluvial Armor,” IAHR Journal of Hydraulic Research, 28:529–544, 1990. Parker, G., M. García, Y. Fukushima, and W. Yu, “Experiments on Turbidity Currents over an Erodible Bed,” Journal of Hydraulic Research, 25(1):123–147, 1987. Press, W. H., B. P. Flannery, S. A. Teulovsky, and W. T. Vetterling, Numerical Recipes, Cambridge University Press, New York, 1986. Ranga Raju, K. G., and Soni, J. P., “Geometry of Ripples and Dunes in Alluvial Channels,” IAHR Journal of Hydraulic Reserach, 14:241–249, 1976. Raudkivi, A. J., Loose Boundary Hydraulics, 3rd ed., Pergamon Press, New York, 1990. Raudkivi, A. J., “Sedimentation: Exclusion and Removal of Sediment fro Diverted Water,” Hydraulic Structures Design Manual, International Association for Hydraulic Research, A. A. Balkema, Rotterdam, 1993. Raudkivi, A. J., and H. H. Witte, “Development of Bed Features,” Journal of Hydraulic Engineering, American Society of Civil Engineer 116:1063–1079, 1990. Renard, K. G., G. R. Foster, G. S. Weesies, D. K. McCool, and D. C. Yoders, coordinators, Predicting Soil Erosion by Water: A Guide to Conservation Planning with the Revised Universal Soil Loss Equation (RUSLE), Agriculture Handbook No. 703, U.S. Department of Agriculture, Washington, DC, 1997. Reynolds, A. J., “Waves on the Bed of an Erodible Channel,” Journal of Fluid Mechanics, 22(1):113-133, 1965. Richards, K. J., “The Formation of Ripples and Dunes on Erodible Bed,” Journal of Fluid Mechanics, 99:597–618, 1980. Rouse, H., “Modern Conceptions of the Mechanics of Turbulence,” Transactions of the American Society of Civil Engineers,102, 1957. Schlichting, H., Boundary Layer Theory, 7th ed., McGraw-Hill, New York, 1979. Sekine, M., and H. Kikkawa, “Mechanics of Saltating Grains II,” Journal of Hydraulic Engineering, American Society of Civil Engineer 118:536–558, 1992. Sekine, M., and G. Parker, “Bedload Transport on a Transverse Slope I” Journal of Hydraulic Engineering, American Society of Civil Engineer 118:513–535, 1992. Shields, A., “Anwendung der Aenlichkeitsmechanik und der Turbulenzforschung auf die Geschiebebewegung,” Mitteilungen der Preussischen Versuchsanstalt fur Wasserbau und Schiffbau, Berlin, Germany by W. P. Ott and J. C. van Uchelen, trans., California Institute of Technology, Pasadena, CA, 1936. Sieben, J., “Modelling of Hydraulics and Morphology in Mountain Rivers,” doctoral dissertation, Delft University of Technology, Netherlands, 1997. Simons, D. B., and E. V. Richardson, “Resistance to Flow in Alluvial Channels,” Professional Paper No. 422J, U.S. Geological Survey, 1966. Simons, D. B., and F. Senturk, Sediment Transport Technology, rev. ed., Waters Resources Publications, Littleton, Co, 1992. Sloff, C. J., “Sedimentation in Reservoirs,” doctoral dissertation, Delft University of Technology, Netherlands, 1997. Smith, J. D., “Stability of a Sand Bed Subjected to a Shear Flow of Low Froude Number,” J. Geophysical Research, 75(30):5928-5940, 1970. Smith, J. D., and S. R. McLean, “Spatially Averaged Flow over a Wavy Surface,” Journal of Geophysical Research, 83:1735–1746, 1977. Takahashi, T., Debris Flow, International Association for Hydraulic Research Monograph Series, A.A. Balkemabuoyancy Rotterdam, 1991. Taylor, B. D., and V. A. Vanoni, “Temperature Effects in Low-Transport, Flat-Bed Flows,” American Society of Civil Engineer Journal of the Hydraulic Division, 98(HY12):2191-2206, 1972. Tubino, M., and G. Seminara, “Free-Forced Interactions in Developing Meanders and Suppression of Free Bars,” Journal of Fluid Mechanics, 214:131-159, 1990. Turner, 1973, “ Bouyancy Effects in Fluids,” Cambridge University Press, Cambridge, 368p. van Rijn, L. C., “Sediment Transport, Part II. Suspended Load Transport,” Journal of Hydraulic
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SEDIMENTATION AND EROSION HYDRAULICS
Sedimentation and Erosion Hydraulics 6.113 Engineering, 110:1613–1641, 1984a. van Rijn, L. C., “Sediment Transport, Part III. Bed Forms and Alluvial Roughness,” American Society of Civil Engineer Journal of Hydraulic Engineering, 110(12):1733-1754, 1984b van Rijn, L. C., “Mathematical Modelling of Morphological Processes in the Case of Suspended Sediment Transport,” Delft Hydraulics Communication No. 382, Delft University of Technology, Netherlands, 1987. Vanoni, V. A. “Factors Determining Bed Forms of Alluvial Streams,” American Society of Civil Engineer Journal of the Hydraulic Division, 100(HY3):363–377, 1974. Vanoni, V. A., ed., Sedimentation Engineering, American Society of Civil Engineer Manuals and Reports on Engineering Practice No. 54, American Society of Civil Engineers, New York, 1975. Wan, Z., and Z. Wang, Hyperconcentrated Flow, International Association for Hydraulic Research Monograph Series, A.A. Balkema, Rotterdam, 1994. White, W. R., H. Milli, and A. D. Crabbe, “Sediment Transport I and II: An Appraisal of Available Methods,” HRS INT 119, Hydraulics Research Station, Wallingford, UK, 1973. Wiberg, P. L., and J. D. Smith, “Model for Calculating Bedload Transport of Sediment,” American Society of Civil Engineer Journal of Hydraulic Engineering, 115(1):101–123, 1989. Wiberg, P., and J. D. Smith, “Calculations of the Critical Shear Stress for Motion of Uniform and Heterogeneous Sediments,” Water Resources Research, 23:1471–1480, 1987. Wiberg, P. L., and J. D. Smith, “A Theoretical Model for Saltating Grains,” Journal of Geophysical Research, 90(C4):7341–7354, 1985. Wilcock, P. R., “Methods for Estimating the Critical Shear Stress of Individual Fractions in MixedSize Sediment,” Water Resources Research, 24:1127–1135, 1988. Wilson, K. C., “Bedload Transport at High Shear Stresses,” American Society of Civil Engineer Journal of Hydraulic Engineering, 92(HY6):49–59, 1966. Yalin, M. S., “An Expression for Bedload Transportation.” American Society of Civil Engineer Journal of the Hydraulics Division, 89(HY3):221–250, 1963. Yalin, M.S., “Geometrical Properties of Sand Waves,” American Society of Civil Engineer Journal of the Hydraulics Division, 90(HY5):105–119, 1964. Yalin, M. S., Mechanics of Sediment Transport, Pergamon, Braunschweig, Germany. 1972. Yalin, M. S., River Mechanics, Pergamon Press, New York, 1992 Yalin, M. S., and E. Karahan, “Steepness of Sedimentary Dunes,” American Society of Civil Engineer Journal of the Hydraulics Division, 105(HY4):381–392, 1979. Yang, C. T. “Incipient Motion and Sediment Transport,” American Society of Civil Engineer Journal of the Hydraulic Division, 99:1679–1704, 1973. Yang, C. T., Sediment Transport: Theory and Practice, McGraw-Hill New York, 1996. Yen, B. C., ed., Channel Flow Resistance: Centennial of Manning’s Formula, Water Resources Publications, Littleton, CO, 1992
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Source: HYDRAULIC DESIGN HANDBOOK
CHAPTER 7
RISK/RELIABILITY-BASED HYDRAULIC ENGINEERING DESIGN Yeou-Koung Tung Department of Civil Engineering Hong Kong University of Science and Technology Clear Water Bay Kowloon, Hong Kong
7.1 INTRODUCTION 7.1.1
Uncertainties in Hydraulic Engineering Design
In designing hydraulic engineering systems, uncertainties arise in various aspects including, but not limited to, hydraulic, hydrologic, structural, environmental, and socioeconomical aspects. Uncertainty is attributed to the lack of perfect knowledge concerning the phenomena and processes involved in problem definition and resolution. In general, uncertainty arising because of the inherent randomness of physical processes cannot be eliminated and one has to live with it. On the other hand, uncertainties, such as those associated with the lack of complete knowledge about processes, models, parameters, data, and so on, can be reduced through research, data collection, and careful manufacturing. Uncertainties in hydraulic engineering system design can be divided into four basic categories: hydrologic, hydraulic, structural, and economic (Mays and Tung, 1992). Hydrologic uncertainty for any hydraulic engineering problem can further be classified into inherent, parameter, or model uncertainties. Hydraulic uncertainty refers to the uncertainty in the design of hydraulic structures and in the analysis of the performance of hydraulic structures. Structural uncertainty refers to failure from structural weaknesses. Economic uncertainty can arise from uncertainties in various cost items, inflation, project life, and other intangible factors. More specifically, uncertainties in hydraulic design could arise from various sources (Yen et al., 1986) including natural uncertainties, model uncertainties, parameter uncertainties, data uncertainties, and operational uncertainties. The most complete and ideal way to describe the degree of uncertainty of a parameter, a function, a model, or a system in hydraulic engineering design is the probability density function (PDF) of the quantity subject to uncertainty. However, such a probability function cannot be derived or found in most practical problems. Alternative ways of expressing the uncertainty of a quantity include confidence intervals or statistical moments. In particular, the second order moment, that is, the variance or standard deviation, is a measure of the dispersion of a random variable. Sometimes, the coefficient of variation, defined as the ratio of standard deviation to the mean, is also used.
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RISK/RELIABILITY-BASED HYDRAULIC ENGINEERING DESIGN
7.2
Chapter Seven
The existence of various uncertainties (including inherent randomness of natural processes) is the main contributor to the potential failure of hydraulic engineering systems. Knowledge of uncertainty features of hydraulic engineering systems is essential for assessing their reliability. In hydraulic engineering design and analysis, the decisions on the layout, capacity, and operation of the system largely depend on the system response under some anticipated design conditions. When some of the components in a hydraulic engineering system are subject to uncertainty, the system’s responses under the design conditions cannot be assessed with certainty. An engineer should consider various criteria including, but not limited to, the cost of the system, failure probability, and consequences of failure, such that a proper design can be made for the system. In hydraulic engineering design and analysis, the design quantity and system output are functions of several system parameters not all of which can be quantified with absolute certainty. The task of uncertainty analysis is to determine the uncertainty features of the system outputs as a function of uncertainties in the system model and in the stochastic parameters involved. Uncertainty analysis provides a formal and systematic framework to quantify the uncertainty associated with the system output. Furthermore, it offers the designer useful insights with regard to the contribution of each stochastic parameter to the overall uncertainty of the system outputs. Such knowledge is essential in identifying the “important” parameters to which more attention should be given to better assess their values and, accordingly, to reduce the overall uncertainty of the system outputs.
7.1.2 Reliability of Hydraulic Engineering Systems All hydraulic engineering systems placed in a natural environment are subject to various external stresses. The resistance or strength of a hydraulic engineering system is its ability to accomplish the intended mission satisfactorily without failure when subject to loading of demands or external stresses. Failure occurs when the resistance of the system is exceeded by the load. From the previous discussions on the existence of uncertainties, the capacity of a hydraulic engineering system and the imposed loads, more often than not, are random and subject to some degree of uncertainty. Hence, the design and operation of hydraulic engineering systems are always subject to uncertainties and potential failures. The reliability, ps, of a hydraulic engineering system is defined as the probability of nonfailure in which the resistance of the system exceeds the load; that is, ps P(L R)
(7.1)
where P(ⴢ) denotes the probability. The failure probability, pf , is the compliment of the reliability which can be expressed as pf P[(L R)] 1 ps
(7.2)
In hydraulic engineering system design and analysis, loads generally arise from natural events, such as floods and storms, which occur randomly in time and in space. A common practice for determining the reliability of a hydraulic engineering system is to assess the return period or recurrence interval of the design event. In fact, the return period is equal to the reciprocal of the probability of the occurrence of the event in any one time interval. For most engineering applications, the time interval chosen is 1 year so that the probability associated with the return period is the average annual probability of the occurrence of the event. Flood frequency analysis, using the annual maximum flow series, is a typical example of this kind of application. Hence, the determination of return period
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RISK/RELIABILITY-BASED HYDRAULIC ENGINEERING DESIGN
Risk/Reliability–Based Hydraulic Engineering Design 7.3
depends on the time period chosen (Borgman, 1963). The main disadvantage of using the return period method is that reliability is measured only in terms of time of occurrence of loads without considering the interactions with the system resistance (Melchers, 1987). Two other types of reliability measures that consider the relative magnitudes of resistance and anticipated load (called design load) are frequently used in engineering practice. One is the safety margin (SM), defined as the difference between the resistance (R) and the anticipated load (L), that is, SM R L
(7.3)
The other is called the safety factor (SF), a ratio of resistance to load, which is defined as SF R/L
(7.4)
Yen (1979) summarized several types of safety factors and discussed their applications to hydraulic engineering system design. There are two basic probabilistic approaches to evaluate the reliability of a hydraulic engineering system. The most direct approach is a statistical analysis of data of past failure records for similar systems. The other approach is through reliability analysis, which considers and combines the contribution of each factor potentially influencing the failure. The former is a lumped system approach requiring no knowledge about the behavior of the facility or structure nor its load and resistance. For example, dam failure data show that the overall average failure probability for dams of all types over 15 m height is around 103 per dam year (Cheng, 1993). In many cases, this direct approach is impractical because (1) the sample size is too small to be statistically reliable, especially for low probability/high consequence events; (2) the sample may not be representative of the structure or of the population; and (3) the physical conditions of the dam may be non–stationary, that is, varying with respect to time. The average risk of dam failure mentioned above does not differentiate concrete dams from earthfill dams, arch dams from gravity dams, large dams from small dams, or old dams from new dams. If one wants to know the likelihood of failure of a particular 10 –year–old double–curvature arch concrete high dam, one will most likely find failure data for only a few similar dams, this is insufficient for any meaningful statistical analysis. Since no dams are identical and dam conditions change with time, in many circumstances, it may be more desirable to use the second approach by conducting a reliability analysis. There are two major steps in reliability analysis: (1) to identify and analyze the uncertainties of each contributing factor; and (2) to combine the uncertainties of the stochastic factors to determine the overall reliability of the structure. The second step, in turn, may proceed in two different ways: (1) directly combining the uncertainties of all factors, or (2) separately combining the uncertainties of the factors belonging to different components or subsystems to evaluate first the respective subsystem reliability and then combining the reliabilities of the different components or subsystems to yield the overall reliability of the structure. The first way applies to very simple structures, whereas the second way is more suitable for complicated systems. For example, to evaluate the reliability of a dam, the hydrologic, hydraulic, geotechnical, structural, and other disciplinary reliabilities could be evaluated separately first and then combined to yield the overall dam reliability. Or, the component reliabilities could be evaluated first, according to the different failure modes, and then combined. Vrijling (1993) provides an actual example of the determination and combination of component reliabilities in the design of the Eastern Scheldt Storm Surge Barrier in The Netherlands. The main purpose of this chapter is to demonstrate the usage of various practical uncertainty and reliability analysis techniques through worked examples. Only the essential theories of the techniques are described. For more detailed descriptions of the methods and applications, see Tung (1996). Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
RISK/RELIABILITY-BASED HYDRAULIC ENGINEERING DESIGN
7.4
Chapter Seven
7.2 TECHNIQUES FOR UNCERTAINTY ANALYSIS In this section, several analytical methods are discussed that would allow an analytical derivation of the exact PDF and/or statistical moments of a random variable as a function of several random variables. In theory, the concepts described in this section are straightforward. However, the success of implementing these procedures largely depends on the functional relation, forms of the PDFs involved, and analyst’s mathematical skill. Analytical methods are powerful tools for problems that are not too complex. Although their usefulness is restricted in dealing with real life complex problems, situations do exist in which analytical techniques could be applied to obtain exact uncertainty features of model outputs without approximation or extensive simulation. However, situations often arise in which analytical derivations are virtually impossible. It is, then, practical to find an approximate solution.
7.2.1 Analytical Technique: Fourier and Exponential Transforms The Fourier and exponential transforms of a PDF, fx(x), of a random variable X are defined, respectively, as Ᏺx(s) E[eisx]
eisxfx(x) dx
(7.5a)
Ᏹx(s) E[esx]
esxfx(x) dx
(7.5b)
∞
–∞
and ∞
–∞
where i 1 and E(·) is the expectation operator; E[eisX] and E[esX] are called, respectively, the characteristic function and moment generating function, of the random variable X. The characteristic function of a random variable always exists for all values of the arguments whereas for the moment generating function this is not necessarily true. Furthermore, the characteristic function for a random variable under consideration is unique. In other words, two distribution functions are identical if and only if the corresponding characteristic functions are identical (Patel et al., 1976). Therefore, given a characteristic function of a random variable, its probability density function can be uniquely determined through the inverse transform as
∞ fx(x) 1 eisxᏲx(s) ds (7.6) 2π –∞ The characteristic functions of some commonly used PDFs are shown in Table 7.1. Furthermore, some useful operational properties of Fourier transforms on a PDF are given in Table 7.2 Using the characteristic function, the rth order moment about the origin of the random variable X can be obtained as
d r Ᏺx(s) E(X r ) µr’ 1r i ds r
s=0
(7.7)
Fourier and exponential transforms are particularly useful when random variables are independent and linearly related. In such cases, the convolution property of the Fourier transform can be applied to derive the characteristic function of the resulting random variable. More specifically, consider that W X1 X2 . . . XN and all Xs are independent
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RISK/RELIABILITY-BASED HYDRAULIC ENGINEERING DESIGN
Risk/Reliability–Based Hydraulic Engineering Design 7.5 TABLE 7.1
Characteristic Functions of Some Commonly Used Distributions
Distribution
PDF, fX(x)
Characteristic Function
Binomial
n
Cx px qnx
(q pis)n
Poisson
ev v x x!
exp{v(eis 1)}
Uniform
1 b a
eibs eias i(b a)s
Normal
1 e 2π σ
Gamma
β (βx)α1 eβx Γ(α)
βα β is
Exponential
βeβx
Extreme– Value I
1 x xξ exp exp β β β
β β is
TABLE 7.2
(
)
1 xµ 2 σ
2
exp {iµs 0.5 s2 σ2}
eiβs Γ(1 iβs)
Operation Properties of Fourier Transform on a PDF
Property
PDF
Random Variable
Fourier Transform
Standard
fX(x)
X
Ᏺx(s)
Scaling
fX(ax)
X
a1Ᏺx(s/a)
Linear
afX(x)
X
a Ᏺx(s)
Translation 1
eaxfX(x)
X
Ᏺx(s ia)
Translation 2
fX(x a)
X
eias Ᏺx(s)
Source: From Springer (1979).
random variables with known PDF, fj(x), j 1, 2, . . . , N. The characteristic function of W then can be obtained as
Ᏺw(s) Ᏺ1(s) Ᏺ2(s) . . . ᏲN(s)
(7.8a)
Ᏹw(s) Ᏹ1(s) Ᏹ2(s) . . . ᏱN(s)
(7.8b)
and
which is the product of the characteristic and moment generating functions of each individual random variable. The resulting characteristic function or moment generating function for W can be used in Eq. (7.7) to obtain the statistical moments of any order for
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RISK/RELIABILITY-BASED HYDRAULIC ENGINEERING DESIGN
7.6
Chapter Seven
the random variable W. Furthermore, the inverse transform of ᏲW(s), according to Eq. (7.6), can be made to derive the PDF of W, if it is analytically possible.
7.2.2 Analytical Technique: Mellin Transform When the random variables in a function W g(X) are independent and nonnegative and the function g(X) has a multiplicative form as N
W g(X) a0
X
ai i
(7.9)
i1
the Mellin transform is particularly attractive for conducting uncertainty analysis (Tung, 1990). The Mellin transform of a PDF fX(x), where x is positive, is defined as
x
MX(s) M[fx(x)]
∞
s1
o
fx(x) dx E (xs1), x 0
(7.10)
where MX(s) is the Mellin transform of the function fX(x) (Springer 1979). Therefore, the Mellin transform provides an alternative way to find the moments of any order for nonnegative random variables. Similar to the convolutional property of the exponential and Fourier transforms, the Mellin transform of the convolution of the PDFs associated with multiple independent random variables in a product form is simply equal to the product of the Mellin transforms of individual PDFs. In addition to the convolution property, the Mellin transform has several useful operational properties as summarized in Tables 7.3 and 7.4. Furthermore, the Mellin transform of some commonly used distributions are summarized in Table 7.5. Example 7.1. Manning’s formula is frequently used for determining the flow capacity TABLE 7.3
Operation Properties of the Mellin Transform on a PDF
Property Standard Scaling Linear Translation Exponentiation
PDF fX(x) fX(x) fX(ax) afX(x) xafX(x) fX(xa)
Random Variable X X X X X
Mellin Transform
Mx(s) asMx(s) a Mx(s) Mx(a s) a1 Mx(s/a)
Source: From Park (1987).
TABLE 7.4
Mellin Transform of Products and Quotients of Random Variables*
Random Variable
PDF Given
MW(s)
WX W X/b W 1/X W XY W X/Y W aXbYc
fX(x) fX(x) fX(x) fX(x), gY(y) fX(x), gY(y) fX(x), gY(y)
MX(s) MX(bs b 1) MX(2s) MX(s)MY(s) MX(s)MY(2s) as 1MX(bs b 1)MY(cs c 1)
Source: From Park (1987). a, b, c: constants ; X, Y, W: random variables.
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RISK/RELIABILITY-BASED HYDRAULIC ENGINEERING DESIGN
Risk/Reliability–Based Hydraulic Engineering Design 7.7 TABLE 7.5
Mellin Transforms for Some Commonly Used Probability Density Functions
Probability
PDF, fX(x)
Mellin Transform
Uniform
1 b a
bs as s (b a)
Standard Normal
2s1 Γs 2 π , for s odd
2 1 ez /2 2π
0
lnX µlnX 2 1 2 σ
(
)
, for s even
Lognormal
1 e 2π xσlnX
Exponential
βeβx
β1sΓ(s)
Gamma
β βx (βx)α1 e Γ(α)
βs1 Γ (α s1) Γ (α)
Triangular
2 x a , a x m b a m a
1 exp (s 1)µlnX (s 1)2 σ2lnX 2
b (bs ms) a (ms as) 2 s(s1) (b a) bm ma
2 b x , m x b b a bm α1
α
x ξ exp β
Weibull
α x ξ β β
Nonstandard beta
(x a)α1 (b x)β1 B(α, β) (b a)α β1
Standard beta
xα1 (1 x) B(α , β)
s1
S 1 k k0
βk ξs1 k Γ k 1 α
Γ (a b) Γ(a s 1)
s1
s 1 k
k0
as1k (b a)k MX(k)
where MX(k) for standard beta
of storm sewer by Q = 0.463 n1 D2.67 S0.5 where Q is flow rate (ft3/s), n is the roughness coefficient, D is the sewer diameter (ft), and S is pipe slope (ft/ft). Assume that all three model parameters are independent random variables with the following statistical properties. Compute the mean and variance of the sewer flow capacity by the Mellin transform.
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RISK/RELIABILITY-BASED HYDRAULIC ENGINEERING DESIGN
7.8
Chapter Seven Parameter
Distribution
n
Uniform distribution with lower bound 0.0137 and upper bound 0.0163
D
Triangular distribution with lower bound 2.853, mode 3.0, and upper bound 3.147
S
Uniform distribution with bounds (0.00457, 0.00543)
Referring to Table 7.4, the Mellin transform of sewer flow capacity, MQ(s), can be derived as MQ(s) 0.463s 1 Mn(s 2) MD(2.67s 1.67) MS(0.5s 0.5) For roughness coefficient n having a uniform distribution, from Table 7.5, one obtains nbs2 nas 2 Mn(s 2) ( s 2)(nb na) For sewer diameter D with a triangular distribution, one obtains 2 MD(2.67s 1.67) (db da)(2.67s 1.67) (2.67s 0.67) db(db2.67s1.67 dm2.67s1.67) da(dm2.67s 1.67 d a2.67s 1.67) . dm da db dm For sewer slope S with a uniform distribution, one obtains (sb)0.5s 0.5 (sa)0.5s 0.5 MS(0.5s 0.5) (0.5s 0.5) (sb sa) Substituting individual terms into MQ(s) results in the expression of the Mellin transform of sewer flow capacity specifically for the distributions associated with the three stochastic model parameters. Based on the information given, the Mellin transforms of each stochastic model parameter can be expressed as 0.0163s 0.0137s Mn(s) 0.0026s 3.147(3.147s 3.00s) 2.853(3.00s 2.853s) MD(s) 2 0.147 0.147 (0.294)s(s 1) 0.00543s 0.00457s MS(s) 0.00086 s The computations are shown in the following table:
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RISK/RELIABILITY-BASED HYDRAULIC ENGINEERING DESIGN
Risk/Reliability–Based Hydraulic Engineering Design 7.9
0.463s 1 0.463
s2
s3
0.463
0.2144
Mn(s 2)
Mn(0) 66.834
Mn(1) 4478.080
MD(2.67s 1.67)
MD(3.67) 18.806
MD(6.34) 354.681
Mn(0.5s 0.5)
MS(1.50) 0.0707
MS(2.00) 0.005
Therefore, the mean sewer flow capacity can be determined as E(Q) MQ(s 2) 0.463 Mn(0) MD(3.67) MS(1.50) 0.463(66.834)(18.806)(0.0707) 41.137 ft3/s The second moment about the origin of the sewer flow capacity is E(Q2) MQ(s 3) 0.4632 Mn(1) MD(6.34) MS(2.00) 0.4632 (4478.08) (354.681) (0.005) 1702.40 ft3/s. The variance of the sewer flow capacity can then be determined as Var(Q) E(Q2) E2(Q) 1702.40 41.1372 10.147 (ft3/s)
2
with the standard deviation being 0.1 47 3.186 ft3/s σQ 1 7.2.3 Approximate Technique: First-Order Variance Estimation (FOVE) Method The FOVE method, also called the variance propagation method (Berthouex, 1975), estimates uncertainty features of a model output based on the statistical properties of the model's random variables. The basic idea of the method is to approximate a model involving random variables by the Taylor series expansion. Consider that a hydraulic or hydrologic design quantity W is related to N random variables X1, X2, . . ., XN as W g(X) g(X1, X2, . . ., XN)
(7.11)
where X (X1, X2, . . ., XN)t, an N–dimensional column vector of random variables, the superscript t represents the transpose of a matrix or vector. The Taylor series expansion of the function g(X) with respect to the means of random variables X µ in the parameter space can be expressed as
N ∂g(X) 1 (Xi µi) ∂Xi µ 2 i=1 i1 (Xi µi) (Xj µj) ε N
W g(µ)
N
j=1
∂2g(X) ∂Xi ∂Xj µ
(7.12)
where µi is the mean of the ith random variable Xi and ε represents the higher order terms. The first–order partial derivative terms are called the sensitivity coefficients, each representing the rate of change of model output W with respect to unit change of each variDownloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
RISK/RELIABILITY-BASED HYDRAULIC ENGINEERING DESIGN
7.10
Chapter Seven
able at µ. Dropping the higher-order terms represented by ε, Eq. (7.12) is a second–order approximation of the model g(X). The expectation of model output W can be approximated as 1 E[W] g(µ) 2
N
N
∂2g(X) ∂Xi ∂Xj µ i1 j1
Cov [Xi, Xj]
(7.13)
and the variance of W g(X) can be expressed as
∂g(X) ∂Xi µ i1 j1 N
Var[W]
1 2
N
N
N
∂g(X) E[(Xi µi) (Xj µj)] ∂Xj
∂g(X) ∂2g(X) E[(Xi µi) (Xj µj) (XK µK)] ∂Xi µ ∂Xj∂XK µ K1 N
i1 j1
(7.14)
As can be seen from Eq. (7.14), when random variables are correlated, the estimation of the variance of W using the second–order approximation would require knowledge about the cross–product moments among the random variables. Information on the cross–product moments are rarely available in practice. When the random variables are independent, Eqs. (7.13) and (7.14) can be simplified, respectively, to 1 E[W] g(µ) 2 and
∂2g(X) E[(Xi µi)2] ∂Xi2 µ i1
N ∂g(X) 2 ∂g(X) 1 σ i2 2 ∂Xi ∂Xi i1 i1
N
Var[W]
N
∂2g(X) E[(Xi µi)3] 2 ∂Xi µ
(7.15)
(7.16)
Referring to Eq. (7.16), the variance of W from a second-order approximation, under the condition that all random variables are statistically independent, would require knowledge of the third moment. For most practical applications where higher order moments and cross–product moments are not easily available, the first order approximation is frequently adopted. In the area of structural engineering, the second order methods are commonly used (Breitung, 1984; Der Kiureghian et al., 1987 Wen, 1987). By truncating the second and higher–order terms of the Taylor series, the first–order approximation of W at X µ is E[W] g(µ) w
and
∂g(X) ∂g(X) Cov (Xi, Xj) stC(X)s ∂Xi µ ∂Xj µ i1 j1 N
Var[W]
(7.17)
N
(7.18)
in which s ∇xW(µ) is an N–dimensional vector of sensitivity coefficients evaluated at µ; C(X) is the variance covariance matrix of the random vector X. When all random variables are independent, the variance of model output W can be approximated as
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RISK/RELIABILITY-BASED HYDRAULIC ENGINEERING DESIGN
Risk/Reliability–Based Hydraulic Engineering Design 7.11 N
Var(W)
si2 σ i2 s’Ds,
(7.19)
i1
in which represents the standard deviation and D diag (σ 21, σ 22, …, σ N2), a diagonal matrix of variances of involved random variables. From Eq. (7.19), the ratio si2 σ i2/Var[W], indicates the proportion of overall uncertainty in the model output contributed by the uncertainty associated with the random variable Xi. Example 7.2. Referring to Example 7.1, the uncertainty associated with the sewer slope due to installation error is 5 percent of its intended value 0.005. Determine the uncertainty of the sewer flow capacity using the FOVE method for a section of 3 ft sewer with a 2 percent error in diameter due to manufacturing tolerances. The roughness coefficient has the mean value 0.015 with a coefficient of variation 0.05. Assume that the correlation coefficient between the roughness coefficient n and sewer diameter D is 0.75. The sewer slope S is uncorrelated with the other two random variables. Solution: The first–order Taylor series expansion of Manning's formula about n µn 0.015, D µD 3.0, and S µS 0.005, according to Eq. (7.12), is
(D 3.0) (S 0.0005) ∂∂QD ∂∂QS
∂Q Q 0.463 (0.015)1 (3)2.67 (0.005)0.5 (n 0.015) ∂n
41.01 [0.463 (1) (0.015)2 (3.0)2.67 (0.005)0.5] (n 0.015) [0.463 (2.67)(0.015)1 (3.0)1.67 (0.005)0.5] (D 3.0) [0.463 (0.5)(0.015)1 (3.0)2.67 (0.005)0.5] (S 0.005) 41.01 2733.99 (n 0.015) 36.50 (D 3.0) 4100.99 (S 0.005) Based on Eq. (7.17), the approximated mean of the sewer flow capacity is µQ 41.01 ft3/s According to Eq. (7.18), the approximated variance of the sewer flow capacity is σ2Q (2733.99)2 Var(n) (36.50)2 Var(D) (4100.99)2 Var(S) 2(2733.99)(36.50) Cov(n, D) 2(2733.99)(4100.99) Cov(n, S) 2(36.50)(4100.99) Cov(D, S) The above expression reduces to σ2Q (2733.99)2 Var(n) (36.50)2 Var(D) (4100.99)2 Var(S) 2(2733.99)(36.50) Cov(n, D) because Cov(n, S) Cov(D, S) 0. Since the standard deviations of roughness, pipe diameter, and slope are Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
RISK/RELIABILITY-BASED HYDRAULIC ENGINEERING DESIGN
7.12
Chapter Seven
σn (0.05)(0.015) 0.00075 σD (0.02)(3.0) 0.06 σS (0.05)(0.005) 0.00025 the variance of the sewer flow capacity can be computed as σQ2 (2733.99)2 (5.625 107) (36.50)2 (3.6 104) (4100.99)2 (6.25 108) 2 (2733.99)(36.50) (3.375 105) 2.052 2.192 1.032 6.74 16.79 ft3/s2 Hence, the standard deviation of the sewer flow capacity is 16.7 9 4.10 ft3/s which is 10.0 percent of the estimated mean sewer flow capacity. Without considering correlation between n and D, σQ2 16.79 6.74 10.05 which underestimates the variance of the sewer flow capacity. The percentages contribution of uncertainty of n, D, and S to the overall uncertainty of the sewer flow capacity under the uncorrelated condition are, respectively, 41.8 percent, 47.7 percent, and 10.5 percent. The uncertainty associated with the sewer slope contributes less significantly to the total sewer flow capacity uncertainty as compared with the other two random variables even though it has the highest sensitivity coefficient among the three. This is because the variance of S, Var(S), is smaller than the variances of the other two random variables.
7.2.4 Approximate Technique: Rosenblueth’s Probabilistic Point Estimation (PE) Method Rosenblueth’s probabilistic point estimation (PE) method is a computationally straightforward technique for uncertainty analysis. It can be used to estimate statistical moments of any order of a model output involving several random variables which are either correlated or uncorrelated. Rosenblueth’s PE method was originally developed for handling random variables that are symmetric (Rosenblueth, 1975). It was later extended to treat nonsymmetric random variables (Rosenblueth, 1981). Consider a model, W g(X), involving a single random variable X whose first three moments or probability density function (PDF)/ probability mass function (PMF) are known. Referring to Fig. 7.1, Rosenblueth’s PE method approximates the original PDF or PMF of the random variable X by assuming that the entire probability mass of X is concentrated at two points x and x. Using the two point approximation, the locations of x and x and the corresponding probability masses p and p are determined to preserve the first three moments of the random variable X. Without changing the nature of the original problem, it is easier to deal with the standardized variable, X' (Xµ)/σ, which has zero mean and unit variance. Hence, in terms of X', the following four simultaneous equations can be established to solve for x', x', p, and p: p p 1
(7.20a)
px’ px’ µX’ 0
(7.20b)
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RISK/RELIABILITY-BASED HYDRAULIC ENGINEERING DESIGN
Risk/Reliability–Based Hydraulic Engineering Design 7.13
fx (x) p–
p+
x–
x+
µ
x
FIGURE 7.1 Schematic diagram of Rosenblueth’s PE method in univariate case (Tung, 1996) 2
2
px’ px’ σ 2X/ 1 3
(7.20c)
3
px’ px’ γ
(7.20d)
in which x.' x µ /σ, x' x µ /σ, and γ is the skew coefficient of the random variable X. Solving Eqs. (7.20a-d) simultaneously, one obtains γ x’ 2
12γ 2
(7.21a)
x’ x’ γ
(7.21b)
x’ p x’ x’
(7.21c)
p 1 p
(7.21d)
When the distribution of the random variable X is symmetric, that is, γ 0, then Eqs. (7.21a–d) are reduced to x' x' 1 and p p 0.5. This implies that, for a symmetric random variable, the two points are located at one standard deviation to either side of the mean with equal probability mass assigned at the two points. From x' and x' the two points in the original parameter space, x and x, can respectively be determined as x µ x' σ
(7.22a)
x µ x' σ
(7.22b)
Based on x and x, the values of the model W g(X) at the two points can be computed, respectively, as w g(x) and w g(x). Then, the moments about the origin of W g(X) of any order can be estimated as E[Wm] µ'W,m p wm p wm
(7.23)
Unlike the FOVE method, Rosenblueth’s PE estimation method provides added capability allowing analysts to account for the asymmetry associated with the PDF of a ranDownloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
RISK/RELIABILITY-BASED HYDRAULIC ENGINEERING DESIGN
7.14
Chapter Seven
dom variable. Karmeshu and Lara Rosano (1987) show that the FOVE method is a first order approximation to Rosenblueth’s PE method. In a general case where a model involves N correlated random variables, the mth moment of the model output W g(X1, X2, …, XN) about the origin can be approximated as E(Wm)
Σp
(δ1, δ2,…, δN)
[w(δ1, δ2,…, δN)]m
(7.24)
in which the subscript δi a sign indicator and can only be or representing the random variable Xi having the value of xi µix'i σi or xi µix'i σi, respectively; the probability mass at each of the 2N points, p(δ1, δ 2,…, δN) can be approximated by
Π pδ N1
N
p(δ1, δ2,…, δN)
i1
with
i i
i1
N
δi δj aij
ji1
ρij / 2N aij N γi 2 1 2 i 1
Π
(7.25)
(7.26)
where ρij is the correlation coefficient between random variables Xi and Xj. The number of terms in the summation of Eq. (7.24) is 2N which corresponds to the total number of possible combinations of and for all N random variables. Example 7.3 Referring to Example 7.2, assume that all three model parameters in Manning’s formula are symmetric random variables. Determine the uncertainty of sewer flow capacity by Rosenblueth’s PE method. Solution: Based on Manning’s formula for the sewer, Q = 0.463 n1 D2.67 S 0.5, the standard deviation of the roughness coefficient, sewer diameter, and pipe slope are σn 0.00075; σD 0.06; σS 0.00025 With N 3 random variables, there are a total of 23 8 possible points to be considered by Rosenblueth’s method. Since all three random variables are symmetric, their skew coefficients are equal to zero. Therefore, according to Eqs. (7.21a–b), n' n' D' D' S' S' 1 and the corresponding values of roughness coefficient, sewer diameter, and pipe slope are n µn n' σn 0.015 (1)(0.00075) 0.01575 n µn n' σn 0.015 (1)(0.00075) 0.01425 D µD D' σD 3.0 (1)(0.06) 3.06 ft D µD D' σD 3.0 (1)(0.06) 2.94 ft S µS S' σS 0.005 (1)(0.00025) 0.00525 S µS S' σS 0.005 (1)(0.00025) 0.00475 Substituting the values of n, n, D, D, S, and S into Manning’s formula to compute the corresponding sewer capacities, one has, for example,
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RISK/RELIABILITY-BASED HYDRAULIC ENGINEERING DESIGN
Risk/Reliability–Based Hydraulic Engineering Design 7.15
Q 0.463 (n)1 (D)2.67 (S)0.5 0.463 (0.01575)1 (3.06)2.67 (0.00525)0.5 42.19 ft3/s Similarly, the values of sewer flow capacity for the other seven points are given in the following table: Point
n
D
S
Q (ft3s)
p
1 2 3 4 5 6 7 8
42.19 40.14 37.92 36.07 46.64 44.36 41.91 39.87
0.03125 0.03125 0.21875 0.21875 0.21875 0.21875 0.03125 0.03125
Because the roughness coefficient and sewer diameter are symmetric, correlated random variables, the probability masses at 23 8 points can be determined, according to Eqs. (7.25–7.26) as p p (1 ρnD ρnS ρDS)/8 (1 0.75 0 0)/8 0.03125 p p (1 ρnD ρnS ρDS)/8 (1 0.75 0 0)/8 0.03125 p p (1 ρnD ρnS ρDS)/8 (1 0.75 0 0)/8 0.21875 p p (1 ρnD ρnS ρDS)/8 (1 0.75 0 0)/8 0.21875 The values of probability masses also are tabulated in the last column of the above table. Therefore, the mth order moment about the origin for the sewer flow capacity can be calculated by Eq. (7.24). The computations of the first two moments about the origin are shown in the following table in which columns (1)–(3) are extracted from the above table.: Point
Q(ft3)
p
Q p
Q2
Q2 p
(1)
(2)
(3)
(4)
(5)
(6)
1 2 3 4 5 6 7 8
42.19 40.14 37.92 36.07 46.64 44.36 41.91 39.87
0.03125 0.03125 0.21875 0.21875 0.21875 0.21875 0.03125 0.03125
1.318 1.254 8.295 7.890 10.203 9.704 1.310 1.246
1780.00 1611.22 1437.93 1301.04 2175.29 1967.81 1756.45 1589.62
55.625 50.351 314.547 284.603 475.845 430.458 54.889 49.676
Sum
——
1.00000
41.219
——
1715.99
2
ft3 From the above table, µQ E(Q) 41.219 ft3 and E(Q2) 1715.994 . Then, the s variance of the sewer flow capacity can be estimated as
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RISK/RELIABILITY-BASED HYDRAULIC ENGINEERING DESIGN
7.16
Chapter Seven
ft3 Var(Q) E(Q2) (µQ)2 1715.994 (41.219)2 16.988 s
2
Hence, the standard deviation of sewer flow capacity is 1 6.9 88 4.12 ft3/s3. Comparing with the results in Example 7.2, one observes that Rosenblueth’s PE method yields higher values of the mean and variance for the sewer flow capacity than those obtained with the FOVE method.
7.2.5 Approximate Technique: Harr’s Probabilistic Point Estimation (PE) Method To avoid the computationally intensive nature of Rosenblueth’s PE method when the number of random variables is moderate or large, Harr (1989) proposed an alternative probabilistic PE method which reduces the required model evaluations from 2N to 2N and greatly enhances the applicability of the PE method for uncertainty analysis of practical problems. The method is a second moment method which is capable of taking into account of the first two moments (that is, the mean and variance) of the involved random variables and their correlations. Skew coefficients of the random variables are ignored by the method. Hence, the method is appropriate for treating normal and other symmetrically distributed random variables. The theoretical basis of Harr's PE method is built on orthogonal transformations of the correlation matrix. The orthogonal transformation is an important tool for treating problems with correlated random variables. The main objective of the transformation is to map correlated random variables from their original space to a new domain in which they become uncorrelated. Hence, the analysis is greatly simplified. Consider N multivariate random variables X (X1, X2, …, XN)t having a mean vector µX (µ1, µ2, …, µN)t and the correlation matrix R(X)
C (X') R (X)
1 ρ12 ρ13 ρ21 1 ρ23
·
·
·
ρ1N ρ2N
·
·
·
·
·
·
·
·
·
·
·
·
·
·
·
·
·
ρN1 ρN2 ρN3
·
·
·
·
·
·
·
·
·
·
1
Note that the correlation matrix is a symmetric matrix, that is, ρij ρij for i j. The orthogonal transformation can be made using the eigenvalue eigenvector decomposition or spectral decomposition by which R(X) is decomposed as R(X) = C(X') = V Λ Vt
(7.27)
where V is an N N eigenvector matrix consisting of N eigenvectors as V = (v1, v2, ..., vN) with vi being the ith column eigenvector and Λ diag(λ1, λ2, …, λN) is a diagonal eigenvalues matrix. In terms of the eigenvectors and eigenvalues, the random vector in the original parameter space can be expressed as X = µ + D1/2 VΛ1/2 Y
(7.28)
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RISK/RELIABILITY-BASED HYDRAULIC ENGINEERING DESIGN
Risk/Reliability–Based Hydraulic Engineering Design 7.17
in which Y is a vector of N standardized random variables having 0 as the mean vector and the identity matrix, I, as the covariance matrix, and D is a diagonal matrix of variances of N random variables. The transformed variables, Y, are linear functions of the original random variables, therefore, if all the original random variables X are normally distributed, then the standardized transformed random variables, Y, are independent standard normal random variables. For a multivariate model W g (X1, X2, …, XN) involving N random variables, Harr’s method selects the points of evaluation located at the intersections of N eigenvector axes with the surface of a hypersphere having a radius of N in the eigenspace as D1/2 vi, i 1, 2, ..., N xi µ N
(7.29)
in which xi represents the vector of coordinates of the N random variables in the parameter space corresponding to the ith eigenvector vi ; µ (µ1, µ2, ..., µN)t, a vector of means of N random variables X. Based on the 2N points determined by Eq. (7.29), the function values at each of the 2N points can be computed. Then, the mth moment of the model output W about the origin can be calculated according to the following equations: wmi wmi gm(xi) gm (xi) wmi = (7.30) = i = 1, 2, …, N; m = 1, 2, … 2 2,
N
λi wmi
E[Wm] = µm(W) =
, for m = 1, 2,… i=1 N
(7.31)
λi
i=1
Alternatively, the orthogonal transformation can be made to the covariance matrix. Example 7.4. Referring to Example 7.2, determine the uncertainty of the sewer flow capacity using Harr’s PE method. Solution: From the previous example, statistical moments of random parameters in Manning’s formula, Q 0.463 n1 D2.67 S0.5 are µn 0.01500; µD 3.00; µS 0.00500 σn 0.00075; σD 0.06; σS 0.00025 From the given correlation relation among the three random variables, the correlation matrix can be established as
1.00 0.75 0.75 1.00 R (n, D, S) 000 000
000
000 1.00
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RISK/RELIABILITY-BASED HYDRAULIC ENGINEERING DESIGN
7.18
Chapter Seven
The corresponding eigenvector matrix and eigenvalue matrix are, respectively,
]
v11 v12 v13 0.7071 0.7071 0.0000 v21 v22 v23 0.7071 0.7071 0.0000 v31 v32 v33 0.0000 0.0000 1.0000
V [v1 v2 v3
and
Λ diag (λ1, λ2, λ3) diag (1.75, 0.25, 1.00)
According to Eq. (7.29), the coordinates of the 2 3 6 intersection points corresponding to the three eigenvectors and the hypersphere with a radius 3 can be determined as
µn
σn 0
0
xi = µD
0 σD
0
µS
0
σS
0.015
0.00075 0
0
0
0.06
0
0
0
0.00025
= 3.0
0.005
3
3
0
vi
vi , for i 1, 2, 3
The resulting coordinates at the six intersection points from the above equation are listed in column (2) of the table given below. Substituting the values of x in column (2) into Manning’s formula, the corresponding sewer flow capacities are listed in column (3). Column (4) lists the value of Q2 for computing the second moment about the origin later. After columns (3) and (4) are obtained, the averaged value of Q and Q2 along each eigenvector are computed and listed in columns (5) and (6), respectively. Point (1)
x (n, D, S) (2)
Q (3)
Q2 (4)
Q (5)
Q2 (6)
1 1 2 2 3 3
(0.01592, 2.9265, 0.00500) (0.01408, 3.0735, 0.00500) (0.01592, 3.0735, 0.00500) (0.01408, 2.9265, 0.00500) (0.01500, 3.00, 0.00543) (0.01500, 3.00, 0.00457)
36.16 46.61 41.22 40.89 42.74 39.20
1307.82 2172.14 1699.09 1671.99 1826.45 1537.17
41.39
1739.98
41.05
1685.54
40.97
1681.81
The mean of the sewer flow capacity can be calculated, according to Eq. (7.31), with m 1, as λ1 Q 1 λ2 Q 2 λ3 Q 3 µQ λ1 λ2 λ3 1.75 (41.39) 0.25 (41.05) 1.00 (40.97) 41.22 ft3/s 3
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RISK/RELIABILITY-BASED HYDRAULIC ENGINEERING DESIGN
Risk/Reliability–Based Hydraulic Engineering Design 7.19
The second moment about the origin is calculated as: 2 λ Q 2 2 λ Q λ1 Q 1 2 3 2 3 2 E(Q ) λ1 λ2 λ3 1.75 (1727.11) 0.25(1673.12) 1.00(1669.54) 1716.05 (ft3/s)2 3 The variance of the sewer flow capacity then can be calculated as 2
Var(Q) E(Q2) (µQ)2 1716.05 (41.22)2 16.82 (ft3/s) . 2 4.10 ft3/s2. Hence, the standard deviation of sewer flow capacity is 16.8 Comparing with the results in Examples 7.2 and 7.3, one observes that the mean and variance of the sewer flow capacity computed with Harr's PE method lie between those computed with the FOVE method and Rosenblueth’s method.
7.3 RELIABILITY ANALYSIS METHODS In a multitude of hydraulic engineering problems uncertainties in data and in theory, including design and analysis procedures, warrant a probabilistic treatment of the problems. The risk associated with the potential failure of a hydraulic engineering system is the result of the combined effects of inherent randomness of external loads and various uncertainties involved in the analysis, design, construction, and operational procedures. Hence, to evaluate the probability that a hydraulic engineering system would function as designed requires performing uncertainty and reliability analyses. As discussed in Sec. 7.1.2, the reliability, ps, is defined as the probability of safety (or non–failure) in which the resistance of the structure exceeds the load, that is, ps P (L R). Conversely, the failure probability, pf, can be computed as pf P (L R) 1 ps. The above definitions of reliability and failure probability are equally applicable to component reliability as well as total system reliability. In hydraulic design, the resistance and load are frequently functions of a number of random variables, that is, L g(XL) g(X1, X2, …, Xm) and R h(XR) h(Xm 1, Xm 2, …,Xn) where X1, X2, …, Xn are random variables defining the load function, g(XL), and the resistance function, h(XR). Accordingly, the reliability is a function of random variables ps P [g(XL) h(XR)]
(7.32)
As discussed in the preceding sections, the natural hydrologic randomness of flow and precipitation are important parts of the uncertainty in the design of hydraulic structures. However, other uncertainties also may be significant and should not be ignored.
7.3.1
Performance Functions and Reliability Index
In the reliability analysis, Eq. (7.32) can alternatively be written, in terms of a performance function, W(X) W(XL, XR), as
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RISK/RELIABILITY-BASED HYDRAULIC ENGINEERING DESIGN
7.20
Chapter Seven
ps P [W(XL, XR) 0] = P [W(X) 0],
(7.33)
in which X is the vector of basic random variables in the load and resistance functions. In reliability analysis, the system state is divided into safe (satisfactory) set defined by W(X) > 0 and failure (unsatisfactory) set defined by W(X) < 0 (Fig. 7.2). The boundary that separates the safe set and failure set is the failure surface, defined by the function W(X) 0, called the limit state function. Since the performance function W(X) defines the condition of the system, it is sometimes called the system state function. The performance function W(X) can be expressed differently as W1(X) R L h(XR ) g(XL )
(7.34)
W2(X) (R/L) 1 [h(XR )/g(XL )] 1
(7.35)
W3 (X) ln(R/L) ln[h(XR )] ln[g(XL )]
(7.36)
Referring to Sec. 7.1.2, Eq. (7.34) is identical to the notion of safety margin, whereas Eqs. (7.35) and (7.36) are based on safety factor representations. Also in the reliability analysis, a frequently used reliability indicator, β, is called the reliability index. The reliability index was first introduced by Cornell (1969) and later formalized by Ang and Cornell (1974). It is defined as the ratio of the mean to the standard deviation of the performance function W(X) µW β (7.37) σW in which µW and σW are the mean and standard deviation of the performance function,
FIGURE 7.2 System states defined by performance function
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RISK/RELIABILITY-BASED HYDRAULIC ENGINEERING DESIGN
Risk/Reliability–Based Hydraulic Engineering Design 7.21
respectively. From Eq. (7.37), assuming an appropriate PDF for the random performance function W(X), the reliability then can be computed as Ps 1 FW(0) 1 FW'(β)
(7.38)
in which FW(·) is the cumulative distribution function (CDF) of the performance function W and W' is the standardized performance function defined as W' (WµW)/σW. The expression of reliability, ps, for some distributions of W(X) are given in Table 7.6. For practically all probability distributions used in the reliability analysis, the value of the reliability, ps, is a strictly increasing function of the reliability index, β. In practice, the normal distribution is commonly used for W(X) in which case the reliability can be simply computed as ps 1 Φ(β) Φ(β)
(7.39)
where Φ(·) is the standard normal CDF whose values can be found in various probability and statistics books.
7.3.2 Direct Integration Method From Eq. (7.1), the reliability can be computed in terms of the joint PDF of the load and resistance as ps
f r
r
1
2
R,L(r,ᐉ) dᐉ dr
ᐉl
l
r
2
ᐉ1
r2
ᐉ
fR,L(r,ᐉ) dr dᐉ
(7.40)
in which fR,L(r, ᐉ) is the joint PDF of random load, L, and resistance, R; r and ᐍ are dummy arguments for the resistance and load, respectively; and (r1, r2) and (ᐉ1, ᐉ2) are the lower and upper bounds for the resistance and load, respectively. This computation of reliability is commonly referred to as the load–resistance interference. When load and resistance are statistically independent, Eq. (7.40) reduces to ps or ps
ᐉ2
ᐉ1
F (r) f (r) dr E [F (R)] r2
r1
L
R
R
L
[1 FR(ᐉ)] fL(ᐉ) dᐉ 1 EL[FR(L)]
(7.41)
(7.42)
in which FL(·) and FR(·) are the marginal CDFs of random load L and resistance R, respectively; and ER[FL(R)] is the expectation of the CDF of random load over the feasible range of the resistance. A schematic diagram illustrating load–resistance interference in the reliability computation, when the load and resistance are independent random variables, is shown in Fig. 7.3. In the case that the PDF of the performance function W is known or derived, the reliability can be computed, according to Eq. (7.33), as ps
∞
0
fW (w) dw
(7.43)
in which fW(w) is the PDF of the performance function. In the conventional reliability analysis of hydraulic engineering design, uncertainty
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amb 3
2 w a , a w m b a ma
1 ,awb b a
ab 2
α a (b a) αβ
(x a)α1 (b x)β1 B(α, β) (b a)α β1
2 b w , m w b b a b m
α β ξ
1 2
ba 1 3 b a
bw b a
ab am bm 1/2 (w a)2 1 , a x m 6µ2W (b a) (m a )
Bu(α, β) a 1 ; u B(α , β) ba
αβ (b a)
1 α β (α β) µ W
1 I [β ξ, α]
eβ(w wo)
µ W Φ ln σlnW
µ W Φ σW
Reliability ps = P(W 0)
α α βξ
1 1 βwo
1 β wo
βeβ(w wo), w wo
β[β(w ξ)]α1eβ(w ξ) , w ξ
ex pσ (2lnW ) 1
1 µW exp µlnW σ2lnW 2
µW σ W
Coefficient of Variation, Ωw
1 1 ln(w ) µlnW 2 , w 0 exp σlnW 2π wσlnW 2
µW
1 wµW 2 1 exp , ∞ w < ∞ 2π σW 2 σw
Source: From Yen et al., 1986.
Uniform
Triangular
Nonstandard Beta
Gamma
Shifted Exponential
Lognormal
Normal
Mean, µW
PDF, fW(w)
Reliability Formulas for Selected Distribution (After Yen et al., 1986)
7.22
Distribution of W
TABLE 7.6
RISK/RELIABILITY-BASED HYDRAULIC ENGINEERING DESIGN
Chapter Seven
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RISK/RELIABILITY-BASED HYDRAULIC ENGINEERING DESIGN
Risk/Reliability–Based Hydraulic Engineering Design 7.23
FIGURE 7.3 Schematic diagram of of load-resistance interference
from the hydraulic aspect often is ignored. Treating the resistance or capacity of the hydraulic structure as a constant reduces Eq. (7.40) to ps
ro
0
fL(ᐉ) dᐉ
(7.44)
in which ro is the resistance of the hydraulic structure, a deterministic constant. If the PDF of the hydrologic load is the annual event, such as the annual maximum flood, the resulting annual reliability can be used to calculate the corresponding design return period. The method of direct integration requires the PDFs of the load and resistance or the performance function be known or derived. This is seldom the case in practice, especially for the joint PDF, due to the complexity of hydrologic and hydraulic models used in the design and of the natural system being approximated with these models. Explicit solution of direct integration can be obtained for only a few PDFs as shown in Table 7.6 for the reliability ps. For most other PDFs the use of numerical integration is unavoidable. For example, the distribution of the safety margin W expressed by Eq. (7.34) has a normal distribution if both load and resistance functions are linear and all random variables are normally distributed. In terms of safety factor expressed as Eqs. (7.35) and (7.36), the distribution of W(X) is lognormal if both load and resistance functions have multiplicative forms involving lognormal random variables. Example 7.5 Referring to Example 7.2, the random variables n, D, and S used in Manning's formula to compute the sewer capacity are independent lognormal random variables with the following statistical properties:
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RISK/RELIABILITY-BASED HYDRAULIC ENGINEERING DESIGN
7.24
Chapter Seven
Parameter
Mean
Coeff. of Variation
n (ft1/6)
0.015
0.05
D (ft)
3.0
0.02
S (ft/ft)
0.005
0.05
Compute the reliability that the sewer can convey the inflow discharge of 35 ft3/s. Solution: In this example, the resistance function is R(n, D, S) 0.463 n1 D2.67 S0.5 and the load is L 35 ft3/s. Since all three stochastic parameters are lognormal random variables, the performance function appropriate for use is W(n, D, S) ln(R) ln(L) [ln(0.463) ln(n) 2.67 ln(D) 0.5 ln(S)] ln(35) ln(n) 2.67 ln(D) 0.5 ln(S) 4.3319 The reliability ps P[W(n,D,S) 0] can then be computed as the following. Since n, D, and S are independent lognormal random variables, hence, ln(n), ln(D), and ln(S) are independent normal random variables. The performance function W(n, D, S) is a linear function of normal random variables, then, by the reproductive property of normal random variables, W(n, D, S) also is a normal random variable with the mean µW µln(n) 2.67 µln(D) 0.5 µln(S) 4.3319 and variance Var(W) Var[ln(n)] 2.672 Var[ln(D)] 0.52 Var[ln(S)] The means and variances of log–transformed variables can be obtained as Var[ln(n)] ln(1 0.052) 0.0025; µln(n) ln(µn) 0.5 Var[ln(n)] 4.201 Var[ln(D)] ln(1 0.022) 0.0004; µln(D) ln(µD) 0.5 Var[ln(D)] 1.0984 Var[ln(S)] ln(1 0.052) 0.0025; µln(S) ln(µS) 0.5 Var[ln(S)] 5.2996 Then, the mean and variance of the performance function, W(n, D, S), can be computed as µW 0.1520; Var(W) 0.005977 The reliability can be obtained as µW 0.1520 ps P(W 0) Φ Φ Φ(1.958) = 0.975 σW 0 .005966
7.3.3 Mean-Value First-Order Second-Moment (MFOSM) Method In the first–order second–moment methods the performance function W(X), defined on the
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RISK/RELIABILITY-BASED HYDRAULIC ENGINEERING DESIGN
Risk/Reliability–Based Hydraulic Engineering Design 7.25
basis of the load and resistance functions, g(XL) and h(XR), are expanded in a Taylor series at a selected reference point. The second and higher order terms in the series expansion are truncated, resulting in an approximation that requires the first two statistical moments of the random variables. This simplification greatly enhances the practicality of the first order methods because in many problems it is rather difficult, if not impossible, to find the PDF of the variables while it is relatively simple to estimate the first two statistical moments. Detailed procedures of the first–order second–moment method are given in Sec. 7.2.3 which describes the FOVE method for uncertainty analysis. The MFOSM method for the reliability analysis first applies the FOVE method to estimate the statistical moments of the performance function W(X). This is done by applying the expectation and variance operators to the first–order Taylor series approximation of the performance function W(X), expanded at the mean values of the random variables. Once the mean and standard deviation of W(X) are estimated, the reliability is computed according to Eqs. (7.38) or (7.39) with the reliability index βMFOSM computed as µW (7.45) βMFOSM st CX ()s where µ and C(X) are the vector of means and covariance matrix of the random variables X, respectively; s ∇xW(µ) is the column vector of sensitivity coefficients with each element representing ∂W/∂Xi evaluated at X µ. Example 7.6 Referring to Example 7.5, compute the reliability that the sewer capacity could convey an inflow peak discharge of 35 ft3/s. Assume that stochastic model parameters n, D, and S are uncorrelated. Solution: The performance function for the problem is W Q 35. From Example 7.2, the mean and standard deviation of the sewer capacity are 40.96 ft3/s and 3.17 ft3/s, respectively. Therefore, the mean and standard deviation of the performance function W are, respectively, µW µQ 35 40.96 35 5.96 ft3/s; σW σQ 3.17 ft3/s The MFOSM reliability index is βMFOSM 5.96/3.17 1.880. Assuming a normal distribution for Q, the reliability that the sewer capacity can accommodate a discharge of 35 ft3/s is ps = P[Q > 35] Φ(βMFOSM) Φ(1.880) 0.9699 The corresponding failure probability is pf Φ (1.880) 0.0301. Ang (1973), Cheng et al. (1986), and Yeng and Ang (1971), indicated that,if the calculated reliability or failure probability is in the extreme tail of a distribution, the shape of the tails of a distribution becomes very critical. In such cases, accurate assessment of the distribution of W(X) should be used to evaluate the reliability or failure probability.
7.3.4 Advanced First-Order Second-Moment (AFOSM) Method The main thrust of the AFOSM method is to mitigate the deficiencies associated with the MFOSM method, while keeping the simplicity of the first–order approximation. The difference between the AFOSM and MFOSM methods is that the expansion point in the first–order Taylor series expansion in the AFOSM method is located on the failure surface
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RISK/RELIABILITY-BASED HYDRAULIC ENGINEERING DESIGN
7.26
Chapter Seven
defined by the limit state equation, W(x) 0. In cases where several random variables are involved in a performance function, the number of possible combinations of such variables satisfying W(x) 0 is infinite. From the design viewpoint, one is more concerned with the combination of random variables that would yield the lowest reliability or highest failure probability. The point on the failure surface associated with the lowest reliability is the one having the shortest distance in the standardized space to the point where the means of the random variables are located. This point is called the design point (Hasofer and Lind, 1974) or the most probable failure point (Shinozuka, 1983). In the uncorrelated standardized parameter space, the design point in x'–space is the one that has the shortest distance from the failure surface W(x') 0 to the origin x' 0. Such a point can be found by solving Minimize
x' = (x't x')1/2
(7.46a)
subject to
W(x') 0.
(7.46b)
in which x' represents the length of the vector x'. Utilizing the Lagrangian multiplier method, the design point can be determined as ∇x'W(x*') x*' x*' α* (7.47) x*' ∇x'W(x*') in which α* x'W(x'*)/ x'W(x'*) is a unit vector eminating from the design point x'* and pointing toward the origin (Fig. 7.4). The elements of α* are called the directional derivatives representing the value of the cosine angle between the gradient vector ∇x'W(x'*) and axes of the standardized variables. Geometrically, Eq. (7.47) shows that the vector x'* is perpendicular to the tangent hyperplane passing through the design point. Recall that xi µi σix'i, for i 1, 2, …, N. By the chain–rule in calculus, the shortest distance, in terms of the original variables x, can be expressed as
FIGURE 7.4 Characteristics of design point in standardized space
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RISK/RELIABILITY-BASED HYDRAULIC ENGINEERING DESIGN
Risk/Reliability–Based Hydraulic Engineering Design 7.27
N
∂W (x) (µi ∂xi x* i1
xi*)
x '
*
N
∂W(x) 2 σ2i ∂xi x j1 *
(7.48)
in which x* (x1*, x2*, …, xN*)t is the point in the original parameter x–space, which can be easily determined from the design point x'* in x'–space as x* µ D1/2 x'*. It is shown in the next subsection that the shortest distance from the origin to the design point, x'* , in fact, is the reliability index based on the first–order Taylor series expansion of the performance function W(X) with the expansion point at x*. 7.3.4.1 First-order approximation of performance function at design point. Referring to Eq. (7.12), the first–order approximation of the performance function, W(X), taking the expansion point xo x*, is
N
W(X)
si* (Xi xi*) s*t (X x*)
(7.49)
i1
in which s* (s1*, s2*, …, sN*)t is a vector of sensitivity coefficients of the performance function W(X) evaluated at the expansion point x*, that is, ∂W(X) si* ∂Xi
X x*
W(x*) is not on the right–hand–side of Eq. (7.49) because W(x*) 0. Hence, at the expansion point x*, the expected value and the variance of the performance function W(X) can be approximated as µW s*t (µ x*)
(7.50)
σW2 s*t C(X) s*
(7.51)
in which µ and C(X) are the mean vector and covariance matrix of the random variables, respectively. If the random variables are uncorrelated, Eq. (7.51) reduces to
N
σW2
si2* σ2i
(7.52)
i1
in which σi is the standard deviation of the ith random variable. When the random variables are uncorrelated, the standard deviation of the performance function W(X) can alternatively be expressed in terms of the directional derivatives as
N
σW
αi* si* σi
(7.53)
i1
where αi* is the directional derivative for the ith random variable at the expansion point x*
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RISK/RELIABILITY-BASED HYDRAULIC ENGINEERING DESIGN
7.28
Chapter Seven
αi*
s σ
i* i , i 1, 2, …, N
sσ
(7.54a)
N
2 j
2 j*
j1
or in matrix form D1/2 ∇xW(x*) α* D1/2 ∇x W (x*)
(7.54b)
which is identical to the definition of α* in Eq. (7.47). With the mean and standard deviation of the performance function W(X) computed at x*, the AFOSM reliability index βAFOSM, given in Eq. (7.37), can be determined as
N
si* (µi xi*) µW i 1 βAFOSM N σW αi* si* σi
(7.55)
i1
Equation (7.55) is identical to Eq. (7.48), indicating that the AFOSM reliability index BAFOSM is identical to the shortest distance from the origin to the design point in the standardized parameter space. This reliability index βAFOSM is also called the Hasofer–Lind reliability index. Once the value of βAFOSM is computed, the reliability can be estimated by Eq. (7.39) as ps Φ(βAFOSM). Since βAFOSM x'*, the sensitivity of βAFOSM with respect to the uncorrelated, standardized random variables is x'* α* (7.56) ∇x' βAFOSM ∇x' x'* x'* Equation (7.56) shows that αi* is the rate of change in βAFOSM due to a 1 standard deviation change in random variable Xi at X x*. Therefore, the relation between ∇x' β and ∇xβ can be expressed as ∇x' βAFOSM D1/2 ∇x' βAFOSM D1/2 α*
(7.57)
It also can be shown that the sensitivity of reliability or failure probability with respect to each random variable can be computed as ∂ps ∂Xi' x'
*
∂p αi*φ(βAFOSM) αi* φ(βAFOSM); s ; i 1, 2, …, N σ ∂Xi x' i
(7.58a)
*
in which φ(·) is the standard normal PDF or in matrix form as ∇x*' ps φ(βAFOSM) α*
(7.58b)
∇x*ps φ(βAFOSM) ∇x βAFOSM φ(βAFOSM) D1\2 α∗ *
These sensitivity coefficients reveal the relative importance of each random variable on reliability or failure probability. 7.3.4.2 Algorithms of AFOSM for independent normal parameters. In the case that X are independent normal random variables, standardization of X reduces them to independent standard normal random variables Z' with mean 0 and covariance matrix I with I
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RISK/RELIABILITY-BASED HYDRAULIC ENGINEERING DESIGN
Risk/Reliability–Based Hydraulic Engineering Design 7.29
being an N N identity matrix. Hasofer and Lind (1974) proposed the following recursive equation for determining the design point z'* W(z' (r)) t z'(r 1) [α(r) z'(r)] α(r) α , r 1, 2, … ∇z' W (z'(r)) (r)
(7.59)
in which subscripts (r) and (r 1) represent the iteration numbers; α denotes the unit gradient vector of the failure surface pointing to the failure region. It would be more covenient to rewrite the above recursive equation in the original x–space as [x(r) µ]t s(r) W(x(r)) x(r 1) µ D s(r) , r 1, 2 3, … st(r) D s(r)
(7.60)
Based on Eq. (7.60), the Hasofer–Lind algorithm for the AFOSM reliability analysis for problems involving uncorrelated, normal random variables can be outlined as follows. Step 1. Select an initial trial solution x(r). Step 2. Compute W(x(r)) and the corresponding sensitivity coefficient vector s(r). Step 3. Revise solution point x(r 1) according to Eq. (7.60). Step 4. Check if x(r) and x(r 1) are sufficiently close. If yes, compute the reliability index βAFOSM according to Eq. (7.55) and the corresponding reliability ps Φ(βAFOSM), then, go to Step 5; otherwise, update the solution point by letting x(r) x(r 1) and return to Step 2. Step 5. Compute the sensitivity of reliability index and reliability with respect to changes in random variables according to Eqs. (7.56), (7.57), and (7.58). Because of the nature of nonlinear optimization, the above algorithm does not necessarily converge to the true design point associated with the minimum reliability index. Therefore, Madsen et al. (1986) suggest that different initial trial points are used and the smallest reliability index is chosen to compute the reliability. Sometimes, it is possible that a system might have several design points. Such a condition could be due to the use of multiple performance functions or the performance function is highly irregular in shape. In the case that there are J such design points, the reliability of the system requires that, at all design points, the system performs satisfactorily. Assuming independence of the occurrence of individual design point, the reliability of the system is the survival of the system at all design points which can be calculated as ps [Φ (βAFOSM)]J
(7.61)
Example 7.7 (uncorrelated, normal). Refer to the data as shown below for the storm sewer reliability analysis in previous examples. Parameter
Mean
Coefficient of Variation
n (ft1/6) D (ft) S (ft/ft)
0.015 3.0 0.005
0.05 0.02 0.05
Assume that all three random variables are independent, normal random variables. Compute the reliability that the sewer can convey an inflow discharge of 35 ft3/s by the Hasofer–Lind algorithm. Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
RISK/RELIABILITY-BASED HYDRAULIC ENGINEERING DESIGN
7.30
Chapter Seven
Solution: The initial solution is taken to be the means of the three random variables, namely, x(1) µ (µn, µD, µS)t (0.015, 3.0, 0.005)t. The covariance matrix for the three random variables are
D=
σ2n
0
0
σ
0
0
0
σ2S
2 D
0
0.000752
0
0
0
0.06
0
0
0
0.000252
2
Because the origin (n, D, S) (0, 0, 0) is located in the failure region compared to the target of 35 ft3/s as opposed to the safe region used in the mathematical derivations, a negative sign must be applied to the performance function QC – QL as W(n, D, S) (QC QL) 0.463 n1 D8/3 S1/2 35 At x(1), the value of the performance function W(n, D, S) 6.010 which is not equal to zero. This implies that the solution point x(1) does not lie on the limit state surface. By Eq. (7.64) the new solution, x(2), can be obtained as x(2) (0.01592, 2.921, 0.004847). Then, one checks the difference between the two consecutive solution points as x(1) x(2) [(0.01592 0.015)2 (2.9213.0)2 (0.004847 0.005)2 ]0.5 0.07857 which is considered large and, therefore, the iteration continues. The following table lists the solution point, x(r), its corresponding sensitivity vector, s(r), and the vector of directional derivatives, α(r), in each iteration. The iteration stops when the difference between the two consecutive solutions is less than 0.001 and the value of the performance function is less than 0.001. Iteration Variable
x(r)
s(r)
α(r)
x(r1)
r1
n 0.1500E 01 0.2734E 04 0.6468E 00 0.1592E 0 01 D 0.3000E 01 0.3650E 02 0.6907E 00 0.2921E 0 01 S 0.5000E 02 0.4101E 04 0.3234E 00 0.4847E 0 02 diff .7857E 01 W .6010E 01 β 0.0000E 00
r2
n 0.1592E 01 0.2226E 04 0.6138E 00 0.1595E 0 01 D 0.2921E 01 0.3239E 02 0.7144E 00 0.2912E 0 01 S 0.4847E 02 0.3656E 04 0.3360E 00 0.4827E 0 02 diff 0.9584E 02 W 0.4421E 00 β 0.1896E 01
r3
n 0.1595E 01 0.2195E 04 0.6118E 00 0.1594E 0 01 D 0.2912E01 0.3209E 02 0.7157E 00 0.2912E 01 S 0.4827E 02 0.3625E 04 0.3369E 00 0.4827E 02 diff 0.1919E 03 W 0.2151E 02 β 0.2056E 01
r4
n 0.1594E 01 0.2195E 04 0.6119E 00 0.1594E 01 D 0.2912E 01 0.3210E 02 0.7157E 00 0.2912E 01 S 0.4827E 02 0.3626E 04 0.3369E 00 0.4827E 02 diff 0.3721E 05 W 0.2544E 06 β 0.2057E 01
After four iterations, the solution converges to the design point x* (n*, D*, S*) (0.01594, 2.912, 0.004827). At the design point x*, the mean and standard deviation of the performance function W can be estimated, by Eqs. (7.50) and (7.53), respectively, as Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
RISK/RELIABILITY-BASED HYDRAULIC ENGINEERING DESIGN
Risk/Reliability–Based Hydraulic Engineering Design 7.31
µW* 5.536 and σW* 2.691 The reliability index can be computed as β* µW* / σW* 2.057 and the corresponding reliability and failure probability can be computed, respectively, as ps Φ(β*) 0.9802; pf 1 ps 0.01983 Finally, at the design point x*, the sensitivity of the reliability index and it reliability with respect to each of the three random variables can be computed by Eqs. (7.57) and (7.58). The results are shown in columns (4)–(7) of the table below. Variable
x
α*
∂β/∂xi'
∂ps/∂xi'
∂β/∂xi
∂ps/∂xi
x∂β/∂βxi xi∂ps/ps∂xi
(1)
(2)
(3)
(4)
(5)
(6)
(7)
n
0.01594
D
2.912
0.7157
0.7157
0.03441
11.9
0.57
16.890
1.703
S
0.00483 0.3369
0.3369
0.01619 1347.0
64.78
3.161
0.319
(8)
0.6119 0.6119 0.02942 815.8 39.22 6.323
(9) 0.638
From the above table, the quantities ∂β/∂x' and ∂ps/∂x' show the sensitivity of the reliability index and the reliability for a one–standard–deviation change in the random variables whereas ∂β/∂x and ∂ps/∂x correspond to a one unit change of random variables in the original space. The sensitivity of β and ps associated with Manning’s roughness is negative whereas those for pipe size and slope are positive. This indicates that an increase in Manning’s roughness would result in a decrease in β and ps, whereas an increase in slope and/or pipe size would increase β and ps.The indication is physically plausible because an increase in Manning’s roughness would decrease the flow carrying capacity of the sewer whereas, on the other hand, an increase in pipe diameter and/or pipe slope would increase the flow carrying capacity of the sewer. Furthermore, one can judge the relative importance of each random variable based on the absolute values of sensitivity coefficients. It is generally difficult to draw meaningful conclusion based on the relative magnitude of ∂β/∂x and βps/∂x because units of different random variables are not the same. Therefore, sensitivity measures not affected by the dimension of the random variables such as ∂β/∂x' and ∂ps/∂x' are generally more useful. With regard to change in β or ps per one standard deviation change in each variable X, for example, pipe diameter is significantly more important than the pipe slope. An alternative sensitivity measure, called the relative sensitivity, is defined as ∂y x ∂y/y (7.62) si% i , i 1, 2, …, N ∂xi/xi ∂xi y in which si% is a dimensionless quantity measuring the percentage change in the dependent variable y due to 1 percent change in the variable xi. The last two columns of the table given above show the percentage change in β and ps due to 1 percent change in Manning’s roughness, pipe diameter, and pipe slope. As can be observed, the pipe diameter is the most important random variable in Manning’s formula affecting the reliability of the flow carrying capacity of the sewer. 7.3.4.3 Treatment of correlated normal random variables. When some of the random variables involved in the performance function are correlated, transformation of correlated variables to uncorrelated ones is made. This can be achieved through the orthogonal transformation such as the spectral decomposition described above.
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RISK/RELIABILITY-BASED HYDRAULIC ENGINEERING DESIGN
7.32
Chapter Seven
Consider that the original random variables are multivariate normal random variables. The original random variables X can be transformed to uncorrelated, standardized normal variables Z' as Z' Λx1/2 Vxt D1/2 (X µx)
(7.63)
in which Λx and Vx are, respectively, the eigenvalue matrix and eigenvector matrix corresponding to the correlation matrix R(X). In the transformed domain as defined by Z', the directional derivatives of the performance function in z'–space, αz', can be computed, according to Eq. (7.47), as ∇z' W(z') αz' ∇z' W(z')
(7.64)
in which the vector of sensitivity coefficients in Z'–space, sz' ∇z'W(z'), can be obtained from ∇xW(x) through the chain–rule of calculus as sz' ∇z' W(z') D1/2 Vx Λx1/2 ∇x W(x) D1/2 Vx Λx1/2 sx
(7.65)
in which sx is the vector of sensitivity coefficients of the performance function with respect to the original random variables X. After the design point is found, one also is interested in the sensitivity of the reliability index and failure probability with respect to changes in the involved random variables. In the uncorrelated, standardized normal Z'–space, the sensitivity of β and ps with respect to Z' can be computed by Eqs. (7.57) and (7.58) with X' replaced by Z'. The sensitivity of β with respect to X in the original parameter space then can be obtained as ∂Z ' ∇xβ i ∇z'β Λ1/2 Vxt Dx1/2 ∇z'β Λ1/2 Vxt Dx1/2 αz' (7.66) ∂Xj from which the sensitivity for ps can be computed by Eq. (7.58b). The procedure for Hasofer–Lind's approach to handle the case of correlated normal variables is given below.
Step 1. Select an initial trial solution x(r). Step 2. Compute W(x(r)) and the corresponding sensitivity coefficient vector sx,(r). Step 3. Revise solution point x(r 1) according to
[x(r) µx]t s(r) W(x(r)) . x(r 1) µx C(X) sx,(r) st x ,(r) C(X) sx, (r)
(7.67)
Step 4: Check if x(r) and x(r 1) are sufficiently close. If yes, compute the reliability index β(r) according to βAFOSM [(x* µx)t C(X)1 (x* µx)]1/2
(7.68)
and the corresponding reliability ps Φ(βAFOSM), then, go to Step 5; otherwise, update the solution point by letting x(r) x(r 1) and return to Step 2. Step 5. Compute the sensitivity of reliability index and reliability with respect to changes in random variables at the design point x*.
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RISK/RELIABILITY-BASED HYDRAULIC ENGINEERING DESIGN
Risk/Reliability–Based Hydraulic Engineering Design 7.33
Example 7.8 (Correlated, normal). Refer to the data in Example 7.7 for the storm sewer reliability analysis problem. Assume that Manning’s roughness (n) and pipe diameter (D) are dependent normal random variables having a correlation coefficient of 0.75. Furthermore, the pipe slope (S) also is a normal random variable but is independent of Manning’s roughness and pipe size. Compute the reliability that the sewer can convey an inflow discharge of 35 ft3/s by the Hasofer–Lind algorithm. Solution. The initial solution is taken as the means of the three random variables, namely, x(1) µ (µn, µD, µS)t ( 0.015, 3.0, 0.005)t. Since the random variables are correlated normal random variables with a correlation matrix as
R (X) =
1.0 ρn,D ρn,S ρn,D 1.0 ρD,s ρn,S ρD,s 1.0
1.00
0.75
0.00
0.75
1.00
0.00
0.00
0.00
1.00
by the spectral decomposition, the eigenvalue matrix associated with the correlation matrix R (X) is Λx diag (1.75, 0.25, 1.00) and the corresponding eigenvector matrix Vx is
VX =
0.7071
0.7071 0.0000
0.7071 0.7071 0.0000 0.0000
0.0000 1.0000
At x(1) ( 0.015, 3.0, 0.005)t, the sensitivity vector of the performance function W(n, D, S) (QC QL) 0.463 n1 D2.67 S1/2 35 is
t
∂W ∂W ∂W sx,(1) , , (2734, 36.50, 4101 )t ∂n ∂D ∂S and the value of the performance function W(n, D, S) 6.010, which is not equal to zero. This indicates that the solution point x(1) does not lie on the limit state surface. Applying Eq. (7.67), the new solution, x(2), can be obtained as x(1) (0.01569, 2.900, 0.004885). Then, checking the difference between the two consecutive solutions as x(1) x(2) [(0.01569 0.015)2 (2.9 3.0)2 (0.004885 0.005)2]0.5 0.1002. This is considered to be large and therefore the iteration continues. The following table lists the solution point, x(r), its corresponding sensitivity vector, sx,(r), and the vector of directional derivatives, αz',(r), in each iteration. The iteration stops when the Euclidean distance between the two consecutive solution points is less than 0.001 and the value of the performance function is less than 0.001. Iter
Var
r1
n D
x(r)
s(r)
α(r)
x(r 1)
0.1500E 01 0.2734E 04 0.1237E 01 0.1569E 01 0.3000E 01 0.3650E 02 0.9999E 00 0.2900E 01
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RISK/RELIABILITY-BASED HYDRAULIC ENGINEERING DESIGN
7.34
Chapter Seven
S 0.5000E 02 0.4101E 04 0.6647E 02 0.4885E 02 diff 7857E 01 W 0.6010E01 β 0.000E00 r2
n
0.1569E 01
0.2256E 04
.1236E 01 0.1572E 01
D
0.2900E 01 0.3259E 02 .9999E 00 0.2891E 01
S 0.4885E 02 0.3623E 04 .7113E 02 0.4872E 02 diff 0.8906E 02 W 0.3972E 00 β 0.1804E 01 r3
n
0.1572E 01
0.2227E 04
.1236E 01 0.1571E 01
D
0.2891E 01 0.3233E 02 .9999E 00 0.2891E 01
S 0.4872E 02 0.3592E 04 .7144E 02 0.4872E 02 diff 0.1120E03 W .1769E 02 β 0.1987E 01 r4
n
0.1571E 01
0.2227E 04
0.1236E 01 0.1571E 01
D
0.2891E 01 0.3233E 02 0.9999E 00 0.2891E 01
S 0.4872E 02 0.3592E 04 0.7144E 02 0.4872E 02 diff 0.9467E 06 W 0.6516E 07 β 0 .1991E 01 After four iterations, the solution converges to the design point x* (n*, D*, S*)t (0.01571, 2.891, 0.004872)t. At the design point x*, the mean and standard deviation of the performance function W can be estimated, by Eqs. (7.50) and (7.53), respectively, as µW* 5.580 and σW* 3.129 The reliability index then can be computed as β* µW*/σW* 1.991 and the corresponding reliability and failure probability can be computed, respectively, as ps Φ(β*) 0.9767; pf 1 ps 0.02326. Finally, at the design point x*, the sensitivity of the reliability index and reliability with respect to each of the three random variables can be computed by Eqs. (7.57), (7.58), (7.66), and (7.62). The results are shown in the following table: Variable xi (1) (2)
α*i (3)
∂β/∂zi (4)
∂ps/∂zi (5)
∂β/∂xi (6)
∂ps/∂xi xi∂β/β∂xi ∂ps xi/ps∂xi (7) (8) (9)
n
0.01571 0.0124 0.0124 0.00068 17.72 0.9746 0.1399 0.01568
D
2.891
0.9999
0.9999
0.05500
0.26
0.0142
0.3734
0.04186
S
0.00487 0.0071
0.0071
0.00039
28.57
1.5720
0.0699
0.00784
The sensitivity analysis indicates similar information about the relative importance of the random variables as in Example 7.7. 7.3.4.4 Treatment of non-normal random variables. When non–normal random variables are involved, it is advisable to transform them into equivalent normal variables. Rackwitz (1976) and Rackwitz and Fiessler (1978) proposed an approach which transforms a non–normal distribution into an equivalent normal distribution so that the value
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RISK/RELIABILITY-BASED HYDRAULIC ENGINEERING DESIGN
Risk/Reliability–Based Hydraulic Engineering Design 7.35
of the CDF of the transformed equivalent normal distribution is the same as that of the original non–normal distribution at the design point x*. Later, Ditlvesen (1981) provided the theoretical proof of the convergence property of the normal transformation in the reliability algorithms searching for the design point. Table 7.7 presents the normal equivalent for some commonly used non–normal distributions in the reliability analysis. By the Rackwitz (1976) approach, the normal transform at the design point x* satisfies the following condition: x µ i* i*N Φ(zi*), i 1, 2, …, N Fi (xi*) Φ σi*N
(7.69)
in which Fi (xi*) is the CDF of the random variable Xi having a value at xi*; µi*N, and σi*N are the mean and standard deviation of the normal equivalent for the ith random variable at Xi xi*; respectively and zi* Φ1[Fi (xi*)] is the standard normal quantile. Equation (7.69) indicates that the cumulative probability of both the original and normal transformed spaces must be preserved. From Eq. (7.69), the following equation is obtained: µi*N xi* zi* σi*N
(7.70)
Note that µi*N and σi*N are functions of the expansion point x*. To obtain the normal equivalent standard deviation, one can take the derivative of both sides of Eq. (7.69) with respect to xi resulting in x µ 1 Φ(zi*) i* i*N fi (xi*) Φ σi*N σi*N σi*N
in which fi(·) and φ(·) are the PDFs of the random variable Xi and the standard normal variable Zi , respectively. From the above equation, the normal equivalent standard deviation σi*N can be computed as Φ(zi*) σi*N fi(xi*)
(7.71)
Therefore, according to Eqs. (7.70) and (7.71), the mean and standard deviation of the normal equivalent of the random variable Xi at any expansion point x* can be calculated. It should be noted that the above normal transformation utilizes only the marginal distributions of the stochastic variables without considering their correlations. Therefore, it is, in theory, suitable for problems involving independent non–normal random variables. When random variables are nonnormal and correlated, additional considerations must be given in the normal transformation and they are described in the next subsection. To incorporate the normal transformation for non–normal, uncorrelated random variables, the iterative algorithms described previously for the AFOSM reliability method can be modified as follows. Step 1: Select an initial trial solution x(r). Step 2: Compute W(x(r)) and the corresponding sensitivity coefficient vector sx,(r). Step 3: Revise solution point x(r 1) according to Eq. (7.60) with the means and standard deviations of non–normal random variables replaced by their normal equivalents, that is, (x(r) µ⺞,(r))t sx,(r) W(x(r)) x(r 1) µ⺞,(r) D⺞,(r) sx,(r) sx,(r) D⺞,(r) sx,(r)
(7.72)
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j
(x* a)2 Φ1 , a w m (b a) (m a) (x b*)2 Φ1 1 , m x b (b a) (b m)
x a * Φ1 ba
2 x* a , a w m b a m a
2 x b* , m x b b a b m
1 f (x) , a x b ba
[β(x* )] j!
x ξ Φ1 exp exp β
j0
1
Φ1 1eβ(x* )
Φ 1 1 e β(x*x0)
ln(x*) µlnX σlnX
x ξ x ξ 1 β exp β exp β, ∞ x ∞
β[β(x* ξ)]α1eβ(x* ξ) , x* ξ Γ(α)
βeβ(x* xo), x xo
Source: From Yen et al., 1986. In all cases µN x* zN σN.
Uniform
Triangular
Type 1 external (max)
Gamma
Shifted exponential
ln(x )µ 2 1 * lnX , x > 0 exp 1 2 2π σ*σlnx σlnX
ZN Φ1 [Fx(X*)
Equivalent Standard Normal Variable
(b a) φ (ZN)
φ(ZN) fx(x *)
φ(ZN) fx(x *)
φ(ZN) fx(x *)
1 exp 2 β(x* xo) β2π
ZN2
s*σlnX
σN
7.36
Lognormal
PDF, fx (x*)
Normal Equivalent for Some Commonly UIsed NonNormal Disatributions.
Distribution of X
TABLE 7.7
RISK/RELIABILITY-BASED HYDRAULIC ENGINEERING DESIGN
Chapter Seven
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RISK/RELIABILITY-BASED HYDRAULIC ENGINEERING DESIGN
Risk/Reliability–Based Hydraulic Engineering Design 7.37
in which the subscript ⺞ stands for statistical properties in equivalent normal space. Step 4: Check if x(r) and x(r 1) are sufficiently close. If yes, compute the reliability index βAFOSM according to Eq. (7.55) and the corresponding reliability ps Φ(βAFOSM), then, go to Step 5; otherwise, update the solution point by letting x(r) x(r 1) and return to Step 2. Step 5: Compute the sensitivity of reliability index and reliability with respect to changes in random variables according to Eqs. (7.57) and (7.58) with D replaced by D⺞ at the design point x*. 7.3.4.5 AFOSM reliability analysis for non-normal, correlated random variables. For most practical engineering problems, parameters involved in load and resistance functions are correlated, non–normal random variables. Such distributional information has important implications on the results of reliability computation, especially on the tail of the probability distribution of the system performance function. The procedures of the Rackwitz normal transformation and orthogonal decomposition described previously can be integrated in the AFOSM reliability analysis. For correlated non normal variables, Der Kiureghian and Liu (1985) and Liu and Der Kiureghian (1986) developed a normal transformation that preserve the marginal probability contents and the correlation structure of multivariate non–normal random variables. More specifically, their approach considers that each non–normal random variable can be transformed to the corresponding standard normal variable as
Zi Φ1 FXi (Xi)
for i 1, 2, …, N
(7.73)
Furthermore, the correlation between a pair of non–normal random variables is preserved in the standard normal space by Nataf's bivariate distribution model as ∞ ∞
x µ x µ i i j ρij j Φij (zi, zj ρ∗ij) dzi dzj σ σj –∞ –∞ i
(7.74)
in which ρij and ρij *are, respectively, the correlation coefficient of random variables Xi and Xj in the original and normal transformed space; and xi F1Xi[Φ(zi)]. For a pair of non–normal random variables, Xi and Xj, with known marginal PDFs and correlation coefficient, ρij, Eq. (7.74) can be applied to solve for ρij*. To avoid the required computation for solving ρij* in Eq. (7.74), Der Kiureghian and Liu (1985) developed a set of semi–empirical formulas as ρ∗ij Tij ρij
(7.75)
in which Tij is a transformation factor depending on the marginal distributions and correlation of the two random variables under consideration. In the case that the pair of random variables considered are both normal, the transformation factor, Tij, has a value of 1. Given the marginal distributions and correlation for a pair of random variables, the formulas of Der Kiureghian and Liu (1985) compute the corresponding transformation factor, Tij, to obtain the equivalent correlation ρij* as if the two random variables were bivariate normal random variables. After all pairs of random variables are treated, the correlation matrix in the correlated normal space, R(Z), is obtained. Ten different marginal distributions commonly used in reliability computations were considered by Der Kiureghian and Liu (1985) and are tabulated in Table 7.8. For each com-
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RISK/RELIABILITY-BASED HYDRAULIC ENGINEERING DESIGN
7.38
Chapter Seven
bination of two distributions, there is a corresponding formula. Therefore, a total of 54 formulas for 10 different distributions were developed which are divided into five categories as shown in Fig. 7.5. The complete forms of these formulas are given in Table 7.9. Due to the semiempirical nature of the equations in Table 7.9, there is a slight possibility that the resulting ρij* may violate its valid range when the original ρij is close to 1 or 1. An algorithm for the AFOSM reliability analysis based on the transformation of Der Kiureghian and Liu for problems involving multivariate non–normal random variables can be found in Tung (1996). The normal transformation of Der Kiureghian and Liu (1985) preserves only the marginal distributions and the second–order correlation structure of the correlated random variables which are partial statistical features of the complete information repesentable by the joint distribution function. Regardless of its approximate nature, the normal transformation of Der Kiureghian and Liu, in most practical engineering problems, represents the best approach to treat the available statistical information about the correlated random variables. This is because, in reality, the choices of multivariate distribution functions for the correlated random variables are few as compared to the univariate distribution functions. Furthermore, the derivation of a reasonable joint probability distribution for a mixture of correlated non–normal random variables is difficult, if not impossible.
7.3.5 Monte Carlo Simulation Methods Monte Carlo simulation is the general purpose method to estimate the statistical properties of a random variable that is related to a number of random variables which may or
FIGURE 7.5 Categories of the Normal Transformation Factor, Tij (From Kiureghian and Liu, 1985)
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RISK/RELIABILITY-BASED HYDRAULIC ENGINEERING DESIGN
Risk/Reliability–Based Hydraulic Engineering Design 7.39 TABLE 7.8
Definitions of Distributions Used in Fig. 7.5 and Table 7.9
Distributions
PDF
Moments and Parameters Rlaios
Normal
1 e 2σ π
Uniform
1 ba
Shifted exponential
βe β (x x0)
(x µ)2 2 σ2
ab (b a)2 µ ; σ2 2 12 1 1 µ x0; σ2 2 β β
( ) α (x x0) e
Shifted Rayleigh
—
2 1 x x 2 α0
µ 1.253 α x0 σ 0.655136 α
2
(x xξ)
Type I largest
1 β e
(x ξ) e β
β µ ξ 0.5772 β πβ σ ;γ 1.1396 6
(Gumbel) (x xξ) β
(x xξ) β
e
Type I Smallest
1 β e
Lognormal
1 e 2π x σlnX
Gamma
βα (x ξ) α1 e β (x ξ)
Type II largest
α β α 1 e β x
µ ξ 0.5772 β πβ σ ; γ 1.1396 6
(
)
1 ln (x) µlnX 2 σln X
µlnX ln (µX) 1 σlnX 2 σ 2 σ2lnX ln 1 X µX
α α µ ξ; σ2 2 β β
α
() β x
2
µ β Γ 1 1 α 2 1 σ2 β2 Γ 1 Γ2 1 α α
Type III smallest
α
( )
α x ξ α 1 e β β
xξ β
1 µ ξ β Γ 1 α 2 1 2 2 σ β Γ 1 Γ2 1 α α
Source: From Der Kiureghian and Liu (1985).
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Semi–empirical Normal Transformation Formulas
0.0%
Max. error
0.0%
1.107
E
0.0%
1.014
R
0.0%
1.031
T1L
0.0%
1.031
T1S
Exact
Max. error
T3S
0.1%
0.1%
1.0300.238Ωj 0.364Ω j 2 1.0310.195Ωj 0.328Ω 2
T2L
Ωj is the coefficient of variation of the j th variable; distribution indices are: G Gamma; L=Lognormal; N Normal; T2L Type2 Largest value; T3S Type3 Smallest Value.
0.0%
1.0011.007 Ωj 0.118 j2
Ωj ln 1 ( Ω2 j)
Tij f (Ωj)
Source: Der Kiureghran and Liu (1985)
*
N
G
L
(b) Category 2 of the transformation factor Tij in Fig. 7.5*
* distribution indices are : E Shifted Exponential; N Normal; R Shifted Rayleigh; T1L Type1 Largest Value; T1S Type 1 Smallest Value. U Uniform.
1.023
N Tij constant
U
7.40
(a) Category 1 of the transformation factor Tij in Fig. 7.5*
TABLE 7.9
RISK/RELIABILITY-BASED HYDRAULIC ENGINEERING DESIGN
Chapter Seven
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(continued)
*
ρij is the correlation coefficient between the ith variable and the jth variable; distribution indices are: E shifted exponential; R shifted Rayleigh; T1L Type 1 Largest Value; T1S Type 1 Smallest Value; U Uniform.
Max. error
T1S Tij f (ρij)
0.0%
0.0%
1.064 0.069ρij2 0.005ρij2
0.0%
1.064 0.069ρij2 0.005 ρij2
1.064 0.069ρij 0.005ρij
Max. error
0.0% 2
1.046 0.045ρij2 0.006 ρij2
1.142 0.154 ρij 0.031 ρij2 0.2%
0.0%
1.055 0.015ρij2
T1S
0.0%
T1L Tij f (ρij)
0.0%
Max. error
1.046 0.045ρij2 0.006ρij2
1.142 0.154 ρij 0.031ρij2 0.2%
0.0%
1.055 0.015ρij2
T1L
2
1.028 0.029ρij2
1.123 0.100 ρij 0.021ρij2 0.1%
0.0%
Tij f (ρij)
1.229 0.367 ρij 0.153ρij2 1.5%
0.0%
1.038 0.008ρij2
R
0.0%
1.133 0.029ρij2
R
Tij f (ρij) Max. error
Max. error
Tij f (ρij) 1.047 0.047ρij2
E
E
U
U
(c) Category 3 of the Transformation Factor Tij in Fig. 7.5
TABLE 7.9
RISK/RELIABILITY-BASED HYDRAULIC ENGINEERING DESIGN
Risk/Reliability–Based Hydraulic Engineering Design 7.41
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(continued)
||
0.3%
Max. error
0.3%
1.031 0.001ρij 0.007Ωj 0.003ρij2 0.131Ωj2 0.132ijΩj
0.3%
0.3%
1.029 0.001ρij 0.014Ωj 0.004ρij2 0.233Ωj2 0.197ijΩj
Max. error
0.9% 1.031 0.001ρij0.007Ωj 0.003ρij2 0.131Ωj2 0.132ijΩ
0.4%
1.029 0.001ρij 0.014 j 0.004 ij2 0.233Ωj2 0.197ijΩj
Max. error
Tij f (ρij,Ωj)
Tij f (ρij,Ωj)
0.9% 1.014 0.001ρij 0.007Ωj 0.002ρij2 2 j 0.126Ωj 0.090ijΩj
1.6%
1.011 0.001ρij 0.014Ωj 0.004ρij2 0.231Ωj2 0.130ijΩj
Max. error
Tij f (ρij,Ωj)
0.1% 1.104 0.003ρij 0.008Ωj 0.014ρij2 0.173Ωj2 0.296ρijΩj
0.7%
1.098 0.003ρij 0.019Ωj 0.025ρij2 0.303Ωj2 0.437ρijΩj
Max. error
Tij f (ρij,Ωj)
Tij f (ρij,Ωj) 1.019 0.014Ωj 0.010ρij2 0.249Ωj2 1.023 0.007Ωj 0.002ρij2 0.127Ωj2
G
1.0%
1.056 0.060ρij 0.263Ωj 0.020ρij2 0.383Ωj2 0.332ijΩj
1.0%
1.056 0.060ρij 0.263Ωj 0.020ρij2 0.383Ωj2 0.332ijΩj
1.2%
1.036 0.038ρij 0.266Ωj 0.028ρij2 0.383Ωj2 0.229ijΩj
0.9%
1.1090.152ρij 0.361 j 0.130ρij2 0.455Ωj2 0.728ρijΩj
2.1%
1.033 0.305Ωj 0.074ρij2 0.405Ωj2
R
0.2%
1.064 0.065ρij 0.210Ωj 0.003ρij2 0.356Ωj2 0.211ijΩj
0.2%
1.064 0.065ρij 0.210Ωj 0.003ρij2 0.356Ωj2 0.211ijΩj
0.2%
1.047 0.042ρij 0.212Ωj 0.353Ωj2 0.136ijΩj
0.4%
1.147 0.145ρij 0.271Ωj 0.010ρij2 0.459Ωj2 0.467ρijΩj
0.5%
1.061 0.237Ωj 0.005ρij2 0.379Ωj2
T2L T3S
ρij is the correlation coefficient between the ith variable and the jth variable; Ωj is the coefficient of variation of the jth variable; distribution indices are: E shifted exponential; G Gamma; L Lognormal; T1L Type, 1 largest value; T1S Type, 1 smallest value; T2L Type, 2 largest value; T3S Type, 3 Smallest Value; U Uniform.
T1S
T1L
R
E
U
L
7.42
(d) Category 4 of the Transformation Factor Ti in Fig. 7.5||
TABLE 7.9
RISK/RELIABILITY-BASED HYDRAULIC ENGINEERING DESIGN
Chapter Seven
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(Continued)
§
Max. error
Tij f(ρij,Ωi,Ωj)
Max. error
Tij f(ρij,Ωi,Ωj)
Max. error
Tij f(ρij,Ωi,Ωj)
Max. error
Tij f(ρij,Ωi,Ωj)
Exact
0.005 ρΩi 0.009 Ωi Ωj0.174 ρΩj
0.104 ρij Ωi 0.126 ρi Ωj0.277 ρij Ωj
3 ij
3 j
2 i
2 j
4.3%
0.257 ij2(Ωi Ωj) 0.141 Ωi Ωj(Ωi Ωj)
3 i
0.218(Ω Ω )0.371 ρij(Ω Ω )
0.570 ij(Ωi Ωj)0.203 Ωi Ωj0.020 ρ
0.055 ρij2 0.662(Ωi2 Ωj2)
1.086 0.054 ρij 0.104(Ωi Ωj)
4.2%
2 j
0.313ρijΩi 0.075ΩiΩj0.182ρijΩj
4.0%
2 i
0.077 ρij(Ωi Ωj) 0.014 Ωi Ωj
2 ij
0.012ρ 0.174Ω 0.379Ω
2 j
0.001 ρ 0.125(Ω Ω ) 2 i
1.029 0.056 ρij0.030Ωi 0.225Ωj
2 ij
2.62%
0.007 ρ(Ωi Ωj)0.007 Ωi Ωj
0.001 ρij20.337(Ωi2 Ωj2)
1.0630.004 ρij0.200(Ωi Ωj)
3.8%
0.005 ρΩi0.034 i j0.481 ρΩj
0.013 ρij2 0.372 Ωi2 0.435 Ωj2
1.065 0.146 ρij 0.241 Ωi0.259 Ωj
4.0%
0.006 ρi 0.003 Ωi Ωj0.111 ρΩj
0.121 Ωi2 0.339 Ωj2
1.032 0.034 ρij0.007 Ωi0.202 Ωj
2.4%
0.002 ρij2 0.220 Ωi2 0.350 Ωj2
4.3%
1.031 0.052ρij 0.011 Ωi0.210 Ωj
0.018 ρij2 0.288 Ωi2 0.379 Ωj2
T3S
1.026 0.082 ρij0.019 ρi0.222 ρj
T2L
1.0020.022 ρij0.012( i j)
4.0%
1.001 0.033ρij 0.004Ωi0.016Ωj ln (1 ρijΩiΩj) 0.002ρij2 0.223Ωi2 0.130Ωj2 ρij (1 Ω2 l1 ( Ω2 i )n j) 0.104ρijΩi 0.029ΩiΩj0.119ρijΩi
G
ρij is the correlation coefficient between the ith variable and the jth variable; Ωi is the coefficient of variation of the ith variable; Ωj is the coefficient of variation of the jth variable; distribution indices are: G Gamma; L Lognormal; T2L Type, 2 Largest Value;T3S Type, 3 Smallest Value. Source: Der Kiureghran and Liu (1985)
T3S
T2L
G
L
L
(e) Category 5 of the Transformation Factor Ti in Fig. 7.5§
TABLE 7.9
RISK/RELIABILITY-BASED HYDRAULIC ENGINEERING DESIGN
Risk/Reliability–Based Hydraulic Engineering Design 7.43
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RISK/RELIABILITY-BASED HYDRAULIC ENGINEERING DESIGN
7.44
Chapter Seven
may not be correlated. In Monte Carlo simulation, the values of stochastic parameters are generated according to their distributional properties. The generated parameter values are used to compute the value of performance function. After a large number of simulated realizations of performance function are generated, the reliability of the structure can be estimated by computing the ratio of the number of realizations with W ≥ 0 to the total number of simulated realizations. The major disadvantage of Monte Carlo simulation is its computational intensiveness. The number of sample realizations required in simulation to accurately estimate the risk depends on the magnitude of the unknown risk itself. In general, as the failure probability value gets smaller, the required number of simulated realizations increases. Therefore, some variations of Monte Carlo simulation to accurately estimate the failure probability while reducing excessive computation time have been developed. They include stratified sampling and Latin hypercubic sampling (McKay et al., 1979), importance sampling (Harbitz, 1983; Schueller and Stix, 1986), and the reduced space approach (Karamchandani, 1987).
7.4 RISK-BASED DESIGN OF HYDRAULIC STRUCTURES Reliability analysis methods can be applied to design hydraulic structures with or without considering risk costs. Risk costs are the cost items incurred due to the unexpected failure of the structures and they can be broadly classified into tangible and intangible costs. Tangible costs are those measurable in terms of monetary units which include damage to properties and structures, loss in business, cost of repair, and so forth. On the other hand, intangible costs are not measurable by monetary units such as psychological trauma, loss of lives, social unrest, and others. Risk-based design of hydraulic structures integrates the procedures of uncertainty and reliability analyses in the design practice. The risk-based design procedure considers trade offs among various factors such as failure probability, economics, and other performance measures in hydraulic structure design. Plate and Duckstein (1987, 1988) list a number of performance measures, called the figures of merit in the risk-based design of hydraulic structures and water resource systems, which are further discussed by Plate (1992). When the risk-based design is embedded into an optimization framework, the combined procedure is called the optimal risk-based design. 7.4.1 Basic Concept The basic concept of risk-based design is shown schematically in Fig. 7.6. The risk function accounting for the uncertainties of various factors can be obtained using the reliability computation procedures described in previous sections. Alternatively, the risk function can account for the potential undesirable consequences associated with the failure of hydraulic structures. For the sake of simplicity, only the tangible damage cost is considered here. Because risk costs associated with the failure of a hydraulic structure cannot be precisely predicted from year to year. A practical way is to quantify these costs using an expected value on an annual basis. The total annual expected cost (TAEC) is the sum of the annual installation cost, operation and maintenance costs, and annual expected damage cost which can be expressed as TAEC(Θ) FC(Θ) CRF E(D Θ)
(7.76)
where FC is the first or total installation cost which is a function of decision vector that
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Risk/Reliability–Based Hydraulic Engineering Design 7.45
FIGURE 7.6 Schematic sketch of risk-based design
may include the size and configuration of the hydraulic structure; E(DΘ) is the annual expected damage cost associated with the structural failure; and CRF is the capital recovery factor, which brings the present worth of the installation costs to an annual basis. The CRF can be computed as (See Section 1.6) (1 i)n1 (7.77) CRF i(1 i)n with n and i being the expected service life of the structure and the interest rate, respectively. Frequently in practice, the optimal risk-based design determines the optimal structural size, configuration, and operation such that the annual total expected cost is minimum. Referring to Fig. 7.6, as the structural size increases, the annual installation cost increases whereas the annual expected damage cost associated with the failure decreases. The optimal risk-based design procedure attempts to determine the minimum point on the total annual expected cost curve. Mathematically, the optimal risk-based design problem can be stated as: Minimize
TAEC(Θ) FC(Θ) CRF E(DΘ)
(7.78a)
subject to
gi(Θ) 0, i 1, 2, …, m
(7.78b)
where gi(Θ) 0, i 1, 2, …, m are constraints representing the design specifications that must be satisfied. In general, the solution to Eqs. (7.78a–b) could be acquired through the use of appropriate optimization algorithms. The selection or development of the solution algorithm is largely problem specific, depending on the characteristics of the problem to be optimized.
7.4.2 Historical Development of Hydraulic Design Methods
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RISK/RELIABILITY-BASED HYDRAULIC ENGINEERING DESIGN
7.46
Chapter Seven
The evolution of hydraulic design methods can be roughly classified into three stages: (1) return period design, (2) conventional risk-based design, and (3) optimal risk-based design with consideration given to various uncertainties. 7.4.2.1 Return-period design. Using the return period design approach, a water resource engineer first determines the design discharge from a frequency discharge relation by selecting an appropriate design frequency or return period. The design discharge then is used to determine the structure size and layout that has a satisfactory hydraulic performance. By the return period design method, the selection of the design return period is crucial to the hydraulic design. Once the design return period is determined, it remains fixed throughout the design process. In the past, the design return period was subjectively selected on the basis of an individuals experience, perceived importance of the structure, and/or legal requirements. The selection of the design return period is a complex procedure which involves consideration of economic, social, legal, and other factors. However, the procedure does not account for these factors explicitly. 7.4.2.2 Conventional risk-based design. The conventional risk-based design considers the inherent hydrologic uncertainty in the calculation of the expected economic losses. In the risk-based design procedure, the design return period is a decision variable instead of being a pre–selected design parameter value as in the return period design procedure. The concept of risk-based design has been recognized for many years. As early as in 1936, the U.S. Congress passed the Flood Control Act (U. S. Statutes 1570) in which consideration of failure consequences in the design procedure was advocated. The economic risks or the expected flood losses were not explicitly considered until the early 1960s. Pritchett’s (1964) work was one of the early attempts to apply the risk-based hydraulic design concept to highway culverts. At four actual locations, Pritchett calculated the investment costs and the expected flood damage costs on an annual basis for several design alternatives among which the most economical was selected. The results indicated that a more economical solution could be reached by selecting smaller culvert sizes compared with the traditional return period method used by the California Division of Highways. The conventional approach has been applied to the design of various hydraulic structures. 7.4.2.3 Risk-based design considering other uncertainties. In the conventional risk–based hydraulic design procedure, economic risks are calculated considering only the randomness of hydrologic events. In reality, there are various types of uncertainties in a hydraulic structure design. Advances were made to incorporate other aspects of uncertainty in various hydraulic structure design.
7.4.3 Tangible Costs in Risk-Based Design of Hydraulic Structures Design of a hydraulic structure, by nature, is an optimization problem consisting of an analysis of the hydraulic performance of the structure to convey flow across or through the structure and a determination of the most economical design alternative. The objective function is to minimize the sum of capital investment cost, the expected flood damage costs, and operation and maintenance costs. For example, the relevant variables and parameters associated with the investment cost and the expected damage costs of highway drainage structures are listed in Tables 7.10 and 7.11, respectively. The maintenance cost over the service life of the structure is generally treated as a yearly constant. Based on Tables 7.10 and 7.11, the information needed for the risk-based design of a highway drainage structure can be categorized into four types: 1. Hydrologic/physiographical data, including flood and precipitation data, drainage Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
RISK/RELIABILITY-BASED HYDRAULIC ENGINEERING DESIGN
Risk/Reliability–Based Hydraulic Engineering Design 7.47
area, channel bottom slope, and drainage basin slope. These data are needed to predict the magnitude of hydrologic events such as streamflow and rainfall by frequency analysis and/or regional analysis. 2. Hydraulic data, including flood plain slopes, geometry of the channel crosssection, roughness coefficients, size of structural opening, and height of embankment. These data are needed to determine the flow carrying capacities of hydraulic structures and to perform hydraulic analysis. 3. Structural data, including material of substructures and layout of structure. 4. Economic data, including (1) type, location, distribution, and economic value of upstream properties such as crops and buildings; (2) unit costs of structural materials, equipment, operation of vehicle, accident, occupancy, and labor fee; (3) depth and duration of overtopping, rate of repair, and rate of accidents; and (4) time of repair and length of detour. In the design of hydraulic structures, the installation cost often is dependent on the environmental conditions such as the location of the structure, geomorphic and geologic conditions, the soil type at the structure site, type and price of construction material, hydraulic conditions, flow conditions, recovery factor of the capital investment, labor and transportation costs. In reality, these factors would result in uncertainties in cost functions used in the analysis. The incorporation of the economic uncertainties in the risk-based design of hydraulic structures can be found elsewhere (U.S. Army Corps of Engineers, 1996).
7.4.4 Evaluations of Annual Expected Flood Damage Cost In reliability–based and optimal risk-based designs of hydraulic structures, the thrust is to evaluate E(DΘ) as the function of the PDFs of load and resistance, damage function, and the types of uncertainty considered. 7.4.4.1 Conventional approach. In the conventional risk-based design where only inherent hydrologic uncertainty is considered, the structural size Θ and its corresponding flow carrying capacity qc, in general, have a one to one, monotonically increasing relation. Consequently, the design variable Θ alternatively can be expressed in terms of design discharge of the hydraulic structure. The annual expected damage cost, in the conTABLE 7.10 Variables and Parameters Relevant in Evaluating Capital Investment Cost of Highway Drainage Structures Pipe Culverts
Box Culverts
Bridges
Parameters
Unit cost of culvert
Unit cost of concrete Unit cost of steel
Unit cost of bridge
Variables
Number of pipes Pipe size Pipe length Pipe materials
Number of barrels Length of barrel Width of barrel Quantity of concrete Quantity of steel
Bridge length Bridge width
Source: From Tung and Bao (1990).
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RISK/RELIABILITY-BASED HYDRAULIC ENGINEERING DESIGN
7.48
Chapter Seven
TABLE 7.11 Damage categories and related economic variables and site characteristics in riskbased design of highway drainage structures Damage Category Floodplain Property Damage: Losses to crops Losses to buildings
Damage to Pavement and Embankment: Pavement damage Embankment damage
Traffic Related Losses: Increased travel cost due to detour Lost time of vehicle occupants Increased risk of accidents on detour Increased risk of accidents on a flooded highway
Economic Variables
Site Characteristics
Type of crops Economic value of crops Economic values of buildings
Location of crop fields Location of buildings Physical layout of drainage structures Roadway geometry Flood characteristics Stream cross–section Slope of channel Channel & floodplain roughness properties
Material cost of pavement Material cost of embankment Equipment costsLabor costs Repair rate for pavement & embankment
Flood magnitude Flood hydrograph Overtopping duration Depth of overtopping Total area of pavement Total volume of embankment Types of drainage structures and layout Roadway geometry
Rate of repair Operational cost of vehicle Distribution of income for vehicle occupants Cost of vehicle accident Rate of accident Duration of repair
Average daily traffic volume Composition of vehicle types Length of normal detour path Flood hydrograph Duration and depth of overtopping
Source: From Tung and Bao (1990).
ventional risk-based hydraulic design, can be computed as
∞
E1(DΘ)
q*c
D(qqc, Θ) f(q) dq
(7.79)
where q*c is the deterministic flow capacity of a hydraulic structure subject to random flood loadings following a PDF, f(q), and D(qq*c,Θ) is the damage function corresponding to the flood magnitude of q and hydraulic structure capacity q*c. Due to the complexity of the
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RISK/RELIABILITY-BASED HYDRAULIC ENGINEERING DESIGN
Risk/Reliability–Based Hydraulic Engineering Design 7.49
damage function and the form of the PDF of floods, the analytical integration of Eq. (7.79), in most practical applications, is difficult, if not impossible. Hence, it is practical to replace Eq. (7.79) by a numerical approximation. Equation (7.79) only considers the inherent hydrologic uncertainty due to the random occurrence of flood events, represented by the PDF, f(q). It does not consider hydraulic and economic uncertainties. Furthermore, a perfect knowledge about the probability distribution of flood flow is assumed. This generally is not the case in reality. 7.4.4.2 Incorporation of hydraulic uncertainty. As described in Sec. 7.1.1, uncertainties also exist in the process of hydraulic computations for determining the flow carrying capacity of the hydraulic structure. In other words, qc is a quantity subject to uncertainty. From the uncertainty analysis of qc, the statistical properties of qc can be estimated. Hence, to incorporate the uncertainty feature of qc in risk-based design, the annual expected damage can be calculated as E2(DΘ)
D(qq ,Θ) f(q) dq g(q Θ) dq = E (Dq ,Θ) g(q Θ) dq ∞
0
∞
0
∞
c
c
c
0
1
c
c
c
(7.80)
in which g(qcΘ) is the PDF of random flow carrying capacity qc . Again, in practical problems, the annual expected damage in Eq. (7.80) would have to be evaluated through the use of appropriate numerical integration schemes. 7.4.4.3 Extension of conventional approach by considering hydrologic parameter uncertainty. Since the occurrence of streamflow is random by nature, the statistical properties such as the mean, standard deviation and skewness of the distribution calculated from a finite sample also are subject to sampling errors. In hydrologic frequency analysis, a commonly used frequency equation for determining the magnitude of a hydrologic event of a specified return period T years is qTR µ KT σ
(7.81)
in which qT is the magnitude of hydrologic event of the return period T years; µ and are the population mean and standard deviation of the hydrologic event under consideration respectively; and KT is the frequency factor depending on the skew coefficient and probability distribution of the hydrologic event of interest. Consider flooding as the hydrologic event that could potentially cause the failure of the hydraulic structure. Due to the uncertainty associated with µ, σ, and KTR in Eq. (7.81), the flood magnitude of a specified return period, qTR, also is a random variable associated with its probability distribution (Fig. 7.7) instead of being a single valued quantity represented by its "average", as commonly done in practice. Sampling distributions for some of the probability distributions frequently used in hydrologic flood frequency analysis have been presented elsewhere (Chowdhury and Stedinger, 1991; Stedinger, 1983). Hence, there is an expected damage corresponding to a flood magnitude of the TR–year return period which can be expressed as E(DTqc,Θ)
∞
qc*
D(qTqc,Θ) h(qT) dqT
(7.82)
where E(DTqc,Θ) is the expected damage corresponding to a T–year flood given a known flow capacity of the hydraulic structure, qc, h(qT) is the sampling PDF of the flood magnitude estimator of a T year return period; and qT is the dummy variable for a
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RISK/RELIABILITY-BASED HYDRAULIC ENGINEERING DESIGN
7.50
Chapter Seven
FIGURE 7.7 Sampling distribution associated with flood estimator (Tung, 1996)
T year flood. To combine the inherent hydrologic uncertainty, represented by the PDF of annual flood, f(q), and the hydrologic parameter uncertainty, represented by the sampling PDF for a flood sample of a given return period, h(qT), the annual expected damage cost can be written as E3(Dqc,Θ)
∞
qc*
∞
qc*
D(qTqc,Θ) h(qTq) dqT f(q) dq
(7.82)
7.4.4.4 Incorporation of hydrologic inherent/parameter and hydraulic uncertainties. To include hydrologic inherent and parameter uncertainties along with the hydraulic uncertainty associated with the flow carrying capacity, the annual expected damage cost can be written as E4(DΘ)
∞
0
∞
qc
∞
qc
D(qT, qcΘ) h(qT) dqT f(q) dq g(qcΘ) dqc
∞
0
E3 (DΘ) g(qcΘ) dqc
(7.83)
Based on the above formulations for computing annual expected damage in the riskbased design of hydraulic structures, one realizes that the mathematical complexity increases as more uncertainties are considered. However, to obtain an accurate estimation of annual expected damage associated with the structural failure would require the consideration of all uncertainties, if such can be practically done. Otherwise, the annual
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RISK/RELIABILITY-BASED HYDRAULIC ENGINEERING DESIGN
Risk/Reliability–Based Hydraulic Engineering Design 7.51
expected damage would, in most cases, be underestimated, leading to inaccurate optimal design. Tung (1987) numerically shows that, without providing full account of uncertainties in the analysis, the resulting annual expected damage is significantly under estimated, even with a 75 years long flood record.
7.4.5 U.S. Army Corps of Engineers Risk-Based Analysis for Flood–Damage Reduction Structures This section briefly summarizes the main features of the U.S. Army Corps of Engineers (USACE, 1996) risk-based analysis procedure applied to flood damage reduction plans, such as levee construction, channel modification, flood detention, or mixed measure plan. The procedure explicitly considers the uncertainties in discharge–frequency relation, stage –discharge function, and stage–damage relation. The performance measures of each flood damage reduction plan include economic indicator, such as annual expected innudation damage reduction, and non–economic measures, such as expected annual exceedance probability, long–term risk, and conditional annual nonexceedance probability, and consequence of capacity exceedance. The long–term failure probability is computed by n
pf(n) 1 [1 pf(1)]
(7.84)
where pf(1) the annual failure probability, and pf(n) the long–term failure probability over a period of n years. Uncertainty in discharge frequency relation, as described in Sec. 7.4.4.3, is mainly arised from the sampling error due to the use of limited amount of flood data in establishing the relation. Statistical procedures for quantifying uncertainty associated with a discharge frequency relation can be found elsewhere (Interagency Advisory Committee on Water Data, 1982; Stedinger et al., 1993). For stage–discharge function, its uncertainty may be contributed from factors like measurement errors from instrumentation or method of flow measurement, bed forms, water temperature, debris or other obstructions, unsteady flow effects, variation in hydraulic roughness with season, sediment transport, channel scour or depoition, changes in channel shape during or as a result of flood events, as well as other factors. Uncertainty associated with stage discharge function for gauged and ungauged reach has been examined by Freeman et al. (1996). Stage damage relation describes the direct economic loss of flood water innudation for a particular river reach. It is an important element in risk-based design and analysis of hydraulic structures. The establishment of stage damage relation requires extensive survey and assessment of economic values of the structures and their contents affected by flood water at different water stages. Components and sources of uncertainty in establishing a stage damage relation is listed in Table 7.12. For example, variation of content to structure value ratios of different types of structure in the United Stated is shown in Table 7.13. In evaluating the performance of different flood damage reduction plans or alternatives within a plan, hydraulic simulations such as backwater computation or unsteady state flow routing, are required to assess the system response before various performance measures can be quantified. Due to this compuational complexity and the presece of large number of uncertainties, the evaluation of various economic and noneconomic performance measures in the risk-based analysis procedure cannot be done analytically. Therefore, the computation procedure adopted in the USACE riskbased analysis for flood damage reduction structures is the Monte Carlo simulation. By Monte Carlo simulation, a large number of plausible discharge–frequency function, stage discharge relation, and stage–damage relation are generated according to the
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RISK/RELIABILITY-BASED HYDRAULIC ENGINEERING DESIGN
7.52
Chapter Seven
TABLE 7.12
Components and Sources of Uncertainty in StageDamage Function
Parameter/Model
Source of Uncertainty
Number of structure in each category
Errors in identifying structures; errors in classifying structures
First-floor elevation of structure
Survey errors, inaccuracies in topographic maps; errors in interpolation of contour lines
Depreciated replacement value of structure
Errors in real estate appraisal; errors in estimation of replacement cost estimation–effective age; errors in estimation of depreciation; errors in estimation of market value
Structure depth-damage function
Errors in post–flood damage survey; failure to account for other critical factors: flood water velocity, duration of flood, sediment load, building material, internal construction, condition, flood warning
Depreciated replacement value of contents
Errors in content–inventory survey, errors in estimates of ratio of content to structure value
Content depth–damage function
Errors in post–flood damage survey, failure to account for other critical factors; floodwater velocity, duration of flood, sediment load, content location, flood warning
Source: From USACE (1996).
TABLE 7.13
ContenttoStructure Value Ratios*,✝
Structure Category
No. of Cases Mean
Standard Deviation
Minimum
Maximum
One story no basement 71,629
0.434
0.250
0.100
2.497
One story basement
8,094
0.435
0.217
0.100
2.457
Two story no basement 16,056
0.402
0.259
0.100
2.492
Two story basement
21,753
0.441
0.248
0.100
2.500
Split level no basement
1,005
0.421
0.286
0.105
2.493
Split level basement
1,807
0.435
0.230
0.102
2.463
Mobil home
2,283
0.636
0.378
0.102
2.474
122,597
0.435
0.253
0.100
2.500
All categories
Source: From USACE (1996). * Note that these are less than ratios commonly used by casualty insurance companies, but those reflect replacement costs rather than depreciated replacement costs. ✝ Research by the Institute of Water Resources suggests that errors may be described best with an asymmetric distribution, such as a log–normal distribution. In that case, the parameters of the error distribution cannot be estimated simply from the values shown in this table.
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RISK/RELIABILITY-BASED HYDRAULIC ENGINEERING DESIGN
Risk/Reliability–Based Hydraulic Engineering Design 7.53
underlying or assume probability distributions for each of the factors with uncertainty involved. Under each generated scenario, necessary hydraulic computations are performed based on which various performance measures of different flood damage reduction plans are calculated. The process is repeated for a large number of possible scenarios and, then, the various performance measures are averaged for comparing the relative merit of different plans. The risk-based analysis procedure is illustrated through an example (see Chap. 9, USACE, 1996) in which the preformance measures of several flood damage reduction plans for the metropolitan Chester Creek Basin in Pennsylvania are examined. Results of the risk-based analysis for each plan are shown in Table 7.14 a–c. Note that fromTable 7.14, there are four alternative levee heights being considered and the mixed measure consists of channel modification and detention. Results of risk–based analysis are shown in Table 7.14a which clearly indicates that the levee plan by building a 8.23m dike is the most cost effective. The median annual exceedance probability shown in second column of Table 7.14b is close to the result of conventional flood frequency analysis without considering any other uncertainties but natural randomness of the floods. Compared with the third column, it is clearly observed that the annual expected exceedance probability is higher than the corresponding one without considering uncertainty. Consequently, the longçterm failure probabilities will be under–estimated if other uncertainties in flood frequency relationship are not accounted for. In Table 7.14c, conditional annual nonexceedance probabilities for each plan under a 50–, 100–, and 250– year event have also indicated the supreiority of levee plan over the other flood damage reduction plans in terms of failure probability. From all economic and non economic indicators used in this risk-based analysis, it appears that the levee height of 8.23m is the most desirable alternative in flood damage reduction for Chester Creek Basin, Pennsylvania. Of course, there may be other issues that may have be to be considered, such as impacts of levee on environment, aesthetics, and
TABLE 7.12 Performance Measures from Risk-Based Analysis of Flood Damage Reduction Plans for Chester Creek Basin, Pennsylvania
(a) Present Economic Benefits of Alternatives Plan
Annual WithProject Residual Damage, $1000's
Annual Innudation Reduction Benefit, $1000's
Annual Cost, $1000's
Annual Net Benefit, $1000's
Without project
78.1
0.0
0.0
0.0
6.68 m levee
50.6
27.5
19.8
7.7
7.32 m levee
39.9
38.2
25.0
13.2
7.77 m levee
29.6
48.5
30.6
17.9
8.23 m levee
18.4
59.7
37.1
22.6
Channel modification
41.2
36.9
25.0
11.9
Detention basin
44.1
34.0
35.8
1.8
Mixed measure
24.5
53.6
45.6
8.0
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7.54
Chapter Seven
(b) Annual Exceedance Probability and Long–Term Risk Plan
Median Estimate of Annual Exceed. Annual Exceed. Probability with Probability Uncertainty Analysis
Long–Term Risk 10 yr
25 yr
50 yr
6.68 m levee
0.010
0.0122
0.12
0.26
0.46
7.32 m levee
0.007
0.0082
0.08
0.19
0.34
7.77 m levee
0.004
0.0056
0.05
0.13
0.25
8.23 m levee
0.002
0.0031
0.03
0.08
0.14
Channel modification
0.027
0.0310
0.27
0.55
0.79
Detention basin
0.033
0.0380
0.32
0.62
0.86
Mixed measure
0.014
0.0160
0.15
0.33
0.55
(c) Conditional Non–Exceedance Probability Plan
Probability of Annual Event 0.02
0.01
0.004
6.68 m levee
0.882
0.483
0.066
7.32 m levee
0.970
0.750
0.240
7.77 m levee
0.990
0.896
0.489
8.23 m levee
0.997
0.975
0.763
Channel modification
0.248
0.019
0.000
Detention basin
0.205
0.004
0.003
Mixed measure
0.738
0.312
0.038
Source: From USACE (1996).
giving the public a false sense of security, before a final decision is made. Irrespect of some incompleteness of the current state–of–the–art of risk-based analysis, the procedure does make an advancement over the conventional procedure by explicitly facing and dealing the uncertainties in design and analysis of hydraulic structures, rather than using a obscure factor of safety. The risk-based procedure provides more useful information for engineers to make better and more scientifically defensible design and analysis.
REFERENCES Ang, A. H. S. “Structural Risk Analysis and Reliability—Basd Design,” Journal of Structural Engineering Division, American Society of Civil Enginners, 99(9):1891—1910, 1973. Ang, A. H. S., and C. A. Cornell, “Reliability Bases of Structural Safety and Design,” Journal of Structural Engineering, American Society of Civil Engineers, 100(9):1755—1769, 1974. Berthouex, P.M. “Modeling Concepts Considering Process Performance, Variability, and Uncertainty,” in: Mathematical Modeling for Water Pollution Control Processes, T. M. Keinath
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RISK/RELIABILITY-BASED HYDRAULIC ENGINEERING DESIGN
Risk/Reliability–Based Hydraulic Engineering Design 7.55 and M. P. Wanielista, eds,. Ann Arbor Science, Ann Arbor, MI., 1975, pp. 405—439. Borgman, L. E., “Risk Criteria.”, Journal of Waterways and Harbors Div., American Society of Civil Engineers, 89(WW3): 1—35, 1963. Breitung, K., “Asymptotic Approximations for Multinormal Integrals.”, Journal of Engineering Mechanics, American Society of Civil Engineers, 110(3): 357—366, 1984. Cheng, S. T. “Statistics on Dam Failures,” in Reliability and Uncertainty Analysis in Hydraulic Design, American Society of Civil Engineers, (ASCE), New York, 1993, pp. 97—106. Cheng, S. T., B. C., Yen, and W. H.,Tang, “Sensitivity of Risk Evaluation to Coefficient of Variation,” Stochastic and Risk Analysis in Hydraulic Engineering, Water Resources Publications, Littleton, CO, 1986, pp. 266—273. Chowdhury, J. U., and J. R. Stedinger, “Confidence Interval for Design Floods with Esitmated Skew Coefficient,” Journal of Hydraulic Engineering, American Society of Civil Engineers, 117(7):811—831, 1991. Cornell, C. A. “A ProbabilityBased Structural Code,” Journal of American Concrete Institute, 66(12): 974—985, 1969. Der Kiureghian, A., Lin, H. Z., and Hwang, S. J., “Second order Reliability Approximation,” Journal of Engineering Mechanics, American Society of Civil Enginners. 113(8): 1208—11225, 1987. Der Kiureghian, A., and P. L., Liu, “Structural Reliability Under Incomplete Probability Information,” Journal of Engineering Mechanics, American Society of Civil Engineers. 112(1):85—104, 1985. Ditlevsen, O., “Principle of Normal Tail Approximation.”, Journal of Engineering Mechanics, American Society of Civil Engineers, 107(6): 1191—1208, 1981. Freeman, G. E., R. R., Copeland, and M. A. Cowan, “Uncertainty in Stage—Dischage Relationships.” in. Goulter and K. Tickle eds, Stochastic Hydraulics, ‘96, A. A. Balkema, The Netherlands, 1996. Harbitz, A., “Efficient and Accurate Probability of Failure Calculation by use of the Importance Sampling Technique,” Proceedings, International Conference on Applications of Statistics and Probability in Soil and Structural Engineering, University de Firenze, Florence, Italy, 1983. Harr, M. E., “Probabilistic Estimates for Multivariate Analyses,” Applied Mathematical Modelling, 13: 313—318, 1989. Hasofer, A. M. and N. C., Lind, “Exact and Invariant Second—Moment Code Format," Journal of Engineering Mechanics Div., American Society of Civil Engineers, 100(1): 111—121, 1974. Interagency Advisory Committee on Water Data, “Guidelines for Determining Flood Flow Frequency,” Bulletin 17B. U.S. Department of Interior, U.S. Geologic Survey, Office of Water Data Coordination, Reston, VA, 1982. Karamchandani, A., “Structural System Reliability Analysis Methods,” Report to Amoco Production Company. Department of Civil Engineering, Stanford University, 1987. Karmeshu and F., Lara Rosano, “Modelling Data Uncertainty in Growth Forecasts,” Applied Mathematical Modelling, 11: 62—68, 1987. Liu, P. L. and A., Der Kiureghian, “Multivariate Distribution Models with Prescribed Marginals and Covariances,” Probabilistic Engineering Mechanics, 1(2):105—112, 1986. Madsen, H. O., S., Krenk, and N. C. Lind, Methods of Structural Safety, Prentice—Hall, Englewood Cliffs, N.J. 1986. Mays, L. W., and Y. K., Tung, Hydrosystems Engineering and Management, McGraw—Hill, New York, 1992. McKay, M. D., R. J., Beckman, and W. J. Conovre, “A Comparison of Three Methods for Selecting Values of Input Variables in the Analysis of Output from a Computer Code,” Technometrics, 21, 1979. Melchers, R. E., Structural Reliability Analysis and Prediction, Ellis Horwood, Ltd., Chichester, UK, 400 pp, 1987. Wen, Y. K. “Approximate Methods for Nonlinear Time—Variant Reliability Analysis.” Journal of Engineering Mechanics, American Society of Civil Engineers, 113(12): 1826—1839, 1987. Park, C. S., "The Mellin Transform in Probabilistic Cash Flow Modeling," The Engineering Economist, 32(2):115134, 1987.
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RISK/RELIABILITY-BASED HYDRAULIC ENGINEERING DESIGN
7.56
Chapter Seven
Patel, J. K., C. H., Kapadia, and D. B. Owen, Handbook of Statistical Distributions, John Wiley Sons, New York, 1976. Plate, E. J. “Stochastic Design in Hydraulics: Concepts for a Broader Application,” in J. T Kuo and G. F. Lin, eds. Stochastic Hydraulics '92, proceedings 6th IAHR Int'l Symp., Taipei, Water Resources Publications, Littleton, CO., 1992, pp. 1—15. Plate, E. J., and L. Duckstein, “Reliability in Hydraulic Design.” in L. Duckstein and E. J. Plate, eds., Engineering Reliability and Risk in Water Resources, Martinus Nijhoff, Dordrecht, The Netherlands, 1987, pp. 2760. Plate, E. J., and L. Duckstein, “Reliability Based Design Concepts in Hydraulic Engineering.” Water Resources Bulletin, American water rexrumen association, 24(2): 234—245, 1988. Pritchett, H. D., “Application of the Principles of Engineering Economy to the Selection of Highway Culverts,” Stanford University, Report EEP—13, 1964. Rackwitz, R., “Practical Probabilistic Approach to Design,” Bulletin 112, Comit’e Europe’en du Beton, Paris, France, 1976. Rackwitz, R., and B., Fiessler, “Structural Reliability Under Combined Random Load Sequence,” Computers and Structures, 9:489—494, 1978. Rosenblueth, E., “Point Estimates for Probability Moments,” Proceedings, National Academy of Science, 72(10):3812—3814, 1975. Rosenblueth, E., “TwoPoint Estimates in Probabilities,” Applied Mathematical Modelling, 5:329—335, 1981. Schueller, G. I. and R., Stix, “A Critical Appraisal of Methods to Determine Failure Probabilities.” Report No. 486, Institute fur Mechanik, Universitat Innsbruck, Austria, 1986. Shinozuka, M., “Basic Analysis of Structural Safety,” Journal of Structrual Engineering Div., American Society of Civil Engineers, 109(3):721740, 1983. Springer, M. D., The Algebra of Random Variables, John Wiley & Sons, New York, 1979. Stedinger, J. R., "Confidence Intervals for Design Events," Journal of Hydraulic Engineering, American Society of Civil Engineers, 109(HY1):13—27, 1983. Tung, Y. K., “Effects of Uncertainties on Optimal Risk-Based Design of Hydraulic Structures,” Journal of Water Resources Planning and Management, American Society of Civil Engineers, 113(5):709—722, 1987. Tung, Y. K., “Mellin Transform Applied to Uncertainty Analysis in Hydrology/Hydraulics,” Journal of Hydraul: Engineer, American Society of Civil Engineers, 116(5):659—674, 1990. Tung, Y. K. “Uncertainty and Reliability Analysis,” in L.W. Mays, el., in Water Resources Handbook, McGraw—Hill, New York, 1996. Tung, Y. K. and Y. Bao, “On the Optimal Risk-Based Designs of Highway Drainage Structures,” Journal of Stochastic Hydrology and Hydraulics, 4(4):311—324, 1990. U. S. Army Corps of Engineers, Ris-Based Ancdysis for Floud damagereduction stadres, em 11102-1619, Washington D.C., Angust 1996. Vrijling, J. K., “Development of Probabilistic Design in Flood Defenses in the Netherlands,” in B. C. Yen and Y. K. Tung, eds., Reliability and Uncertainty Analysis in Hydraulic Design, American Society of Civil Engineers, New York, 1993, pp. 133—178. Yen, B. C., "Safety Factor in Hydrologic and Hydraulic Engineering Design," in E. A. McBean, K. W. Hipel, and T. E. Unny, eds., Reliability in Water Resources Management, Water Resources Publications, Littleton, CO, 1979, pp. 389—407. Yen, B. C., and A. H.—S., Ang, “Risk Analysis in Design of Hydraulic Projects," in C. L. Chiu, ed., Stochastic Hydraulics, Proceedings of First International Symposium, University of Pittsburgh, Pittsburgh, PA, 1971, pp. 694701. Yen, B. C., S. T., Cheng, and C. S. Melching, “First Order Reliability Analysis,” in B. C. Yen, ed., Stochastic, pp. 1.36 and Risk Analysis in Hydraulic Engineering, Water Resources Publications, Littleton, CO, 1986.
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Source: HYDRAULIC DESIGN HANDBOOK
CHAPTER 8
HYDRAULIC DESIGN FOR ENERGY GENERATION H. Wayne Coleman C. Y. Wei James E. Lindell Harza Company Chicago, Illinois
8.1 INTRODUCTION This chapter describes the design aspects of hydraulic structures related to the production of hydroelectric power. These structures include headrace channels; intakes; conveyance tunnels; surge tanks; penstocks; penstock manifolds; draft-tube exits; tailtunnels, including tail-tunnel surge tanks and outlets; and tailrace channels. The procedures provided in this chapter are most suitable for developing the preliminary designs of hydraulic structures related to the development of the hydroelectric projects. To finalize designs, detailed studies must be conducted: for example, economic analysis for the determination of penstock diameters, computer modeling of hydraulic transients for surge tank design, and studies of physical models of intake and its approach.
8.2 HEADRACE CHANNEL An open-channel called the headrace channel or power channel (canal) is sometimes required to connect a reservoir with a power intake when the geology or topography is not suitable for a tunnel or when an open-channel is more economical. The channel can be lined or unlined, depending on the suitability of the foundation material and the projects economics. Friction factors for various linings used for design are as follows: Manning’s n Lining
Minimum.
Maximum
Unlined rock
0.030
0.035
Shotcrete
0.025
0.030
Formed concrete
0.012
0.016
Grassed earth
0.030
0.100
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HYDRAULIC DESIGN FOR ENERGY GENERATION
8.2
Chapter Eight
Headrace channels are generally designed and sized for a velocity of about 2 m/s (6.6 ft/s) at design flow conditions. Economic considerations may result in some variation from this velocity, depending on actual project conditions. Channel sections are normally trapezoidal because this shape is easier to build for many different geologic conditions. The bottom width should be at least 2 m (6.6 ft) wide. Side slopes are determined according to geologic stability as follows: earth, 2H:1V or flatter; and rock, 1H:1V or steeper. The channel’s proportions—bottom width versus depth— are largely a matter of construction efficiency. In general, the minimum bottom width reduces excavation, but geologic conditions may require a wider, shallower channel. The channel slope will result from the conveyance required to produce design velocity for design flow. Channel bends should have a center-line radius of 3W to 5W or more, where W is the water surface width of the design flow. For this radius, head loss and the rise in the water surface at the outer bank (superelevation) will be minimal. If the radius must be reduced, the following formula can be used to estimate head loss hL: 2 hL KbV 2g
(8.1)
where Kb 2 (W/Rc), W channel width, Rc center-line radius, and V mean velocity. Superelevation will be as follows (Chow, 1959): 2W V2 Z Rc 2g
(8.2)
where Z rise in water surface above mean flow depth. Freeboard must include allowances for the following conditions: (1) static conditions with maximum reservoir level (unless closure gates are provided to isolate the channel from the reservoir), (2) water surface rise (superelevation) caused by flow around a curve, and (3) surge resulting from shut-off of flow downstream or sudden increase of flow upstream. A forebay is provided at the downstream end of the headrace channel to facilitate one or more of the following: (1) low approach velocity to intake, (2) surge reduction, (3) sediment removal (desanding), or (4) storage. The forebay should be designed to maintain the approach flow conditions to the intake as smoothly as possible. As the minimum requirement, a small forebay should be provided to facilitate good entrance conditions to the intake. It should include a smooth transition to a section with a velocity not exceeding 0.5 m/s (1.64 ft/s) at the face of the intake structure A larger forebay could be required for upsurge protection during rapid closure of turbine gates for load rejection. The size would be determined on the basis of the freeboard allowance for the entire headrace channel and on a hydraulic transient analysis of the channel, if necessary. Surge calculations should consider maximum and minimum friction factors, depending on which is more critical for the case under study. Hydraulic transient (surge) studies are generally performed using a one-dimensional, unsteady open-channel-flow simulation program. The computer model developed should be capable of simulating the operation of various hydraulic structures, the effect of the forebay, and operation of the power plant. Several advanced open-channel flow-simulation programs have been described by Brater et al. (1996).
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HYDRAULIC DESIGN FOR ENERGY GENERATION
Hydraulic Design for Energy Generation 8.3
(a)
(b) Exhibit 8.1 Sun Koshi hydroelectric project, Nepal. (a) A view of the desanding basin (looking upstream) showing concrete guide vanes.( (b) Layout Of the desanding basin.
A large forebay is required if it will be used for diurnal storage–say, for a power peaking operation. In such a case, maximum and minimum operating levels would include the required water volume, with the intake located below the minimum level. Such a forebay also could accommodate the other three functions described above. When the flow carries too much sediment and its removal is required to protect the turbines, a still larger forebay would be provided to function as a desanding basin (also known as a desilting basin or desander). However, the desanding basin is more likely to be located at the upstream end of the headrace channel. Exhibit 8.1 Illustrates a desending basin. The basin can be sized using the following equation (Vanoni, 1977):
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HYDRAULIC DESIGN FOR ENERGY GENERATION
8.4
Chapter Eight
FIGURE 8.1 Settling velocity as a function of particle diameter. (Dingman, 1984)
LV
s
P (1 e VD ) 100%
(8.3)
where P percentage of sediment of a particular size to be retained by the basin, L basin length, Vs settling (fall) velocity of suspended particles, V mean flow velocity, and D depth of the desanding basin. The settling velocity Vs for each particular sand particle size can be estimated from Fig. 8.1. A separate sluicing outlet (or outlets) would be provided to flush the desanding basin intermittently.
8.3 INTAKES Most power intakes are horizontal, a few are vertical, and very few are inclined. Figures 8.2, 8.3, and 8.4 are examples of the three types of intakes. Exhibit 8.2 illustrates the layout of a hydroelectric project with the intakes. The horizontal intake is usually connected to a tunnel or penstock on a relatively small slope (up to 2–3 percent). The vertical intake is frequently used in pumped-storage projects when the upper reservoir is on high ground, such as a mountain top, and a vertical shaft-tunnel is the obvious choice. An inclined intake is used when the topography, geology, or type of dam dictate a steeper slope for the downstream tunnel or penstock. A variation on the three basic intake types is a tower structure, sometimes required for
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HYDRAULIC DESIGN FOR ENERGY GENERATION
FIGURE 8.2 A typical horizontal intake. (Harza Engineering Co.)
Hydraulic Design for Energy Generation 8.5
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HYDRAULIC DESIGN FOR ENERGY GENERATION
8.6
Chapter Eight
FIGURE 8.3 A typical vertical intake. (Harza Engineering Co.)
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HYDRAULIC DESIGN FOR ENERGY GENERATION
FIGURE 8.4 A typical inclined intake. (Harza Engineering Co.)
Hydraulic Design for Energy Generation 8.7
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HYDRAULIC DESIGN FOR ENERGY GENERATION
8.8
Chapter Eight
Exhibit 8.2 (a)
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HYDRAULIC DESIGN FOR ENERGY GENERATION
Hydraulic Design for Energy Generation 8.9
(b) Exhibit 8.2 Karun hydroelectric project, Iran (a) A vew of the dam and control structure (looking donwnstream showing spillway crest, radial gates, power intakes, and diversion tunnel entrace structure. (b) Layout of dam showing spillway, intake and powerhouse.
selective withdrawal of water. The tower includes openings with trashracks and bulkheads at various levels, which permit water to be withdrawn from different depths to control temperature or water quality. Computer modeling of a reservoir’s temperature and waterquality structure is generally required to finalize the required opening sites. Descriptions of several reservoir-simulation models can be found in Brater et al. (1996). Figure 8.5 is an example of a multilevel intake tower structure for selective withdrawal.Exhibit 8.3 illustrates the intake structure for a pumped storage project. Trashracks for power intakes are designed for a velocity of about 1 m/s (3.3 ft/s) when the intake is accessible for cleaning. If a trashrack is not accessible for cleaning, the allowable velocity is approximately 0.5 m/s (1.6 ft/s). Trashrack bar spacing is dictated by turbine protection requirements, but clear spacing of 5cm (2 in) is typical. Although head loss through trashracks depends heavily on the amount of clogging, the following can be used for a clean trashrack, (U.S. Bureau of Reclamation, 1987);
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HYDRAULIC DESIGN FOR ENERGY GENERATION
8.10
Chapter Eight
FIGURE 8.5 A typical multi-level intake tower structure for selective withdrawal. (Harza Engineering Co.)
V2 hL Kt n 2g
(8.4)
2 A A where Vn velocity based on the net area, Kt 1.45 0.45 n n , An net area of Ag Ag trashrack and support structure, and Ag gross area of trashrack and support structure.
An intake gate is generally provided when the power tunnel or penstock is long or when a short penstock does not have a turbine inlet valve. This gate is provided for emergency closure against flow in case of runaway conditions at the turbine. The effective area of the gate is usually about the same as that of the power tunnel or penstock, but it is rectangular in shape, with a height that is the same as the conduit’s diameter and a width that is 0.8 the conduit diameter. A bulkhead (or stop log) is provided upstream of the intake gate for servicing the gate. The trashrack slot might be used for this function by first pulling the trashrack.
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HYDRAULIC DESIGN FOR ENERGY GENERATION
Hydraulic Design for Energy Generation 8.11
(a)
(b) Exhibit 8.3 Rocky mountain pumped storage project, Georgia. (a) Intake structure of the upper reservoir. (b) Closed up view of the upper reservoir intake structure. (c) General layout of the project including upper reservoir intake, power tunnel, and power house.
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HYDRAULIC DESIGN FOR ENERGY GENERATION
Chapter Eight
Exhibit 8.3 (c)
8.12
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HYDRAULIC DESIGN FOR ENERGY GENERATION
Hydraulic Design for Energy Generation 8.13
A hydraulic study is generally conducted for emergency closure of the intake gate. The maximum turbine flow or runaway flow should be considered. The runaway flow may be 50 percent higher than the normal turbine flow for a propeller turbine. In the hydraulic study, the water levels and pressures, as well as flow into and from the gate well, as a function of gate position are investigated (Fig. 8.6). With this information, critical gate loads can be determined for the gate and hoist. The gate also may be used for penstock filling. A minimum gate opening of 10 to 15 cm (4-6 in) is usually specified for this, but a special hydraulic study must be made to determine potential gate load and vibration if the gate opens continuously by accident. In such cases, a generous gate well or air vent must be provided downstream of the gate to provide relief once the tunnel or penstock fills. The head loss for a bulkhead or gate slot, including top opening, is generally about 0.1 of the local velocity head at the slot. The transition length (m or ft) Lt from gate section to tunnel or penstock should be approximately:
FIGURE 8.6 A typical intake gate arrangement. (Harza Engineering Co.)
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HYDRAULIC DESIGN FOR ENERGY GENERATION
8.14
Chapter Eight
VD Lt C
(8.5)
where V tunnel/penstock velocity (m/s or ft/s), D tunnel/penstock diameter (m or ft), and C 3.00 for units in metric systems or 9.84 for units in English systems. The variation of velocity in the transition section should be as close to linear as practicable. Overall head loss for an intake includes trashrack, bellmouth (0.1 V2/2g), gate slots, and transition. The potential vortex formation for an intake should be checked using Fig. 8.7. Note that when the intake Froude number (V/兹g 苶D 苶) exceeds 0.5, submergence requirements increase dramatically, and the vortex formation is difficult to predict. In this case, a physical model study should be carried out.
8.4 TUNNELS When the powerhouse is situated a considerable distance from the intake and when geologic conditions permit, a tunnel is often used to convey the flow for power generation. The size of the tunnel is dictated by economics: that is, construction cost is added to the cost of head loss (loss of generating revenue) to obtain the minimum combined cost. This determination is usually obtained by trial and error because the process does not lend itself
FIGURE 8.7 Intake submergence and vortex formation. (Gulliver and Arndt, 1991)
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HYDRAULIC DESIGN FOR ENERGY GENERATION
Hydraulic Design for Energy Generation 8.15
to a simple formula. The resulting tunnel velocity with the economic diameter is usually in the range of 3 to 5 m/s (10 to 17 ft/s). The shape of the excavated tunnel normally will approximate a square bottom and a circular top. The diameter of the circular top (or the width of the square bottom) should be larger than the required diameter. If the tunnel is lined with concrete, its cross section is likely to be circular or have a square or trapezoidal bottom. If it is unlined or lined with shotcrete, the excavated shape will remain, with some smoothing by filling the larger overbreak sections. Lining is an economic consideration, balancing the cost of the lining with the power loss caused by friction. Even an unlined tunnel will have lined sections, such as portals, and sections where rock needs extra support for geologic stability. Friction factors for design are as follows: Manning’s n Lining
Minimum Maximum
Unlined
0.030
0.035
Shotcrete
0.025
0.030
Formed concrete
0.012
0.016
Minimum friction corresponds to new conditions and is used for turbine-rating and pressure-rise calculations. Maximum friction corresponds to aging and is used for economic-diameter and pressure-drop calculations. Tunnel slope is dictated by construction suitability and geology, with a minimum of 1:1000 for drainage during dewatered condi-
(a)
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HYDRAULIC DESIGN FOR ENERGY GENERATION
Chapter Eight
Exhibit 8.4 Bath County pumped storage project, Virginia. (a) Surge tank openings during construction (44-ft inside diamenter and 300-ft deep) (b) Layout of the project including upper reservor intake, control structure, surge tanks, power tunnel and powerhouse
8.16
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HYDRAULIC DESIGN FOR ENERGY GENERATION
Hydraulic Design for Energy Generation 8.17
tions. Tunnel bends generally have large radii for convenience of construction. Vertical bends at shafts usually have a minimum radius of 3D to minimize head loss and to provide constructibility.
8.5 SURGE TANKS Surge tanks generally are used near the downstream end of tunnels or penstocks to reduce changes in pressure caused by hydraulic transients (waterhammer) resulting from load changes on the turbines (ASCE, 1989; Chaudhry, 1987; Gulliver and Arndt, 1991; Moffat et al., 1990; Parmakian, 1955; Rich, 1951; Wylie and Streeter, 1993; Zipparro and Hasen, 1993). A surge tank should be provided if the maximum rise in speed caused by maximum load rejection cannot be reduced to less than 60 percent of the rated speed by other practical methods, such as increasing the generator’s inertia or the penstock’s diameter or by decreasing the effective closing time of the wicket gates. In general, the provision of a surge tank should be investigated if
冘 0
LiVi
0 3 to 5 for units in m/s and m or H n
10 to 20 for units in ft/s and ft,
(8.6)
where Li is the length of a penstock segment and Vi is the velocity for the segment (Dingman, 1984). The term 冱LiVi is computed from the intake to the turbine and Hn is the minimum net head. Surge tanks normally are located as close as possible to the powerhouse for maximum effectiveness and may be free-standing or excavated in rock. The tanks are usually vented to atmosphere or can be pressurized as air chambers. The latter is not used frequently because of requirements of size, air compressors, and air tightness. Exhibit 8.4 illustrates a pumped storage project with a surge tank. Figure 8.8 shows typical installations of surge tanks for controlling hydraulic transients. Surge tanks usually are simple cylindrical vertical shafts or towers, but other geometric designs are used when the surge amplitude is to be limited. For instance, an enlarged chamber can be used at the top if upsurge might cause the water level to rise above the ground surface. Similarly, an enlargement or lateral tunnel or chamber is sometimes used near the bottom of the shaft if downsurge would caused the water level to drop below the tunnel crown. When the geometry is a cylinder, analysis is relatively simple and can be performed using design charts. If the geometry is more complicated, a hydraulic transient simulation model is required to carry out the study (Chaudhry, 1987; Wylie and Streeter, 1993; Brater et al., 1996). Hydraulic stability for a surge tank assures that surging is limited and brief after load changes (Rich, 1951; Parmakian, 1955; U.S. Bureau of Reclamation 1980; Zipparro and Hasen, 1993). The minimum cross-sectional area of a simple cylindrical surge tank required for stability can be determined using the Thoma formula: AL AST 2gcH
(8.7)
where AST minimum tank area, A tunnel area between reservoir and surge tank, L tunnel length between reservoir and surge tank, g gravitational acceleration, c
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HYDRAULIC DESIGN FOR ENERGY GENERATION
Chapter Eight
FIGURE 8.8A Typical vented surge tank installation. Bath County powerplant (1985): 2100 MW pumped storage development on Back Creek, Virginia. Moose River powerplant (1987): 12 MW development on Moose River, New York. (Harza Engineering Co.)
8.18
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HYDRAULIC DESIGN FOR ENERGY GENERATION
Hydraulic Design for Energy Generation 8.19
FIGURE 8.8B Typical pressurized surge tank installation. Moose River powerplant (1987): 12 MW development on Moose River, New York. (Harza Engineering Co.)
H 1 H head loss coefficient H minimum head loss from reservoir to V2 2g V2 / 2g surge tank, including tunnel velocity head V2/2g, and H = minimum net operating head on turbine. For a simple surge tank (without an orifice), increase the diameter obtained from the Thoma formula by 50 percent. For a typical surge tank with a restricted orifice, increase the diameter by 25 percent. These increases are necessary to provide damping of the oscillation in a reasonable period of time. Maximum upsurge in a cylindrical surge tank can be determined from Fig. 8.9. For a given tank size, the optimum size of the orifice is based on the balanced head design so that the maximum tunnel pressure below the surge tank equals the maximum upsurge level. Maximum downsurge in a cylindrical surge tank can be determined from Fig. 8.10. Here again, the size of the orifice should be based on balanced head design as a first attempt. However, since downsurge may differ from upsurge, and the required orifice size may be different for the two purposes, shaping the orifice (i.e., changing the discharge coefficient) by rounding the top or bottom may satisfy the two area requirements approximately. For maximum upsurge, use the maximum normal headwater, minimum head loss between reservoir and surge tank, and maximum plant flow. Assume full plant load-rejection (tripout) in the shortest reasonable time. For maximum downsurge, use the minimum normal headwater, maximum head loss, and accept load from 50 percent to 100 percent in the shortest reasonable time. At some projects, such as pumped-storage plants, the load acceptance is criterion is more extreme; full load acceptance, is 0 percent to 100 percent in the shortest reasonable time. The controlling criterion will be used to design the orifice on downsurge. When the surge tank geometry is complex (noncylindrical), a computer model should be used to determine the limiting surge levels. (See Brater et al., 1996, for available computer models). Freeboard
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HYDRAULIC DESIGN FOR ENERGY GENERATION
8.20
Chapter Eight
FIGURE 8.9 Maximum surge in surge tank due to instantaneous stopping of flow. (Parmakian, 1955)
FIGURE 8.10 Maximum surge in a surge tank resulting from instantaneous starting of flow. (Parmakian, 1955)
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HYDRAULIC DESIGN FOR ENERGY GENERATION
Hydraulic Design for Energy Generation 8.21
for the surge tank is 10 percent of the computed rise in the water level in the surge tank for upsurge and 15 percent of the drop in the water level for downsurge to maintain. submergence of the tank invert or the orifice to avoid admitting air into the penstock. Pressurized air chambers are often used in pumping plants for surge protection. They are used occasionally for power plants when the generating flow is not excessive. The hydraulic characteristics of the chambers are complicated by the compressibility effects of air and temperature, and the analysis does not lend itself to simple formulas and charts. A computer model is required to verify performance. Fig. 8.8(B) shows a typical air chamber design for a hydropower plant.
8.6 PENSTOCK A penstock generally refers to a steel conduit or steel-lined tunnel connecting a reservoir or surge tank to a powerhouse (ASCE, 1989, 1993; U.S. Bureau of Reclamation, 1967; Chaudhry, 1987; Gulliver, and Arndt, 1991; Warnick et al., 1984; Wylie and Streeter, 1993; Zipparro and Hasen, 1993). It is used when the internal pressure is high enough to make a concrete-lined tunnel or unlined rock tunnel uneconomical, particularly where cover is low. Penstock size is usually governed by project economics. The economical diameter is determined by the minimum combined cost of construction and energy reduction caused by head loss in the penstock. The energy loss decreases as the diameter of the penstock increases while construction cost increase. As with tunnels, the most economical diameter can be determined more accurately by a trial-and-error procedure. The following variables are generally considered (U.S. Bureau Reclamation, 1967; Gulliver and Arndt, 1991): 1. Cost of pipe
7. Surface roughness (friction factor)
2. Value of energy loss
8. Weight of steel penstock
3. Plant efficiency
9. Design discharge
4. Minor loss factor
10.Allowable hoop stress
5. Average head 6. Waterhammer effect For the assessment of a preliminary design or a feasibility level, the most economical diameter can be estimated using the following formula (Moffat et al., 1990). CP0.43 De 0 H .60
(8.8)
where De the most economical penstock diameter (m or ft), H = the rated head (m or ft), P = the rated capacity of the plant (kW or hp), and C = 0.52 (for metric units) or 3.07 (for English units). If the project is a small hydropower installation, the following simple equation can be used (Warnick et al., 1984). De CQ0.5
(8.9)
where De = the most economical penstock diameter (m or ft), Q = the design discharge (m3/s or ft3/sec), and C = 0.72 (for metric units) or 0.40 (for English units).
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HYDRAULIC DESIGN FOR ENERGY GENERATION
8.22
Chapter Eight
For large hydroelectric projects with heads varying from approximately 60 m (190 ft) to 315 m (1,025 ft) and power capacities ranging from 154 MW to 730 MW, the following equation can be used (Warnick et al., 1984). Cp0.43 De 0. h 63
(8.10)
where De the most economical penstock diameter (m or ft), p the rated turbine capacity (kW or hp), h the rated net head (m or ft), and C 0.72 ( for metric units) or 4.44 (for English units). The maximum velocity in the penstock is normally kept lower than 10 m/s (33 ft/s). To determine the minimum thickness of the penstock, based on the need for stiffness, corrosion protection, and handling requirements, the following formula can be used (U.S. Bureau of Reclamation, 1967; Warnick et al., 1984). DK tmin 400
(8.11)
where tmin the minimum thickness of the penstock (mm or in), D = penstock diameter (mm or in), and K = 500 (for metric units) or 20 (for English units). After determining the economic diameter, check for the operating stability of the generating unit-penstock combination using the following steps (Chaudhry, 1987; U.S. Bureau of Reclamation, 1980; Warnick et al., 1984). 1. Determine the mechanical starting time in seconds for the unit Tm as (GD2) N2 Tm 36 104 P
(8.4)
(WR2)N2 Tm 1.6 106 P1
(8.13)
or
where GD2 flywheel effect of the turbine and generator rotating parts used in metric system (kg-m2), WR2 flywheel effect of the turbine and generator rotating parts in English system (lb-ft2) 5.932 GD2, G weight of rotating parts (kg), D 2 radius of gyration of the rotating parts (m), W weight of rotating parts (lb), R radius of gyration of the rotating parts (ft), N turbine speed (rpm), P maximum turbine output (kW), and P1 maximum turbine output (hp). Tm is the time for torque to accelerate the rotating mass from zero to rotational speed. Together, the turbine runner in water, connecting shafts, and the generator develop the flywheel effect WR2 or GD2. The WR2 can be determined using on the following formulas:
冢 冣
Pd WR 2turbine 23,800 3 N /2
5/4
(8.14)
and
冢
冣
KVA WR2normal 356,000 3/ N 2 generator
5/4
(8.15)
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HYDRAULIC DESIGN FOR ENERGY GENERATION
Hydraulic Design for Energy Generation 8.23
where Pd turbine rated output (hp) and kVA generator rated output (kilovoltamperes). 2. Determine the water column starting time for the penstock TW as follows: 冱(LV) Tw gH
(8.16)
where 冱(LV) summation of product of length (measured from nearest open water surface) and velocity for each segment of penstock from intake or surge tank to tailrace (m2/s or ft2/sec), g gravitational acceleration (m/s2 or ft/sec2), and H minimum net operating head (m or ft). 3. In general, Tm/Tw2 should be maintained greater than 2 for good operating stability and to have reasonably good responses to load changes. If Tm/Tw2 is less than 2, there are three possible solutions: • Increase WR2 or GD2 for the generator; this is relatively inexpensive for increases of up to 50. • Increase the penstock diameter; this is probably not economical, except for a narrow range. • Add a surge tank or move the surge tank closer to the powerhouse. A combination of these three possible solutions may be the most cost-effective solution. The following friction factors are recommended for designing steel penstocks:
Exhibit 8.5 A typical steel penstock branch structure being fabricated
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HYDRAULIC DESIGN FOR ENERGY GENERATION
8.24
Chapter Eight
Penstock Age
Manning’s n
New
0.012
Old
0.016
Use the value for new penstock to calculating turbine-rating and pressure-rise. To calculate pressure drop use the higher values. Design pressure is determined on the basis of the turbine’s characteristics and the closure rates of the wicket gates or needle valves. For Pelton turbines, closure rates are slow, and design pressure rise is usually of the order of 20 percent of the static pressure head. For Francis turbines, design pressure rise is usually 30 to 40 percent of the static pressure head, depending on the cost of steel lining required. A fast closure is desirable to minimize speed rise and the potential for runaway conditions in the turbine. Detailed pressure conditions are determined by a computer model that includes the water conductors and surge tank as well as the turbine discharge-speed characteristics and generator inertia. Many computer programs capable of simulating hydraulic transients are described in Wylie and Streeter, 1993. Such computer simulation studies are often required of turbine or governor manufacturers now as a part of the specifications. Ultimately, the predicted pressure conditions are verified in the load rejection tests during unit start-up. The profile for a free-standing penstock is based on the topographic and geologic conditions of the ground. In other cases, the penstock may consist of shaft and tunnel sections that are largely lined with concrete, with a relatively short section of steel-lined penstock near the powerhouse. If the penstock is free-standing, the risk of penstock rupture is greater than it is for the shaft and tunnel system. If there is a long tunnel section upstream of the free-standing penstock, an emergency closure valve is often added near the tunnel outlet. A hydraulic transient study is necessary to determine closure conditions (by accident or because of penstock rupture). A vent must be provided to admit air just downstream of the valve for penstock rupture and must be large enough to prevent collapse of the penstock from internal subatmospheric pressure caused by water-column separation. A free-standing penstock also requires small air inlet-outlet valves at local high points to remove air during filling and admit air during dewatering.
8.6.1 Penstock Branches A penstock often delivers water to more than one turbine. In such cases, the penstock is branched in various ways to subdivide the flow.Exhibit 8.5 illustrates a typical steel penstock branch structure. When the powerhouse is normal to the penstock, several configurations are possible (Fig. 8.11). If the powerhouse is at an angle with the penstock, a manifold is used (Fig. 8.12). Head losses in branches and manifolds depend on precise geometry and often are developed by model studies. However, for a typical well-designed layout, the following head loss coefficients can be used to estimate the head loss hb from the main into a branch: V2 hb Kb 2g
(8.17)
where V = branch velocity (m/s or ft/s); g = gravitational acceleration (m/s2 or ft/sec2); and Kb = head loss coefficient 0.2 for symmetrical bifurcation, 0.3 for symmetrical trifurcation, and 0.2 for manifold branch.
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HYDRAULIC DESIGN FOR ENERGY GENERATION
Hydraulic Design for Energy Generation 8.25
FIGURE 8.11A Example penstock branch configurations for powerhouse normal to the penstock.
The diameters of branched penstocks are usually determined so that the velocity is increased significantly relative to the main penstock. Here again, the branch size is determined by economics so that construction and material costs added to cost of energy loss are at a minimum. The lower limit for the size of the branch is the size of the turbine inlet that is normally provided by the turbine manufacturer. If a turbine inlet valve is provided, its diameter will either be equal to the inlet diameter or be between the inlet diameter and the penstock branch diameter. This valve is usually a spherical type, and, as such, no head loss occurs in the fully open position. Friction losses in the branch penstocks are calculated using the same friction factors used for the main penstock and the conduit lengths up to the net head taps in the turbine inlet.
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HYDRAULIC DESIGN FOR ENERGY GENERATION
8.26
Chapter Eight
FIGURE 8.11B Configurations for single bifurcated, double y-branching, and trifurcated penstocks. (Harza Engineering Co.).
8.7 DRAFT-TUBE EXITS Draft-tubes are designed by considering the turbine’s characteristics. The net head for the turbine is based on pressure taps at the spiral-case inlet and near the draft-tube exit. Therefore, any head losses which occur after the draft-tube pressure tap are subtracted from the turbine net head. Because the exit head loss is generally considered to be the average velocity head at the end of the draft-tube, a longer draft-tube with expansion to a larger area would, in theory, reduce this loss. In actuality, however the flow is not uniform
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HYDRAULIC DESIGN FOR ENERGY GENERATION
Hydraulic Design for Energy Generation 8.27
FIGURE 8.12A Examples penstock manifold configurations for a powerhouse oriented at an angle with the penstock.
FIGURE 8.12B Penstock manifold for an installation with six units. (Harza Engineering Co.)
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HYDRAULIC DESIGN FOR ENERGY GENERATION
8.28
Chapter Eight
at this point; it is highly turbulent and swirling, and the true exit loss is difficult to define. Current thinking is that further extension of the draft-tube is not economical. The rule of thumb is to end the draft-tube when the mean velocity is about 2m/s and to base the exit head loss on this velocity. A trashrack is usually provided at the end of the draft-tube at a pumped-storage project to prevent entry of coarse debris during the pumping mode. However, during the generating mode with the trashrack in place, the trashrack is subject to vibration caused by the concentration of flow and by swirling. This complicates the design of the trashrack and increases its cost. The analysis of the rack is a combined hydraulic and structural one. The hydraulic loadings consist of drag forces on rack bars that are dependent on velocity patterns along with pulsation of pressure caused by swirling flow. The data on hydraulic conditions can be obtained from a physical model (usually the model from the pump-turbine manufacturer) because fully developed mathematical models are not readily available to predict these forces. A structural mathematical model is then applied using the hydraulic loadings obtained from the hydraulic model tests. By trial and error, the trashrack is designed to withstand the flow-induced vibrations.
8.8 TAIL-TUNNELS An underground power plant will have a tail-tunnel to deliver the flow to the downstream river or lake. For a pumped-storage project, this tunnel provides flow both ways, because it acts as the inlet tunnel during pumping. For a conventional hydroelectric plant with generating only, the tunnel is usually pressurized.. However, if the turbines are the Pelton type, the tunnel is likely to be free flow to maintain freeboard on the turbine. For a pumped-storage plant, the tunnel is most likely to be pressurized, because it must deliver water both ways. If the tunnel is pressurized and is long enough, a surge chamber will be required to prevent large fluctuations of pressure on the turbines during load changes. The number of tail-tunnels, usually one or two, is based on economics and constructability. From an operational standpoint, two tunnels are desirable to allow partial operation of the plant even during maintenance or inspection of one of the tunnels. However, two tunnels are usually more expensive than one, and usually only one will be used unless its size becomes unmanageable. The limiting size is dictated by available equipment and tunneling methods. These factors must be evaluated carefully when estimating the costs of one tunnel versus two tunnels. A manifold is used to collect the flow from the individual draft-tubes and guide the flow through a transition section to the tail-tunnel proper. This manifold is similar in concept to the penstock manifold, but generally the velocities are much lower. The velocity at the end of the draft-tube is typically 2 m/s (7 ft/s) and 3 m/s (10 ft/s) at the tail-tunnel. Therefore, head losses are not significant and the flow conditions are generally acceptable. A typical tail-tunnel manifold design is shown on Fig. 8.13.
8.8.1 Tail-Tunnel Surge Tanks When an underground power plant has a significant length of pressurized tail-tunnel, a surge tank is likely to be required. The procedures for sizing and determining extreme surges are similar to the procedures used for surges in the head-tunnel, using the hydraulic characteristics of the tail-tunnel instead of the head-tunnel. (Refer to Sec. 8.5). Figure 8.14 shows a typical tail-tunnel surge chamber.
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HYDRAULIC DESIGN FOR ENERGY GENERATION
Hydraulic Design for Energy Generation 8.29
FIGURE 8.13 A typical tail-tunnel manifold arrangement. (Harza Engineering Co.)
8.8.2 Tail-Tunnel Outlet Structures The tail-tunnel outlet structure is typically a bulkhead structure, which might incorporate some flow spreading for energy recovery. The spreading of the flow is an economic decision based on construction costs and the value of energy loss. Figure 8.15 shows a typical structure of a tail-tunnel outlet. If the project is the pumped-storage type, the outlet structure will incorporate trashracks at the face of the structure, and the velocity at the trashracks will be approximately 1.0 m/s (3.3 ft/s), because the racks tend to be self-cleaning during the generating mode.
8.9 TAILRACE CHANNELS If the outlet structure is a significant distance from the receiving waterway, a tailrace channel will be required (Fig. 8.16). The sizing of the channel will be similar to that of the headrace channel. (Refer to Sec. 82).
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HYDRAULIC DESIGN FOR ENERGY GENERATION
Chapter Eight
FIGURE 8.14 A typical tail-tunnel surge chamber. (Harza Engineering Co.)
8.30
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HYDRAULIC DESIGN FOR ENERGY GENERATION
Hydraulic Design for Energy Generation 8.31
FIGURE 8.15 A typical tail-tunnel outlet structure. (Harza Engineering Co.)
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HYDRAULIC DESIGN FOR ENERGY GENERATION
8.32
Chapter Eight
FIGURE 8.16 Tailrace channels of the Guri Project. (EDELCA, Venezuela)
REFERENCES American Society of Civil Engineer (ASCE), Civil Engineering Guidelines for Planning and Designing Hydroelectric Developments: Vol. 2 Waterways, American Society of Civil Engineers, New York, 1989. American Society of Civil Engineer (ASCE), Steel Penstock, ASCE Manuals and Reports on Engineering Practice No. 79, American Society of Civil Engineers, New York, 1993. Brater, E. F., King, H. W., J. E. Lindell, and C. Y. Wei, Handbook of Hydraulics, 7th ed., McGraw-Hill, New York, 1996. Chaudhry, M. H., Applied Hydraulic Transients, 2nd ed., Van Nostrand Reinhold, New York, 1987. Chow, V. T., Open-Channel Hydraulics, McGraw-Hill, New York, 1959. Dingman, S. L., Fluvial Hydrology, W. H. Freeman, New York, 1984. Gulliver, J. S., and R. E. A. Arndt, Hydropower Engineering Handbook, McGraw-Hill, New York, 1991. Henderson, F. M., Open Channel Flow, Macmillan, New York, 1966. Moffat, A. I. B., C. Nalluri, and R. Narayanan, Hydraulic Structures, Unwin Hyman, London, UK, 1990. Parmakian, J., Waterhammer Analysis, Dover Publications, New York, 1955. Rich, G. R., Hydraulic Transients, Dover Publications, New York, 1951. U. S. Army Corps of Engineer (USACE), Hydraulic Design Criteria, U.S. Army Corps of Engineers, Waterways Experiment Station, Vicksburg, MS, 1988. U.S. Bureau of Reclamation, Selecting Hydraulic Reaction Turbines, Engineering Monograph No.20, Department of the Interior, 1980.
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HYDRAULIC DESIGN FOR ENERGY GENERATION
Hydraulic Design for Energy Generation 8.33 U. S. Bureau of Reclamation, Design of Small Dams, U.S. Department of the Interior, Denver, Co, 1987. U. S. Bureau of Reclamation Welded Steel Penstocks, Engineering Monograph No.3, U.S. Department of the Interior, Denver, Co, 1967. Vanoni, V. A., ed., Sedimentation Engineering, American Society of Civil Engineers, New York 1977. Warnick, C. C., H. A. Mayo Jr., J. L. Carson, and L. H. Sheldon, Hydropower Engineering, Prentice-Hall, NJ, 1984. Wylie, E. B., and V. L. Streeter, Fluid Transients in Systems, Prentice-Hall, Englewood Cliffs, NJ, 1993. Zipparro, V. J., and H. Hasen, Davis' Handbook of Applied Hydraulics, 4th ed., McGraw-Hill, New York, 1993.
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Source: HYDRAULIC DESIGN HANDBOOK
CHAPTER 9
HYDRAULICS OF WATER DISTRIBUTION SYSTEMS Kevin Lansey Department of Civil Engineering and Engineering Mechanics University of Arizona Tucson, Arizona
Larry W. Mays Department of Civil and Enviromental Engineering Arizona State University Tempe, Arizona
9.1 INTRODUCTION In developed countries, water service is generally assumed to be reliable and utility customers expect high-quality service. Design and operation of water systems require an understanding of the flow in complex systems and the associated energy losses. This chapter builds on the fundamental flow relationships described in Chap. 2 by applying them to water distribution systems. Flow in series and parallel pipes is presented first and is followed by the analysis of pipe networks containing multiple loops. Water-quality modeling is also presented. Because solving the flow equations by hand for systems beyond a simple network is not practical, computer models are used. Application of these models is also discussed.
9.1.1 Configuration and Components of Water Distribution Systems A water distribution system consists of three major components: pumps, distribution storage, and distribution piping network. Most systems require pumps to supply lift to overcome elevation differences and energy losses due to friction. Pump selection and analysis is presented in Chap. 10. Storage tanks are included in systems for emergency supply or for balancing storage to reduce energy costs. Pipes may contain flow-control devices, such as regulating or pressure-reducing valves. A schematic of a distribution system is shown in Fig. 9.1. The purpose of a distribution system is to supply the system’s users with the amount of water demanded under adequate pressure for various loading conditions. A loading condition is a spatial pattern of demands that defines the users’ flow requirements. The flow rate in individual pipes results from the loading condition and is one variable that describes the networks hydraulic condition. The piezometric and pressure heads are other descriptive variables. The piezometric or hydraulic head is the surface of the hydraulic grade line or the pressure head (p/) plus the elevation head (z): 9.1 Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
HYDRAULICS OF WATER DISTRIBUTION SYSTEMS
9.2
Chapter Nine
Tank 2 Reservoir 9
10 Pump 9
11 10
12 11
111
13 12
112
113 22
21 21 121
31
22
23
122
32
FIGURE 9.1 Network schematic (from EPANET User’s Menual, Rossman, 1994)
p h z
(9.1)
Because the velocity is relatively small compared to the pressure in these systems, the velocity head typically is neglected. Heads are usually computed at junction nodes. A junction node is a connection of two or more pipes or a withdrawal point from the network. A fixed-grade node (FGN) is a node for which the total energy is known, such as a tank. The loading condition may remain constant or vary over time. A distribution system is in steady state when a constant loading condition is applied and the system state (the flow in all pipes and pressure head at all nodes) does not vary in time. Unsteady conditions, on the other hand, are more common and hold when the system’s state varies with time. Extended-period simulation (EPS) considers time variation in tank elevation conditions or demands in discrete time periods. However, within each time period, the flow within the network is assumed to be in steady state. The only variables in the network that are carried between time steps of an EPS are the tank conditions that are updated by a conservation of mass relationship. Dynamic modeling refers to unsteady flow conditions that may vary at a point and between points from instant to instant. Transient analysis is used to evaluate rapidly varying changes in flow, such as a fast valve closure or switching on a pump. Gradually varied conditions assume that a pipe is rigid and that changes in flow occur instantaneously along a pipe so that the velocity along a pipe is uniform but may change in time. Steady, extended period simulation, and gradually temporally varied conditions are discussed in this chapter. Transient analysis is described in Chap. 12.
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HYDRAULICS OF WATER DISTRIBUTION SYSTEMS
Hydraulics of Water Distribution Systems 9.3
9.1.2
Conservation Equations for Pipe Systems
The governing laws for flow in pipe systems under steady conditions are conservation of mass and energy. The law of conservation of mass states that the rate of storage in a system is equal to the difference between the inflow and outflow to the system. In pressurized water distribution networks, no storage can occur within the pipe network, although tank storage may change over time. Therefore, in a pipe, another component, or a junction node, the inflow and outflow must balance. For a junction node, Qin Qout qext
(9.2)
where Qin and Qout are the pipe flow rates into and out of the node and qext is the external demand or supply. Conservation of energy states that the difference in energy between two points is equal to the frictional and minor losses and the energy added to the flow in components between these points. An energy balance can be written for paths between the two end points of a single pipe, between two FGNs through a series of pipes, valves and pumps, or around a loop that begins and ends at the same point. In a general form for any path,
∑h
L,i
iIp
∑ hp, j ∆E
(9.3)
jJp
where hL,i is the head loss across component i along the path, hp, j is the head added by pump j, and E is the difference in energy between the end points of the path. Signs are applied to each term in Eq. (9.3) to account for the direction of flow. A common convention is to determine flow directions relative to moving clockwise around the loop. A pipe or another element of energy loss with flow in the clockwise direction would be positive in Eq. (9.3), and flows in the counterclockwise direction are given a negative sign. A pump with flow in the clockwise direction would have a negative sign in Eq. (9.3), whereas counterclockwise flow in a pump would be given a positive sign. See the Hardy Cross method in Sec. 9.2.3.1 for an example.
9.1.3 Network Components The primary network component is a pipe. Pipe flow (Q) and energy loss caused by friction (hL) in individual pipes can be represented by a number of equations, including the Darcy-Weisbach and Hazen-Williams equations that are discussed and compared in Sec. 2.4.2. The general relationship is of the form hL KQn
(9.4)
where K is a pipe coefficient that depends on the pipes diameter, length, and material and n is an exponent in the range of 2. K is a constant in turbulent flow that is commonly assumed to occur in distribution systems. In addition to pipes, general distribution systems can contain pumps, control valves, and regulating valves. Pumps add head hp to flow. As shown in Fig. 9.2, the amount of pump head decreases with increasing discharge. Common equations for approximating a pump curve are hp AQ2 BQ hc
(9.5)
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HYDRAULICS OF WATER DISTRIBUTION SYSTEMS
9.4
Chapter Nine 200
Pump head
150 Pump curve Pump
100 Horsepower curve
50
0 0
1
2
3
4
Flow rate FIGURE 9.2 Typical pump curve
or hp hc CQm
(9.6)
where A, B, C, and m are coefficients and hc is the maximum or cutoff head. A pump curve can also be approximated by the pump horsepower relationship (Fig. 9.2) of the form γQh Hp p 550
(9.7)
where Hp is the pump’s water horsepower. Further details about pumps and pump selection are discussed in Chap. 10. Valves and other fittings also appear within pipe networks. Most often, the head loss in these components is related to the square of the velocity by 2 Q2 hm Kv V Kv 2 2g A 2g
(9.8)
where hm is the head loss, and Kv is an empirical coefficient. Table 2.2 lists Kv values for a number of appurtenances. Pressure-regulating valves (PRVs) are included in many pipe systems to avoid excessive pressure in networks covering varying topography or to isolate pressure zones for reliability and maintaining pressures. Pressure regulators maintain a constant pressure at the downstream side of the valve by throttling flow. Mathematical representation of PRVs may be discontinuous, given that no flow can pass under certain conditions.
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HYDRAULICS OF WATER DISTRIBUTION SYSTEMS
Hydraulics of Water Distribution Systems 9.5 A
Pipe 1 D 10’’, C 100, L 100’
Pipe 2 D 8’’, C 120, L 200’
b
a
Pipe 3 D 6’’, C 80, L 50’
d
c B
Pipe 1 L 120m; D 25 cm; C 100 Pipe 2 L 100m; D 40; C 80
Q 0.2 m3/s
B
A Pipe 3 L 150m; D 30 cm; C 120
FIGURE 9.3 Pipe systems. A: Series pipe system (not to scale) B: Branched pipe
9.2 STEADY–STATE HYDRAULIC ANALYSIS 9.2.1 Series and Parallel Pipe Systems The simplest layouts of multiple pipes are series and parallel configurations (Fig. 9.3). To simplify analysis, these pipes can be converted to an equivalent single pipe, that have the same relationship between head loss and flow rate as the original complex configuration. Series systems, as shown in the Fig. 9.3A, may consist of varying pipe sizes or types. However, because no withdrawals occur along the pipe, the discharge through each pipe is the same. Since the pipes are different, head losses vary between each segment. The total head loss from a to b is the sum of the head losses in individual pipes, hL =
冘 冘 hL,i =
iIp
KiQni
(9.9)
iIp
where Ip are the set of pipes in the series of pipes. Assuming turbulent flow conditions and a common equation, with the same ni for all pipes, a single equivalent pipe relationship can be substituted: hL KeQn (9.10) where Ke is the pipe coefficient for the equivalent pipe. Ke can be determined by combining Eqs. (9.9) and (9.10): Ke = K1 K2 K3 ...
冘
Ki
(9.11)
iIp
Note that no assumption was made regarding Q, so Ke is independent of the flow rate.
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HYDRAULICS OF WATER DISTRIBUTION SYSTEMS
9.6
Chapter Nine
Problem. For the three pipes in series in Fig. 9.3, (1) find the equivalent pipe coefficient, (2) calculate what the discharge in the pipes is if the total head loss is 10 ft, and (3) determine what the piezometric head is at points b, c, and d if the total energy at the inlet (pt. a) is 95 ft? Solution. For English units, the K coefficient for the Hazen-Williams equation is φL K C1.85 D4.87
(9.12)
where φ is a unit constant equal to 4.73, L and D are in feet, and C is the Hazen-Williams coefficient. Substituting the appropriate values gives K1 0.229, K2 0.970, and K3 2.085. The equivalent Ke is the sum of the individual pipes (Eq. 9.11), or Ke 3.284. Using the equivalent loss coefficient, the flow rate can be found by Eq. 9.10, or hL KeQ1.85. For hL equals 10 feet and Ke equals 3.284, the discharge is 1.83 cfs. This relationship and Ke can be used for any flow rate and head loss. Thus, if the flow rate was 2.2 cfs, the head loss by Eq. 9.10 would be hL 3.284 (2.2)1.85 14.1 ft. The energy at a point in the series pipes can be determined by using a path head-loss equation of the form Eq. (9.3). The total energy at Point. b is the total energy at the source minus the head loss in the first pipe segment, or Hb Ha hL,I 95 K1Q1.85 95 0.229(1.83)1.85 94.3 ft Similarly, the head losses in the second and third pipes are 2.97 and 6.38 ft., respectively. Thus, the energy at B and C are 91.33 and 84.95 ft, respectively. Two or more parallel pipes (Fig. 9.3B) can also be reduced to an equivalent pipe with a similar Ke. If the pipes are not identical in size, material, and length, the flow through each will be different. The energy loss in each pipe, however, must be the same because they have common end points, or hA hB hL,1 hL,2 hL,j
(9.13)
Since flow must be conserved, the flow rate in the upstream and downstream pipes must be equal to the sum of the flow in the parallel pipes, or Q Q1 Q2 ...
冘
Qm
(9.14)
mMp
where pipe m is in the set of parallel pipes, Mp. Manipulating the flow equation (Eq. 9.4), the flow in an individual pipe can be written in terms of the discharge by Q (hL /K)1/n Substituting this in Eq. (9.14) gives h ,1 1/n1 h ,2 1/n2 h ,3 1/n3 Q L L L … K1 K2 K3
(9.15)
As is noted in Eq. (9.13), the head loss in each parallel pipe is the same. If the same n is assumed for all pipes, Eq. (9.15) can be simplified to 1/n 1/n 1/n Q hL1/n 1 1 1 ... hL1/n K1 K2 K3
冘
1 1/n K mMp i
1/n hL1/n 1 (9.16) Ke
Dividing by hL1/n isolates the follwing equivalent coefficient: Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
HYDRAULICS OF WATER DISTRIBUTION SYSTEMS
Hydraulics of Water Distribution Systems 9.7
1/n
1 冱 K mM p
i
1/n 1 Ke
(9.17)
Because the K values are known for each pipe based on their physical properties, Ke can be computed, then substituted in Eq. (9.10) to determine the head loss across the parallel pipes, given the flow in the main pipe. Problem. Determine the head loss between points A and B for the three parallel pipes. The total system flow is 0.2 m3/s. Also find the flow in each pipe. The head-loss coefficient K for each pipe is computed by Eq. (9.12), with equal to 10.66 for SI units and L and D in meters, or K1 218.3, K2 27.9, and K3 80.1. The equivalent Ke is found from Eq. 9.17: 1
1.85 1 K1
1
1
1
1.85 1.85 1.85 1 1 1 0.313 K2 K3 Ke
or Ke 8.58. By Eq. (9.10), the head loss is hL KeQ1.85 8.58*(0.2)1.85 0.437 m. The flow in each pipe can be computed using the individual pipe’s flow equation and K. For example, Q1 (hL/K1)1/1.85 (0.437/218.3)1/1.85 0.035 m3/s. Similarly, Q2 and Q3 are 0.105 and 0.060 m3/s, respectively. Note that the sum of the flows is 0.2 m3/s, which satisfies conservation of mass.
9.2.2 Branching Pipe Systems The third basic pipe configuration consists of branched pipes connected at a single junction node. As shown in Fig. 9.4, a common layout is three branching pipes. Under steady conditions, the governing relationship for this system is conservation of mass applied at the junction. Since no water is stored in the pipes, the flow at the junction must balance Reservoir 1 H 100 m Reservoir 2 H 60 m Pipe 1 Pipe 2
Junction with pressure, P Pipe 3 Reservoir 3 H 40 m
FIGURE 9.4 Branched pipe system.
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HYDRAULICS OF WATER DISTRIBUTION SYSTEMS
9.8
Chapter Nine
Q1 Q2 Q3 0
(9.18)
where the sign on the terms will come from the direction of flow to or from the node. In addition to satisfying continuity at the junction, the total head at the junction is unique. Given all the pipe characteristics for each system in Fig. 9.4, the seven possible unknowns are the total energy at each source (3), the pipe flows (3), and the junction node’s total head P (1). Four equations relating these variables are available: conservation of mass (Eq. 9.18) and the three energy loss equations. Thus, three of the seven variables must be known. Two general problems can be posed. First, if a source energy, the flow from that source and one other flow or source energy is known, all other unknowns can be solved directly. For example, if the flow and source head for reservoir 1 and pipe 1 are known, the pipe flow equation can be used to find P by the following equation (when flow is toward the junction): Hs1 P hL,1 K1Q1n
(9.19)
If a flow is the final known (e.g., Q2), Q3 can be computed using Eq. (9.18). The source energies can then be computed using the pipe flow equations for Pipes 2 and 3, in the form of Eq. (9.19), with the computed P. If the final known is a source head, the discharge in the connecting pipe can be computed using the pipe equation in the form of Eq. (9.19). The steps in the previous paragraph are then repeated for the last pipe. In all other cases when P is unknown, all unknowns can be determined after P is computed. P is found most easily by writing Eq. 9.18 in terms of the source heads. From Eq. (9.19), |Hs1 P| 1/n Q1 sign(Hs1 P) K1
(9.20)
where a positive sign indicates flow to the node. Substituting Eq. (9.20) for each pipe in Eq. (9.18) gives 1 1 |Hs1 P| n |Hs2 P| n F(P) sign(Hs1 P) sign(Hs2 P) K1 K2 1 n (9.21) |Hs3 P| sign(Hs3 P) 0 K3 If a pipe’s flow rate is known, rather than the source head, the flow equation is not substituted; instead the actual flow value is substituted in Eq. (9.21). The only unknown in this equation is P and it can be solved by trial and error or by a nonlinear equation solution scheme, such as the Newton-Raphson method. The Newton-Raphson method searches for roots of an equation F(x). At an initial estimate of x, a Taylor series expansion is used to approximate F: ∂F | ∆x ∂2F | ∆x2 …. 0 = F(x) 2 x ∂x x ∂x
(9.22)
where x is the change in x required to satisfy F(x). Truncating the expansion at the firstorder term gives a linear equation for ∆x: F(x) x ∂F/∂x|x
(9.23)
The estimated x is updated by x x ∆x. Since the higher order terms were dropped from Eq. (9.22), the correction may not provide an exact solution to F(x). So several iterDownloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
HYDRAULICS OF WATER DISTRIBUTION SYSTEMS
Hydraulics of Water Distribution Systems 9.9
ations may be necessary to move to the solution. Thus, if ∆x is less than a defined criteria, the solution has been found and the process ends. If not, the updated x is used in Eq. (9.23) and another ∆x is computed. In the three reservoir case, xP and the required gradient ∂F/∂P is:
冤
( ∂F 1 |Hs1 P| ∂P
n
K1
1 n
)
1
|Hs2 P| (
K2
1 n
)
1
|Hs3 P| (
K3
1 n
冥
)
1
1 1 1 n1 n1 n1 nK2|Q2| nK3|Q3| nK1|Q1|
(9.24)
F(P) is computed from Eq. (9.21) using the present estimate of P. ∆P is then computed using P F(P)/(∂F/∂P), and P is updated by adding ∆P to the previous P. The iterations continue until ∆P is less than a defined value. The Newton-Raphson method also can be used for multiple equations, such as the nodal equations (Sec. 9.2.3.3). A matrix is formed of the derivatives of each equation and the update vector is calculated. Problem. Determine the flow rates in each pipe for the three-pipe system shown in Fig. 9.4. The friction factors in the table below assume turbulent flow conditions through a concrete pipe (ε 0.08 cm). Solution. Using the Darcy-Weisbach equation (n 2), the K coefficients are computed using 8fL K
2D5g
Pipe D (cm) L (m) f[]
1 80
2 40
3 40
1000
600
700
0.0195
0.0235
0.0235
K
4.9
113.8
132.7
Reservoir elevation H (m)
100
60
40
Iteration 1 In addition to the three discharges, the energy at the junction P also is unknown. To begin using the Newton-Raphson method, an initial estimate of P is assumed to be 80 m, and Eq. (9.21) is evaluated as follows:
1
1
|100 80| 2 |60 80| 2 F(P 80m) sign(100 80) sign(60 80) 4.9 pipe1 113.8 pipe2
|40 80| 2 (40 80) 2.020 0.419 0.549 1.052 m3/s 132.7 pipe3 1
which states that flow enters the node at more than 1.052 m3/s, than leaves through pipes 2 and 3 with P 80 m. Therefore, P must be increased. The correction is computed by Eq. (9.23) after computing ∂F/∂P using Eq. (9.24): Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
HYDRAULICS OF WATER DISTRIBUTION SYSTEMS
9.10
Chapter Nine
1 1 1 ∂F (21) (21) 2*132.7*|0.549| (21) 2*113.8*|0.419| ∂P 2*4.9*|2.020| (0.0505 0.0105 0.0069) 0.0679 The correction is then F(P) 1.052 P 15.5 m. ∂F 0.0679 冫P ∂P The P for the next iteration is then P 80 15.5 95.5 m. The following iterations give Iteration 2: F(P 95.5 m) 0.247; ∂F冫∂P冨P 95.5 0.120; ∆P 2.06 m, P 93.44 m Iteration 3:F(P 93.44 m) 0.020; ∂F冫∂P冨P 93.44 0.102; ∆P 0.20 m, P 93.24 m. Iteration 4: F(P 93.24 m) 7.x104; ∂F冫∂P冨P 93.24 0.101; ∆P 0.006 m, P 93.25 m. Stop based on F(P) or P, with P 93.25 m. Problem. In the same system, the desired flow in pipe 3 is 0.4 m3/s into the tank. What are the flows in the other pipes and the total energy required in Tank 3? Solution. First, P is determined with Q1 and Q2 using Eq. 9.21. Then Hs3 can be calculated by the pipe flow equation. Since Q3 is known, Eq. (9.21) is 1
1
|H P| n |H P| n s1 s2 F(P) Q1 Q2 Q3 sign(Hs1 P) sign(Hs2 P) 0.4 0 K1 K2
Iteration 1 Using an initial trial of P equal to 90 m, F(P) 0.514 m3/s. When evaluating Eq. (9.24), only the first two terms appear since the flow in pipe 3 is defined, or
冢
冣冨
1 ∂F 1 = ∂P nK1|Q1| nK2|Q2|
P = 90m
0.080
The correction for the first iteration is then – (0.514/0.080) 6.42 m, and the new P is 96.42 m. The next two iterations are Iteration 2: F(P 96.42 m) 0.112; ∂F冫∂P冨P 96.42m 0.127; P 0.88 m, P 95.54 m Iteration 3: F(P 95.54 m) 0.006; ∂F冫∂P冨P 95.54m 0.115; P 0.05 m, P 95.49 m To determine Hs3, the pipe flow equation (Eq. 9.20) is used with the known discharge, or |Hs3 95.49| 1/2 Q3 0.4 sign(Hs3 95.49) ⇒ Hs3 74.26 m 132.7
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HYDRAULICS OF WATER DISTRIBUTION SYSTEMS
Hydraulics of Water Distribution Systems 9.11
9.2.3 Pipe Networks A hydraulic model is useful for examining the impact of design and operation decisions. Simple systems, such as those discussed in Secs. 9.2.1 and 9.2.2, can be solved using a hand calculator. However, more complex systems, require more effort even for steady state conditions, but, as in simple systems, the flow and pressure-head distribution through a water distribution system must satisfy the laws of conservation of mass and energy (Eqs. 9.2 and 9.3). These relationships have been written in different ways to solve for different sets of unknowns. Using the energy loss-gain relationships for the different components, the conservation equations can be written in three forms: the node, loop, and pipe equations. All are nonlinear and require iterative solution schemes. The form of the equations and their common solution methods are described in the next four sections. Programs that implement these solutions are known as network solvers or simulators and are discussed in Sec. 9.5. 9.2.3.1 Hardy Cross method. The Hardy Cross method was developed in 1936 by Cross before the advent of computers. Therefore, the method is amenable to solution by hand but, as a result, is not computationally efficient for large systems. Essentially, the method is an application of Newton’s method to the loop equations. Loop equations. The loop equations express conservation of mass and energy in terms of the pipe flows. Mass must be conserved at a node, as discussed in Sec. 9.2.2 for branched pipes. For all Nj junction nodes in a network, it can be written as
冘
Qi qext
(9.25)
iIj
Conservation of energy (Eq. 9.3) can be written for closed loops that begin and end at the same point (∆E 0) and include pipes and pumps as
冘
KiQin
iIL
冘
(AipQ2ip BipQip Cip) 0
(9.26)
ipIp
This relationship is written for Nl independent closed loops. Because loops can be nested in the system, the smallest loops, known as primary loops, are identified, and each pipe may appear twice in the set of loops at most. The network in Fig. 9.1 contains 3 primary loops. Energy also must be conserved between points of known energy (fixed-grade nodes). If Nf FGNs appear in a network, Nf 1 independent equations can be written in the form of
冘 iIL
KiQni
冘
(AipQ2ip BipQip Cip) ∆EFGN
(9.27)
ipIp
where EFGN is the difference in energy between the two FGNs. This set of equations is solved by the Hardy Cross method (Cross, 1936) by successive corrections to the pipe flows in loops and by the linear theory method by solving for the pipe flows directly (Sec. 9.2.3.2). Solution method. To begin the Hardy Cross method, a set of pipe flows is assumed that satisfies conservation of mass at each node. At each step of the process, a correction ∆QL is determined for each loop. The corrections are developed so that they maintain conservation of mass (Eq. 9.25), given the initial set of flows. Since continuity will be preserved, those relationships are not included in the next steps. The method then focuses on determining pipe flows that satisfy conservation of energy. When the initial flows are substituted in Eqs. (9.26) and (9.27), the equations are not
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HYDRAULICS OF WATER DISTRIBUTION SYSTEMS
9.12
Chapter Nine
likely to be satisfied. To move toward satisfaction, a correction factor ∆QL is determined for each loop by adding this term to the loop equation or for a general loop
冘
Ki(Qi ∆QL)n
iIL
冘
(Aip(Qip QL)2 Bip(Qip ∆QL) Cip) ∆E (9.28)
ipIp
Note that ∆E equals zero for a closed loop and signs on terms are added as described in Sec. 9.1.3. Expanding Eq. 9.28 and assuming that ∆QL is small so that higher order terms can be dropped gives
冘
冘冨 冘冨
KiQ ni n
iIL
KiQn1
iIL
∆QL冨
冘
(AipQ2ip BipQip Cip)
ipIp
(2AipQip∆QL Bip∆QL)冨 ∆E
(9.29)
ipIp
Given Qi,k the flow estimates at iteration k, Eq. (9.29) can be solved for the correction for loop L as
) (冘 冘 冨 冘 冨 冨 冘冨 ∆QL
KiQ i,k
iIL
(AipQ2ip,k BipQip,k Cip) ∆E
ipIp
KiQ n1i,k
n
iIL
(2AipQip,k Bip)
(9.30)
ipIp
In this form, the numerator of Eq. (9.30) is the excess head loss in the loop and should equal zero by conservation of energy. The terms are summed to account for the flow direction and component. The denominator is summed arithmetically without concern for direction. Most texts present networks with only closed loops and no pumps. Equation (9.30) simplifies to this case by dropping the pump terms and setting ∆E to zero, or
冘 冘冨
iIL
KiQni,k
冘
hL,i
iIL
F(Qk)
/
∆QL ∂F n1 n Ki Qi,k 冨 n 冨hL,i /Qi,k冨 ∂Q 冨Qi,k iIL
冘
(9.31)
iIL
Comparing Eq. (9.31) with Eq. (9.23) shows that the Hardy Cross correction is essentially a Newton’s method. The ∆QL corrections can be computed for each loop in sequence and can be applied before moving to the next loop (Jeppson, 1974) or corrections for all loops can be determined and applied simultaneously. Once the correction has been computed, the estimates for the next iteration are computed by Qi,k1 Qi,k ∆QL
(9.32)
Qk1 is then used in the next iteration. The process of determining corrections and updating flows continues until the ∆QL for each loop is less than some defined value. After the flows are computed, to determine the nodal heads, head losses or gains are computed along a path from fixed-grade nodes to junction nodes.
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HYDRAULICS OF WATER DISTRIBUTION SYSTEMS
Hydraulics of Water Distribution Systems 9.13
The Hardy Cross method provides an understanding of principles and a tool for solving small networks by hand. However, it is not efficient for large networks compared with algorithms presented in the following sections. Problem. List the loop equations for the network shown in Fig. 9.5 using the direction of flow shown. Then determine the flow in each pipe and the total energy at Nodes 4 and 5. Solution The loop equations consist of conservation of mass at the five junction nodes and the loop equations for the two primary loops and one pseudo-loop. In the mass balance equations, inflow to a node is positive and outflow is negative. Node 1.
Q1 Q2 Q5 0
Node 2.
Q2 Q3 Q6 2
Node 3.
Q3 Q4 Q7 0
Node 4.
Q5 Q8 1
Node 5. Loop I.
Q6 Q7 Q8 2 hL,2 hL,6 hL,8 hL,5 0 K2Q22 K6Q26 K8Q28 K5Q25
Loop II.
hL,7 hL,6 hL,3 0 K7Q27 K6Q26 K3Q23
Pseudo-loop.
hL,4 hp hL,3 hL,2 hL,1 EFGN,2 EFGN,1 0 K4Q24 (ApQ24 BpQ4 Cp) K3Q23 K2Q22 K1Q21 1 EFGN,2 EFGN,1 n
Because the Darcy-Weisbach equation is used, n equals 2. The loop equations assume that flow in the clockwise direction is positive. Flow in Pipe 5 is moving counterclockwise and is given a negative sign for loop I. Flow in pipe 6 is moving clockwise relative to loop I (positive sign) and counter clockwise relative to Loop II (negative sign). Although flow is moving clockwise through the pump in the pseudo-loop, hp is given a negative sign because it adds energy to flow. To satisfy conservation of mass, the initial set of flows given below is assumed, where the values of K for the Darcy-Weisbach equation are given by fL 8fL KDW 2 A Dg π2D5g
(9.33)
The concrete pipes are 1 ft in diameter and have a friction factor of 0.032 for turbulent flow. Pipe 1
2
3
4
5
6
7
8
K
1.611
2.417
2.417
1.611
3.222
3.222
4.028
2.417
Q
2.5
1.0
1.5
2.5
1.5
0.5
1.0
0.5
Also, Ap 6, Bp 0, and Cp 135’. Iterahtion 1. To compute the correction for the pseudo-loop, the numerator of Eq. (9.30) is K4Q24 (ApQ24 Cp) K3Q23 K2Q22 K1Q21 EFGN,2 EFGN,1 = 1.611(2.5)2 (6(2.5)2 135) 2.417(1.5)2 2.417(1.0)2 1.611(2.5)2 (10 100) 4.48
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HYDRAULICS OF WATER DISTRIBUTION SYSTEMS
Chapter Nine
5 L 3000’ 4
5
8
L 3000’
2 2
1 cfs
L 4000’
1 1 L 2000’
EPGN,1 100’
FIGURE 9.5 Example network (Note all pipes have diameters of 1 ft and friction factors equal to 0.032).
2 cfs
L 5000’
6
L 4000’
2 cfs
3
L 3000’
7
3
4
L 2000’
hp 135 6Q2
EFGN ;2 10’
9.14
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HYDRAULICS OF WATER DISTRIBUTION SYSTEMS
Hydraulics of Water Distribution Systems 9.15
The denominator is nK4Q4 2ApQ4 nK3Q3 nK2Q2 nK1Q1 2 (1.611(2.5))|2(-6)2.5| 2(2.417(1.5)) 2 冨2.417(1.0)冨2 冨1.611(2.5)冨 58.20. Thus, the correction for the pseudo-loop QPL is (4.48) QPL 0.077 58.20 The correction for Loop I is computed next. The numerator of Eq. (9.30) is K2Q22 K6Q26 K8Q28 K5Q25 2.417(1.0)2 3.222(0.5)2 –2.417(0.5)2 – 3.222(1.5)2 4.63 and the denominator is nK2Q2 nK6Q6 nK8Q8 2(2.417(1.0)) 2(3.222(0.5))2(2.417(0.5))2(3.222(1.5)) 20.14 Thus, the correction for Loop 1, ∆QI is (4 .63) 0.230 QI 20.14 Finally to adjust loop II from the numerator of Eq. (9.30) is K7Q27 K6Q26 K3Q23 4.028(1.0)2 3.222(0.5)2 2.417(1.5)2 2.22 and the denominator is nK7Q27 nK6Q26 nK3Q23 2(4.028(1.0)) 2(3.222(0.5)) 2(2.417(1.5)) 18.53 Thus, the correction for the loop II, ∆QII, is (2.22) ∆QII 0.120 18.53 The pipe flows are updated for Iteration 2 as follows: Pipe
1
2
3
4 and pump
5
6
7
8
∆Q
0.077
0.230
0.077
0.077
0.230
0.230
0.120
0.230
(0.077)
0.120
1.15
1.46
1.12
0.27
Q
2.42
0.120 2.58
1.27
0.61
Because the flow direction for Pipe 1 is counterclockwise relative to the pseudo-loop, the correction is given a negative sign. Similarly, Pipe 2 receives a negative correction for the pseudo-loop. Pipe 2 is also in Loop I and is adjusted with a positive correction for that loop since flow in the pipe is in the clockwise direction for Loop I. Pipes 3 and 6 also appear in two loops and receive two corrections.
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HYDRAULICS OF WATER DISTRIBUTION SYSTEMS
9.16
Chapter Nine
Iteration 2. The adjustment for the pseudo–loop is K4Q24 (ApQ24 Cp) K3Q23 K2Q22 K1Q21 (EFGN,2 EFGN,1) ∆QpL nK4Q4 2ApQ4 nK3Q3 nK2Q2 nK1Q1 1.611(2.58)2 (6(2.58)2 135) 2.417(1.46)2 2.417(1.15)2 1.611(2.42)2 10 100 2(1.611(2.58)) 2ⱍ 6(2.58)ⱍ 2(2.417(1.46)) 2(2.417(1.15)) 2(1.611(2.42)) 1.82 0.030 59.69
In the correction for loop I, the numerator of eq. (9.30) is K2Q22 K6Q26 K8Q28 K5Q25 2.417(1.15)2 3.222(0.61)2 2.417(0.27)2 3.222(1.27)2 0.978 and the denominator is nK2Q2 nK6Q6 nK8Q8 nK5Q5 2 (2.417(1.15)) 2 (3.222(0.61)) 2 (2.417(0.27)) 2(3.222(1.27)) 18.98 Thus the correction for loop, ∆QI is ( 0.978) 0.052 ∆QI 18.98 Finally, to correct loop II, the numerator of Eq. (9.30) is K7Q27 K6Q26 K3Q23 4.028(1.12)2 3.222(0.61)2 -2.417(1.46)2 1.30 and the denominator is nK7Q27 nK6Q26 nK3Q23 2*[4.028(1.12)] 2[3.222(0.61)] 2[2.417(1.46)] 20.01 Thus, the correction for the pseudo-loop, ∆QII is ( 1.30) 0.065 ∆QII 20.01 The pipe flows are updated for iteration 3 as follows: Pipe Q Q
1 0.030 2.39
2
3
4 and pump
5
6
7
8
0.030
0.052
0.052
0.065
0.052
1.18
0.22
0.052
0.030
(0.030)
0.065
1.17
1.43
0.065 2.61
1.22
0.60
Iteration 3. The corrections for iteration 3 are 0.012, 0.024, and 0.021 for the pseudo-loop, loop I and loop II, respectively. The resulting flows are as follows:
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HYDRAULICS OF WATER DISTRIBUTION SYSTEMS
Hydraulics of Water Distribution Systems 9.17 Pipe
1
Q
0.012
Q
2.38
2
3
4 and pump
5
6
7
8
0.012
0.024
0.024
0.021
0.024
1.20
0.20
0.024
0.012
(0.012)
0.024
1.18
1.42
0.024 2.62
1.20
0.60
After two more iterations, the changes become small, and the resulting pipe flows are as follows. Note that the nodal mas balance equations are satisfied at each iteration. Pipe Q
1
2
3
4 and pump
5
6
7
8
2.37
1.19
1.41
2.63
1.18
0.60
1.21
0.18
The total energy at Nodes 4 and 5 can be computed by path equations from either FGN to the nodes. For example, paths to Node 4 consist of Pipes 1 and 5 or of pipes 4 (with the pump), 7, and 8. For the path with pipes 1 and 5, the part equation is 100 K1Q 12 K5Q 52 100 1.611(2.37)2 3.222(1.19)2 100 9.05 4.56 86.39m For the path containing pipes 4, 7 and 8 the result is 10 (135 6(2.63)2) 1.611(2.63)2 4.028(1.22)2 2.417(0.19)2 10 93.50 11.14 6.00 0.09 86.45m This difference can be attributed to rounding errors. Note that pipe 8 received a positive sign in the second path equation. Because the flow in Pipe 8 is the opposite of the path direction, the energy along the path is increasing from Nodes 5 to 4. The total energy at Node 5 can be found along pipes 4 and 7 or 86.36 m or along the path of Pipes 1-2-6, giving (100 9.05 3.42 1.16 86.37m). 9.2.3.2 Linear theory method. Linear theory solves the loop equations or Q equations (Eqs. 9.25 to 9.27). Np equations (Nj Nl Nf –1) can be written in terms of the Np unknown pipe flows. Since these equations are nonlinear in terms of Q, an iterative procedure is applied to solve for the flows. Linear theory, as described in Wood and Charles (1972), linearizes the energy equations (Eqs. 9.26 and 9.27) about Qi,k1, where the subscript k1 denotes the current iteration number using the previous iterations Qi,k as known values. Considering only pipes in this derivation, these equations are
冘
冘
Qi,k1 qext for all Nj nodes
(9.34)
iIj
n1 Q KiQi,k i,k1 0
for all Nl closed loops
(9.35)
iIL
and
冘
n1 Q KiQi,k for all Nf 1 independent pseudoloops i,k1 ∆EFGN
(9.36)
iIL
These equations form a set of linear equations that can be solved for the values of Qi,k1. The absolute differences between successive flow estimates are computed and compared to a convergence criterion. If the differences are significant, the counter k is updated and the process is repeated for another iteration. Because of oscillations in the flows around the final solution, Wood and Charles (1972) recommended that the average of the flows from the previous two iterations should be used as the estimate for the
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HYDRAULICS OF WATER DISTRIBUTION SYSTEMS
9.18
Chapter Nine
next iterations. Once the pipe flows have been determined, the nodal piezometric heads can be determined by following a path from a FGN and accounting for losses or gains to all nodes. Modified linear theory Newton method. Wood (1980) and his collaborators at the University of Kentucky developed the KYPIPE program but essentially modified the original linear theory to a Newton’s method. However, rather than solve for the change in discharge (∆Q), Qk1 is determined. To form the equations, the energy equations (Eq. 9.3) are written in terms of the current estimate of Qk, including pipes, minor losses and pumps, as f(Qk)
冘 iIL
KiQnk
冘
imIm
Ki Q2km
冘
(AipQ2k BipQk Cip) E
(9.37)
ipIp
where for simplicity the subscripts i, im, and ip denoting the pipe, minor loss component, and pump, respectively, are dropped from the flow terms and k again denotes the iteration counter. This equation applies to both closed loops (∆E 0) and pseudo-loops (∆E ∆EFGN), but, in either case, f(Qk) should equal zero at the correct solution. To move toward the solution, the equations are linearized using a truncated Taylor series expansion: ∂f f(Qk1) f(Qk) Q (Qk1 Qk) f(Qk) Gk(Qk1 Qk) (9.38) ∂Q k Note that f and Q are now vectors of the energy equations and pipe flow rates, respectively, and Gk is the matrix of gradients that are evaluated at Qk. Setting Eq. (9.38) to zero and solving for Qk1 gives
0 f(Qk) Gk(Qk1 Qk) or
GkQk1 GkQk f(Qk)
(9.39)
This set of (Nl Nf –1) equations can be combined with the Nj junction equations in Eq. (9.34) that also are written in terms of Qk1 to form a set of Np equations. This set of linear equations is solved for the vector of Np flow rates using a matrix procedure. The values of Qk1 are compared with those from the previous iteration. If the largest absolute difference is below a defined tolerance, the process stops. If not, Eq. (9.39) is formed using Qk1 and another iteration is completed. 9.2.3.3 Newton-Raphson method and the node equations. The node equations are the conservation of mass relationships written in terms of the unknown nodal piezometric heads. This formulation was described in Sec. 9.2.2 for branching pipe system. In Fig. 9.4, if P and the pipe flows are unknown, the system is essentially a network with one junction node with three FGNs. In a general network, Nj junction equations can be written in terms of the Nj nodal piezometric heads. Once the heads are known, the pipe flows can be computed from the pipe’s head-loss equation. Other network components, such as valves and pumps, are included by adding junction nodes at each end of the component. Node equations are then written using the flow relationship for the component. Solution method. Martin and Peters (1963) were the first to publish an algorithm using the Newton-Raphson method for solving the node equations for a pipe network. Shamir and Howard (1968) showed that pumps and valves could be incorporated and unknowns other than nodal heads could be deternined by the method. Other articles have been published that attempt to exploit the sparse matrix structure of this problem.
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HYDRAULICS OF WATER DISTRIBUTION SYSTEMS
Hydraulics of Water Distribution Systems 9.19
At Iteration k, the Newton-Raphson method is applied to the set of junction equations F(hk) for the nodal heads hk. After expanding the equations and truncating higher order terms, the result is ∂F F(hk) ∆hk 0 (9.40) ∂h hk where F is the set of node equations evaluated at hk, the vector of nodal head estimates at iteration k. ∂F/∂h is the Jacobian matrix of the gradients of the node equations with respect to the nodal heads. This matrix is square and sparse because each nodal head appears in only two nodal balance equations. The unknown corrections ∆hk can be determined by solving the set of linear equations: ∂F F(hk) ∆hk (9.41) ∂h hk The nodal heads are then updated by:
hk1 hk ∆hk
(9.42)
As in previous methods, the magnitude of the change in nodal heads is examined to determine whether the procedure should end. If the heads have not converged, Eq. (9.41) is reformulated with hk1 and another correction vector is computed. If the final solution has been found, the flow rates are then computed using the component relationships with the known heads. As in all formulations, at least one FGN must be hydraulically connected to all nodes in the system. Some convergence problems have been reported if poor initial guesses are made for the nodal heads. However, the node equations result in the smallest number of unknowns and equations of all formulations. Problem. Write the node equations for the system in Fig. 9.5. Node 1: 1
1
1
|100 h | n |h h | n |h4 h1| n 1 2 1 sign(100 h1) sign(h2 h1) sign(h4 h1) 0 K5 K1 K2
Node 2 (note that the right-hand side is equal to the external demand): 1
sign(h1
|h1 h2| n h2) K2
1
sign(h3
|h h | n 3 2 h2) K3
1
sign(h5
|h5 h2| n h2) K6
2
Node 3: 1
1
1
|h2 h3| n |h5 h3| n |h h | n pd 3 sign(h2 h3) 0 sign(h5 h3) sign(hpd h3) K3 K7 K4
Node 4: 1
1
|h1 h4| n |h4 h5| n sign(h1 h4) sign(h5 h4) 1 K5 K8
Node 5: 1
1
1
|h h | n |h4 h5| n |h5 h3| n 2 5 sign(h2 h5) sign(h4 h5) sign(h3 h5) 2 K8 K7 K6
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HYDRAULICS OF WATER DISTRIBUTION SYSTEMS
9.20
Chapter Nine
New node for the pump: 1
1
|h3 hpd| n (h 10) 135 2 pd sign(h3 hpd) 0 K4 6
The first term in the pump node equation is the outflow from the pump toward Node 3 in Pipe 4. The second term is the discharge relationship for the pump, written in terms of the total energy at the outlet of the pump hpd. Because the pump relationship is different from that for Pipe 4, this new node with zero demand was added at the outlet of the pump (assuming that the pump inlet is the tank). This type of node must be added for every component (valve, pipe, or pump); therefore, one must know the precise location of the component. For example, if a valve, appears within a pipe, to be exact in system representation, new nodes would be added on each side of the valve, and the pipe would be divided into sections upstream and downstream of the valve. In summary, six equations can be written for the system to determine six unknowns (the total energy for Nodes 1 to 5 and for the pump node). Using the solution from the Hardy Cross method gives the following nodal heads, the values of which can be confirmed to satisfy the node equations: Node Total head (m)
1
2
3
4
5
Pump
90.95
87.54
92.35
86.45
86.38
103.50
Pipe
1
2
3
4
5
6
7
8
Pump
Pipe flow (m3/s)
2.37
1.19
1.41
2.63
1.18
0.60
1.22
0.18
2.63
9.2.3.4 Gradient algorithm Pipe equations. Unlike the node and loop equations, the pipe equations are solved for Q and h simultaneously. Although this requires a larger set of equations to be solved, the gradient algorithm by Todini and Pilati (1987) has been shown to be robust to the extent that this method is used in EPANET (see Sec. 9.5.3). To form the pipe equations, conservation of energy is written for each network component in the system in terms of the nodal heads. For example, a pipe equation is ha hb KQn
(9.43)
and, using a quadratic approximation, a pump equation is hb ha AQ2 BQ C
(9.44)
where ha and hb are the nodal heads at the upstream and downstream ends of the component. These equations are combined with the nodal balance relationships (Eq. 9.2) to form Nj Np equations with an equal number of unknowns (nodal heads and pipe flows). Solution method. Although conservation of mass at a node is linear, the component flow equations are nonlinear. Therefore, an iterative solution scheme, known as the gradient algorithm, is used. Here the component flow equations are linearized using the previous flow estimates Qk. For pipes, KQ n1 Qk1 (ha hb) 0 k
(9.45)
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HYDRAULICS OF WATER DISTRIBUTION SYSTEMS
Hydraulics of Water Distribution Systems 9.21
In matrix form, the linearized equations are
and
A12h A11Q A10h0 0
(9.46)
A21Q qext 0
(9.47)
where Eq. 9.46 is the linearized flow equations for each network component and Eq. 9.47 is the nodal flow balance equations. A12 ( A21T) is the incidence matrix of zeros and ones that identify the nodes connected to a particular component and A10 identifies the fixed grade nodes. A11 is a diagonal matrix containing the linearization coefficients (e.g.,|KQkn1 |). Differentiating eqs. 9.46 and 9.47 gives: NA11 A21
A12 dQ dE 0 dh dq
(9.48)
where dE and dq are the residuals of equations 9.2 and 9.43-44 evaluated at the present solution, Qk and hk. N is a diagonal matrix of the exponents of the pipe equation (n). Eq. 9.48 is a set of linear equations in terms of dQ and dh. Once solved 1Q and h are updated by and
Qk1 Qk dQ
(9.49)
hk1 hk dh
(9.50)
Convergence is checked by evaluating dE and dq and additional iterations are completed as necessary. Todini and Pilati (1987) applied an alternative efficient recursive scheme for solving for Qk1 and hk1. The result is 1 A )1{A N1(Q A1 A H )(q A Q )} hk1 (A21N1A11 11 12 12 k 10 0 ext 21 k
(9.51)
then using hk1, Qk1 by is determined: Qk1 (1N1)Qk N1 A111 (A12Hk1 A10H0)
(9.52)
where A11 is computed at Qk. Note that N and A11 are diagonal matrices so the effort for inversion is negligible. Yet, one full matrix must be inverted in this scheme. Problem. Write the pipe equations for the network in Fig. 9.5. Solution. The pipe equations include mass balance equations for each node in the system. The network contains five junction nodes plus an additional node downstream of the pump. The pump is considered to be a link and is assumed to be located directly after the FGN. Conservation of energy equations are written for each pipe and pump link. Eight pipe equations and one pump equation are written. The total number of equations is then 15, which equals the 15 unknowns, including 8 pipe flows, 1 pump flow, and 6 junction node heads, including the additional nodal head at the pump outlet hp. Node 1:
Q1 Q2 Q5 0
Pipe 1:
100 h1 K1Qn1
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HYDRAULICS OF WATER DISTRIBUTION SYSTEMS
9.22
Chapter Nine
Node 2:
Q2 Q3 Q6 2
Pipe 2:
h1 h2 K2Qn2
Node 3:
Q3 Q4 Q7 0
Pipe 3:
h3 h2 K3Q3
Node 4:
Q5 Q8 1
Pipe 4:
hp h3 K4Qn4
Node 5:
Q6 Q7 Q8 2
Pipe 5:
h1 h4 K5Qn5
Pump Node:
Qp Q4 0
Pipe 6:
h2 h5 K6Qn6
Pump:
hp 10 135 6Q2p
Pipe 7:
h3 h5 K7Qn7
Pipe 8:
h4 h5 K8Qn8
n
9.2.3.5 Comparison of solution methods. All four methods are capable of solving the flow relationships in a system. The loop equations by the Hardy Cross method are inefficient compared with the other methods and are dropped from further discussion. The Newton-Raphson method is capable of solving all four formulations, but because the node equations result in the fewest equations, they are likely to take the least amount of per iteration. In applications to the node equations, however, possible convergence problems may result if poor initial conditions are selected (Jeppson, 1974). Linear theory is reportedly best for the loop equations and should not be used for the node or loop equations with the Q corrections, as used in Hardy Cross (Jeppson, 1974). Linear theory does not require initialization of flows and, according to Wood and Charles (1972), always converges quickly. A comparative study of the Newton-Raphson method and the linear theory methods was reported by Holloway (1985). The Newton-Raphson scheme was programmed in two codes and compared with KYPIPE that implemented the linear theory. For a 200-pipe network, the three methods converged in eight or nine iterations, with the Newton-Raphson method requiring the least amount of computation time. Salgado, Todini, and O’Connell (1987) compared the three methods for simulating a network under different levels of demand and different system configurations. Four conditions were analyzed and are summarized in Table 9.1. Example A contains 66 pipes and 41 nodes but no pumps. Example B is similar to Example A, but 6 pumps are introduced and a branched connection has been added. Example C is the same network as in Example B with higher consumptions, whereas Example D has the same network layout but the valves are closed in two pipes. Closing these pipes breaks the network into two systems. The results demonstrate that all methods can simulate the conditions, but the gradient method for solving the pipe equations worked best for the conditions analyzed. All comparisons and applications in this chapter are made on the basis of assuming reasonably sized networks. Given the speed and memory available in desktop computers, it is likely that any method is acceptable for these networks. To solve extremely large systems with several thousand pipes, alternative or tailored methods are necessary. Discussion of these approaches is beyond the scope of this chapter. However, numerical simulation of these systems will become possible, as discussed in Sec. 9.5 on network calibration, but good representation of the system with accurate parameters may be difficult. 9.2.3.6 Extended-period simulation. As noted earlier, time variation can be considered in network modeling. The simplest approach is extended-period simulation, in which a sequence of steady-state simulations are solved using one of the methods described earlier in this section. After each simulation period, the tank levels are updated and demand and operational changes are introduced.
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HYDRAULICS OF WATER DISTRIBUTION SYSTEMS
Hydraulics of Water Distribution Systems 9.23 TABLE 9.1 Example
Comparison of solution methods Special conditions Node equations
Solution method: Loop equations
Pipe equations
A
Low velocities
Converged Iterations 16, T 70 s
Converged Iterations 17, T 789 s
Converged Iterations 16, T 30 s
B
Pumps and branched network
Converged Iterations 12, T 92 s
Slow convergence Iterations 13, T 962 s
Converged Iterations 10, T 34 s
C
Example B with high demand
Converged Iterations 13, T 100 s
Slow convergence Iterations 15, T 1110 s
Converged Iterations 12, T 39 s
D
Closed pipes
Converged Iterations 21, T 155 s Some heads not available
Converged Iterations 21, T 1552 s Some heads not available
Converged Iterations 19, T 57 s
Source: Modified from Todini and Pilati (1987).
Tank levels or water-surface elevations are used as known energy nodes. The levels change as flow enters or leaves the tank. The change in water height for tanks with constant geometry is the change in volume divided by the area of the tank, or V Q ∆t ∆HT T T (9.53) AT AT where QT and VT are the flow rate and volume of flow that entered the tank during the period, respectively; ∆t is the time increment of the simulation; AT is the tank area; and ∆HT is the change in elevation of the water surface during period T. More complex relationships are needed for noncylindrical tanks. With the updated tank levels, the extended-period simulation continues with these levels as known energy nodes for the next time step. The process continues until all time steps are evaluated. More complex unsteady analysis are described in the next section.
9.3 UNSTEADY FLOW IN PIPE NETWORK ANALYSIS In steady state analysis or within an extended-period simulation, changes in the distributions of pressure and flow are assumed to occur instantaneously after a change in external stimulus is applied. Steady conditions are then reached immediately. In some cases, the time to reach steady state and the changes during this transition may be important. Recently, work has proceeded to model rapid and gradual changes in flow conditions. Rapid changes resulting in transients under elastic column theory are discussed in Chap. 12. Two modeling approaches for gradually varied unsteady flow under a rigid-column assumption are described in this section.
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HYDRAULICS OF WATER DISTRIBUTION SYSTEMS
9.24
Chapter Nine
9.3.1 Governing Equations In addition to conservation of mass, the governing equations for unsteady flow under rigid pipe assumptions are developed from conservation of momentum for an element (Fig. 9.6). Conservation of momentum states that the sum of the forces acting on the volume of fluid equals the time rate of change of momentum, or
冘
d(mv) F F1 F2 Ff (9.54) dt a where F1 and F2 are the forces on the ends of the pipe element, Ff is the force caused by friction between the water and the pipe, and m and v are the mass and velocity of the fluid in the pipe element. The end forces are equal to the force of the pressure plus the equivalent force caused by gravity or for the left-hand side of the element: p F1 γA1 z1 γAh1 γ
(9.55)
The friction force is the energy loss times the volume of fluid, or Ff γAhL.
(9.56)
The change of momentum can be expanded to γALv d g d(mv) d(ρVv) γL d(Av) γL dQ (9.57) dt dt dt g dt g dt where the mass is equal ρV AL, in which all terms are constants with respect to time g and can be taken out of the differential. Note that under the rigid-water-column assumption, the density is a constant as opposed to elastic-water-column theory. Substituting these terms in the momentum balance gives
EGL
p2 z2 h2 γ
p1 z1 h1 γ
Ff F1
Q
F2
FIGURE 9.6 Force balance on a pipe element
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HYDRAULICS OF WATER DISTRIBUTION SYSTEMS
Hydraulics of Water Distribution Systems 9.25
γL dQ γA(h1 h2 hL) (9.58) g dt Assuming that a steady state friction loss relationship can be substituted for hL and dividing each side by A, L dQ h1 h2 KQn (9.59) gA dt With conservation of mass (Eq. 9.2), this ordinary differential equation and its extensions for loops have been used to solve for time-varying flow conditions.
9.3.2 Solution Methods 9.3.2.1 Loop formulation. Holloway (1985) and Chaudhry and Islam (1994) extended the momentum equation (Eq. 9.59) to loops as follows:
冘
冘
(h1i h2i)
iIp
冘
KiQni
iIp
iIp
Li dQi gAi dt
(9.60)
Separating variables and integrating over time gives
冕 冘 t+∆t
t
iI
p
冕 冘 t+∆t
(h1i h2i)dt
t
iI
p
KiQnidt =
冕 冘 Qt+∆t
Qt
iIp
Li dQi gAi
(9.61)
At any instant in time, the head loss around a closed loop must equal zero, so the first term can be dropped. Dropping this term also eliminates the nodal piezometric heads as unknowns and leaves only the pipe flows. One of several approximations for the friction loss term can be used: KQt∆t|Qt|n1∆t
(9.62)
K[(Qt∆t Qt)|Qt∆t Qt|n1/2n]∆t
(9.63)
K[(Qt∆t |Qt∆t|n1 Qt|Qt|n1) / 2n]∆t
(9.64)
Holloway (1985) obtained results using Eq. (9.62), known as the integration approximation that compared favorably with the other two nonlinear forms. Using this form in Eq. (9.61),
冘 iIp
Li Qit gAi
冘 iIp
t n1 ∆t KiQt∆t i |Qi |
冘 iIp
Li Qt∆t gAi i
(9.65)
This equation is written for each loop and is used with the nodal conservation of mass equations to given Np equations for the Np unknown pipe flows. Note that these equations are linear in terms of Qt∆t can be solved at each time step in sequence using the previous time step for the values in the constant terms. 9.3.2.2 Pipe formulation with gradient algorithm. An alternative solution method developed by Ahmed and Lansey (1999) used the momentum equation for a single pipe (Eq. 9.59) and the nodal flow balance equations to form a set of equations similar to those developed in the gradient algorithm. An explicit backward difference is used to solve the equations. The rigt–hand side of Eq. 9.59 is written in finite difference form as.
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HYDRAULICS OF WATER DISTRIBUTION SYSTEMS
9.26
Chapter Nine
Li dQ Li (Qt∆ti Qti ) i gAi dt gAi t
(9.66)
The left–hand side of Eq. 9.9 is written in terms on the unknowns h and Q at time step t t. After substituting and rearranging a general algebraic equation for pipe between two nodes results in Ki|Qit |n1
Li t∆t Li t t∆t ∆t Q [ht∆t 1i h2i ] ∆t Qi gAi i gAi
(9.67)
Np equations of this form can be written for each pipe or other component. With the Nj nodal flow balance equations, a total of Nj Np equations can be written in terms of an equal number of unknown pipe flows and nodal heads. Given an initial condition at time t, the pipe flows and nodal heads at time t t by solving Eq. 9.67 and Eq. 9.2 The new values are then used for the next time step until all times have been evaluated. Unlike the loop formulation, in the form above, Eq. 9.67 is nonlinear with respect to the unknowns. In addition, like the loop equation, the time step will influence the accuracy of the results.
9.4 WATER-QUALITY MODELING Interest in water quality in distribution systems heightened with the passing of the 1986 amendment to the Safe Drinking Water Act. This amendment required that standards must be developed for chlorine levels not only at the point of disinfection but also at the most distant point of withdrawal. Thus, modeling the fate and transport of dissolved substances in networks with emphasis on chlorine became necessary. As a result, methods of analysis and computer programs implementing these methods, such as EPANET (Rossman, 1994), have been developed. Since the velocity in pipes is relatively high, constituents in the water are assumed to move completely with the flow that is, by advective transport. This assumption allows the use of explicit numerical modeling schemes to solve for constituent movement within the system. As in hydraulic analysis, steady and unsteady transport models have been developed. Both models use conservation of mass as the basic governing equation describing mixing and movement. Because advective transport dominates, the pipe flow rates are critical in estimating transport in the system. In most unsteady water-quality models, extended-period simulation has been used to account for demand and operational changes (Sec. 9.2.3.6) Although water quality analysis considering slow transients using rigidwater-column theory for the flow analysis has been performed by Chaudhry and Islam (1998), it will not be discussed here. As water moves through the network, constituent (with emphasis on chlorine) decay is generally assumed to follow first-order kinetics, or ct c0ektt
(9.68)
where c0 and ct are the constituent concentrations at times 0 and t, respectively, t is time and kt is the first-order decay coefficient, which is defined by k kf kt kb w (9.69) RH(kw kf) where RH is the hydraulic radius of the pipe, and kb, kw, and kf are the bulk flow-decay constant, the wall reaction rate constant, and a mass transfer coefficient that is dependent on the Reynold’s number, respectively.
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HYDRAULICS OF WATER DISTRIBUTION SYSTEMS
Hydraulics of Water Distribution Systems 9.27
9.4.1 Steady State Modeling Given a steady flow distribution, the contribution from different sources or the concentration of a constituent at withdrawal nodes can be determined by solving a set of linear algebraic equations. Under the assumption that complete mixing occurs at a junction node, the general conservation of mass equation under these conditions states that the mass of constituent entering the junction equals the mass leaving the junction, or
冘
QjCIj QSCS=
jIk
冘
QiCo
(9.70)
iIe
where CIj is the constituent concentration in incoming pipe j, Co is the concentration in all outgoing pipes, and Cs is the constituent concentration in the incoming source water. Qj is the volumetric flow rate in incoming pipe j and Qs is the external-source flow rate. Qi is the outgoing flow from the node in pipe i. If the junction is a demand node, the external demand is included in set Ie. Given steady flow, the total inflow must equal the total outflow. Substituting the flow balance and solving for the concentration in all flows leaving the node, Co gives
冘 jIk
QjCIj QsCs
冘
Co Qi
(9.71)
iIe
One constituent mass balance equation can be written for each node. Since the flow rates are defined by the hydraulic relationships, Cs is known, and the CI for one node is the outflow from another node, the system of equations can be solved for the Nj unknown Co’s. A steady-state model provides the concentrations at all points in the network under steady flow and concentrations. By modeling each source concentration independently in a series of simulations, the model also can be used to determine the relative source contribution at any point under the same conditions.
9.4.2 Dynamic Analysis Steady flow conditions for water quality provide information regarding movement of dissolved substances but are likely to be less useful for predicting point concentrations under normal operations. Unsteady analysis, also known as dynamic modeling, provides a more realistic picture and better estimates of constituent movement under time-varying flow conditions. Dynamic modeling can solve several types of problems. In addition to determining the variation in concentration at a point over time, it can be used to determine the age of or average travel time for water at some location and time. Finally, as with steady models, the relative source contributions providing flow to a point can be computed. 9.4.2.1 Governing equations. To determine the fate and transport of dissolved substances under unsteady conditions, the primary governing equation is the one-dimensional advection equation that is solved in conjunction with the assumption of complete mixing at a node. The advection equation is
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HYDRAULICS OF WATER DISTRIBUTION SYSTEMS
9.28
Chapter Nine
∂Ci ∂Ci (9.72) ui R(Ci) ∂t ∂x where Ci is the constituent concentration in pipe i at location x and time t; ui is the velocity in pipe i, and R(Ci) is the reaction/decay function. The decay relationship for firstorder kinetics R ktc is used when modeling chlorine and possibly other nonconservative substances. For conservative substances, such as fluoride, the reaction relationship is simply zero. Finally, when modeling water age, R is equal to one and the concentration C is interpreted as the water age with new water entering the system having concentration equal to zero. Tanks act as sources or sinks in the system with variable water quality, depending on the history of inflow and outflow as well as on the reactions in the tank. The simplest water-quality relationship for a tank assumes that the water is mixed completely. In this case, the variation in constituent concentration is
冘
冘
∂(VTCT) QiCEi QjCT R(CT) = ∂t iI jO T
(9.73)
T
where VT and CT are the storage volume and constituent concentration within the tank at time t, respectively. Pipes in the set of IT provide inflows Qi to the tank, and pipes in the set OT receive flows Qj from the tank. CE is the concentration at the exit of the pipe as it enters the tank. R is the reaction relationship for water in the tank. 9.4.2.2 Solution methods Eulerian methods. Rossman and Boulos (1996) compared the different solution methods for solving the unsteady water-quality problem. This section generally follows their notation and terminology. Dynamic models can be classified spatially as Eulerian or Lagrangian models and temporally as time driven or event driven. Eulerian methods define a grid of either points or volume segments within a pipe. Flow and the associated constituents are tracked through this fixed grid. Chaudhry and Islam (1998) used a finite-difference method with a fixed-point grid, and Grayman et al. (1988), and its extension by Rossman et al. (1993), have developed the discrete-volume method (DVM). The following discussion focuses on the DVM as it has been implemented in the EPANET model (Rossman, 1994). For a given hydraulic condition, the DVM divides each pipe into equally sized, completely mixed, volume segments. The number of segments for a particular pipe is computed by Li t ni ti (9.74) ui∆t ∆t where Li and ui are the length of and flow velocity in pipe i, respectively; tti is the travel time for water to pass through pipe i; and t is the duration of the water quality time step. A small t provides the highest numerical accuracy at the expense of higher computation times. When the flow conditions change in the network (i.e., u changes), the grid must be redefined. At each water-quality step, four operations are completed, as shown in Fig. 9.7. First, the present constituent masses are reduced to account for the decay reactions. Next, the elements from each segment are advanced to the next downstream segment. Third, if the segment is the most downstream in a pipe, the flow is mixed with the flow from other pipes that enter the node using Eq. (9.71). Finally, the flow from the node is passed to the first segments of pipes leaving the node. These operations are repeated for each water-quality time step until the flow distribution changes. Pipes are then resegmented, and the process is repeated for that hydraulic condi-
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HYDRAULICS OF WATER DISTRIBUTION SYSTEMS
Hydraulics of Water Distribution Systems 9.29 Original mass
After reaction
Transport to downstream node
Transport along link
Transport out of node
FIGURE 9.7 Computational steps of discrete volume method (From Rossman and Boulos (1996))
tion. When the pipes are divided for different flow conditions, the number of segments may be different and some numerical blending occurs. As a result, the accuracy of DVM and finite difference methods depends on the selection of the water-quality time step ∆t. Lagrangian methods. Unlike Eulerian methods which use a fixed grid, Lagrangian methods track segments of water as they move through a network. As the front or leading edge of the segment reaches a node, it is combined with other incoming segments. The segments leaving the node are developed with constituent levels determined by Eq. (9.71) (Fig. 9.8). Two approaches have been used to define when segments are combined and transported through a pipe. Liou and Kroon (1987) applied this type of model using a defined time step to determine when to combine segments. During each time step, the total mass of constituent and volume of water that reaches a node is computed. The average nodal concentration is computed, and new segments emanating from the node are introduced. To avoid adding too many new segments, they are created only when the concentration difference between the new and the previous segment in a link is above a threshold. When more than one segment in a link reaches a downstream node in one time step, artificial mixing will occur. Rather than combine segments at defined time intervals, the second Lagrangian approach is an event-driven method (Boulos et al., 1994, 1995; Hart et al., 1987; El-Shorbagy and Lansey, 1994; and Shah and Sinai, 1985). Event methods combine segments each time a front reaches a node, thus avoiding artificial mixing. Since defined times are not used, the projected times when a front reaches a downstream node are comDownloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
HYDRAULICS OF WATER DISTRIBUTION SYSTEMS
9.30
Chapter Nine
3
2 1 6
5
2.5
4
A
4
3 1 2 5
6
3
4
2.5
B
4
3 9
6
5
4 3.5
3
2.5
C FIGURE 9.8 Water quality transport for the Lagrangian methods for a conservative substance at three different times. The flowrates in the two inflow pipes are equal and the flowrate in the outgoing pipe is then twice the flow in either inflow pipe. A: water quality at time t: flow is to the left, and the constituent level equals the average of the inflow concentrations, or (4+1)/2 = 2.5 B: The water quality at time t + t some time after the front concentration 2 in the vertical pipe reached the node. For some time, the inflow concentrations were 2 and 4, or an average outflow concentration of 3, C: Water quality at some later time: Two elements have developed downtream. The element with a concentration of 3.5 developed when the inflows of 13 and 4 mixed at node. The final element closest to the node with concentration 4 developed when the inflow with concentrations of 3 and 5 mixed at the node.
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HYDRAULICS OF WATER DISTRIBUTION SYSTEMS
Hydraulics of Water Distribution Systems 9.31
puted for the present flow condition. The water-quality conditions at nodes remain constant until the next segment front reaches a node. At that time, new segments are generated in pipes that carry flow from the node that the first front reaches. The concentration in these segments is computed by Eq. (9.71) and is recorded with the transition time. Projection times are then updated, and the process continues when the next closest front reaches a node or the hydraulic condition changes. If the flow condition changes, new projection times are computed. Event-driven models avoid numerical dispersion; however, the method can result in a large number of segments. To save computer memory, segments can be combined according to the difference in concentration between adjacent segments. Further error may result during flow reversals for reactive constituents. Comparison of methods. Rossman and Boulos (1996) conducted numerical experiments comparing the alternative methods described in the previous sections, and reached the following conclusions: 1. The numerical accuracy of all methods is similar, except that the Eulerian methods had occasional problems. All methods can represent observed behavior adequately in real systems. 2. Network size is not always an indicator of solution time and computer memory requirements. 3. Lagrangian methods are more efficient in both time and memory requirements than Eulerian methods when modeling chemical constituents. 4. The time-dependent Lagrangian method are most efficient in computation time for modeling water age, whereas the Eulerian methods are the most memory efficient. Overall, Rossman and Boulos concluded that the time-based Lagrangian method was the most versatile unless computer memory was limiting for modeling water age for large networks. In which case, Eulerian methods were preferable.
9.5 COMPUTER MODELING OF WATER DISTRIBUTION SYSTEMS Because the numerical approaches for analyzing distribution systems cannot be completed by hand except for the smallest systems, computer-simulation models have been developed. These models solve the system of nonlinear equations for the pipe flows and nodal heads. In addition to the equation solver, many modeling packages have sophisticated input preprocessors, which range from spreadsheets to tailored full-page editors, and output postprocessors, including links with computer-aided drafting software and geographic information systems. Although these user interfaces ease the use of the simulation models, a dependable solver and proper modeling are crucial for accurate mathematical models of field systems. An array of packages is available, and the packages vary in their level of sophistication. The choice of a modeling package depends on the modeling effort. Modeling needs range from designing subdivisions with fewer than 25 pipes to modeling large water utilities that possibly involve several thousand links and nodes. Users should select the package that best suit their objectives.
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HYDRAULICS OF WATER DISTRIBUTION SYSTEMS
9.32
Chapter Nine
9.5.1 Applications of Models Clark, et al. (1988) identified a series of seven steps that are necessary to develop and apply a water distribution simulation model: 1. Model selection: Definition of modeling requirements including the model’s purpose. The desired use of a model imprtant must be understood when selecting one (hydraulic or water quality) because the necessary accuracy of the model and the level of detail required will vary, depending on its expected use. 2. Network representation: Determination of how the components of a system will be represented in the numerical model. Step 2 includes skeletonizing the piping system by not including some pipes in the model or making assumptions regarding the parameter values for pipes, such as assuming that all pipes of a certain type have the same roughness value. The degree of model simplification depends on what problems the model will be used to help address. 3. Calibration: Adjustment of nonmeasurable model parameters, with emphasis on the pipe roughness coefficients, so that predicted model results compare favorably with observed field data (see Sec. 9.5.2). This step also may require reexamination the network representation. 4. Verification: Comparison of model results with a second set of field data (beyond that used for calibration) to confirm the adequacy of the network representation and parameter estimates. 5. Problem definition: Identification of the design or operation problem and incorporation of the situation in the model (e.g., demands, pipe status or operation decisions). 6. Model application: Simulation of the problem condition. 7. Display/analysis of results: Presentation of simulation results for modeler and other decision-makers in graphic or tabular form. Results are analyzed to determine whether they are reasonable and the problem has been resolved. If the problem is not resolved, new decisions are made at step 5 and the process continues.
9.5.2 Model Calibration Calibration, step 3 above, is the process of developing a model that represents field conditions for the range of desired conditions. The time, effort, and money expended for data collection and model calibration depend on the model’s purpose. For example, a model for preliminary planning may not be extremely accurate because decisions are at the planning level and an understanding of only the major components is necessary. At the other extreme, a model used for engineering decisions regarding a system that involves pressure and water-quality concerns may reguire significant calibration efforts to provide precise predictions. All models should be calibrated before they are used in the decision-making process. The calibration process consists of data collection, model calibration, and model assessment. Data collection entails gathering field data, such as tank levels, nodal pressures, nodal elevations, pump head and discharge data, pump status and flows, pipe flows, and, when possible, localized demands. These data are collected during one or more loading conditions or over time through automated data logging. Rossman et al., (1994) discussed using water-quality data for calibration. To ensure that a calibration will be successful, the number of measurements must exceed the number of parameters to be estimated in the model. If this condition is not satisfied, multiple sets of parameters that match
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HYDRAULICS OF WATER DISTRIBUTION SYSTEMS
Hydraulics of Water Distribution Systems 9.33
the field observations can be found: that is, a unique solution may not be determined. Each set may give dramatically different results when predicting under other conditions. During model calibration, field data are compared with model estimates and model parameters are adjusted so that the model predictions match the field observations. Two stages of model calibration are desirable. The first stage is a gross study of the data and the model predictions. The intent is to insure that the data are reasonable and that major modeling assumptions are valid. For example, this level would determine if valves assumed to be open are actually closed or if an unexpectedly high withdrawal, possibly caused by leakage, is occurring. Walski (1990) discussed this level of calibration. [TITLE] EPANET Example network 1 [JUNCTIONS] Elevation ft
ID 10 11 12 13 21 22 23 31 32
[CONTROLS] LINK 9 OPEN IF NODE 2 BELOW 110 LINK 9 CLOSED IF NODE 2 ABOVE 140
Demand gpm
710 710 700 695 700 695 690 700 710
0 150 150 100 150 200 150 100 100
[PATTERNS] ID Multipliers 1 1
1.0 1.0
1.2 0.8
1.4 0.6
1.6 0.4
1.4 0.6
1.2 0.8
[QUALITY] [TANKS] ID
Elev. ft
Init. Level
Min. Level
Max. Level
Diam ft
2 9
850 800
120
100
150
50.5
ID
Head Node
Tail Length Node ft
10 11 12 21 22 31 110 111 112 113 121 122
10 11 12 21 22 31 2 11 12 13 21 22
[PIPES]
11 12 13 22 23 32 12 21 22 23 31 32
10530 5280 5280 5280 5280 5280 200 5280 5280 5280 5280 5280
Diam. Rough in. Coeff. 18 14 10 10 12 6 18 10 12 8 8 6
100 100 100 100 100 100 100 100 100 100 100 100
Initial Concen. mg/l
Nodes 2 9 2
32
0.5 1.0 1.0
[REACTIONS] GLOBAL BULK .5 GLOBAL WALL 1
; Bulk decay coeff. ; Wall decay coeff.
[TIMES[ DURATION 24 PATTERN TIME STEP
;24 hour simulation period ;2 hour pattern time period
[OPTIONS] QUALITY MAP
Chlorine Net1.map
; Chlorine analysis ; Map coordinates file
[PUMPS] ID
Head Node
Tail Node
Design ft
H–Q gpm
9
9
10
250
1500
[END]
FIGURE 9.9 EPANET input file for example network (Figure 9.1)
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HYDRAULICS OF WATER DISTRIBUTION SYSTEMS
9.34
Chapter Nine EPANET Hydraulic and Water Quality Analysis for Pipe Networks Version 1.0
EPANET Example Network 1 Input data File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .net 1. inp Verification File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hydraulics File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Map File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Net 1. map Number of Pipes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12 Number of Nodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11 Number of Tanks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2 Number of Pumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1 Number of Valves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .0 Headloss Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Hazen–Williams Hydraulic Timestep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.00 hrs Hydraulic accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .0.001000 Maximum Trials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .40 Quality Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Chlorine Minimum Travel Time . . . . . . . . . . . . . . . . . . . . . . . . . . . .6.00 min Maximum Segments per Pipe . . . . . . . . . . . . . . . . . . . . . .100 Specific Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.00 Kinematic Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.10e–005 sq ft/sec Chemical Diffusivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.3e–008 sq ft/sec Total Duration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .24.00 hrs Reporting Duration All Nodes All Links
Node Results at 0.00 hrs: Node
Elev. ft.
Demand gpm
Grade ft
Pressure psi
Chlorine mg/L
10
710.00
0.00
1004.50
127.61
0.50
11
710.00
150.00
985.31
119.29
0.50
12
700.00
150.00
970.07
117.02
0.50
13
695.00
100.00
968.86
118.66
0.50
21
700.00
150.00
971.55
117.66
0.50
22
695.00
200.00
969.07
118.75
0.50
23
690.00
150.00
968.63
120.73
0.50
31
700.00
100.00
967.35
115.84
0.50
32
710.00
100.00
965.63
110.77
0.50
2
850.00
765.06
970.00
52.00
1.00
Tank
9
800.00
1865.06
800.00
0.00
1.00
Reservoir
FIGURE 9.10 EPANET output file for example network (figure 9.1)
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HYDRAULICS OF WATER DISTRIBUTION SYSTEMS
Hydraulics of Water Distribution Systems 9.35 Link results at 0.00 hrs. Link
Start Node
End Node
Diameter in
Flow gpm
Velocity fps
10 11 12 21 22 31 110 11 112 113 121 122 9
10 11 12 21 22 31 2 11 12 13 12 22 9
11 12 13 22 23 32 12 21 22 23 31 32 10
18.00 14.00 10.00 10.00 12.00 6.00 18.00 10.00 12.00 8.00 8.00 6.00
1865.06 1233.57 129.41 190.71 120.59 40.77 765.06 481.48 189.11 29.41 140.77 59.23 1865.06
2.35 2.57 0.53 0.78 0.34 0.46 0.96 1.97 0.54 0.19 0.90 0.67 96 hp
Headloss /1000ft 1.82 2.89 0.23 0.47 0.08 0.33 0.35 2.61 0.19 0.04 0.79 0.65 204.50
Pump
Node Results at 1.00 hrs node 10 11 12 13 21 22 23 31 32 2 9
Elev. demand ft 710.00 710.00 700.00 695.00 700.00 695.00 690.00 700.00 710.00 850.00 800.00
Grade gpm 0.00 150.00 150.00 100.00 150.00 200.00 150.00 100.00 100.00 747.57 1847.57
Pressure ft
Chlorine psi
mg/l
1006.92 988.05 973.13 971.91 974.49 972.10 971.66 970.32 968.63 973.06 800.00
128.65 120.48 118.35 119.98 118.94 120.07 122.04 117.13 112.06 53.32 0.00
1.00 0.45 0.44 0.44 0.43 0.44 0.45 0.41 0.40 0.97 1.00
Flow gpm
Tank Reservoir
Link Results at 1.00 hrs. Link
Start Node
End Node
Diameter in
10 11 12 21 22 31 110 111
10 11 12 21 22 31 2 11
11 12 13 22 23 32 12 21
18.00 14.00 10.00 10.00 12.00 6.00 18.00 10.00
1847.49 1219.82 130.19 187.26 119.81 40.42 747.49 477.68
Velocity fps
Headloss /1000ft
2.33 2.54 0.53 0.76 0.34 0.46 0.94 1.95
1.79 2.83 0.23 0.45 0.08 0.32 0.34 2.57
FIGURE 9.10 (Continued)
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HYDRAULICS OF WATER DISTRIBUTION SYSTEMS
9.36
Chapter Nine
Link Results at 1.00 hrs (continued) Link
Start Node
End Node
Diameter in
112 113 121 122 9
12 13 21 22 9
22 23 31 32 10
12.00 8.00 8.00 6.00
Flow gpm 192.14 30.19 140.42 59.58 1847.49
Velocity fps 0.55 0.19 0.90 0.68 97 hp
Headloss /1000ft 0.20 0.05 0.79 0.66 206.92
Pump
Node Results at 2.00 hrs Node
Elev. ft
10 11 12 13 21 22 23 31 32 2 9
710.00 710.00 700.00 695.00 700.00 695.00 690.00 700.00 710.00 850.00 880.00
Demand gpm 0.00 180.00 180.00 120.00 180.00 240.00 180.00 120.00 120.00 516.44 1836.44
Grade ft
Pressure psi
1008.43 989.77 976.09 974.02 975.41 973.81 973.33 969.96 968.13 976.06 800.00
129.31 121.22 119.63 120.90 119.34 120.81 122.77 116.98 111.85 54.62 0.00
Chlorine mg/L 1.00 0.87 0.81 0.37 0.76 0.38 0.40 0.34 0.31 0.94 1.00
Tank Reservoir
Link results at 2.00 hrs. Link
Start Node
End Node
Diameter in
10 11 12 21 22 31 110 111 112 113 121 122 9
10 11 12 21 22 31 2 11 12 13 21 22 9
11 12 13 22 23 32 12 21 22 23 31 32 10
18.00 14.00 10.00 10.00 12.00 6.00 18.00 10.00 12.00 8.00 8.00 6.00
Flow gpm 1836.44 1163.77 173.00 150.47 127.00 42.20 516.44 492.67 294.33 53.00 162.20 77.80 1836.44
Velocity fps 2.32 2.43 0.71 0.61 0.36 0.48 0.65 2.01 0.83 0.34 1.04 0.88 97 hp
Headloss /1000ft 1.77 2.59 0.39 0.30 0.09 0.35 0.17 2.72 0.43 0.13 1.03 1.08 208.43
Pump
FIGURE 9.10 (Continued)
After the model representation is determined to be reasonable, the second stage of model calibration begins with the adjustment of individual model parameters. At this level, the two major sources of error in a model are the demands and the pipe roughness coefficients. The demands are uncertain because water consumption is largely unmonitored in the short term, is highly variable, and because the water is consumed along a pipe, whereas it is modeled as a point of withdrawal. Because pipe roughnesses vary over time and are not directly measurable, they must be inferred from field measurements. Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
HYDRAULICS OF WATER DISTRIBUTION SYSTEMS
Hydraulics of Water Distribution Systems 9.37
Adjustment of these terms and others, such as valve settings and pump lifts, can be made by trial and error or through systematic approaches. Several mathematical modeling methods have been suggested for solving the model calibration problem (Lansey and Basnet, 1991). Once a model is believed to be calibrated, an assessment should be completed. The assessment entails a sensitivity analysis of model parameters to identify which parameters have a strong impact on model predictions and future collection should emphasize improving. The assessment also will identify the predictions (nodal pressure heads or tank levels) that are sensitive to calibrated parameters and forecasted demands. Model assessments can simply be plots of model predictions versus parameter values or demand levels, or they can be more sophisticated analyses of uncertainty, as discussed in Araujo (1992) and Xu and Goulter (1998). 9.5.3 Model Results Water-distribution simulation models require the model parameters, such as pipe and pump characteristics, nodal demands, and valve settings, to solve the appropriate set of equations and display the nodal peizometric heads, pipe flow rates water quality predictions, and other results, such as pipe head loss and pipe velocities. No standard format is used between models. Abbreviated input and output files are shown in Figs. 9.9 and 9.10 for a sample system shown in Fig. 9.1. These files are for the EPANET code and are used because the EPANET program is in the public domain and it models both flow and water quality in an extended-period simulation format. The constituent, chlorine, is reactive and results are shown for a selected subgroup of nodes. As in most models, the constituent levels along a pipe are not provided. Finally, tank concentrations, although not shown directly, can be found by examining the concentration closest to the tank node when flow is exiting the tank.
REFERENCES Ahmed, I., Application of the gradient method for analysis of water networks, Master’s thesis (Civil Engineering), University of Arizona, Tucson, 1997. Araujo, J.V., A statistically based procedure for calibration of water systems, Doctoral dissertation, Oklahoma State University, Stillwater, 1992. Ahmed, I., and K. Lansey, “Analysis of unsteady flow in networks using a gradient algorithm based method,” ASCE Specialty Conference on Water Resoures, Tempe, AZ, June, 1999. Boulos, P. F., T. Altman, P. A. Jarrige, and F. Collevati, “An event-driven method for modeling contaminant propagation in water networks,” Journal of Applied Mathematical Modeling, 18(2): 8492, 1994. Boulos, P. F., Altman, T., Jarrige, P. A., and Collevati, F. “Discrete simulation approach for network water quality models,” Journal of Water Resources Planning and Management, 121(1): 49-60, 1995. Clark, R., Grayman, W. and R. M. ales, “Contaminant propagation in distribution systems” J. of Environmental Eng., 114(4): 1988. Cross, H. “Analysis of flow in networks of conduits or conductors,” Bulletin No. 286, University of Illinois Engineering Experimental Station, Urbana, IL 1936. El-Shorbagy, W., and K. Lansey, “Non-conservative water quality modeling in water systems,” Proceedings of the AWWA Specialty Conference on Computers in the Water Industry, Los Angeles, April 1994. Grayman, W. M., R.M. Clark, and R.M. Males, “Modeling Distribution System Water Quality: Dynamic Approach,” Journal of Water Resources Planning and Management, 114(3),: 295312, 1988. Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
HYDRAULICS OF WATER DISTRIBUTION SYSTEMS
9.38
Chapter Nine
Hart, F. L., J. L. Meader, and S.-M. Chiang, “CLNET—A simulation model for tracing chlorine residuals in a potable water distribution network,” AWWA Distribution System Symposium Proceedings, American Water Works Association, Denver, CO, 1987. Holloway, M. B., Dynamic Pipe Network Computer Model, Doctoral dissertation, Washington State University, Pullman, WA, 1985. Islam R., and M. H. Chaudhry, “Modeling of constituent transport in unstead flows in pipe networks,” J. of Hydraulics Division, 124 (11): 1115–1124, 1998. Jeppson, R. W., Analysis of Flow in Pipe Networks, Ann Arbor Science, Ann Arbor, MI, 1974. Lansey, K., and C. Basnet, “Parameter Estimation for Water Distribution Systems,” Journal of Water Resources Planning and Management, 117(1): 126-144, 1991. Liou, C. P., and J. R. Kroon, “Modeling the Propagation of Waterborne Substances in Distribution networks,” Journal of the American Water Works Association, 79(11): 54-58, 1987. Martin, D. W., and G. Peters, “The Application of Newton's Method to Network Analysis by Digital computer,” Journal of the Institute of Water Engineers, 17: 115-129, 1963. Rossman, L. A., “EPANET—users manual,” EPA-600/R-94/057, U.S. Environmental Protection Agency, Risk Reduction Engineering Laboratory, Cincinnati, OH, 1994. Rossman, L., and P. Boulos, “Numerical methods for modeling water quality in distribution systems: A comparison,” Journal of Water Resources Planning and Management, 122(2),: 137146, 1996. Rossman, L. A., P. F. Boulos, and T. Altman, “Discrete Volume Element Method for Network Water Quality Models,” Journal of Water Resources Planning and Management, 119(5): 505517, 1993. Rossman, L. A., R. M. Clark, and W. M. Grayman, “Modeling Chlorine Residuals in DrinkingWater Distribution Systems,” Journal of Environmental Engineering, American Society of Civil Engineers, 120(4): 303-320, 1994. Salgado, R., E. Todini, and P. E. O’Connell, “Comparison of the gradient method with some traditional methods for the analysis of water supply distribution networks,” Proceeediigs International Conference on Computer Applications for Water Supply and Distribution 1987, Leicester Polytechnic, UK, September 1987. Shah, M. and G. Sinai, “Steady State Model for Dilution in Water Networks.” Journal of Hydraulics Division, 114(2), 192-206, 1988. Shamir, U., and C. D. Howard, “Water Distribution System Analysis,” Journal of Hydraulics. Division, 94(1). 219-234, 1965. Todini, E., and S. Pilati, “A gradient method for the analysis of pipe networks,” International Conference on Computer Applications for Water Supply and Distribution 1987, Leicester Polytechnic, UK, September 1987. Walski, T., “Hardy–Cross meets Sherlock Holmes or model calibration in Austin, Texas,” Journal of the American Water Works Association, 82:34–38, March, 1990. Wood, D. J., User's Manual—Computer Analysis of Flow in Pipe Networks Including Extended Period Simulations, Department of Civil Engineering, University of Kentucky, Lexington, KY, 1980. Wood, D., and C. Charles, “Hydraulic Network Analysis Using Llinear Theory,” Journal of Hydraulic Division, 98, (HY7): 1157-1170, 1972. Xu, C., and I. Goulter, “Probabilistic Model for Water Distribution Reliability,” Journal of Water Resources Planning and Management, 124(4): 218-228, 1998.
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Source: HYDRAULIC DESIGN HANDBOOK
CHAPTER 10
PUMP SYSTEM HYDRAULIC DESIGN B. E. Bosserman Boyle Engineering Corporation Newport Beach, CA
10.1 PUMP TYPES AND DEFINITIONS 10.1.1 Pump Standards Pump types are described or defined by various organizations and their respective publications: • Hydraulics Institute (HI), American National Standard for Centrifugal Pumps for Nomenclature, Definitions, Application and Operation [American National Standards Institute (ANSI)/HI 1.1-1.5-1994] • American Petroleum Institute (API), Centrifugal Pumps for Petroleum, Heavy Duty Chemical, and Gas Industry Services, Standard 610, 8th ed., August 1995 • American Society of Mechanical Engineers (ASME), Centrifugal Pumps, Performance Test Code PTC 8.2–1990 In addition, there are several American National Standards Institute (ANSI) and American Water Works Associations (AWWA) standards and specifications pertaining to centrifugal pumps: • ANSI/ASME B73.1M-1991, Specification for Horizontal End Suction Centrifugal Pumps for Chemical Process. •
ANSI/ASME B73.2M-1991, Specification for Vertical In-Line Centrifugal Pumps for Chemical Process.
• ANSI/ASME B73.5M-1995, Specification for Thermoplastic and Thermoset Polymer Material Horizontal End Suction Centrifugal Pumps for Chemical Process. • ANSI/AWWA E 101-88, Standard for Vertical Turbine Pumps—Lineshaft and Submersible Types.
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PUMP SYSTEM HYDRAULIC DESIGN
10.2
Chapter Ten
10.1.2 Pump Definitions and Terminology Pump definitions and terminology, as given in Hydraulics Institute (HI) 1.1-1.5-1994 (Hydraulics Institute, 1994), are as follows: Definition of a centrifugal pump. A centrifugal pump is a kinetic machine converting mechanical energy into hydraulic energy through centrifugal activity. Allowable operating range. This is the flow range at the specified speeds with the impeller supplied as limited by cavitation, heating, vibration, noise, shaft deflection, fatigue, and other similar criteria. This range to be defined by the manufacturer. Atmospheric head (hatm). Local atmospheric pressure expressed in ft (m) of liquid. Capacity. The capacity of a pump is the total volume throughout per unit of time at suction conditions. It assumes no entrained gases at the stated operating conditions. Condition points • Best efficiency point (BEP). The best efficiency point (BEP) is capacity and head at which the pump efficiency is a maximum. • Normal condition point. The normal condition point applies to the point on the rating curve at which the pump will normally operate. It may be the same as the rated condition point. • Rated condition point. The rated condition applies to the capacity, head, net positive suction head, and speed of the pump, as specified by the order. • Specified condition point. The specified condition point is synonymous with rated condition point. Datum. The pump's datum is a horizontal plane that serves as the reference for head measurements taken during test. Vertical pumps are usually tested in an open pit with the suction flooded. The datum is then the eye of the first–stage impeller (Fig. 10.1). Optional tests can be performed with the pump mounted in a suction can. Regardless of the pump's mounting, its datum is maintained at the eye of the first-stage impeller. Elevation head (Z). The potential energy of the liquid caused by its elevation relative to a datum level measuring to the center of the pressure gauge or liquid surface. Friction head. Friction head is the hydraulic energy required to overcome frictional resistance of a piping system to liquid flow expressed in ft (m) of liquid. Gauge head (hg). The energy of the liquid due to its pressure as determined by a pressure gauge or other pressure measuring device. Head. Head is the expression of the energy content of the liquid referred to any arbitrary datum. It is expressed in units of energy per unit weight of liquid. The measuring unit for head is ft (m) of liquid. High-energy pump. High-energy pump refers to pumps with heads greater than 650 ft (200 m) per stage and requiring more than 300 hp (225 KW) per stage. Impeller balancing • Single–plane balancing (also called static balancing). Single–plane balancing refers to correction of residual unbalance to a specified maximum limit by removing or
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PUMP SYSTEM HYDRAULIC DESIGN
Pump System Hydraulic Design 10.3
FIGURE 10.1 Terminology for a pump with a positive suction head.
adding weight in one correction plane only. This can be accomplished statically using balance rails or by spinning. • Two–plane balancing (also called dynamic balancing). Two plane–balancing referes to correction of residual unbalance to a specified limit by removing or adding weight in two correction planes. This is accomplished by spinning on appropriate balancing machines. Overall efficiency (ηOA). This is the ratio of the energy imparted to the liquid (Pw) by the pump to the energy supplied to the (Pmot); that is, the ratio of the water horsepower to the power input to the primary driver expressed in percent. Power • Electric motor input power (Pmot). This is the electrical input power to the motor. Pump input power (Pp). This is the power delivered to the pump shaft at the driver to pump coupling. It is also called brake horsepower. Pump output power (Pw). This is the power imparted to the liquid by the pump. It is also called water horsepower. QHs (U.S. units) (10.1) Pw 3960
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PUMP SYSTEM HYDRAULIC DESIGN
10.4
Chapter Ten
QHs Pw 366
(S.I. units)
(10.2)
where Q flow in gal/min (U.S.) or m3/hr (SI) H head in feet (U.S.) or meters (SI) S specific gravity Pw power in a horsepower (U.S.) or kilowatt (SI) Pump efficiency (ηp). This is the ratio of the energy imported to the liquid (Pw) to the energy delivered to the pump shaft (Pp) expressed in percent. Pump pressures • Field test pressure. The maximum static test pressure to be used for leak testing a closed pumping system in the field if the pumps are not isolated. Generally this is taken as 125 percent if the maximum allowable casing working pressure. In cases where mechanical seals are used, this pressure may be limited by the pressure-containing capabilities of the seal. Note: Seesure of the pump to 125 percent of the maximum allowable casing working pressure on the suction splitcase pumps and certain other pump types. • Maximum allowable casing working pressure. This is the hcase pumps and certain other pump types. • Maximum allowable casing working pressure. This is the highest pressure at the specified pumping temperature for which the pump casing is designed. This pressure shall be equal to or greater than the maximum discharge pressure. In the case of some pumps (double suction, vertical turbine, axial split case can pumps, or multistage, for example), the maximum allowable casing working pressure on the suction side may be different from that on the discharge side. • Maximum suction pressure. This is the highest section pressure to which the pump will be subjected during operation. • Working pressure (pd). This is the maximum discharge pressure that could occur in the pump, when it is operated at rated speed and suction pressure for the given application. Shut off. This is the condition of zero flow where no liquid is flowing through the pump, but the pump is primed and running. Speed. This is the number of revolutions of the shaft is a given unit of time. Speed is expressed as revolutions per minute. Suction conditions • Maximum suction pressure. This is the highest suction pressure to which the pump will be subjected during operation. • Net positive suction head available (NPSHA). Net positive suction head available is the total suction head of liquid absolute, determined at the first-stage impeller datum, less the absolute vapor pressure of the liquid at a specific capacity:
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PUMP SYSTEM HYDRAULIC DESIGN
Pump System Hydraulic Design 10.5
NPSHA hsa hvp
(10.3)
where hsa total suction head absolute hatm hs
(10.4)
or NPSHA hatm hs hvp
(10.5)
• Net positive suction head required (NPSHR). This is the amount of suction head, over vapor pressure, required to prevent more than 3 percent loss in total head from the first stage of the pump at a specific capacity. • Static suction lift (Is). Static suction lift is a hydraulic pressure below atmospheric at the intake port of the pump. • Submerged suction. A submerged suction exists when the centerline of the pump inlet is below the level of the liquid in the supply tank. • Total discharge head (hd). The total discharge head (hd) is the sum of the discharge gauge head (hgd) plus the velocity head (hvd) at point of gauge attachment plus the elevation head (Zd) from the discharge gauge centerline to the pump datum: • Total head (H). This is the measure of energy increase per unit weight of the liquid, imparted to the liquid by the pump, and is the difference between the total discharge head and the total suction head. This is the head normally specified for pumping applications since the complete characteristics of a system determine the total head required. hd hgd hvd Zd
(10.6)
• Total suction head (hs), closed suction test. For closed suction installations, the pump suction nozzle may be located either above or below grade level. • Total suction head (hs), open suction. For open suction (wet pit) installations, the first stage impeller of the bowl assembly is submerged in a pit. The total suction head (hs) at datum is the submergence (Zw). If the average velocity head of the flow in the pit is small enough to be neglected, then: hs Zw
(10.7)
where Zw vertical distance in feet from free water surface to datum. The total suction head (hs), referred to the eye of the first-stage impeller is the algebraic sum of the suction gauge head (hvs) plus the velocity head (hvs) at point of gauge attachment plus the elevation head (Zs) from the suction gauge centerline (or manometer zero) to the pump datum: hs hgs hvs Zs
(10.8)
The suction head (hs) is positive when the suction gauge reading is above atmospheric pressure and negative when the reading is below atmospheric pressure by an amount exceeding the sum of the elevation head and the velocity head. Velocity head (hv). This is the kinetic energy of the liquid at a given cross section. Velocity head is expressed by the following equation:
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PUMP SYSTEM HYDRAULIC DESIGN
10.6
Chapter Ten
v2 hv 2g
(10.9)
where v is obtained by dividing the flow by the crosssectional area at the point of gauge connection.
10.1.3 Types of Centrifugal Pumps The HI and API standards do not agree on these definitions of types of centrifugal pumps (Figs. 10.2 and 10.3). Essentially, the HI standard divides centrifugal pumps into two types (overhung impeller and impeller between bearings), whereas the API standard divides them into three types (overhung impeller, impeller between bearings, and vertically suspended). In the HI standard, the “vertically suspended” type is a subclass of the “overhung impeller” type.
FIGURE 10.2 Kinetic type pumps per ANSI/HI-1.1-1.5-1994.
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PUMP SYSTEM HYDRAULIC DESIGN
FIGURE 10.3 Pump class type identification per API 610.
Pump System Hydraulic Design 10.7
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PUMP SYSTEM HYDRAULIC DESIGN
Chapter Ten
FIGURE 10.4 Typical discharge curves.
10.8
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PUMP SYSTEM HYDRAULIC DESIGN
Pump System Hydraulic Design 10.9
10.2 PUMP HYDRAULICS 10.2.1 Pump Performance Curves The head that a centrifugal pump produces over its range of flows follows the shape of a downward facing or concave curve (Fig. 10.4). Some types of impellers produce curves that are not smooth or continuously decreasing as the flow increases: that is, there may be dips and valleys in the pump curve.
10.2.2
Pipeline Hydraulics and System Curves
A system curve describes the relationship between the flow in a pipeline and the head loss produced; see Fig. 10.5 for an example. The essential elements of a system curve include: • The static head of the system, as established by the difference in water surface elevations between the reservoir the pump is pumping from and the reservoir the pump is pumping to, • The friction or head loss in the piping system. Different friction factors representing the range in age of the pipe from new to old should always be considered. The system curve is developed by adding the static head to the headlosses that occur as flow increases. Thus, the system curve is a hyperbola with its origin at the value of the static head. The three most commonly used procedures for determining friction in pipelines are the following: 10.2.2.1 Hazen-Williams equation. The Hazen-Williams procedure is represented by the equation: V 1.318C R0.63S0.54 (U.S. units)
(10.10a)
where: V velocity, (ft/s), C roughness coefficient, R hydraulic radius, (ft), and S friction head loss per unit length or the slope of the energy grade line (ft/ft). In SI units, Eq. (10.10a) is V 0.849CR0.63S0.54
(10.10b)
where V velocity (m/s), C roughness coefficient, R hydraulic radius, (m) and, S friction head loss per unit length or the slope of the energy gradeline in meters per meter. A more convenient form of the Hazen-Williams equation for computing headloss or friction in a piping system is
4.72 Q HL 4 D .86 C
1.85
(10.11a)
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PUMP SYSTEM HYDRAULIC DESIGN
10.10
Chapter Ten
FIGURE 10.5 Typical system head-capacity curves.
where HLheadloss, (ft), L length of pipe, (ft), D pipe internal diameter, (ft), Q flow, (ft3/s), and C roughness coefficient or friction factor. In SI units, The Hazen-Williams equation is. CD 151Q
HL L 1000
1.85
2.63
10.74L Q D4.86 C
1.85
(10.11b)
where HL head loss, (m), Q flow, (m3/s), D pipe diameter, (m), and L pipe length, (m). The C coefficient typically has a value of 80 to 150; the higher the value, the smoother the pipe. C values depend on the type of pipe material, the fluid being conveyed (water or sewage), the lining material, the age of the pipe or lining material, and the pipe diameter. Some ranges of values for C are presented below for differing pipe materials in Table 10.1. TABLE 10.1 Hazen-Williams Coefficents Pipe Material PVC Steel (with mortar lining) Steel (unlined) 120 to 140 Ductile iron (with mortar lining)
C Value for Water
C Value for Sewage
135 –150 120–145 110–130 100–140
130 –145 120–140 110–130 100–130
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PUMP SYSTEM HYDRAULIC DESIGN
Pump System Hydraulic Design 10.11 TABLE 10.1 (Continued) Pipe Material Asbestos cement Concrete pressure pipe Ductile iron (unlined)
C Value for Water
C Value for Sewage
120–140 130–140 80–120
110–135 120–130 80–110
AWWA Manual M11, Steel Pipe—A Guide for Design and Installation (AWWA, 1989), offers the following relationships between C factors and pipe diameters for water service: C 140 0.17d for new mortar-lined steel pipe (U.S. units)
(10.12)
140 0.0066929d (SI units, d in (mm) C 130 0.16d (U.S. units) for long-term considerations of lining
(10.13)
deterioration, slime buildup, and so on. 130 0.0062992d (SI units, d in mm), where C roughness coefficient or friction factor (See Table 10.1) d pipe diameter, inches or millimeters, as indicated above. 10.2.2.2 Manning’s equation. Manning’s procedure is represented by the equation 1.486 V R2/3S1/2 (U.S. units) n
(10.14)
1 V R2/3S1/2 (SI units), n where Vvelocity, (f/s or m/s), nroughness coefficient, R Hydraulic radius, (ft or m), and S friction head loss per unit length or the slope of the energy grade line in feet per foot or meters per meter. A more convenient form of the Manning equation for computing head loss or friction in a pressurized piping system is 4.66 L (nQ)2 HL 16 D /3 10.29L(nQ)2 16 D /3
(U.S. units)
(10.15)
(SI units)
where nroughness coefficient, HL head loss (ft or m), L length of pipe (ft or m), Dpipe internal diameter (ft or m), and Qflow (cu3/s or m3/s). Values of n are typically in the range of 0.010 – 0.016, with n decreasing with smoother pipes.
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PUMP SYSTEM HYDRAULIC DESIGN
10.12
Chapter Ten
10.2.2.3 Darcy-Weisbach equation. The Darcy-Weisbach procedure is represented by the equation L V2 HL f (10.16) D 2g where f friction factor from Moody diagram, g acceleration due to gravity 32.2 (ft/s) (U.S. units) 9.81 m/s2 (SI units), HL head loss (ft or m), L length of pipe (ft or m), D pipe internal diameter (ft or m), and V velocity (ft/s or m/sec). Sanks et al., (1998) discuss empirical equations for determining f values. A disadvantage of using the Darcy-Weisbach equation is that the values for f depend on both roughness (E/D) and also on the Reynolds number (Re): VD R (10.17) v where R = reynolds number (dimensionless), V = fluid velocity in the pipe (ft/s or m/s), D = pipe inside diameter (ft or m), and v = kinematic viscosity (ft2/s or m2/s) Values for f as a function of Reynold’s number can be determined by the following equations: 64 R less than 2000: f (10.18) R
2.51 E/ D R 2000–4000: 1 2 log10 3.7 f Rf
(10.19)
0.25 R greater than 4000: f (10.20) E/D 5.74 2 log10 0. 9 3.7 R where E/D roughness, with E absolute roughness, feet or meters, and D pipe diameter, (ft or m).
Equation 10.19 is the Colebrook-White equation, and Eq. 10.20 is an empirical equation developed by Swamee and Jain, in Sanks et al., (1998). For practical purposes, f values for water works pipelines typically fall in the range of 0.016 to about 0.020. 10.2.2.4 Comparisons of f, C, and n. The Darcy-Weisbach friction factor can be compared to the Hazen-Williams C factor by solving both equations for the slope of the hydraulic grade line and equating the two slopes. Rearranging the terms gives, in SI units, v D
1 f 1.85 C
134
0.15
0.167
(10.21a)
where v is in m/s and D is in m. In U.S. customary units, the relationship is v D
1 f 1.85 C
194
0.15
0.167
(10.21b)
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PUMP SYSTEM HYDRAULIC DESIGN
Pump System Hydraulic Design 10.13
where v is in fps and D is in feet (Sanks et al., 1998). For pipes flowing full and under pressure, the relationship between C and n is D0.037 n 1.12 CS0.04
(10.22a)
in SI units, where D is the inside diameter ID in m. In U.S. customary units, the equation is D0.037 n 1.07 CS0.04
(10.22b)
where D is the (ID) in ft.
10.2.3 Hydraulics of Valves The effect of headlosses caused by valves can be determined by the equation for minor losses: 2 hL K V zg
(10.23)
where hL minor loss (ft or m), K minor loss coefficient (dimensionless), V fluid velocity (ft/s or m/s), and g acceleration due to gravity ( 32.2 fts /s or 9.81 ms/s). Headloss or pressure loss through a valve also is determined by the equation Q Cv P (U.S. units) 0.3807Cv P
(S.I. units)
(10.24)
where Q flow through valve (gal/m or m3/s), CV valve capacity coefficient, and P pressure loss through the valve (psi or kPa) The coefficient CV varies with the position of the valve plug, disc, gate, and so forth. CV indicates the flow that will pass through the valve at a pressure drop of 1 psi. Curves of CV versus plug or disc position (0–90,with 0 being in the closed position) must be obtained from the valve manufacturer’s catalogs or literature. CV and K are related by the equation 2 CV 29.85 d K
(U.S. units)
(10.25)
where d valve size, (in). Thus, by determining the value for CV from the valve manufacturer’s data, a value for K can then be calculated from Eq. (10.25). This K value can then be used in Eq. (10.23) to calculate the valve headloss.
10.2.4 Determination of Pump Operating Points—Single Pump The system curve is superimposed over the pump curve; (Fig. 10.6). The pump operating points occur at the intersections of the system curves with the pump curves. It should be Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
PUMP SYSTEM HYDRAULIC DESIGN
10.14
Chapter Ten
observed that the operating point will change with time. As the piping ages and becomes rougher, the system curve will become steeper, and the intersecting point with the pump curve will move to the left. Also, as the impeller wears, the pump curve moves downward. Thus, over a period of time, the output capacity of a pump can decrease significantly. See Fig. 10.7. for a visual depiction of these combined effects.
10.2.5 Pumps Operating in Parallel To develop a composite pump curve for pumps operating in parallel, add the flows together that the pumps provide at common heads (Fig. 10.8). This can be done with identical pumps (those having the same curve individually) as well as with pumps having different curves.
10.2.6 Variable–Speed Pumps The pump curve at maximum speed is the same as the one described above. The point on a system-head curve at which a variablespeed pump will operate is similarly determined by the intersection of the pump curve with the system curve. What are known as the pump affinity laws or homologous laws must be used to determine the pump curve at reduced speeds. These affinity laws are described in detail in Chap 12. For the discussion here, the relevant mathematical relationships are Sanks et al., (1998). Q n 1 1 Q2 n2
(10.26)
H n 1 (1 )2 H2 n2
(10.27)
P n 1 (1 )3 P2 n2
(10.28)
where Qflow rate, Hhead, Ppower, nrotational speed, and subscripts 1 and 2 are only for corresponding points. Equations (10.26) and (10.27) must be applied simultaneously to ensure that Point 1 “corresponds” to Point 2. Corresponding points fall on parabolas through the original. They do not fall on system H-Q curves. These relationships, known collectively as the affinity laws, are used to determine the effect of changes in speed on the capacity, head, and power of a pump. The affinity laws for discharge and head are accurate because they are based on actual tests for all types of centrifugal pumps, including axial-flow pumps. The affinity law for power is not as accurate because efficiency increases with an increase in the size of the pump. When applying these relationships, remember that they are based on the assumption that the efficiency remains the same when transferring from a given point on one pump curve to a homologous point on another curve. Because the hydraulic and pressure characteristics at the inlet, at the outlet, and through the pump vary with the flow rate, the errors produced by Eq. (10.28) may be excessive, although errors produced by Eqs. (10.26) and (10.27) are extremely small. See Fig. 10.9 for an illustration of the pump curves at different speeds. Example. Consider a pump operating at a normal maximum speed of 1800 rpm, having a head-capacity curve as described in Table 10.2. Derive the pump curve for operating speeds of 1000—1600 rpm at 200-rpm increments. The resulting new values for capacity (Q) and head (H) are shown in Table 10.2. The
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PUMP SYSTEM HYDRAULIC DESIGN
FIGURE 10.6 Determining the operating point for a single-speed pump with a fixed value of hstat
Pump System Hydraulic Design 10.15
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PUMP SYSTEM HYDRAULIC DESIGN
10.16
Chapter Ten
FIGURE 10.7 Effect of impeller wear
values are derived by taking the Q values for the 1800 rpm speed and multiplying them by the ratio (n1/n2) and by taking the H values for the 1800 rpm speed and multiplying them by the ratio (n1/n2)2 .
10.3 CONCEPT OF SPECIFIC SPEED 10.3.1 Introduction: Discharge–Specific Speed The specific speed of a pump is defined by the equation: nQ0.50 Ns (10.29) H0.75 where Ns = specific speed (unitless), n = pump rotating speed (rpm), Q = pump discharge flow (gal/mm, m3/s, L/s, m3h) (for double suction pumps, Q is one-half the total pump flow, and H = total dynamic head (ft or m) (for multistage pumps, H is the head per stage), The relation between specific speeds for various units of discharge and head is given in Table 10.3, wherein the numbers in bold type are those customarily used (Sank et al., 1998). Pumps having the same specific speed are said to be geometrically similar. The specific speed is indicative of the shape and dimensional or design characteristics of the pump impeller (HI, 1994). Sanks et al. (1998) also gives a detailed description and discussion of impeller types as a function of specific speed. Generally speaking, the various types of
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1,00
0,790
0,605
0,444
0,309
0,889
0,777
0,667
0,555
1,600
1,400
1,200
1,000
0,309
0,555
1,000
1,00
0,444
0,667
1,200
1,800
0,605
0,777
1,400
Ratio (n1/n2)2
0,790
0,889
1,600
Ratio n1/n2
1,00
1,00
1,800
Speed (rpm)
Ratio (n1/n2)2
Ratio n1/n2
Speed (rpm)
62
89
121
158
200
0
0
0
0
0
19
27
36
47
60
Point 1 Q (gpm) H(feet)
0
0
0
0
0
Point 1 Q (gpm) H(feet)
56
80
109
142
180
1,111
1,333
1,556
1,778
2,000
49
71
97
126
160
33
42
49
56
63
17
24
33
44
55
70
84
98
112
126
15
22
30
39
49
Head—Capicity at Varios Points Point 2 Point 3 Q (gpm) H(feet) Q (gpm) H(feet)
556
667
778
889
1,000
Head—Capicity at Varios Points Point 2 Point 3 Q (gpm) H(feet) Q (gpm) H(feet)
40
58
79
103
130
105
126
147
168
189
12
18
24
32
40
Point 4 Q (gpm) H(feet)
1,667
2,000
2,333
2,667
3,000
Point 4 Q (gpm) H(feet)
PUMP SYSTEM HYDRAULIC DESIGN
Pump System Hydraulic Design 10.17
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PUMP SYSTEM HYDRAULIC DESIGN
Chapter Ten
FIGURE 10.8 Pumps operating in parallel
10.18
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PUMP SYSTEM HYDRAULIC DESIGN
Pump System Hydraulic Design 10.19
FIGURE 10.9 Typical Discharge Curves for a Variable Speed Pump
impeller designs are as follows: Type of Impeller Radial-vane Mixed-flow Axial-flow
Specific Speed Range (U.S. Units) 500 –4200 4200–9000 9000–15,000
10.3.2 Suction-Specific Speed Suction-specific speed is a number similar to the discharge specific and is determined by the equation 0.50 nQ S NPSHR0.75
(10.30)
where S suction-specific speed (unitless) n pump rotating speed (rpm) Q pump discharge flow as defined for Eq. (10.29). NPSHR net positive suction head required, as described in Sec. 10.4 The significance of suction-specific speed is that increased pump speed without proper suction head conditions can result in excessive wear on the pump’s components (impeller, shaft, bearings) as a result of excessive cavitation and vibration (Hydraulics Institute, 1994). That is, for a given type of pump design (with a given specific speed), there is an equivalent maximum speed (n) at which the pump should operate. Rearranging Eq. (10.30) results in S NPSHA0.75 n 0. Q 50
(10.31)
Equation (10.31) can be used to determine the approximate maximum allowable pump
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PUMP SYSTEM HYDRAULIC DESIGN
10.20
Chapter Ten
TABLE 10.3 Equivalent Factors for Converting Values of Specific Speed Expressed in One Set of Units to the Corresponding Values in Another Set of Units Quantity N Q H
Expressed in Units of (rev/min, L/s, m)
(rev/min, m3/s, m)
(rev/min, m3/h, m)
(rev/min, gal/mn, ft)
(rev/min, ft3/s, ft)
1.0
0.0316
1.898
1.633
0.0771
31.62
1 .0
60.0
51.64
2.437
0.527
0.0167
1.0
0.861
0.0406
0.612
0.0194
1.162
1.0
0.0472
12.98
0.410
24.63
21.19
1.0
Source: Sanks, et al 1998 For example, if the specific speed is expressed in metric units (e.g., N rev/min, Q m3/s, and H m), the corresponding value expressed in U.S. customary units (e.g., N rev/min, Q gal/min, and H feet) is obtained by multiplying the metric value by 51.64.
speed as a function of net positive suction head available and flow for a given type of pump (i.e., a given suction-specific speed). Inspection of Eq. (10.31) reveals that, for a given specific speed, the following pump characteristics will occur: • The higher the desired capacity (Q), the lower the allowable maximum speed. Thus, a properly selected high-capacity pump will be physically larger beyond what would be expected due solely to a desired increased capacity. • The higher the NPSHA, the higher the allowable pump speed.
10.4 NET POSITIVE SUCTION HEAD Net positive suction head, or NPSH, actually consists of two concepts: • the net positive suction available (NPSHA), and • the net positive suction head required (NPSHR). The definition of NPSHA and NPSHR, as given by the Hydraulics Institute (1994), were presented in Sec. 10.1.
10.4.1 Net Positive Suction Head Available Figure 10.1 visually depicts the concept of NPSHA. Since the NPSHA is the head available at the impeller, friction losses in any suction piping must be subtracted when making the calculation. Thus, the equation for determining NPSHA becomes NPSHA hatm hs hvp hL
(10.32)
where:
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PUMP SYSTEM HYDRAULIC DESIGN
Pump System Hydraulic Design 10.21
hatm atmospheric pressure (ft or m). hs
static head of water on the suction side of the pump (ft or m) (hs is negative if the water surface elevation is below the eye of the impeller).
hvp vapor pressure of water, which varies with both altitude and temperature (ft or m), and hL
friction losses in suction piping (ft or m), typically expressed as summation of velocity heads (KV2/2g) for the various fittings and pipe lengths in the suction piping.
Key points in determining NPSHA are as follows (Sanks et al., 1998): • the barometric pressure must be corrected for altitude, • storms can reduce barometric pressure by about 2 percent, and • the water temperature profoundly affects the vapor pressure. Because of uncertainties involved in computing NPSHA, it is recommended that the NPSHA be at least 5 ft (1.5 m) greater than the NPSHR or 1.35 times the NPSHR as a factor of safety (Sanks et al., 1998). An example of calculating NPSHA is presented in Section 10.5.
10.4.2 Net Positive Suction Head Required by a Pump Hydraulics Institute (1994) and Sanks et al., (1998) have discussed the concept and implications of NPSHR in detail. Their discussions are presented or summarized as follows. The NPSHR is determined by tests of geometrically similar pumps operated at constant speed and discharge but with varying suction heads. The development of cavitation is assumed to be indicated by a 3 percent drop in the head developed as the suction inlet is throttled, as shown in Fig. 10.10. It is known that the onset of cavitation occurs well before the 3 percent drop in head (Cavi, 1985). Cavitation can develop substantially before any drop in the head can be detected, and erosion indeed, occurs more rapidly at a 1 percent change in head (with few bubbles) than it does at a 3 percent change in head (with many bubbles). In fact, erosion can be inhibited in a cavitating pump by introducing air into the suction pipe to make many bubbles. So, because the 3 percent change is the current standard used by most pump manufacturers to define the NPSHR, serious erosion can occur as a result of blindly accepting data from catalogs. In critical installations where continuous duty is important, the manufacturer should be required to furnish the NPSHR test results. Typically, NPSHR is plotted as a continuous curve for a pump (Fig. 10.11). When impeller trim has a significant effect on the NPSHR, several curves are plotted. The NPSH required to suppress all cavitation is always higher than the NPSHR shown in a pump manufacturer's curve. The NPSH required to suppress all cavitation at 40 to 60 percent of a pump's flow rate at BEP can be two to five times as is necessary to meet guaranteed head and flow capacities at rated flow (Fig. 10.10; Taylor, 1987). The HI standard (Hydraulics Institute, 1994) states that even higher ratios of NPSHA to NPSHR may be required to suppress cavitation: It can take from 2 to 20 times the NPSHR to suppress incipient cavitation completely, depending on the impeller's design and operating capacity. If the pump operates at low head at a flow rate considerably greater than the capacity at the BEP, Eq. (10.33) is approximately correct:
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PUMP SYSTEM HYDRAULIC DESIGN
10.22
Chapter Ten
NPSHR at operating point Q at operating point Q at BEP NPSHR at BEP
n
(10.33)
where the exponent n varies from 1.25 to 3.0, depending on the design of the impeller. In most water and wastewater pumps, n lies between 1.8 and 2.8. The NPSHR at the BEP increases with the specific speed of the pumps. For high-head pumps, it may be necessary either to limit the speed to obtain the adequate NPSH at the operating point or to lower the elevation of the pump with respect to the free water surface on the suction side i to increase the NPSHA. 10.4.3 NPSH Margin or Safety Factor Considerations Any pump and piping system must be designed such that the net positive suction head available (NPSHA) is equal to, or exceeds, the net positive suction head required (NPSHR) by the pump throughout the range of operation. The margin is the amount by which NPSHA exceeds NPSHR (Hydraulics Institute, 1994). The amount of margin required varies, depending on the pump design, the application, and the materials of construction. Practical experience over many years has shown that, for the majority of pump applications and designs, NPSHR can be used as the lower limit for the NPSH available. However, for highenergy pumps, the NPSHR may not be sufficient. Therefore, the designer should consider an appropriate NPSH margin over NPSHR for high-energy pumps that is sufficient at all flows to protect the pump from damage caused by cavitation.
FIGURE 10.10 Net positive suction head criteria as determined from pump test results.
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PUMP SYSTEM HYDRAULIC DESIGN
Pump System Hydraulic Design 10.23
FIGURE 10.11 NPSH Required to suppress visible cavitation.
10.4.4 Cavitation Cavitation begins to develop in a pump as small harmless vapor bubbles, substantially before any degradation in the developed head can be detected (Hydraulics Institute, 1994). This is called the point of incipient cavitation (Cavi, 1985; Hydraulics Institute, 1994). Studies on high-energy applications show that cavitation damage with the NPSHA greater than the NPSHR can be substantial. In fact, there are studies on pumps that show the maximum damage to occur at NPSHA values somewhere between 0 and 1 percent head drop (or two to three times the NPSHR), especially for high suction pressures as required by pumps with high impeller-eye peripheral speeds. There is no universally accepted relationship between the percentage of head drop and the damage caused by cavitation. There are too many variables in the specific pump design and materials, properties of the liquid and system. The pump manufacturer should be consulted about NPSH margins for the specific pump type and its intended service on high-energy, low-NPSHA applications. According to a study of data contributed by pump manufacturers, no correlation exists, between the specific speed, the suction specific speed, or any other simple variable and the shape of the NPSH curve break-off. The design variables and manufacturing variables are too great. This means that no standard relationship exists between a 3, 2, 1 or 0 percent head drop. The ratio between the NPSH required for a 0 percent head drop and the NPSHR is not a constant, but it generally varies over a range from 1.05 to 2.5. NPSH for a 0, 1, or 2 percent head drop cannot be predicted by calculation, given NPSHR. A pump cannot be constructed to resist cavitation. Although a wealth of literature is available on the resistance of materials to cavitation erosion, no unique material property or combination of properties has been found that yields a consistent correlation with cavitation damage rate (Sanks et al., 1998). Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
PUMP SYSTEM HYDRAULIC DESIGN
10.24
Chapter Ten
10.5 CORRECTED PUMP CURVES Figures 10.6 and 10.9 depict “uncorrected” pump curves. That is, these curves depict a pump H-Q curve, as offered by a pump manufacturer. In an actual pumping station design, a manufacturer's pump must be “corrected” by subtracting the head losses that occur in the suction and discharge piping that connect the pump to the supply tank and the pipeline system. See Table 10.4 associated with Fig. 10.12 in the following sample problem in performing these calculations. The example in Table 10.4 uses a horizontal pump. If a vertical turbine pump is used, minor losses in the pump column and discharge elbow also must be included in the analysis. This same example is worked in U.S. units in Appendix 10.A to this chapter. Problem
1. Calculation of minor losses. The principal headloss equation for straight sections of pipe is: 151Q CD
L HL 1000
1.85
(10.11a)
2.63
where L length (m), D pipe diameter (m), Q flow (m3/s), C Hazen-Williams friction factor.
TABLE 10.4 Item in Fig. 10.12 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Calculate Minor Losses Description
Pipe Size mm
m
Entrance 90º elbow 4.5 m of straight pipe 30º elbow 2 m of straight pipe Butterfly valve 1.2 m of straight pipe 300 mm 200 mm reducer 150 mm 250 mm increaser 1 m of straight pipe Pump control valve 1 m of straight pipe Butterfly valve 0.60 m of straight pipe 90º elbow 250 1.5 m of straight pipe Tee connection
300 300 300 300 300 300 300 200 250 250 250 250 250 250 0.25 250 250
0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.20 0.25 0.25 0.25 0.25 0.25 0.25 0.30 0.25 0.25
Friction Factor K* C+ 1.0 0.30 140 0.20 140 0.46 140 0.25 0.25 140 0.80 140 0.46 140 140 0.50
Typical K values. Different publications present other values. Reasonable value for mortar-lined steel pipe. Value can range from 130 to 145.
* †
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PUMP SYSTEM HYDRAULIC DESIGN
Pump System Hydraulic Design 10.25
FIGURE 10.12 Piping system used in example in Table 10.4.
The principal headloss equation for fittings is HL
0 0
V2 K 2g
where K fitting friction coefficient, V velocity (m/s), g acceleration due to gravity (ms/s) Sum of K values for various pipe sizes: K300 1.96 K200 0.25 K250 2.31 Sum of C values for various pipe sizes:
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PUMP SYSTEM HYDRAULIC DESIGN
10.26
Chapter Ten
Pipe lengths for 300-mm pipe: L = 7.7 m Pipe lengths for 250-mm pipe: L = 4.1 m Determine the total headloss: V2300 mm V2250 mm V2200 mm HL HL 300 mm HL 250 mm K300 K250 mm K200 mm 2g 2g 2g 151Q 140 0.30
1.85
7. 7 HL 300 mm 1000
2.63
151 Q 140 0.25
1.85
4. 1 HL 250 mm 1000
2.63
3.10 Q1.85
4.00 Q1.85
Convert V 2 /2g terms to Q2 terms: 16 0.0826 1 1 Q Q Q 21g A1 Q 21g D πD /4 2g π D
V2 1 Q 2g 2g A
2
2
2
2
2
2
2
2
2
4
2
4
Therefore,
V2300 mm 0.0826 K300 mm 1.96 4 Q2 = 19.99 Q2 (0.30) 2g V2200 mm 0.0826 K200 mm 0.25 4 Q2 12.90 Q2 (0.20) 2g K250
TABLE 10.5
V2250 0.0826 Q2 21.15 Q2 2g 2.31 (0.25)4
Convert Pump Curve Head Values to Include Minor Piping Losses Q
L/s
H (m) m3/s
Uncorrected
Corrected
0
0
60
60
63
0.063
55
54.74
126
0.126
49
47.99
189
0.189
40
37.74
252
0.252
27
23.01
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PUMP SYSTEM HYDRAULIC DESIGN
Pump System Hydraulic Design 10.27 2
V 0 mm V 2 0 mm V 2 0 mm Total HL HL 300 mm HL 250 mm K300 mm 30 K250 mm 25 K200 mm 20 2g 2g 2g 3.10 Q1.85 4.00 Q1.85 19.99 Q2 12.90 Q2 21.15 Q2 7.10 Q1.85 54.04 Q2 2. Modification of pump curve. Using the above equation for HL, a “modified” pump curve can then be developed (see Table 10.5) The H values as corrected must then be plotted. The operating point of the pump is the intersection of the corrected H-Q curve with the system curve. 3. Calculation of NPSHA. Using the data developed above for calculating the minor losses in the piping, it is now possible to calculate the NPSHA for the pump. Only the minor losses pertaining to the suction piping are considered: items 1-8 in Fig. 10.12. For this suction piping, we have: K300 mm 1.96 K200 mm 0.25 Sum of the C values: pipe length for 300-mm pipe is L 7.7 m. Determine the headloss in2 the suction2 piping: V300 V200 HL HL 300 mm K300 2g K200 2g V2200 V3200 HL300 mm 1.96 2g 0.25 2g
Tabla 10.6 Computation of NPSHA: Condition
Flow (m3/s)
Highstatic suction head 0
Low-static suction head
hs (m)
hatm (m)
hvp (m)
HL at Flow NPSH at Flow (m) (m)
9.0
10.35
0.24
0.00
19.11
0.06
9.0
10.35
0.24
0.12
18.99
0.12
9.0
10.35
0.24
0.53
18.58
0.18
9.0
10.35
0.24
1.20
17.91
0.24
9.0
10.35
0.24
2.12
16.99
0.30
9.0
10.35
0.24
3.29
15.81
0
1.0
10.35
0.24
0.00
11.11
0.06
1.0
10.35
0.24
0.12
10.99
0.12
1.0
10.35
0.24
0.53
10.58
0.18
1.0
10.35
0.24
1.20
9.91
0.24
1.0
10.35
0.24
2.12
8.99
0.30
1.0
10.35
0.24
3.29
7.82
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PUMP SYSTEM HYDRAULIC DESIGN
10.28
Chapter Ten
3.10 Q1.85 19.99 Q2 12.90 Q2 3.10 Q1.85 32.89 Q2 For Fig. 10.12, assume that the following data apply: High-water level elevation 683 m Low-water level elevation 675 m Pump centerline elevation 674 m Therefore: Maximum static head 683 674 9 m Minimum static head 675 674 1 m Per Eq. (10.32), with computation of NPSHA shown in table 10.5 NPSHA hatm hs hvp hL For this example, use hatm 10.35 m hvp 0.24 m at 15°C hs
= 9 m maximum
hs
= 1 m minimum
10.6 HYDRAULIC CONSIDERATIONS IN PUMP SELECTION 10.6.1 Flow Range of Centrifugal Pumps The flow range over which a centrifugal pump can perform is limited, among other things, by the vibration levels to which it will be subjected. As discussed in API Standard 610 (American Petroleum Institute, n.d.), centrifugal pump vibration varies with flow, usually being a minimum in the vicinity of the flow at the BEP and increasing as flow is increased or decreased. The change in vibration as flow is varied from the BEP depends on the pump's specific speed and other factors. A centrifugal pump's operation flow range can be divided into two regions. One region is termed the best efficient or preferred operating region, over which the pump exhibits low vibration. The other region is termed the allowable operating range, with its limits defined as those capacities at which the pump’s vibration reaches a higher but still “acceptable” level. ANSI/HI Standard 1.1–1.5 (Hydraulics Institute, 1994) points out that vibration can be caused by the following typical sources: 1. Hydraulic forces produced between the impeller vanes and volute cutwater or diffuser at vane-passing frequency. 2. Recirculation and radial forces at low flows. This is one reason why there is a definite minimum capacity of a centrifugal pump. The pump components typically are not designed for continuous operation at flows below 60 or 70 percent of the flow that occurs at the BEP.
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PUMP SYSTEM HYDRAULIC DESIGN
Pump System Hydraulic Design 10.29
3. Fluid separation at high flows. This is one reason why there is also a definite maximum capacity of a centrifugal pump. The pump components typically are not designed for continuous operation of flows above about 120 to 130 percent of the flow that occurs at the BEP. 4. Cavitation due to net positive suction head (NPSH) problems. There is a common misconception that if the net positive suction head available (NPSHA) is equal to or greater than the net positive suction head required (NPSHR) shown on a pump manufacturer's pump curve, then there will be no cavitation. This is wrong! As discussed in ANSI/HI 1.1–1.5-1994 (Hydraulics Institute, 1994 and also discussed by Taylor (1987), it takes a suction head of 2 to 20 times the NPSHR value to eliminate cavitation completely. 5. Flow disturbances in the pump intake due to improper intake design. 6. Air entrainment or aeration of the liquid. 7. Hydraulic resonance in the piping. 8. Solids contained in the liquids, such as sewage impacting in the pump and causing momentary unbalance, or wedged in the impeller and causing continuous unbalance. The HI standard then states: The pump manufacturer should provide for the first item in the pump design and establish limits for low flow. The system designer is responsible for giving due consideration to the remaining items. The practical applications of the above discussion by observing what can happen in a plot of a pump curve-system head curve as discussed above in Fig. 10.6. If the intersection of the system curve with the pump H-Q curve occurs too far to the left of the BEP (i.e., at less than about 60 percent of flow at the BEP) or too far to the right of the BEP (i.e., at more than about 130 percent of the flow at the BEP), then the pump will eventually fail as a result of hydraulically induced mechanical damage.
10.6.2 Causes and Effects of Centrifugal Pumps Operating Outside Allowable Flow Ranges As can be seen in Fig. 10.6, a pump always operates at the point of intersection of the system curve with the pump H-Q curve. Consequently, if too conservative a friction factor is used in determining the system curve, the pump may actually operate much further to the right of the assumed intersection point so that the pump will operate beyond its allowable operating range. Similarly, overly conservative assumptions concerning the static head in the system curve can lead to the pump operating beyond its allowable range. See Fig. 10.13 for an illustration of these effects. The following commentary discusses the significance of the indicated operating points 1 through 6 and the associated flows Q1 through Q6. • Q1 is the theoretical flow that would occur, ignoring the effects of the minor head losses in the pump suction and discharge piping. See Fig. 10.12 for an example. Q1 is slightly to the right of the most efficient flow, indicated as 100 units. • Q2 is the actual flow that would occur in this system, with the effects of the pump suction and discharge piping minor losses included in the analysis. Q2 is less than Q1, and Q2 is also to the left of the point of most efficient flow. As shown in Fig. 10.7, as the impeller wears, this operating point will move even further to the left and the pump will become steadily less efficient.
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PUMP SYSTEM HYDRAULIC DESIGN
10.30
•
Chapter Ten
Q1 and Q2 are the flows that would occur assuming that the system head curve that is depicted is “reasonable,” that is, not unrealistically conservative. If, in fact, the system head curve is flatter (less friction in the system than was assumed), then the operating point will be Q3 (ignoring the effects of minor losses in the pump suction and discharge piping). If these minor losses are included in the analysis, then the true operating point is Q4. At Q3, the pump discharge flow in this example is 130 percent of the flow that occurs at the BEP. A flow of 130 percent of flow at the BEP is just at edge of, and may even exceed, the maximum acceptable flow range for pumps (see discussion in Sec. 10.6.1). With most mortar-lined steel or ductile-iron piping systems, concrete pipe, or with plastic piping, reasonable C values should almost always be in the range of 120 145 for water and wastewater pumping systems. Lower C usually would be used only when the pumping facility is connected to existing, old unlined piping that may be rougher.
• If the static head assumed was too conservative, then the actual operating points would be Q5 or Q6. Q5 is 150 percent of the flow at the BEP. Q6 is 135 percent of the flow at the BEP. In both cases, it is most likely that these flows are outside the allowable range of the pump. Cavitation, inadequate NPSHA, and excessive hydraulic loads on the impeller and shaft bearings may likely occur, with resulting poor pump performance and high maintenance costs.
10.6.3 Summary of Pump Selection In selecting a pump, the following steps should be taken: 1. Plot the system head curves, using reasonable criteria for both the static head range and the friction factors in the piping. Consider all feasible hydraulic conditions that will occur: a. Variations in static head b. Variations in pipeline friction factor (C value) Variations in static head result from variations in the water surface elevations (WSE) in the supply reservoir to the pump and in the reservoir to which the pump is pumping. Both minimum and maximum static head conditions should be investigated: • Maximum static head. Minimum WSE in supply reservoir and maximum WSE in discharge reservoir. • Minimum static head. Maximum WSE in supply reservoir and minimum WSE in discharge reservoir. 2. Be sure to develop a corrected pump curve or modified pump curve by subtracting the minor losses in the pump suction and discharge piping from the manufacturer's pump curve (Table 10.5 and Fig. 10.13). The true operating points will be at the intersections of the corrected pump curve with the system curves. 3. Select a pump such that the initial operating point (intersection of the system head curve with pump curve) occurs to the right of the BEP. As the impeller wears, the pump output flow will decrease (Fig. 10.7), but the pump efficiency will actually increase until the impeller has worn to the level that the operating point is to the left of the BEP. For a system having a significant variation in static head, it may be necessary to select a pump curve such that at high static head conditions the operating point is to the left of the BEP. However, the operating point for the flows that occur a majority of the time should be at or to the right of the BEP. Bear in mind that high static head Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
PUMP SYSTEM HYDRAULIC DESIGN
FIGURE 10.13 Determining the operating point for a single-speed pump.
Pump System Hydraulic Design 10.31
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PUMP SYSTEM HYDRAULIC DESIGN
10.32
Chapter Ten
conditions normally only occur a minority of the time: the supply reservoir must be at its low water level and the discharge reservoir must simultaneously be at its maximum water level—conditions that usually do not occur very often. Consequently, select a pump that can operate properly at this condition—but also select the pump that has a BEP which occurs at the flow that will occur most often. See Fig. 10.14 for an example. 4. In multiplepump operations, check the operating point with each combination of pumps that may operate. For example, in a two-pump system, one pump operating alone will produce a flow that is greater than 50 percent of the flow that is produced with both pumps operating. This situation occurs because of the rising shape of the system head curve; see Fig. 10.8. Verify that the pump output flows are within the pump manufacturer's recommended operating range; see Fig. 10.13. 5. Check that NPSHA exceeds the NPSHR for all the hydraulic considerations and operating points determined in Steps 1 and 3.
10.7 APPLICATION OF PUMP HYDRAULIC ANALYSIS TO DESIGN OF PUMPING STATION COMPONENTS 10.7.1 Pump Hydraulic Selections and Specifications 10.7.1.1 Pump operating ranges Identify the minimum, maximum, and design flows for the pump based on the hydraulic analyses described above. See Fig. 10.14 as an example. • The flow at 100 units would be defined as the design point. • There is a minimum flow of 90 units. • There is a maximum flow of 115 units. In multiple–pump operation, the combination of varying static head conditions and the different number of pumps operating in parallel could very likely result in operating points as follows (100 units flow at BEP; see accompaning Table 10.7). Table 10.7 Pump Operating Ranges Operating Flow Condition
Flow (per Pump)
Comments
Minimum
70
Maximum static head condition, all pumps operating
Normal 1
100
Average or most frequent operating condition: fewer than all pumps operating, average static head condition. Might also be the case of all pumps operating, minimum static head condition.
Normal 2
110
Fewer than all pumps operating, minimum static head condition
Maximum 1
115
Maximum static head condition, one pump operating
Maximum 2
125
Minimum static head condition, one pump operating
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PUMP SYSTEM HYDRAULIC DESIGN
FIGURE 10.14 Determining the operating points for a single-speed pump with variation in values of hstat.
Pump System Hydraulic Design 10.33
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PUMP SYSTEM HYDRAULIC DESIGN
10.34
Chapter Ten
Some observations of the above example are: • The flow range of an individual pump is about 1.8:1 (125 70). • The pump was deliberately selected to have its most efficient operating point (Q 100) at the most frequent operating condition, not the most extreme condition. This will result in the minimum power consumption and minimum power cost for the system. • The pump was selected or specified to operate over all possible conditions, not just one or two conditions. In variablespeed pumping applications, the minimum flow can be much lower than what is shown in these examples. It is extremely important that the minimum flow be identified in the pump specification so that the pump manufacturer can design the proper combination of impeller type and shaft diameter to avoid cavitation and vibration problems. 10.7.1.2 Specific pump hydraulic operating problems. Specific problems that can occur when operating a centrifugal pump beyond its minimum and maximum capacities include (Hydraulics Institute, 1994): • Minimum flow problems. Temperature buildup, excessive radial thrust, suction recirculation, discharge recirculation, and insufficient NPSHA. • Maximum flow problems. Combined torsional and bending stresses or shaft deflection may exceed permissible limits; erosion drainage, noise, and cavitation may occur because of high fluid velocities. 10.7.2 Piping Having selected a pump and determined its operating flows and discharge heads or pressures, it is then desirable to apply this data in the design of the piping. See Fig. 10.12 for typical piping associated with a horizontal centrifugal pump. 10.7.2.1 Pump suction and discharge piping installation guidelines. Section 1.4 in the Hydraulic Institute (HI) publication ANSI/HI 1.1–1.5 (1994) and Chap. 6 in API Recommended Practice 686 (1996) provide considerable discussion and many recommendations on the layout of piping for centrifugal pumps to help avoid the hydraulic problems discussed above. 10.7.2.2 Fluid velocity. The allowable velocities of the fluid in the pump suction and discharge piping are usually in the following ranges: Suction:
3–9 ft/s (4–6 ft/s most common) 1.0–2.7 m/s (1.2–1.8 m/s most common)
Discharge:
5–15 ft/s (7–10 ft/s most common) 1.5–4.5 m/s (2–3 m/s most common)
Bear in mind that the velocities will vary for a given pump system as the operating point on a pump curve (i.e., intersection of the pump curve with the system curve) varies for the following reasons:
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PUMP SYSTEM HYDRAULIC DESIGN
Pump System Hydraulic Design 10.35
1. Variation in static heads, as the water surface elevations in both the suction and discharge reservoirs vary 2. Long-term variations in pipeline friction factors (Fig. 10.5) 3. Long-term deterioration in impeller (Fig. 10.7) 4. Variation in the number of pumps operating in a multipump system (Fig. 10.8). A suggested procedure for sizing the suction and discharge piping is as follows: 1. Select an allowable suction pipe fluid velocity of 3–5 ft/s (1.0–1.5 m/s) with all pumps operating at the minimum static head condition. As fewer pumps are used, the flow output of each individual pump will increase (typically by about 20 to 40 percent with one pump operating compared to all pumps operating) with the resulting fluid velocities in the suction piping also increasing to values above the 3–5 ft/s (1.0–1.5 m/s) nominal criteria; 2. Select an allowable discharge pipe fluid velocity of 5–8 ft/s (1.5–2.4 m/s) also with all pumps operating at the minimum static head condition. As discussed above, as fewer pumps are used, the flow output of each individual pump will increase with the resulting fluid velocities in the discharge piping also increasing in values above the 5–8 ft/s (1.5–2.4 m/s) nominal criteria. 10.7.2.3 Design of pipe wall thickness (pressure design) Metal pipes are designed for pressure conditions by the equation for hoop tensile strength: PD t 2SE
(10.34)
where t
wall thickness, in or mm
D inside diameter, in or mm (although in practice, the outside diameter is often conservatively used, partly because the ID is not known initially and because it is the outside diameter (OD) that is the fixed dimension: ID then varies with the wall thickness) P design pressure (psi or kPa) S allowable design circumferential stress (psi or kPa) E longitudinal joint efficiency The design value for S is typically 50 percent of the material yield strength, for “normal” pressures. For surge or transient pressures in steel piping systems, S is typically allowed to rise to 70 percent of the material yield strength (American Water Works Association 1989). The factor E for the longitudinal joint efficiency is associated with the effective strength of the welded joint. The ANSI B31.1 (American Society for Mechanical Engineers, 1995) and B31.3 (American Society for Mechanical Engineers, 1996) codes for pressure piping recommend the values for E given in Table 10.8
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PUMP SYSTEM HYDRAULIC DESIGN
10.36
Chapter Ten
TABLE 10.8 Weld Joint Efficiencies Type of Longitudinal Joint
Weld Joint Efficiency Factor (E)
Arc or gas weld (steel pipe) Single-butt weld
0.80
Double-butt weld
0.90
Single-or double-butt weld with 100% radiography
1.00
Electric resistance weld (steel pipe)
0.85
Furnace butt weld (steel pipe)
0.60
Most steel water pipelines
0.85
Ductile iron pipe
1.0
The wall thickness for plastic pipes [polyvinyl chloride (PVC), high-density polyethylene (HDPE), and FRP] is usually designed in the United States on what is known as the hydrostatic design basis or HDB: 2t HDB Pt (10.35) D–t F where Pt total system pressure (operating surge), t minimum wall thickness (in), D average outside diameter (in), HDB hydrostatic design basis (psi) anh F factor of safety (2.50–4.00) 10.7.2.4 Design of pipe wall thickness (vacuum conditions). If the hydraulic transient or surge analysis (see Chap. 12) indicates that full or partial vacuum conditions may occur, then the piping must also be designed accordingly. The negative pressure required to collapse a circular metal pipe is described by the equation:
2E e 3 ∆P (10.36) 2 (1 µ )SF D where ∆P = difference between internal and external pipeline pressures (psi or kPa) , E = modulus of elasticity of the pipe material (psi or kPa), µ = Poisson's ratio, SF = safety factor (typically 4.0), e = wall thickness (in or m) anh D = outside diameter (in or m) Because of factors such as end effects, wall thickness variations, lack of roundness, and other manufacturing tolerances, Eq. (10.3b) for steel pipe is frequently adjusted in practice to
50,000,000 e ∆P SF D
3
(10.37)
10.7.2.5 Summary of pipe design criteria. The wall thickness of the pump piping system is determined by consideration of three criteria: 1. Normal operating pressure [Eq. (10.34)], with S 50 percent of yield strength 2. Maximum pressure due to surge (static dynamic transient rise), using Eq. (10.34) with S 70 percent of yield strength (in the case of steel pipe) 3. Collapsing pressure, if negative pressures occur due to surge conditions (Eq. 10.36).
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PUMP SYSTEM HYDRAULIC DESIGN
Pump System Hydraulic Design 10.37
10.8 IMPLICATIONS OF HYDRAULIC TRANSIENTS IN PUMPING STATION DESIGN Hydraulic transient, or surge, analysis is covered in detail in Chap. 12. Surge or hydraulic transient effects must be considered in pump and piping systems because they can cause or result in (Sanks et al., 1998): • rupture or deformation of pipe and pump casings, • pipe collapse, • vibration, • excessive pipe or joint displacements, or • pipe fitting and support deformation or even failure. The pressures generated due to hydraulics, thus, must be considered in the pipe design, as was discussed in Sec. 10.7, above.
10.8.1 Effect of Surge on Valve Selection At its worst, surges in a piping can cause swing check valves to slam closed violently when the water column in the pipeline reverses direction and flows backward through the check valve at a significant velocity before the valve closes completely. Consequently, in pump and piping systems in which significant surge problems are predicted to occur, check valves or pump control valves are typical means to control the rate of closure of the valve. Means of controlling this rate of closure include • Using a valve that closes quickly, before the flow in the piping can reverse and attain a high reverse velocity. • Providing a dashpot or buffer on the valve to allow the valve clapper or disc to close gently. • Closing the valve with an external hydraulic actuator so that the reverse flowing water column is gradually brought to a halt. This is frequently done with ball or cone valves used as pump control valves. The pressure rating of the valve (both the check valve or pump control valve and the adjacent isolation) should be selected with a pressure rating to accommodate the predicted surge pressures in the piping system.
10.8.2 Effect of Surge on Pipe Material Selection Metal piping systems, such as steel and ductile iron, have much better resistance to surge than do most plastic pipes (PVC, HDPE, ABS, and FRP). The weakness of plastic pipes with respect to surge pressures is sometimes not adequately appreciated because the wave velocity (a) and, hence, the resulting surge pressures are significantly lower than is the case with metal piping systems. Since the surge pressures in plastic piping are lower than those in metal piping systems, there is sometimes a mistaken belief that the entire surge problem can then be neglected. However, plastic piping systems inherently offer less resistance to hydraulic transients than do metal piping systems, even with the lower pressures. This is particularly the case with solvent or adhesive welded plastic fittings. Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
PUMP SYSTEM HYDRAULIC DESIGN
10.38
Chapter Ten
HDPE has better resistance to surge pressures than other plastic piping systems. In addition, the joints are fusion butt welded, not solvent welded, which results in a stronger joint. However, HDPE is still not as resistant to surge effects as a properly designed steel or ductile iron piping system.
REFERENCES American Petroleum Institute,Centrifugal Pumps for Petroleum, Heavy Duty Chemical, and Gas Industry Services, API Standard 610, 8th ed American Petroleum Institute, Washington, DC. American Society of Mechanical Engineers (ASME), B31.1, Power Piping, ASME, NewYork, 1995. American Society of Mechanical Engineers (ASME), B31.3, Process Piping, ASME, NewYork, 1996 American Water Works Association, Steel Pipe—A Guide for Design and Installation, AWWA M11, 3rd ed., American Water Works Association, Denver, CO 1989. Cavi, D., “NPSHR Data and Tests Need Clarification,” Power Engineering, 89:47–50, 1985. Hydraulics Institute, American National Standard for Centrifugal Pumps for Nomenclature, Definitions, Applications, and Operation, ANSI/HI 1.1–1.5-1994, Hydraulics Institute, Parsippany, NJ, 1994. American Petroleum Institute, Recommended Practices for Machinery Installation and Installation Design, Practice 686, 1st ed. Washington, DC, 1996. Sanks, R. L., et al., Pumping Station Design, 2nd ed., Butterworths, 1998. Taylor., “Pump Bypasses Now More Important,” Chemical Engineering, May 11, 1987.
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PUMP SYSTEM HYDRAULIC DESIGN
Pump System Hydraulic Design 10.39
APPENDIX 10. A
PUMP SYSTEM HYDRAULIC DESIGN Calculation of minor losses and NPSHA in piping and modification of a pump curve (U.S. units) Part 1. Calculation of Minor Losses Principal headloss equations
C
1.85
4.72L Q • For straight sections of pipe: HL 4 .86 D
[Sec Eq. (10.11)]
where L = length in feet, D = pipe diameter (ft), Q = flow (ft3/s) anh C = Hazen-Williams friction factor
0
• For fittings: HL
0
2 K V 2g
[Sec Eq. (10.23)]
where K fitting friction coefficient, V velocity in (ft/s), anh g acceleration due to gravity [(fts)/s]
Sum of K values for various pipe sizes: • K12 1.96 • K8 0.25 • K10 2.31 Sum of C values for various pipe sizes: • Pipe lengths for 12-in pipe: L 26 ft • Pipe lengths for 10-in pipe: L 13 ft. Determine the total headloss:
V28 V212 V210 HL HL 12in HL 10in K12 K10 K8 2g 2g 2g
4.72(26) Q HL 12in (12/12)4.86 140
4.72(13) Q HL 10in (10/12)4.86 140
1.85
1.85
0.013139Q1.85
0.015936Q1.85
V2 Convert terms to Q2 terms 2g
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PUMP SYSTEM HYDRAULIC DESIGN
10.40
Chapter Ten Friction Factor
Item in Fig. 10.12
Description
Pipe Size (in)
K*
1
Entrance
12
1.0
2
90º elbow
12
0.30
3
15 ft of straight pipe
12
4
30º elbow
12
5
7 ft of straight pipe
12
6
Butterfly valve
12
7
4 ft of straight pipe
12
12 in 8 in reducer 6 in 10 in increaser 3 ft of straight pipe Pump control valve 3 ft of straight pipe Butterfly valve 2 ft of straight pipe 90º elbow 5 ft of straight pipe Tee connection
8 10 10 10 10 10 10 10 10 10
8 9 10 11 12 13 14 15 16 17
C+
140 0.20 140 0.46 140
0.25 0.25 140 0.80 140 0.46 140 0.30 140 0.50
Typical K values. Different publications present other values. Reasonable value for mortar-lined steel pipe. Value can range from 130 to 145.
* †
2g A Q
V2 1 Q 2g 2g A
1
2
1
2
2
Q 2
1 1 2g D2/4
Q
16 1 2g 2D4
2
2
2
0.025173 2 Q D4 Therefore, V122 0.025173 Q2 K12 2g 1.96 (12/12)4
0.04933 Q2
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PUMP SYSTEM HYDRAULIC DESIGN
Pump System Hydraulic Design 10.41
V28 0.025173 Q2 K8 2g 0.25 (8/12)4
0.031859 Q2 V120 0.025173 Q2 K10 2g 2.31 (10/12)4
0.12058 Q2
V28 V122 V120 Total HL HL 12in HL 10in K12 2g K10 2g K8 2g 0.013139 Q1.85 0.015936 Q1.85 0.04933 Q2 0.12058 Q2 0.031859 Q2 0.0291 Q1.85 0.202 Q2 Part 2: Modification of Pump Curve Using the above equation for HL, a “modified” pump curve can then be developed by converting pump curve head values to include minor piping losses: Q
H (ft)
GPM
CFS
0
0
Uncorrected 200
Corrected 200
1000
2.228
180
178.87
2000
4.456
160
151.52
3000
6.684
130
120.0
4000
8.912
90
72.29
The H values as corrected must then be plotted. The operating point of the pump is the intersection of the corrected H-Q curve with the system curve.
Part 3: Calculation of NPSHA Using the data developed above for calculating the minor losses in the piping, it is now possible to calculate the NPSHA for the pump. Only the minor losses pertaining to the suction piping are considered: Items 1–8 in Fig. 10.12. For this suction piping, we have: K12 1.96, K8 0.25, sum of C values, Pipe length for 12–in pipe: L 26 ft. Determine the headloss in the suction piping V122 V28 HL HL 12in K12 2g K8 2g V122 V28 HL 12in 1.96 2g 0.25 2g 0.013139 Q1.85 0.04933Q2 0.031859Q2 Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
PUMP SYSTEM HYDRAULIC DESIGN
10.42
Chapter Ten
Compute NPSHA: Condition High-static suction head
Low-static suction head
Flow (ft3/s)
hs (ft)
hatm (ft)
hvp (ft)
HL at Flow NPSHAat Flow (ft) (ft)
0
29
33.96
0.78
0.00
62.18
2
29
33.96
0.78
0.37
61.81
4
29
33.96
0.78
1.47
60.71
6
29
33.96
0.78
3.28
58.90
8
29
33.96
0.78
5.81
56.37
10
29
33.96
0.78
9.05
53.13
0
5
33.96
0.78
0.00
38.18
2
5
33.96
0.78
0.37
37.81
4
5
33.96
0.78
1.47
36.71
6
5
33.96
0.78
3.28
34.90
8
5
33.96
0.78
5.81
32.37
10
5
33.96
0.78
9.05
29.13
0.013139 Q1.85 0.081189Q2 For Fig. 10.12, assume that the following data apply High-water level Elevation 2241 ft Low-water level Elevation 2217 ft Pump centerline elevation 2212 ft Therefore: Maximum static head 2241 2212 29 ft. Minimum static head 2217 2212 5 ft. Per Eq. (10.31), NPSHA hatm hs hvp hL For this example, use hatm 33.96 ft hvp 0.78 ft at 60°F hs 29 ft maximum hs 5 ft minimum
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Source: HYDRAULIC DESIGN HANDBOOK
CHAPTER 11
WATER DISTRIBUTION SYSTEM DESIGN Mark A. Ysusi Montgomery Watson Fresno, California
11.1 INTRODUCTION The primary purpose of a water distribution system is to deliver water to the individual consumer in the required quantity and at sufficient pressure. Water distribution systems typically carry potable water to residences, institutions, and commercial and industrial establishments. Though a few municipalities have separate distribution systems, such as a highpressure system for fire fighting or a recycled wastewater system for nonpotable uses, most municipal water distribution systems must be capable of providing water for potable uses and for nonpotable uses such as fire suppression and irrigation of landscaping. The proper function of a water distribution system is critical to providing sufficient drinking water to consumers as well as providing sufficient water for fire protection. Because these systems must function properly, the principals of their planning, design, and construction need to be understood. This chapter focuses on the critical elements of planning and design of a water distribution system. The information presented primarily discusses typical municipal water distribution systems; however, the hydraulic and design principles presented can be easily modified for the planning and design of other types of pressure distribution systems such as fire protection and recycled wastewater.
11.1.1
Overview
Municipal water systems typically consist of one or more sources of supply, appropriate treatment facilities, and a distribution system. Sources of supply include surface water, such as rivers or lakes, groundwater, and in some instances, brackish or sea water. The information contained in this chapter is limited to the planning and design of distribution systems and does not address issues related to identifying and securing sources of supply or designing and constructing appropriate water treatment facilities. Water distribution systems usually consist of a network of interconnected pipes to transport water to the consumer, storage reservoirs to provide for fluctuations in demand, and pumping facilities.
11.1.2 Definitions Many of the frequently used terms in water distribution system planning and design are defined here. 11.1 Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
WATER DISTRIBUTION SYSTEM DESIGN
11.2
Chapter Eleven
Average day demand. The total annual quantity of water production for an agency or municipality divided by 365. Maximum day demand. The highest water demand of the year during any 24-h period. Peak hour demand. The highest water demand of the year during any 1-h period. Peaking factors. The increase above average annual demand, experienced during a specified time period. Peaking factors are customarily used as multipliers of average day demand to express maximum day and peak hour demands. Distribution pipeline or main. A smaller diameter water distribution pipeline that serves a relatively small area. Water services to individual consumers are normally placed on distribution pipelines. Distribution system pipelines are normally between 150 and 400 mm (6–16 in.). Transmission pipeline or main. A larger diameter pipeline, designed to transport larger quantities of water during peak demand periods. Water services for small individual consumers are normally not placed on transmission pipelines. Transmission mains are normally pipelines larger than 400 mm (16 in.).
11.2 DISTRIBUTION SYSTEM PLANNING The basic question to be answered by the water distribution system planner/designer is, “How much water will my system be required to deliver and to where?” The answer to this question will require the acquisition of basic information about the community including historical water usage, population trends, planned growth, topography, and existing system capabilities, to name just a few. This information can then be used to plan for logical extension of the existing system and to determine improvements necessary to provide sufficient water at appropriate pressure. 11.2.1 Water Demands The first step in the design of a water distribution system is the determination of the quantity of water that will be required, with provision for the estimated requirements for the future. In terms of the total quantity, the water demand in a community is usually estimated on the basis of per capita demand. According to a study published by the U.S. Geological Survey, the average quantity of water withdrawn for public water supplies in 1990 was estimated to about 397 L per day per capita (Lpdc) or 105 gal per day per capita (gpdc). The withdrawals by state are summarized in Table 11.1. The reported water usage shown in Table 11.1 illustrates a wide variation. Per capita water use varies from a low use in Pennsylvania of just over 60 gpcd to over 200 gpcd in Nevada. These variations depend on geographic location, climate, size of the community, extent of industrialization, and other influencing factors unique to most communities. Because of these variations, the only reliable way to estimate future water demand is to study each community separately, determining exiting water use characteristics and extrapolating future water demand using population trends. In terms of how the total water use is distributed within a community throughout the day, perhaps the best indicator is land use. In a metered community, the best way to determine water demand by land use is to examine actual water usage for the various types of land uses. The goal of examining actual water usage is to develop water “duties” for the
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WATER DISTRIBUTION SYSTEM DESIGN
Water Distribution System Design 11.3 TABLE 11.1
Estimates Use of Water in the United States in 1990
State Alabama Alaska Arizona Arkansas California Colorado Connecticut Delaware Distrit of Columbia Florida Geogia Hawaii Idho Illinois Indiana Iowa Kansas Kentucky Louisiana Maine Maryland Massachusetts Michigan Minnesota Mississipi Missouri Montana Nebraska Nevada New Hampshire New Jersey New Mexico New York North Carolina North Dakota Ohio Oklahoma Oregon Pennsylvania Rhode Island South Carolina South Dakota
L /Capita /per day 379 299 568 401 556 549 265 295 678 420 435 450 704 341 288 250 326 265 469 220 397 250 291 560 466 326 488 435 806 269 284 511 450 254 326 189 322 420 235 254 288 307
gal /Capita /day 100 79 150 106 147 145 70 78 179 111 115 119 186 90 76 66 86 70 124 58 105 66 77 148 123 86 129 115 213 71 75 135 119 67 86 50 85 111 62 67 76 81
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WATER DISTRIBUTION SYSTEM DESIGN
11.4
Chapter Eleven TABLE 11.1
(Continued)
State Tennessee Texas Utah Vermont Virginia Washington West Virginia Wisconsin Wyoming Puerto Rico Virgin Islands United Stated total
L /Capita/per day 322 541 825 303 284 522 280 197 617 182 87 397
gal /Capita /day 85 143 218 80 75 138 74 52 163 48 23 105
Source: Solley et al (1993)
various types of land uses that can be used for future planning. Water duties are normally developed for the following land uses: • Single, family residential (some communities have low–medium–and, high–density zones) • Multifamily residential • Commercial (normally divided into office and retail categories) • Industrial (normally divided into light and heavy categories and separate categories for very high users • Public (normally divided into park, or open space, and schools) Water duties are normally expressed in gallons per acre per day. Table 11.2 shows typical water duties in the western United States. It should be noted that the definitions of land use terms like “low–density residential,” “medium–density residential,” and so on, will vary by community and should be examined carefully. Another method of distributing water demand is to examine the water usage for individual users. This is particularly the case when an individual customer constitutes a significant portion of the total system demand. Table 11.3 presents water use for many different establishments. Although the rates vary widely, they are useful in estimating total water use for individual users when no other data are available.
11.2.2 Planning and Design Criteria To effectively plan and design a water distribution system, criteria must be developed and adopted against which the adequacy of the existing and planned system can be compared. Typical criteria elements include the following: • Supply • Storage
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WATER DISTRIBUTION SYSTEM DESIGN
Water Distribution System Design 11.5 TABLE 11.2
Typical Water Duties*
Land Use
Low
Low–density residential
400
Medium–density residential
Water Duty. (gal/day/acre) High Average 3,300
1,670 2,610
900
3,800
High–density residential
2,300
12,000
4,160
Single family residential
1,300
2,900
2,300
Multifamily residential
2,600
6,600
4,160
Office commercial
1,100
5,100
2,030
Retail commercial
1,100
5,100
2,040
Light industrial
200
4,700
1,620
Heavy industrial
200
4,800
2,270
Parks
400
3,100
2,020
Schools
400
2,500
1,700
Source: Adapted from Montgomery Watson study of data of 28 western U.S. cities. Note: gal 3.7854 L.
TABLE 11.3
Typical Rates of Water Use for Various Establishments
User Airport, per passenger Assembly hall, per seat Bowling alley, per alley Camp Pioneer type Children’s, central toilet and bath Day, no meals Luxury, private bath Labor Trailer with private toilet and bath, per unit (2 1/2 persons) Country clubs Resident type Transient type serving meals Dwelling unit, residential Apartment house on individual well Apartment house on public water supply, unmetered Boardinghouse Hotel
Range of Flow L/person gal/person or unit/day or unit/day 10–20 6–10 60–100
3–5 2–3 16–26
80–120 160–200 40–70 300–400 140–200 500–600
21–32 42–53 11–18 79–106 37–53 132–159
300–600 60–100
79–159 16–26
300–400 300–500
79–106 79–132
150–220 200–400
40–58 53–106
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WATER DISTRIBUTION SYSTEM DESIGN
11.6
Chapter Eleven TABLE 11.3. (Continued)
User Lodging house and tourist home Motel Private dwelling on individual well or metered supply Private dwelling on public water supply, unmetered Factory, sanitary wastes, per shift Fairground (based on daily attendance)
Range of Flow L/person gal/person or unit/day or unit/day 120–200
32–53
400–600
106–159
200–600
53–159
400–800
106–211
40–100
11–26
2–6
1–2
Institution Average type
400–600
106–159
700–1200
185–317
Office
40–60
11–16
Picnic park, with flush toilets
20–40
5–11
Average
25–40
7–11
Kitchen wastes only
10–20
3–5
Short order
10–20
3–5
4–8
1–2
Hospital
Restaurant (including toilet)
Short order, paper service Bar and cocktail lounge
8–12
2–3
Average type, per sear
120–180
32–48
Average type 24 h, per seat
160–220
42–58
Tavern, per seat Service area, per counter seat (toll road) Service area, per table seat (toll road)
60–100
16–26
1000–1600
264–423
600–800
159–211
School Day, with cafeteria or lunchroom
40–60
11–16
Day, with cafeteria and showers
60–80
16–21
Boarding
200–400
53–106
1000–3000
264–793
First 7.5 m (⬇ 25 ft) of frontage
1600–2000
423–528
Each additional 7.5 m of frontage
1400–1600
370–423
40–60
11–16
Indoor, per seat, two showings per day
10–20
3–5
Outdoor, including food stand, per car (3 1/3 persons)
10–20
3–5
Self-service laundry, per machine Store
Swimming pool and beach, toilet and shower Theater
Source: Adapted from Metcalf and Eddy (1979).
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WATER DISTRIBUTION SYSTEM DESIGN
Water Distribution System Design 11.7
• Fire demands • Distribution system analysis • Service pressures 11.2.2.1 Supply. In determining the adequacy of water supply facilities, the source of supply must be large enough to meet various water demand conditions, and be able to meet at least a portion of normal demand during emergencies such as power outages and disasters. At a minimum, the source of supply should be capable of meeting the maximum day system demand. It is not advisable to rely on storage to make up any shortfall in supply at maximum day demand. The fact that maximum day demand may occur several days consecutively must be considered by the system planner/designer. It is common for communities to provide a source of supply that meets the maximum day demand, with the additional supply to meet peak hour demand coming from storage. Some communities find it more economical to develop a source of supply that not only meets maximum day but also peak hour demand. It is also good practice to consider standby capability in the source of supply. If the system has been designed where the entire capacity of the supply is required to meet the maximum demand, any portion of the supply that is placed out of service due to malfunction or maintenance will result in a deficient supply. For example, a community that relies primarily on groundwater for its supply should, at a minimum, be able to meet its maximum day demand with at least one of its largest wells out of service. 11.2.2.2 Storage. The principal function of storage is to provide reserve supply for (1) operational equalization, (2) fire suppression reserves, and (3) emergency needs. Operational storage is directly related to the amount of water necessary to meet peak demands. The intent of operational storage is to make up the difference between the consumers’ peak demands and the system’s available supply. It is the amount of desirable stored water to regulate fluctuations in demand so that extreme variations will not be imposed on the source of supply. With operational storage, system pressures are typically improved and stabilized. The volume of operational storage required is a function of the diurnal demand fluctuation in a community and is commonly estimated at 25 percent of the total maximum day demand. Fire storage is typically the amount of stored water required to provide a specified fire flow for a specified duration. Both the specific fire flow and the specific time duration varies significantly by community. These values are normally established through the local fire marshall and are typically based on guidelines established by the Insurance Service Office, a nonprofit association of insurers that evaluate relative insurance risks in communities. Emergency storage is the volume or water recommended to meet demand during emergency situations such as source of supply failures, major transmission main failures, pump failures, electrical power outages, or natural disasters. The amount of emergency storage included with a particular water system is an owner option, typically based on an assessment of risk and the desired degree of system dependability. In considering emergency storage, it is not uncommon to evaluate providing significantly reduced supplies during emergencies. For example, it is not illogical to assume minimal demand during a natural disaster. 11.2.2.3 Fire demands. The rate of flow to be provided for fire flow is typically dependent on the land use and varies by community. The establishment of fire flow criteria should always be coordinated with the local fire marshall. Typical fire flow requirements are shown in Table 11.4. 11.2.2.4 Distribution system analysis. In evaluating an existing system or planning a proposed system, it is important to establish the criteria of operational scenarios against
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WATER DISTRIBUTION SYSTEM DESIGN
11.8
Chapter Eleven TABLE 11.4
Typical Fire Flow Requirements
Land Use Single–family residential Multi family residential Commercial
Fire Flow Requirements, gal/m* 500–2,000 1,500–3,000 2,500–5,000
Industrial
3,500–10,000
Central business district
2,500–15,000
Note: gal 3.7854 L.
which the system will be compared. Any system can be shown to be inadequate if the established criteria is stringent enough. Most systems are quite capable of meeting the average day conditions. It is only when the system is stressed that deficiencies begin to surface. The degree to which the system will be realistically stressed is the crux of establishing distribution system analysis criteria. In evaluating a system it is common to see how the system performs under the following scenarios: • Peak hour demand • Maximum day demand plus fire flow Evaluating the system at peak hour demand gives the designer a look at system-wide performance. Placing fire flows at different locations in the system during a “background” demand equivalent to maximum day demand will highlight isolated system deficiencies. Obviously, it is possible for fires to occur during peak hour demand, but since this simultaneous occurrence is more unlikely than for a fire to occur sometime during the maximum day demand, this is not usually considered to be an appropriate criteria for design of the system. 11.2.2.5 Service pressures. There are differences in the pressures customarily maintained in the distribution systems in various communities. It is necessary that the water pressure in a consumer’s residence or place of business be neither too low nor to high. Low pressures, below 30 psi, cause annoying flow reductions when more than one waterusing device is in service. High pressures may cause faucets to leak, valve seats to wear out quickly, or hot water heater pressure relief valves to discharge. Additionally, abnormally high pressures can result in water being wasted in system leaks. The Uniform Plumbing Code requires water pressures not exceed 80 psi at service connections, unless the service is provided with a pressure reducing device. Another pressure criteria, related to fire flows, commonly requires a minimum of 20 psi at the connecting fire hydrant used for fighting the fire. Table 11.5 presents typical service pressure criteria. 11.2.3 Peaking Coefficients Water consumption changes with the seasons, the days of the week, and the hours of the day. Fluctuations are greater in (1) small than in large communities, and (2) during short rather than during long periods of time. Variations in water consumption are usually expressed as ratios to the average day demand. These ratios are commonly called peaking coefficients. Peaking coefficients should be developed from actual consumption data for an individual community, but to assist the reader, Table 11.6 presents typical peaking coefficients.
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WATER DISTRIBUTION SYSTEM DESIGN
Water Distribution System Design 11.9 TABLE 11.5
Typical Service Pressure Criteria
Condition
Service Pressure Criteria (psi)
Maximum pressure
65–75
Minimum pressure during maximum day
30–40
Minimum pressure during peak hour
25–35
Minimum pressure during fires
20
Note: psi 6.895 kPa.
TABLE 11.6
Typical Peaking Coefficients
Ratio of Rates
U.S. Range
Common Range
Maximum day: average day
1.5–3.5:1
1.8–2.8:1
Peak hour: average day
2.0–7.0:1
2.5–4.0:1
11.2.4 Computer Models and System Modeling Modeling water distribution systems with computers is a proven, effective, and reliable technology for simulating and analyzing system behavior under a wide range of hydraulic conditions. The network model is represented by a collection of pipe lengths interconnected in a specified topological configuration by node points, where water can enter and exit the system. Computer models utilize laws of conservation of mass and energy to determine pressure and flow distribution throughout the network. Conservation of mass dictates that for each node the algebraic sum of flows must equal zero. Conservation of energy requires that along each closed loop the accumulated energy loss must be zero. These laws can be expressed as nonlinear algebraic equations in terms of either pressures (node formulation) or volumetric flow rates (loop and pipe formulation). The nonlinearity reflects the relationship between pipe flow rate and the pressure drop across its length. Due to the presence of nonlinearity in these equations, numerical solution methods are iterative. Initial estimated values of pressure or flow are repeatedly adjusted until the difference between two successive iterates is within an acceptable tolerance. Several numerical iterative solution techniques have been suggested, from which the newtonian method is the most widely used. See chapter 9 for more details on modeling. 11.2.4.1 History of computer models. Prior to computerization, tedious, and time–consuming manual calculations were required to solve networks for pressure and flow distribution. These calculations were carried out using the Hardy-Cross numerical method of analysis for determinate networks. Only simple pipeline systems consisting of a few loops were modeled and under limited conditions because of the laborious effort required to obtain a solution. The first advent of computers in network modeling was with electric analogues, followed by large mainframe digital computers to smaller microcomputers. The computational power of a laptop computer today is vastly superior to the original computing machines that would fill several floors in an office building and at a fraction of the cost. Many of the early computer models did not have interactive on-screen graphics, thus limiting the ability of engineers to develop and interpret model runs. The user interface was very rudimentary and often an afterthought. Input was either by punch cards or formatted American Standard Code for Information Interchange (ASCII) files created with a
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WATER DISTRIBUTION SYSTEM DESIGN
11.10
Chapter Eleven
text editor. Errors were commonplace and just getting a data file that would run could involve days if not weeks of effort, depending on the size and complexity of the network being modeled. Model output was usually a voluminous tabular listing of key network results. Interpretation of the results was time consuming and typically involved hand plotting of pressure contours on system maps. Because of the widespread use of microcomputers during the last two decades, network modeling has taken on new dimensions. Engineers today rely on computer models to solve a variety of hydraulic problems. The use of interactive on-screen graphics to enter and edit network data and to color code and display network maps, attributes, and analysis results has become commonplace in the water industry. This makes it much easier for the engineer to construct, calibrate, and manipulate the model and visualize what is happening in the network under various situations such as non-compliance with system performance criteria. The engineer is now able to spend more time thinking and evaluating system improvements and less time on flipping through voluminous pages of computer printouts, thus leading to improved operation and design recommendations. The new generation of computer models have greatly simplified the formidable task of collecting and organizing network data and comprehending massive results. 11.2.4.2 Software packages. Many of the software packages available offer additional capabilities beyond standard hydraulic modeling such as water quality assessment (both conservative and reactive species), multiquality source blending, travel time determination, energy and power cost calculation, leakage and pressure management, fire flow modeling, surge (transient) analysis, system head curve generation, automated network calibration, real-time simulation, and network optimization. Some sophisticated models can even accommodate the full library of hydraulic network components including pressure regulating valves, pressure sustaining valves, pressure breaker valves, flow control valves, float valves, throttle control valves, fixed– and variable speed pumps, turbines, cylindrical and variable cross-sectional area tanks, variable head reservoirs, and multiple inlet/outlet tanks and reservoirs. Through their predictive capabilities, computer network models provide a powerful tool for making informed decisions to support many organizational programs and policies. Modeling is important for gaining proper understanding of system dynamic behavior, operator training, optimizing the use of existing facilities, reducing operating costs, determining future facility requirements, and addressing water quality distribution issues. There is an abundance of network modeling software in the marketplace today. Some are free and others can be purchased at a nominal cost. Costs can vary significantly between models depending on the range of the features and capabilities provided. The four major sources of computer models include consulting firms, commercial software companies, universities, and government agencies. Many of the programs available from these sources have been on the market for several years with established track records. Most of the recent computer models however, provide very sophisticated and intuitive graphical user interfaces and results presentation environment, as well as direct linkages with information management systems such as relational databases and geographical information systems. Table 11.7 lists the names, addresses, and phone numbers of network modeling software vendors, along with their primary modeling product. 11.2.4.3 Development of a system model. As indicated above, the computerized tools available to the engineer today are impressive and powerful. Once an appropriate software is selected, data must be then input to the software in order to develop a computer model of the water system under study. Input data includes the physical attributes of the system including pipe sizes and lengths, topography, reservoir and pump characteristics, as well as the anticipated nodal demands.
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WADISO — Water Distribution Systems: Simulation and Sizing Water Works WATER/WGRAPH WaterMax WATNET WATSYS
PIPE-FLO PIPES RINCAD RJN CASS WORKS SDP Stoner Workstation TDHNET USU-NETWK
PICCOLO
Vancouver Ladysmith Chicago Trevose Victoria
800-1188 West Georgia Street 2474 Pylades Drive RR#3 20 N. Wacker Drive, Suite 1530 8 Neshaminy Interplex, Suite 219 27 Linden Avenue
Syntex Systems Corporation Municipal Hydraulics Ltd. The Pitometer Associates WRc Expertware Dev. Corp.
Lacey Bexley, NSW 2217 Laval Wheaton Victoria Carlisle Laurel Logan
NANTERRE Cedex
Pasadena Truckee Huntsville Lexington
Cuyahoga Falls Rochester Madison Waterbury Cincinnati Livermore Baton Rouge
City
Boca Raton
2121 Front Street 65 West Broad St. 6612 Mineral Point Road 37 Brookside Road 26 W. Martin Luther King Drive 3062 East Avenue 11814 Coursey Blvd. Suite 220
Address
300 N. Lake Ave. Suite 1200 P.O. Box 8128 One Madison Industrial Park Civil Engineering Software Center, University of Kentucky SAFEGE Consulting Engrs. P.O. Box 727 Parc de l'Ile, 15-27 rue du Port Engineered Software, Inc. 4531 Intelco Loop Watercom Pty Ltd. 105 Queen Victoria St. CEDEGER 1417 rue Michelin RJN Computer Services, Inc. 200 W. Front Street Charles Howard & Assoc. Ltd. 852 Fort St. 2nd Floor Stoner Associates, Inc. P.O. Box 86 TDH Engineering 607 Ninth St. Utah State University Utah State University, Dept of Civil & Environmental Engineering Lewis Publishers - 1990 2000 Corporate Blvd. NW
Computer Modeling, Inc. CEDRA BOSS International Haestad Methods, Inc. US EPA Faast Software Kelix Software Systems
AQUA AVWater BOSS EMS CYBERNET EPANET FAAST FLOW NETWORK ANALYSIS H2ONET HYDRONET InWater KYPIPE
MW Soft, Inc. Tahoe Designs Software Intergraph University of Kentucky
Vendor
Distribution System Modeling Software
Software
TABLE 11.7
B.C. B.C. IL PA B.C.
FL
Quebec IL BC PA MD UT
WA
CA CA AL KY
OH NY WI CT OH CA LA
State
Canada Canada USA USA Canada
USA
USA Australia Canada USA Canada USA USA USA
France
USA USA USA USA
USA USA USA USA USA USA USA
Country
V6E 4A2 VOR2EO 60606 19053 V8V 4C9
33431-9868
H7L4S2 60187 V8W 2H7 17013 20707 84322-4100
98503
92007
91101 92162 35898 40506
44221 14614 53705-4200 06708 45268 94550-4738 70816
Zip Code
(604) 688-8271 (250) 722-3810 (312) 236-5655 (215) 244-9972 (250) 384-5955
1-800-272-7737
(360) 412-0702 612-9587-5384 (514) 629-8888 (630) 682-4700 (250) 385-0206 (717) 243-1900 (301) 490-4515 (801) 797-2943
01133146147181
(626) 568-6868 (530) 582-1525 1-800-345-4856 (606) 257-3436
(330) 929-7886 (716) 232-6998 1-800-488-4775 1-800-727-6555 (513) 569-7603 (510) 455-8086 (504) 769-6785
Telephone
(604) 688-1286 (250) 722-3088 (312) 580-2691 (215) 244-9977 (250) 383-1692
1-800-374-3401
(360) 412-0672 612-9587-5384 (514) 382-3077 (630) 682-4754 (250) 385-7737 (717) 243-5564 (301) 490-4515 (801) 750-1185
01133147247202
(626) 568-6619 (530) 582-8579 (205) 730-6109 (606) 257-8005
(330) 929-2756 (716) 262-2042 (608) 258-9943 (203) 597-1488 (513) 569-7185 (510) 455-8087 —
Fax
WATER DISTRIBUTION SYSTEM DESIGN
Water Distribution System Design 11.11
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WATER DISTRIBUTION SYSTEM DESIGN
11.12
Chapter Eleven
Development of the nodal demands normally involves distributing the average day flow throughout the system in proportion to land use. This is commonly accomplished by determining a demand area for each node, measuring the area of each different land use within the demand area, multiplying the area of each land use within the demand area by its respective average day water duty (converted to gal/min or L/sec), and summing the water duties for each land use within the demand area and applying the sum at the node. In the past this effort required extensive mapping and determining the land use areas by planimeter or hand measurement. Today, with the advent of graphical information system software (GIS), development of nodal demands is normally an activity involving computer based mapping. The elements of the system, the demand areas, and the land uses are all mapped in separate layers in the GIS software. The GIS software capability of “polygon processing” intersects the different layers and automatically computes the land use sums with the various demand areas. When the water duties are multiplied by their respective land use, the result is the average day system demand, proportioned to each node by land use. The water system computer model is then used to apply global peaking factors as described above.
11.3 PIPELINE PRELIMINARY DESIGN The purpose of performing the water system planning tasks as outlined above is to develop a master plan for correcting system deficiencies and providing for future growth. Normally the system improvements are prioritized and a schedule or capital improvement program is developed based upon available (now or future) funding. As projects leave the advanced planning stage, they begin the process of preliminary design. During preliminary design, the considerations of pipeline routing (alignment), subsurface conflicts, and rights-of-way are considered.
11.3.1 Alignment In deciding upon an appropriate alignment for a pipeline, important considerations include right-of-way (discussed further below), constructability, access for future maintenance, and separation from other utilities. Many communities adopt standardized locations for utility pipelines (such as water lines will generally be located 15 ft north and east of the street centerline). Such standards compliment alignment considerations.
11.3.2
Subsurface Conflicts
A critical element of developing a proposed pipeline alignment is an evaluation of subsurface conflicts. To properly evaluate subsurface conflicts it will be necessary for the designer to identify the type, size, and accurate location of all other underground utilities along the proposed pipeline alignment. This information must be considered in the design and accurately placed on the project plans so that the contractor (or whoever is constructing the line) is completely aware of potential conflicts. It is good practice to thoroughly investigate potential utility conflicts. For example, it is not enough to simply determine that the proposed pipeline route will cross an electrical conduit. The exact location and dimensions of electrical conduits also need to be deter-
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WATER DISTRIBUTION SYSTEM DESIGN
Water Distribution System Design 11.13
mined and the proposed water pipeline designed accordingly. What is shown on a utility company plat as a single line representing an electrical conduits may turn out to be a major electrical line with several conduits encased in concrete having a cross section 2 ft wide and 4 ft deep! Or, what is shown as a buried 3/4-inch telephone line may turn out to be a fiber-optic telecommunications cable that, if severed during construction, will result in exorbitant fines being levied by the communications utility. Another water pipeline alignment consideration is the lateral separation of the line from adjacent sanitary sewer lines. Many state and local health officials require a minimum of 10 ft of separation (out-to-out) between potable water and sanitary sewer lines. 11.3.3 Rights-of-Way The final location of a pipeline can only be selected and construction begun once appropriate rights-of-way are acquired. Adequate right-of-way both for construction and for future access are necessary for a successful installation. Water lines are commonly located in streets and roadways dedicated to the public use. On occasion, it is necessary to obtain rights-of-way for transmission-type pipelines across private lands. If this is the case, it is very important to properly evaluate the width of temporary easement that will be required during construction and the width of permanent easement that will be required for future access. If a pipeline is to be installed across private property, it is also very important for the entity that will own and maintain the pipeline to gain agreements that no permanent structures will be constructed within the permanent easements and to implement a program of monitoring construction on the private property to ensure that access to the pipeline is maintained. Otherwise, as the property changes hands in the future, the pipeline stands a good chance of becoming inaccessible.
11.4 PIPING MATERIALS The types of pipe and fittings commonly used for pressurized water distribution systems are discussed in this section. The types of pipeline materials are first presented and then factors effecting the types of materials selected by the designer are presented in Section 11.4.7. The emphasis throughout this section is on pipe 100 mm (4 in.) in diameter and larger. References to a standard or to a specification are given here in abbreviated form - code letters and numbers only such as American National Standards Institute (ANSI) B36.10. Double designations such as ANSI/AWWA C115/A21.15 indicate that American Waterworks Association (AWWA) C115 is the same as ANSI A21.15. Most standards are revised periodically, so it is advisable for the designer to obtain the latest edition.
11.4.1 Ductile Iron Pipe (DIP) Available in sizes 100–1350 mm (4–54 in.), DIP is widely used throughout the United States in water distribution systems. On the East Coast and in the Midwest, DIP is commonly used for both smaller distribution mains and larger transmission mains. On the West Coast, DIP is generally used for distribution pipelines 40 mm (16 in) and smaller, with alternative pipeline materials often selected for larger pipelines due to cost. Detailed descriptions of DIP, fittings, joints, installation, thrust restraint, and other factors relating to design as well as several important ANSI/AWWA specifications are contained in the Ductile Iron Pipe Research Association (DIPRA) handbook (DIPRA, 1984). Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
WATER DISTRIBUTION SYSTEM DESIGN
11.14
Chapter Eleven
11.4.1.1 Materials. DIP is a cast-iron product. Cast-iron pipe is manufactured of an iron alloy centrifugally cast in sand or metal molds. Prior to the early 1970s, most castiron pipe and fittings were gray iron, a brittle material that is weak in tension. But now all cast-iron pipe, except soil pipe (which is used for nonpressure plumbing applications) is made of ductile iron. Ductile iron is produced by the addition of magnesium to molten low-sulfur base iron, causing the free graphite to form into spheroids and making it about as strong as steel. Regular DIP (AWWA C151) has a Brinell hardness (BNH) of about 165. Tolerances, strength, coatings and linings, and resistance to burial loads are given in ANSI/AWWA C151/A21.51. 11.4.1.2 Available sizes and thicknesses. DIP is available in sizes from 100 to 1350 mm (4–54 in). The standard length is 5.5 m (18 ft) in pressure ratings from 1380 to 2400 kPa (200–350 lb/in2). Thickness is normally specified by class, which varies from Class 50 to Class 56 (see DIPRA, 1984 or ANSI/AWWA C150/A21.50). Thicker pipe can be obtained by special order. 11.4.1.3 Joints. For DIP, rubber gasket push-on and mechanical are the most commonly used for buried service. These joints allow for some pipe deflection (about 2–5° depending on pipe size) without sacrificing water tightness. Neither of these joints are capable of resisting thrust across the joint and require thrust blocks or some other sort of thrust restraint at bends and other changes in the flow direction. Flanged joints (AWWA C115 or ANSI B16.1) are sometimes used at fitting and valve connections. Grooved end joints (AWWA C606) are normally used for exposed service and are seldom used for buried service. Flanged joints are rigid and grooved end joints are flexible. Both are restrained joints and do not typically require thrust restraint. Other types of restrained joints such as restrained mechanical joints are also available for buried service. Various types of ductile iron pipe joints are shown in Fig. 11.1. 11.4.1.4 Gaskets. Gaskets for ductile iron push-on and mechanical joints described in AWWA C111 are vulcanized natural or vulcanized synthetic rubber. Natural rubber is suitable for water pipelines but deteriorates when exposed to raw or recycled wastewater. Gaskets for DIP flanges should be rubber, 3.2 mm (1/8 in) thick. Gaskets for grooved end joints are available in ethylene propylene diene monomer (EPDM), nitrile (Buna-N), halogenated butyl rubber, Neoprene™, silicone, and
FIGURE 11.1 Couplings and joints for ductile iron pipe: (a) flexible coupling; (b) mechanical joint; (c) push-on joint; (d) ball joint. Adapted from Sanks et al,. (1989).
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WATER DISTRIBUTION SYSTEM DESIGN
Water Distribution System Design 11.15
fluorelastomers. EPDM is commonly used in water service and Buna-N in recycled wastewater. 11.4.1.5 Fittings. Some standard ductile or gray iron fittings are shown in Fig. 11.2. A list of standard and special fittings is also given in Table 11.8. Ductile iron fittings are normally only available in standard configurations as described in AWWA C110. Greater cost and longer delivery times can be expected for special fittings. Fittings are designated by the size of the openings, followed (where necessary) by the deflection angle. A 90° elbow for 250 mm (10 in) pipe would be called a 250 mm (10 in) 90° bend (or elbow). Reducers, reducing tees, or reducing crosses are identified by giving the pipe diameter of the largest opening first, followed by the sizes of other openings in sequence. Thus, a reducing tee on a 300 mm (12 in) line for a 150 mm (6 in) fire hydrant run might be designated as a 300 mm 150 mm 300 mm (12 in 6 in 12 in ) tee. Standard ductile iron fittings are commonly available in flanged, mechanical joint, and push-on ends. It is considered good practice to include sufficient detail in construction plans and specifications to illustrate the type of joints that are expected at connections. Failure to detail a restrained joint when one is required by the design, could result in an unstable installation. 11.4.1.6 Linings. Considering its low cost, long life, and sustained smoothness, cementmortar lining for DIP in water distribution systems is the most useful and common. Standard thicknesses for shop linings specified in AWWA C104 are given in Table 11.9. Pipe can also be lined in place with the thicknesses given in Table 11.10. Because the
FIGURE 11.2 Ductile iron flanged fittings. Adopted from Sanks et al,. (1989).
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WATER DISTRIBUTION SYSTEM DESIGN
11.16
Chapter Eleven TABLE 11.8 Ductile Iron and Gray Cast-Iron Fittings, Flanged, Mechanical Joint, or Bell and Spigot* Standard Fittings
Special Fittings
Bends (90°, 45°, 22.5°, 11.25°)
Reducing bends (90°)
Base bends
Flared bends (90°, 45°)
Caps
Flange and flares
Crosses
Reducing tees
Blind flanges
Side outlet tees
Offsets
Wall pipes
Plugs
True wyes
Reducers
Wye branches
Eccentric reducers Tees Base tees Side outlet tees Wyes Size from 100 to 350 mm (4–54 in).
*
standard, shop-applied mortar linings are relatively thin, some designers prefer to specify shop linings in double thickness. The designer should also be careful in specifying mortar lining thickness to match the pipe inside diameter (ID) with system valve IDs, particularly with short-body butterfly valves where the valve vane protrudes into the pipe. If the pipe ID is too small, the valve cannot be fully opened.
TABLE 11.9
Thickness of Shop-Applied Cement-Mortal Linings Lining Thickness Ductile Iron Pipe* mm in
Steel Pipe✝ mm in
4–10
1.6
1/16
6.4
1/4
Nominal Pipe Diameter mm in 100–250 300
1–2
1.6
1/16
7.9
5/16
350–550
14–22
2.4
3/32
7.9
5/16
600
24
2.4
3/32
9.5
3/8
750–900
30–36
3.2
1/8
9.5
3/8
1050–1350
42–54
3.2
1/8
12.7
1/2
Single thickness per AWWA C104. Linings of double thickness are also readliy available. Per AWWA C205.
* ✝
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WATER DISTRIBUTION SYSTEM DESIGN
Water Distribution System Design 11.17 TABLE 11.10
Thickness of Cement-Mortal Linings of Pipe in Place per AWWA C602
Nominal Pipe Diameter mm in 100–250 300 350–550 600–900 1050–1350 1500 1650–2250 2250
4–10 12 14–22 24–36 42–54 60 66–90 90
DIP or Gray Cast Iron (new or Old Pipe) mm in 3.2 4.8 4.8 4.8 6.4 — — —
1/8 3/16 3/16 3/16 1/4 — — —
Steel Pipe Old Pipe New Pipe mm in mm in 6.4 6.4 7.9 9.5 9.5 9.5 12.7 12.7
1/4 1/4 5/16 3/8 3/8 3/8 1/2 1/2
4.8 4.8 6.4 6.4 9.5 9.5 11.1 12.7
3/16 3/16 1/4 1/4 3/8 3/8 7/16 1/2
Although cement-mortar lining is normally very durable, it can be slowly attacked by very soft waters with low total dissolved solids content (less than 40 mg/L), by high–sulfate waters, or by waters undersaturated in calcium carbonate. For such uses, the designer should carefully investigate the probable durability of cement mortar and consider the use of other linings. Other linings and uses are shown in Table 11.11. In general, the cost of cement mortar is about 20 percent of that of other linings, so other linings are not justified except where cement mortar would not provide satisfactory service. 11.4.1.7 Coatings. Although DIP is relatively resistant to corrosion, some soils (and peat, slag, cinders, muck, mine waste, or stray electric current) may attack the pipe. In these applications, ductile iron manufacturers recommend that the pipe be encased in loose-fitting, flexible polyethylene tubes 0.2 mm (0.008 in) thick (see ANSI/AWWA C105/A21.5). These are commonly known as “baggies.” An asphaltic coating approximately 0.25 mm (0.001 in) thick is a common coating for ductile iron pipe in noncorrosive soils. In some especially corrosive applications, a coating such as adhesive, hotapplied extruded polyethylene wrap may be required.
TABLE 11.11
Linings for Ductile Iron and Steel Pipe
Lining Material
Reference Standard
Recommended Service
Cement mortar
AWWA C104, C205
Glass Epoxy Fusion-bonded epoxy Coal-tar epoxy Coal-tar enamel Polyurethane Polyethylene
None AWWA C210 AWWA C213 AWWA C210 AWWA C203 None ASTM D 1248
Potable water, raw water and sewage, activated and secondary sludge Primary sludge, very aggressive fluids Raw and potable water Potable water, raw water and sewage Not recommended for potable water Potable water Raw sewage, water Raw sewage
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WATER DISTRIBUTION SYSTEM DESIGN
11.18
Chapter Eleven
In corrosive soils, the following coatings may be appropriate for protecting the pipe: • Adhesive, extruded polyethylene wrap • Plastic wrapping (AWWA C105) • Hot-applied coal-tar enamel (AWWA C203) • Hot-applied coal-tar tape (AWWA C203) • Hot-applied extruded polyethylene [ASTM D 1248 (material only)] • Coal-tar epoxy (MIL-P-23236) • Cold-applied tape (AWWA C209) • Fusion-bonded epoxy (AWWA C213) Each of the above coatings is discussed in detail in the referenced specifications. Each coating system has certain limited applications and should be used in accordance with the NACE standards or as recommended by a competent corrosion engineer.
11.4.2 Polyvinyl Chloride (PVC) Pipe In the United States, where it is used in both water and wastewater service, polyvinyl chloride (PVC) is the most commonly used plastic pipe for municipal water distribution systems. Because of its resistance to corrosion, its light weight and high strength to weight ratio, ease of installation, and its smoother interior wall surface, PVC has enjoyed rapid acceptance for use in municipal water distribution systems since the 1960s. There are several other types of plastic pipe, but PVC is the most common plastic pipe selected for use in municipal systems and will be the only type of plastic pipe addressed in this section. There are also several different PVC pipe specifications. Only those having AWWA approval will be addressed in this section, since only those should be used for municipal water distribution systems. Highdensity polyethylene pipe (HDPE) is discussed in Sec. 11.4.5. 11.4.2.1 Materials. PVC is a polymer extruded under heat and pressure into a thermoplastic that is nearly inert when exposed to most acids, alkalis, fuels, and corrosives, but it is attacked by ketones (and other solvents) sometimes found in industrial wastewaters. Basic properties of PVC compounds are detailed in ASTM D 1784. ASTM D 3915 covers performance characteristics of concern, or cell classification, for PVC compounds to be used in pressure pipe applications. Generally, PVC should not be exposed to direct sunlight for long periods. The impact strength of PVC will decrease if exposed to sunlight and should not be used in above-ground service. In North America, PVC pipe is rated for pressure capacity at 23ºC (73.4ºF). The pressure capacity of PVC pipe is significantly related to its operating temperature. As the temperature falls below 23ºC (73.4ºF), such as in normal buried service, the pressure capacity of PVC pipe increases to a level higher than its pressure rating or class. In practice, this increase is treated as an unstated addition to the working safety factor but is not otherwise considered in the design process. On the other hand, as the operating temperature rises above 23ºC (73.4ºF), the pressure capacity of PVC pipe decreases to a level below its pressure rating or class. Thermal derating factors, or multipliers, are typically used if the PVC pipe will be used for higher temperature services. Recommended thermal derating factors are shown in Table 11.12. The pressure rating or class for PVC pipe at service temperature of 27ºC (80ºF) would need to multiplied
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WATER DISTRIBUTION SYSTEM DESIGN
Water Distribution System Design 11.19 TABLE 11.12 Thermal Derating Factors for PVC Pressure Pipes and Fittings* Maximum Service Temperature °C °F
Multiply the Pressure Rating or Pressure Class at 73.4°F (23°C) by These Factors
27
80
0.88
32
90
0.75
38
100
0.62
43
110
0.50
49
120
0.40
54 60
130 140
0.30 0.22
Source: Handbook of PVC Pipe, 1991.
by a thermal derating factor of 0.88. The pressure rating or class for PVC pipe at service temperature of 60°C (140ºF) would need to multiplied by a thermal de-rating factor of 0.22. 11.4.2.2 Available sizes and thicknesses. AWWA C900 covers PVC pipe in sizes 100 to 300 mm (412 in.). AWWA C905 covers PVC pipe in sizes 350–900 mm (14–36 in). There are important differences in these two specifications that should be understood by the designer. AWWA C900 PVC pipe is manufactured in three “pressure classes” (100, 150, and 200). The pressure class selected is typically the highest normal operating system pressure in psi. AWWA C900 PVC pipe design is based on a safety factor of 2.5 plus an allowance for hydraulic transients (surge). AWWA C905 does not provide for “pressure classes” but refers to PVC pressure pipe in terms of “pressure rating.” As with pressure class, pressure rating also refers to system pressure in psi. While AWWA C905 covers six pressure rating categories (100, 125, 160, 165, 200, and 235), the most commonly available pressure ratings are 165 and 235. The design of AWWA C905 PVC pipe is based on a safety factor of 2.0 and does not include an allowance for surge. In view of this important difference between the two specifications, designers often specify higher pressure ratings of C905 PVC pipe than system pressure would tend to indicate in order to allow for the reduced factor of design safety. Both C900 and C905 contain required pipe dimension ratios. Dimension ratios define a constant ratio between the outside diameter and the wall thickness. For a given dimen-
TABLE 11.13
Pressure Class versus DR-AWWA C900
DR
Pressure Class at Safety Factor = 2.5, psi (kPa)
14
200 (1380)
18
150 (1030)
25
100 (690)
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WATER DISTRIBUTION SYSTEM DESIGN
11.20
Chapter Eleven TABLE 11.14
Pressure Rating versus DR-AWWA C905
DR
Pressure Rating at Safety Factor 2.0 psi (kPa)
18
235*
*
21
200
25
165*
26
160
32.5 41
125 100
Most commonly used ratings for municipal systems.
sion ratio, pressure capacity and pipe stiffness remain constant, independent of pipe size. Table 11.13 presents dimension ratios (DR) with corresponding pressure classes as defined in AWWA C900. Table 11.14 presents dimension ratios with corresponding pressure ratings as defined in AWWA C905. 11.4.2.3 Joints. For PVC pipe, a rubber gasket bell and spigot type joint is the most commonly joint used for typical, municipal buried service. The bell and spigot joint allows for some pipe deflection (Handbook of PVC Pipe, 1991) without sacrificing water tightness. This joint is not capable of resisting thrust across the joint and requires thrust blocks or some other sort of thrust restraint at bends and other changes in the flow direction. Mechanical restraining devices are commonly used to provide restraint at PVC pipe joints where necessary. PVC pipe joints are specified in ASTM D 3139. At connections to fittings and other types of piping, it is also common to detail a plain end (field-cut pipe) PVC pipe. Plain end pipes are used to connect to mechanical joint ductile iron fittings and to flange adapters. 11.4.2.4 Gaskets. Gaskets for PVC joints are specified in ASTM F 477. As with gaskets for DIP, gaskets for PVC pipe are vulcanized natural or vulcanized synthetic rubber. Natural rubber is suitable for water pipelines but deteriorates when exposed to raw or recycled wastewater. EPDM is commonly used in water service and nitrile (Buna N), in recycled wastewater. 11.4.2.5 Fittings. AWWA C900 and C905 PVC pipe for municipal use are manufactured in ductile iron pipe OD sizes, so ductile iron fittings, conforming to AWWA C110, are used in all available sizes. See Sec. 11.4.1.5 for discussion on ductile iron fittings. Although not widely used, PVC fittings, in configurations similar to ductile iron fittings, are also available for smaller line sizes. AWWA C907 covers PVC pressure pipe fittings for pipe sizes 100–200 mm (48 in.) in pressure classes 100 and 150. 11.4.2.6 Linings and Coatings. PVC pipe does not require lining or coating.
11.4.3 Steel Pipe Steel pipe is available in virtually any size from 100 m through 3600 mm (4144 in) for use in water distribution systems. Though rarely used for pipelines smaller than 400 mm (16 in.), it is widely used in the western United States for transmission pipelines in sizes
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WATER DISTRIBUTION SYSTEM DESIGN
Water Distribution System Design 11.21
larger than 600 mm (24 in.). The principal advantages of steel pipe include high strength, the ability to deflect without breaking, the ease of installation, shock resistance, lighter weight than ductile iron pipe, the ease of fabrication of large pipe, the availability of special configurations by welding, the variety of strengths available, and the ease of field modification. 11.4.3.1 Materials. Conventional nomenclature refers to two types of steel pipe: (1) mill pipe and (2) fabricated pipe. Mill pipe includes steel pipe of any size produced at a steel pipe mill to meet finished pipe specifications. Mill pipe can be seamless, furnace butt welded, electric resistance welded, or fusion welded using either a straight or spiral seam. Mill pipe of a given size is manufactured with a constant outside diameter and a variable internal diameter depending on the required wall thickness. Fabricated pipe is steel pipe made from plates or sheets. It can be either straight or spiral-seam fusion welded pipe, and it can be specified in either internal or external diameters. Spiral-seam, fusion-welded pipe may either mill pipe or fabricated pipe. Steel pipe may be manufactured from a number of steel alloys with varying yield and ultimate tensile strengths. Internal working pressure ratings vary from 690 to 17,000 kPa (100–2500 lb/in2) depending on alloy, diameter, and wall thickness. Steel piping in water distribution systems should conform to AWWA C200, in which there are many ASTM standards for materials (see ANSI B31.1 for the manufacturing processes). 11.4.3.2 Available sizes and thicknesses. Sizes, thicknesses, and working pressures for pipe used in water distribution systems range from 100 m to 3600 mm (4144 i) as shown in Table 42 of AWWA M11 (American Water Works Association). The standard length of steel water distribution pipe is 12.2 m (40 ft). Manufacturers should be consulted for the availability of sizes and thicknesses of steel pipe. Table 4-2 of AWWA M11 allows a great variety of sizes and thicknesses. According to ANSI B36.10, • Standard weight (STD) and Schedule 40 are identical for pipes up to 250 mm (10 in). Standard weight pipe 300 mm (12 in) and larger have walls 0.5 mm (3/8 in) thick. For standard weight pipe 300 mm (12 in) and smaller, the ID approximately equals the nominal pipe diameter. For pipe larger than 300 mm (12 in), the outside diameter (OD) equals the nominal diameter. • Extra strong (XS) and Schedule 80 are identical for pipes up to 200 mm (8 in). All larger sizes of extra strong-weight pipe have walls 12.7 mm (1/2 in) thick. • Double extra strong (XXS) applies only to steel pipe 300 mm (12 in) and smaller, There is no correlation between XXS and schedule numbers. For wall thicknesses of XXS, which (in most cases) is twice that of XS, see ANSI B36.10. For sizes of 350 mm (14 in) and larger, most pipe manufacturers use spiral welding machines and, in theory, can fabricate pipe to virtually and desired size, In practice, however, most steel pipe manufacturers have selected and built equipment to produce given ID sizes. Any deviation from manufacturers’ standard practices is expensive so it is always good practice for the designer to consult pipe manufacturers during the design process. To avoid confusion, the designer should also show a detail of the specified pipe size on the plans or tabulate the diameters in the specifications. For cement-mortar lined steel pipelines, AWWA C200, C205, C207, and C208 apply. Steel pipe must sometimes either be reinforced at nozzles and openings (tees, wye branches) or a greater wall thickness must be specified. A detailed procedure for deter-
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WATER DISTRIBUTION SYSTEM DESIGN
11.22
Chapter Eleven
FIGURE 11.3 Reinforcement for steel pipe openings. (a) collar plate; (b) wrapper plate; (c) crotch plates. Adopted from Sanks et al,. (1989)
mining whether additional reinforcing is required is described in Chap. II and Appendix H of ANSI B31.3. If additional reinforcement is necessary, it can be accomplished by a collar or pad around the nozzle or branch, a wrapper plate, or crotch plates. These reinforcements are shown in Figure 11.3 and the calculations for design are given in AWWA M11 (American Water Works Association, 1989).
FIGURE 11.4 Welded and rubber-gasketed joints for steel pipe. AWWA MII (1989)
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WATER DISTRIBUTION SYSTEM DESIGN
Water Distribution System Design 11.23
11.4.3.3 Joints. For buried service, bell and spigot joints with rubber gaskets or mechanical couplings (with or without thrust harnesses) are common. Welded joints are also common for pipe 600 mm (24 in) and larger. Linings are locally destroyed by the heat of welding, so the ends of the pipe must be bare and the linings field applied at the joints. The reliability of field welds is questionable without careful inspection, but when properly made they are stronger than other joints. A steel pipeline project specification involving field welding should always include a carefully prepared section on quality assurance and testing of the welds. Different types of steel pipe joints are shown in Fig. 11.4. 11.4.3.4 Gaskets. Gaskets for steel flanges are usually made of cloth-inserted rubber either 1.6 mm (1/16 in) or 3.2 mm (1/8 in) thick and are of two types: • ring (extending from the ID of the flange to the inside edge of the bolt holes); • full face (extending from the ID of the flange to OD). Gaskets for mechanical and push-on joints for steel pipe are the same as described in Section 11.4.1.4 for ductile iron pipe. 11.4.3.5 Fittings. For steel pipe 100 mm (4 in) and larger, specifications for steel fittings can generally be divided into two classes, depending on the joints used and the pipe size: • Flanged, welded (ANSI B16.9)
TABLE 11.15 Steel Fittings Mitered Fittings
Wrought Fittings
Crosses Two–piece elbows, 0–30° bend Three–piece elbows, 31–60° bend Four–piece elbows, 61–90° bend Four–piece, long radius elbows Laterals, equal diameters Laterals, unequal diameters Reducers Eccentric reducers Tees Reducing tees True wyes
Caps 45° elbows 90° elbows, long radius 90° elbows, short radius 90° reducing elbows, long radius Multiple-outlet fittings Blind flanges Lap joint flanges Slip-on flanges Socket-type welding flanges Reducing flanges Threaded flanges Welding neck flanges Reducers Eccentric reducers 180° returns, long radius Saddles Reducing outlet tees Split tees Straight tees True wyes
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WATER DISTRIBUTION SYSTEM DESIGN
11.24
Chapter Eleven
FIGURE 11.5 Typical mitered steel fittings. Adopted from Sanks et al,. (1989)
FIGURE 11.6 Wrought (forged) steel fittings for use with welded flanges. Adopted from Sanks et al,. (1989)
• Fabricated (AWWA C208). Fittings larger than 100 mm (4 in) should conform to ANSI B16.9 (“smooth” or wrought) or AWWA C208 (mitered). Threaded fittings larger than 100 mm (4 in.) should be avoided. The ANSI B16.9 fittings are readily available up to 300–400 mm (12–16 in) in diameter. Mitered fittings are more readily available and cheaper for larger fittings. The radius of a mitered elbow can range from 1 to 4 pipe diameters. The hoop tension
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WATER DISTRIBUTION SYSTEM DESIGN
Water Distribution System Design 11.25
concentration on the inside of elbows with a radius less than 2.5 pipe diameters may exceed the safe working stress. This tension concentration can be reduced to safe levels by increasing the wall thickness, as described in ANSI B31, AWWA C208, and Piping Engineering (Tube Turns Division, 1974). Design procedures for mitered bends are described in ANSI B31.1 and B31.3. Types of steel fittings are shown in Table 11.15 and in Figures 11.5 and 11.6. 11.4.3.6 Linings and coatings. Cement mortar is an excellent lining for steel pipe. Tables 11.9 and 11.10 show required thicknesses for steel pipe. Steel pipe can also be coated with cement mortar. Recommended mortar coating thicknesses are shown in AWWA C205. These thicknesses, however, are often thinner than those required to provide adequate protection. Many designers specify a minimum cement mortar coating thickness of at least 19 mm (3/4 in). In corrosive soils, the following coatings may be appropriate for protecting steel pipe: • Hot-applied coal-tar enamel (AWWA C203) • Cold-applied tape system (AWWA C214) • Fusion-bonded epoxy (AWWA C213) • Coal-tar epoxy (AWWA C210) • Hot-applied extruded polyethylene [ASTM D 1248 (material only)] As another alternative, epoxy-lined and -coated steel pipe can be used. Because this lining is only 0.3–0.6 mm (0.12–0.20 in) thick, the ID of the bare pipe is only slightly reduced by such linings. Epoxy-lined steel pipe is covered by AWWA C203, C210, and C213 standards. Before specifying epoxy lining and coating, pipe suppliers must be consulted to determine the limitations of sizes and lengths of pipe that can be lined with epoxy. Flange faces should not be coated with epoxy if flanges with serrated finish per AWWA C207 are specified.
11.4.4 Reinforced Concrete Pressure Pipe (RCPP) Several types of RCPP are manufactured and used in North America. These include steel cylinder (AWWA C300), prestressed, steel cylinder (AWWA C301), noncylinder (AWWA
TABLE 11.16
*
General Description of Reinforced Concrete Pressure Pipe
Type of Pipe
AWWA Standard
Steel Cylinder
Reinforcement
Design Basis*
Steel cylinder Prestressed, steel cylinder Noncylinder Pretensioned, steel cylinder
C300 C301 C302 C303
X X None X
Mild reinforcing steel Prestressed wire Mild reinforcing steel Mild reinforcing steel
Rigid Rigid Rigid Semirigid
“Rigid” and “cemirigid” are terms used in AWWA M9 and are intended to differentiate between two design theories. Rigid pipe does not depend on the passive resistance of the soil adjacent to the pipe for support of vertical loads. Semirigid pipe requires passive soil resistance for vertical load support. The terms “rigid” and “semirigid” as used here should not be confused with the definitions stated by Marston in Iowa State Experiment Station Bulletin No. 96.
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WATER DISTRIBUTION SYSTEM DESIGN
11.26
Chapter Eleven
FIGURE 11.7 Cross section of AWWA C300 pipe. AWWA M9 (1995)
C302), and pretensioned, steel cylinder (AWWA 303). Some of these types are made for a specific type of service condition and others are suitable for a broader range of service conditions. A general description of RCPP types is shown in Table 11.16. The designer should be aware that not all RCPP manufactures make all of the pipe types listed. 11.4.4.1 Steel cylinder pipe, AWWA C300. Prior to the introduction of prestressed steel cylinder pipe (AWWA C301), in the early 1940s, most of the RCPP in the United States was steel cylinder type pipe. New installations of steel cylinder pipe have been declining over the years as AWWA C301 and C303 pipes have gained acceptance. Steel cylinder pipe is manufactured in diameters of 750–3600 mm (30–44 in). Standard lengths are 3.6–7.2 meters (12–24 ft). AWWA C300 limits the reinforcing steel furnished in the cage(s) to no less 40 percent of the total reinforcing steel in the pipe. The maximum loads and pressures for this type of pipe depend on the pipe diameter, wall thickness, and strength limitations of the concrete and steel. The designer should be aware that this type of pipe can be designed for high internal pressure, but is limited in external load capacity. A cross section of AWWA C300 pipe and a typical joint configuration is shown in Fig. 11.7. 11.4.4.2 Prestressed steel cylinder pipe, AWWA C301. Prestressed steel cylinder pipe has been manufactured in the United States since 1942 and is the most widely used type of concrete pressure pipe, except in the western United States. Due to cost considerations, AWWA C301 pipe is often used for high–pressure transmission mains, but it also has been for distribution mains and for many other low and high pressure uses. A distressing number of failures of this pipe occurred in the United States primarily during the 1980s. The outer shell of the concrete cracked, allowing the reinforcement to corrode and subsequently fail. These failures have resulted in significant revisions to the standards covering this pipe’s design. Even so, the designer should not necessarily depend solely on AWWA specifications or on manufacturers’ assurances, but should make a careful analysis of internal pressure (including waterhammer) and external loads. Make certain that the tensile strain in the outer concrete is low enough so that cracking will either not occur at all or will not penetrate to the steel under the worst combination of external and internal loading.
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WATER DISTRIBUTION SYSTEM DESIGN
Water Distribution System Design 11.27
FIGURE 11.8 Cross section of AWWA C301 pipe. AWWA M9 (1995)
Prestressed cylinder pipe has the following two general types of fabrication: (1) a steel cylinder lined with a concrete core, or (2) a steel cylinder embedded in concrete core. Lined cylinder pipe is commonly available in IDs from 400 to 1200 mm (16–48 in). Sizes through 1500 mm (60 in) are available through some manufacturers. Embedded cylinder pipe is commonly available in inside diameters 1200 mm (48 in) and larger. Lengths are generally 4.9–7.3 m (16–24 ft), although longer units can be furnished. AWWA C304, Standard for Design of Prestressed Concrete Cylinder Pipe covers the design of this pipe. The maximum working pressure for this pipe is normally 2758 kPa (400 psi). The design method is based on combined loading conditions (the most critical type of loading for rigid pipe) and includes surge pressure and live loads. Cross sections of AWWA C301 pipe (lined and embedded) and typical joint configurations are shown in Fig. 11.8.
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WATER DISTRIBUTION SYSTEM DESIGN
11.28
Chapter Eleven
FIGURE 11.9 Cross section of AWWA C302 pipe. AWWA M9 (1995)
11.4.4.3 Noncylinder pipe, AWWA C302. The maximum working pressure of noncylinder pipe is 379 kPa (55 psi) and is generally not suitable for typical municipal systems. Noncylinder pipe is commonly furnished in diameters of 300 to 3600 mm (12–144 in), but larger diameters can be furnished if shipping limitations permit. Standard lengths are 2.4–7.3 m (8–24 ft) with AWWA C302 limiting the maximum length that can be furnished for each pipe size. Cross sections of AWWA C302 pipe with steel and concrete joint ring configurations are shown in Fig. 11.9. 11.4.4.4 Pretensioned steel cylinder, AWWA C303. Pretensioned steel cylinder, or commonly called concrete cylinder pipe (CCP), is manufactured in Canada and in the western and southwestern areas of the United States. It is commonly available in diameters of 300–1350 mm (12–54 in). Standard lengths are generally 7.3 to 12.2 m(24–40 ft). With maximum pressure capability up to 2758 kPa (400 psi), the longer laying length, and the overall lighter handling weight, AWWA C303 is a popular choice among many designers for various applications including municipal transmission and distributions mains. Manufacture of CCP begins with a fabricated steel cylinder with joint rings which is hydrostatically tested. A cement-mortar lining is then placed by the centrifugal process inside the cylinder. The nominal lining thickness is 13 mm (1/2 in) for sizes up to and including 400 mm (16 in), and 19 mm (3/4 in) for larger sizes. After the lining is cured, the cylinder is wrapped, typically in a helical pattern, with a smooth, hot-rolled steel bar, using a moderate tension in the bar. The size and spacing of the bar, as well as the thickness of the steel cylinder, are proportioned to provide the required pipe strength. The cylinder and bar wrapping are then covered with a cement slurry and a dense mortar coating that is rich in cement. Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
WATER DISTRIBUTION SYSTEM DESIGN
Water Distribution System Design 11.29
FIGURE 11.10 Cross section of AWWA C303 pipe. AWWA M9 (1995)
The design of CCP is based on a semirigid pipe theory in which internal pressure and external load are designed for separately but not in combination. Since the theory of semirigid pipe design for earth loads above the pipe is based on the passive soil pressure adjacent to the sides of the pipe, the design must be closely coordinated with the installation conditions. A cross section of AWWA C303 pipe and a typical joint configuration is shown in Fig. 11.10. 11.4.5 High–Density Polyethylene (HDPE) Pipe Polyethylene pressure pipe has been used in the United States by various utilities in urban environments for several years. Nearly all natural gas distribution pipe installed in the United States since 1970 is polyethylene. It has only recently, however, become available as an AWWA–approved transmission and distribution system piping material. AWWA Standard C906, Polyethylene Pressure Pipe and Fittings, 4 in. through 63 in., for Water Distribution, became effective March 1, 1992. AWWA Standard C901, Polyethylene Pressure Pipe, Tubing, and Fittings, 1/2 in. through 3 in., for Water Service, has been in effect since 1978. Prior to 1992, polyethylene pipe use in municipal water distribution systems was normally limited to water services. Since AWWA approval in 1992, however, polyethylene pipe is now being used in transmission and distribution system applications. Because of its resistance to corrosion, its light weight and high strength to weight ratio, resistance to cracking, smoother interior wall surface, and its demonstrated resistance to damage during seismic events, HDPE pipe is gaining acceptance for use in municipal water systems 11.4.5.1 Materials. Low density polyethylene was first introduced in the 1930’s and 1940s in England and then in the United States. This first material was commonly used for cable coatings. Pipe grade resins were developed in the 1950s and have evolved to today’s high density, extra high molecular weight materials. AWWA C906 specifies several different resins but, today all HDPE water pipe, manufactured in the United States, is made with a material specified in ASTM D 3350 by a cell classification 345434C.
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WATER DISTRIBUTION SYSTEM DESIGN
11.30
Chapter Eleven TABLE 11.17 DR* 11
Pressure Class versus DR–AWWA C906 Pressure Class, Safety Factor 2.0,psi (kPa) 160 (1100)
13.5
130
17
100
(900) (690)
21
80
(550)
These DRs are from the standard dimension ratio series established by ASTM F 412. *
HDPE pipe is rated for pressure capacity at 23ºC (73.4ºF). Being a thermoplastic, the pressure capacity of HDPE pipe is related to its operating temperature. Through the normal range of municipal water system temperatures, 0ºC–24º (32º F–75º F), the pressure rating of HDPE remains relatively constant. As the operating temperature rises above 23ºC (73.4ºF), however, the pressure capacity of HDPE pipe decreases to a level below its pressure class. The pressure rating for HDPE pipe at service temperature of 60°C (140°F) would be about half its rating at 23ºC (73.4ºF). 11.4.5.2 Available sizes and thicknesses. AWWA C906 covers HDPE pipe in sizes 100 –1600 mm (4–63 in). The design, according to AWWA C906, of HDPE pipe is similar to AWWA C900 PVC pipe in that HDPE pipe is rated according to “pressure classes.” The pressure classes detailed in AWWA C906 include allowance for pressure rises above working pressure due to occasional positive pressure transients not exceeding two times the nominal pressure class and recurring pressure surges not exceeding one and one-half times the nominal pressure class. AWWA C906 lists HDPE pipe sizes according to the IPS (steel pipe) and the ISO (metric) sizing systems. Ductile iron pipe sizes are also available. As with AWWA C900 and C905 (PVC pipe), AWWA C906 contains dimension ratio (outside diameter to wall thickness) specifications. Table 11.17 presents dimension ratios with corresponding pressure classes as defined in AWWA C906 for commonly available HDPE pipe. 11.4.5.3 Joints. HDPE pipe can be joined by thermal butt-fusion, flange assemblies, or mechanical methods as may be recommended by the pipe manufacturer. HDPE is not to be joined by solvent cements, adhesives (such as epoxies), or threaded-type connections. Thermal butt-fusion is the most widely used method for joining HDPE piping. This procedure uses portable field equipment to hold pipe and/or fittings in close alignment while the opposing butt-ends are faced, cleaned, heated and melted, fused together, and then cooled under fusion parameters recommended by the pipe manufacturer and fusion equipment supplier. For each polyethylene material there exists an optimum range of fusion conditions, such as fusion temperature, interface pressure, and cooling time. Thermal fusion should only be conducted by persons who have received training in the use of the fusion equipment according to the recommendations of the pipe manufacturer and fusion equipment supplier. In situations where different polyethylene piping materials must be joined by thermal butt-fusion process, both pipe manufacturers should be consulted to determine the appropriate fusion procedures. ASTM D 2657 covers thermal butt-fusion of HDPE pipe.
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WATER DISTRIBUTION SYSTEM DESIGN
Water Distribution System Design 11.31
HDPE pipe is normally joined above ground and then placed in the pipeline trench. The thermal butt-fusion joint is not subject to movement due to thrust and does not require thrust restraint such as thrust blocks. Flanged and mechanical joint adapters are available for joining HDPE pipe to valves and ductile iron fittings. The designer should always consult the pipe manufacturer to insure a proper fit between pipe and fittings. The designer should also make sure that, when connecting to a butterfly valve, the valve disc will freely swing to the open position without hitting the face of the stub end or flange adapter. 11.4.5.4 Gaskets. Gaskets are not necessary for HDPE pipe using thermal butt-fusion joints. 11.4.5.5 Fittings. AWWA C906 HDPE pipe for municipal use is manufactured in ductile iron pipe OD sizes, so ductile iron fittings, conforming to AWWA C110, can be used in all available sizes. See Sec. 11.4.1.5 for discussion on ductile iron fittings. Although not widely used, HDPE fittings, in configurations similar to ductile iron fittings, are also available. AWWA C906 covers HDPE pressure pipe fittings. 11.4.5.6 Linings and coatings. HDPE pipe does not require lining or coating.
11.4.6 Asbestos-Cement Pipe (ACP) Asbestos-cement pipe (ACP), available in the United States since 1930, is made by mixing Portland cement and asbestos fiber under pressure and heating it to produce a hard, strong, yet machinable product. It is estimated that over 480,000 km (300,000 mi) of ACP is now in service in the United States. In the late 1970s, attention was focused on the hazards of asbestos in the environment and, particularly, in drinking water. There was significant debate of the issue with one set of experts advising of the potential dangers and a second set of experts claiming that pipes made with asbestos do not result in increases in asbestos concentrations in the water. Studies have shown no association between water delivered by ACP and any general disease, however, the general fear that resulted from the controversy had a tremendous negative impact on ACP use in the United States. The debate on the health concerns of using ACP along with the introduction of PVC pipe into the municipal water system market, has reduced the use of ACP significantly in the past several years. 11.4.6.1 Available sizes and thicknesses. ACP is available in diameters of 100–1050 mm (4–42 in). Refer to ASTM C 296 and AWWA C401, C402, and C403 for thickness and pressure ratings and AWWA C401 and C403 for detailed design procedures. AWWA C401 for 100–400 mm (4–16 in) pipe is similar to AWWA C403 for 450–1050 mm (18 –42 in) pipe. The properties of asbestos-cement for distribution pipe (AWWA C400) and transmission pipe (AWWA C402) are identical. However, under AWWA C403 (transmission pipe) the suggested minimum safety factor is 2.0 for operating pressure and 1.5 for external loads, whereas the safety factors under AWWA C402 (distribution pipe) are 4.0 and 2.5, respectively. So the larger pipe has the smaller safety factors. Section 4 in AWWA C403 justifies this on the basis of surge pressure in large pipe tending to be less than those in small pipes. However, surge pressures are not necessarily a function of pipe diameter (Chap. 12). The operating conditions, including surge pressures, for any proposed pipeline installation should be closely evaluated before the pipe class is selected. It is the engineer’s design prerogative to select which of the safety factors should apply. AWWA C400 specifies that safety factors should be no less than
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WATER DISTRIBUTION SYSTEM DESIGN
11.32
Chapter Eleven
4.0 and 2.5 if no surge analysis is made. The low safety factors given in AWWA C403 should only be used if all loads (external, internal, and transient) are carefully and accurately evaluated. 11.4.6.2 Joints and fittings. The joints are usually push-on, twin-gasketed couplings, although mechanical and rubber gasket push-on joints can be used to connect ACP to ductile iron fittings. Ductile iron fittings conforming to ANSI/AWWA C110/A21.10 are used with ACP, and adapters are available to connect ACP to flanged or mechanical ductile iron fittings. Fabricated steel fittings with rubber gasket joints can also be used.
11.4.7 Pipe Material Selection Buried piping for municipal water transmission and distribution must resist internal pressure, external loads, differential settlement, and corrosive action of both soils and, potentially, the water it carries. General factors to be considered in the selection of pipe include the following: • Service conditions – Pressure (including surges and transients) – Soil loads, bearing capacity of soil, potential settlement – Corrosion potential of soil – Potential corrosive nature of some waters • Availability – Local availability and experienced installation personnel – Sizes and thicknesses (pressure ratings and classes) – Compatibility with available fittings • Properties of the pipe – Strength (static and fatigue, especially for waterhammer) – Ductility – Corrosion resistance – Fluid friction resistance (more important in transmission pipelines) • Economics – Cost (installed cost, including freight to job site and installation) – Required life – Cost of maintenance and repairs The items listed above are general factors relating to pipe selection to be considered during the design of any pipeline. Since most municipal water system projects are either let out to competitive bid or are installed as a part of private land development,
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WATER DISTRIBUTION SYSTEM DESIGN
Water Distribution System Design 11.33 TABLE 11.18. Pipe
Comparison of Pipe for Municipal System Service Advantages
Disavantages/limitations
Ductile iron (DIP)
Yield strength: 290,000 kPa (42,000 lb/in2); E 166 3 106 kPa (24 106 lb/in2); ductile, elongation ⬇ 10%; good corrosion resistance, wide variety available fittings and joints; available sizes: 100–1350 mm (4–54 in); ID, wide range of available thicknesses, good resistance to waterhammer, high strength for supporting earth loads
Maximum pressure 2400 kPa (350 lb/in2); high cost especially for long freight hauls, no diameters above 1350 mm (54 in); difficult to weld, may require wrapping or cathodic protection in corrosive soils
Steel
Yield strengths: 207,000–414,000 kPa (30,000–60,000 lb/in2); ultimate strengths: 338,000–518,000 kPa (49,000–75,000 lb/in2); E 207 106 kPa (30 106 lb/in2); ductile, elongation varies from 17 to 35%, pressure rating to 17,000 kPa (2500 lb/in2); diameters to 3.66 m (12 ft); widest variety of available fittings and joints, custom fittings can be mitered and welded, excellent resistance to waterhammer, low cost, high strength for supporting earth loads
Poor corrosion resistance unless both lined and coated or wrapped, may require cathodic protection in corrosive soils, higher unit cost in smaller diameters
Polyvinyl chloride (PVC)
Tensile strength (hydrostatic design basis) 26,400 kPa (4000 lb/in2); E 2,600,000 kPa (400,000 lb/in2); light weight, very durable, very smooth, liners and wrapping not required, can use ductile iron fittings with adapters, diameters from 100 to 375 mm (4–36 in)
Maximum pressure 2400 kPa (350 lb/in2); waterhammer not included in AWWA C905; limited resistance to cyclic loading, unsuited for outdoor use above ground
High-density polyethylene (HDPE)
Tensile strength (hydrostatic design basis) 11,000 kPa (1600 lb/in2); E 896,000 kPa (130,000 lb/in2); lightweight, very durable, very smooth, liners and wrapping not required, can use ductile iron fittings, diameters from 100 to 1600 mm (4 to 63 in)
Maximum pressure 1750 kPa (250 lb/in2); relatively new product, 750 mm (30 in) is largest size available for municipal system pressures, thermal butt-fusion joints, requires higher laborer skill
Reinforced concrete pressure (RCPP)
Several types available to suit different conditions, high strength for supporting earth loads, wide variety of sizes from 300 to 3600 mm (24–144 in)
Attacked by soft water, acids, sulfides, sulfates, and chlorides, often requires protective coatings; waterhammer can crack outer shell, exposing reinforcement to corrosion and destroying its strength with time; maximum pressure 1380 kPa (200 lb/in2)
AsbestosYield strength: not applicable; design based cement (ACP) on crushing strength, see ASTM C 296 and C 500; E 23,500,000 kPa (3,400,000 lb/in2); rigid, lightweight in long lengths, low cost; diameters from 100 to 1050 mm (4–42 in), compatible with cast-iron fittings, pressure ratings from 1600 to 3100 kPa (225– 450 lb/in2) for large pipe 450 mm (18 in) or more
Attacked by soft water, acids, sulfates; requires thrust blocks at elbows tees, and dead ends; maximum pressure 1380 kPa (200 lb/in2) for pipe up to 400 mm (16 in); health hazards of asbestos in potable water service are controversial
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WATER DISTRIBUTION SYSTEM DESIGN
11.34
Chapter Eleven
the designer will find that installed cost, lacking specific service conditions that require otherwise, will tend to dictate pipe selection. For example, steel pipe and reinforced concrete pressure pipe are both available in 300 mm (12 in) diameter. However, the installed cost of ductile iron pipe or PVC pipe is typically lower (typical municipal use) in the 300 mm (12 in) size. Therefore, if the service conditions do not require the high pressure capabilities of steel or reinforced concrete pressure pipe, the logical choice for 300 mm pipe (12 in) will optionally be ductile iron or PVC. Conversely, if the proposed pipeline is 900 mm (36 in) in diameter, the installed cost of both steel and reinforced concrete pressure pipe, depending on location, tend to be much more competitive. A general comparison of the various types of pipe used in municipal water systems is shown in Table 11.18.
11.5 PIPELINE DESIGN This section will address typical issues that are addressed during the design of water distribution and transmission pipelines. Pressure pipelines must primarily be able to resist internal pressures, external loads (earth and impact loads), forces transferred along the pipe when pipe-to-soil friction is used for thrust restraint, and handling during construction. Each of these design issues will be discussed and appropriate formulas presented. 11.5.1 Internal Pressures The internal pressure of a pipeline creates a circumferential tension stress, frequently termed hoop stress, that governs the pipeline thickness. In other words, the pipe must be thick enough to withstand the pressure of the fluid within. The internal pressure used in design should be that to which the pipe may subjected during its lifetime. In a distribution system this pressure may be the maximum working pressure plus and allowance for surge. It may be also the pipeline testing pressure or the shutoff head of an adjacent pump. In a transmission pipeline, the pressure is measured by the vertical distance between the pipe centerline and the hydraulic grade line. Potential hydraulic grade lines on transmission pipelines should be carefully considered. The static hydraulic grade line is potentially much higher than the dynamic grade line if a downstream valve is closed. Hoop tensile stress is given by the equation pD s = (11.1) 2t 2 where s allowable circumferential stress in kPa (lb/in ) p pressure in kPa (lb/in2) D outside diameter of pipe in mm (in) t thickness of the pipe in mm (in) It should be noted that this equation is the basis for determining the circumferential stress in steel and reinforced pressure pipe and for determining the pressure classes and pressure ratings for virtually all other different types of pressure pipe. 11.5.2 Loads on Buried Pipe Buried pipes must support external superimposed loads, including the weight of the soil above plus any live loads such as wheel loads due to vehicles or equipment. The two broad categories for external structural design are rigid and flexible pipe. Rigid pipe supports external loads because of the strength of the pipe itself. Flexible pipe distributes the exter-
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WATER DISTRIBUTION SYSTEM DESIGN
Water Distribution System Design 11.35
nal loads to surrounding soil and/or bedding material. For rigid pipe, the soil between the pipe and the trench wall is more compressible than the pipe. This causes the pipe to carry most of the load across the width of the trench. For a flexible pipe, the fact that the pipe deflects causes the soil directly over the pipe to settle more than the adjacent soil. This settlement produces shearing forces which tend to reduce the load on the flexible pipe. DIP, steel pipe, PVC pipe, and HDPE, pipe should be considered flexible and designed accordingly. AWWA C300, C301, and C302 pipe and AC pipe should be considered rigid. AWWA C303 pipe is designed for external loads, according to AWWA M9 (AWWA, 1995), as “semirigid” using rigid pipe formulas to determine the pipe load and controlled pipe deflection as with a flexible pipe. Supporting strengths for flexible conduits are generally given as loads required to produce a deflection expressed as a percentage of the diameter. Ductile iron pipe may be designed for deflections up to 3 percent of the pipe diameter according to ANSI A21.50. Historically, plastic pipe manufacturers generally agreed that deflections up to 5 percent of the diameter were acceptable. Some manufacturers suggest, however, that deflections up to 7 percent are permissible. Many engineers, however, believe these values are much too liberal and use 2 to 3 percent. Recommended design deflections for flexible pipe are shown in Table 11.19. The following generally describes the analysis of superimposed loads on buried pipes. As will be seen, the design involves the stiffness of the pipe, the width and depth of the trench, the kind of bedding, the kind of surrounding soil, and the size of pipe. There are several different types of pipeline installation conditions that should be recognized by the design engineer, because different installation conditions will result in different loads on the pipeline. In this text, the only type of installation condition addressed is commonly referred to as a trench condition, where the width of the trench for the pipeline is no larger than two times the width of the pipe. This condition, naturally, requires that the surrounding soil will hold a vertical (or nearly vertical) wall. The subject of how a buried pipe resists earth loads is a subject that should be thoroughly understood by the pipeline design engineer. A complete presentation of this subject is outside this text, however, further discussions are given in AWWA M11 (American Water Works Association, 1984), AWWA M9 (American Water Works Association, 1995), the Handbook of PVC Pipe (Uni-Bell PVC Pipe Association, 1991), the DIPRA handbook; (Ductile Iron Pipe Research Association, 1984), and in many other publications. 11.5.2.1 Earth loads. The Marston theory is generally used to determine the loads
TABLE 11.19
Recommended Maximum Deflections for Flexible Pipe
Type of Pipe DIP PVC HDPE Steel, mortar lined and coated Steel, mortar lined and flexible coated (tape) Steel, flexible coating and lining AWWA C303
Maximum Deflection*,✝ 2–3% 3–5% 3–5% 1.5–2% 2–3% 3–5% D2/4000
Percentages are of pipe diameter. D in AWWA C303 is pipe diameter.
* ✝
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WATER DISTRIBUTION SYSTEM DESIGN
11.36
Chapter Eleven
imposed on buried pipe by the soil surrounding it. This theory is applicable to both flexible and rigid pipes installed in a variety of conditions. Trench conduits are installed in relatively narrow excavations in passive or undisturbed soil and then covered with earth backfill to the original ground surface. The trench load theory is based on the following assumptions: • Load on the pipe develops as the backfill settles because the backfill is not compacted to the same density as the surrounding earth. • The resultant load on an underground structure is equal to the weight of the material above the top of the conduit minus the shearing or friction forces on the sides of the trench. These shearing forces are computed in accordance with Rankine’s theory. • Cohesion is assumed to be negligible because (1) considerable time must elapse before effective cohesion between the backfill material and the sides of the trench can develop, and (2) the assumption of no cohesion yields the maximum probable load on the conduit. • In the case of rigid pipe, the side fills may be relatively compressible and the pipe itself will carry practically all the load developed over the entire width of the trench. When a pipe is placed in a trench, the prism of backfill placed above it will tend to settle downward. Frictional forces will develop along the sides of the trench walls as the backfill settles and act upward against the direction of the settlement. The fill load on the pipe is equal to the weight of the mass of fill material less the summation of the frictional load transfer. 11.5.2.2 Rigid pipe. The load on buried rigid pipe is expressed by the following formula: Wd = CdwBd2
(11.2)
where Wd = trench fill load, pounds per linear foot (lb/Lft) Cd = trench load coefficient w = unit weight of fill material, (lb/ft3) Bd = width of trench at the top of the pipe, in ft Cd is further defined as: 1e2Ku(H/Bd) Cd (11.3) 2Ku' where Cd = trench load coefficient e = base of natural logarithms K = tan (45º - '2) Rankine’s ratio of active lateral unit pressure to vertical unit pressure, with ' = friction angle between backfill and soil u' tan ' friction coefficient of friction between fill material and sides of trench H height of fill above top of pipe (ft) Bd width of trench at the top of the pipe (ft) Recommended values for the product of Ku' for various soils are: Ku 0.1924 for granular materials without cohesion Ku' 0.1650 maximum for sand and gravel Ku' 0.1500 maximum for saturated top soil Ku' 0.1300 maximum for ordinary clay Ku' 0.1100 maximum for saturated clay
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WATER DISTRIBUTION SYSTEM DESIGN
Water Distribution System Design 11.37
For very deep trenches, the load coefficient Cd approaches a value of Ku'/2, so an accurate selection of the appropriate Ku' value becomes more important. The design engineer can benefit greatly from the expert services of an experienced geotechnical engineer who can provide this data to the designer. Generally, though, when the character of the soil is uncertain, it is adequate to assume, for preliminary design, that Ku' 0.150 and w 120 lb/ft3 (1922 kg/m3). Study of the load formula shows an increase in trench width, Bd (Bd is measured at the top of the pipe), will cause a marked increase in load. Consequently, the value of B should be held to the minimum that is consistent with efficient construction operations and safety requirements. If the trench sides are sloped back, or if the width of the trench is large in comparison with the pipe, Bd and the earth load on the pipe can be decreased by constructing a narrow subtrench at the bottom of the wider trench. As trench width increases, the upward frictional forces become less effective in reducing the load on the pipe until the installation finally assumes the same properties as a positive projecting embankment condition, where a pipe is installed with the top of the pipe projecting above the surface of the natural ground (or compacted fill) and then covered with earth fill. This situation is common when Bd is approximately equal to or greater than H. The positive projection embankment condition represents the severest load to which a pipe can be subjected. Any further increase in trench width would have no effect on the trench load. The maximum effective trench width, where transition to a positive projecting embankment condition occurs, is referred to as the “transition trench width.” The trench load formula does not apply when the transition trench width has been exceeded. 11.5.2.3 Flexible pipe. For a flexible pipe, the ability to deflect without cracking produces a situation where the central prism of soil directly over the pipe settles more than the adjacent soil prisms between the pipe and the trench wall. This differential settlement produces shearing forces which reduce the load on a flexible pipe. If the flexible pipe is buried in a trench less than two times the width of the pipe, the load on the pipe may be computed as follows: B Wd = CdwBd2c = CdwBd Bc (11.4) Bd where Wd = trench fill load, in pounds per linear foot Cd = trench load coefficient as defined above in Eq. 11.4 w = unit weight of fill material (lb/ft3) Bd = width of trench at the top of the pipe (ft) Bc = outside diameter of the pipe (ft) The deflection of a properly designed flexible pipe installation is limited by the pipe stiffness and the surrounding soil. Under soil loads, the pipe tends to deflect and develop passive soil support at the sides of the pipe. Recommended deflection limits for various types of pipe are shown above. The Iowa deflection formula was first proposed by M. G. Spangler. It was later modified by Watkins and Spangler and has been frequently rearranged by others. In one of its most common forms, deflection is calculated as follows: KWr3 ∆x D 3 EI 0.061E'r
(11.5)
where ∆x horizontal deflection of pipe (in) D deflection lag factor, see further definition below (1.01.5) K bedding constant (0.1) W load per unit of pipe length, in pounds per linear inch (lb/Lin) r radius (in) E' modulus of elasticity of pipe (lb/in2) I transverse moment of inertia per unit length of pipe wall, (in4/(Lin) in3) E' modulus of soil reaction, see further definition below (lb/in2) In pipe soil systems, as with all engineering systems involving soil, the soil consoli-
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WATER DISTRIBUTION SYSTEM DESIGN
11.38
Chapter Eleven
dation at the sides of the pipe continues with time after the maximum load reaches the top of the pipe. Spangler recognized that some pipe deflections increased by as much as 30 percent over a period of 40 years. For this reason, he recommended addition of a deflection lag factor (Dl )of 1.5 as a conservative design procedure. Others recommend using an ultimate load with a Dl equal to unity. One attempt to develop information on values of E' was conducted by Amster K. Howard of the U. S. Bureau of Reclamation. Howard reviewed both laboratory and field data from many sources. Using information from over 100 laboratory and field tests, he compiled a table of average E' values for various soil types and densities. Howard’s data are reproduced in Table 11.20. These data can be used in design of pipe soil installations. 11.5.3 Thrust Restraint Thrust forces are unbalanced forces in pressure pipelines that occur at changes in direction (such as in bends, wyes, tees, etc.), at changes in cross-sectional area (such as in reducers), or at pipeline terminations (such as at bulkheads). If not adequately restrained, these forces tend to disengage nonrestrained joints. Two types of thrust forces are (1) hydrostatic thrust due to internal pressure of the pipeline, and (2) hydrodynamic thrust due to changing momentum of flowing water. Since most water lines operate at relatively low velocities, the dynamic force is insignificant and is usually ignored when computing thrust. For example, the dynamic force created by water flowing at 2.4 m/s (8 ft/s) is less than the static force created by 6.9 kPa (1 psi). Typical examples of hydrostatic thrust are shown in Fig. 11.11. The thrust in dead ends, outlets, laterals, and reducers is a function of the internal pressure P and the crosssectional area A at the pipe joint. The resultant thrust at a bend is also a function of the deflection angle ∆ and is given by the following: T 2PA sin ( /2)
(11.6)
where T hydrostatic thrust, in pounds P internal pressure, in pounds per square inch A cross-sectional area of the pipe joint, in square inches deflection angle of bend, in degrees For buried pipelines, thrust resulting from small angular deflections at standard and beveled pipe with rubber-gasket joints is resisted by dead weight or frictional drag of the pipe, and additional restraint is not usually needed. Thrust at in-line fittings, such as valves and reducers, is usually restrained by frictional drag on the longitudinally compressed downstream pipe. Other fittings subjected to unbalanced horizontal thrust have the following two inherent sources of resistance: (1) frictional drag from the dead weight of the fitting, earth cover, and contained water, and (2) passive resistance of soil against the against the back of the fitting. If frictional drag and/or passive resistance is not adequate to resist the thrust involved, then it must be supplemented either by increasing the supporting area on the bearing side of the fitting with a thrust block or by increasing frictional drag of the line by “tying” adjacent pipe to the fitting. Unbalanced uplift thrust at a vertical deflection is resisted by the dead weight of the fitting, earth cover, and contained water. If that is not adequate to resist the thrust involved, then it must be supplemented either by increasing the dead weight with a gravity-type thrust block or by increasing the dead weight of the line by “tying” adjacent pipe to the fitting. When a high water table or submerged conditions are encountered, the effects of buoyDownloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
WATER DISTRIBUTION SYSTEM DESIGN
Water Distribution System Design 11.39 TABLE 11.20 Average Values of Modulus of Soil Reaction, E′ (for Initial Flexible Pipe Deflection)* E′ for Degree of Compaction of Bedding, (psi)
Soil type–pipe bedding material (Unified Classification System)✝ (1)
dumped (2)
Slight,
85% Proctor,
40% relative density (3)
Moderate 85%–95% Proctor, 40–70% relative density (4)
High. 95% Proctor 70% relative density (5)
Fine-grained soils (LL50) Soils with medium to high plasticity, CH, MH, CH, MH Fine-grained soils (LL 50) Soils with medium to no plasticity, CL. ML, ML–CL, with less than 25% coarse–grained particles Fine-grained soils (LL 50) Soils with medium to no plasticity, CL. ML, ML-CL, with less than 25% coarse–grained particles
50
200
400
1000
100
400
1000
2000
Coarse–grained soils with fines GM, GC, SM, SC contains more than 12% fines Coarse-grained soils with litle or no fines GW, GP, SW, SP contains less than 12% fines
200
1000
2000
3000
Crushed rock
1000
3000
3000
3000
2
2
1
.05
Accuracy in terms of percentage deflection
No data available, consult a competent Soils engineer, otherwise use E′ 0
✝
ASTM Designation D 2487, USBR Designation E-3 Or any borderline soil beginning with one of these symbols (i.e. GM-GC, GC0SC). For 1% accuracy and predicted deflection of 3`%, actual deflection would be between 2% and 4% *Note: Values applicable only for fills less than 50 ft (15 m). Table does not include any safery factor. For use in predicting initial deflections only, appropriate defelction lag factor must be applied for long-term deflections. If bedding falls on the borderline between two compaction categories, select lower E′ value or average the two values. Percentage Proctor based on laboratory maximun dry density from test standards using about 12,500 ft-lb/ft3 (598,000 J/m3) American Society for Testing of Material (ASTM D698, AASHTO T-99, USBR Designation E-11). 1 psi 6.9 kPa. Source: Howard, A. K., Soil Reaction for Buried Flexible Pipe, U.S. Bureau of Reclamation, Denver, CO. Reprinted with permission from American Society of Civil Engineers. Abbreviations: CH, ;CH-MH, ;CL, ;GC, ;GM, ;GP, ;GW, ;LL, liquid limit; MH, ;SC, ;SM, ;SP, ;SW. *
ancy on all materials should be considered. 11.5.3.1 Thrust blocks. Thrust blocks increase the ability of fittings to resist movement by increasing the bearing area. Typical thrust blocking is shown in Fig. 11.12. Thrust block size can be calculated based on the bearing capacity of the soil as follows:
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WATER DISTRIBUTION SYSTEM DESIGN
11.40
Chapter Eleven
FIGURE 11.11 Hydrostatic thrust for typical fittings. (AWWA m9, 1995)
Area of block LB HB (T/σ)
(11.7)
where LB HB T
area of bearing surface of thrust block (ft2)
thrust force (lb)
safe bearing value for soil (lb/ft2) If it is impractical to design the block for the thrust force to pass through the geometric center of the soil bearing area, then the design should be evaluated for stability. Determining the safe bearing value is the key to “sizing” a thrust block. Values can vary from less than 1000 lb/ft2 (49.7 kN/m2) for very soft soils to several tons per square foot (kN/m2) for solid rock. Determining the safe bearing value of soil is beyond the scope of this text. It is recommended that a qualified geotechnical expert, knowledgeable of local conditions, be consulted whenever the safe bearing value of a soil is in question. Most thrust block failures can be attributed to improper construction. Even a correctly sized block can fail if it is not properly constructed. The thrust block must be placed against undisturbed soil and the face of the block must be perpendicular to the direction of and centered on the line of action of the thrust. Many people involved in construction do not realize the magnitude of the thrusts involved. As an example, a thrust block behind a 36-in (900 mm), 90º bend operating at 100 psi (689 kPa) must resist a thrust force in excess of 150,000 lb (667 kN). Another factor frequently overlooked is that thrust increases in proportion to the square of pipe diameter. A 36–in (900 mm) pipe produces about four times the thrust produced by an 18-in (450 mm) pipe operating at the same internal pressure. Even a properly designed and constructed thrust block can fail if the soil behind it is disturbed. Thrust blocks of proper size have been poured against undisturbed soil only to fail because another excavation immediately behind the block collapsed when the line was pressurized. The problems of later excavation behind thrust blocks and simply having “chunks” of buried concrete in the pipeline right-of-way have led some engineers to use tied joints only.
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WATER DISTRIBUTION SYSTEM DESIGN
Water Distribution System Design 11.41
FIGURE 11.12 Typical thrust block details. (Sanks, et al., 1989)
11.5.3.2 Restrained joints. Many engineers choose to restrain thrust from fittings by tying adjacent pipe joints. This method fastens a number of pipe joints on each side of the fitting to increase the frictional drag of the connected pipe and resist the fitting thrust. Much has been written about the length of pipe that is necessary to resist hydrostatic thrust. Different formulas are recommended in various references, particularly with respect to the length of pipe needed at each leg of a horizontal bend. AWWA M9 (AWWA, 1995) provides the following: PA sin( /2) L f(2We Wp Ww)
(11.8)
where L = length of pipe tied to each bend leg (ft) P = internal pressure (lb/in2) A = crosssectional area of first unrestrained pipe joint (in2) = deflection angle of bend (°) f = coefficient of friction between pipe and soil We = weight of soil prism above pipe (lb/Lft) Wp = weight of pipe (lb/Lft) Ww = weight of water in pipe (lb/Lft) This formula assumes that the thrust is resolved by a frictional force directly acting directly opposite to the hydrostatic thrust. It also assumes that the weight of the earth on top of the pipe can be included in the frictional calculation no matter what the installation depth. The factor 2We appearing in the denominator indicates that the weight of the earth is acting both on the top and the bottom of the pipe. AWWA M11 (AWWA, 1989) presents the following alternative formula: PA(1cos ) L f(We+Wp+Ww)
(11.9)
This formula assumes that the thrust is resolved by frictional forces acting along the length the pipe and does not assume that the weight of the earth acts both on the top and bottom of the pipe. The resolution of hydrostatic thrust, acting on a bend of angle ∆, by a length of
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WATER DISTRIBUTION SYSTEM DESIGN
11.42
Chapter Eleven
restrained-joint pipe on each side of the bend is governed by the following: • Restraint of the pipe is accomplished by a combination of indeterminate forces including friction between the pipe and soil along the pipe and passive soil pressure perpendicular to the pipe. • When the pipe is pressurized, the thrust T is not counteracted until the elbow (and length of restrained pipe) moves an amount sufficient (albeit very small) to mobilize friction and passive pressures. • To develop the frictional resistance at the top of the pipe, it is necessary that the prism of earth above the pipe be restrained from movement. • Given the above statements, the reader is directed to Fig. 11.13. The following should be noted: • The method proposed in AWWA M9 (AWWA, 1995) does not directly relate the resolution of thrust to a frictional force acting along a length of pipe. • The method proposed in AWWA M11 (AWWA, 1989) relates resolution of thrust to a frictional force acting along one leg of the pipe to be restrained. In practice, both legs on each side of the bend are restrained. • The resolved forces method presents a rational distribution of forces acting to resolve the thrust. According to Fig. 11.13, the resolution of thrust T by frictional forces acting along the pipe is given by the following equation: Rf = PA sin2 /2
(11.10)
where Rf frictional forces acting along each leg of pipe from bend of angle Inspection of Fig. 11.13 also indicates the following: • The PA sin /2 cos /2 force is a shearing force acting across the pipe joint. These forces can be significant, even for small ’s and should be considered in the design. • The length of pipe to be restrained using AWWA M9 (AWWA, 1995) or AWWA M11 (AWWA, 1989) formulas, is always greater than the lengths using the resolved forces method, PA sin2 /2, so either AWWA method can be used as a conservative analysis. As stated above, in order to develop the frictional resistance at the top of the pipe, it is necessary that the prism of earth above the pipe be restrained from movement. This can be assumed to be true and the frictional resistance at the top of the pipe included in the calculation if the following can be shown: 2Po tan Wf
(11.11)
Po is further defined as: Po (/2) H2 ko where Po force available from earth prism to provide frictional resistance (lb) W weight of soil prism above pipe (lb) f coefficient of friction between pipe and soil H
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WATER DISTRIBUTION SYSTEM DESIGN
Water Distribution System Design 11.43
T 2PA sin ∆/2
PA
PA
T
F F 2Lf(Wp Ww 2We) T
AWWA M9 FORCE DIAGRAM
AWWA M9 FORCES
T 2PA sin ∆/2
PA
PA
PA sin ∆
T
PA(I cos∆)
F PA sin ∆
PA cos ∆ PA (I cos ∆)
AWWA MI FORCES
AWWA MII FORCE DIAGRAM
T 2PA sin ∆/2
3 PA sin ∆/2 cos ∆/2
1
2 PA sin /2 cos /2 ∆
∆
PA
PA 2
T/2
T 3 I PA sin /2 Rf 2∆
T 2PA sin ∆/2
4
4 PA sin /2 2 ∆
RESOLVED FORCES
RESOLVED FORCE DIAGRAM
FIGURE 11.13 Frictional thrust restraint.
height of cover over top of pipe (ft) ko coefficient of soil at rest: 0.4 for crushed rock, 0.6 for saturated silty sands ϕ soil internal angle of friction, varies with soil type, consult geotechnical expert Therefore, if the soil prism above the pipe is restrained from movement and can be included in the restraint calculation, the formula for the length of pipe with restrained joints, in accordance with Fig. 11.13, becomes:
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WATER DISTRIBUTION SYSTEM DESIGN
11.44
Chapter Eleven
PA sin2 2 L (11.12) f(2We Wp Ww) If the soil prism above the pipe is not restrained from movement, and cannot be included in the restraint calculation, the formula for the length of pipe with restrained joints, in accordance with Figure 11.13, becomes: PA sin2 2 L f(We Wp Ww)
(11.13)
In all of the above equations, the value of the coefficient of friction f between the pipe and soil affects the length of pipe that will be required to be restrained. Tests and experience indicate that the value of f is not only a function of the type of soil, but it is also greatly affected by the degree of compaction and moisture content of the backfill, the pipe exterior, and even the pipe joint configuration. Therefore, care should be exercised in the selection of f. Coefficients of friction are generally in the range of 0.2 for PVC or a polyethylene bag to 0.35– 0.4 for a cement mortar coating.
11.6 DISTRIBUTION AND TRANSMISSION SYSTEM VALVES There are many different types of valves used in municipal water systems, particularly when water treatment plants and pumping station plants are included. There are, however, relatively few different types of valves common to water distribution systems and transmission mains. The discussion in this section will be limited to those valves normally used in municipal distribution and transmission systems.
11.6.1 Isolation Valves Isolation valves, as their name indicates, are placed into the system to isolate a portion of the system for repair, inspection, or maintenance. They are normally either fully closed or fully opened. Valves that remain in one position for extended periods become difficult (or even impossible) to operate unless they are “exercised” from time to time. Valves should be exercised at least once each year (more often if the water is corrosive or dirty). In a distribution system, isolation valves are normally installed at junctions. The normal “rule of thumb” for how many valves to install at a junction is one less valve than there are legs at the junctions. In other words, a cross junction (four legs) would require three valves, and a tee junction (three legs) would require two valves, and so on. The designer is encouraged, however, to seriously evaluate the need to isolate a critical line segment and require appropriate valving. For example, isolation of a critical line segment between two cross junctions could require the closing of six isolation valves if the “rule of thumb” is used in design. This obvious inconvenience has caused some designers to adopt a more conservative approach and specify one valve for each leg at junctions. On large-diameter transmission pipelines, it is common to require the installation of isolation valves at periodic points on the pipeline to minimize the amount of pipeline that must be drained for inspection or maintenance. Depending on the size of the pipeline, isolation valve spacing of up to 5 mi is not uncommon. In municipal water distribution and transmission systems, the two most common types of isolation valves are gate valves and butterfly valves.
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WATER DISTRIBUTION SYSTEM DESIGN
Water Distribution System Design 11.45
11.6.1.1 Gate valves. For distribution systems, where line sizes are typically between 150 and 400 mm (6–16 in), gate valves are very common. The gate valve has a disc sliding in a bonnet at a right angle to the direction of flow. Common gate valve types include the following subtypes: • Double disc • Solid wedge resilient seated • Knife The double disc gate valve is one of the most popular types for municipal distribution systems. after the discs drop into their seats, further movement of the stem wedges the discs outward to produce a leak proof shut-off even at pressure exceeding 1700 kPa (250 lb/in2). Opening the valve reverses the procedure. Hence, the discs do not slide until the wedging is relaxed, and sliding and grinding between the disc rings and body rings are minimized. Solid wedge resilient seated gate valves are a very popular valve choice and are gaining acceptance throughout the United States, particularly when the water contains even small amounts of sand or sediment. The seat of a gate valve is a pocket that can entrap solids and prevent the valve from fully closing. The resilient seat type greatly reduces this problem because it has no pocket in the body in which the gate seats (see AWWA C590). Instead, the rubber edge of the disc seats directly on the valve body. The disc is encapsulated with a resilient material (usually vulcanized rubber) that presses against the smooth, prismatic body of the valve. Because there is not pocket for the disc at the bottom of the valve to collect grit, the resilient seated gate valve is suitable for grit-laden waters as well as clean water service. The knife gate valve is lighter than other types of gate valves and is capable of handling more debris than other gate valve types; but does not shut off as effectively and is subject to leakage around the stem packing. The knife gate valve is only suitable when some leakage can be tolerated and when the maximum pressures are around 170–350 kPa (25–50 lb/in2). This valve is not often seen in municipal water system service. Gate valves are available in rising stem and nonrising stem designs. Most gate valves for buried service are nonrising stem, furnished with a 2 in2 nut that can be accessed from the ground surface to operate the valve. 11.6.1.2 Butterfly valves. A butterfly valve is a quarter-turn valve in which a disc is rotated on a shaft so that the disc seats on a ring in the valve body. The seat is usually an elastomer bonded or fastened either to the disc or to the body. Most, if not all, manufacturers have now standardized on the short body style (see AWWA C504). In the long body style, the disc is contained entirely within the valve body when the disc is in the fully open position. In the short body style, the disc protrudes into the adjacent piping when in the open position. In using the short body style butterfly valve, the designer must make sure that the pipe, including interior lining, is large enough to accept the disc. The designer should pay close attention to the seat design if the valve will be subject to throttling. AWWA C504 standards alone do not ensure butterfly valve seats that are adequate for severe throttling (as in pump control valves) where seats must be very rugged for longevity.
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WATER DISTRIBUTION SYSTEM DESIGN
11.46
Chapter Eleven
Some agencies use butterfly valves for all isolation valve sizes. Because the disc of the butterfly valve prevents passage of a line pig and pigging distribution system lines is considered by many to be a normal maintenance activity, many agencies only use gate valves for 300 mm (12 in) and smaller distribution system lines. Due to economic considerations, butterfly valves are much more commonly used than gate valves on transmission mains where the valve size is larger than 300 mm (12 in.). 11.6.2 Control Valves Special control valves are sometimes used to modulate flow or pressure by operating in a partly open position, creating headloss or pressure differential between upstream and downstream locations. Some control valves are manually operated (e.g., needle valves used to control the flow of a fluid in a valued actuator). Some control valves are poweroperated by programmed controllers. These are several different varieties of special control valves commonly used in distribution and transmission systems: pressure reducing, pressure sustaining, flow control, altitude, and pressure relief. Control valves are selected on the basis of the requirements of the hydraulic system. The designer should use great care in selecting both the type and size of control valve that will be used by carefully evaluating the range of flows that will be handled by the valve. If a valve that is too large is selected, the headloss through the valve may not be enough for proper function of the valve. On the other hand, a large differential pressure across the valve may cause cavitation which will cause noise, vibration, fluttering of the valve disc, and excessive wearing of the valve seats. While some valves can accommodate sustained velocities of up to 6.1 m/s (20 ft/s), designs for flow velocities between 2.4 and 3.7 m/s (8–12 ft/s) are common. If the designer finds that the expected range of flows is too great to be handled by one size valve, installations incorporating two of more different sizes of valves are common. With this type of installation, the smaller valves are set to operate at low flows with the larger valves only becoming active during higher flow periods. Most special control valves have the same body. Only the exterior piping (or more appropriately tubing) to the hydraulic actuator (diaphragm or piston) in the bonnet is changed to effect the type of control wanted, whether it be constant flow, constant pressure, or proportional flow. Most control valves are either angle or globe pattern. Angle valves and globe valves are similar in construction and operation except that in an angle valve, the outlet is at 90º to the inlet and the headloss is typically half as great as in the straight-through globe valve. An angle valve is useful if it can serve the dual purpose of a 90º elbow and a valve. Conversely, an angle valve should not be used in a straight piping run where a globe valve should be used. As in the angle valve, a globe valve has a disc or plug that moves vertically in a bulbous body. Flow through a globe valve is directed through two 90º turns (upward then outward) and is controlled or restricted by the disc or plug. The pressure drop across a globe valve is higher than a comparably sized angle valve. Globe valves are either diaphragm or piston operated. 11.6.2.1 Pressure reducing valve. Pressure reducing valves are often used to establish lower pressure in systems with more than one pressure zone. The pressure reducing valve will modulate to maintain a preset downstream pressure independent of the upstream pressure. As upstream pressure increases, the valve will close, creating more headloss across the valve, until the target pressure is obtained. As upstream pressure decreases, the valve will, conversely, open. If the upstream pressure decreases to a point lower than the target pressure, the valve would be wide open.
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WATER DISTRIBUTION SYSTEM DESIGN
Water Distribution System Design 11.47
11.6.2.2 Pressure sustaining valves. Pressure sustaining valves serve a unique purpose and their application is limited. In essence they operate in an opposite fashion than a pressure reducing valve. The pressure sustaining valve functions to maintain a minimum upstream pressure, closing as the upstream pressure drops and opening as the upstream pressure raises. They are sometimes used in multiple pressure zone systems when the downstream zone demand can create pressures that are too low in the upstream zone if not controlled. In these cases, the upstream pressure can be maintained by a pressure sustaining valve. Naturally, the demand in the downstream zone would need to be met by another source. 11.6.2.3 Flow control valves. Flow control valves, like the pressure reducing valve, modulate to maintain a downstream flow characteristic but, rather than pressure, this valve will modulate to maintain a preset flow. The flow can be determined by a number of alternatives. Completely hydraulic valves can be operated in response to an orifice plate (factory sized for the design flow) in the piping or the valve can be operated by an electric operator with some type of flow meter driving the electric operator. Venturi, magnetic, and propeller meter installations are common. As upstream pressure varies, the flow control valve will open or close to deliver the preset flow. 11.6.2.4 Altitude valves. Altitude valves are used to add water to reservoirs and to oneway tanks used in surge control (Chap. 12). Altitude valves are made in many variations of two functional designs: • One, in which the valve closes upon high water level in the tank and does not open again until the water leaves through a separate line and the water level in the tank falls. • Two, in which the valve closes upon high water level in the tank and opens to allow water to flow out of the tank when pressure on the valve inlet falls below a preset level or below the reservoir pressure on the downstream side of the valve. 11.6.2.5 Pressure relief valves. As the name implies, pressure relief valves serve to release fluid in a pressure system before a high pressure can develop and overstress piping and valves. They are often used in pumping station piping or in other locations where valve operation may induce higher pressures than can be tolerated by the system. Pressure relief valves are set to open at a preset high pressure. They normally vent to atmosphere. To function properly, they must be positioned so that their discharge is handled in a safe and environmentally sound manner.
11.6.3 Blow-offs Most water distribution systems, in spite of the best planning efforts, have deadends where the water can become stagnant. These locations may include cul-de-sacs beyond the last customer service, or portions of the distribution system not yet connected to the remainder of the system. At such dead-end locations, it is common to install a blow-off that can be periodically opened to allow the stagnant water to be removed from the system. Blow–offs often consist of a small diameter pipe, extended to the surface and terminated in a valve box with a valve than can be operated to allow removal of water from the system. Blow-offs sizes between 50 and 100 mm (2—4 in) are common. The function of blow-offs on transmission pipelines is often to allow draining of the line for maintenance or inspection and for flushing of the pipeline during construction. To function as a flushing element, the blow-off must be sized to allow cleansing velocity in the main pipeline.
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WATER DISTRIBUTION SYSTEM DESIGN
11.48
Chapter Eleven
11.6.4 Air Release and Vacuum Relief Valves Air valves are installed with pipelines to admit or vent air. There are basically two types: air-release valves and air-and-vacuum valves. In addition, a combination air valve is available that combines the functions of an air-release and an air-and-vacuum valve. Air-release valves are used to release air entrained under pressure at high points of a pipeline where the pipe slopes are too steep for the air to be carried through with the flow. The accumulation of air can become so large that it impairs the pipelines flow capacity. Air-release valves are installed at high points to provide for the continuous venting of accumulated air. An air-release valve consists of a chamber in which a float operates through levers to open a small air vent in the chamber top as air accumulates and to close the vent as the water level rises. The float must operate against an air pressure equal to the water pressure and must be able to sustain the maximum pipeline pressure. Air-and-vacuum valves are used to admit air into a pipe to prevent the creation of a vacuum that may be the result of a valve operation, the rapid draining or failure of a pipe, a column separation, or other causes. Although uncommon, a vacuum in a pipeline can cause the pipe to collapse from atmospheric pressure. Air-and-vacuum valves also serve to vent air from the pipeline while it is filling with water. An air-and-vacuum valve consists of a chamber with a float that is generally center guided. The float opens and closes against a large air vent. As the water level recedes in the chamber, air is permitted to enter; as the water level rises air is vented. Air-and-vacuum valves are often used as surge control devices and must be carefully sized when used for this purpose (Chap. 12). The airand-vacuum valve does not vent air under pressure. Air-release valves and air-and-vacuum valves, if not installed directly over the pipe, may be located adjacent to the pipeline. A horizontal run of pipe connects the air valve and the pipeline. The connecting pipe should rise gradually to the air valve to permit flow of the air to the valve for venting. The performance requirements of the valves are based on the venting capacity and the pressure differential across the valves (system water pressure less atmospheric pressure). The valves must be protected against freezing and the vents from these valves must be located above ground or positioned so as to prevent contamination when operating. Manufacturers’ catalogs should be consulted for accurate sizing information.
REFERENCES American Water Works Association, AWWA M11, Steel Pipe—A Guide for Design and Installation, 3rd ed., American Water Works Association, Denver, CO, 1989. American Water Works Association, AWWA M9, Concrete Pressure Pipe, 2d ed., American Water Works Association, Denver, CO, 1995. Ductile Iron Pipe Research Association, DIPRA, Handbook of Ductile Iron Pipe, 6th ed., Ductile Iron Pipe Research Association, Birmingham, AL, 1984. Handbook of PVC Pipe, Design and Construction, 3rd ed., Uni-Bell PVC Pipe Association, Dallas, TX, 1991. Lyne, C., Updates on Polyethylene Pipe Standard AWWA C906, CA/NV AWWA Spring Conference, 1997. Manganaro, C. A., Harnessed Joints for Water Pipe, AWWA Annual Conference, San Diego, CA, 1969. Metcalf & Eddy, Inc., Water Resources and Environmental Engineering, 2d ed., McGraw-Hill, New York, 1979.
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WATER DISTRIBUTION SYSTEM DESIGN
Water Distribution System Design 11.49 Piping Engineering, Tube Turns Division, Louisville, KY, 1974. Sanks, R. L., et al., Pumping Station Design, Butterworths, 1989. Spangler, M. G., Soil Engineering, 2d ed., International Textbook Company, Scranton, PA, 1960. Solley, W. B., R. R. Pierce, and H. A. Perlman, Estimated use of water in the United States in 1990 U. S. Geological Survey Circular 1081, Washington, D. C. 1993.
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Source: HYDRAULIC DESIGN HANDBOOK
CHAPTER 12
HYDRAULIC TRANSIENT DESIGN FOR PIPELINE SYSTEMS C. Samuel Martin School of Civil and Environmental Engineering Georgia Institute of Technology Atlanta, Georgia
12.1 INTRODUCTION TO WATERHAMMER AND SURGING By definition, waterhammer is a pressure (acoustic) wave phenomenon created by relatively sudden changes in the liquid velocity. In pipelines, sudden changes in the flow (velocity) can occur as a result of (1) pump and valve operation in pipelines, (2) vapor pocket collapse, or (3) even the impact of water following the rapid expulsion of air out of a vent or a partially open valve. Although the name waterhammer may appear to be a misnomer in that it implies only water and the connotation of a “hammering“ noise, it has become a generic term for pressure wave effects in liquids. Strictly speaking, waterhammer can be directly related to the compressibility of the liquid—primarily water in this handbook. For slow changes in pipeline flow for which pressure waves have little to no effect, the unsteady flow phenomenon is called surging. Potentially, waterhammer can create serious consequences for pipeline designers if not properly recognized and addressed by analysis and design modifications. There have been numerous pipeline failures of varying degrees and resulting repercussions of loss of property and life. Three principal design tactics for mitigation of waterhammer are (1) alteration of pipeline properties such as profile and diameter, (2) implementation of improved valve and pump control procedures, and (3) design and installation of surge control devices. In this chapter, waterhammer and surging are defined and discussed in detail with reference to the two dominant sources of waterhammer—pump and/or valve operation. Detailed discussion of the hydraulic aspects of both valves and pumps and their effect on hydraulic transients will be presented. The undesirable and unwanted, but often potentially possible, event of liquid column separation and rejoining are a common justification for surge protection devices. Both the beneficial and detrimental effects of free (entrained or entrapped) air in water pipelines will be discussed with reference to waterhammer and surging. Finally, the efficacy of various surge protection devices for mitigation of waterhammer is included.
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HYDRAULIC TRANSIENT DESIGN FOR PIPELINE SYSTEMS
12.2
Chapter Twelve
12.2 FUNDAMENTALS OF WATERHAMMER AND SURGE The fundamentals of waterhammer, an elastic process, and surging, an incompressible phenomenon, are both developed on the basis of the basic conservational relationships of physics or fluid mechanics. The acoustic velocity stems from mass balance (continuity), while the fundamental waterhammer equation of Joukowsky originates from the application of linear momentum [see Eq. (12.2)].
12.2.1 Definitions Some of the terms frequently used in waterhammer are defined as follows. • Waterhammer. A pressure wave phenomenon for which liquid compressibility plays a role. • Surging. An unsteady phenomenon governed solely by inertia. Often termed mass oscillation or referred to as either rigid column or inelastic effect. • Liquid column separation. The formation of vapor cavities and their subsequent collapse and associated waterhammer on rejoining. • Entrapped air. Free air located in a pipeline as a result of incomplete filling, inadequate venting, leaks under vacuum, air entrained from pump intake vortexing, and other sources. • Acoustic velocity. The speed of a waterhammer or pressure wave in a pipeline. • Joukowsky equation. Fundamental relationship relating waterhammer pressure change with velocity change and acoustic velocity. Strictly speaking, this equation only valid for sudden flow changes.
12.2.2 Acoustic Velocity For wave propagation in liquid-filled pipes the acoustic (sonic) velocity is modified by the pipe wall elasticity by varying degrees, depending upon the elastic properties of the wall material and the relative wall thickness. The expression for the wave speed is ao K /ρ a (12.1) 1 D K 1 D K e E e E where E is the elastic modulus of the pipe wall, D is the inside diameter of the pipe, e is the wall thickness, and ao is the acoustic velocity in the liquid medium. In a very rigid pipe or in a tank, or in large water bodies, the acoustic velocity a reduces to the wellknown relationship a ao (K /ρ ). For water K 2.19 GPa (318,000 psi) and ρ 998 kg/m3 3 (1.936 slug/ft ), yielding a value of ao 1483 m/sec (4865 ft/sec), a value many times that of any liquid velocity V.
12.2.3 Joukowsky (Waterhammer) Equation There is always a pressure change ∆p associated with the rapid velocity change ∆V across a waterhammer (pressure) wave. The relationship between ∆p and ∆V from the basic physics of linear momentum yields the well-known Joukowsky equation Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
HYDRAULIC TRANSIENT DESIGN FOR PIPELINE SYSTEMS
Hydraulic Transient Design for Pipeline Systems 12.3
∆p ρa∆V
(12.2)
where ρ is the liquid mass density, and a is the sonic velocity of the pressure wave in the fluid medium in the conduit. Conveniently using the concept of head, the Joukowsky head rise for instantaneous valve closure is ∆p ρa∆V aV ∆H o (12.3) ρg ρg g The compliance of a conduit or pipe wall can have a significant effect on modification of (1) the acoustic velocity, and (2) any resultant waterhammer, as can be shown from Eq. (12.1) and Eq. (12.2), respectively. For simple waterhammer waves for which only radial pipe motion (hoop stress) effects are considered, the germane physical pipe properties are Young's elastic modulus (E) and Poisson ratio (µ). Table 12.1 summarizes appropriate values of these two physical properties for some common pipe materials. The effect of the elastic modulus (E) on the acoustic velocity in water-filled circular pipes for a range of the ratio of internal pipe diameter to wall thickness (D/e) is shown in Fig. 12.1 for various pipe materials.
12.3 HYDRAULIC CHARACTERISTICS OF VALVES Valves are integral elements of any piping system used for the handling and transport of liquids. Their primary purposes are flow control, energy dissipation, and isolation of portions of the piping system for maintenance. It is important for the purposes of design and final operation to understand the hydraulic characteristics of valves under both steady and unsteady flow conditions. Examples of dynamic conditions are direct opening or closing of valves by a motor, the response of a swing check valve under unsteady conditions, and the action of hydraulic servovalves. The hydraulic characteristics of valves under either noncavitating or cavitating conditions vary considerably from one type of valve design to another. Moreover, valve characteristics also depend upon particular valve design for a special function, upon absolute size, on manufacturer as well as the type of pipe fitting employed. In this section the fundamentals of valve hydraulics are presented in terms of pressure drop (headloss) characteristics. Typical flow characteristics of selected valve types of control—gate, ball, and butterfly, are presented. TABLE 12.1
Physical Properties of Common Pipe Materials
Material
Young's Modulus E (GPa)
Poisson's Ratio µ
Asbestos cement
23–24
–
Cast iron
80–170
0.25–0.27
Concrete
14–30
0.10–0.15
Concrete (reinforced)
30–60
–
Ductile iron
172
0.30
Polyethylene
0.7–0.8
0.46
PVC (polyvinyl chloride)
2.4–3.5
0.46
Steel
200–207
0.30
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HYDRAULIC TRANSIENT DESIGN FOR PIPELINE SYSTEMS
12.4
Chapter Twelve
FIGURE 12.1 Effect of wall thickness of various pipe materials on acoustic velocity in water pipes.
12.3.1 Descriptions of Various Types of Valves Valves used for the control of liquid flow vary widely in size, shape, and overall design due to vast differences in application. They can vary in size from a few millimeters in small tubing to many meters in hydroelectric installations, for which spherical and butterfly valves of very special design are built. The hydraulic characteristics of all types of valves, albeit different in design and size, can always be reduced to the same basic coefficients, notwithstanding fluid effects such as viscosity and cavitation. Figure 12.2 shows cross sections of some valve types to be discussed with relation to hydraulic performance.
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HYDRAULIC TRANSIENT DESIGN FOR PIPELINE SYSTEMS
Hydraulic Transient Design for Pipeline Systems 12.5
b.) Globe valve a.) Gate valve (circular gate)
c.) Needle valve
e.) Butterfly valve
d.) Gate valve (square gate)
f.) Ball valve
FIGURE 12.2 Cross sections of selected control valves: (From Wood and Jones, 1973).
12.3.2 Definition of Geometric Characteristics of Valves The valve geometry, expressed in terms of cross-sectional area at any opening, sharpness of edges, type of passage, and valve shape, has a considerable influence on the eventual hydraulic characteristics. To understand the hydraulic characteristics of valves it is useful, however, to express the projected area of the valve in terms of geometric quantities. With reference to Fig. 12.2 the ratio of the projected open area of the valve Av to the full open valve Avo can be related to the valve opening, either a linear measure for a gate valve, or an angular one for rotary valves such as ball, cone, plug, and butterfly types. It should be noted that this geometric feature of the valve clearly has a bearing on the valve hydraulic performance, but should not be used directly for prediction of hydraulic performance— either steady state or transient. The actual hydraulic performance to be used in transient calculations should originate from experiment.
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HYDRAULIC TRANSIENT DESIGN FOR PIPELINE SYSTEMS
12.6
Chapter Twelve
12.3.3
Definition of Hydraulic Performance of Valves
The hydraulic performance of a valve depends upon the flow passage through the valve opening and the subsequent recovery of pressure. The hydraulic characteristics of a valve under partial to fully opened conditions typically relate the volumetric flow rate to a characteristic valve area and the head loss ∆H across the valve. The principal fluid properties that can affect the flow characteristics are fluid density ρ, fluid viscosity µ, and liquid vapor pressure pv if cavitation occurs. Except for small valves and/or viscous liquids or both, Reynolds number effects are usually not important, and will be neglected with reference to water. A valve in a pipeline acts as an obstruction, disturbs the flow, and in general causes a loss in energy as well as affecting the pressure distribution both upstream and downstream. The characteristics are expressed either in terms of (1) flow capacity as a function of a defined pressure drop or (2) energy dissipation (headloss) as a function of pipe velocity. In both instances the pressure or head drop is usually the difference in total head caused by the presence of the valve itself, minus any loss caused by regular pipe friction between measuring stations. The proper manner in determining ∆H experimentally is to measure the hydraulic grade line (HGL) far enough both upstream and downstream of the valve so that uniform flow sections to the left of and to the right of the valve can be established, allowing for the extrapolation of the energy grade lines (EGL) to the plane of the valve. Otherwise, the valve headloss is not properly defined. It is common to express the hydraulic characteristics either in terms of a headloss coefficient KL or as a discharge coefficient Cf where Av is the area of the valve at any opening, and ∆H is the headloss defined for the valve. Frequently a discharge coefficient is defined in terms of the fully open valve area. The hydraulic coefficients embody not only the geometric features of the valve through Av but also the flow characteristics. Unless uniform flow is established far upstream and downstream of a valve in a pipeline the value of any of the coefficients can be affected by effects of nonuniform flow. It is not unusual for investigators to use only two pressure taps—one upstream and one downstream, frequently 1 and 10 diameters, respectively. The flow characteristics of valves in terms of pressure drop or headloss have been determined for numerous valves by many investigators and countless manufacturers. Only a few sets of data and typical curves will be presented here for ball, butterfly, and gate, ball, butterfly, and gate valves CD. For a valve located in the interior of a long continuous pipe, as shown in Fig. 12.3, the presence of the valve disturbs the flow both upstream and downstream of the obstruction as reflected by the velocity distribution, and the pressure variation, which will be non— hydrostatic in the regions of nonuniform flow. Accounting for the pipe friction between
V2 2g
FIGURE 12.3 Definition of headloss characteristics of a valve.
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HYDRAULIC TRANSIENT DESIGN FOR PIPELINE SYSTEMS
Hydraulic Transient Design for Pipeline Systems 12.7
upstream and downstream uniform flow sections, the headloss across the valve is expressed in terms of the pipe velocity and a headloss coefficient KL 2 ∆H KLV (12.4) 2g Often manufacturers represent the hydraulic characteristics in terms of discharge coefficients
Q Cf Avo2g∆ H CF Avo2gH ,
(12.5)
where 2 (12.6) H ∆H V 2g Both discharge coefficients are defined in terms of the nominal full-open valve area Avo and a representative head, ∆H for Cf and H for CQ, the latter definition generally reserved for large valves employed in the hydroelectric industry. The interrelationship between Cf , CF, and KL is 1 CF2 1 (12.7) KL 2 CF2 Cf Frequently valve characteristics are expressed in terms of a dimensional flow coefficient Cv from the valve industry
p Q Cv∆
(12.8)
where Q is in American flow units of gallons per minute (gpm) and ∆p is the pressure loss in pounds per square inch (psi). In transient analysis it is convenient to relate either the loss coefficient or the discharge coefficient to the corresponding value at the fully open valve position, for which Cf Cfo. Hence, Q Cf Qo Cfo
∆∆HH τ ∆∆HH o
(12.9)
o
Traditionally the dimensionless valve discharge coefficient is termed τ and defined by C C C KLo τ f v f (12.10) Cfo Cvo Cfo KL
12.3.4 Typical Geometric and Hydraulic Valve Characteristics The geometric projected area of valves shown in Fig. 12.2 can be calculated for ball, butterfly, and gate valves using simple expressions. The dimensionless hydraulic flow coefficient is plotted in Fig. 12.4 for various valve openings for the three selected valves along with the area ratio for comparison. The lower diagram, which is based on hydraulic measurements, should be used for transient calculations rather than the upper one, which is strictly geometric.
12.3.5 Valve Operation The instantaneous closure of a valve at the end of a pipe will yield a pressure rise satisfy-
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HYDRAULIC TRANSIENT DESIGN FOR PIPELINE SYSTEMS
12.8
Chapter Twelve
FIGURE 12.4 Geometric and hydraulic characteristics of typical control valves
ing Joukowsky's equation—Eq. (12.2) or Eq. (12.3). In this case the velocity difference ∆V 0 Vo , where Vo is the initial velocity of liquid in the pipe. Although Eq. (12.2) applies across every wavelet, the effect of complete valve closure over a period of time greater than 2L/a, where L is the distance along the pipe from the point of wave creation to the location of the first pipe area change, can be beneficial. Actually, for a simple pipeline the maximum head rise remains that from Eq. (12.3) for times of valve closure tc 2L/a, where L is the length of pipe. If the value of tc 2L/a, then there can be a considerable reduction of the peak pressure resulting from beneficial effects of negative wave reflections from the open end or reservoir considered in the analysis. The phenomenon can still be classified as waterhammer until the time of closure tc 2L/a, beyond which time there are only inertial or incompressible deceleration effects, referred to as surging, also known as rigid column analysis. Table 12.2 classifies four types of valve closure, independent of type of valve. Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
HYDRAULIC TRANSIENT DESIGN FOR PIPELINE SYSTEMS
Hydraulic Transient Design for Pipeline Systems 12.9 TABLE 12.2
Classification of Valve Closure
Time of Closure tc
Type of Closure
Maximum Head ∆Hmax
Phenomenon
0 2L/a 2L/a 2L/a
Instantaneous Rapid Gradual Slow
aVo /g aVo /g aVo /g aVo /g
Waterhammer Waterhammer Waterhammer Surging
Using standard waterhammer programs, parametric analyses can be conducted for the preparation of charts to demonstrate the effect of time of closure, type of valve, and an indication of the physical process—waterhammer or simply inertia effects of deceleration. The charts are based on analysis of valve closure for a simple reservoir-pipe-valve arrangement. For simplicity fluid friction is often neglected, a reasonable assumption for pipes on the order of hundreds of feet in length.
12.4 HYDRAULIC CHARACTERISTICS OF PUMPS Transient analyses of piping systems involving centrifugal, mixed-flow, and axial-flow pumps require detailed information describing the characteristics of the respective turbomachine, which may pass through unusual, indeed abnormal, flow regimes. Since little if any information is available regarding the dynamic behavior of the pump in question, invariably the decision must be made to use the steady-flow characteristics of the machine gathered from laboratory tests. Moreover, complete steady-flow characteristics of the machine may not be available for all possible modes of operation that may be encountered in practice. In this section steady-flow characteristics of pumps in all possible zones of operation are defined. The importance of geometric and dynamic similitude is first discussed with respect to both (1) homologous relationships for steady flow and (2) the importance of the assumption of similarity for transient analysis. The significance of the eight zones of operation within each of the four quadrants is presented in detail with reference to three possible modes of data representation. The steady-flow characteristics of pumps are discussed in detail with regard to the complete range of possible operation. The loss of driving power to a pump is usually the most critical transient case to consider for pumps, because of the possibility of low pipeline pressures which may lead to (1) pipe collapse due to buckling, or (2) the formation of a vapor cavity and its subsequent collapse. Other waterhammer problems may occur due to slam of a swing check valve, or from a discharge valve closing either too quickly (column separation), or too slowly (surging from reverse flow). For radial-flow pumps for which the reverse flow reaches a maximum just subsequent to passing through zero speed (locked rotor point), and then is decelerated as the shaft runs faster in the turbine zone, the head will usually rise above the nominal operating value. As reported by Donsky (1961) mixed-flow and axial-flow pumps may not even experience an upsurge in the turbine zone because the maximum flow tends to occur closer to runaway conditions. 12.4.1 Definition of Pump Characteristics The essential parameters for definition of hydraulic performance of pumps are defined as • Impeller diameter. Exit diameter of pump rotor DI . • Rotational speed. The angular velocity (rad/s) is ω, while N = 2 πω/60 is in rpm. Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
HYDRAULIC TRANSIENT DESIGN FOR PIPELINE SYSTEMS
12.10
Chapter Twelve
• Flow rate. Capacity Q at operating point in chosen units. • Total dynamic head (TDH). The total energy gain (or loss) H across pump, defined as P P V2 V2 H d zd s zs d s 2g 2g γ γ
(12.11)
where subscripts s and d refer to suction and discharge sides of the pump, respectively,
12.4.2 Homologous (Affinity) Laws Dynamic similitude, or dimensionless representation of test results, has been applied with perhaps more success in the area of hydraulic machinery than in any other field involving fluid mechanics. Due to the sheer magnitude of the problem of data handling it is imperative that dimensionless parameters be employed for transient analysis of hydraulic machines that are continually experiencing changes in speed as well as passing through several zones of normal and abnormal operation. For liquids for which thermal effects may be neglected, the remaining fluid-related forces are pressure (head), fluid inertia, resistance, phase change (cavitation), surface tension, compressibility, and gravity. If the discussion is limited to single-phase liquid flow, three of the above fluid effects—cavitation, surface tension, and gravity (no interfaces within machine)—can be eliminated, leaving the forces of pressure, inertia, viscous resistance, and compressibility. For the steady or even transient behavior of hydraulic machinery conducting liquids the effect of compressibility may be neglected. In terms of dimensionless ratios the three forces yield an Euler number (ratio of inertia force to pressure force), which is dependent upon geometry, and a Reynolds number. For all flowing situations, the viscous force, as represented by the Reynolds number, is definitely present. If water is the fluid medium, the effect of the Reynolds number on the characteristics of hydraulic machinery can usually be neglected, the major exception being the prediction of the performance of a large hydraulic turbine on the basis of model data. For the transient behavior of a given machine the actual change in the value of the Reynolds number is usually inconsequential anyway. The elimination of the viscous force from the original list reduces the number of fluid-type forces from seven to two–pressure (head) and inertia, as exemplified by the Euler number. The appellation geometry in the functional relationship in the above equation embodies primarily, first, the shape of the rotating impeller, the entrance and exit flow passages, including effects of vanes, diffusers, and so on; second, the effect of surface roughness; and lastly the geometry of the streamline pattern, better known as kinematic similitude in contrast to the first two, which are related to geometric similarity. Kinematic similarity is invoked on the assumption that similar flow patterns can be specified by congruent velocity triangles composed of peripheral speed U and absolute fluid velocity V at inlet or exit to the vanes. This allows for the definition of a flow coefficient, expressed in terms of impeller diameter DI and angular velocity ω: Q CQ 3 ωDI
(12.12)
The reciprocal of the Euler number (ratio of pressure force to inertia force) is the head coefficient, defined as
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HYDRAULIC TRANSIENT DESIGN FOR PIPELINE SYSTEMS
Hydraulic Transient Design for Pipeline Systems 12.11
gH CH 2 2 ω DI
(12.13)
P CP 35 ρω DI
(12.14)
A power coefficient can be defined
For transient analysis, the desired parameter for the continuous prediction of pump speed is the unbalanced torque T. Since T P/ω, the torque coefficient becomes T CT 25 ρω DI
(12.15)
Traditionally in hydraulic transient analysis to refer pump characteristics to so-called rated conditions—which preferably should be the optimum or best efficiency point (BEP), but sometimes defined as the duty, nameplate, or design point. Nevertheless, in terms of rated conditions, for which the subscript R is employed, the following ratios are defined; Q Flow: v QR
speed: ω N ωR NR
head: h H HR
torque: β T TR
Next, for a given pump undergoing a transient, for which DI is a constant, Eqs. (12.12–12.15) can be written in terms of the above ratios CQ QR ω v α CQR Q ωR
2 CH h H ωR 2 2 α CHR HR ω
2 β CT T ωR 2 α CTR TR ω2
12.4.3 Abnormal Pump (Four–Quadrant) Characteristics The performance characteristics discussed up to this point correspond to pumps operating normally. During a transient, however, the machine may experience either a reversal in flow, or rotational speed, or both, depending on the situation. It is also possible that the torque and head may reverse in sign during passage of the machine through abnormal zones of performance. The need for characteristics of a pump in abnormal zones of operation can best be described with reference to Fig. 12.5, which is a simulated pump power failure transient. A centrifugal pump is delivering water at a constant rate when there is a sudden loss of power from the prime move—rin this case an electric motor. For the postulated case of no discharge valves, or other means of controlling the flow, the loss of driving torque leads to an immediate deceleration of the shaft speed, and in turn the flow. The three curves are dimensionless head (h), flow (v), and speed (α). With no additional means of controlling the flow, the higher head at the final delivery point (another reservoir) will eventually cause the flow to reverse (v 0) while the inertia of the rotating parts has maintained positive rotation (α 0). Up until the time of flow reversal the pump has been operating in the normal zone, albeit at a number of off-peak flows. To predict system performance in regions of negative rotation and/or negative flow the analyst requires characteristics in these regions for the machine in question. Indeed, any peculiar characteristic of the pump in these regions could be expected to have an influence on the hydraulic transients. It is important to stress that the results of such analyses are critically governed by the following three factors: (1) availability of complete pump characteristics in zones the pump will operate, (2) complete reliance on dynamic similitude (homologous) laws during transients, and (3) assumption that steady-flow derived pumpcharacteristics are valid for transient analysis. In vestigations by Kittredge (1956) and Knapp (1937) facilitated the understanding of Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
HYDRAULIC TRANSIENT DESIGN FOR PIPELINE SYSTEMS
12.12
Chapter Twelve
FIGURE 12.5 Simulated pump trip without valves in a single-pipeline system.
abnormal operation, as well as served to reinforce the need for test data. Following the work by Knapp (1941) and Swanson (1953), and a summary of their results by Donsky (1961), eight possible zones of operation, four normal and four abnormal, will be discussed here with reference to Fig. 12.6, developed by Martin (1983). In Fig. 12.6 the head H is shown as the difference in the two reservoir elevations to simplify the illustration. The effect of pipe friction may be ignored for this discussion by assuming that the pipe is short and of relatively large diameter. The regions referred to on Fig. 12.6 are termed zones and quadrants, the latter definition originating from plots of lines of constant head and constant torque on a flow-speed plane (v α axes). Quadrants I (v 0, α 0) and III (v 0, α 0) are defined in general as regions of pump or turbine operation, respectively. It will be seen, however, that abnormal operation (neither pump nor turbine mode) may occur in either of these two quadrants. A very detailed description of each of the eight zones of operation is in order. It should be noted that all of the conditions shown schematically in Fig. 12.6 can be contrived in a laboratory test loop using an additional pump (or two) as the master and the test pump as a slave. Most, if not all, of the zones shown can also be experienced by a pump during a transient under the appropriate set of circumstances. Quadrant I. Zone A (normal pumping) in Fig. 12.6 depicts a pump under normal operation for which all four quantities— Q, N, H, and T are regarded as positive. In this case Q 0, indicating useful application of energy. Zone B (energy dissipation) is a condition of positive flow, positive rotation, and positive torque, but negative head—quite an abnormal condition. A machine could operate in Zone B by (1) being overpowered by another pump or by a reservoir during steady operation, or (2) by a sudden drop in head during a
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HYDRAULIC TRANSIENT DESIGN FOR PIPELINE SYSTEMS
Hydraulic Transient Design for Pipeline Systems 12.13
FIGURE 12.6 Four quadrants and eight zones of possible pump operation. (From Martin, 1983)
transient caused by power failure. It is possible, but not desirable, for a pump to generate power with both the flow and rotation in the normal positive direction for a pump, Zone C (reverse turbine), whichis caused by a negative head, resulting in a positive efficiency because of the negative torque. The maximum efficiency would be quite low due to the bad entrance flow condition and unusual exit velocity triangle. Quadrant IV. Zone H, labeled energy dissipation, is often encountered shortly after a tripout or power failure of a pump, as illustrated in Fig. 12.5. In this instance the combined inertia of all the rotating elements—motor, pump and its entrained liquid, and shaft—has maintained pump rotation positive but at a reduced value at the time of flow reversal caused by the positive head on the machine. This purely dissipative mode results Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
HYDRAULIC TRANSIENT DESIGN FOR PIPELINE SYSTEMS
12.14
Chapter Twelve
in a negative or zero efficiency. It is important to note that both the head and fluid torque are positive in Zone H, the only zone in Quadrant IV. Quadrant III. A machine that passes through Zone H during a pump power failure will then enter Zone G (normal turbining) provided that reverse shaft rotation is not precluded by a mechanical ratchet. Although a runaway machine rotating freely is not generating power, Zone G is the precise mode of operation for a hydraulic turbine. Note that the head and torque are positive, as for a pump but that the flow and speed are negative, opposite to that for a pump under normal operation (Zone A). Subsequent to the tripout or load rejection of a hydraulic turbine or the continual operation of a machine that failed earlier as a pump, Zone F (energy dissipation) can be encountered. The difference between Zones F and G is that the torque has changed sign for Zone F, resulting in a braking effect, which tends to slow the freewheeling machine down. In fact the real runaway condition is attained at the boundary of the two zones, for which torque T 0. Quadrant II. The two remaining zones—D and E—are very unusual and infrequently encountered in operation, with the exception of pump/turbines entering Zone E during transient operation. Again it should be emphasized that both zones can be experienced by a pump in a test loop, or in practice in the event a machine is inadvertently rotated in the wrong direction by improper wiring of an electric motor. Zone D is a purely dissipative mode that normally would not occur in practice unless a pump, which was designed to increase the flow from a higher to lower reservoir, was rotated in reverse, but did not have the capacity to reverse the flow (Zone E, mixed or axial flow), resulting in Q 0, N 0, T 0, for H 0. Zone E, for which the pump efficiency 0, could occur in practice under steady flow if the preferred rotation as a pump was reversed. There is always the question regarding the eventual direction of the flow. A radial-flow machine will produce positive flow at a much reduced capacity and efficiency compared to N 0 (normal pumping), yielding of course H 0. On the other hand, mixed and axial-flow machines create flow in the opposite direction (Quadrant III), and H 0, which corresponds still to an increase in head across the machine in the direction of flow.
12.4.4 Representation of Pump Data for Numerical Analysis It is conventional in transient analyses to represent h/α2 and β/α2 as functions of v/α, as shown in Fig. 12.7 and 12.8 for a radial-flow pump. The curves on Fig. 12.7 are only for positive rotation (α 0), and constitute pump Zones A, B, and C for v 0 and the region of energy dissipation subsequent to pump power failure (Zone H), for which v 0. The remainder of the pump characteris-tics are plotted in Fig. 12.8 for α 0. The complete characteristics of the pump plotted in Figs. 12.7 and 12.8 can also be correlated on what is known as a Karman-Knapp circle diagram, a plot of lines of constant head (h) and torque (β) on the coordinates of dimensionless flow (v) and speed (α). Fig. 12.9 is such a correlation for the same pump. The complete characteristics of the pump require six curves, three each for head and torque. For example, the h/α2 curves from Figs. 12.7 and 12.8 can be represented by continuous lines for h l and h l, and two straight lines through the origin for h 0. A similar pattern exists for the torque (β) lines. In addition to the eight zones A–H illustrated in Fig. 12.6, the four Karman-Knapp quadrants in terms of v and, are well defined. Radial lines in Fig. 12.9 correspond to constant values for v/α in Figs. 12.7 and 12.8, allowing for relatively easy transformation from one form of presentation to the other. In computer analysis of pump transients, Figs. 12.7 and 12.8, while meaningful from
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HYDRAULIC TRANSIENT DESIGN FOR PIPELINE SYSTEMS
Hydraulic Transient Design for Pipeline Systems 12.15
FIGURE 12.7 Complete head and torque characteristics of a radial-flow pump for positive rotation. (From Martin, 1983)
FIGURE 12.8 Complete head and torque characteristics of a radialflow pump for negative rotation. (From Martin, 1983)
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HYDRAULIC TRANSIENT DESIGN FOR PIPELINE SYSTEMS
12.16
Chapter Twelve
the standpoint of physical understanding, are fraught with the difficulty of |v/α| becoming infinite as the unit passes through, or remains at, zero speed ( = 0). Some have solved that problem by switching from h/α2 versus v/α to h/v2 versus α/v, and likewise for β, for |v/α| l. This technique doubles the number of curves on Figs. 12.7 and 12.8, and thereby creates discontinuities in the slopes of the lines at |v/α| 1, in addition to complicating the storing and interpolation of data. Marchal et al. (1965) devised a useful transformation which allowed the complete pump characteris-tics to be represented by two single curves, as shown for the same pump in Fig. 12.10. The difficulty of v/α becoming infinite was eliminated by utilizing the function tan1 (v/α) as the abscissa. The eight zones, or four quadrants can then be connected by the continuous functions. Although some of the physical interpretation of pump data has been lost in the transformation, Fig. 12.10 is now a preferred correlation for transient analysis using a digital computer because of function continuity and ease of numerical interpolation. The singularities in Figs. 12.7 and 12.8 and the asymptotes in Fig. 12.9 have now been avoided.
12.4.5 Critical Data Required for Hydraulic Analysis of Systems with Pumps Regarding data from manufacturers such as pump curves (normal and abnormal), pump and motor inertia, motor torque-speed curves, and valve curves, probably the most critical for pumping stations are pump-motor inertia and valve closure time. Normal pump curves are
FIGURE 12.9 Complete four-quadrant head and torque characteristics of radial-flow pump. (From Martin, 1983)
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HYDRAULIC TRANSIENT DESIGN FOR PIPELINE SYSTEMS
Hydraulic Transient Design for Pipeline Systems 12.17
FIGURE 12.10 Complete head and torque characteristics of a radial-flow pump in Suter diagram. (From Martin, 1983)
usually available and adequate. Motor torque-speed curves are only needed when evaluating pump startup. For pump trip the inertia of the combined pump and motor is important.
12.5 SURGE PROTECTION AND SURGE CONTROL DEVICES There are numerous techniques for controlling transients and waterhammer, some involving design considerations and others the consideration of surge protection devices. There must be a complete design and operational strategy devised to combat potential waterhammer in a system. The transient event may either initiate a low-pressure event (downsurge) as in the case of a pump power failure, or a high pressure event (upsurge) caused by the closure of a downstream valve. It is well known that a downsurge can lead to the undesirable occurrence of water-column separation, which itself can result in severe pressure rises following the collapse of a vapor cavity. In some systems negative pressures are not even allowed because of (1) possible pipe collapse or (2) ingress of outside water or air. The means of controlling the transient will in general vary, depending upon whether the initiating event results in an upsurge or downsurge. For pumping plants the major cause of unwanted transients is typically the complete outage of pumps due to loss of electricity to the motor. For full pipelines, pump startup, usually against a closed pump discharge valve for centrifugal pumps, does not normally result in significant pressure transients. The majority of transient problems in pumping installations are associated with the potential (or realized) occurrence of water-column separation and vapor-pocket collapse, resulting from the tripout of one or more pumps, with or without valve action. The pumpdischarge valve, if actuated too suddenly, can even aggravate the downsurge problem. To
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HYDRAULIC TRANSIENT DESIGN FOR PIPELINE SYSTEMS
12.18
Chapter Twelve
FIGURE 12.11 Schematic of various surge protection devices for pumping installations
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HYDRAULIC TRANSIENT DESIGN FOR PIPELINE SYSTEMS
Hydraulic Transient Design for Pipeline Systems 12.19
combat the downsurge problem there are a number of options, mostly involving the design and installation of one or more surge protection devices. In this section various surge protection techniques will be discussed, followed by an assessment of the virtue of each with respect to pumping systems in general. The lift systems shown in Fig. 12.11 depict various surge protection schemes.
12.5.1 Critical Parameters for Transients Before discussing surge protection devices, some comments will be made regarding the various pipeline, pump and motor, control valve, flow rate, and other parameters that affect the magnitude of the transient. For a pumping system the four main parameters are (1) pump flow rate, (2) pump and motor WR2, (3) any valve motion, and (4) pipeline characteristics. The pipeline characteristics include piping layout—both plan and profile— pipe size and material, and the acoustic velocity. So-called short systems respond differently than long systems. Likewise, valve motion and its effect, whether controlled valves or check valves, will have different effects on the two types of systems. The pipeline characteristics—item number (4)—relate to the response of the system to a transient such as pump power failure. Clearly, the response will be altered by the addition of one or more surge protection device or the change of (1) the flow rate, or (2) the WR2, or (3) the valve motion. Obviously, for a given pipe network and flow distribution there are limited means of controlling transients by (2) WR2 and (3) valve actuation. If these two parameters can not alleviate the problem than the pipeline response needs to be altered by means of surge protection devices.
FIGURE 12.12 Cross sectional view of surge tanks and gas–n related surge protection devices
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HYDRAULIC TRANSIENT DESIGN FOR PIPELINE SYSTEMS
12.20
Chapter Twelve
FIGURE 12.13 Cross sections of vacuum breaker, air release and surge relief valves.
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HYDRAULIC TRANSIENT DESIGN FOR PIPELINE SYSTEMS
Hydraulic Transient Design for Pipeline Systems 12.21
12.5.2 Critique of Surge Protection For pumping systems, downsurge problems have been solved by various combinations of the procedures and devices mentioned above. Details of typical surge protection devices are illustrated in Figs. 12.12 and 12.13. In many instances local conditions and preferences of engineers have dictated the choice of methods and/or devices. Online devices such as accumulators and simple surge tanks are quite effective, albeit expensive, solutions. One-way surge tanks can also be effective when judiciously sized and sited. Surge anticipation valves should not be used when there is already a negative pressure problem. Indeed, there are installations where surge anticipation functions of such valves have been deactivated, leaving only the surge relief feature. Moreover, there have been occasions for which the surge anticipation feature aggravated the low pressure situation by an additional downsurge caused by premature opening of the valve. Regarding the consideration and ultimate choice of surge protection devices, subsequent to calibration of analysis with test results, evaluation should be given to simple surge tanks or standpipes, one-way surge tanks, and hydropneumatic tanks or air chambers. A combination of devices may prove to be the most desirable and most economical. The admittance of air into a piping system can be effective, but the design of air vacuum-valve location and size is critical. If air may be permitted into pipelines careful analysis would have to be done to ensure effective results. The consideration of air-vacuum breakers is a moot point if specifications such as the Ten State Standards limit the pressures to positive values.
12.5.3 Surge Protection Control and Devices Pump discharge valve operation. In gravity systems the upsurge transient can be controlled by an optimum valve closure—perhaps two stage, as mentioned by Wylie and Streeter (1993). As shown by Fleming (1990), an optimized closing can solve a waterhammer problem caused by pump power failure if coupled with the selection of a surge protection device. For pump power failure a control valve on the pump discharge can often be of only limited value in controlling the downsurge, as mentioned by Sanks (1989). Indeed, the valve closure can be too sudden, aggravating the downsurge and potentially causing column separation, or too slow, allowing a substantial reverse flow through the pump. It should also be emphasized that an optimum controlled motion for single-pump power failure is most likely not optimum for multiple-pump failure. The use of microprocessors and servomechanisms with feedback systems can be a general solution to optimum control of valves in conjunction with the pump and pipe system. For pump discharge valves the closure should not be too quick to exacerbate downsurge, nor too slow to create a substantial flow back through the valve and pump before closure. Check valves. Swing check valves or other designs are frequently employed in pump discharge lines, often in conjunction with slow acting control valves. As indicated by Tullis (1989), a check valve should open easily, have a low head loss for normal positive flow, and create no undesirable transients by its own action. For short systems, a slowresponding check valve can lead to waterhammer because of the high reverse flow generated before closure. A spring or counterweight loaded valve with a dashpot can (1) give the initial fast response followed by (2) slow closure to alleviate the unwanted transient. The proper selection of the load and the degree of damping is important, however, for proper performance.
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HYDRAULIC TRANSIENT DESIGN FOR PIPELINE SYSTEMS
12.22
Chapter Twelve
Check valve slam is also a possibility from stoppage or failure of one pump of several in a parallel system, or resulting from the action of an air chamber close to a pump undergoing power failure. Check valve slam can be reduced by the proper selection of a dashpot. Surge anticipator valves and surge relief valves. A surge anticipation valve, Fig. 12.13c frequently installed at the manifold of the pump station, is designed to open initially under (1) pump power failure, or (2) the sensing of underpressure, or (3) the sensing of overpressure, as described by Lescovitch (1967). On the other hand, the usual type of surge relief valve opens quickly on sensing an overpressure, then closes slowly, as controlled by pilot valves. The surge anticipation valve is more complicated than a surge relief valve in that it not only embodies the relief function at the end of the cycle, but also has the element of anticipation. For systems for which water-column separation will not occur, the surge anticipation valve can solve the problem of upsurge at the pump due to reverse flow or wave reflection, as reported in an example by White (1942). An example of a surge relief valve only is provided by Weaver (1972). For systems for which watercolumn separation will not occur, Lundgren (1961) provides charts for simple pipeline systems. As reported by Parmakian (1968,1982a-b) surge anticipation valves can exacerbate the downsurge problem inasmuch as the opening of the relief valve aggravates the negative pressure problem. Incidents have occurred involving the malfunctioning of a surge anticipation valve, leading to extreme pressures because the relief valve did not open. Pump bypass. In shorter low-head systems a pump bypass line (Fig. 12.11) can be installed in order to allow water to be drawn into the pump discharge line following power failure and a downsurge. As explained by Wylie and Streeter (1993), there are two possible bypass configurations. The first involves a control valve on the discharge line and a check valve on the bypass line between the pump suction or wet well and the main line. The check valve is designed to open subsequent to the downsurge, possibly alleviating column separation down the main line. The second geometry would reverse the valve locations, having a control valve in the bypass and a check valve in the main line downstream of the pump. The control valve would open on power failure, again allowing water to bypass the pump into the main line. Open (simple) surge tank. A simple on-line surge tank or standpipe (Fig. 12.11) can be an excellent solution to both upsurge and downsurge problems, These devices are quite common in hydroelectric systems where suitable topography usually exists. They are practically maintenance free, available for immediate response as they are on line. For pumping installations open simple surge tanks are rare because of height considerations and the absence of high points near most pumping stations. As mentioned by Parmakian (1968) simple surge tanks are the most dependable of all surge protection devices. One disadvantage is the additional height to allow for pump shutoff head. Overflowing and spilling must be considered, as well as the inclusion of some damping to reduce oscillations. As stated by Kroon et al. (1984) the major drawback to simple surge tanks is their capital expense. One-way surge tank. The purpose of a one-way surge tank is to prevent initial low pressures and potential water-column separation by admitting water into the pipeline subsequent to a downsurge. The tank is normally isolated from the pipeline by one or more lateral pipes in which there ok one or more check valves to allow flow into the pipe if the HGL is lower in the pipe than the elevation of the water in the open tank. Under normal operating conditions the higher pressure in the pipeline keeps the check valve closed. The
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HYDRAULIC TRANSIENT DESIGN FOR PIPELINE SYSTEMS
Hydraulic Transient Design for Pipeline Systems 12.23
major advantage of a one-way surge tank over a simple surge tank is that it does not have to be at the HGL elevation as required by the latter. It has the disadvantage, however, on only combatting initial downsurges, and not initial upsurges. One-way surge tanks have been employed extensively by the U.S. Bureau of Reclamation in pump discharge lines, principally by the instigation of Parmakian (1968), the originator of the concept. Another example of the effective application of one-way surge tanks in a pumping system was reported by Martin (1992), to be discussed in section 12.9.1. Considerations for design are: (1) location of high points or knees of the piping, (2) check valve and lateral piping redundancy, (3) float control refilling valves and water supply, and other appurtenances. Maintenance is critical to ensure the operation of the check valve(s) and tank when needed. Air chamber (hydropneumatic surge tank). If properly designed and maintained, an air chamber can alleviate both negative and positive pressure problems in pumping systems. They are normally located within or near the pumping station where they would have the greatest effect. As stated by Fox (1977) and others, an air chamber solution may be extremely effective in solving the transient problem, but highly expensive. Air chambers have the advantage that the tank–sometimes multiple–can be mounted either vertically or horizontally. The principal criteria are available water volume and air volume for the task at hand. For design, consideration must be given to compressed air supply, water level sensing, sight glass, drains, pressure regulators, and possible freezing. Frequently, a check valve is installed between the pump and the air chamber. Since the line length between the pump and air chamber is usually quite short, check valve slamming may occur, necessitating the consideration of a dashpot on the check valve to cushion closure. The assurance of the maintenance of air in the tank is essential—usually 50 percent of tank volume, otherwise the air chamber can be ineffective. An incident occurred at a raw water pumping plant where an air chamber became waterlogged due to the malfunctioning of the compressed air system. Unfortunately, pump power failure occurred at the same time, causing water column separation and waterhammer, leading to pipe rupture. Air vacuum and air release valves. Another method for preventing subatmospheric pressures and vapor cavity formation is the admittance of air from air—vacuum valves (vacuum breakers) at selected points along the piping system. Proper location and size of air—vacuum valves can prevent water-column separation and reduce waterhammer effects, as calculated and measured by Martin (1980). The sizing and location of the valves are critical, as stated by Kroon et al. (1984). In fact, as reported by Parmakian (1982a,—b) the inclusion of air-vacuum valves in a pipeline did not eliminate failures. Unless the air-vacuum system is properly chosen, substantial pressures can still occur due to the compression of the air during resurge, especially if the air is at extremely low pressures within the pipeline when admitted. Moreover, the air must be admitted quickly enough to be effective. Typical designs are shown in Fig. 12.13 As shown by Fleming (1990) vacuum breakers can be a viable solution. The advantage of an air-vacuum breaker system, which is typically less expensive than other measures such as air chambers, must be weighed against the disadvantages of air accumulation along the pipeline and its subsequent removal. Maintenance and operation of valves is critical in order for assurance of valve opening when needed. Air removal is often accomplished with a combined air—release air-vacuum valve. For finished water systems the admittance of air is not a normal solution and must be evaluated carefully. Moreover, air must be carefully released so that no additional transient is created.
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HYDRAULIC TRANSIENT DESIGN FOR PIPELINE SYSTEMS
12.24
Chapter Twelve
Flywheel. Theoretically, a substantial increase in the rotating inertia (WR2) of a pumpmotor unit can greatly reduce the downsurge inasmuch as the machine will not decelerate as rapidly. Typically, the motor may constitute from 75 to 90 percent of the total WR2. Additional WR2 by the attachment of a flywheel will reduce the downsurge. As stated by Parmakian (1968), a 100 percent increase in WR2 by the addition of a flywheel may add up to 20 percent to the motor cost. He further states that a flywheel solution is only economical in some marginal cases. Flywheels are usually an expensive solution, mainly useful only for short systems. A flywheel has the advantage of practically no maintenance, but the increased torque requirements for starting must be considered. Uninterrupted power supply (UPS). The availability of large uninterrupted power supply systems are of potential value in preventing the primary source of waterhammer in pumping; that is, the generation of low pressures due to pump power failure. For pumping stations with multiple parallel pumps, a UPS system could be devised to maintain one or more motors while allowing the rest to fail, inasmuch as there is a possibility of maintaining sufficient pressure with the remaining operating pump(s). The solution usually is expensive, however, with few systems installed.
12.6 DESIGN CONSIDERATIONS Any surge or hydraulic transient analysis is subject to inaccuracies due to incomplete information regarding the systems and its components. This is particularly true for a water distribution system with its complexity, presence of pumps, valves, tanks, and so forth, and some uncertainty with respect to initial flow distribution. The ultimate question is how all of the uncertainties combine in the analysis to yield the final solution. There will be offsetting effects and a variation in accuracy in terms of percentage error throughout the system. Some of the uncertainties are as follows. The simplification of a pipe system, in particular a complex network, by the exclusion of pipes below a certain size and the generation of equivalent pipes surely introduces some error, as well as the accuracy of the steady-state solution. However, if the major flow rates are reasonably well known, then deviation for the smaller pipes is probably not too critical. As mentioned above incomplete pump characteristics, especially during reverse flow and reverse rotation, introduce calculation errors. Valve characteristics that must be assumed rather than actual are sources of errors, in particular the response of swing check valves and pressure reducing valves. The analysis is enhanced if the response of valves and pumps from recordings can be put in the computer model. For complex pipe network systems it is difficult to assess uncertainties until much of the available information is known. Under more ideal conditions that occur with simpler systems and laboratory experiments, one can expect accuracies when compared to measurement on the order of 5 to 10 percent, sometimes even better. The element of judgment does enter into accuracy. Indeed, two analyses could even differ by this range because of different assumptions with respect to wave speeds, pump characteristics, valve motions, system schematization, and so forth. It is possible to have good analysis and poorer analysis, depending upon experience and expertise of the user of the computer code. This element is quite critical in hydraulic transients. Indeed, there can be quite different results using the same code. Computer codes, which are normally based on the method of characteristics (MOC), are invaluable tools for assessing the response based of systems to changes in surge protection devices and their characteristics. Obviously, the efficacy of such an approach is
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HYDRAULIC TRANSIENT DESIGN FOR PIPELINE SYSTEMS
Hydraulic Transient Design for Pipeline Systems 12.25
enhanced if the input data and network schematization is improved via calibration. Computer codes have the advantage of investigating a number of options as well as optimizing the sizing of surge protection devices. The ability to calibrate a numerical analysis code to a system certainly improves the determination of the proper surge protection. Otherwise, if the code does not reasonably well represent a system, surge protection devices can either be inappropriate or under- or oversized. Computer codes that do not properly model the formation of vapor pockets and subsequent collapse can cause considerable errors. Moreover, there is also uncertainty regarding any free or evolved gas coming out of solution. The effect on wave speed is known, but this influence can not be easily addressed in an analysis of the system. It is simply another possible uncertainty. Even for complicated systems such as water distribution networks, hydraulic transient calculations can yield reasonable results when compared to actual measurements provided that the entire system can be properly characterized. In addition to the pump, motor, and valve characteristics there has to be sufficient knowledge regarding the piping and flow demands. An especially critical factor for a network is the schematization of the network; that is, how is a network of thousands of pipes simplified to one suitable for computer analysis, say hundreds of pipes, some actual and some equivalent. According to Thorley (1991), a network with loops tends to be more forgiving regarding waterhammer because of the dispersive effect of many pipes and the associated reflections. On the other hand, Karney and McInnis (1990) show by a simple example that wave superposition can cause amplification of transients. Since water distribution networks themselves have not been known to be prone to waterhammer as a rule, there is meager information as to simplification and means of establishing equivalent pipes for analysis purposes. Large municipal pipe networks are good examples wherein the schematization and the selection of pipes characterizing the networks need to be improved in orde to represent the system better.
12.7 NEGATIVE PRESSURES AND WATER COLUMN SEPARATION IN NETWORKS For finished water transmission and distribution systems the application of 138 kPa (20 psig) as a minimum pressure to be maintained under all conditions should prevent column separation from occurring provided analytical models have sufficient accuracy. Although water column separation and collapse is not common in large networks, it does not mean that the event is not possible. The modeling of water column separation is clearly difficult for a complicated network system. Water column separation has been analytically modeled with moderate success for numerous operating pipelines. Clearly, not only negative pressures, but also water column separation, are unwanted in pipeline systems, and should be eliminated by installation of properly designed surge protection devices. If the criterion of a minimum pressure of 138 kPa (20 psig) is imposed then the issue of column separation and air-vacuum breakers are irrelevant, except for prediction by computer codes. Aside from research considerations, column separation is simulated for engineering situations mainly to assess the potential consequences. If the consequences are serious, as they often are in general, either operational changes or more likely surge protection devices are designed to alleviate column separation. For marginal cases of column separation the accuracy of pressure prediction becomes difficult. If column separation is not to be allowed and the occurrence of vapor pressure can be adequately predicted, then the simulation of column separation itself is not necessary.
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HYDRAULIC TRANSIENT DESIGN FOR PIPELINE SYSTEMS
12.26
Chapter Twelve
Some codes do not simulate water column separation, but instead only maintain the pressure at cavity location at vapor pressure. The results of such an analysis are invalid, if indeed an actual cavity occurred, at some time subsequent to cavity formation. This technique is only useful to know if a cavity could have occurred, as there can be no assessment of the consequences of column separation. The inability of any code to model water column separation has the following implications: (1) the seriousness of any column separation event, if any, can not be determined, and (2) once vapor pressure is attained, the computation model loses its ability to predict adequately system transients. If negative pressures below 138 kPa (20 psig) are not to be allowed the inability of a code to assess the consequences of column separation and its attendant collapse is admittedly not so serious. The code need only flag pressures below 138 kPa (20) psig and negative pressures, indicating if there is a need for surge protection devices. The ability of any model to properly simulate water column separation depends upon a number of factors. The principal ones are • Accurate knowledge of initial flow rates • Proper representation of pumps, valves, and piping system • A vapor pocket allowed to form, grow, and collapse • Maintenance of vapor pressure within cavity while it exists • Determination of volume of cavity at each time step • Collapse of cavity at the instant the cavity volume is reduced to zero
12.8 TIME CONSTANTS FOR HYDRAULIC SYSTEMS • Elastic time constant 2L te a
(12.16)
LV tf o o gH
(12.17)
• Flow time constant
• Pump and motor inertia time constant Iω IωR2 tm R TR ρgQRHRηRT
(12.18)
• Surge tank oscillation inelastic time constant ts 2p
LgAA T
t
(12.19)
T
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HYDRAULIC TRANSIENT DESIGN FOR PIPELINE SYSTEMS
Hydraulic Transient Design for Pipeline Systems 12.27
12.9 CASE STUDIES For three large water pumping systems with various surge protection devices waterhammer analyses and site measurements have been conducted. The surge protection systems in question are (1) one-way and simple surge tanks, (2) an air chamber, and (3) airvacuum breakers.
12.9.1
Case Study with One-way and Simple Surge Tanks
A very large pumping station has been installed and commissioned to deliver water over a distance of over 30 kilometers. Three three-stage centrifugal pumps run at a synchronous speed of 720 rpm, with individual rated capacities of 1.14 m3/sec, rated heads of 165 m, and rated power of 2090 kw. Initial surge analysis indicated potential water-column separation. The surge protection system was then designed with one-way and simple surge tanks as well as air-vacuum valves strategically located. The efficacy of these various surge protection devices was assessed from site measurements. Measurements of pump speed, discharge valve position, pump flow rate, and pressure at seven locations were conducted under various transient test conditions. The site measurements under three-pump operation allowed for improvement of hydraulic transient calculations for future expansion to four and five pumps. Figure 12.14 illustrates the profile of the ground and the location of the three pairs of surge tanks. The first and second pair of surge tanks are of the one-way (feed tank) variety, while the third pair are simple open on-line tanks.
FIGURE 12.14 Case study of pump power failure at pumping station with three pair of surge tanks— two pair one way and one pair simple surge tanks Martin(1992).
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HYDRAULIC TRANSIENT DESIGN FOR PIPELINE SYSTEMS
12.28
Chapter Twelve
Figure 12.15. The test program and transient analysis clearly indicated that the piping system was adequately protected by the array of surge tanks inasmuch as there were no negative pressures.
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HYDRAULIC TRANSIENT DESIGN FOR PIPELINE SYSTEMS
Hydraulic Transient Design for Pipeline Systems 12.29
FIGURE 12.16 Case study of air chamber performance for raw water supply.
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HYDRAULIC TRANSIENT DESIGN FOR PIPELINE SYSTEMS
12.30
Chapter Twelve
Pump trip tests were conducted for three-pump operation with cone valves actuated by the loss of motor power. For numerical analysis a standard computer program applying the method of characteristics was employed to simulate the transient events. Figure 12.15 shows the transient pressures for three pump power failure. The transient pressures agree reasonably well for the first 80 seconds. The minimum HGL's in Fig. 12.14 also show good agreement, as well as the comparison of measured and calculated pump speeds in
12.9.2
Case Study with Air chamber
Hydraulic transients caused by simultaneous tripping of pumps at the pumping station depicted on Fig. 12.16 were evaluated to assess the necessity of surge protection. Without the presence of any protective devices such as accumulators, vacuum breakers, or surge suppressors, water hammer with serious consequences was shown to occur due to depressurization caused by the loss of pumping pressure following sudden electrical outage. In the case of no protection a large vapor cavity would occur at the first high point above the pumping station, subsequently collapsing after the water column between it and the reservoir stops and reverses. This phenomenon, called water-column separation, can be mitigated by maintaining the pressures above vapor pressure. The efficacy of the 11.6 m (38 ft) diameter air chamber shown in Fig. 12.16 was investigated analytically and validated by site measurements for three-pump operation. The envelope of the minimum HGL drawn on Fig. 12.16 shows that all pressures remained positive. The lower graph compares the site measurement with the calculated pressures obtained by a standard waterhammer program utilizing MOC.
12.9.3
Case Study with Air-vacuum Breaker
Air-inlet valves or air-vacuum breakers are frequently installed on liquid piping systems and cooling water circuits for the purpose of (1) eliminating the potential of water-column separation and any associated waterhammer subsequent to vapor pocket collapse; (2) protecting the piping from an external pressure of nearly a complete vacuum; and (3) providing an elastic cushion to absorb the transient pressures. A schematic of the pumping and piping system subject to the field test program is shown in Fig. 12.17. This system provides the cooling water to a power plant by pumping water from the lower level to the upper reservoir level. There are five identical vertical pumps in parallel connected to a steel discharge pipe 1524 mm (60 in) in diameter. On the discharge piping of each pump there are 460 mm (18 in) diameter swing check valves. Mounted on top of the 1524 mm (60 in) diameter discharge manifold is a 200 mm (8 in) diameter pipe, in which is installed a swing check valve with a counter weight. Air enters the vacuum breaker through the tall riser, which extends to the outside of the pump house. Transient pressures were measured in the discharge header for simultaneous tripout of three, four, and five pumps. The initial prediction of the downsurge caused by pump power failure was based on the method of characteristics with a left end boundary condition at the pumps, junction boundary condition at the change in diameter of the piping, and a constant pressure boundary condition at the right end of the system. The predicted pressure head variation in the pump discharge line is shown in Fig. 12.17 for a simulated five pump tripout. The predicted peak pressure for the five pump
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HYDRAULIC TRANSIENT DESIGN FOR PIPELINE SYSTEMS
Hydraulic Transient Design for Pipeline Systems 12.31
Figure 12.17. Case Study of Vacuum Breaker Performance for River Water System of Nuclear Plant, Martin (1980).
tripout compares favorably with the corresponding measured peak, but the time of occurrence of the peaks and the subsequent phasing vary considerably. Analysis without a vacuum breaker or other protective device in the system predicted waterhammer pressure caused by collapse of a vapor pocket to exceed 2450 kPa (355 psi). The vacuum breaker effectively reduced the peak pressure by 60 per cent. Peak pressures can be adequately predicted by a simplified liquid column, orifice, and air spring system. Water-column separation can be eliminated by air-vacuum breakers of adequate size.
REFERENCES Chaudhry, M. H., Applied Hydraulic Transients, 2d ed., Van Nostrand Reinhold, New Yorle 1987. Donsky, B., “Complete Pump Characteristics and the Effects of Specific Speeds on Hydraulic Transients,” Journal of Basic Engineering, Transactions, American Society of Mechanical Engineers, 83: 685–699, 1961. Fleming, A. J., “Cost-Effective Solution to a Waterhammer Problem,” Public Works, 42—44, 1990. Fox, J. A., Hydraulic Analysis of Unsteady Flow in Pipe Networks, John Wiley & Sons, New Yorle 1977. Karney, B. W., and McInnis, D., “Transient Analysis of Water Distribution Systems,” Journal American Water Works Assoriation, 82: 62–70, 1990. Kittredge, C. P., “Hydraulic Transients in Centrifugal Pump Systems,” Transactions,American Society of Mechanical Engineers, 78: 1307–1322, 1956. Knapp, R. T., “Complete Characteristics of Centrifugal Pumps and Their Use in Prediction of Transient Behavior,” Transactions, American Society of Mechanical Engineers , 59:683–689, 1937. Knapp, R. T., “Centrifugal-Pump Performance Affected by Design Features,” Transactions, American Societiy of Mecharnal Engineers, 63:251–260, 1941. Kroon, J. R., Stoner, M. A., and Hunt, W. A., “Water Hammer: Causes and Effects,” Journal American Water Works Assoriation, 76:39–45, 1984. Lescovitch, J. E., “Surge Control of Waterhammer by Automatic Valves,” Journal American Water Works Assoriation, 59:632-644, 1967.
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HYDRAULIC TRANSIENT DESIGN FOR PIPELINE SYSTEMS
12.32
Chapter Twelve
Lundgren, C. W., “Charts for Determining Size of Surge Suppressor for Pump-Discharge Lines,” Journal of Engineering for Power, Transactions, American Society of Mechanical Engineers, 93:43–47, 1961. Marchal, M., Flesh, G., and Suter, P., “The Calculation of Waterhammer Problems by Means of the Digital Computer,” Proceedings, International Symposium on Waterhammer in Pumped Storage Projects, American Society of Mechanical Engineers (ASME), Chicago, 1965. Martin, C. S., “Entrapped Air in Pipelines,” Paper F2 Second BHRA International Conference on Pressure Surges, The City University, London, September 22–24, 1976. Martin, C. S., “Transient Performance of Air Vacuum Breakers,” Fourth International Conference on Water Column Separation, Cagliari, November 11–13, 1979. “Transient Performance Air Vacuum Breakere,” L'Energia Elettrica, Proceedings No. 382, 1980, pp. 174–184. Martin, C. S., “Representation of Pump Characteristics for Transient Analysis,” ASME Symposium on Performance Characteristics of Hydraulic Turbines and Pumps, Winter Annual Meeting, Boston, November 13–18, pp. 1–13,1983. Martin, C. S., “Experience with Surge Protection Devices,” BHr Group International Conference on Pipelines, Manchester, England, March pp. 171–178 24–26, 1992. Martin, C. S., “Hydraulics of Valves,” in J. A. Schetz and A. E. Fuhs, eds. Handbook of Fluid Dynamics and Fluid Machinery, Vol. III, McGraw-Hill, pp. 2043–2064. 1996, Parmakian, J., Water Hammer Analysis, Prentice-Hall, New York, 1955. Parmakian, J., “Unusual Aspects of Hydraulic Transients in Pumping Plants,” Journal of the Boston Society of Civil Engineers, 55:30–47, 1968. Parmakian, J., “Surge Control,” in M. H. Chaudhry, ed., Proceedings, Unsteady Flow in Conduits, Colorado State University, pp. 193–207 1982, Parmakian, J., “Incidents, Accidents and Failures Due to Pressure Surges,” in M. H. Chaudhry ed., Proceedings, Unsteady Flow in Conduits, Colorado State University, pp. 301–311 1982. Sanks, R. L., Pumping Station Design, Butterworths, Bestar, 1989. Stepanoff, A. I., Centrifugal and Axial Flow Pumps, John Wiley & Sons, New York, 1957. Swanson, W.M., “Complete Characteristic Circle Diagrams for Turbomachinery,” Transactions, American Society of Mechanical Engineers, 75:819–826, 1953. Thorley, A. R. D., Fluid Transients in Pipeline Systems, D. & L. George Ltd., 1991. Tullis, J. P., Hydraulics of Pipelines, John Wiley & Sons, New Yorh1989. Watters, G. Z., Modern Analysis and Control of Unsteady Flow in Pipelines, Ann Arbor Science, Ann Arbor, MI, 1980. Weaver, D. L., “Surge Control,” Journal American Water Works Association, 64: 462–466, 1972. White, I. M., “Application of the Surge Suppressor in Water Systems,” Water Works Engineering, 45,304-306, 1942. Wood, D.J., and Jones, S.E., “Waterhammer Charts for Various Types of Valves”, ASCE, Journal of Hydraulics Division, HY1, 99:167-178, 1973. Wylie, E. B., and Streeter, V. L., Fluid Transients in Systems, Prentice-Hall, 1993.
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Source: HYDRAULIC DESIGN HANDBOOK
CHAPTER 13
HYDRAULIC DESIGN OF DRAINAGE FOR HIGHWAYS G. Kenneth Young, JR. Stuart M. Stein GKY and Associates, Inc. Springfield, VA
13.1 INTRODUCTION The objective of this chapter is to provide design guidance for highway hydraulic elements: gutters, roadside conveyance, inlets, and bridge scuppers. Proper hydraulic design of highway drainage is essential to avoid disruption of the highways transportation function, maintain safe travel conditions, and sustain infrastructure. When rain falls on a sloped pavement surface, it forms a thin film of water that increases in thickness as it flows to the edge of the pavement and concentrates in gutters or roadside ditches; overflow from gutters and ditches spreads out onto the pavement. Factors that influence the depth of water on the pavement and its spread in gutters are the length of the flow path, the texture and slope of the surface, and rainfall intensity. As the depth of water on the pavement or the gutter spread increases, the potential for vehicular hydroplaning or disruption of the highway’s transportation function increases. With design methods and guidance, the surface drainage elements are sized to function at predetermined reliability thresholds. The design of highway surface drainage is a critical component of the highway system (Johnson and Chang, 1984). The hydraulics of cross drainage handled by culverts and bridges is not considered in this chapter, and subsurface storm drains that accept surface water removed at inlets also are excluded. This chapter provides information on the following aspects of highway drainage design: highway geometrics that influence hydraulic design, design event selection, design flow estimation, gutter design, ditch hydraulic and stability design, inlet design, and bridge deck hydraulics. The reader also is directed to documents of the Federal Highway Administration, including Hydraulic Engineering Circular No. 12 (HEC–12), “Drainage of Highway Pavement” (Johnson and Chang, 1984), HEC–15, “Design of Roadside Channels with Flexible Linings” (Chen and Cotton, 1986), HEC–21, “Design of Bridge Deck Drainage” (Young et al., 1993), and HEC–22, “Urban Drainage Design Manual” (Brown et al., 1996). Information on the physics of surface drainage can be found in Anderson et al. (1995). 13.1 Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
HYDRAULIC DESIGN OF DRAINAGE FOR HIGHWAYS
13.2
Chapter Thirteen
13.2 GENERAL GEOMETRIC AND PAVEMENT GUIDELINES THAT INFLUENCE DRAINAGE Types of pavement materials and highway geometrics can influence drainage. Placement of drainage structures can have a secondary adverse effect with respect to safety.
13.2.1 Pavement Sheet flow on the pavement can cause hydroplaning, and pavement materials can influence sheet flow. Smooth pavements can cause water to flow faster and reduce the thickness of the sheet flow film; however, smooth pavements can decrease the coefficient of friction of pavement and tires and lead to longer skids as well. As the depth of water flowing over a roadway’s surface increases, the potential for hydroplaning increases. When a rolling tire encounters a film of water on the roadway, the water is channeled through the tire tread pattern and through the surface roughness of the pavement. Hydroplaning occurs when the drainage capacity of the tire tread pattern and the pavement surface is exceeded and the water begins to build up in front of the tire. As the water builds up, a water wedge is created, producing a hydrodynamic force that can lift the tire off the pavement surface. This is considered full dynamic hydroplaning, and, because the water offers little shear resistance, the tire loses its tractive ability and the driver loses control of the vehicle. Hydroplaning is a function of the water depth, roadway geometrics, vehicle speed, treat depth, tire inflation pressure, and conditions of the pavement surface. The following are several pavement design factors: 1. Design the highway to reduce the drainage path lengths of the water flowing over the pavement. Crown sections split flow to both sides and are superior to long sloping runs of pavement from one side to the other. This will prevent build-up of sheet flow thickness. 2. Increase the pavement surface texture depth by such methods as grooving of Portland cement concrete. An increase of pavement surface texture will increase the drainage capacity at the tire-pavement interface and reduce hydroplaning. 3. Use open-graded pavements, which have been shown to reduce greatly the hydroplaning potential of the roadway surface. This reduction is the result of the water’s ability to be forced through the pavement under the tire and to enter the surbase rather than run off. Surface texture also increases the coefficient of tire-to-pavement friction. Such open grading also involves provision of subsurface drainage details to the pavement design. 4. Use intercepting drainage structures along the roadway to capture the sheet of water over the pavement, which will reduce the thickness of the film of water and reduce the hydroplaning potential of the roadway surface. Long slotted inlets perpendicular to the flow and open expansion joints on bridges can accomplish this.
13.2.2 Grade The recommended minimum values of roadway longitudinal slope given in the Policy on Geometric Design of the American Association of State Highway and Transportation Officials (AASHTO, 1990) provide safe, acceptable pavement drainage. In addition, the following general guidelines are presented:
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HYDRAULIC DESIGN OF DRAINAGE FOR HIGHWAYS
Hydraulic Design of Drainage for Highways 13.3
1. A minimum longitudinal gradient is more important for a curbed pavement than for an uncurved pavement because the water is constrained by the curb on the one hand, and runs off the shoulder, on the other, with crowned cross sections. 2. Desirable gutter grades should not be less than 0.5 percent for curbed pavements, with an absolute minimum of 0.3 percent. Minimum grades can be maintained in flat terrain by using of a rolling profile or by warping the cross slope to achieve rolling gutter profiles. 3. In sag vertical curves, a minimum grade of 0.3 percent should be maintained within 15 m (50 ft) of the low point of the curve. This is accomplished where the length of the curve in meters divided by the algebraic difference in grades in percent (K) is equal to or greater than 50 (156 in English units). This is represented as L K G2 G1
(13.1)
where K = vertical curve constant m/percent (ft/percent), L = horizontal length of curve m (ft), G1 = grade of roadway on one side of point of vertical intersection, percent, G2 = grade of roadway on other side of point of vertical intersection, percent.
13.2.3 Cross Slope Table 13.1 indicates an acceptable range of cross slopes (AASHTO, 1990). These cross slopes are a compromise between the need for reasonably steep cross slopes for drainage and relatively flat cross slopes for driver comfort and safety. These cross slopes represent standard practice. AASHTO (1990) should be consulted before deviating from these values. Cross slopes of 2 percent have little effect on driver effort in steering or on friction demand for vehicle stability (Gallaway et al., 1979). Use of a cross slope steeper than 2 percent on pavements with a central crown line is not desirable; however, in areas of intense rainfall, a somewhat steeper cross slope (2.5 percent) can be used to facilitate drainage. On multilane highways where three or more lanes are sloped in the same direction, it is desirable to counter the resulting increase in sheet-flow depth by increasing the cross
TABLE 13.1
Normal Pavement Cross Slopes
Surface Type
Range in Rate or Surface Slope
High-type surface 2 lanes
0.015 0.020
3 or more lanes, each direction
0.015 minimum, increase 0.005 to 0.010 per lane, 0.040 maximum
Intermediate surface
0.015 0.030
Low-type surface
0.020 0.060
Shoulders Bituminous or concrete
0.020 0.060
With curbs
0.040
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HYDRAULIC DESIGN OF DRAINAGE FOR HIGHWAYS
13.4
Chapter Thirteen
slope of the outermost lanes. The two lanes adjacent to the crown line should be pitched at the normal slope, and successive lane pairs, or portions thereof outward, should be increased by approximately 0.5 to 1 percent. The maximum pavement cross slope should be limited to 4 percent. The following are additional guidelines related to cross slope: 1. Inside lanes can be sloped toward the median if conditions warrant. This is not widely encouraged. 2. Median areas should not be drained across travel lanes. 3. The number and length of flat pavement sections in cross-slope transition areas should be minimized. Consideration should be given to increasing cross slopes in sag vertical curves and crest vertical curves and in sections of flat grade. 4. Shoulders should be sloped to drain away from the pavement, except with raised, narrow medians and superelevations.
13.2.4 Safety The placement of drainage structures in the traveled way can constitute an obstacle to moving traffic. Median inlets, storm drain inlets, and junctions need to have top elevations that do not project significantly above the surface. Culvert entrances and end walls need to be situated away from likely trajectories of errant vehicles that have departed traffic lanes because a driver must maneuver to avoid an accident or is weary. As a general rule, drainage devices should provide a low silhouette to oncoming traffic.
13.3 DESIGN FREQUENCY AND SPREAD The most significant design decisions considered in sizing pavement drainage facilities are the frequency of the design runoff event and the allowable spread of water on the pavement. A related consideration is the use of an event of lesser frequency (greater storm magnitude) to check the drainage design (AASHTO, 1991; Johnson and Chang, 1984). Spread and design frequency are not independent. The implications of the use of a criteria for spread of half a traffic lane is considerably different for one design frequency than for a lesser frequency. Spread also has different implications for a low-traffic, lowspeed highway than for a higher classification highway. Balancing risks and flooding are central to the issue of highway pavement drainage and are important to highway safety.
13.3.1 Risk Balancing The objective of highway storm-drainage design is to provide safe passage for vehicles during the storm event. The design of a drainage system for a section of highway pavement is to collect runoff in the gutter or roadside ditch and convey it to inlets or culverts in a manner that provides a reasonable degree of safety for traffic and pedestrians at a reasonable cost. As spread from the curb or roadside ditch increases, the risks of traffic accidents and delays and the nuisance and possible hazard to pedestrian traffic increase. The recurrence interval and allowable spread for design implies acceptable risks of accidents and traffic delays and acceptable costs for the drainage system. Risks associated with water on traffic lanes are greater with high-volume traffic, high speeds, and higher highway classifications than with lower-volumes traffic, speeds, and highway classifications.
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HYDRAULIC DESIGN OF DRAINAGE FOR HIGHWAYS
Hydraulic Design of Drainage for Highways 13.5
The major considerations that enter into the selection of design frequency and design spread are as follows: 1. Functional classification of the highway drives risks: higher functions imply lower acceptable risks, which translate to low intervals of recurrence with a typical limit of 1 in 100 years. Because ponding on traffic lanes of high-speed, high-volume highways is contrary to the public’s expectations, the risks of accidents and the costs of traffic delays are high. 2. Design speed is important to the selection of design criteria. (Also, at speeds greater than 70 km/h (44 mi/h), water on the pavement can cause hydroplaning; in this case, rain intensity, is more significant than spread.) 3. Projected traffic volumes are an indicator of the economic importance of keeping the highway open to traffic. The opportunity costs of lost driver and passenger times associated with traffic delays increase rapidly with increasing volumes of traffic. 4. The intensity of rainfall may significantly affect the selection of design frequency and spread. Risks associated with the spread of water on pavements may be less in arid areas that are subject to high-intensity thunderstorms than in areas that have frequent but less intense storms. 5. Capital costs are the other side of the equation. Cost considerations make it necessary to balance the approach to the selection of design criteria to be essentially risk based. “Tradeoffs” between desirable and practicable criteria are necessary because of costs. Other considerations include inconvenience, hazards, and nuisances to pedestrian traffic. These considerations should not be minimized; in some locations, such as commercial areas, they may assume major importance. Local design practice also may be a major consideration because it can affect the feasibility of designing to higher standards and it influences the public’s perception of acceptable practice. The relative elevation of the highway and surrounding terrain is an additional consideration where water can be drained only through a storm drainage system, as in underpasses and depressed sections. The potential for ponding to hazardous depths should be considered when selecting the frequency and spread criteria and when checking the design against storm runoff events of lesser frequency than in the design event. Spread on traffic lanes can be tolerated to greater widths when traffic volumes and speeds are low. Spreads of one-half of a traffic lane or more are usually considered to be a minimum-type design for low-volume local roads. The selection of design criteria for intermediate types of facilities can be difficult. For example, some arterials with relatively high traffic volumes and speeds may not have shoulders that will convey the design runoff without encroaching on the traffic lanes.
13.3.2 Design Guidance Regarding Frequency and Spread Table 13.2 provides suggested minimum design frequencies and spread based on the type of highway and the traffic speed. The recommended design frequency for depressed sections and underpasses where ponded water can be removed only through the storm drainage system is a 50-yr frequency event. The use of a lesser frequency event, such as a 100-yr storm, to assess hazards at critical locations where water can pond to appreciable depths is commonly referred to as a check storm or check event.
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HYDRAULIC DESIGN OF DRAINAGE FOR HIGHWAYS
13.6
Chapter Thirteen
TABLE 13.2
Suggested Minimum Design Frequency and Spread
Road Classification
Design Frequency
Design Spread
High volume
70 km/h (45 mph)
10–yr
Shoulder 1 m (3 ft)
Divided
70 km/h (45 mph)
10–yr
Shoulder
Bi-directional
Sag point
50–yr
Shoulder 1 m (3 ft)
Collector
70 km/h (45 mph)
10–yr
1
70 km/h (45 mph)
10–yr
Shoulder
Sag point
10–yr
1
/2 driving lane
Low ADT
5–yr
1
/2 driving lane
High ADT
10–yr
1
/2 driving lane
Sag point
10–yr
1
/2 driving lane
Local streets
/2 driving lane
13.3.3 Selection of Check Storm and Spread A check storm should be used any time that runoff could cause unacceptable flooding during less frequent events. In addition, inlets should always be evaluated for a check storm when a series of inlets terminates at a sag vertical curve where ponding to hazardous depths could occur. The frequency selected for the check storm should be based on the same considerations used to select the design storm: i.e., the consequences of spread exceeding that chosen for design and the potential for ponding. Where no significant ponding can occur, check storms are normally unnecessary. Criteria for spread during the check event are one lane open to traffic during the check storm event or one lane free of spread during the check storm event.
13.4 SELECTION OF DESIGN HYDROLOGY The typical highway drainage area is small, and design flows are linked to design rainfall intensity. Three methods are discussed for the selection of design rainfall intensity: (1) the rational method that pegs rainfall intensity to a design frequency, (2) the avoidance of the hydroplaning method, and (3) driver vision-impairment method. The first method uses established drainage policy (such as that implied in Table 13.2), to select a return period and calculate time of concentration to avoid ponding. The second and third methods consider vehicle safety directly, either from the standpoint of avoidance of hydroplaning films or from driver vision being impaired because of heavy rain. All methods lead to designs for which selected spread is a design requirement. Method 1 involves rainfall return period. Methods 2 and 3 select rainfall intensity on the basis of physical limits of the vehicle (skid avoidance) or the driver’s vision; in these two cases, the design frequency is imputed by the physical limits rather than selected by the analyst or set by the prevailing policy. 13.4.1 Design Flow Calculation The commonly used equation for the calculation of peak flow from the small areas associated with highway drainage is a simple mass balance which is considered to be valid for areas less than 80 ha (200 ac) (ASCE, 1992): Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
HYDRAULIC DESIGN OF DRAINAGE FOR HIGHWAYS
Hydraulic Design of Drainage for Highways 13.7
CIA Q Kc
(13.2)
where Q flow, m3/s (ft3/s), C dimensionless runoff coefficient; outflow volume/rainfall volume, I rainfall intensity, mm/h (in/h); to be selected by rational method, avoidance of hydroplaning method, or driver vision impairment method, A drainage area ha (ac), and Kc units conversion equal to 360 (1 in English units). Assumptions inherent in Eq. (13.2) are as follows (McCuen, et al., 1996): • Peak flow occurs as the entire area contributes to flow. • Rainfall intensity is uniform over a time duration equal to the time of concentration Tc. The time of concentration is the time required for water to travel from the most remote point of the basin to the design element. • Peak flow frequency is the same as that of the rainfall intensity: i.e., the 10-yr intensity of rainfall is assumed to produce the 10-yr peak flow. • The coefficient of runoff is independent of intensity and duration. Selection of the runoff coefficient C is a function of the ground cover (i.e., land use). Typical values for C are given in Table 13.3. If the area contains varying types of land, a composite coefficient is calculated through areal weighting (McCuen et al., 1996):
冘 冘
Ci Ai
Weighted C Ai i
(13.3)
i
where i is the subscript designating values for different types of land. Highway rights-ofway typically have C 0.5 because of high pavement coverage.
TABLE 13.3
Runoff Coefficients
Type of Drainage Area Business Downtown areas Neighborhood areas Residential Single-family areas Multi-units, detached Multi-units, attached Suburban Apartment dwelling areas Industrial Light areas Heavy areas Parks, cemeteries Playgrounds
Runoff Coefficient C*
0.70 – 0.95 0.50 – 0.70 0.30 – 0.50 0.40 – 0.60 0.60 – 0.75 0.25 – 0.40 0.50 – 0.70 0.50 – 0.80 0.60 – 0.90 0.10 – 0.25 0.20 – 0.40
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HYDRAULIC DESIGN OF DRAINAGE FOR HIGHWAYS
13.8
Chapter Thirteen TABLE 13.3
(Continued)
Type of Drainage Area
Runoff Coefficient C*
Railroad yard areas Unimproved areas Lawns Sandy soil, flat, 2% Sandy soil, average, 2 7 % Sandy soil, steep, 7% Heavy soil, flat, 2% Heavy soil, average, 2 7% Heavy soil, steep, 7% Streets Asphaltic Concrete Brick Drives and walks
0.20 – 0.40 0.10 – 0.30 0.05 – 0.10 0.10 – 0.15 0.15 – 0.20 0.13 – 0.17 0.18 – 0.22 0.25 – 0.35 0.70 – 0.95 0.80 – 0.95 0.70 – 0.85 0.75 – 0.85
Roofs
0.75 – 0.95
Source: ASCE, 1960. *Higher values are usually appropriate for steeply sloped areas and longer return periods because infiltration and other losses have a proportionally smaller effect on runoff in these cases.
13.4.2 Rainfall Intensity by the Rational Method The intensity, duration, and frequency (IDF) curves of rainfall are necessary to select the intensity of rainfall by the rational method. Regional IDF curves are available in most state highway agency manuals and also are available from the National Oceanic and Atmospheric Administration and the FHWA HYDRAIN computer program. If IDF curves are not available, they need to be developed. The assumption in the rational method is that the time of concentration for a drop of water to move from the high point to the low point in an area equals the rainfall duration to realize the maximum flow for a given intensity of rainfall. Thus, the strategy is to calculate the time of concentration as the sum of the travel times of the droplet as it moves along its path: overland or sheet flow, shallow concentrated flow, gutter or ditch flow, and pipe flow. The sum of times is accumulated in a downhill direction until the droplet reaches the drainage element to be sized. Thus, time of concentration is calculated as the sum of the travel times within the various consecutive flow segments. This is called the segment method. 13.4.2.1 Sheet flow travel time. Sheet flow is the shallow mass of runoff on a planar surface with a uniform depth across the sloping surface. This usually occurs at the headwater of streams over relatively short distances, rarely more than about 90 m (300 ft) and possibly less than 25 m (80 ft). Sheet flow is commonly estimated with a version of the kinematic wave equation, a derivation of Manning’s equation, as follows (McCuen et al., 1996):
冢 冣
Kc nL Ts 0 I .4 兹苶 S
0.6
(13.4)
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HYDRAULIC DESIGN OF DRAINAGE FOR HIGHWAYS
Hydraulic Design of Drainage for Highways 13.9
where Ts = sheet flow travel time, min; n = roughness coefficient as given in Table 13.4; L = flow length, m (ft); I = rainfall intensity, mm/h (in/h); S = surface slope, m/m (ft/ft); and Kc = empirical coefficient equal to 6.943 (0.933 in English units). Because I depends on Ts and Ts is not initially known, the computation of Ts is an iterative process. An initial estimate of Ts is assumed and used to obtain I from the IDF curve for the locality. The Ts is then computed from Eq. (13.4) and used to check the initial value of I. If they are not the same, the process is repeated until two successive Ts estimates are the same to within a reasonable tolerance (McCuen et al., 1996). For the following types of segments downhill from the sheet flow, the velocity is calculated and the segment travel time computed. To arrive at the travel time, the segment length is divided by the velocity. 13.4.2.2 Shallow concentrated flow velocity. After short distances of, at most, 90 m (300 ft), sheet flow tends to concentrate in rills and then gullies of increasing proportions. Such flow is usually referred to as shallow concentrated flow. The velocity of such flow can be estimated using a relationship between velocity and slope as follows (McCuen et al., 1996): Vc Kc kSp0.5
TABLE 13.4
(13.5)
Manning’s Roughness Coefficient (n) for Overland Sheet Flow
Surface Description
n
Smooth asphalt Smooth concrete Ordinary concrete lining Good wood Brick with cement mortar Vitrified clay Cast iron Corrugated metal pipe Cement rubble surface Fallow (no residue) Cultivated soils Residue cover 20% Residue cover 20% Range (natural) Grass Short grass, prairie Dense grasses Bermuda grass Woods* Light underbrush Dense underbrush
0.011 0.012 0.013 0.014 0.014 0.015 0.015 0.024 0.024 0.05 0.06 0.17 0.13 0.15 0.24 0.41 0.40 0.80
Source Mc Cuen et al 1996*When selecting n, consider cover to a height of about 30 mm. This is the only part of the plant cover that will obstruct sheet flow.
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HYDRAULIC DESIGN OF DRAINAGE FOR HIGHWAYS
13.10
Chapter Thirteen
where Vc velocity, m/s (ft/s); k intercept coefficient, as given in Table 13.5; Sp slope, percent; and Kc units conversion equal to 1 (0.3048 in English units). 13.4.2.3 Gutter flow velocity. To find the time of flow in the gutter-flow component of the time of concentration, a method for estimating the average velocity in a reach of gutter is needed. The time of flow in a triangular channel with uniform inflow per unit of length can be estimated accurately by using of an average velocity of flow in the gutter. Integration of the Manning equation for a right triangular channel with respect to time and distance yields an average velocity for the channel length at the point where spread is equal to 65 percent of the maximum spread for channels with zero flow at the upstream end, as discussed in HEC-12, “Drainage of Highway Pavement” (Johnson and Chang, 1984). The velocity in a gutter with a curb (shallow cross slope, 1 to 10 percent) is K Vg c G0.5 Sx0.67 T0.67 (13.6) n where Vg velocity, m/s (ft/s); Kc units conversion equal to 0.752 (1.12 in English units); G grade, m/m (ft/ft); Sx cross slope, m/m (ft/ft); T spread, m (ft) and, n Manning’s friction factor (typically taken to be 0.016 for highway gutters). Thus, if the allowable spread is 5 m, then 0.65 5 3.25 m of spread is used in Eq. (13.6). 13.4.2.4 Open channel and pipe flow velocity. Flow in gullies empties into channels or pipes. Open channels are assumed to begin where either the blue streamline shows on the U.S. Geological Survey quadrangle sheets or the channel is visible on aerial photographs. Cross-section al geometry and roughness should be obtained for all channel reaches in the watershed. Manning’s equation can be used to estimate average flow velocities in pipes and open channels as follows: K VP c R2/3S1/2 n
(13.7)
where
TABLE 13.5
Intercept Coefficients for Velocity Versus Slope Relationship of Eq. (13.5)
Land Cover/Flow Regime
k
Forest with heavy ground litter; hay meadow (overland flow)
0.076
Trash fallow or minimum tillage cultivation; contour or strip cropped; woodland (overland flow)
0.152
Short grass pasture (overland flow)
0.213
Cultivated straight row (overland flow)
0.274
Nearly bare and untilled (overland flow); alluvial fans in western mountain regions
0.305
Grassed waterway (shallow concentrated flow)
0.457
Unpaved (shallow concentrated flow)
0.491
Paved area (shallow concentrated flow); small upland gullies
0.619
Source: McCuen et al., 1996
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HYDRAULIC DESIGN OF DRAINAGE FOR HIGHWAYS
Hydraulic Design of Drainage for Highways 13.11
n roughness coefficient as given in Table 13.6; VP velocity, m/s (ft/s); R hydraulic radius (defined as the flow area divided by the wetted perimeter), m (ft); S slope, m/m (ft/ft); and Kc units conversion factor equal to 1 (1.49 in English units). For a circular pipe flowing full, the hydraulic radius is one-half the radius. For a wide rectangular channel (w 10 d), the hydraulic radius is approximately equal to the depth. 13.4.2.5 Combined shallow, gutter, open-channel, and pipe travel time. The travel time Tc is calculated as follows:
TABLE 13.6
Values of Manning Coefficient (n) for Channels and Pipes
Conduit Material Closed conduits Asbestos-cement pipe Brick Cast iron pipe: Cement lined & seal coated Concrete (monolithic) Concrete pipe Corrugated–metal pipe (13 mm 65 mm [1/2in 2 1/2in corrugations]) Plain Paved invert Spun asphalt lined Plastic pipe (smooth) Vitrified clay Pipes Liner plates Open Channels Lined channels Asphalt Brick Concrete Rubble or riprap Vegetal Excavated or dredged channels Earth, straight and uniform Earth, winding, fairly uniform Rock Unmaintained Natural channels (minor streams, top width at flood stage 30 m [100 ft]) Fairly regular section Irregular section with pools
Mannings n*
0.0110.015 0.0130.017 0.0110.015 0.0120.014 0.0110.015
0.0220.026 0.0180.022 0.0110.015 0.0110.015 0.0110.015 0.0130.017
0.0130.017 0.0120.018 0.0110.020 0.0200.035 0.0300.40 0.0200.030 0.0250.040 0.0300.045 0.0500.14
0.030.07 0.030.10
Source: Jennings et al., 1994. *Lower values are usually for well-constructed and maintained (smoother) pipes and channels.
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HYDRAULIC DESIGN OF DRAINAGE FOR HIGHWAYS
13.12
Chapter Thirteen
冘
Tc Ts
i
Li 60Vi
(13.8)
where Tc time of concentration, min; Ts sheet flow travel time, min; Li flow length for segment i, m (ft); and Vi velocity for segment i, m/s (ft/s) (Vc, Vg, or Vp). 13.4.2.6 Rainfall intensity as a function of duration and return period. Once Tc is determined, it is assumed to be equal to the rainfall’s duration. With the rational method, the selected return period, as tabulated in the Table 13.2 guidance, has an associated rainfall intensity-versus-duration curve. This curve is used to establish the rational method rainfall intensity i for use with Eq. (13.2).
13.4.3 Rainfall Intensity by Avoidance of the Hydroplaning Method HEC–21 (Young et al., 1993) developed an alternative method for selecting rainfall intensity that is not dependent on rainfall frequency. (Note that once an intensity is selected, a frequency is implied.) The method assumes that if drivers have a hydroplaning slip sensation at the design speed (plus a reasonable allowance), they will typically slow down to 90 km/h (55 mph) or less and this behavior governs highway function more than traffic reaction to gutter flooding. The method selects values of vehicle speed, tire tread depth, pavement texture, and tire pressure and calculates the thickness of the sheet flow film at incipient hydroplaning (assumed at 10 percent spindown: e.g., the tire rolls 1.1 times the circumference to move one circumference). The empirical equation for the vehicle speed which initiates hydroplaning is V K1SD0.04(K2P)0.3 [(K3TD) K3] 0.06A
(13.9)
where A Texas Transportation Institute empirical curve fitting relationship which is the greater of: 10.409 3.507 A1 (K4d)0.06 or
冤
冥
28.952 7.817 K4TXD0.14 A2 (K4d)0.06
(13.10)
where V vehicle speed in km/h (mph); TD tire tread depth in mm (1/32 in); TXD pavement texture depth, mm (in); d water film depth, mm (in); P tire pressure, kPa (psi); K1 unit conversion equal to 0.3048 (1 in English units); K2 unit conversion equal to 6.894 (1 in English units); K3 unit conversion equal to 0.794 (1 in English units) K4 unit conversion equal to 25.4 (1 in English units) and SD spindown (percent); hydroplaning is assumed to begin at 10 percent spin down. This occurs when the tire rolls 1.1 times the circumference to achieve a forward progress distance equal to one circumference. The method determines a film depth, d, associated with selected values for V, TD, TXD, P, and with SD 10 percent, by solving the above equation. An estimate of design
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HYDRAULIC DESIGN OF DRAINAGE FOR HIGHWAYS
Hydraulic Design of Drainage for Highways 13.13
d for V 88 km/h, TD 5.55 mm (median tire tread), TXD 0.97 mm (mean pavement texture), P 186 kPA (median tire pressure), and SD 10 percent (by definition) is d 1.87 mm. This is suggested as a sound design value because it represents the combination of the mean or median of all the above parameters. Manipulation of Eq. (13.9), using typical values, gives the following sensitivity information. An increase of 1 percent in pavement texture increases the hydroplaning depth by 1.6 percent, an increase of 1 percent in tread depth increases the hydroplaning depth by 0.8 percent, and an increase of 1 percent in tire pressure increases the hydroplaning depth by 2.4 percent. Study of Eq. (13.9) indicates that 90 km/h (55 mph) is the speed value of concern for practical control of hydroplaning. At this speed, a decrease of 1 percent in speed increases the hydroplaning depth by 25 percent. Speeds below 90 km/h (55 mph) tend to be safe from the threat of hydroplaning because heavy rainfall is insufficient to generate hydroplaning depth. An increase in speed of 1 percent decreases the hydroplaning depth by 25 percent. Above 90 km/h (55 mph), hydroplaning can occur on extremely thin surface films associated with light rainfall intensities–intensities of 25 mm/h (1 in/h) and less, which are rainfalls that are usually smaller than those used to design gutters, inlets, and storm sewers. Once a design d is determined, it is assumed that the thickness of the water film on the pavement should be less than d. Water flows in a sheet across the surface to the edge of the gutter flow. The width of sheet flow is the width W of the deck area minus the design spread T, or (W–T). At the edge of the gutter flow, the design sheet flow depth d is obtained from Eq. (13.9), with a suggested default value of d 1.867 mm (0.0735 in). Using Manning’s equation and continuity and solving for i gives the hydroplaning design rainfall intensity in mm/h (in/h), as K d1.67 Sx i c 2 2 0.25 Cn (Sx G ) (W T)
(13.11)
where i rainfall intensity, mm/h (in/h); C dimensionless runoff coefficient; n Manning’s roughness coefficient; Sx cross slope, m/m (ft/ft); G grade, m/m (ft/ft); d thickness of water film, mm (in);Kc constant equal to 2289.4612 (64,904.4 in English units); W width of deck area, m (ft); and T design spread, m (ft). This hydroplaning design rainfall is used in Eq. (13.2) and is determined without respect to rainfall frequency. However, once established, a frequency is imputed from the associated time of concentration.
13.4.4 Rainfall Intensity by the Driver Vision-Impairment Method HEC–21 developed another alternative method for selection of rain intensity (Young et al., 1993). The method is also not dependent on rainfall frequency. The method assumes that if drivers cannot see, then they will slow down or stop, and this behavior governs highway function more than reaction to gutter flooding: Kc Sv i0.68 V
(13.12)
where Sv driver visibility, m; i rainfall intensity, mm; V vehicle speed, km/h; and Kc constant equal to 143,587.88 (338,721.26 in English units).
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HYDRAULIC DESIGN OF DRAINAGE FOR HIGHWAYS
13.14
Chapter Thirteen
This empirical relationship was developed based on the Texas A&M test. At 90 km/h (55 mph), the nonpassing minimum stopping sight distance is 150 m (500 ft) (this is the lower value of a range given by AASHTO). Substituting 90 km/h and 150 m into Eq. (13.12) gives a rainfall intensity of 125 mm/h. Note that cars in a travel corridor generate splash and spray that increase the density of water droplets. Therefore, a lower rainfall of 100 mm/h may be more realistic as a threshold value that will cause slight impairment because spray will increase the density of rain particles at roadway eye level. That is, design intensities i above 100 mm/h will probably obscure drivers’ visibility in heavy traffic and decrease sight distances to less than minimum stopping sight distance recommended by AASHTO. Therefore, 100 to 150 mm/h is a suggested threshold design rain fall intensity range for avoiding drivers’ vision impairment; this intensity range is determined without respect to frequency, but, depending on time of concentration, it imputes a frequency. Rainfall intensities below this range should not obscure a drivers view through a windshield with functioning windshield wipers. However, night driving in the rain is highly dependent on vision, and data supporting the predictive relationship were obtained in daylight.
13.5 GUTTER DESIGN A pavement gutter is adjacent to the roadway and conveys storm water. Gutter spread (top flow width) may include a portion or all of a travel lane. Gutter sections can be categorized as conventional or as the shallow swale type illustrated in Fig. 13.1. The most representative conventional gutters have a uniform cross slope (Fig. 13.1 A1) or a composite cross slope, where the gutter slope varies from the pavement cross slope (Fig. 13.1 A2). To compute triangular, shallow depth, gutter flow, the Manning equation is integrated for an increment of width across the section (Izzard, 1946). The resulting equation is K Q c Sx1.67GL0.5T2.67 n
(13.13)
FIGURE 13.1 Typical gutter sections (Source: Brown, Stein and Warner, 1996)
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HYDRAULIC DESIGN OF DRAINAGE FOR HIGHWAYS
Hydraulic Design of Drainage for Highways 13.15
where Kc empirical coefficient equal to 0.376 (0.56 in English units); n Manning’s coefficient, as given in Table 13.7; Q flow rate, m3/s (ft3/s); T width of flow (spread), m (ft); Sx cross slope, m/m ft/ft; and Sx cross slope, m/m (ft/ft). GL grade, m/m (ft/ft).Equation (13.13) neglects the resistance of the curb face. The design of composite gutter sections requires additional consideration of flow in the depressed segment of the gutter Qw. The depressed flow Qw. relates to the total flow as Qw Q Qs
(13.14)
where Qw = flow rate in the depressed section of the gutter, m3/s (ft3/s ); Q = gutter flow rate, m3/s (ft3/s ); and Qs = flow capacity of the gutter section above the depressed section, calculated using Eq. (13.13), m3/s (ft3/s). The cross-sectional area above the depressed section can be divided into Qs to get the gutter flow velocity Vg. Then Qw. is equal to Vg times the cross-sectional area of the depressed section. The use of depressed sections can put more depth of water above inlet entrances and increase their efficiency. When the pavement cross section is curved, gutter capacity varies with the configuration of the pavement. For this reason, discharge-spread- or discharge-depth-at-the-curb relationships developed for straight cross slopes are not applicable unless approximations are made. When curbs are not needed for traffic control, a small swale section with a V-shape or circular shape can be used to convey runoff from the pavement. Also, the control of pavement runoff on fills may be needed to protect the embankment from erosion. Small swale sections are sized to have sufficient capacity to convey the flow to a location suitable for interception. Shallow symmetric V-ditches, with side slopes of 10 percent or less, can be evaluated using Eq. (13.13) and doubling the result. Flow in shallow, circular gutter sections can be represented by the relationship Qn 0.488 d D Kc 2.67 0.5 D SL
TABLE 13.7
(13.15)
Manning’s n for Street and Pavement Gutters
Type of Gutter or Pavement Concrete gutter, troweled finish Asphalt pavement Smooth texture Rough texture Concrete gutter, asphalt pavement Smooth Rough Concrete pavement Float finish Broom finish For gutters with a small slope, where sediment may accumulate, increase above values of n by
Manning’s n 0.012 0.013 0.016 0.013 0.015 0.014 0.016 0.02
Source: Federal Highway Administration, 1977.
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HYDRAULIC DESIGN OF DRAINAGE FOR HIGHWAYS
13.16
Chapter Thirteen
where d = depth of flow in circular gutter, m (ft); D = diameter of circular gutter, m (ft); and Kc = empirical coefficient equal to 1.179 (0.972 in English units). The width of circular gutter section Tw is represented by the chord of the arc which can be computed using Eq. (13.16): Tw 2(r2 (2 d)2)0.5
(13.16)
where Tw = width of circular gutter section, m (ft), and r = radius of flow in circular gutter, m (ft). As gutter flow approaches the low point in a sag vertical curve, the flow can exceed the allowable design spread values as a result of the continually decreasing gutter slope. The spread in these areas should be checked to insure it remains within allowable limits. If the computed spread exceeds design values, additional flanking inlets should be provided to reduce the flow as it approaches the low point. Sag vertical curves and inlets are discussed further in a later section. The effects of spread on gutter capacity are greater than the effects of cross slope and grade, as would be expected because of the larger exponent of the spread term in Eq. (13.13). The magnitude of the effect is demonstrated by the gutter capacity of a 3-m (9.8-ft) spread being 18.8 times greater than with a 1-m (3.3-ft) spread, and three times greater than with a spread of 2 m (6.6 ft). The effects of cross slope also are relatively great, as illustrated by a comparison of gutter capacities with different cross slopes. At a cross slope of 4 percent, a gutter has 10 times the capacity of a gutter of 1 percent cross slope. A gutter at 4 percent cross slope has 3.2 times the capacity of a gutter at 2 percent cross slope. Little latitude is generally available to vary grade to increase the gutter capacity, but grade changes that change the gutter capacity are frequent. A change from GL 0.04 to 0.02 will reduce the gutter capacity to 71 percent of the capacity at GL 0.04. However, grade is not typically within the hydraulic designer’s set of responsibilities, whereas the design problem is: for a given grade, size the gutter and space the inlets to control spread. The flow time in gutters is an important component of the time of concentration for the contributing drainage area to an inlet. To find the gutter flow component of the time of concentration, Eq. (13.6) for estimating the average velocity in a reach of gutter is used.
13.6 ROADSIDE DITCH DESIGN Roadside and median channels are open-channel systems that collect and convey storm water from the pavement surface, roadside, and median areas. These channels may outlet to a storm drain piping system via a drop inlet, to a detention or retention basin or other storage component, or to an outfall channel. Roadside and median channels are normally trapezoidal in cross section and are lined with grass or other protective linings. The design or analysis of roadside and median channels follows the basic principles of open channel flow. A more complete coverage of open channel flow concepts can be found in other chapters in this handbook and in Chow (1959) and Richardson et al., (1990). Open channel flow is classified using the following characteristics: (1) steady (discharge passing a cross section is constant) or unsteady, (2) uniform (flow rate and depth remain constant along a reach) or varied, and (3) subcritical (Froude number less than 1) or supercritical–sometimes called tranquil or rapid. Most natural flow conditions are neither steady nor uniform. However, in most cases of small element drainage design, it can be assumed that the flow will vary gradually in Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
HYDRAULIC DESIGN OF DRAINAGE FOR HIGHWAYS
Hydraulic Design of Drainage for Highways 13.17
time and space and can be described as steady, uniform flow for short periods and distances. Gradually varied flows are nonuniform flows in which the depth and velocity change gradually enough in the direction of flow that vertical accelerations can be neglected. Subcritical flow is distinguished from supercritical flow by a dimensionless number called the Froude number (Fr), which represents the ratio of inertial forces to gravitational forces and is defined for rectangular channels by the following equation: V Fr (13.17) (gy)0.5 where V = mean velocity, m/s (ft/s); g = acceleration of gravity, 9.8 m/s2 (32.2 ft/s2)’ and y = flow depth, m (ft) Critical flow occurs when the Fr has a value of one (1.0). The flow depth at critical flow is referred to as critical depth. Subcritical flow occurs when the Fr is less than one (Fr 1). Subcritical flow is characterized by slower velocities, deeper depths, and mild slopes, whereas supercritical flow is represented by faster velocities, shallower depths, and steeper slopes. Supercritical flow occurs when the Fr is greater than one (Fr 1). Supercritical flows are apt to be erosive and may require energy dissipation measures to slow them down to manageable velocities. Most small-element open-channel flows are subcritical. However, supercritical flows are not uncommon for smooth-lined ditches on steep grades; if such elements transition to flat grades or channels with higher friction, a hydraulic jump may result. It is important to evaluate the Fr in open channel flows to determine how close a particular flow is to the critical condition. In other words, a hydraulic jump occurs as an abrupt transition from supercritical to subcritical flow. Significant changes in depth and velocity occur in the jump and energy is dissipated. The potential for a hydraulic jump to occur should be considered in all cases where the Fr is close to one (1.0) or where the slope of the channel bottom changes abruptly from steep to mild. The characteristics and analysis of hydraulic jumps are covered in detail in Chow (1959) and in HEC–14 (Federal Highway Administration, 1983).
13.6.1 Steady Uniform Flow Design Small element design is based on the depth of steady uniform flow. This flow is called the normal flow and is computed with Manning’s equation. The general form of Manning’s equation is: Kn A R0.67 So0.5 Q n
(13.18)
where Kn = empirical conversion equal to 1.0 (1.486 in English units); Q = discharge rate, m3/s (ft3/s); A = cross sectional flow area, m2 (ft2); R = hydraulic radius, m (ft); = A/P, m (ft); P = wetted perimeter, m (ft); So = energy grade line slope, m/m (ft/ft); and n = Manning’s roughness coefficient. The selection of an appropriate Manning’s n value for design purposes is often based on observation and experience. Manning’s n values also vary with normal flow depth since n is a friction factor and friction is a boundary effect that is averaged into the water column. Table 13.8 provides a tabulation of Manning’s n values for various lining materials for open channels Manning’s roughness coefficient for vegetative and other linings vary significantly, depending on the amount of submergence. The classification of vegetal covers by degree of retardance is provided in Table 13.9; Table 13.10 provides a list of Manning’s n relationships for five classes of vegetation defined by their degree of retardance. Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
HYDRAULIC DESIGN OF DRAINAGE FOR HIGHWAYS
13.18
13.6.2
Chapter Thirteen
Water Surface Superelevation in Bends
Flow around a bend in an open channel induces centrifugal forces because of the change in the direction of flow (Chow, 1959). This results in a superelevation of the water surface at the outside of bends and can cause the flow to splash over the side of the channel if adequate freeboard is not provided. This superelevation can be estimated by the following equation: V2T ∆d (13.19) gRc TABLE 13.8
Manning’s Roughness Coefficients for Open Channels n Values for Given Depth Ranges
Lining Category
Lining Type
Rigid
Concrete Grouted riprap Stone masonry Soil cement Asphalt Bare soil Rock cut Woven paper net Jute net Fiberglass roving Straw with net Curled wood mat Synthetic mat 25 mm (1 in), D50 50 mm (2 in), D50 150 mm (6 in), D50 300 mm (12 in), D50
Unlined Temporary*
Gravel riprap Rock riprap
0015 m (0.05 ft)
0.150.60 m (0.52.0 ft)
0.60 m ( 2.0 ft)
0.015 0.040 0.042 0.025 0.018 0.023 0.045 0.016 0.028 0.028 0.065 0.066 0.036
0.013 0.030 0.032 0.022 0.016 0.020 0.035 0.015 0.022 0.021 0.033 0.035 0.025
0.013 0.028 0.030 0.020 0.016 0.020 0.025 0.015 0.019 0.019 0.025 0.028 0.021
0.044 0.066 0.104 --–
0.033 0.041 0.069 0.078
0.030 0.034 0.035 0.040
Source: Chen and Cotton, (1986). Note: Values listed are representative values for the respective depth ranges. Manning’s roughness coefficients,(n) vary with the flow depth. *Some “temporary” linings become permanent when buried.
TABLE 13.9
Classification of Vegetal Covers According to Degree of Retardance.
Retardance Class Cover
Condition
A
Weeping lovegrass Yellow bluestem schaemum
Excellent stand, tall, average 0.76 m (2.5 ft) Excellent stand, tall, average 0.91 (3.0 ft)
B
Kudzu Bermuda grass Native grass mixture (little bluestem, bluestem, blue gamma, and other long and short midwest grasses)
Very dense growth, uncut Good stand, tall, average 0.30 m (1.0 ft) Good stand, unmowed
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HYDRAULIC DESIGN OF DRAINAGE FOR HIGHWAYS
Hydraulic Design of Drainage for Highways 13.19 TABLE 13.9
(Continued)
Retardance Class Cover Weeping lovegrass Lespedeza sericea Alfalfa Weeping lovegrass Kudzu Blue gamma C
D
E
Condition Good stand, tall, average 0.61 m (2.0 ft) Good stand, not woody, tall, average 0.48 m (1.6 ft) Good stand, uncut, average 0.28 m (0.91 ft) Good stand, unmowed, average 0.33 m (1.1 ft) Dense growth, uncut Good stand, uncut, average 0.33 m (1.1 ft)
Crabgrass Bermuda grass Common lespedeza Grass-legume mixture: summer (orchard grass, redtop Italian ryegrass, and common lespedeza)
Fair stand, uncut, avg. 0.25–1.20 m (0.9–4.0 ft) Good stand, mowed, average 0.15 m (0.5 ft) Good stand, uncut, average 0.28 m (0.91 ft) Good stand, uncut average 0.15–0.20 m (0.5–1.5 ft)
Centipedegrass Kentucky bluegrass Bermuda grass Common lespedeza Buffalo grass Grass-legume mixture: fall, spring (orchard grass, redtop Italian ryegrass, and common lespedeza) Lespedeza sericea
Very Dense cover, average 0.15 m (0.5 ft) Good stand, headed, avg. 0.15 to 0.30 m (0.5 to 1.0 ft) Good stand, cut to .06 m (0.2 ft) Excellent stand, uncut, average Good stand, uncut, average 0.08 tp 0.15 m (0.3 to 0.5 ft) Good stand, uncut, 0.10 to 0.13 (0.3 to 0.4 ft)
Bermuda grass Bermuda grass
After cutting to 0.05 m (0.2 ft) height. Good stand before cutting Good stand, cut to average 0.04 m (0.1 ft) Burned stubble
Source: Reproduced from HEC-15 (Chen and Cotton, 1986) Note: Covers classified have been tested in experimental channels. Covers were green and generally uniform.
TABLE 13.10
Manning’s n Relationships for Vegetal Degree of Retardance
Retardance Class
Manning’s n Equation*
A
1.22 R1/6 / [30.2 19.97 log (R1.4 So0.4)]
B
1.22 R1/6 / [37.4 19.97 log (R1.4 So0.4)]
C
1.22 R1/6 / [44.6 19.97 log (R1.4 So0.4)]
D
1.22 R1/6 / [49.0 19.97 log (R1.4 So0.4)]
E
1.22 R1/6 / [52.1 19.97 log (R1.4 So0.4)]
* Equations are valid for flows less than 1.42 m3/s (50 ft3/s). Nomograph solutions for these equations are contained in HEC-15 (Chen and Cotton, 1986).
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HYDRAULIC DESIGN OF DRAINAGE FOR HIGHWAYS
13.20
Chapter Thirteen
where ∆d = difference in water surface elevation between the inner and outer banks of the channel in the bend, m (ft); V = average velocity, m/s (ft/s); T = surface width of the channel, m (ft); g = gravitational acceleration, 9.8 m/s2 (32.2 ft/s2); and Rc = radius to the center line of the channel, m (ft). Equation (13.19) is valid for subcritical flow conditions. The elevation of the water surface at the outer channel bank will be ∆d/2 higher than the center line water surface elevation (the average water surface elevation immediately before the bend) and the elevation of the water surface at the inner channel bank will be ∆d/2 lower than the elevation of the water surface at the center line. Flow around a channel bend imposes higher shear stress on the channel’s bottom and banks and is discussed in more detail in the next section. The increased stress requires additional design considerations within and downstream of the bend.
13.6.3 Shear Stresses in Open Channels Stable channel design concepts presented in HEC-15 (Chen and Cotton, 1986) provide a means of evaluating and defining channel configurations that will perform within acceptable limits of stability. Stability is achieved when the material forming the channel boundary effectively resists the erosive forces of the flow. Principles of rigid boundary hydraulics can be applied to evaluate this type of system. A pragmatic approach is to limit channel velocities to the range of 0.3 to 0.6 m/s (1 to 2 ft/s) to avoid erosion. Better estimates are possible by making tractive force calculations that consider actual physical processes occurring at the channel boundary. In uniform flow, the shear stress exerted on the bed is equal to the effective component of the gravitational force acting on the body of water parallel to the channel bottom. The average shear stress is equal to τ γxS
(13.20)
where τ average shear stress, Pa (lb/ft2); γ unit weight of water, 9810 N/m3 (62.4 lb/ft2) at 15.6 oC (60 oF); x height of water column, m (ft); and S average bed slope (or energy slope in varied flow conditions) m/m (ft/ft).The maximum shear stress on the channel bed is computed as follows (Chow, 1959): τd γdS
(13.21)
where τd maximum shear stress, Pa (lb/ft2) and d maximum depth of flow, m (ft). Shear stress in channels is not distributed uniformly along the wetted perimeter of a channel. A typical distribution of shear stress in a trapezoidal channel tends toward zero at the corners with a maximum on the bed of the channel at its center line, and the maximum for the side slopes occurs around the lower third of the slope, as illustrated in Fig. 13.2. Flow around bends creates higher shear stresses on the channel sides and bottom compared to straight reaches. The maximum shear stress in a bend is a function of the ratio of the channel’s curvature to its bottom width. This ratio increases as the bend becomes sharper and the maximum shear stress in the bend increases. The bend shear stress can be computed using the following relationship: τb Kbτd
(13.22)
where τb bend shear stress, Pa (lb/ft2); Kb function of Rc/B as shown in Figure 13.3; Rc radius to the centerline of the channel, m (ft); B bottom width of channel, m (ft); Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
HYDRAULIC DESIGN OF DRAINAGE FOR HIGHWAYS
Hydraulic Design of Drainage for Highways 13.21
FIGURE 13.2 Distribution of shear stress. (Source: Brown, Stein, Warner, 1996)
FIGURE 13.3 Kb Factor for maximun shear stress on chanel bends. (Source: Brown, Stein, Warner, 1996)
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HYDRAULIC DESIGN OF DRAINAGE FOR HIGHWAYS
13.22
Chapter Thirteen
τd maximum channel shear stress, Pa (lb/ft2) calculated with Eq. (13.21). The increased shear stress produced by the bend persists downstream of the bend a distance Lp, which is computed using the following relationship: K R7/6 Lp c nb
(13.23)
where Lp length of protection (length of increased shear stress due to the bend) from the point of tangency downstream, m (ft); nb Manning’s roughness in the channel bend; R hydraulic radius, m (ft); and Kc coefficient equal to 0.897 (0.736 in English units). 13.6.4 Parameters for Stable Channel Design Parameters required for the evaluation and design of stable roadside and median channels include discharge frequency, channel geometry, safety, channel slope, vegetation type, freeboard, and shear stress. This section provides criteria relative to the selection or computation of these design elements. The applied shear stress is calculated and compared with the permissible alternative shear stresses in the next section. Discharge frequency. Roadside and median drainage channels are designed to accommodate the hydrologic design flow. However, when designing temporary channel linings, a 2-yr return period is appropriate. Channel geometry. Highway drainage channels are typically trapezoidal in shape. Several typical shapes with equations for determining channel properties are illustrated in Fig. 13.4. The depth, bottom width, and top width of the channel must be selected to provide the necessary flow area. The side slopes for triangular and trapezoidal channels should not exceed the angle of repose of the soil, the lining material, or both and should generally be 3:1 or flatter (Chen and Cotton, 1986) without consideration of the stability of the side slopes. Safety. Design of roadside and median channels should be integrated with the highway’s geometric and pavement-design to insure proper consideration of safety and pavement drainage needs. In areas where traffic safety may be a concern, the side slopes of the channel should be 4:1 or flatter. Channel slope. Channel bottom slopes are generally dictated by the road profile. However, if channel stability conditions warrant, it may be feasible to adjust the channel gradient slightly to achieve a more stable condition. Channel gradients greater than 2 percent may require the use of flexible linings to maintain stability. Most flexible lining materials are suitable for protecting channel gradients of up to 10 percent, with the exception of some grasses. Linings, such as riprap and wire-enclosed riprap, are more suitable for protecting steep channels with gradients in excess of 10 percent. Rigid linings, such as concrete paving, are stable to erosion but can be susceptible to failure from other causes, such as overtopping, freeze-thaw cycles, swelling, and excessive soil pore water pressure. Freeboard. The freeboard of a channel is the vertical distance from the water’s surface to the top of the channel. The importance of this factor depends on the consequence of overflow of the channel bank. At a minimum, the freeboard should be sufficient to prevent waves, changes in superelevation, or fluctuations in water surface that result in the sides overflowing. In a permanent roadside or median channel, about 150 mm (0.5 ft) of freeboard is generally considered to be adequate. For temporary channels, no freeboard is nec-
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HYDRAULIC DESIGN OF DRAINAGE FOR HIGHWAYS
Hydraulic Design of Drainage for Highways 13.23
FIGURE 13.4 Channel geometries. (Source: Brown, Stein, Warner, 1996)
essary. However, Fr’s greater than one warrant freeboard height equal to the flow depth to compensate for the large variations in flow caused by waves, splashing, and surging. Transitions from steep to mild also should have increased freeboard. Permissible shear stress. The exerted shear stress is calculated with Eq. (13.21) or (13.22) using the design parameters. Then, the permissible shear stress is used to find an acceptable design alternative. The permissible or critical shear stress in a channel defines the force required to initiate movement of the channel bed or lining material. Table 13.11 presents permissible values of shear stress for manufactured, vegetative, and riprap channel lining. The permissible shear stress values for noncohesive soils is a function of the mean diameter of the channel material (Fig. 13.5). For larger sized stones not shown in Fig. 13.5 and rock riprap, the permissible shear stress is given by the following equation: τp KpD50
(13.24)
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HYDRAULIC DESIGN OF DRAINAGE FOR HIGHWAYS
13.24
Chapter Thirteen
FIGURE 13.5 Permissible shear for non-cohesive soils. (Source: Brown, Stein, Warner, 1996, English units preserved from source document)
TABLE 13.11
Permissible Shear Stresses for Lining Materials. Permissible Unit Shear Stress
Lining Category
Lining Type
Temporary*
Woven paper net Jute net Fiberglass roving: Single Double Straw with net Curled wood mat Synthetic mat Class A Class B Class C Class D Class E 25 mm (1 in) 50 mm (2 in) 150 mm (6 in) 300 mm (12 in) Noncohesive Cohesive
Vegetative
Gravel riprap Rock riprap Bare soil
Pa
lb/ft2
7.2 21.6
0.15 0.45
28.7 40.7 69.5 74.3 85.7 177.2 100.6 47.9 228.7 16.8 15.7 31.4 95.7 191.5 See Fig. 13.5 See Fig. 13.6
0.60 0.85 1.45 1.55 2.00 3.70 2.10 1.00 0.60 0.35 0.33 0.67 2.00 4.00
*Some “temporary” linings become permanent when buried. Source: From HEC-15 (Chen and Cotton, 1986).
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HYDRAULIC DESIGN OF DRAINAGE FOR HIGHWAYS
Hydraulic Design of Drainage for Highways 13.25
FIGURE 13.6 Permissible shear stress for cohesive soils. (Source: Brown, Stein, Warner, 1996, English units preserved from source document)
where τp permissible shear stress, Pa (lb/ft2); D50 mean riprap size, m (ft); and Kp empirical constant equal to 628 (4.0). For cohesive materials, the plasticity index provides a good guide for determining the permissible shear stress illustrated in Fig. 13.6. For trapezoidal channels protected with gravel or riprap having side slopes steeper than 3:1, stability of the side slopes must be considered. This analysis is performed by comparing the ratio of the tractive force between side slopes and channel bottom with the ratio of shear stresses exerted on the channel sides and bottom. The ratio of shear stresses on the sides and bottom of a trapezoidal channel K1 is given in Fig.13.7, and the tractive force ratio K2 is given in Fig. 13.8. The angle of repose for different rock shapes and sizes is provided in Fig. 13.9. The required rock size for the side slopes is found using the following equation:
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HYDRAULIC DESIGN OF DRAINAGE FOR HIGHWAYS
13.26
Chapter Thirteen
FIGURE 13.7 Channel side stress to botton shear stress ratio, K1. (Source: Brown, Stein, Warner, 1996)
FIGURE 13.8 Tractive force ratio, K2. (Source: Brown, Stein, Warner, 1996)
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HYDRAULIC DESIGN OF DRAINAGE FOR HIGHWAYS
Hydraulic Design of Drainage for Highways 13.27
FIGURE 13.9 Angle of repose of riprap in terms of mean size and shape of stone. (Source, Brown, Stein, Warner, 1996, English units preserved from source document)
K (D50)sides 1 (D50)bottom K2
(13.25)
where D50 the mean riprap size, m (ft.); K1 ratio of shear stresses on the sides and bottom of a trapezoidal channel as shown in Fig. 13.7; and K2 ratio of tractive force on the sides and bottom of a trapezoidal channel shown in Fig. 13.8.
13.7 DRAINAGE INLET DESIGN As water accumulates and flows in the gutters and ditches, spread increases. At the point where spread reaches the design value, an inlet is provided to capture some or all of the
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HYDRAULIC DESIGN OF DRAINAGE FOR HIGHWAYS
13.28
Chapter Thirteen
flow to control the spread. Letting some gutter flow pass the gutter (“flow by”) for subsequent capture can be efficient because it is uneconomical to make the inlets as wide as the design spread. The hydraulic capacity of a storm drain inlet depends on its geometry as well as on the characteristics of the gutter flow. Inlet capacity governs both the rate of water removal from the gutter and the amount of water that can enter the storm drainage system. Inadequate inlet capacity or poor inlet location may cause flooding on the roadway, resulting in a hazard to the traveling public.
13.7.1 Inlets Storm drain inlets are used to collect runoff and discharge it to an underground storm drainage system. Inlets are typically located in gutter sections, paved medians, and roadside and median ditches. Inlets used for draining of highway surfaces can be divided into the following four classes: (1) Grate inlets (grate-covered openings); (2) Curb-opening inlets (vertical curb openings); (3) Combination inlets (grate plus curb opening), and (4) Slotted inlets (under surface pipe with slotted crown). Grate inlets, as a class, perform satisfactorily over a wide range of gutter grades but lose capacity with increase in grade. A disadvantage is clogging. In addition, where bicycle traffic occurs, grates should be bicycle safe (narrow opening widths or transverse grating with the latter being hydraulically inefficient). Curb-opening inlets are most effective on flatter slopes (less than 3 percent grade), in sags, and with flows which typically carry significant amounts of floating debris. The interception capacity of curb-opening inlets significantly decreases as the gutter grade steepens. Combination inlets provide a high-capacity inlet that offers the advantages of both grate and curb-opening inlets. The curb opening precedes the grate in a “sweeper” configuration, and it acts as a trash interceptor. Used in a sag configuration, the sweeper inlet can have a curb opening on both sides of the grate. Slotted inlets can be used in areas where it is desirable to intercept sheet flow before it crosses onto a section of roadway. Their principal advantage is their ability to intercept sheet flow over a wide section of roadway. Slotted inlets are highly susceptible to clogging. Inlet interception capacity Qi is the flow intercepted by an inlet. The efficiency (E) of an inlet is the fraction of total flow that the inlet will intercept. The efficiency changes with changes in cross slope, longitudinal slope, total gutter flow, and, to a lesser extent, pavement roughness. Efficiency is defined by Q E i Q
(13.26)
where E inlet efficiency; Q total gutter flow, m3/s (ft3/s); and Qi intercepted flow, m3/s (ft3/s). Flow that is not intercepted by an inlet is termed carryover, bypass, or flow by and is equal to Qb Q Qi
(13.27)
when Qb bypass flow, m3/s (ft3/s). The interception capacity of all inlet configurations increases and the efficiency decreases as flow rates increase. The depth of water next to the curb is the major factor in the interception capacity of both grate inlets and curb-opening inlets. The interception
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HYDRAULIC DESIGN OF DRAINAGE FOR HIGHWAYS
Hydraulic Design of Drainage for Highways 13.29
capacity of a grate inlet depends on the amount of water flowing over the gate, the size and configuration of the grate, and the velocity of flow in the gutter. The interception capacity of a curb-opening inlet is largely dependent on flow depth at the curb and curbopening length. Flow depth at the curb and, consequently, the interception capacity and efficiency of the curb-opening inlet is increased by the use of a local gutter depression at the curb-opening or a continuously depressed gutter to increase the proportion of the head and flow adjacent to the curb. For curb inlets, top slab supports placed flush with the curb line can substantially reduce the interception capacity of curb openings and therefore should be avoided or be recessed and rounded in shape. Slotted inlets function in essentially the same manner as curb opening inlets: i.e., as weirs with flow entering from the side. Interception capacity depends on the depth of flow and the length of the inlet. Efficiency depends on the depth of flow, the inlet length of the inlet, and the total flow in the gutter. A combination inlet consisting of a curb-opening inlet placed upstream of a grate inlet has a capacity equal to that of the curb-opening length upstream of the grate plus that of the grate, taking into account the reduced spread and depth of flow over the grate because of the interception by the curb opening. This inlet configuration has the added advantage of intercepting debris that might otherwise clog the grate and deflect water away from the inlet. Grate inlets in sag vertical curves operate as weirs for shallow ponding depths and as orifices at greater depths. Between weir and orifice flow depths, a transition from weir to orifice flow occurs. The capacity at a given depth can be affected severely if debris collects on the grate. Curb-opening inlets operate as weirs in sag vertical curve locations up to a ponding depth equal to the opening height. At depths above 1.4 times the opening height, the inlet operates as an orifice and the transition between weir and orifice flow occurs between these depths. The curb-opening height and length and the water depth at the curb affect inlet capacity. The effective water depth at the curb can be increased by using a continuously depressed gutter, a locally depressed curb opening, or an increased cross slope, thus decreasing the width of spread at the inlet. Slotted drains are not recommended in sag locations because they are susceptible to clogging by debris.
13.7.2 Grate Inlet Design Grates are effective highway pavement-drainage inlets when clogging by debris is not a problem. Typical grate configurations are shown in Fig. 13.10. If clogging may be a problem, see Table 13.12, where grates are ranked for their susceptibility to clogging based on laboratory tests using simulated leaves. When the velocity approaching the grate is less than the “splash-over” velocity, the grate will intercept essentially all the frontal flow. Conversely, when the gutter flow velocity exceeds the “splash-over” velocity for the grate, only part of the flow will be intercepted. A part of the flow along the side of the grate will be intercepted, depending on the cross slope of the pavement, the length of the grate, and the velocity of flow. The ratio of frontal flow to total gutter flow Eo for a uniform cross slope is expressed by Q W 2.67 Eo w 1 1 Q T
(13.28)
where Q total gutter flow, m3/s (ft3/s); Qw flow in width, W, m3/s (ft3/s); W width of depressed gutter or grade, m (ft); and T total spread of water, m (ft). The ratio of side flow Qs to total gutter flow is
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HYDRAULIC DESIGN OF DRAINAGE FOR HIGHWAYS
13.30
Chapter Thirteen
FIGURE 13.10 Typical grates. Source: (HEC-12: Johnson and Chang, 1984, English units preserved from source document)
TABLE 13.12
Average Debris Handling Efficiencies of Grates Tested Longitudinal Slope
Rank
Grate
0.005
0.040
1
Curved vane
46
61
2
30º85 tilt bar
44
55
3
45º85 tilt bar
43
48
4
P50
32
32
5
P50 100
18
28
6
45º60 tilt bar
16
23
7
Reticuline
12
16
8
P30
9
20
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HYDRAULIC DESIGN OF DRAINAGE FOR HIGHWAYS
Hydraulic Design of Drainage for Highways 13.31
Q Q s 1 w 1 Eo Q Q
(13.29)
The ratio of frontal flow intercepted to total frontal flow Rf is expressed by Rf 1 Kc (V Vc)
(13.30)
where Kc constant equal to 0.295 (0.09 in English units); V velocity of flow in the gutter, m/s; and Vo gutter velocity where splash-over first occurs, m/s. (Note: Rf cannot exceed 1.0.) The splash-over velocity and Rf factor are solved by the chart in Fig. 13.11 as a function of the type and length of the grate and average gutter velocity approaching the inlet. The ratio of side flow intercepted to total side flow Rs, or side flow interception efficiency, is expressed by 1 Rs K V1.8 1 c SxL2/3 where Kc = a constant equal to 0.0828 (0.15 in English units). The overall efficiency E of a grate is expressed by the following equation: E Rf Eo Rs (1 Eo)
(13.31)
(13.32)
FIGURE 13.11 Grate inlet frontal flow interception efficiency. Source: HEC-12 (Johnson and Chang, 1984, English units preserved from source document)
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HYDRAULIC DESIGN OF DRAINAGE FOR HIGHWAYS
13.32
Chapter Thirteen
The interception capacity of a grade inlet on a grade is equal to the efficiency of the grate multiplied by the total gutter flow: Qi EQ Q[Rf Eo Rs (1 Eo)]
(13.33)
13.7.3 Curb-Opening Inlet Design Curb opening heights vary in dimension, however, a typical maximum height is approximately 100 to 150 mm (4 to 6 in). The length of the curb-opening inlet required for total interception of gutter flow on a pavement section with a uniform cross slope is expressed by the following equation: 1 0.6 (13.34) LT Kc Q0.42 SL0.3 nSx where Kc constant equal to 0.817 (0.6 in English units); LT curb opening length required to intercept 100 percent of the gutter flow, m (ft); SL grade; and Q gutter flow, m3/s (ft3/s). The efficiency of curb-opening inlets shorter than the length required for total interception is expressed by L 1.8 E 1 1 L
(13.35)
where L curb-opening length, m (ft); The length of inlet required for total interception by depressed curb-opening inlets or curb openings in depressed gutter sections can be found by the use of an equivalent cross slope Se in Equation (13.34) in place of Sx. Se can be computed as Se Sx S’wEo
(13.36)
where S’w cross slope of the depressed gutter and Eo is the same ratio used to compute the frontal-flow interception of a grate inlet (Eq. (13.28)). The length of curb opening required for total interception can be reduced significantly by increasing the cross slope or the equivalent cross slope. Increasing the equivalent cross slope can be accomplished by using of a continuously depressed gutter section or locally depressed gutter sections. To compute efficiency, E for curb inlets shorter than the length needed for 100 percent capture of flow, Eq. (13.35) is applicable with either straight cross slopes or composite cross slopes.
13.7.4 Slotted Inlet Design Placed at right angles to the flow, the slot acts as a short grate. Assuming a splash-over velocity of 0.3 m/s (1 ft/s) and no side flow, the grate-inlet efficiency equations can be used. Placed parallel to the flow, in the gutter notch, the flow interception by slotted inlets and curb-opening inlets is similar. Analysis of data from the Federal Highway Administration’s tests of slotted inlets with slot widths 45 mm (1.75 in) indicates that the length of slotted inlets required for total interception can be computed with the curb-
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HYDRAULIC DESIGN OF DRAINAGE FOR HIGHWAYS
Hydraulic Design of Drainage for Highways 13.33
opening equations. It should be noted that it is less expensive to add length to a slotted inlet to increase its interception capacity than it is to add length to a curb-opening inlet. Wide experience with the debris-handling capabilities of slotted inlets is not available. Deposition of debris in the pipe is a commonly encountered problem. However, the configuration of slotted inlets make them accessible for cleaning with a high-pressure water jet.
13.7.5 Combination Inlet Design Because the interception capacity of a combination inlet consisting of a curb opening and a grate placed side by side is no greater than that of the grate alone, capacity is computed by neglecting the curb opening. Typical designs place the curb opening upstream of the grate. The curb opening intercepts debris that might clog the grate and is called a “sweeper” inlet. A sweeper combination inlet has an interception capacity equal to the sum of the curb opening upstream of the grate plus the grates capacity, except that the frontal flow, and thus the interception capacity of the grate, is reduced by interception by the curb opening.
13.7.6 Design Adjustments for Sag Locations Inlets in sag locations operate as weirs under low head conditions and as orifices at greater depths and are more susceptible to clogging. Orifice flow begins at depths dependent on the grate size, the height of the curb opening, or the slot width of the inlet. At depths between those at which weir flow definitely prevails and those at which orifice flow prevails, flow is in a transition stage. Grate inlets alone are not recommended for use in sag locations because of the tendencies of grates to become clogged. Combination inlets or curb-opening inlets are recommended for use in these locations. The capacity of grate inlets operating as weirs is Qi CwPd1.5
(13.37)
where P perimeter of the grate in m (ft) disregarding the side against the curb, Cw weir coefficient 1.66 (3.0 English units) and d flow depth, m (ft). The capacity of a grate inlet operating as an orifice is Qi CoAg (2gd)0.5
(13.38)
where Co orifice coefficient 0.67; Ag clear opening area of the grate, m2 (ft2); g 9.90 m/s2 (32.16 ft/s2); and d flow depth, m (ft). Use of Eq. (13.38) requires the clear area of opening of the grate. Tests of three grates for the Federal Highway Administration (Burgi, 1978) showed that for flat bar gates, the clear opening is equal to the total area of the grate less the area occupied by longitudinal and lateral bars. However, curved vane grate performed about 10 percent better than one would expect by calculating a net opening equal to the total area less the area of the bars projected on a horizontal plane. Tilt bar and curved vane grates are not recommended for sump locations. The capacity of a depressed curb-opening inlet operating as a weir (curb water depths less than opening height) is Qi Cw (L 1.8W)d1.5
(13.39)
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HYDRAULIC DESIGN OF DRAINAGE FOR HIGHWAYS
13.34
Chapter Thirteen
where Cw weir coefficient 1.25 (2.3 in English units); L length of curb opening, m (ft); W lateral width of depression, m (ft); and d depth at curb measured from the normal cross slope, m (ft): i.e., d TSx. For curb-opening inlets with a continuously depressed gutter, it is reasonable to expect that the effective weir length would be as great as that for an inlet in a local depression. The weir equation for curb-opening inlets without any depression simplifies to Qi Cw L d1.5
(13.40)
Without depression of the gutter section, the weir coefficient Cw becomes 1.60 (3.0, English system). At curb-opening lengths greater than 3.6 m (12 ft), Eq. (13.40) for nondepressed inlets produces captured flows that exceed the values for depressed inlets computed using Eq. (13.39). Since depressed inlets will perform at least as well as nondepressed inlets of the same length, Eq. (13.40) should be used for all curb-opening inlets longer than 3.6 m (12 ft). Curb-opening inlets operate as orifices at curb depths greater than approximately 1.4 times the opening height. The capacity of curb-opening inlets operating as orifices is Qi C0 hL(2gd)0.5
(13.41)
or h Qi C0Ag 2g di 2
0.5
(13.42)
where Co orifice coefficient (0.67); do effective head on the center of the orifice throat, m (ft); L length of orifice opening, m (ft); Ag clear area of opening, m2 (ft2); di depth of lip of curb opening, m (ft); and h height of curb-opening orifice, m (ft). The above variables are detailed in Fig. 13.12 as a function of typical variations in geometric configuration. For curb-opening inlets with other than vertical faces (see Fig. 13.12), Eq. (13.41) can be used with h orifice throat width, m (ft) and do effective head on the center of the orifice throat, m (ft). Slotted inlets in sag locations perform as weirs to depths of above 0.06 m (0.2 ft), depending on slot width. At depths greater than approximately 0.12 m (0.4 ft), they perform as orifices. Between these depths, flow is in a transition stage. The capacity of a slotted inlet operating as a weir can be computed by Qi CwL(d1.5)
(13.43)
where Cw weir coefficient 1.4 (2.2248 for English units); L length of slot, m (ft); and d depth at curb measured from the normal cross slope, m (ft). The capacity of a slotted inlet operating as an orifice can be computed by Qi 0.8 L W(2gd)0.5
(13.44)
where W width of slot, m (ft); L length of slot, m (ft); d depth of water at slot for d 0.12 m (0.4 ft), m (ft); and g 9.81 m/s2 (32.16 ft/s2). Combination inlets consisting of a grate and a curb opening are considered advisable for use in sags. Equal-length inlets refer to a grate inlet placed alongside a curb-opening
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HYDRAULIC DESIGN OF DRAINAGE FOR HIGHWAYS
Hydraulic Design of Drainage for Highways 13.35
FIGURE 13.12 Curb-opening inlets. Source: (Brown, Stein, and Warner, 1996)
inlet, both of which have the same length. A sweeper inlet refers to a grate inlet placed at the downstream end of a curb-opening inlet; the curb-opening inlet is longer than the grate inlet and intercepts the flow before the flow reaches the grate. The sweeper is more efficient than the equal-length combination inlet and the curb opening and has the ability to intercept any debris which may clog the grate inlet. The capture capacity of the equallength combination inlet in weir flow is essentially equal to that of a grate alone. In orifice flow, the capacity of the equal-length combination inlet is equal to the capacity of the grate plus the capacity of the curb opening. A typical assumption is to size sweeper inlets at the sag, assuming that the grate in between is 100 percent clogged.
13.7.7 Inlet Locations The location of inlets is determined by geometric controls that require inlets at specific locations, the use and location of flanking inlets in sag vertical curves, and the criterion of spread on the pavement at the design rainfall intensity. To design the location of the inlets adequately for a given project, the following information is needed: • A layout or plan sheet suitable for outlining drainage areas. • Road profiles. • Typical cross sections.
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HYDRAULIC DESIGN OF DRAINAGE FOR HIGHWAYS
13.36
Chapter Thirteen
• Grading cross sections. • Superelevation diagrams. • Contour maps. • Intensity-duration-frequency rainfall curves. Because the typical inlet is located at a gutter location having sufficient drainage area to generate the design spread, full or partial removal of spread is accomplished. Bypass of some of the gutter flow, although leading to more frequent spacing, can result in much smaller inlets at considerable cost savings. There are a number of locations where inlets may be necessary with little regard to contributing drainage area. These locations should be marked on the plans before any computations regarding discharge, water spread, inlet capacity, or flow bypass. Examples of such locations follow: • at all sag points in the gutter grade • immediately upstream of median breaks, entrance/exit ramp gores, crosswalks, and street intersections: i.e., at any location where water could flow onto the travelway • immediately upgrade of bridges (to prevent pavement drainage from flowing onto bridge decks) • immediately downstream of bridges (to intercept bridge deck drainage) • immediately upgrade of cross-slope reversals • at the end of channels in cut sections • on side streets immediately upgrade from intersections • behind curbs, shoulders, or sidewalks to drain low areas In addition to the areas identified above, runoff from areas draining toward the highway pavement should be intercepted by roadside channels or inlets before it reaches the roadway. This applies to drainage from cut slopes, side streets, and other areas alongside the pavement.
13.8 BRIDGE-DECK DRAINAGE DESIGN From a hydraulic standpoint, the ideal bridge has no sag point and bridge deck drainage is not needed (Young et al., 1993). Such a bridge has off-bridge inlets at the upper end to remove gutter flow before it reaches to the bridge and at the lower end to collect bridge deck drainage. If the bridge is too long, spread on the bridge deck may necessitate inlets. On a long bridge, typical inlets are small and are sometimes called scuppers: a 3-m (10ft) shoulder is desirable for maintenance of scuppers. The off-bridge inlets are typically curb inlets and are designed using methods presented in Sec. 13.7. The on-bridge inlets, if rectangular, also can be designed using equations in Sec. 13.7; the dimensions typically are less than 1 m to a side. If the inlets are circular, their diameter is typically low. Figure 13.13 gives the capture efficiency for circular inlets. The dimensions of inlets are relatively constrained for placement and integration into bridge decks. Many decks are pre- or post-tensioned structural slabs, and inlets are detailed that may interfere with structural continuity. Thus, the surface grate and the recessed collection chamber must be considered.
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HYDRAULIC DESIGN OF DRAINAGE FOR HIGHWAYS
FIGURE 13.13 Efficiency curves for circular scuppers. (Source: Young, Walker and Chang, 1993, English units preserved from source document)
Hydraulic Design of Drainage for Highways 13.37
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HYDRAULIC DESIGN OF DRAINAGE FOR HIGHWAYS
13.38
Chapter Thirteen
The constraints lead to small inlets and are associated with reinforcing bar schedules or post-tensioned cable spacings. Typical dimensions need to be less than 0.5 m (18 in); these details need to be structurally designed to transfer loads into the slab. Large inlet spans cause a need for special designs and reinforcing details. The hydraulic problem is that spreads of 2.5 to 3.0 m (8 to 10 ft) of water in gutters are not reduced effectively with small inlets having capture efficiencies of 5 to 10 percent. Numerous closely spaced small inlets are necessary to control spread. Ideally, a bridge should have 3-m (10-ft) shoulders and the inlet boxes should be placed at the outside edge of the shoulder. In this position, the maintenance crew can park on the shoulder and work on the side away from traffic in reasonable safety. These drains have a good chance of being maintained regularly. Unfortunately, accidents may happen when lanes must be blocked to service the drains or when the maintenance crew must work on the edge of the stream of traffic. This can result in poorly placed inlets receiving inadequate maintenance. The larger the inlet, the fewer inlets to maintain. Hydraulically, larger inlets handle larger volumes of flow and are more apt to clean themselves. The larger the inlet, the easier it is to clean with a shovel. Inlets should be sized as large as possible: Practically speaking, 1 m (36 in) is probably an upper limit for inlets placed within deck slabs. Round vertical scuppers should not be less than 155 mm (6 in) in diameter, with a diameter of 200 m (8 in) preferred. Scuppers with a diameter of 100 mm (4-in) are not uncommon in practice. However, their limited hydraulic capacity, coupled with their tendency to plug with debris, mitigates recommending their use. Although such features are easy to place, they are relatively ineffective with capture efficiencies on the order of 5 percent. Nonetheless, they may be convenient when drainage can fall directly to underlying surfaces without under-deck pipe collection and downspouts. With direct fall of water, their small size is an advantage. The water collected at inlets either falls directly to surfaces beneath bridges or is collected. Collected storm water is conveyed to bridge support columns and downspouts that are affixed to vertical bridge members. The collectors are typically cast iron. The collector pipes should be sloped at least 2 percent (20 mm/m [1/4 in/ft] or more–preferably 8 percent–to provide sufficient velocities at low flow to move silt and small debris to avoid clogging.
13.8.1 Inlet Design for Constant-Grade Bridges The general procedure is to start at the high end of the bridge and work downslope from inlet to inlet. First, the designer selects a rainfall intensity and design spread. General bridges dimensions, bridge grade, roughness and runoff coefficients, and inlet specifications are assumed to be known. The procedure is as follows: 1. Find the appropriate rainfall intensity. If the rational method is used, assume a time of concentration of 5 min. 2. Find the flow on the deck Q at design spread T using Eq. (13.13). 3. Start at the high end of the bridge and compute the inlet spacing using Eqs. (13.45) and (13.46), the derivations of which are given by Young et al. (1993): KQ Lo c for the first inlet CiWp
(13.45)
and
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HYDRAULIC DESIGN OF DRAINAGE FOR HIGHWAYS
Hydraulic Design of Drainage for Highways 13.39
KQ Lc c E, for the general case CiWp
(13.46)
where i design rainfall intensity, mm/h (in/h) (Step 1) Q gutter flow, m3/s Lc constant distance between inlets, m (ft). Lo distance to first inlet, m (ft). C Rational runoff coefficient. Wp width of pavement contributing to gutter flow, m (ft). E constant, which is equal to 1 for first inlets in all cases and is equal to capture efficiency for subsequent inlets of constant-slope bridges and Kc unit conversion equal to 79,662.5 (43,560 in English units). Because the first inlet receives virtually no bypass from upslope inlets, the constant E can be assumed to be equal to 1. If the computed distance Lo is greater than the length of the bridge, then inlets are not needed and only bridge-end treatment design need be considered. 4. If inlets are required, the designer should proceed to calculate the constant inlet spacing Lc for the subsequent inlets. 5. If the bridge slope is nearly flat (less than about 0.003 m/m [0.003 ft/ft]), then the procedures for flat bridges should also be followed as a check.
13.8.2 Inlet Design for Flat Bridges Four steps are involved when designing inlets for flat bridges: 1. Rainfall intensity, design spread (T), pavement width (Wp), bridge length (LB), Manning’s n (n), rational runoff coefficient (C), and gutter cross slope (Sx) are assumed to be known. If the rational method is used, assume a time of concentration of 5 min and determined the intensity from the intensity-duration-frequency (IDF) curves. 2. Constant inlet spacing Lc can then be computed using Equ. (13.47), the derivation of which is given by Young, et al. (1993): Kc S 1.44 T 2.11 Lc (n Ci Wp)0.67 x
(13.47)
where Kc constant equal to 63,415 (1312 in English units). 3. The computed spacing is then compared with the known bridge length, If Lc is greater than the length of the bridge, then there is no need for inlets and the designer need be concerned only with the design of bridge end treatments. If Lc is less than the bridge length, then the total needed inlet perimeter (P) can be computed using Eq.(13.48), which is based on critical depth along the perimeter of the inlet (weir flow) (CiWp)0.33 T 0.61 P Kc Sx0.06 n0 .67
(13.48)
where Kc 320.1 (102.5 in English units). 4. Adapt spacing to accommodate structural and aesthetic constraints. Bridges with vertical curves, having either sags or high points, are nearly flat at their low-or high-point stations. Bridges with grades less than about 0.3 percent are nearly flat.
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HYDRAULIC DESIGN OF DRAINAGE FOR HIGHWAYS
13.40
Chapter Thirteen
For nearly flat stations on vertical curves or constant grades, the designer should set spacing assuming tthat he bridge is flat.
13.9 SAMPLE PROBLEMS Example 1. (Brown et al., 1996) Given: The following characteristics of the flow path Flow segment
Length (m)
Slope (m/m)
Segment description
1
25
0.005
Short-grass prairie
2
43
0.005
Short-grass pasture
3
79
0.006
Grassed waterway
4
146
0.008
380-mm concrete pipe
Find: Rainfall intensity I 60 mm/hr the time of concentration Tc for the area. Solution Step 1. Calculate time of concentration for each segment. Segment 1: Obtain Manning's n roughness coefficient from Table 13.4: n 0.15. then determine the sheet flow travel time using Eq. (13.4): T1 (6.943/10.4) (nL/S0.5)0.6 [6.943/(60)0.4] [(0.15) (25)/(0.005)0.5]0.6 14.6 min Segment 2: Obtain intercept coefficient k from Table 13.5: k 0.213. Then determine the concentrated flow velocity from Eq. 13.5: Vc KckSp0.5 (1)(0.213)(0.5)0.5 0.15 m/s Segment 3: Obtain intercept coefficient k from Table 13.5: k 0.457. Then determine the concentrated flow velocity from Eq. (13.5): Vc KckSp0.5 (1)(0.457)(0.6)0.5 0.35 m/s Segment 4: Obtain Manning's n roughness coefficient from Table 13.6: n = 0.011. Then determine the pipe flow velocity from Eq. (13.7): V = (1.0/0.011)(0.38/4)0.67 (0.008)0.5 1.7 m/s Step 2. Determine the total travel time from Eq. 13.8: Tc T1 T2 T3 T4 Tc 4.6 43/[(60)(0.15)] + 79/[(60)(0.35)] + 146/[(60)(1.7)] = 24.5 min. use 25 min.
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HYDRAULIC DESIGN OF DRAINAGE FOR HIGHWAYS
Hydraulic Design of Drainage for Highways 13.41
Example 2. (Brown et al., 1996) Given: The following existing and proposed land uses. Existing conditions (unimproved): Land use Unimproved Grass Grass Total
Area, ha
Runoff coefficient C
8.95 8.60 17.55
0.25 0.22
Proposed conditions (improved): Land use
Area, ha
Paved Lawn Unimproved Grass Grass Total
Runoff coefficient C
2.20 0.66 7.52 7.17 17.55
0.90 0.15 0.25 0.22
Times of concentration: Time of concentration Tc (min) Existing condition (unimproved):
88
Proposed condition (improved):
66
Partial 10-yr IDF information: Duration (min)
Rainfall intensity (mm/hr)
60 65 70 75 80 85 90
60 57 55 53 51 49 47
Find: The 10-yr peak flow using the rational formula. Solution Step 1. Determine weighted C for existing (unimproved) conditions using Eq. 13.3: Weighted C (CxAx)/A [(8.95)(0.25) (8.60)(0.22)]/(17.55) 0.235 Step 2. Determine weighted C for proposed (improved) condition using Eq. 13.3: Weighted C [(2.2) (0.90) (0.66)(0.15) (7.52) (0.25) (1.17)(0.22)]/(17.55) 0.315 Step 3. Determine rainfall intensity, I, from the 10yr IDF curve for each time of concentration.
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HYDRAULIC DESIGN OF DRAINAGE FOR HIGHWAYS
13.42
Chapter Thirteen
Rainfall intensity I Existing condition (unimproved):
48 mm/hr
Proposed condition (improved):
58 mm/hr
Step 4. Determine peak flow rate Q using equation 13.2: Existing condition (unimproved):
Q CIA/Kc (0.235)(48)(17.55)/360 = 0.55 m3/s
Proposed condition (improved):
Q CIA/Kc (0.315)(58)(17.55/360 = 0.88 m3/s
Example 3 Given: Gutter flow with a spread of 3 m, n = 0.016, Sx = 0.02, and 1 percent grade. Find: Flow Solution Determine the flow using Eq. 13.13: 2.67
Q = (0.376/0.016)(0.02)1.67(0.01)0.5 (3)
= 0.064 m3/s
Example 4 Given: A curved vane grate inlet which is 0.3 m wide and 0.5 m long with a full flow of 0.064 m3/s and gutter characteristics from Example 3. Find: The inlet capture efficiency. Solution Step 1. Determine the ratio of frontal flow to gutter flow using Eq. 13.28: Eo 1 [1 (0.3/3)]2.67 0.245 Step 2. Determine the velocity using Eq. 13.6: 0.67
Vg = (0.752/0.016) (0.01)0.5 (0.02)0.67 (3) = 0.71 m/s Step 3. Determine the frontal flow interception ratio using Fig. 13.11: For a 0.5 m (1.6 ft) long grate, the splashover velocity is .4 m/s (4.6 ft/s), which is greater than the gutter velocity. This implies an Rf of 1.0. Step 4. Determine the side flow interception ratio using Eq. 13.31: Rs 1/(1 ([(0.0828)(0.71)1.8]/[(0.02)(0.5)2/3])) 0.22 Step 5. Calculate the overall efficiency using Eq. 13.32: E (1.0)(0.245) (0.22) (1 0.245) 0.411 Example 5 Given: the configuration described in Example 4 and pavement width of 11 m, a rainfall intensity of 75 mm/hr, and a rational runoff coefficient of 0.7 (pavement and median).
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HYDRAULIC DESIGN OF DRAINAGE FOR HIGHWAYS
Hydraulic Design of Drainage for Highways 13.43
Find: The inlet spacing. Step 1. Calculate the distance to the first inlet using Eq. 13.45: Lo = (3 600 000) (0.064)/[(0.7) (75)(11)] = 400 m Step 2. Calculate general inlet spacing using Eq. 13.46: Lc = LoE = (40)(0.411) = 165 m
REFERENCES AASHTO, “Storm Drainage Systems”, Model Drainage Manual Chapter 13. American Association of State Highway and Transportation Officials, Washington, DC, 1991. AASHTO, Policy on Geometric Design of Highways and Streets, American Association of State Highway and Transportation Officials, Washington, DC, 1990. Anderson, D. A., J. R., Reed, R. S., Huebner, J. J., Henry, W. P., Kilareski, and J. C., Warner, Improved Surface Drainage of Pavements, Hydraulic Engineering Circular No. 12, Federal Highway Administration, U.S. Department of Transportation, Washington, DC, 1995. ASCE, “Design and Construction of Urban Stormwater Management Systems”, ASCE Manuals and Reports of Engineering Practice No. 77, WEF Manual of Practice RD–20, American Society of Civil Engineers, New York, 1992. ASCE, Design Manual for Storm Drainage, American Society of Civil Engineers, New York, 1960. Brown, S. A., S. M., Stein, and J. C., Warner, Urban Drainage Design Manual, Hydraulic Engineering Circular No. 22, FHWA–SA–96–078, Federal Highway Administration, U.S. Department of Transportation, Washington, DC, 1996. Burgi, P. H., D. E. Gober, June 1977. Bicycle-Safe Grates Inlets Study, Volume 1 Hydraulic and Safety Characteristics of Three Selected Grate Inlets on Continous Grades. Report No. FHWARD-77-24, Federal Highway Administration. Burgi, P. H., May 1978. Bicycle-Safe Grate Inlets Study, Volume 2 Hydraulic Characteristics of three Selected Grate Inlets on Continous Grades. Report No. FHWA-RD-78-4, Federal Highway Administration. Burgi, P. H., Bicycle-Safe Grate Inlets Study: Vol. 3 – Hydraulic Characteristics of Three Selected Grate Inlets in a Sump Condition, Report No. FHWA–RD–78–70, Federal Highway Administration, Washington, DC, September 1978. Chen, Y. H., and G. K.,Cotton, Design of Roadside Channels with Flexible Linings, Hydraulic Engineering Series No. 15, Publication No. FHWA–IP–87–7, prepared for the Federal Highway Administration, U.S. Department of Transportation, Washington, DC, 1986. Chow, V. T., Open-Channel Hydraulics, McGraw-Hill, New York, 1959. Federal Highway Administration, Design Charts for Open-Channel Flow, Hydraulic Design Series No. 3, U.S. Department of Transportation, Washington, DC, 1977 (reprint). Federal Highway Administration, Hydraulic Design of Energy Dissipators, Hydraulic Engineering Circular No. 14, U.S. Department of Transportation, Washington, DC, 1983. Gallaway, B. C., et al, December 1979.Pavement and Geometric Design Criteria for Minimizing Hidroplaning. Texas Transportation Institute, Texas A&M University, Federal highway Administratuion, Report No. FHWA-RD-79-30, A Technical Summary. Izzard, C. F., “Hydraulics of Runoff from Developed Surfaces,” Proc. Highway Research Board, Vol. 26, p. 129-150, Highway Research Board, Washington, DC, 1946. Jennings, M. E., W. O., Thomas, Jr., and H. C., Riggs, Nationwide Summary of U.S. Geological Survey Regional Regression Equations for Estimating Magnitude and Frequency of Floods for Ungaged Sites, 1993, U.S. Geological Survey, Water-Resources Investigations Report No.944002, prepared in cooperation with the Federal Highway Administration and the Federal Emergency Management Agency, Reston, VA, 1994.
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HYDRAULIC DESIGN OF DRAINAGE FOR HIGHWAYS
13.44
Chapter Thirteen
Johnson, F. L., and F. M., Chang, Drainage of Highway Pavements, Hydraulic Engineering Circular No. 12, FHWA–TS–84–202, Federal Highway Administration, Washington, DC, 1984. McCuen, R. H., P. A., Johnson, and R. M., Ragan, Hydrology, Hydraulic Design Series No. 2, FHWA–SA–96–067, Federal Highway Administration, U.S. Department of Transportation, Washington, DC, 1996. M. E. Jennings, W. O. Thomas, Jr., and H. C. Riggs, 1994. Nationwide Summary of U.S. Geological Survey Regional Regression Equations for Estimating Magnitude and Frequency of Floods for Ungaged Sites, 1993. US Geological Survey, Water-Resources Investigations Report 94-4002, prepared in cooperation with the federal Highway Administration and the Federal Emergency Managment Agency, Reston, Virginia. Richardson, E. V., D. B., Simons, and P. Y., Julien, Highways in the River Environment, prepared for the Federal Highway Administration, Washington, D.C, by the Department of Civil Engineering, Colorado State University, Fort Collins, Colorado, 1990. Young, G. K., S. M., Walker, and F., Chang, Design of Bridge Deck Drainage, Hydraulic Engineering Circular No. 21, FHWA–SA–92–010, Federal Highway Administration, U.S. Department of Transportation, Washington, D.C, 1993. V. B. Sauer, W. O. Thomas Jr., V. A. Stricker, and K. V. Wilson, 1983. Flood Characteristics of Urban Watersheds in the United States. Prepared in cooperation with U. S. Department of Transportation, Federal Highway Administration, U.S. Geological Survey Water-Supply Paper 2207. V. B. Sauer, 1989. Dimensionless Hydrograph Method of Simulating Flood Hydrographs. Preprint, 68th Annual Meeting of the Transportation Research Board, WEashington, D.C., (January) pp. 22-26.
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Source: HYDRAULIC DESIGN HANDBOOK
CHAPTER 14
HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS Ben Chie Yen Department of Civil & Environmental Engineering University of Illinois at Urbana-Champaign Urbana, Illinois
A.Osman Akan Department of Civil and Enviromental Engineering Old Dominion University Norfolk, Virginia
14.1. INTRODUCTION Generally speaking urban drainage systems consist of three parts: the overland surface flow system, the sewer network, and the underground porous media drainage system. Some elements of these components are shown schematically in Fig. 14.1. Traditionally no design is considered for the urban porous media drainage part. Recently porous media drainage facilities such as infiltration trenches have been designed for flood reduction or pollution control in cities with high land costs. For example, preliminary work on this aspect of urban porous media drainage design can be found in, Fujita (1987), Morita et al. (1996), Takaaki and Fujita (1984) and Yen and Akan (1983). Much has yet to be developed to refine and standardize on such designs; no further discussion on this underground subject will be given in this chapter. From a hydraulic engineering viewpoint, urban drainage problems can be classified into two types: (1) design and (2) prediction for forecasting or operation. The required hydraulic level of the latter is often higher than the former. In design, a drainage facility is to be built to serve all future events not exceeding a specified design hydrologic level. Implicitly the size of the apparatus is so determined that all rainstorms equal to and smaller than the design storm are presumably considered and accounted for. Sewers, ditches, and channels in a drainage network each has its own time of concentration and hence its own design storm. In the design of a network all these different rainstorms should be considered. On the other hand, in runoff prediction the drainage apparatus has already been built or predetermined, its dimensions known, and simulation of flow from a particular single rainstorm event is made for the purpose of real-time forecasting to be used for operation and runoff control, or sometimes for the determination of the flow of a past event for legal purposes. The hydrologic requirements for these two types of problems are different. In the case of prediction, a given rainstorm with its specific temporal and spatial distributions is considered. For design purposes, hypothetical rainstorms with assigned design return period or acceptable risk level and assumed temporal and spatial
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HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
14.2
Chapter Fourteen
FIGURE 14.1 Schematic of components of urban catchment. (From Metcalf & Eddy, Inc. et al., 1971).
distributions of the rainfall are used. Table 14.1 lists some of these two types of design and prediction problems. In the case of sanitary sewers, for design purposes the problem becomes the estimation of the critical runoffs in both quantity and quality, from domestic, commercial, and TABLE 14.1
Types of Urban Drainage Problems (a) Design Problems
Type
Design Purpose
Hydro Information Sought
Required Hydraulic Level
Sewers
Pipe size (and slope) determination Channel dimensions
Peak discharge, Qp for design return period Peak discharge, Qp for design return period Design hydrograph, Q(t)
Low
Design hydrograph, Q(t)
Low to moderate
Design peak discharge, Qp Design peak discharge, Qp Design hydrograph Design hydrograph
Low to moderate
Drainage channels Detention/retention storage ponds Manholes and junctions
Geometric dimensions (and outlet design) Geometric dimensions
Roadside gutters
Geometric dimensions
Inlet catch basins
Geometric dimensions
Pumps Control gates or valves
Capacity Capacity
Low to moderate Low to moderate
Low Moderate to high Moderate to high
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HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
Hydraulic Design of Urban Drainage Systems 14.3 TABLE 14.1 (continued) Types of Urban Drainage Problems (b) Prediction Problems Type
Purpose
Hydro Input
Hydro Information Sought
Required Hydraulic Level
Real-time operation
Real-time regulation of flow
Predicted and/or just measured rainfall, network data
Hydrographs, Q(t, xi)
High
Performance evaluation
Simulation for evaluation of a system
Specific storm event, network data
Hydrographs, Q(t, xi)
High
Storm event simulation
Determination of runoff at specific locations for particular past or specified events
Given past storm event or specified input hyetographs, network data
Hydrographs, Q(t, xi)
Moderate-high
Flood level Determination determination of the extent of flooding
Specific storm hyetographs, netwark data
Hydrographs and stages
High
Storm runoff quality control
Reduce and control of water pollution due to runoff from rainstorms
Event or continuous rain and pollutant data, network data
Hydrographs Q(t, xi) Pollutographs, c(t, xi)
Moderate to high
Storm runoff master planning
Long-term, usually Long–term data large spatial scale planning for stormwater management
Runoff volume Pollutant volume
Low
industrial sources over the service period in the future. For real-time control problems it involves simulation and prediction of the sanitary runoff in conjunction with the control measures. The basic hydraulic principles useful for urban drainage have been presented in Chapter 3 for free surface flows, Chapters 2 and 12 for pipe flows, and Chapter 10 for pump systems. In the following, more specific applications of the hydraulics to urban drainage components will be described. However, the hydraulic design for drainage of highway and street surfaces, roadside gutters, and inlets has been described in Chapter 13, design of stable erodible open channels in Chapter 16, and certain flow measurement structures adaptable to urban drainage in Chapter 21; therefore they are not included in this chapter.
14.2 HYDRAULICS OF DRAINAGE CHANNELS Flows in urban drainage channels usually are open-channel flows with a free water surface. However, sewer pipes, culverts, and similar conduits under high flow conditions could become surcharged, and pressurized conduit flows do occur. Strictly speaking, the flow is
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HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
14.4
Chapter Fourteen
always unsteady, that is, changing with time. Nevertheless, in a number of situations, such as in most cases of flow in sanitary sewers and for some rainstorm runoffs, change of flow with time is slow enough that the flow can be regarded as approximately steady.
14.2.1 Open-Channel Flow Open-channel flow occurs on overland, ditches, channels, and sewers in urban areas. Unsteady flow in open channels can be described by a momentum equation given below in both discharge (conservative) and velocity (nonconservative) forms together with its various simplified approximate models: 2 1 ∂Q 1 ∂ βQ 1 gA ∂t gA ∂x A gA
U q dσ ∂∂Yx S S 0. o
σ
x 1
0
f
(14.1)
dynamic wave quasi-steady dynamic wave noninertia kinematic wave
1 ∂V (2β 1) V ∂V (β 1) V2 ∂A ∂Y S S 0. o f g ∂t g ∂x ∂x gA ∂x
(14.2)
where x flow longitudinal direction measured horizontally (Fig. 14.2); A flow crosssectional area normal to x; y vertical direction; Y depth of flow of the cross section, measured vertically; Q discharge through A; V Q/A, cross-sectional average velocity along x direction; So channel slope, equal to tan θ, θ angle between channel bed and horizontal plane; Sf friction slope; σ perimeter bounding the cross section A; q1 lateral flow rate (e.g., rain or infiltration) per unit length of channel and unit length of perimeter σ, being positive for inflow; Ux x-component velocity of lateral flow when
FIGURE 14.2 Schematic of open, channel flow.
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HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
Hydraulic Design of Urban Drainage Systems 14.5
o
joining the main flow; g gravitational acceleration; t time; M (gA)1 (Ux σ V)g1 dσ and β Boussinesq momentum flux correction coefficient for velocity distribution: A β 2 Q
u dA o
(14.3)
2
σ
u x-component of local (point) velocity averaged over turbulence. The continuity equation is ∂A ∂Q ∂t ∂x
q dσ o
σ
1
(14.4)
If the channel is prismatic or very wide, such as the case of overland flow, Eq. (14.4) can be written as ∂Y 1 ∂ (VY) ∂t b ∂x
q dσ o
σ
1
(14.5)
where b is the water surface width of the cross section. In practice, it is more convenient to set the x and y coordinates along the horizontal longitudinal direction and gravitational vertical direction, respectively, when applied to flow on overland surface and natural channels for which So tan θ. For human-made straight prismatic channels, sewers, pipes, and culverts, it is more convenient to set the x-y directions along and perpendicular to the longitudinal channel bottom. In this case, the flow depth h is measured along the y direction normal to the bed and it is related to Y by Y h cos θ, whereas the channel slope So sin θ. The friction slope Sf is usually estimated by using a semiempirical formula such as Manning’s formula n2VV 4/3 n2QQ 4/3 Sf R 22 R Kn2 Kn A
(14.6)
or the Darcy-Weisbach formula QQ Sf f VV f 8gR 8gR A2
(14.7)
where n Manning’s roughness factor, Kn 1.486 for English units and 1.0 for SI units; f the Weisbach resistance coefficient; and R the hydraulic radius, which is equal to A divided by the wetted perimeter. The absolute sign is used to account for the occurrence of flow reversal. Theoretically, the values of n and f for unsteady nonuniform open-channel and pressurized conduit flows have not been established. They depend on the pipe surface roughness and bed form if sediment is transported, Reynolds number, Froude number, and unsteadiness and nonuniformity of the flow (Yen, 1991). One should be careful that for unsteady nonuniform flow, the friction slope is different from either the pipe slope, the dissipated energy gradient, the total-head gradient, or the hydraulic gradient. Only for steady uniform flow without lateral flow are these different gradients equal to one another. At present, we can only use the steady uniform flow values of n and f given in the literature as approximations. The advantage of f is its theoretical basis from fluid mechanics and its being nondimensional. Its values for steady uniform flow can be found from the Moody diagram or the Colebrook-White formula given in Chap. 2, as well as in stan-
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HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
14.6
Chapter Fourteen
dard hydraulics and fluid mechanics references. Its major disadvantage is that for a given pipe and surface roughness, the value of f varies not merely with the Reynolds number but also with the flow depth. In other words, as the flow depth in the sewer changes during a storm runoff, f must be recomputed repeatedly. Manning’s n was originally derived empirically. Its major disadvantage is its troublesome dimension of length to one-sixth power that is often misunderstood. Its main advantage is that for flows with sufficiently high Reynolds number over a rigid boundary with a given surface roughness in a prismatic channel, the value of n is nearly constant over a wide range of depth (Yen, 1991). Values of n can be found in Chow (1959) or Chap. 3. Other resistance coefficients and formulas, such as Chezy’s or Hazen-Williams’s, have also been used. They possess neither the direct fluid mechanics justification as f nor independence of depth as n. Therefore, they are not recommended here. In fact, HazenWilliams’s may be considered as a special situation of Darcy-Weisbach’s formula. A discussion of the preference of the resistance coefficients can be found in Yen (1991). Equations (14.6) and (14.7) are applicable to both surcharged and open-channel flows. For the open-channel case, the pipe is flowing partially filled and the geometric parameters of the flow cross section are computed from the geometry equations given in Fig. 14.3. The pair of momentum and continuity equations [Eqs. (14.1) and (14.4) or Eqs. (14.2) and (14.5)] with β 1 and no lateral flow is often referred to as the Saint-Venant equations or full dynamic wave equations. Actually, they are not an exact representation of the unsteady flow because they involve at least the following assumptions: hydrostatic pressure distribution over A, uniform velocity distribution over A (hence β 1), and negligible spatial gradient of the force due to internal stresses. Those interested in the more exact form of the unsteady flow equations should refer to Yen (1973b, 1975, 1996). Conversely, simplified forms of the momentum equation, namely, the noninertia (misnomer diffusion wave) and kinematic wave approximations of the full momentum equation [Eq. (14.1)] are often used for the analysis of urban drainage flow problems. Among the approximations shown in Eqs. (14.1) or (14.2), the quasi-steady dynamic wave equation is usually less accurate and more costly in computation than the noninertia equation, and hence, is not recommended for sewer flows. Akan and Yen (1981), among others, compared the application of the dynamic wave, noninertia, and kinematic wave equations for flow routing in networks and found the noninertia approximation generally agrees well with the dynamic wave solutions, whereas the solution of the kinematic wave approximate is clearly different from the dynamic wave solution, especially when the downstream backwater effect is important. Table 25.2 of Yen (1996) gives the proper form of the equations to be used for different flow conditions.
FIGURE 14.3 Sewer pipe flow geometry. (From Yen, 1986a)
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HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
Hydraulic Design of Urban Drainage Systems 14.7 TABLE 14.2
Theoretical Comparison of Approximations to Dynamic Wave Equation Kinematic wave
Noninertia
Quasi-steady dynamic wave
Dynamic wave
Boundary conditions required
1
2
2
2
Account for downstream backwater effect and flow reversal
No
Yes
Yes
Yes
Damping of flood peak
No
Yes
Yes
Yes
Account for flow acceleration
No
No
Only convective acceleration
Yes
Analytical solutions do not exist for Eqs. (14.1) and (14.2) or their simplified forms except for very simple cases of the kinematic wave and noninertia approximations. Solutions are usually sought numerically as described in Chap. 12. In solving the differential equations, in addition to the initial condition, boundary conditions should also be properly specified. Table 14.2 shows the boundary conditions required for the different levels of approximations of the momentum equation. It also shows the abilities of the approximations in accounting for downstream backwater effects, flood peak attenuation, and flow acceleration. For flows that can be considered as invariant with time the steady flow momentum equations which are simplified from Eq. (14.2) for different conditions are given in Table 14.3. The lateral flow contribution, mq, can be from rainfall (positive) or infiltration (negative) or both. Instead of these equations, the following Bernoulli total head equation is often used for flow profile computations:
TABLE 14.3 Cross-Section-Averaged One-Dimensional Momentum Equations for Steady Flow of Incompressible Homogeneous Fluid Prismatic channel
dβ
K (K K′) DY F ddyx S S Vg dx Y ddKx m 2
2
o
f
q
b
Constant piezometric pressure distribution K K′ 1
β constant K K′ 1 Prismatic or wide channel Definitions:
dY S (1 F2) dys F2S (1 F2) o o dx dx 2 dβ D ∂ B Sf F2 b V mq g dx B ∂x 2 dYs (F So Sf mq) (1 F2) dx
dY (So Sf mq) (1 F2) dx
or
V F gD bβ /
mq 1 gA
Db water surface width;
(U 2βV)qdσ 0
x
σ
K and K′ piezometric pressure distribation cor rection factors for main an lateral flows;
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HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
14.8
Chapter Fourteen
α2V22 α1V12 Y2 yb2 Y1 yb1 he hq , 2g 2g
(14.8)
where the subscripts 1 and 2 the cross sections at the two ends of the computational reach, ∆x, of the channel, Y yb the stage of the water surface where the channel bed elevation at section 1 is yb1 and that at section 2 is yb2 yb1 So ∆x; hq is the energy head from the lateral flow, if any; the energy head loss he Se ∆x where Se is the slope of the energy line; and α the Coriolis convective kinetic energy flux correction coefficient due to nonuniform velocity distribution over the cross section (Chow, 1959; Yen, 1973). If there are other energy losses, they should be added to the right-hand side of the equation. Methods of backwater surface profile computation using these equations are discussed in Chap. 3. If the flow is steady and uniform, Eqs. (14.1) and (14.4) or Eqs. (14.2) and (14.5) reduce to So Sf and Q AV. Hence, for steady uniform flow using Manning’s formula, K (φ sinφ)5/3 , Q 0.0496 n So1/2 D8/3 2 n φ /3
(14.9)
where φ is in radians (Fig. 14.3). Correspondingly, the Darcy-Weisbach formula yields 1 Q 8
2gfS D o
5/2
(φ sinφ)5/3 . 2 φ /3
(14.10)
Figure 14.4 is a plot of these two equations that can be used to find φ. 14.2.2 Surcharge Flow Sewers, culverts, and other drainage pipes sometimes flow full with water under pressure, often known as surcharge flow (Fig. 14.5). Such pressurized conduit flow occurs under extreme heavy rainstorms or under designed pipes. There are two ways to simulate unsteady surcharge flow in urban drainage: (1) The standard transient pipe flow approach and (2) the hypothetical piezometric open slot approach. 14.2.2.1 Standard transient pipe flow approach. In this approach, the flow is considered as it is physically, that is, pressurized transient pipe flow. For a uniform size pipe, the flow cross-sectional area is constant, being equal to the full pipe area Af; hence ∂A/∂x 0. The continuity and momentum equations [Eqs. (14.4) and (14.2) with q1 0] can be rewritten as Q AfV
(14.11)
1 ∂V ∂ βV2 Pa Sf , g ∂t ∂x g γ
(14.12)
where Pa the piezometric pressure of the flow and –the specfic weight of the fluid. If the pipe has a constant cross section and is flowing full with an incompressible fluid throughout its length, then ∂V/∂x 0. By further neglecting the spatial variation of β, integration of Eq. (14.12) over the entire length, L, of the sewer pipe yields P a γ
exit
entrance
1 ∂V V2 V2 Hu Ku Hd Kd LSf 2g 2g g ∂t
(14.13)
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HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
Hydraulic Design of Urban Drainage Systems 14.9
FIGURE 14.4 Central angle of water surface in circular pipe (from Yen, 1986a).
FIGURE 14.5 Surcharge flow in a sewer. (After Pansic, 1980).
or L ∂Q Q2 Hu Hd (Ku Kd) 2 SfL , gA ∂t 2gAf
(14.14)
where Hu the total head at the entrance of the pipe, Hd the water surface outside the pipe exit, and Ku and Kd the entrance and exit loss coefficients, respectively (Fig. 14.5). Equations (14.11) and (14.12) can also be derived as a special case of the commonly used general, basic, closed conduit transient flow continuity equation for waterhammer and pressure surge analysis, see, e.g., Chaudhry, 1979; Stephenson, 1984; Wood, 1980; Wylie and Streeter, 1983.
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HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
14.10
or
Chapter Fourteen
1 dA 1 dρ ∂V 0 A dt ρ dt ∂x
(14.15)
1 ∂H c2 ∂V ∂H sin θ 0 V ∂t gV ∂x ∂x
(14.16)
and the momentum equation 1 ∂V V ∂V ∂H Sf 0 g ∂t g ∂x ∂x
(14.17)
where ρ the bulk density of the fluid, H Pa/γ the piezometric head above the reference datum, and c the celerity of the pressure surge. The fact that Eqs. (14.11) and (14.12) can be derived from Eqs. (14.2) and (14.5) is the theoretical basis of the Preissmann hypothetical slot concept, which will be discussed below. 14.2.2.2 Hypothetical slot approach. This approach introduces hypothetically a continuous, narrow, piezometric slot attached to the pipe crown and over the entire length of the pipe as shown in Fig. 14.6. The idea is to transform the pressurized conduit flow situation into a conceptual open-channel flow situation by introducing a virtual free surface to the flow. The idea was suggested by Preissmann (Cunge and Wegner, 1964). The hypothetical open-top slot should be narrow so that it would not introduce appreciable error in the volume of water. Conversely, the slot cannot be too narrow, with the aim of avoiding the numerical problem associated with a rapidly moving pressure surge. A theoretical basis for the determination of the width of the slot is to size the width such that the wave celerity in the slotted sewer is the same as the surge celerity of the compressible water in the actual elastic pipe. The celerity c1 of the slot pipe is c1 gA /b
(14.18)
where b the slot width and A the flow cross-sectional area. Neglecting the area contribution of the slot and hence A Af πD2/4 for a circular pipe, and equating c1 to the pressure wave speed c in the elastic pipe without the hypothetical slot, the theoretical slot width is
FIGURE 14.6 Preissmann hypothetical piezometric open slot.
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HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
Hydraulic Design of Urban Drainage Systems 14.11
b πgD2/4c2
(14.19)
The surge speed in a pipe usually ranged from a few hundred feet per second to a few thousand feet per second. For an elastic pipe with a wall thickness e and Young’s modulus of elasticity Ep, assuming no pressure force from the soil acting on the pipe, the surge speed c is E ηEfD c2 f / 1 (14.20) ρf Epe
where Ef is the bulk modulus of elasticity and ρf is the bulk density, respectively, of the flowing water (Wylie and Streeter, 1983). Special conditions of pipe anchoring against longitudinal expansion or contraction and elasticity relevant to the surge speed c are given in Table 14.4 where ω Poisson’s ratio for the pipe wall material, that is, –ω is the ratio of the lateral unit strain to axial unit strain, and α is a constant to account for the rigidity with respect to axial expansion of the pipe. For small pipes, Eq. (14.19) may give too small a slot width, which would cause numerical problems. Cunge et al. (1980) recommend a width of 1 cm or larger. The transition between part-full pipe flow and slot flow is by no means computationally smooth and easy, and assumptions are necessary (Cunge and Mazadou, 1984). One approach is to assume a gradual width transition from the pipe to the slot. Sjöberg (1982) suggested two alternatives for the slot width based on two different values of the wave speed c in Eq. (14.19). For the alternative applicable to h/D 0.9999, his suggested slot width b can be expressed as b (14.21) 106 0.05423 exp[(h/D)24] D He further proposed to compute the flow area A and hydraulic radius R when the depth h is greater than the pipe diameter D as TABLE 14.4
Special Conditions of Surge Speed in Full Pipe, Eq. (14.20)
Factor Pipe Anchor
Condition e (1 ω) α D η 2 De D Freedom of pipe longitudinal expansion
Entirely free Only one (expansion joints at both ends) end anchored
Value of axial expansion factor α
1
1 0.5 ω
Entire length anchored 1 - ω2
Elasticity E Rigid pipe
Air entrainment
No air
Ep ∞
ρf ρwV –w + ρaV –a
ρf ρw
c2 Ef /ρf
Ew Ef 1V –a[(Ew/Ea) 1
Ef Ew
Subscript w denotes water (liquid); Subscript a air; subscript f fluid mixture; V – volume.
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HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
14.12
Chapter Fourteen
A (πD2/4) (h D)b
(14.22)
R D/4
(14.23)
A slight improvement to Sjöberg’s suggestion to provide a smoother computational transition is to use A A9999 b(h 0.9999D) (14.24) for h/D 0.9999 and assume that the transition starts at h/D 0.91. Between h/D 0.91 and 0.9999, real pipe area A and surface width B are used. However, for h/D 0.91, R is computed from Manning’s formula using pipe slope and a discharge equal to the steady uniform flow at h 0.91D, Q91; thus, for h/D 0.91 R (A91/A)R91
(14.25)
Because of the lack of reliable data, neither the standard surcharge sewer solution method nor the Preissmann hypothetical open-slot approach has been verified for a single pipe or a network of pipes. Past experiences with waterhammer and pressure surge problems in closed conduits may provide some indirect verification of the applicability of the basic flow equations to unsteady sewer flows. Nevertheless, direct verification is highly desirable. Jun and Yen (1985) performed a numerical testing and found there is no clear superiority of one approach over the other. Nevertheless, specific comparison between them is given in Table 14.5. They suggested that if the sewers in a network are each divided into many computational reaches and a significant part of the flow duration is under surcharge, the standard approach saves computer time. Conversely, if transition between open-channel and pressurized conduit flows occurs frequently and the transitional stability problem is important, the slot model would be preferred.
TABLE 14.5
Comparison Between Standard Surcharge Approach and Slot Approach
Item
Standard Surcharge Approach
Hypothetical Slot Approach
Concept
Direct physical
Conceptual
Flow equations
Two different sets, one equation for surcharge flow, two equations for open-channel flow
Discretization for solution
Whole pipe length for surcharge flow Constant
Same set of two equations (continuity and momentum) for surcharge and openchannel flows Divide into ∆x’s
Water volume within pipe
Discharge in pipe at given time
Same
Transition between open channel flow and surcharge flow
Specific criteria
Varies slightly with slot size, inaccurate if slot is too wide, stability problems if slot is too narrow Varies slightly with ∆x, thus allows transition to progress within pipe Slot width transition to avoid numerical instability
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HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
Hydraulic Design of Urban Drainage Systems 14.13 TABLE 14.5
(Continued)
Item
Standard Surcharge Approach
Part full over pipe length
Assume entire pipe length full or free Time accounting for transition Yes, specific inventory of surcharged pipes at different times Programming efforts More complicated because of two sets of equations and time accounting and computer storage for transition
Computational effort
Depending mainly on accounting for transition times
Hypothetical Slot Approach Assume full or free ∆x by ∆x No, implicit Relatively simple because of one equation set and no specific accounting and storage for transition between open-channel and full-pipe flows Depending mainly on space discretization ∆x
14.3 FLOW IN A SEWER 14.3.1 Flow in a Single Sewer Open-channel flow in sewers and other drainage conduits are usually unsteady, nonuniform, and turbulent. Subcritical flows occur more often than supercritical. For slowly time varying flow such as the case of the flow traveling time through the entire length of the sewer much smaller than the rising time of the flow hydrograph, the flow can often be treated approximately as stepwise steady without significant error. The flow in a sewer can be divided into three regions: the entrance, the pipe flow, and the exit. Figure 14.7 shows a classification of 10 different cases of nonuniform pipe flow
FIGURE 14.7 Classification of flow in a sewer pipe. (After Yen, 1986a).
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HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
14.14
Chapter Fourteen
FIGURE 14.8 Types of sewer entrance flow. (After Yen, 1986a).
based on whether the flow at a given instant is subcritical, supercritical, or surcharge. There are four cases of pipe entrance condition, as shown in Fig. 14.8 and below: Case
Pipe entrance hydraulic condition
I
Nonsubmerged entrance, subcritical flow
II
Nonsubmerged entrance, supercritical flow
III
Submerged entrance, air pocket
IV
Submerged entrance, water pocket
Case I is associated with downstream control of the pipe flow. Case II is associated with upstream control. In Case III, the pipe flow under the air pocket may be subcritical, supercritical, or transitional. In Case IV, the sewer flow is often controlled by both the upstream and downstream conditions. Pipe exit conditions also can be grouped into four cases as shown in Fig. 14.9 and below: Case
Pipe exit hydraulic condition
A
Nonsubmerged, free fall
B
Nonsubmerged, continuous
C
Nonsubmerged, hydraulic jump
D
Submerged
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HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
Hydraulic Design of Urban Drainage Systems 14.15
FIGURE 14.9 Types of sewer exit flow. (After Yen, 1986a).
In Case A, the pipe flow is under exit control. In Case B, the flow is under upstream control if it is supercritical and downstream control if subcritical. In Case C, the pipe flow is under upstream control while the junction water surface is under downstream control. In Case D, the pipe flow is often under downstream control, but it can also be under both upstream and downstream control. The possible combinations of the 10 cases of pipe flow with the entrance and exit conditions are shown in Table 14.6 for unsteady nonuniform flow. Some of these 27 possible combinations are rather rare for unsteady flow and nonexistent for steady flow, for example, Case 6. For steady flow in a single sewer, by considering the different mild-slope M and steep-slope S backwater curves (Chow, 1959) as different cases, there are 27 possible cases in addition to the uniform flow, of which six types were reported by Bodhaine (1968). TABLE 14.6
Pipe Flow Conditions
Case
Pipe Flow
Possible Entrance conditions
Possible Exit Conditions
1
Subcritical
I, III
A, B
2
Supercritical
II, III
B, C
3
Subcritical → hydraulic drop → supercritical
I, III
B, C
4
Supercritical → hydraulic jump → subcritical
II, III
A, B
5
Supercritical → hydraulic jump → surcharge
II, III
D
6
Supercritical → surcharge
II, III
D
7
Subcritical → surcharge
I, III
D
8
Surcharge → supercritical
IV
B, C
9
Surcharge → subcritical
IV
A, B
IV
D
10 Surcharge Source: From Yen (1986a).
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HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
14.16
Chapter Fourteen
The nonuniform pipe flows shown in Fig. 14.7 are classified without considering the different modes of air entrainment. The types of the water surface profile, equivalent to the M, S, and A (adverse slope) types of backwater curves for steady flow, are also not taken into account. Additional subcases of the 10 pipe flow cases can also be classified according to rising, falling, or stationary water surface profiles. For the cases with a hydraulic jump or drop, subcases can be grouped according to the movement of the jump or drop, be it moving upstream or downstream or stationary. Furthermore, flow with adverse sewer slope also exists because of flow reversal. During runoff, the change in magnitude of the flow in a sewer can range from only a few times dry weather low flow in a sanitary sewer to as much as manyfold for a heavy rainstorm runoff in a storm sewer. The time variation of storm sewer flow is usually much more rapid than that of sanitary sewers. Therefore, the approximation of assuming steady flow is more acceptable for sanitary sewers than for storm and combined sewers. In the case of a heavy storm runoff entering an initially dry sewer, as the flow enters the sewer, both the depth and discharge start to increase as illustrated in Fig. 14.10 at times t1, t2, and t3 for the open-channel phase. As the flow continues to rise, the sewer pipe becomes completely filled and surcharges as shown at t4 and t5 in Fig. 14.10. Surcharge flow occurs when the sewer is underdesigned, when the flood exceeds that of the design return period, when the sewer is not properly maintained, or when storage and pumping occur. Under surcharge conditions, the flow-cross-sectional area and depth can no longer increase because of the sewer pipe boundary. However, as the flood inflow continues to increase, the discharge in the sewer also increases due to the increasing difference in head between the upstream and downstream ends of the sewer, as sketched in the discharge hydrograph in Fig. 14.10. Even under surcharge conditions while the sewer
FIGURE 14.10 Time variation of flow in a sewer. (After Yen, 1986a).
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HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
Hydraulic Design of Urban Drainage Systems 14.17
diameter remains constant, the flow is usually nonuniform. This is due to the effects of the entrance and exit on the flow inside the sewer, and hence, the streamlines are not parallel. As the flood starts to recede, the aforementioned flow process is reversed. The sewer will return from surcharged pipe flow to open-channel flow, shown at t6 and t7 in Figure 14.1. Since the recession is usually—but not always—more gradual than the rising of the flood, the water surface profile in the sewer is usually more gradual during flow recession than during rising. The differences in the gradient of the water surface profiles during the rising and recession of the flood bear importance in the self-cleaning and pollutant-transport abilities of the sewer. During the rising period, with relatively steep gradient, the flow can carry not only the sediment it brings into the sewer but also erodes the deposit at the sewer bottom from previous storms. For a given discharge and gradient, the amount of erosion increases with the antecedent duration of wetting and softening of the deposit. During the recession, with a flatter water surface gradient and deceleration of the flow, the sediment being carried into the sewer by the flow tends to settle onto the sewer bottom. If the storm is not heavy and the flood is not severe, the rising flow will not reach surcharge state. The flood may rise, for example, to the stage at t3 shown in Fig. 14.10 and then starts to recede. The sewer remains under open-channel flow throughout the storm runoff. For such frequent small storms, the flow in the sewer is so small that it is unable to transport out the sediment it carries into the sewer, resulting in deposition to be cleaned up by later heavy storms or through artificial means. For a single-peak flood entering a long circular sewer having a diameter D and pipe surface roughness k, Yen (1973a) reported that for open-channel flow, the attenuation of the flood peak, Qpx, at a distance x downstream from the pipe entrance (x 0) and the corresponding occurrence time of this peak, tpx, can be described dimensionlessly as x k 0.17 Rb 0.42 Qp0 0.16 Qpx exp0.0771 Qp0 D D D D2.5 g g1.32 4 tp01.64 (tg tp0) 1 D .32
(14.26)
Qpx k 0.11 Rb 0.66 V (tpx tp0) w 6.03 log10 0.18 520 D Qp0 D D Qp4.04 0.1 g0.82 0.5 , 0.68 tp0 (tg tp0) 0 4 0 . 2 D .82 Qb g D
(14.27)
where Qp0 and tp0 the peak discharge and its time of occurrence at x 0, respectively; Qb is the steady base flow rate and Rb hydraulic radius of the base flow; tg the time to the centroid of the inflow hydrograph at x 0 above the base flow; g the gravitational acceleration; and Vw (Qb/Ab) (gAb/Bb)1/2 the wave celerity of the base flow, where Ab the base flow cross sectional area and Bb the corresponding water-surface width. In both equations, the second nondimensional parameter in the right-hand side k/D is a pipe property parameter; the third parameter Rb/D is a base flow parameter; the fourth nondimensional parameter represents the influence of the flood discharge; whereas the fifth and last nondimensional parameter reflects the shape of the inflow hydrograph. The single – peak hydrograph shown in Fig. 14.10 is an ideal case for the purpose of illustration. In reality, because the phase shift of the peak flows in upstream sewers and the time–varying nature of rainfall and inflow, usually the real hydrographs are multipeak.
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HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
14.18
Chapter Fourteen
Because the flow is nonuniform and unsteady, the depth-discharge relationship, also known as the rating curve in hydrology, is nonunique. Even if we are willing to consider the flow to be steady uniform as an approximation, the depth-discharge relation is nonlinear, and within a certain range, nonunique, as shown nondimensionally and ideally in Fig. 14.11 for a circular pipe. The nonunique depth-discharge relationship for nonuniform flow, aided by the poor quality of the water and restricted access to the sewer, makes it difficult to measure reliably the time-varying flow in sewers. Among the many simple and sophisticated mechanical or electronic measurement devices that have been attempted on sewers and reported in the literature, the simple, mechanical Venturi-type meter, which has side constriction instead of bottom constriction to minimize the effect of sediment clogging, still appears to be the most practical measurement means, that is, if it is properly designed, constructed, and calibrated and if it is located at a sufficient distance from the entrance and exit of the sewer. On the other hand, the hydraulic performance graph described in Sec. 14.6.1 can be used to establish the rating curve for a steady nonuniform flow.
FIGURE 14.11 Rating curve for steady uniform flow in circular pipe.
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HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
Hydraulic Design of Urban Drainage Systems 14.19
Flow in sewers is perhaps one of the most complicated hydraulic phenomena. Even for a single sewer, there are a number of transitional flow instability problems. One of them is the surge instability of the flow in pipes of a network. The other four types of instabilities that could occur in a single sewer pipe are the following: The instability at the transition between open-channel flow and full conduit flow, the transitional instability between supercritical flow and subcritical flow in the open-channel phase, the water-surface roll-wave instability of supercritical open-channel flow, and a near dry-bed flow instability. Further discussion on these instabilities can be found in Yen (1978b, 1986a). It is important to realize the existence of these instabilities in flow modeling. 14.3.2. Discretization of Space-Time Domain of a Sewer for Simulation No analytical solutions are known for the Saint—Venant equations or the surcharged sewer flow equation. Therefore, these equations for sewer flows are solved numerically with appropriate initial and boundary conditions. The differential terms in the partial differential equations are approximated by finite differences of selected grid points on a space and time domain, a process often known as discretization. Substitution of the finite differences into a partial differential equation transforms it into an algebraic equation. Thus, the original set of differential equations can be transformed into a set of finite difference algebraic equations for numerical solution. Theoretically, the computational grid of space and time need not be rectangular. Neither need the space and time differences ∆x and ∆t be kept constant. Nonetheless, it is usually easier for computer coding to keep ∆x and ∆t constant throughout a computation. For surcharge flow, Eq. (14.14) dictates the application of the equation to the entire length of the sewer, and the discretization applies only to the time domain. In an open-channel flow, it is normally advisable to subdivide the length of a sewer into two or three computational reaches of ∆x, unless the sewer is unusually long or short. One computational reach tends to carry significant inaccuracy due to the entrance and exit of the sewer and is usually incapable of sufficiently reflecting the flow inside the sewer. Conversely, too many computational reaches would increase the computational complexity and costs without significant improvement in accuracy. The selection of the time difference ∆t is often an unhappy compromise of three criteria. The first criterion is the physically significant time required for the flow to pass through the computational reach. Consider a typical range of sewer length between 100 and 1000 ft and divide it into two or three ∆x, and a high flow velocity of 5–10 ft/s, a suitable computational time interval would be approximately 0.2–2 min. For a slowly varying unsteady flow, this criterion is not important and larger computational ∆t will suffice. For a rapidly varying unsteady flow, this criterion should be taken into account to ensure the computation is physically meaningful. The second criterion is a sufficiently small ∆t to ensure numerical stability. An often-used guide is the Courant criterion ∆x/∆t V gA /B
(14.28)
In sewers, which usually have small ∆x compared to rivers and estuaries, this criterion often requires a ∆t less than half a minute and sometimes 1 or 2 s. The third criterion is the time interval of the available input data. It is rare to have rainfall or corresponding inflow hydrograph data with a time resolution as short as 2, 5, or even 10 min. Values for ∆t smaller than the data time resolution can only be interpolated. This criterion becomes important if the in-between values cannot be reliably interpolated.
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HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
14.20
Chapter Fourteen
In a realistic application, all three criteria should be considered. Unfortunately, in many computations only the second numerical stability is considered. There are many, many numerical schemes that can be adopted for the solution of the Saint-Venant equations or their approximate forms [Eqs. (14.1)–(14.5)]. They can be classified as explicit schemes, implicit schemes, and the method of characteristics. Many of these methods are described in Chap. 12, as well as in Abbott and Basco (1990), Cunge et al. (1980), Lai (1986), and Yen (1986a).
14.3.3 Initial and Boundary Conditions As discussed previously and indicated in Table 14.1, boundary conditions, in addition to initial conditions, must be specified to obtain a unique solution of the Saint-Venant equations or their approximate simplified equations. The initial condition is, of course, the flow condition in the sewer pipe when computations start, t 0, that is, either the discharge Q(x, 0), or the velocity V(x, 0), paired with the depth h(x, 0). For a combined sewer, this is usually the dry-weather flow or base flow. For a storm sewer, theoretically, this initial condition is a dry bed with zero depth, zero velocity, and zero discharge. However, this zero initial condition imposes a singularity in the numerical computation. To avoid this singularity problem, either a small depth or a small discharge is assumed so that the computation can start. This assumption is justifiable because physically there is dry-bed film flow instability, and the flow, in fact, does not start gradually and smoothly from dry bed. Based on dry-bed stability consideration, an initial depth on the order of 0.25 in., or less than 5 mm, appears reasonable. However, in sewers, this small initial depth usually is unsatisfactory because negative depth is obtained at the end of the initial time step of the computation. The reason is that the continuity equation of the reach often requires a water volume much bigger than the amount of water in the sewer reach with a small depth. Hence, an initial discharge, or base flow, that permits the computation to start is assumed. For a storm sewer, the magnitude of the base flow depends on the characteristics of the inflow hydrograph, the sewer pipe, the numerical scheme, and the size of ∆t and ∆x used. For small ∆x and ∆t, a relatively large base flow is required, but may cause a significant error in the solution. In either case, it is not uncommon that in the first few time steps of the computation, the calculated depth and discharge decrease as the flood propagates, a result that contradicts the actual physical process of rising depth and discharge. Nonetheless, if the base flow is reasonably selected and the numerical scheme is stable, this anomaly would soon disappear as the computation progresses. An alternative to this assumed base flow approach to avoid the numerical problem is to use an inverted Priessmann hypothetical slot throughout the pipe bottom and assigning a small initial depth, discharge or velocity to start the computation. Currey (1998) reported satisfactory use of slot width between 0.001 and 0.01 ft. As to boundary conditions, when the Saint-Venant equations are applied to an interior reach of a sewer not connected to its entrance or exit, the upstream condition is simply the depth and discharge (or velocity) at the downstream end of the preceding reach, which are identical with the depth and discharge at the upstream of the present reach. Likewise, the downstream condition of the reach is the shared values of depth and discharge (or velocity) with the following reach. Therefore, the boundary conditions for an interior reach need not be explicitly specified because they are implicitly accounted for in the flow equations of the adjacent reaches. For the exterior reaches containing either the sewer entrance or the exit, the upstream boundary conditions required depend on whether the flow is subcritical or supercritical as indicated in Table 14.7.
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HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
Hydraulic Design of Urban Drainage Systems 14.21 TABLE 14.7 Some Types of Specified Boundary Conditions for Simulation of Exterior Reaches of Sewers Location
Upstream End of Sewer Entrance Reach (x 0)
Downstream End of Sewer Exit Reach (x L)
Subcritical flow
One of Q(0, t) h(0, t) V(0, t)
One of h(L, t); e.g. ocean tides, lakes Q(L, t); release hydrograph Q(h); rating curve V(h); storage-velocity relation
Supercritical flow
for all t to be simulated
for all t to be simulated
Two of the above
None
For a sewer that is divided into M computational reaches and M 1 stations, there is a continuity equation and a momentum equation written in finite difference algebraic form for each reach. There are 2(M 1) unknowns, namely, the depth and discharge (or velocity) at each station. The 2(M 1) equations required to solve for the unknowns come from M continuity equations and M momentum equations for the M reaches, plus the two boundary conditions. If the flow is subcritical, one boundary condition is at the sewer entrance (x 0) and the other is at the sewer exit (x L). If the flow is supercritical, both boundary conditions are at the upstream end, the entrance, one of them often is a critical depth criterion. If at one instant a hydraulic jump occurs in an interior reach inside the sewer, two upstream boundary conditions at the sewer entrance and one downstream boundary condition at the sewer exit should be specified. If a hydraulic drop occurs inside the sewer, one boundary condition each at the entrance and exit of the sewer is needed; the drop is described with a critical depth relation as an interior boundary condition. Handling the moving surface discontinuity, shown schematically in Fig. 14.12, is not a simple matter. The moving front may travel from reach to reach slowly in different ∆t, or it may move through the entire sewer in one ∆t. If, for any reason, it is desired to compute the velocity of the moving front Vw between two computational stations i and i 1 in a sewer, the following equation can be used as an approximation;
FIGURE 14.12 Moving water surface discontinuity in a sewer. (After Yen, 1986a).
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HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
14.22
Chapter Fourteen
AiVi Ai 1Vi 1 Vw Ai 1 Ai
(14.29)
14.3.4 Storm Sewer Design with Rational Method The most important components of an urban storm drainage system are storm sewers. A number of methods exist for designing the size of such sewers. Some are highly sophisticated, using the Saint-Venant equations, whereas others are relatively simple. In contrast to storm runoff prediction/simulation models, sophisticated storm sewer design methods do not necessarily provide a better design than the simpler methods, mainly because of the discrete sizes of commercially available sewer pipes. If the peak design discharge Qp for a sewer is known, the required sewer dimensions can be computed by using Manning’s formula such that n Qp AR2/3 (14.30) Kn So which can be obtained from Eq. (14.6) by assuming the friction slope Sf is equal to the sewer slope So. All other symbols in the equation have been defined previously. For a circular sewer pipe, the minimum required diameter dr is n Q 3/8 dr 3.208 p Kn So
(14.31a)
where kn 1 for SI units and 1.486 for English units. If the Darcy-Wesibach formula (Eq. 14.7) is used, 1/5 f dr 0.811 Qp2 (14.31b) gS o These two equations are plotted in Fig. 14.13 for design applications. The assumption So Sf essentially implies that around the time of peak discharge, the flow can well be regarded approximately as steady uniform flow for the design, despite the fact that the actual spatial and temporal variations of the flow are far more complicated as described in Sec. 14.3.1. In sewer designs, there are a number of constraints and assumptions that are commonly used in engineering practice. Those pertinent to sewer hydraulic design are as follows: 1. Free surface flow exists for the design discharge, that is, the sewer is under “gravity flow” or open-channel flow. The design discharge used is the peak discharge of the total inflow hydrograph of the sewer. 2. The sewers are commercially available circular sizes no smaller than, say, 8 in. or 200 mm in diameter. In the United States, the commercial sizes in inches are usually 8, 10, 12, and from 15 to 30 inches with a 3-in. increment, and from 36 to 120 in. with an increment of 6 in. In SI units, commercial sizes, depending on location, include most if not all of the following: 150, 175, 200, 250, 300, 400, 500, 600, 750, 1000, 1250, 1500, 1750, 2000, 2500, and 3000 mm. 3. The design diameter is the smallest commercially available pipe that has a flow capacity equal to or greater than the design discharge and satisfies all the appropriate constraints. 4. To prevent or reduce permanent deposition in the sewers, a nominal minimum permissible flow velocity at design discharge or at nearly full-pipe gravity flow is speci-
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HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
Hydraulic Design of Urban Drainage Systems 14.23
FIGURE 14.13 Required sewer diameter. (m or ft)
fied. A minimum full-pipe flow velocity of 2 ft/s or 0.5 m/s at the design discharge is usually recommended or required. 5. To prevent the occurrence of scour and other undesirable effects of high velocity flow, a maximum permissible flow velocity is also specified. The most commonly used value is 10 ft/s or 3 m/s. However, recent studies indicate that due to the improved quality of modern concrete and other sewer pipe materials, the acceptable velocity can be considerably higher. 6. Storm sewers must be placed at a depth that will allow sufficient cushioning to prevent breakage due to ground surface loading and will not be susceptible to frost. Therefore, minimum cover depths must be specified. 7. The sewer system is a tree-type network, converging toward downstream. 8. The sewers are joined at junctions or manholes with specified alignment, for example, the crowns aligned, the inverts aligned, or the centerlines aligned. 9. At any junction or manhole, the downstream sewer cannot be smaller than any of the upstream sewers at that junction, unless the junction has significantly large detention
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HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
14.24
Chapter Fourteen
storage capacity or pumping. There also is evidence that this constraint is unnecessary for very large sewers. Various hydrologic and hydraulic methods exist for the determination of the design discharge Qp. Among them the rational method is perhaps the most widely and simplest used method for storm sewer design. With this method, each sewer is designed individually and independently, except that the upstream sewer flow time may be used to estimate the time of concentration. The design peak discharge for a sewer is computed by using the rational formula Qp i Cjaj,
(14.32)
where i the intensity of the design rainfall; C the runoff coefficient (see Chap. 5 for its values); and a is surface area. The subscript j represents the jth subarea upstream to be drained. Note that aj includes all the subareas upstream of the sewer being designed. Each sewer has its own design i because each sewer has its own flow time of concentration and design storm. The only information needed from upstream sewers for the design of a current sewer is the upstream flow time for the determination of the time of concentration. The rational formula is dimensionally homogenous and is applicable to any consistent measurement units. The runoff coefficient C is dimensionless. It is a peak discharge coefficient but not a runoff volume fraction coefficient. However, in English units usually the formula is used with the area aj in acres and rain intensity i in inches per hour. The conversion factor 1.0083 is approximated as unity. The procedure of the rational method is illustrated in the following in English units for the design of the sewers of the simple example drainage basin A shown schematically in Fig. 14.14. The catchment properties are given in Table 14.8. For each catchment, the length Lo and slope So of the longest flow path—or better, the largest Lo/S o —should first be identified. As discussed Sec. 14.7, a number of formulas are available to estimate the inlet time or time of concentration of the catchment to the inlet. In this example, Eq. (14.86) is used with K 0.7 for English units and heavy rain, that is, to 0.7(nLo/So)0.6. The catchment overland surface texture factor N is determined from Table 14.16 The design rainfall intensity is computed from the intensity-duration-frequency relation for this location, 100Tr0.2 i(in./h) (14.33) td 25 TABLE 14.8
Characteristics of Catchments of Example Drainage Basin A
Catchment
Area (Acres)
Longest Overland Path Length Lo Slope Surface Texture (ft) N
Inlet Time to (min)
Runoff Coefficient C
I
2
250
0.010
0.015
6.2
0.8
II
3
420
0.0081
0.016
9.3
0.7
III
3
400
0.012
0.030
11.7
0.4
IV
5
640
0.010
0.020
12.9
0.6
V
5
660
0.010
0.021
13.1
0.6
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HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
Hydraulic Design of Urban Drainage Systems 14.25
FIGURE 14.14 Sewer design example drainage basin A. (a) Layout (b) Profiles.
where td the rain duration (min) which is assumed equal to the time of concentration, tc, of the area described, and Tr the design return period in years. For this example, Tr 10 years. Determination of i for the sewers is shown in Table 14.9a. The entries in this table are explained as follows: Column 1. Sewer number identified by the inlet numbers at its two ends. Column 2. The sewer number immediately upstream, or the number of the catchment that drains directly through manhole or junction into the sewer being considered. Column 3. The size of the directly drained catchment. Column 4. Value of the runoff coefficient for each catchment. Column 5. Product of C and the corresponding catchment area. Column 6. Summation of Cjaj for all the areas drained by the sewer; it is equal to the sum of contributing values in Column 5.
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HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
14.26
Chapter Fourteen
Column 7. Values of inlet time to the sewer for the catchments drained, that is, the overland flow inlet time for directly drained catchments, or the time of concentration for the immediate upstream connecting sewers. Column 8. The sewer flow time of the immediate upstream connecting sewer as given in Column 9 in Table 14.9b. Column 9. The time of concentration tc for each of the possible critical flow paths, tc inlet time (Column 7) sewer flow time (Column 10) for each flow path. Column 10. The design rainfall duration td is assumed equal to the longest of the different times of concentration of different flow paths to arrive at the entrance of the sewer being considered, for example, for Sewer 31, td is equal to 13.9 min from Sewer 21, which is longer than that from directly contributing Catchment V (13.1 min). Column 11. The rainfall intensity i for the duration given in Column 10 is obtained from the intensity-duration relation for the given location, in this case, Eq. (14.33) for the 10-year design return period. Table 14.9b shows the design of the sewers for which the Manning n 0.015, minimum soil cover is 4.0 ft, and minimum nominal design velocity is 2.5 ft/s. The exit sewer of the system (Sewer 31) flows into a creek for which the bottom elevation is 11.90 ft, the ground elevation of its bank is 21.00 ft, and its 10-year flood water level is 20.00 ft. Column 1. Sewer number identified by its upstream inlet (manhole) number. Column 2. Ground elevation at the upstream manhole of the sewer. Column 3. Length of the sewer. Column 4. Slope of the sewer, usually follows approximately the average ground slope along the sewer. Column 5. Design discharge Qp computed according to Eq. (14.32); thus, the product of Columns 6 and 11 in Table 14.9a. Column 6. Required sewer diameter, as computed by using Eq. (14.31) or Fig. 14.13; for Manning’s formula with n 0.015 and dr in ft, Eq. (14.31a) yields Qp 3/8 dr 0.0324 So
in which Qp, in ft3/s, is given in Column 5 and So is in Column 4. Column 7. The nearest commercial nominal pipe size that is not smaller than the required size is adopted. Column 8. Flow velocity computed as V Q/Af; that is, it is calculated as Column 5 multiplied by 4/π and divided by the square of Column 7. As discussed in Yen (1978b), there are several ways to estimate the average velocity of the flow through the length of the sewer. Since the flow is actually unsteady and nonuniform, usually the one used here, using full pipe cross section, is a good approximation. Column 9. Sewer flow time is computed as equal to L/V, that is, Column 3 divided by Column 8 and converted into minutes.
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31–41
21
V
5
0.6 7.9
3.0
1.2
3.0
12
0.6
1.2
2.1
1.6
(5)
Cjaj
3.7
5
0.4
0.7
0.8
Cj (4)
Runoff Coefficient
11
IV
21–31
3
3
II
III
2
I
Directly Drained Area Catchment or aj Contributing Upstream Sewer (Acres) (2) (3)
12–21
11–21
(1)
Sewer
10.9
7.9
1.2
3.7
(6)
∑ Cjaj
TABLE 14.9 Rational Method Design of Sewers of Example Drainage Basin A (a) Design Rain Intensity
12.9
13.1
11.7
9.3
12.9
11.7
9.3
6.2
(min) (7)
Inlet Time
1.0
—
0.9
1.4
—
—
—
—
13.9
13.1
12.6
10.7
12.9
11.7
9.3
6.2
13.9
12.9
11.7
9.3
4.07
4.18
4.32
4.62
Upstream Time of Design Rain Design Rain Sewer Flow Concentration Duration Intensity Time tc td i (min) (min) (min) (in./h) (8) (9) (10) (11)
HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
Hydraulic Design of Urban Drainage Systems 14.27
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400
500
41.50
31.90
28.70
28.70
12
21
31
(31
500
360
450
35.00
11
(3)
(2)
(ft)
L
Length
(1)
Upstream Manhole Ground Elev. (ft)
0.0156
0.0144
0.0100
0.0290
0.0081
(4)
S
Slope
44.4
44.4
33.0
5.2
17.1
(5)
(ft3/s)
2.50
2.53
2.43
0.99
1.98
(6)
(ft)
2.50
2.75
2.50
1.00
2.00
(7)
(ft)
Design Required Diameter Discharge Diam. Used Qp dr dn
6.7
6.6
5.4
(8)
(ft/s)
Flow Velocity V
1.0
0.9
1.4
(9)
(min)
Sewer Flow Time
7.80
7.20
4.00
10.44
3.65
(10)
(ft)
SL
23.85
23.85
27.85
37.50
31.00
(11)
(ft)
21.35
21.10
25.35
36.50
29.00
(12)
(ft)
16.05
16.65
23.85
27.06
27.35
(13)
(ft)
13.55)
13.90
21.35
26.06
25.35
(14)
(ft)
Upstream Upstream Downstream Downstream Crown Elev. Invert Elev. Crown Elev. Invert Elev.
14.28
Sewer
(b) Sewer Design
HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
Chapter Fourteen
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HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
Hydraulic Design of Urban Drainage Systems 14.29
Column 10. Product of Columns 3 and 4; this is the elevation difference between the two ends of the sewer. Column 11. The upstream pipe crown elevation of Sewer 11 is computed from the ground elevation minus the minimum soil cover, 4.0 ft, to save soil excavation cost. In this example, sewers are assumed invert aligned except the last one (Sewer 31), which is crown aligned at its upstream (23.85 ft for upstream of Sewer 31 and downstream of Sewer 21) to reduce backwater influence from the water level at sewer exit. Column 12. Pipe invert elevation at the upstream end of the sewer, equal to Column 11 minus Column 7. Column 13. Pipe crown elevation at the downstream end of the sewer, equal to Column 11 minus Column 10. Column 14. Pipe invert elevation at the downstream end of the sewer, equal to Column 13 minus Column 7. For the last sewer, the downstream invert elevation should be higher than the creek bottom elevation, 11.90 ft. The above example demonstrates that, in the rational method, each sewer is designed individually and independently, except the computation of sewer flow time for the purpose of rainfall duration determination for the next sewer, that is, the values of tf in Column 8 of Table 14.9a are taken from those in Column 9 of Table 14.9b. The profile of the example designed sewers are shown as the solid lines in Fig. 14.14b. If the water level of the creek downstream of Sewer 31 is ignored, theoretically a cheaper design could be achieved by putting the exit Sewer 31 on a slightly steeper slope, from 0.0144 to 0.0156 to reduce the pipe diameter from 2.75 to 2.50 ft. The new slope can be estimated from 410/3 n2 So 2 Qp2 /d16/3. 2 π Kn
(14.34)
This alternative is shown with the parentheses in Table 14.9b and as dashed lines in Fig. 14.14b. However, one should be aware that the water level of a 10-year flood in the creek is 20.00 ft and hence, the last sewer is actually surcharged and its exit is submerged. The sewer will not achieve the design discharge unless its upstream manhole is surcharged by almost 4 ft (20.00–16.05). Therefore, the original design of 2.75 ft diameter is a safer and preferred option in view of the backwater effect from the tailwater level in the creek. In fact, Sewer 21-31 may also be surcharged due to the downstream backwater effect. Sometimes, a backwater profile analysis is performed on the sewer network to assess the degree of surcharge in the sewers and manholes. In such an analysis, energy losses in the pipes and manholes should be realistically accounted for. However, the intensity-duration-frequency-based design rainfall used in the rational method design is an idealistic, conceptual, simplistic rain and the probability of its future occurrence is nil. The actual performance of the sewer system varies with different actual rainstorms, each having different temporal and spatial rain distributions. But it is impossible to know the distributions of these future rainstorms, whereas the ideal rainstorms adopted in the design of the rational method are used as a consistent measure of protection level. Although designing sewers using the rational method is a relatively simple and straightforward matter, checking the performance of the sewer system is a far more complex task requiring thorough understanding of the hydrology and hydraulics of watershed runoff. For instance, checking the network performance by using an unsteady flow simulation model would require simulation of the unsteady flow in various locations in the network accounting for losses
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HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
14.30
Chapter Fourteen
in sewer pipes as well as in manholes and junctions (the latter will be discussed in the next section). Moreover, for a given sewer network layout, by using different sewer slopes, alternative designs of the network sewers can be obtained. A cost analysis should be conducted to select the most economic feasible design. This can be done with a system optimization model such as Illinois Least-Cost Sewer System Design Model (ILSD) (Yen et al., 1984).
14.4 HYDRAULICS OF SEWER JUNCTIONS There are various auxiliary hydraulic structures such as junctions, manholes, weirs, siphons, pumps, valves, gates, transition structures, outlet controls, and drop shafts in a sewer network. Information relevant to design of most of these apparatuses are well described in standard fluid mechanics textbooks and references, particularly in the German text by Hager (1994) and Federal Highway Administration (FHWA, 1996). In this section, the most important auxiliary component in modeling the sewer junctions are discussed. For sewers of common size and length, the headloss for the flow through a sewer is usually two to five times the velocity head. Thus, the head loss through a junction is comparable to the sewer pipe loss, and is not a minor loss.
14.4.1 Junction Classifications A sewer junction usually has three or four sewer pipes joined to it. Under normal flow conditions, one downstream pipe receives the outflow from the junction and other pipes flow into the junction. However, junctions with only two or more than four joining pipes are not uncommon. The most upstream junctions of a sewer network are usually one-way junctions having only one sewer connected to a junction. The horizontal cross section of the junction can be circular or square or may be another shape. The diameter or horizontal dimension of a junction normally is not smaller than the largest diameter of the joining sewers. To allow the workers room to operate, usually junctions are not smaller than 3 ft (1 m) in diameter. For large sewers, the access to the junction can be smaller than the diameter of the largest joining sewer. Sewers may join a junction with different vertical and horizontal alignments, and they may have different sizes and slopes. Vertically, the pipes may join at the junction with their centerlines or inverts or crowns aligned, or with any line of alignment in between. There is no clearly preferred alignment that could simultaneously satisfy the requirements of good hydraulics at low and high flows without complicating either construction cost or design. The bottom of the junction is usually at or slightly lower than the lowest invert of the joining sewers. In the horizontal alignment, often the outflow sewer is aligned with one (usually the major) inflow sewer in a straight line with other sewers joining at an angle. For cities with square blocks, right-angle junctions are most common. Typical sewer benching and flow guides in junctions are shown in Fig. 14.15. With the alignment of the joining pipes and the shape and dimensions of junctions not standardized, the precise, quantitative hydraulic characteristics of the junctions vary considerably. As a result, there are many individual studies of specified junctions, but a general comprehensive quantitative description of junctions is yet to be produced. For the purpose of hydraulic analysis, junctions can be classified according to the following scheme (Yen, 1986a):
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HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
Hydraulic Design of Urban Drainage Systems 14.31
FIGURE 14.15 Junction benching of sewers and flow guides.
1. According to the geometry: (a) one-way junction, (b) two-way junction, (c) threeway junction—merging (two pipes flow into one pipe) or dividing (one pipe flows into two pipes), and (d) four- or more-way junction—merging, dividing, or merging and dividing. 2. According to the flow in the joining pipes: (a) open-channel junction (with openchannel flow in all joining pipes), (b) surcharge junction (with all joining pipes surcharged), and (c) partially surcharged junction (with some, but not all, joining pipes surcharged). 3. According to the significance of the junction storage on the flow: storage junction or point junction. Hydraulically, the most important feature of a junction is that it imposes backwater effects to the sewers connected to it. A junction provides, in addition to a volume—however small—of temporal storage, redistribution and dissipation of energy, and mixing and transfer of momentum of the flow and of the sediments and pollutants it carries. The precise, detailed hydraulic description of the flow in a sewer junction is rather complicated because of the high degree of mixing, separation, turbulence, and energy losses. However, correct representation of the junction hydraulics is important in realistic simulation and reliable computation of the flow in a sewer system (Sevuk and Yen, 1973).
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HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
14.32
Chapter Fourteen
14.4.2 Junction Hydraulic Equations The continuity equation of the water in a junction is ds
0
∑ Qi Qj dt
(14.35)
0
where Qi the flow into or out from the junction by the i-th joining sewer, being positive for inflow and negative for outflow; Qj the direct, temporally variable water inflow into (positive) or the pumpage or overflow or leakage out from (negative) the junction, if any; s the storage in the junction; and t time. For a two-way junction, the index i 1, 2; for a three-way junction, i 1, 2, 3, and so on. The energy equation in a one-dimensional analysis form is 0
0
V2
P
dY
0
V2
∑Qi 2gi γi Zi QjHj s dt ∑QiKi 2gi ,
(14.36)
0
where Zi, Pi, Vi the pipe invert elevation above the reference datum, piezometric pressure above the pipe invert, and velocity of the flow at the end of the section of the ith sewer where it meets the junction, respectively; Hj the net energy input per unit volume of the direct inflow expressed in water head; Ki the entrance or exit loss coefficient for the ith sewer; Y the depth of water in the junction; and g the gravitational acceleration. The first summation term in Eq. (14.36) is the sum of the energy input and output by the joining pipes. The second term at the left-hand side of the equation is the net energy brought in by the direct inflow. The first term to the right of the equal sign is the energy stored in the junction as its water depth rises. The last term is the energy loss. The momentum equations for the two horizontal orthogonal directions x and z are 0
gPγ dA
(14.37)
gPγ dA ,
(14.38)
∑(QiVix) 0
0
x
0
and 0
∑(QiViz) 0
0
z
0
where px and pz the x and z components of the pressure acting on the junction boundary, respectively, and A the solid and water boundary surface of the junction. The direct flow Qj is assumed entering the junction without horizontal velocity component. The right-hand side term of Eqs. (14.37 and 14.38) is the x or z component force, where the integration is over the entire junction boundary surface A. The left-hand side term is the sum of momentum of the inflow and outflow of the joining pipes. Note that for a threeway merging junction, two of the Qi’s are positive and one Qi is negative, whereas for a three-way dividing junction, two of the Qi’s are negative. Joliffe (1982), Kanda and Kitada (1977), Taylor (1944), and others suggested the use of momentum approach to deal with high velocity situations. To illustrate this approach, consider the three-way junction shown in Fig. 14.16. The control volume of water at the junction enclosed by the dashed line is regarded as a point, and there is no volume change associated with a change of depth within it. One of the two merging sewers is along the direction of the downstream sewer, whereas the branch sewer makes an angle ϕ with it. When one assumes that the pressure distribution is hydrostatic and the flow is steady, the force-momentum relation can be written as
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HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
Hydraulic Design of Urban Drainage Systems 14.33
FIGURE 14.16 Control volume of junction for momentum analysis.
γh 2A2 γh 3A3 cos ϕ γh bAb sin ϕ γh 1A1 F
(14.39)
ρQ1V1 ρQ2V2 ρQ3V3 cos ϕ , where A the flow cross-sectional area in a sewer; h depth of the centroid of A; γ the specific weight; ρ the density of water; Q the discharge; V Q/A the cross-sectional mean velocity; and F the sum of other forces that are normally neglected. Some of these neglected forces are the component of the water weight in the control volume along the small bottom slope, the shear stresses on the walls and bottom, and the force due to geometry of the junction if the sewers are not invert aligned or the longitudinal sewers are of different dimensions. The subscripts 1, 2, and 3 identify the sewers shown in Fig. 14.16, and b represents the exposed wall surface of the branch in the control volume shown as ab in the figure. For the special case of invert aligned sewers with the branch (pipe 3) joining at right angle, ϕ 90º, Eq. (14.39) can be simplified as A2(gh 2 V22) A1(gh 1 V12)
(14.40)
Q2 A2 (gh1/V12) 1 1/2 Q1 A1 (gh2/V22) 1
(14.41)
or
Based on experimental results of invert-aligned equal-size pipes merging with ϕ 90º, Joliffe (1982) observed that the upstream depth h1 h2 and proposed that h3 h2 ξF3b hc1 hc1
(14.42)
where hc1 the critical depth in the downstream sewer, F3 the Froude number of the flow in the branch sewer, and Q Q 2. ξ 0.999 0.482 2 0.381 2 Q1 Q1
(14.43)
Q Q 2 Q 3 b 0.514 0.067 2 0.197 2 0.122 2 Q1 Q1 Q1
(14.44)
The equation describing the load of sediment or pollutants, expressed in terms of concentration c, can be derived from the principle of conservation as d 0 (14.45) cds ∑ Qici Qjcj G , dt s
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HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
14.34
Chapter Fourteen
where G a source (positive) or sink (negative) of the sediment or pollutants in the junction. Equations (14.35–14.44) are the theoretical basic equations for sewer junctions. They are applicable to junctions under surcharge as well as open-channel flows in the joining pipes. However, more specific equations can be written for the point-type and storage-type junctions.
14.4.3 Experiments on Three-Way Sewer Junctions and Loss Coefficients Proper handling of flow in sewer networks required information on the loss coefficients at the junctions. Unfortunately, there exists practically no useful quantitative information on energy and momentum losses of unsteady flow passing through a junction. Therefore, steady flow information on sewer junction losses are commonly used as an approximation. Table 14.10 summarizes the experimental conditions of three-way merging, surcharging, top-open sewer junctions conducted by Johnston and Volker (1990), Lindvall (1984), and Sangster et al. (1958, 1961). Also listed in the table are the experiments by Blaisdell and Mason (1967), Serre et al. (1994), and Ramamurthy and Zhu (1997); these experiments were not conducted on open-top sewer junctions but on three-way merging closed pipes. They are listed in the table as an example because these tests were conducted with different branch and main diameter ratios and with different pipe alignments. Hence, they may provide helpful information for sewer junctions. There exists considerably more information on merging or dividing branched closed conduits than on sewer junctions. The reader may look elsewhere (e.g., Fried and Idelchik, 1989, Miller, 1990) for information about centerline-aligned three-way joining pipes as an approximation to sewer junctions. The loss coefficients K2 1 and K3 1 for the merging flow are defined as Vi2 Vj2 hi Zi hj Zj 2g 2g . (14.46) kij V12 2g Figure 14.17 shows the experimental results of (1987) and Sangster et al. (1958) and Lindvall for the case of identical pipe size of the main and 90º merging lateral. The corresponding curves suggested by Miller (1990) and Fried and Idelchik (1989) for threeway identical closed pipe junctions are also shown as a reference. The values of the loss coefficients in a sewer junction that is open to air on its top are expected to be slightly higher than the enclosed pipe junction cases given by Miller because of the water volume at the junction above the pipes. The effect of the relative size of the joining branch pipe is shown in Fig. 14.18. The experimental data of Sangster et al. (1961) have identical upstream pipe sizes, D2 D3 for four different values of lateral branch to downstream main pipe area ratio, A3/A1. The data of Johnston and Volker (1990) on surcharged circular open-top sewer junction are not plotted in Fig. 14.18 because the mainline pipe area ratio A2/A1 0.41 instead of unity in the figure. Conversely, as a comparison, the smoothed curve of K21 for A3/A1 0.5 of the three-way pipe junction of Serre et al. (1994) with A1 A2 is plotted in Fig. 14.18a, and their experimental curves of K3 1 for A3/A1 0.21 and 0.118 are plotted in Fig. 14.18b. Also shown in the figure, as reference, are the three-way pipe junction curves for different values of A3/A1 suggested by Fried and Idelchik (1989) and Miller (1990) for identical size of main pipes, A2 A1. The experiments of Sangster et al. (1961) indicated that for a given A3/A1, the effect of A2/A1 on the loss coefficients is minor. Therefore, their curves should be comparable with those of Fried and Idelchik, Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
Square, rectangular, or round box
Round box
Square box
Enclosed pipe junction
Enclosed pipe junction
Enclosed Rectangular, rectangular 4.14 mm high, conduit junction main width 91.5 mm, branch width 20.4 mm,70.5 mm, or 91.5 mm
Lindvall (1984, 1987)
Johnston and Volker (1990)
Blaisdell and Mason (1967)
Serre et al. (1994)
Ramamurthy and Zhu (1997)
Horizontal
Horizontal
Horizontal
(Horizontal)
Circular, (Horizontal) Dmain 444 mm, Dbr/Dmain 0.14, 0.23, 0.34, or 0.46
Circular, Dbr/Dmain 0.25 1.0
Circular, Dmain d 70 mm, Dmain up/Dmain d 0.64, Dbr/Dmain d 0.91
Circular, Dmain 144 mm, Dbr/Dmain 1.0, 0.686, or 0.389
Circular, D 3.0 in.3.75 in. 4.75 in. or 5.72 in.
Same height
Center aligned
Center or top aligned
Flushed bottom
Center aligned
Flushed bottom
Straight through and one 90º emerging branch
Straight through and one 90º merging channel
Straight through and one mergin channel at 15º–165º by 15º increments
Centerline aligned with slight deflector for lateral in junction
Straight through and one 90º merging channel
Straight through and one 90º merging channel
Pipe Alignment at Junction Vertical Longitudinal
Sangster et al. (1958, 1961)
Channel Slope
Type of Junction
Reference
Shape of Channels
Experimental Studies on Three-way Junction of Merging Surcharged Channels
TABLE 14.10
Steady
Steady
Steady
Steady
Steady
Steady
Type of Flow
Reynolds number effect insignificant
Loss coefficient dependent on junction diameter, lateral pipe diameter, and flow ratio
Also tests of opposed lateral pipes; tests with grate inflow into junction
Remarks
HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
Hydraulic Design of Urban Drainage Systems 14.35
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HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
14.36
Chapter Fourteen
FIGURE 14.17 Experimental headloss coefficients for surcharged 3-way sewer junction with identical pipe sizes and 90o merging lateral. (a) Mainline loss coefficient K21; (b) Branch loss coefficient K31.
Miller, and Serre et al. However, Fig. 14.18 depicts considerable disagreement among the different sources, indicating the need for more reliable investigations. The joining angle of the lateral branch is a significant factor affecting the loss coefficients, particular on K3 1. The values of the loss coefficients decrease if the joining angle more or less aligns with the flow direction of the main, and increase if the lateral flow is directed against the main. The references of Fried and Idelchik (1989) and Miller (1990) provide some idea on the adjustment needed for the K values due to the joining angle.
FIGURE 14.18 Headloss coefficients for surcharged 3-way junction with 90o merging lateral of different sizes. (a) Mainline loss coefficient K21. (b) Branch loss coefficient K31.
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HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
Hydraulic Design of Urban Drainage Systems 14.37
FIGURE 14.19 Headloss coefficients for 3-way open-channel sewer junction with identical pipe sizes and 90E merging lateral. (a) Mainline loss coefficient K-21 (B) Branch loss coefficient K-31 (After Yen, 1987).
Listed in Table 14.11 is a summary of experiments on steady flow in three-way merging open-channel junctions. Most of the studies were done with point-type junctions. The experimental subcritical flow results of storage-type junctions by Marsalek (1985) and Townsend and Prins (1978) are plotted in Fig. 14.19 for lateral joining 90º to the same size mainline pipes. Yevjevich and Barnes (1970) gave the combined main and lateral loss coefficient but not the separate coefficients, making the result difficult to be used in routing simulation. The points in the figure scatter considerably, but they are generally in the same range of the loss coefficient values for surcharged three-way 90º merging junction except K3 1 for Townsend and Prins’ data. It is interesting to note that the most frequently encountered sewer junctions are three- and four–way box junctions with unsteady subcritical flow in the joining circular sewers. None of the open-channel experiments was conducted under these conditions. All were tested with steady flow. It is obvious that existing experimental evidence and theory do not yield reliable quantitative information on the loss coefficients of three-way sewer junctions. Before more reliable information is obtained, provincially for design and simulation of three joining identical size sewers, for K2 1 a curve drawn between that of Lindvall and that of Sangster et al. can be used as an approximation. For K3 1, the curve of Lindvall can be used. For joining pipes of unequal sizes, the curves of Sangster et al. appear to be tentatively acceptable.
14.4.4 Loss Coefficient for Two-Way Sewer Junctions Two-way junctions are used for change of pipe slope, pipe alignment, or pipe size. Experimental studies on two-way, surcharged, top-open sewer junctions are listed in Table
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Type of Junction
Point
Point
Point
Point
Square box
Taylor (1944)
Bowers (1950)
Behlke and Pritchett (1966)
Webber and Greated (1966)
Yevjevich and Barnes (1970)
Circular, Dmain 6.25 in., Dbr 1.87 in.
Rectangular, B 5 in.
Rectangular or trapezoidal (side slope 1:1)
Trapezoidal, identical width, B 7.2 in.
Rectangular, identical width, B 4 in.
Shape of Channels
0.00008 0.00054 0.00107
Horizontal
Each channel slope varied independently
0.0062, 0.012
Horizontal
Channel Slope
Flushed bottom or crown aligned
Flushed bottom
Flushed bottom
Flushed bottom
Flushed bottom
Vertical
Straight through and one 90º merging pipe
Straight through and one merging channel at 30º, 60º, or 90º
Straight through and one merging channel at 15º, 30º, or 45º
Straight through and one merging channel at 51º
Straight through and one merging channel at 45º or 135º
Longitudinal
Pipe Alignment at Junction
Experimental Studies on Three-Way Junction of Merging Open Channels
Subcritical
Also theoretical analysis based on momentum, good agreement with 45º merging but not with 135º merging
Subcritical
Subcritical
Subcritical
Subcritical
Greater loss for the case of crown aligned lateral
Greater losses associated with increasing merging angles of branch channel
Supercritical Supercritical Use of tapered wall in the junction to diminish diagonal wave and pile-up problems
Supercritical Supercritical Structure P7, hydraulic jumps formed upstream of junction, other structures with lateral bottom up to 3 ft above main
Subcritical
Type of Flow Upstream Downstream Remarks Pipes Pipe
14.38
References
TABLE 14.11
HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
Chapter Fourteen
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Horizontal
Point
Point
Point
Square box Circular, or round box identical diameter
Lin and Soong (1979)
Joliffe (1982)
Best and Reid (1984)
Marsalek (1985)
Rectangular, identical width, B 0.5 ft
Circular, equal diameter, D 69 mm
Rectangular, B 457 mm
Circular, Dmain 160 mm, Dbr 102 mm
Rectangular box
Townsend and Prins (1978)
Flushed bottom
Flushed bottom
Straight through and one merging channel
Straight through and one merging channel at 30o 60º, or 90º
Invert drop (15 or 18 mm)
Horizontal
Channel slope adjustable to achieve equilibrium water depth
Straight through and one 90º merging channel
Straight through and one 90º merging channel
Flushed bottom
Not available Straight through and one merging channel at 15º, 45º, 70º, or 90º
Horizontal, Flushed 0.0001, 0.0075, bottom 0.005, or 0.01
Horizontal
Less than 0.01 Invert drop Straight through across and one merging junction box channel at 45º or 90º
Horizontal
Bbr100mm; circular different sizes
Rectangular, Bmain 100, 200, 400mm
Circular, different sizes
Point
Kanda and Kitada (1977)
Radojkovic and Point Maksimovic (1977)
(Continued)
TABLE 14.11
Subcritical
Subcritical
Subcritical
Subcritical
Subcritical
Subcritical
Subcritical
Sub- or supercritical
Subcritical
Subcritical
Systems operated for 0.1 F 0.3
Upstream flow depth depends on critical depth in downstream pipe
Energy loss coefficient as a function of lateral to total flow rate ratio
Simple junction box and special junction box with flow deflector
Supercritical Supercritical Junction zone with expansion and without expansion
Supercritical Supercritical Also theoretical analysis based on momentum
HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
Hydraulic Design of Urban Drainage Systems 14.39
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Type of Junction
Square, rectangular, or round box
Rectangular box
Rectangular box or round box
Rectangular box or round box
Round box
Square box or round box
Square box
Round box
Sangster et al. (1958, 1961)
Ackers (1959)
Archer et al. (1978)
Howarth and Saul (1984)
Lindvall (1984)
Marsalek (1984)
Johnston and Volker (1990)
Bo Pedersen and Mark (1990)
Circular, identical diameter, D 90 mm
Circular, identical diameter, D 88 mm
Circular, identical diameter, D 6 in
Circular, identical diameter, D 144 mm
Circular, identical diameter, D 88 mm
Circular, identical diameter, D 102 mm
Circular, identical diameter, D 6 in.
Horizontal
Horizontal
Horizontal
Horizontal
Horizontal
0.002 and 0.010
0.0094–0.0192
Horizontal
Channel Slope
Center aligned
Center aligned
Flushed bottom
Center aligned
Flushed bottom
Flushed bottom
Flushed bottom
Flushed bottom
Straight through
Straight through
Straight through
Straight through
Straight through
Straight through, or 30º or 60º bend in junction
Straight through or 45º bend in junction or 52º bend downstream of junction
Straight through or 90º bend
Pipe Alignment at Junction Vertical Longitudinal
Steady
Steady
Steady
Steady
Steady or unsteady
Steady
Steady
Steady
Type of Flow
Four types of benching
Three types of benching
Manhole diameter 2.26D, headloss coefficient constant for given junction geometry
Loss coefficient increases as junction diameter increases
Also studied part–full supercritical flow
Also test of grate inflow into junction
Remarks
14.40
Circular, D 3.0, 3.75, 4.75, or 5.72 in.
Shape of Channels
Experimental Studies on Straight-Through Two-Way Open-Top Junction of Surcharged Channels
References
TABLE 14.12
HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
Chapter Fourteen
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HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
Hydraulic Design of Urban Drainage Systems 14.41
14.12. All the experiments were conducted with the same size upstream and downstream pipes joining the junction. Only Sangster et al. (1958, 1961) tested also the effect of different joining pipe sizes. These experimental results show that for a straight-through, twoway junction, the value of the loss coefficient is usually no higher than 0.2. Alignment of the joining pipes and benching in the junction are also important factors to determine the value of the loss coefficient. Figure 14.20a shows the headloss coefficiet of a surcharged two-way open-top junction connecting two pipes of identical diameters aligned centrally given by the experiments of Archer et al. (1978), Howarth and Saul (1984), Johnston and Volker (1990) and Lindvall (1984). Noticeable is the swirl and instability phenomena when the junction submergence (junction depth to pipe diameter ratio) is close to two and the corresponding high head loss coefficient. The ranges of loss coefficient given by Ackers (1959), Marsalek (1984), and Sangster et al. (1958) are also indicated in Fig. 14.20a, but the data on the variation with the pipe-to-junction–size ratio was not given by these investigators. Sangster et al. (1958) also tested the effect of different sizes of joining pipes for surcharged two-way junction. Some of their results are plotted in Fig. 14.20b. They did not indicate a clear influence of the effect of the size of the junction box. However, Bo Pedersen and Mark (1990) demonstrated that the loss coefficient of a two–way junction can be estimated as a combination of the exit headloss due to a submerged discharging jet and the entrance loss of flow contraction. They suggested that the loss coefficient K depends mainly on the size ratio between the junction and the joining pipes of identical size. For an infinitely large storage junction, the theoretical limit of K is 1.5. For the junction–diameter to pipe-diameter ratio, DM/D less than 4, they proposed to estimate the K values according to benching as shown in Fig. 14.21.
14.4.5 Storage Junctions For a storage- (or reservoir-) type junction, the storage capacity of the junction is relatively large in comparison to the flow volume and hydraulically it behaves like a reservoir. A water surface, and hence, the depth in the junction can be defined without great difficulty. A significant portion of the energy carried in by the flows from upstream sewers is dissipated in the junction. If the horizontal cross-sectional area of the junction Aj remains constant, independent of the junction depth Y, the storage is s AjY. Hence, ds/dt Aj(dY/dt) Aj(dH/dt), where H Y Z the water surface elevation above the reference datum, and Z the elevation of the junction bottom. Therefore, from Eq. (14.35), dH (14.47) ∑Qi Qj Aj dt Either the energy equation (Eq. 14.36) or the momentum equations [Eqs. (14.37) and (14.38)] can be used as the dynamic equation of the junction. If the energy loss coefficient Ki in Eq. (14.36) can be determined, use of the energy equation is appropriate. On the other hand, if the pressure on the junction boundary can be determined, the momentum equation is also applicable. If the junction were truly a large reservoir, both the loss coefficients and the pressure could reasonably be estimated on the basis of information on steady flow entering or leaving a reservoir. Customarily for the convenience of computation, instead of Eq. (14.36), the junction energy relationship is divided for each joining sewer by relating the total head of the sewer flow to the total head in the junction. Assuming that the energy contribution from the direct lateral inflow Qj is negligible, the component of Eq. (14.36) for each joining sewer i can be written as
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HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
14.42
Chapter Fourteen
FIGURE 14.20 Headloss coefficient for surcharged 2way open-top straight-through sewer junction. (a) Same size sewers upstream and downstream. (b) Different joining pipe sizes. (After Yen, 1987).
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HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
Hydraulic Design of Urban Drainage Systems 14.43
FIGURE 14.21 Effect of benching on loss coefficient of surcharged two-way sewer junction according to Bo Pedersen and Mark (1990).
H (1 Ki)(Vi2/2g) (Pi/γ) Zi
(14.48)
For open-channel flow in the joining pipes, the piezometric term Pi/γ is Pi/γ hi
(14.49)
where hi is the open-channel flow depth of the ith pipe at the junction. It should be cautioned that Eq. (14.48) is applicable only when there is no free surface discontinuity between the junction and the sewer. In other words, they are applicable to Cases B and D in Fig. 14.9 and all four cases in Fig. 14.8. The flow equations for these pipe exit and entrance cases are given in Table 14.13.
14.4.6 Point Junctions A point-type junction is the one whose storage capacity is negligible, and the junction is treated as a single confluence point. Hence, Eq. (14.35) is reduced to
∑Qi Qj 0
(14.50)
For subcritical flow in the sewers emptying into the point junction, the flow can discharge freely into and without the influence of the junction only when a free fall exists over a nonsubmerged drop at the end of the pipe (Case A in Fig. 14.9). Otherwise, the subcritical flow in the sewer is subject to backwater effect from the junction. Since the junction is treated as a point, the dynamic condition of the junction is usually represented by a kinematic compatibility condition of common water surface at the junctions for all the joining pipes without a free fall (Harris, 1968; Larson et al., 1971; Roesner et al., 1984; Sevuk and Yen, 1973; and Yen, 1986a). Thus, by neglecting the junction storage and for subcritical sewer flow into the junction, hi hic
if Zi hic Zo ho
(Case A in Fig. 14.9)
(14.51)
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Condition
Free fall at sewer exit
Subcritical flow in sewer
Supercritical flow in sewer
Supercritical flow in sewer, hydraulic jump in junction
A
B
B
C
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Supercritical flow in sewer
Submerged sewer entrance, openchannel flow in sewer
Surcharged sewer
II
III
IV
Source: From Yen (1986a).
Subcritical flow in sewer
I
For outflow from junction into sewer
Submerged sewer exit
D h hc H (1 K)(V2/2g) h Z
HhZ
H Z hc F 1
ZD H H Z hc F 1
ZD H Z hc H F 1
H (1 K)(V2/2g) h Z
H (1 - K)(Vc2/2g) hc Z Q AvVv Vv Cv 2g(H ) Z H (1 K)(V2/2g) (P/) Z
h hc H Z D δ(V2/2g) H Z hc (Vc2/2g)
H Z D δ(Vc2/2g) (h hc)
H Z D δ(V2/2g) h 30,000 with ks/R < 0.05 as R1/6 Kn ks 1.95 1 0. n log R 9 4 2 g 12R
(14.65)
For the third region of fully developed turbulent flow (Eq. 14.65), it is well known that n can be regarded approximately as a constant (Yen, 1991) and its value can be estimated from standard tables such as in Chow (1959) or Table 3.3. For shallow overland flow under rainfall, raindrops bring in mass, momentum, and energy input into the flow, and hence, the resistance coefficient is modified. Based on the result of a regression analysis by Shen and Li (1973), the values of n and f for R 900 can be estimated from the following nondimensional relationship g i 0.4 n 2 f 8 24 660 3 1/6 K n R g v
(14.66)
For a higher Reynolds number, Eq. (14.62) or constant n applies. One of the frequent purposes for overland flow simulation is to determine the peak discharge and its time of occurrence. For a continuous rainfall, the time to reach the peak discharge when all the areas within the watershed contribute is one definition of the time of concentration. A popular method of solving such overland flow problems is the kinematic wave approximation of the Saint-Venant equations because it is relatively simple, easy to solve, and requires only one boundary condition for solution, whereas the other two higher level approximations require two boundary conditions (Sec. 14.3.4). Its biggest drawback is its incapability to account for the backwater effect from downstream. Such backwater effect does exist in urban subcritical overland flow, for example, when the street surface flow joins the gutter flow, and the backwater effect from inlet catch basin. Despite the heterogeneous nature of urban catchments, a fundamental understanding of the overland surface runoff process can be gained through the consideration of the runoff of rainfall excess on a sloped, homogeneous, relatively smooth, plane surface. After the initial losses are satisfied, rain water starts to accumulate on the surface. Initially when the amount of water is small and the surface tension effect is predominant, the water may be held as isolated pots without occurrence of flow, as one would observe on a glass surface with a small amount of water. As rain water supply continues, the surface tension can no longer overcome the gravity force and the momentum input of the raindrops along the slope of the surface. The individual water pools merge and flow starts downslope. One should be wise and extremely careful to select the appropriate simplified equations to solve overland flow problems. For instance, if the geometry of a short street gutter is well defined, the hydraulic characteristics of the inlet catch basin downstream of the gutter are known, and a relatively reasonable accurate result is desired, the kinematic wave approximation will not be acceptable and at least the noninertia approximation should be used. Conversely, when simulating a whole block conceptually as a flow plane, there is no reason to use the Saint–Venant or noninertia equations because the grossly aggregated information of the block is incompatible with the sophisticated equations. Likewise, when
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HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
Hydraulic Design of Urban Drainage Systems 14.65
representing a long flow surface as an impervious or pervious surface which is described merely by its length, width, slope, and overall average surface roughness type, the kinematic wave approximation usually suffices, whereas the noninertia or Saint-Venant equations overkill because the downstream backwater effect is insignificant except for a small stretch at the very downstream.
14.7.2 Kinematic Wave Modeling of Overland Flow Despite its heterogeneous nature, urban overland surface is often hypothetically conceived as a collection of wide planes in modeling. For most overland flows, the depth is relatively small compared to flow length and the downstream backwater effect is insignificant; hence, the kinematic water approximation is applicable. For a wide open channel where the hydraulic radius R is equal to the flow depth Y, the kinematic wave momentum equation, So Sf, can be simplified as V aYm 1
(14.67)
or in terms of discharge per unit width of the channel q1 as q1 aYm
(14.68)
1/2 where m 5/3 and a K nS o /n for the Manning formula, m 3/2 and a (8gSo/f) for the Darcy-Weisbach formula, and m 3/2 and a C So for the Chezy formula. If either Eq. (14.67) or Eq. (14.68) is solved together with the continuity relationship [Eqs. (14.4) or (14.5)] in a nonlinear form, the simplification is a nonlinear kinematic wave approximation, often simply referred to as kinematic wave. If Eq. (14.67) or Eq. (14.68) is solved together with a simplified, linear form of Eqs. (14.4) or (14.5), the simplification is a linear kinematic wave approximation. Combining Eq. (14.67) with Eq. (14.5) and assuming a to be independent of x, we obtain
∂Y ∂Y ∂Y ∂ (aYm) maYm 1 ie ∂t ∂x ∂x ∂t
(14.69)
where the rainfall excess ie is ie i f'/
(14.70)
where i the rain intensity on the water surface and f' infiltration rate at channel bottom, that is, land surface. Since from Eq. (14.68) with a independent of x, ∂q1/∂t maYm1(∂Y/∂t). Substitution of this relation into the continuity equation yields another popular form of the kinematic wave approximation used in modeling: /
∂q1 1 ∂q1 ie ∂t c ∂x c
(14.71)
where c maYm1
(14.72)
Assuming that a and m are both constants, Eqs. (14.69) and (14.71) yield dx c maYm1 mV dt
(14.73)
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HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
14.66
Chapter Fourteen
dq1 ie dx dq1 iemV dt dY ie dt dY ie dx mV
and
(14.74) (14.75) (14.76) (14.77)
For an initially dry surface (Y 0 for 0 x L at t 0) under constant rainfall excess, ie and zero depth at the upstream end, integration of Eq. (14.76) yields Y iet
(14.78)
Substituting this equation into Eq. (14.73) and integrating, one has x x0 aiem1 tm
(14.79)
Let xO 0, the equilibrium peak discharge time, te, can be obtained with x L where L is the total length of the overland surface: L 1/m te m1 aie
(14.80)
At this time, the discharge per unit width from the overland surface is q1L ieL
(14.81)
q1L a(iet)m
(14.82)
and the discharge for 0 t te is
FIGURE 14.41 Sketch of kinematic-wave water surface profile during buildup times.
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HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
Hydraulic Design of Urban Drainage Systems 14.67
The water surface profile during the buildup period 0 t te based on the above kinematic wave analysis is shown in Fig. 14.41. Generalized charts are available in the literature to determine the peak kinematic overland flow rate from infiltrating surfaces for which ie is not constant (Akan, 1985a, 1985b, 1988). 14.7.3 Time of Concentration The equilibrium time given in Eq. (14.80) for Manning’s formula is often referred to as the kinematic wave time of concentration:
nL tc Kn So
0.6
i0.4
(14.83)
For infiltrating overland flow planes, Akan (1989) obtained numerical solution to the kinematic overland flow and the Green and Ampt infiltration equations and fitted the following equation to the numerical results by regression:
nL tc Kn So
0.6
(i K)0.4 3.10K1.33Pf φ(1 Si)i 2.33
(14.84)
where K is the soil hydraulic conductivity, φ is porosity, Pf is characteristic suction head, and Si is the initial degree of saturation of the soil. Note that this equation reduced to Eq. (14.83) for impervious surfaces with K 0. Morgali and Linsley (1965) numerically solved the Saint-Venant equations instead of the kinematic wave approximation for a number of runoffs from idealized catchment surface, and the results were regressed to give the following equation for the time of concentration in minutes, n0.605 L0.593 tc K (14.85) So0.38 ie0.388 where K 0.99 for English units with L in feet, and i in in./h. Izzard (1946) provided the following equation based on his experiments of artificial rain on sloped surfaces: L 1/3 tc 410.0007i1/3 2k 2 /3 i C So
(14.86)
for English units with iL 500, L in ft and i in in./h, and C is the rational formula runoff coefficient. In practical applications, often the rain intensity i is unknown a priori. Hence, tc of Eqs. (14.83)–(14.86) is computed iteratively with the aid of a rainfall intensity relationship. For overland surfaces of regular geometry beyond the two-dimensional surface just discussed, formulas for peak discharge and time of equilibrium estimation can be found in Akan (1985c) and Singh (1996). In addition to the hydraulic-based equations for time of concentration, a number of hydrologic-based empirical time of concentration formulas also exist (Kibler, 1982). Equations (14.83) and (14.85), when applied to actual catchments, usually yield a tc value smaller than found empirically. There are a number of possible reasons to cause this discrepancy (Yen, 1987), including the following: 1. The catchment surface is usually not homogeneous as is assumed in the derivation of Eqs. (14.83) and (14.85), the surface undulation is far more than the sand-equivalent roughness assumed in the derivation. Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
14.68
Chapter Fourteen
2. For shallow depth Manning’s n is not a constant (Yen, 1991) and raindrop impact increases n. 3. The sensitivity to rain input i0.4 is far more than reality. In the derivation, the input i is assumed as evenly distributed over the surface and without momentum, a pattern different from real rainfall. 4. Equations (14.83) and (14.85) are based on the time reaching the peak flow considering the influence of the flood wave propagation, different from the water particle travel time along the longest (or largest L/So) flow path. 5. The hydraulic time of peak flow measured from the commencement of rainfall excess, whereas the hydrologic time of concentration measured from the commencement of rainfall. Considering the aforementioned factors and to eliminate the rain intensity iteration process, Yen and Chow (1983) proposed the following formula for the overland flow time of concentration: NL 0.6 tc K o S
(14.87)
where K is a constant and N is an overland texture factor, similar to Manning’s n but modified for heterogeneous nature of overland surfaces. The values of K and N, modified slightly from their originally proposed values, are given in Tables 14.15 and 14.16, respectively.
14.8 MODELING OF CATCHMENT RUNOFF 14.8.1 Scientific Fineness versus Practical Simplicity Urban catchment runoff comes mostly from rainfall excess, that is, rainfall minus abstractions. The contribution of prompt subsurface return flow is usually negligible. (This is not necessarily the case for sewers, where leakage through joints and cracked pipes could be
TABLE 14.15
Values of K for Yen and Chow Formula Light rain
Rain intensity
For Lo in feet with For Lo in metres with
Moderate rain
Heavy Rain
(in./h)
0.8
0.8–1.2
1.2
(mm/h)
20
20–30
30
to in hours
0.025
0.018
0.012
to in min
1.5
1.1
0.7
to in hours
0.050
0.036
0.024
to in min
3.0
2.2
1.4
Source: From Yen and Chow (1983).
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HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
Hydraulic Design of Urban Drainage Systems 14.69 TABLE 14.16
Overland Texture Factor N for Eq. (14.86)
Overland Surface
Low
Medium
High
Smooth asphalt pavement
0.010
0.012
0.015
Smooth impervious surface
0.011
0.013
0.015
Tar and sand pavement
0.012
0.014
0.016
Concrete pavement
0.012
0.015
0.017
Tar and gravel pavement
0.014
0.017
0.020
Rough impervious surface
0.015
0.019
0.023
Smooth bare packed soil
0.017
0.021
0.025
Moderate bare packed soil
0.025
0.030
0.035
Rough bare packed soil
0.032
0.038
0.045
Gravel soil
0.025
0.032
0.045
Mowed poor grass
0.030
0.038
0.045
Average grass, closely clipped sod
0.040
0.050
0.060
Pasture
0.040
0.055
0.070
Timberland
0.060
0.090
0.120
Dense grass
0.060
0.090
0.120
Shrubs and bushes
0.080
0.120
0.180
Business
0.014
0.022
0.035
Semibusiness
0.022
0.035
0.050
Industrial
0.020
0.035
0.050
Dense residential
0.025
0.040
0.060
Land use
Suburban residential
0.030
0.055
0.080
Parks and lawns
0.040
0.075
0.120
Source: From Yen and Chow (1983).
considerable.) Among the abstractions, infiltration is by far the most significant. On a rainstorm-event basis, evapotranspiration is relatively negligible. Interception varies with land use and seasons. Depression storage is a matter of definition and subsequent method of estimation. Quantitative information on the initial losses—interception and depression storage—can be found in, for example, Chow (1964) or Maidment (1993). At any rate, for a heavy rainstorm, the amount of initial losses is relatively small. However, it should be noted that in terms of pollution, or runoff on an annual basis, the contributions of light rainstorms are also significant. The hydrologic characteristics of urban catchments vary with land uses and seasons. The surface may range from the relatively impervious surfaces such as streets, sidewalks, driveways, parking lots, and roofs to pervious surfaces such as lawns, gardens, bare soil,
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HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
14.70
Chapter Fourteen
FIGURE 14.42. Elements of urban catchment. (After Yen, 1987).
and parks. Rainfall excess on these surfaces are drained directly or indirectly through adjacent different types of surfaces and gutters into inlet catch basins, and then into sewers or channels (Fig. 14.42)]. The geometric composition of these different types of surfaces in forming a city block or catchment is usually random. This random heterogeneous nature of urban catchment surface imposes great difficulty in precise simulation of rainstorm runoff. Essentially, each surface requires a special, individual, “custom made” treatment which is costly and impractical in terms of both data and computation requirements. From the scientific viewpoint, existing knowledge appears to allow detailed scientific and quantitative simulation of each of the rainfall abstraction and surface flow processes than the current practice in urban drainage. Such simulation has not been incorporated in engineering practice mainly due to the conflicts between the detailed truthfulness in the scientific approach and the need for efficiency, simplicity, and tolerable accuracy in the practical procedures. In practice, various assumptions are explicitly or implicitly made so that some degree of simplification can be achieved for the sake of practical application and analysis. In the design of the size of most drainage facilities, usually knowing the design peak discharge, Qp, suffices. Conversely, for operation, planning, stormwater quality control and design involving runoff volume (such as detention storage), the discharge or stage hydrograph of the design rainstorm is needed. For the former, Qp, traditionally simple hydrologic methods such as the rational method can be used. For the latter, the runoff hydrograph can be determined using a hydrologic or hydraulic simulation model. Hydraulic-based simulation models employ a momentum or energy equation [either Eqs. (14.1) or (14.2), or any of the simplified approximations], together with the continuity equation; whereas hydrologic models do not consider momentum or energy equations.
14.8.2 Modeling Procedure Physically based simulation of the catchment rainfall-runoff process considers the water transport processes in the elements or components (Fig. 14.42) and their relative distribution within the catchment. Because of the number of elements in a catchment and the
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HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
Hydraulic Design of Urban Drainage Systems 14.71
amount of computations involved, such simulations are usually done using a computerbased model. In formulating a physically based model, the following process phases are considered after the rainfall input has been determined: 1. Decomposition of the catchment. In this phase, one determines the level of subdivision of the catchment and components used to represent a subcatchment. Should the catchment be divided merely into subcatchments? Should the pervious and impervious surfaces be considered separately? Should the street gutters and inlet catch basins be considered specifically? How should the roof contribution in the model be treated? How are the different types of surfaces in a subcatchment related to one another? Should the detention ponds be treated separately and individually? The more the different surfaces and components are aggregated as a unit, the simpler the model, but the greater the loss of physical reality. 2. Methods for abstractions. In this phase, the methods to calculate the losses due to interception, depression storage, and infiltration are selected. One should consider if the abstraction values are assigned catchment wide or if they should be allowed to vary for different subcatchments and different types of surfaces. The latter is more physically satisfactory, but it also requires more input information. It should be decided if the water detained on the overland surface contributes to infiltration when rain supply is insufficient. If multiple-event continuous modeling is being considered, methods to calculate evapotranspiration and infiltrability recovery should also be included. 3. Runoff from subcatchments. In this phase, the method of transforming rain excess water to runoff and the routing of runoff on the surfaces and subcatchments to the inlet catch basins is selected in accordance with the level of subdivision of the catchment. In hydraulically based models, for routing runoff in a catchment, in addition to the continuity equation, a flow momentum equation of some form is used. The continuity equation can be based on the cross-sectional averaged form [Eqs. (14.4) or (14.5)]. The momentum equation can be the full dynamic wave equation or any of its simplifications shown in Eqs. (14.1) or (14.2). These equations are applied to the elements of an urban catchment, step by step in sequence as shown in Fig. 14.42 for each time step to yield the runoff hydrograph of the catchment. It is not necessary to use the same routing method for the different elements and types of surfaces in a catchment. For example for an aggregated pervious surface, the time-area method, at most a kinematic wave routing usually would suffice because of the gross representation of its hydraulic characteristics. Conversely, for a street pavement, gutter, and inlet catch basin system, a kinematic wave routing may not be sufficient to provide realistic results because of its relatively welldefined geometric properties and the mutual backwater effects, and hence a noninertia routing may be desirable. It is obviously impractical to apply the highly sophisticated routing methods to each of the overland surfaces and gutters. The construction cost per unit length of such surfaces and gutters is small. And the cost of collection of data needed for such sophisticated computation methods is high. Moreover, the hydraulic characteristics of the overland surface change with time, depending on cleaning and season. On the other hand, the total length of streets and gutters in a catchment and in a city may be considerable. It is also desirable to have reasonably accurate inlet hydrographs as the input to the sewer system. Besides, for pollution control, the estimate of pollutant transport depends on the runoff estimation. Therefore, selection of the most appropriate overland routing method for a model and a catchment is a difficult task requiring delicate balance. Hydrologic simulation models for catchment runoff are not discussed here. They can be found in Chow (1964), Maidment (1993), and many other hydrology books. They
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HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
14.72
Chapter Fourteen
range from distributed system model similar to the hydraulic-based model described above in Steps 1–3, but the flow velocity in Step (3) is estimated by using some empirical techniques; or in the lumped hydrologic system models such as the conceptual reservoir-channel models of the Nash/Dooge type or the unit hydrograph methods the runoff is estimated from an assumed relationship without considering the physical process (Yen, 1986b). In most cases, synchronized runoff and rainfall data are needed for derivation of the catchment unit hydrograph or calibration of the lumped system model; singleevent data sets are difficult to obtain and if available often are not sufficiently reliable. For a city or a region, the urban surfaces tend to have some degree of similarity, especially in the United States where many cities have standardized square or rectangular blocks. It is, therefore, possible to group the urban surfaces into typical blocks. Reliable simulations can be made to establish unit hydrographs for the typical blocks or subcatchments. A few of such typical unit hydrographs should be sufficient for a catchment. In later applications, the use of the unit hydrographs provides relatively accurate results avoiding the repetitive costly sophisticated routing computation for individual rainstorms and blocks. Yen et al. (1977) first proposed this approach for a catchment in San Francisco using a nonlinear kinematic wave routing for the surface and gutter flow. Akan and Yen (1980) developed nondimensional unit hydrographs for street-gutter-inlet systems using dynamic wave routing and considering specifically the inlet capacity allowing by pass flow. Harms (1982) took a similar concept using a semiempirical approach to establish 1–min unit hydrographs. However, the unit hydrograph theory suffers from the linearity assumption between rainfall excess and surface runoff, making it inaccurate when the depth of the simulated rainstorm is significantly different from that of the rainstorm the unit hydrograph is derived. Lee and Yen (1997) introduced a hydraulic element of kinematic-wave based flow time determination on geomorphologically represented catchment subdivision for derivation of the catchment instantaneous unit hydrograph, making it a hydraulic distributed model and allowing derivation of unit hydrographs for ungaged catchments.
14.8.3 Selected Catchment Hydraulic Simulation Models There exist many urban rainfall-runoff models. A summary of the important features of selected hydraulic-based urban catchment models, mostly nonproprietary, is given in Table 14.17. Some models cover both catchment surface and sewer network parts; only the catchment part is summarized in this table. The sewer part is given in Tables 14.21 and 14.22. One should refer to the original references for details and objectives of these models. Similar hydrology-based watershed models that have also been applied to urban catchments, such as SCS-TR55 and TR-20, Hydrologic Engineering Center (HEC-1) (or its replacement HEC-HMS), and RORB are not presented here.
14.8.4 Verification and Calibration of Models Models should never be used without being tested and verified. It has happened again and again that in the enthusiasm in model development, models are used without verification. All models have their own assumptions and simplifications. Moreover, most urban runoff models contain coefficients, exponents, or adjustment factors that require calibration with data to determine their values. Besides verification and application for predictions, there are other operational modes of models such as those shown in Table 14.18. In calibration, we try to determine the most
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No
Single hyetograph
Single hyetograph
Hyetographs
Cincinnati
ILLUDAS
IUSR
Yes
No
No
Horton’s formula, from rain only
Horton’s formula
Depression storage
Area and entry line of direct contributing surface
Strips
Yes
Pervious Area Contribution
Horton’s Divided into strips with formula, from input length, width, slope, rain only and roughness
Area and entry line of direct contributing surface, area of supplemental surface
Strips
Yes
Abstractions Pervious Impervious Area Area Infiltration Contribution
Different Horton’s constants for formula, from impervious rain only and previous surfaces
Depression storage by exponential function
Depression storage by exponential function
Rainfall Input Allow Initial Areal Losses Distribution
Summary of Selected Urban Catchment Surface Runoff Models
Chicago Single hydrograph hyetograph
Model
TABLE 14.17
Nonlinear kinematic wave routing
Time-area with Izzard’s time formula or kinematic wave
Storage routing with constant depth detention storage function and Manning’s formula
Modified Izzard’s
Surface Runoff Routing Method
Nonlinear kinematic wave routing with Manning’s formula
No
Continuity eq. of steady spatially varied flow
Linear kinematic wave storage routing with Manning’s formula
Street Gutter
Yes
No
No
No
Selected References
Yes Chow and Yen (1976)
Yes Terstreip and Stall (1974)
Yes Papadakis and Preul (1972); Univ. of Cincinnati (1970)
Yes Keifer et al. (1978); Tholin and Keifer (1960)
User’s Inlet Manual Catch Basin
HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
Hydraulic Design of Urban Drainage Systems 14.73
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No
Yes
Yes
Single hyetograph
Single hyetograph
Hyetographs
CTH
Belgrade
MOUSE
WALLRUS/ Hyetographs HYDROWORKS
No
No
Yes
Ehime Single Stormwater hyetograph Runoff
Hyetographs
Depression storage for impervious surface by exponential function
Depression storage by graph
Depression storage
Rainfall Input Allow Initial Areal Losses Distribution
(Continued)
Horton’s en formula or coupled with subsurface flow
Horton’s formula
By graph based on Horton’s formula
Horton’s or Green and Ampt’s formulas
Pervious Area Contribution
Strips
Strips
Conceptual two-linearreservoir simulation
Strips
Strips
Divided into unit-width strips with input length, slope, and roughness
Divided into strips
Divided into strips with input length, width, slope, and roughness n
Abstractions Pervious Impervious Area Area Infiltration Contribution
Nonlinear kinematic wave routing
Nonlinear kinematic wave routing
Kinematic wave
Kinematic wave
Linear kinematic wave, storage routing with uniform depth continuity equation and Manning’s formula
Surface Runoff Routing Method
No
No
No
No
No
No
No
Selected References
Yes Wallinq ford software (1997)
Yes DHI (1994)
No Radojkovic and Maksimovic (1984)
No Arnell (1980)
No Toyokuni and Watanabe (1984)
Yes Huber and Dickinson (1988); Huber and Heaney (1982); Metcalf & Eddy, et al. (1971)
User’s Inlet Manual Catch Basin
Nonlinear kinematic No wave routing
Continuity
No
Linear kinematic wave, storage equation with Manning’s formula and continuity equations
Street Gutter
14.74
SWMM
Model
TABLE 14.17
HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
Chapter Fourteen
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HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
Hydraulic Design of Urban Drainage Systems 14.75 TABLE 14.18 Mode Prediction
Modes of Operation of Models Input Known
Transformation Parameters Coefficient Values Known
Known
Output ?
Calibration
Known
Known
?
Known
Verification
Known
Known
Known
?(/Known?)
Validation
Known
Known
Known
Applicable ?
Known
Known
Known (?)
Known
Detection
?
Known
Parameter identification
Known
?
Sensitivity
Known
Known
Known
?(Per unit change of parameter or coefficient)
Reliability
Known
Known
Known
?(Over likely ranges of parameters)
Source: From Yen (1986b).
suitable values of the coefficients of the parameters (variables) knowing the input and output from observed data. In verification, we have the parameters and their coefficient values all determined for the model, and we have the data on both input and output. The input is run through the model to produce output, which is compared to the known output in the data set to verify the agreement between the computed and observed outputs. On the other hand, verification is different from validation. Validation is to ascertain if the correct equation or model is used to solve the problem. Verification is to find out if the equation or model is solved correctly. No model can do everything. For example, a good flow simulation model may not produce a good design of the drainage system. Conversely, a good design model may not—and often need not—be an accurate flow simulation model. Therefore, models should be verified and validated according to their objectives and their applications. In verifying a model, the verification criteria should be setup to confirm with the model objectives. For example, the verification can be made according to the peak discharge, time to peak, or to the fitting of the hydrograph as desired by the objective. Various verification fitting error measures have been suggested in the literature (ASCE Task Committee, 1993; Yen, 1982). Some measures are listed in Table 14.19. In the table, the magnitude parameter Q can be discharge, depth, velocity, or concentration as appropriate to the problem investigated. The subscript p denotes the peak magnitude of the time graph. The subscript m represents the measured or true values used as the gauge for the curve fitting and verification of the model simulation. The selection of the error measures to evaluate the merit of simulation models depends on the objective of the simulation. For example, if the accuracy of the peak rate and peaking time are of paramount importance, εQp and εtp would be the most appropriate error measures. If the overall fitting of the curves is the main objective, εRMS would be the most important measure, while ετ1, εva, εtp, and εQp could be used as auxiliary measures. In calibration, since the reliability of a single set of data is uncertain, the more sets of
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V (V Vm) / Vm
Cumulative volume error
1\2
1/2
(Q Qm)2 dt
0
t
i
0
t
0
Source: From Yen (1982). *(V or Vm becomes total volume if t flow duration considered, T. Subscript m 5 measured or reference base values; subscript i summation index; subscrit p peak magnitude of the time graph.
Gm 1 Vm
0
T
Qm(tm tpm)2 dt
Second moments with respect to tpm: T G 1 Q(t tpm)2 dt V 0
1
g (G Gm)/Gm
Graph dispersion error
t
tQdt 1 Qi ti t t V i 2 0
t
τi 1 V
Qrm 1 1 Qm2 dt 1 Q2mi t Vm 0 2 2Vm i
τ (τ1 τ1m)/τ1m
tp (tp tpm)/tpm
0
i
Qmdt Qmit
t
Qr 1 1 Q2dt 1 Qi2t V 02 2V i
Vm
0
0
Qdt Qit
t
V
Peak time first-moment error
Time errors Peak-rate time error
t
0
T
T (Qi Qmi)2t Vm i
RMS T 1 Vm T
Qr (Q Qr Qrm) / Qrm
Root-mean-square
0
1 Q Qmdt Qi Qmit Vm i
Rate moment error
0
1 va Vm
Absolute volume error
t
d (Q Qm) / Qm
Qp (Qp Qpm) / Qpm
Remarks*
14.76
Mean rate error
Definition
Magnitude errors Peak rate error
Simulation or Measurement Error Measures
Error measure
TABLE 14.19
HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
Chapter Fourteen
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HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
Hydraulic Design of Urban Drainage Systems 14.77
data used, the better. Different sets of data would produce different sets of coefficient values. Normally, a weighted average (e.g., through optimization) of the values is adopted for each of the coefficients. It is not infrequent to see a model misused or abused. Sometimes this is due to the lack of understanding about how the model works. Sometimes it is due to the lack of appreciation of the operational modes. For example, data used for calibration should not be used again for verification. Yet, this situation happens again and again. In such a case of using the same data for calibration and verification, the difference between the model output and the recorded data is simply a reflection of the numerical errors and the deviation of the particular data set from the weighted average situation. Not all models require calibration. Presumably, some strictly physically based models have their coefficient values assigned based on available information and no calibration is needed. However, in rainfall-runoff modeling, some degree of spatial and temporal aggregation of the physical process is unavoidable. Therefore, calibration is desirable, if not necessary.
14.9 DETENTION AND RETENTION STORAGE Detention and retention basins are widely used to control the increased runoff due to urbanization of undeveloped areas. These basins can also offer excellent water quality benefits since pollutants are removed from the stormwater runoff through sedimentation, degradation, and other mechanisms, as the runoff is temporarily stored in a basin. Detention basins are sometimes called dry ponds, because they store runoff only during wet weather. The outlet structures are designed to completely empty the basin after a storm event. Retention basins are sometimes called wet ponds since they retain a permanent pool.
Post development hydrograph (Pond inflow)
Flow rate
Required pond volume
Pre–development peak flow rate Routed post–development hydrograph (pond outflow)
Time FIGURE 14.43 Routing of runoff through detention basin.
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Anti–seep collar Pipe bedding
Over–size barrel
FIGURE 14.44 Basic elements of detention basin.
Riprap energy dissipator
Clay core
Stormwater management stracture with hinged grate
Water surface
Protection
Flow direction
Maintenance Shoulder
14.78
Design Tailwater
Endwall
Geotextile Treatment
Emergency spillway
HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
Chapter Fourteen
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Hydraulic Design of Urban Drainage Systems 14.79
FIGURE 14.45 Detention–outlet structures.
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HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
14.80
Chapter Fourteen
14.9.1 Detention Basins The primary function of a detention basin is to control the quantity of stormwater runoff. Most stormwater management policies require that the postdevelopment peak flow rates be reduced to predevelopment peak flow rates for one or more specified design return periods such as 2, 10, and 25 years. Peak flow reduction is achieved by routing the postdevelopment runoff through a detention basin, that is by detaining the runoff temporarily in a basin. Figures 14.43 illustrates the effect of a detention basin on storm water runoff. The schematic diagram given in Fig. 14.44 shows the basic elements of a detention basin. In addition, sediment forebays are often used for partial removal of sediments from the stormwater runoff before it enters the detention basin. Energy dissipating structures such as baffle chutes are used at inlets. Most detention basins also have a trickle flow ditch or gutter in the bottom sloped towards the outlet to provide drainage of the pond bottom. 14.9.1.1 Detention basin design guidelines. Specific design criteria for detention basins vary in different local ordinances. Some general guidelines are summarized herein. Similar guidelines can be found elsewhere in the literature [ASCE, 1996; Federal Highway Administration (FHWA), 1996; Loganathan et al. 1993; Stahre and Urbonas, 1990; Urbonas and Stahre, 1993; Yu and Kaighn, 1992]. The main objective of a detention basin is to control the peak runoff rates. The outfall structures should be designed to limit the peak outflow rates to allowable rates. A detention basin should also provide sufficient volume for temporary storage of runoff. The inlet, outlet, and side slopes should be stabilized where needed to prevent erosion. The side slopes should be 3H/1V or flatter. The channel bottom should be sloped no less than 2 percent toward the trickle ditch. Detention basin length to width ratio should be no less than 3.0. Outlets should have trash racks. Coarse gravel packing should be provided if a perforated riser outlet is used. An emergency spillway should be built to provide controlled overflow relief for large storms. A 100-year storm event can be used for the emergency spillway design. 14.9.1.2 Outlet structures. Detention basin outlet structures can be of orifice-type, weirtype, or combinations of the two. Schematics of basic outlet structures are shown in Fig. 14.45. Discharge through an orifice outlet is calculated as Q koao2gH o,
(14.88)
where ao the orifice area, ko the orifice discharge coefficient, and Ho the effective head. If the orifice is submerged by the tailwater, Ho is the difference between the headwater and tailwater elevations. If the orifice is not submerged by the tailwater, it is assumed that Q O if the headwater is below the centroid of the orifice. Otherwise, Ho is set equal to the difference between the headwater elevation and the orifice centroid. This approximation is acceptable for small orifices. To account for partial flow in lanrge orifices, the inlet control culvert flow formulation discussed in Chap. can be used to determine the orifice flow rates. Short outflow pipes smaller than 0.3 m (1.0 ft) in diameter can also be treated as an orifice provided that Ho is greater than 1.5 times the diameter. Typical values of ko are 0.6 for square-edge uniform entrance conditions, and 0.4 for ragged edge orifices (FHWA, 1996). Weir-type structures include sharp-crested weirs, broadcrested weirs, spillways, and v-notch weirs. Flow over spillways, broad–crested weirs, and sharp crested weirs with no end contractions is expressed as Q kwLc2gHo1.5
(14.89)
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HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
Hydraulic Design of Urban Drainage Systems 14.81
where kw the weir discharge coefficient, Lc the effective crest length, and Ho the head over the weir crest. The weir discharge coefficient depends on the type of the weir and the head. Discharge coefficients for various types and structures are tabulated in Chapter. The head over the weir is the difference between the water surface elevation in the detention basin and the weir crest. For sharp crested weirs with end contractions Q kw (Lc 2Ho)2 gHo1.5
(14.90)
Q kv (185)2 g(tan φ2) Ho2.5,
(14.91)
and for V-notch weirs
where kv the V-notch discharge coefficient, φ the notch angle, and Ho the head over the notch bottom. Riser pipes act like a weir at low heads and like an orifice at higher heads. It is also possible that the flow will be controlled by the outflow barrel at even higher heads. In many applications, the outflow barrel is oversized to avoid the flow control by the barrel. In that case the outflow through the structure is calculated for a given head both using the weir and orifice flow equations, and the smaller of the two is used. If a trash rack is installed, the clear water area should be used in the calculations. It should be noted that many engineers design riser pipes so that orifice-type flow will not occur, because it is often observed that vortices form in the structure under orifice flow conditions. Sometimes antivortex structures are installed to avoid this problem. Multiple outlets are used if the design criteria require that more than one design storm be considered. Figure 14.46 displays schematics of several multipleoutlet structures. 14.9.1.3 Stage–storage relationships. The stage-storage relationship is an important detention basin characteristic. For regularshaped basins, this relationship is obtained from the geometry of the basin. For instance, for trapezoidal detention basins that has a rectangular base of W L and a side slope of z, the relationship between the volume, S, and the flow depth d is S L W d (L W) zd2 43 z2d3
(14.92)
FIGURE 14.46 Example multiple outlet structures.
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HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
14.82
Chapter Fourteen
For irregular-shaped detention basins, first the surface area, As, versus elevation, h, relationship is obtained from the contour maps of the detention basin site. Then As1 As2 S2 S1 (h2 h1) (14.93) 2 where S1 and As1 correspond to elevation h1, and S2 and As2 correspond to h2. Equation (14.93) is applied to sequent elevations. A more accurate relationship is h2 h1 S2 S1 s1 As2 (14.94) AS1 As2 A 3 The stage-storage relationship for most human-made and natural basins can also be approximated by
S bhc
(14.95)
where b and c fitting parameters. Figure 14.47 displays approximate relationships between the parameter b and c, the base area, length to width ratio, and the side slope for trapezoidal basins. 14.9.1.4 Detention pond design aids. The conventional procedure for the hydraulic design of a detention basin is a trial-and-error procedure, and it consists of the following steps: 1. Calculate the detention basin inflow hydrograph(s) for the design return period(s) being considered. A rainfall-runoff model, such as HEC1, TR-20, or SCSHYDRO, can be used for this purpose. For urbanizing areas, the inflow hydrograph(s) are normally those calculated for postdevelopment conditions.
Side slope (H/V) 12
34
Side slope (H/V) 12
80,000 70,000
34
80,000 70,000
bottom area (m2 or ft2)
60,000 50,000
bottom area (m2 or ft2)
60,000 50,000
40,000
40,000
30,000
30,000
20,000 10,000
20,000 10,000
FIGURE 14.47 Detention basin stage-storage parameters. (After Currey and Akan, 1998).
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Smax Qp 0.872 0.861 SR Ip
Smax Qp 0.932 0.792 SR Ip
td Tc Smax Iptd Qp( ) 2
Qp Qp 2 Qp 3 Smax 0.682 1.43 1.64 0.804 SR Ip Ip Ip
Qp Qp 2 Qp 3 Smax 0.660 1.76 1.96 0.730 SR Ip Ip Ip
Smax Qp 1 SR Ip
1.291(1 Qp / Ip)0.753 Smax (Tb/tp)0.411 SR Qp 2 Smax 1 SR Ip
Equation
Single
Single
Not specified
Not specified
Not specified
Orifice type
Weir type
Not specified
Not specified
Not specified
Not specified
Not specified
Not specified Not specified
Not specified
Outlet Types(s)
Not specified
Number of Outlets
Table 14.20 Design Aid Equations for Definition and Retention Storage
Constant reservoir surface area, valid for Q 0.2 p 0.9 Ip
Constant reservoir surface area, valid for Q 0.2 p 0.9 Ip
Trapezoidal inflow hydrograph, rising limb of outflow hydrograph is linear,
td storm duration, Tc time of concentration
For SCS 24-h Types II and III rainfall
For SCS 24-h Types I and IA rainfall
Triangular inflow and outflow hydrograph
Triangular inflow hydrograph, trapezoid outflow hydrograph
Based on numerical simulations Tb time base of inflow hydrograph
Remarks
Kessler and Diskin (1991)
Kessler and Diskin (1991)
Aron and Kibler (1990)
Soil Conservation Service (1986)
Soil Conservation Service (1986)
Baker (1979)
Abt and Grigg (1978)
Wycoff and Singh (1976)
Reference
HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
Hydraulic Design of Urban Drainage Systems 14.83
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Qp
ao
R
w
hmax
p
Qp
R
R
p
1/c
R 1/c
o.847 S 8.841 I S b
R
p
1.5/c
p
R
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p
p
R
0.5/c
b Q 0.847 S o.841 I S
p
Qp kO2g
hydrograph, S Currey and Akan Q max p 0.847 0.841 SR Ip
Lc
max
R
0.922 S 0.787 I S h b Q b k 2g Q 0.922 S 0.787 I S
Qp Smax 0.922 0.787 SR Ip
Qp 2 Qp 3 Smax Qp 0.97 1.42 0.82 0.46 SR Ip Ip Ip
Single
Single
Single
Number of Outlets
type
Single
Weir type
Orifice type
Weir type
Outlet Types(s)
acceleration
h stage, ao orifice area, ko = orfice discharge coefficient, g gravitational
stage-storage relationship: S bhc,
Orifice
Gamma function inflow hydrograph, stage-storage relationship: S bhc, h stage, Lc weir crest length, kw weir discharge coefficient, g gravitional acceleration
Gamma function inflow hydrograph
Gamma function inflow hydrograph
Remarks
(1998)
Gamma function inflow
Currey and Akan (1998)
McEnroe (1992)
McEnroe (1992)
Reference
14.84
Qp 2 Qp 3 Smax Qp 0.98 1.17 0.77 0.46 SR Ip Ip Ip
Equation
Table 14.20 (Continued)
HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
Chapter Fourteen
HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
Hydraulic Design of Urban Drainage Systems 14.85
2. Set the hydraulic design criteria. In most applications, the postdevelopment peak(s) are required to be reduced to the magnitude(s) of the predevelopment peak(s) for the design return period(s). If predevelopment peak(s) are not available, a rainfall–runoff model can be used to calculate them. The hydraulic design criteria may also restrict the maximum water surface elevation in the detention basin. 3. Trial design a detention basin. A trial design consists of the stage-storage relationship, and the types, sizes and elevations of the outlet structures. 4. Route the inflow hydrograph(s) through the trial-designed detention basin and check if the design criteria set are met. If not go back to Step 3. Also, if the criteria are met, but the outflow peak(s) are much smaller than the allowable value(s), then the trial basin is overdesigned. Again, go back to Step 3. The level-pool routing procedure is adequate for most detention basin design situations. This procedure is based on the solution of the hydrologic storage routing equation dS (14.96) I Q, dt where I inflowrate and t time. Unless a computer program is used, Eq. (14.96) is solved by employing a semigraphical method like the storage indication method, which can be found in any standard hydrology textbook. Obviously a good trial design is the key in this procedure. Designing a detention basin can become a tedious and lengthy task if the trial designed basins are not chosen carefully. Various charts and equations are available in the literature that can be used as trial design aids. Most of these aids are based on predetermined solutions to Eq. (14.96) in dimensionless form (Akan, 1989, Akan, 1990; Currey and Akan, 1998; Kessler and Diskin, 1991; McEnroe, 1992). Others are based on assumed inflow and outflow shapes (Abt and Grigg, 1978; Aron and Kibler, 1990) or results of numerous routings for many detention basins (Soil Conservation Service, 1986; Wycoff and Singh, 1976). Table 14.20 presents various design aid equations, where Ip the peak inflow rate (peak discharge of postdevelopment hydrograph), Qp the allowable peak outflow rate, Smax the required storage volume, and SR the volume of runoff. The use of these design aids can be illustrated through a simple-example. Suppose the rainfall excess resulting from a design rainfall is 3.5 in over a 2,178,000 ft2 urban watershed, and the runoff hydrograph has a peak of Ip 212ft3/s occurring at tp 30 min 1800 s. A detention basin is to be designed to reduce the peak flow rate to Qp 120 ft3/s. A weir-type outlet will be used that has kw 0.40. It is also required that the depth of water above the weir crest not to exceed 6.50 ft. A trapezoidal detention basin is suggested width a length-to-width ratio of 4 and sideslopes of 3H/1V. To size the required basin, let the surface area of the detention basin at the weir crest elevation be 40,000 ft2. Then from Fig. 14.47, b 42,500 and c 1.055. By definition, SR (3.5/12)(2,178,000) 635,250 ft3. Using the equations suggested by Currey and Akan (1998) from Table 14.20, we obtain Smax 302,715 ft3, hmax 6.43 ft, and Lc 2.29 ft. Note that hmax 6.50 ft., so the suggested basin with a base area of 40,000, ft2 should work. The Soil Conservation Service (SCS), (1986) equations given Table 14.20 are recommended if the standard SCS design rainfall hyetographs are to be used. Also, these equations are not restricted to single outlet detention basins. Suppose a detention basin is required to control the stormwater runoff for 2-year and 25-year events. Given for the 2year event are Ip 91 ft3/s, Qp 50 ft3/s, SR 408,480 ft3, and for the 25 year event Ip 360 ft3/s, Qp 180 ft3/s, SR 928,750 ft3. A two-stage weir outlet is suggested with kw 0.40, and the maximum water depth above the lower weir crest is not allowed to exceed 5 ft. The design is to be based on SCS 24-h Type II rainfall. Suppose a trapezoidal deten-
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HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
14.86
Chapter Fourteen
tion basin with a length-to-width ratio of 4 and side slopes of 3H/1V is suggested. From Table 14.22, Smax 928,750 [0.682 1.43(180/360) 1.64(180/360)2 0.804 (180/360)3] 256,750 ft3. Likewise, Smax 105,400 ft3 for the 2–year event. To determine the base area (or the surface area of the detention basin at the lower crest elevation), use Eq. (14.92) with S 256,800 ft3, L 4W, z 3, and d 5 ft. Solving the equation for W, we obtain approximately W 104 ft, and then L 416 ft. To size the lower crest for the 2-year event use Smax 105,400 ft3. Now substituting S 105,400 ft3, W 104 ft, L 416 ft and z 3 in Eq. (14.92) and solving for d, we obtain the maximum head over the lower crest for the 2-year event as being 2.25 ft. Next, using the weir equation (Eq. 14.89) with Q 50 ft3/s, kw 0.40, h 2.25 ft, and g 32.2 ft/s2, we obtain Lc 4.61 ft for the lower crest. Let the upper crest be placed 2.30 ft above the lower crest. To size the upper crest, the 25-year event is considered. The maximum head over the lower crest will be 5 ft. At this head the lower crest will discharge 165 ft3/s [from Eq. (14.89)]. Therefore, the upper crest should be sized to pass (180 165) 15 ft3/s under a head of (5.00 2.30) 2.70 ft. From the weir formula [Eq. (14.89)] we obtain Lc 1.05 ft for the upper crest.
14.9.2 Extended Detention Basins Extended detention basins are effective means of removing particulate pollutants from urban storm water runoff. As shown in Fig. 14.48, an extended detention pond has two stages. The bottom stage is expected to be inundated frequently. The top stage remains dry except during large storms.
FIGURE 14.48 Extended detention basin. (After Schueler, 1987).
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HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
Hydraulic Design of Urban Drainage Systems 14.87
14.9.2.1 Detention volume and time. An extended detention basin is designed to detain a certain quantity of runoff, sometimes referred to as the water quality volume, for a certain period of time to achieve the targeted level of pollutant removal. The volume to be detained and the duration over which this volume to be released vary in different stormwastet management policies. For example, Hampton Roads Planning District Commission (1992) requires that a quantity of runoff calculated as 0.5 inch times the impervious watershed area be released over 30 h in southeastern Virginia. Prince George County Department of Environmental Resources (1984) requires the runoff volume generated from the 1year, 24-hour storm be released over a minimum of 24 h. American Society of Civil Engineers (1998) outlines a procedure to size extended basins serving up to 1.0 km2 (0.6 m3) watersheds. In this procedure, the volume of water to be detained per unit watershed area, Po, is estimated as Po ar(0.858i3 0.78i2 0.774i 0.04)P6,
(14.97)
where ar a regression coefficient, i the watershed imperviousness expressed as a fraction, and P6 the mean storm precipitation depth that can be obtained from Fig. 14.49. The value of the regression coefficient ar is 1.109, 1.299, and 1.545 for detention volume release times of 12, 24, and 48 h, respectively. Interpolation of these values is allowed for durations between 24 and 48 h. The use of this procedure can be illustrated by a simple example. Suppose an extended detention basin is to be designed for a 150-acre watershed in Norfolk, Virginia that is 40 percent impervious. Determine the required size if the detained runoff is to be released over 36 hours. From Fig. 14.49, P6 0.67 in for Norfolk, Virginia. Because the watershed is 40 percent impervious, i 0.40. Also, interpolating the ar values between 24 and 48 h, ar 1.422 for 36 h. Then, from Eq. (14.97), we obtain Po 0.27 in. Therefore, the volume of runoff to be detained is (150) (0.27/12) 3.38 aq-ft 147,233 ft3. It is advisable to increase this volume by about 20 percent for sedimentation.
FIGURE 14.49 Mean storm precipitation depth in inches. (After ASCE, 1998).
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HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
14.88
Chapter Fourteen
FIGURE 14.50 Extended detention pond outlets. (After Schueler, 1987).
14.9.2.2 Extended detention outlet structures. The outlets for extended detention basins are designed to slowly release the captured runoff from the basin over the specified emptying time to allow settling of particulate pollutants. We sometimes refer to these outlets as water quality outlets. Low-flow orifices are often used as outlet structures. Figure 14.50 displays various methods for extending detention times. As pointed out by Schueler (1987) and ASCE (1998), however, extended detention outlet structures are generally prone to clogging. This makes the design of outlet structures difficult since the hydraulic performance of a clogged outlet will be uncertain and different from what it is designed for. Regular cleanouts must be performed. A hydrograph routing approach is probably the best way to size an extended detention basin and the water quality outlet. However, this requires an inflow hydrograph. In practice, as discussed in the preceding section, only the volume of captured runoff is considered for pollutant removal. There are no broadly accepted procedures to convert this volume to an inflow hydrograph. Therefore, the water quality outlets are often sized by using approximate hydraulics. This can be illustrated by a simple example. Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
Hydraulic Design of Urban Drainage Systems 14.89
Suppose an extended detention basin has a bottom length of 80 ft, a width of 20 ft, and side slopes of 3:1(H:V). The outlet is to be sized so that it will release a water quality volume of 10,200 ft3 over a period of 40 h. To determine the depth of water corresponding to this volume, Eq. (14.92) is written as 10,200 (80)(20)d (80 20)3d2 (4/3)32d3. By trial and error, d 3.6 ft. Let the outlet structure be comprised of 1/2-in circular ragged edge orifice holes cut around a riser pipe. Let the average elevation of the holes be 1 ft above the pond bottom. Therefore, the average head over the orifice holes is (3.6 1.0)/2 1.3 ft. To empty 10,200 ft3 over 40 h 144,000 s, the average release rate is 10,200/144,000 0.0708 ft3/s. Noting that the orifice area of a 1/2in hole is 0.00136 ft2, and ko 0.40 for ragged edge orifices, we can write Eq. 14.88 as 0.0708 N(0.40)(0.00136)2 (3 2.2 )1.3 , where N is the number of orifice holes. Solving for N we obtain N 14.22. Therefore, we use 14 holes evenly distributed. 14.9.2.3 Extended detention basin design considerations. Additional design considerations for extended detention basins can be found in various publications (ASCE, 1998; FHWA, Schueler, 1987; Urbonas and Stahre, 1993). Briefly, the basin should gradually expand from the inlet, toward the outlet. A length-to-width ratio of 2 or higher is recommended. Side slopes should not be steeper than 3:1 (H:V) and flatter than 20:1 (H:V). A riprap, concrete, or paved low-flow channel is required to convey trickle flows. A twostage design is recommended with a 1.5- to 3.0 ft-deep bottom stage and a 2.0- to 6.0-ft-deep upper stage. A wetland marsh created in the bottom stage will help remove soluble pollutants that cannot be removed by settling. The detention basin inlet should be protected to prevent erosion. If the outlet is not protected by a gravel pack, some form of trash rack should be used. A sediment forebay is recommended to encourage sediment deposition to occur near the point of inflow
14.9.3 Retention Basins Retention basins or wet ponds retain a permanent pool during dry weather as shown in Fig. 14.51. A high removal rate of sediment, biological oxygen demand (BOD), organic nutrients, and trace metals can be achieved if stormwater is retained in the wet pond long enough. During wet weather, the incoming runoff displaces the old stormwater from the permanent pool from which significant amounts of pollutants have been removed. The new runoff is retained until it is displaced by subsequent storms. The permanent pool therefore will capture and treat the small and frequently occurring stormwater runoff which generally contain high levels of pollutant loading. The storage volume provided above the permanent pool is used to control the runoff peaks caused by the specified design storm events. 14.9.3.1 Permanent pool volume. Among all the factors influencing the pollutant removal efficiency of a retention basin, the size of the permanent pool is the most important. As pointed out by Schueler (1987), in general, “bigger is better.” However, after a threshold size is reached, further removal by sedimentation is negligible.
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HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
14.90
Chapter Fourteen
FIGURE 14.51 Retention basin. (After Yu and Kaighn, 1992).
The required size of the permanent pool in relation to the contributing watershed area varies in different stormwater management policies. For example, FHWA (1996) and Yu and Kaign (1992) recommend a permanent pool size three times the water quality volume defined for extended detention basins. Montgomery County Department of Environmental Protection (1984), Maryland, requires a volume greater than 0.5 in times the total watershed area. ASCE (1998) recommends that Eq. (14.97), be used with a drain time of 12 hours to determine the permanent pool volume. It is also recommended that a surcharge extended detention volume, equal to the permanent pool volume, be provided above the permanent pool. U.S. Environmental Protection Agency (1986) provides geographically based design curves to determine the permanent pool surface area as percent of the contributing watershed area (see Fig. 14.52). Hartigan (1989) and Walker (1987) treat a retention basin as a small euthrophic lake and employ empirical models to size the retention pond. This procedure is outlined by ASCE (1998). U.S. Environmental Protection Agency (1986) presented a procedure to evaluate the long-term pollutant removal efficiency of retention basins depending on the basin size and the rainfall statistics of the project area. This procedure was developed by DiToro and Small (1979), and is outlined in various publications (Akan, 1993; Stahre and Urbonas, 1990; Urbonas and Stahre, 1993). 14.9.3.2 Retention basin design considerations. Wet ponds can be designed to control the peak runoff rates from rare and large storm events if additional storage volume is provided above the permanent pool. The size of the additional volume can be determined by using the procedures described for detention basins. The outlet structures for retention basins include a low flow outlet to control the runoff from frequent storm events and overflow devices to control the runoff from larger storms.
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HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
Hydraulic Design of Urban Drainage Systems 14.91
FIGURE 14.52 Design curves for solids settling. for low–density residential land use. (After USEPA, 1986).
FIGURE 14.53 Retention basin outlet structures. (After Schueler, 1987).
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HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
14.92
Chapter Fourteen
Typical outflow structures are shown in Fig. 14.53 (Schueler, 1987). Additional design considerations have been presented by Schueler (1987). In summary, the pond should be wedge-shaped, narrowest at the inlet and widest at the outlet. A minimum length to width ratio of 3:1 should be used. The pond depth should average 3–6 ft, with a shallow underwater bench around the pond’s perimeter. Side-slopes should be no steeper than 3:1 (H:V) and not flatter than 20:1 (H:V). If the soils at the pond site are highly permeable, the pond’s bottom should be lined by impervious geotextile or a 6-in clay liner. The inlets and outfalls should be protected by riprap or other means to prevent erosion. Wet ponds should be surrounded by a 25-ft buffer strip planted with water-tolerant grasses and shrubs. A sediment forebay should be constructed near the inlet of the pond with extra storage equal to the projected sediment trapping over a 20 to 40-year period.
14.9.4 Computer Models for Detention and Retention Basin Design As discussed in the preceding sections, a trial-and-error procedure is used for hydraulic design of retention and detention basins. A basin is first trial-designed, and then the design hydrographs are routed through the basin to verify if the design criteria are met. Therefore, any reservoir routing computer program can be useful for designing detention and retention basins. The widely known TR-20 (Soil Conservation Service, 1986) and HEC-1 (Hydrologic Engineering Center, 1990), for instance, have reservoir routing schemes and can be used for pond design. These models are in public domain. Commercially available pond routing software are a lot more user-friendly, and they include Watershed Modeling Standalone (www.eaglepoint.com). The commercially available models allow a variety of different outlet structures and simulation of multiple storm events. Also available are PONDOPT (www.cahh.com) and BASINOPT (www.cahh.com) which include an analysis option for reservoir routing as in the other pond models. These two models also have a unique design option which performs all the iterations internally. The ponds are sized and the outlet invert elevations and sizes are determined by the program for multiple–return periods.
14.10 SEWER HYDRAULIC SIMULATION MODELS A model is defined here as a method or simulation algorithm that has been coded into a computer program for computations and applications. Numerous models have been developed for sewer networks. These sewer models can be classified in different ways as follows: 1. According to the purpose of the model: (a) design models—hydraulic design, or Optimal design, risk-based design; (b) evaluation/predictions models, (c) planning models. 2. According to the objective of the project: (a) flood control or (b) pollution control. 3. According to the extent of space consideration: (a) overland surface only, (b) sewer system only, or (c) sewer system and overland surface. 4. According to the nature of wastewater: (a) sanitary sewer models or (b) storm and combined sewer models. 5. According to water-quality considerations: (a) quantity only, (b) quality only (rare), or (c) quantity and quality. 6. According to time considerations of rainfall input: (a) single-event models or (b) multiple-event continuous models. Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
Hydraulic Design of Urban Drainage Systems 14.93
7. According to probability considerations: (a) deterministic or (b) probabilistic—pure statistical or stochastic. 8. According to systems concept: (a) lumped system or (b) distributed system. 9. According to hydrologic principles considered: (a) hydrologic (principle of mass conservation) or (b) hydraulic (principles of conservation of mass and momentum or energy). In the first classification, the design models are for the determination of the size of the sewers and perhaps also their slope and layout of either a new sewer systems or an extension or modification of an existing system. The evaluation/prediction models are those used to simulate the flow in an existing or predetermined sewer system for which the size, slope, and layout are already specified. Their use is to compute the flow in the sewers to check the adequacy of sewer capacity, system performance, operation, management of pollution abatement, flood mitigation, and so forth. Or, the model may be incorporated as part of a real-time operation system. The planning models are those models used for strategy planning and decision making for urban or regional storm and waste water management, usually applied to a larger time and spatial frame than the design or evaluation models. The design models design the sewers in a network for a hypothetical future event which is represented by the design storms of specified return period or risk level. The evaluation/prediction models simulate the runoff produced by a rainstorm of the past, present, future, or the flow from other sources. The planning models usually consider a relatively long continuous period of time covering many rainstorms and dry periods in between. The planning models utilize the least hydraulic consideration of flow on overland areas and in sewers. Often, a simple water budget balance suffices. A typical example is the STORM model (Hydrologic Engineering Center, 1974). Supposedly, for the purpose of reliable flow simulation, the evaluation/prediction models require the highest level of hydraulic sophistication and accuracy. However, many lower level models do exist. Due primarily to the discrete sizes of commercial pipes, usually a moderate level of hydraulics is adequate for the design models (Yen and Sevuk, 1975; Yen et al., 1976). Most of the existing sewer models are evaluation/prediction models. Aside from the design models derived from the rational method, there are actually very few true sewer design models; among them only two models, ILSD (Yen et al., 1976, 1984) and WASSP (Price, 1982b), have published user’s guides and arrangements for release of programs. Some of the evaluation/prediction models have the ability to compute the diameter required for gravity flow of a specific discharge. However, they are not true design models because different sewers should be designed for different rainstorms of different durations corresponding to the different time of concentration of the sewers. Hence, many computer runs are required to complete the design of a network using these models. In the last classification, the hydraulic models can further be classified according to the level of hydraulics shown in Eq. (14.1) or (14.2) as follows: dynamic wave models, noninertia models, nonlinear kinematic wave models, and linear kinematic wave models. It is impossible to summarize and report the hydraulic properties of all the exiting sewer models in this chapter. Therefore, only selected models are made in this presentation. Since this article deals with the hydraulics of sewers, in the following section, only the hydraulics of selected models are discussed. For models that allow more than one hydraulic level for flow routing, they are presented according to their respective highest hydraulic level. For information and comparison of the nonhydraulic aspects of the models, the reader should refer to other references such as those by Brandstetter (1976), Chow and Yen (1976), and Colyer and Pethick (1976) in addition to the original model developers’ reports or papers. Models without hydraulic consideration of the sewer flow, such as
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HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
14.94
Chapter Fourteen
the rational method models, are excluded. When water quality transport simulation is sought, nearly all the models perform the flow routing first and allow another pollutant transport model—usually in concentration form—coupled with the routing result for simulation. Only the Storm Water Management Model (SWMM) has the quality portion integrated in the model as a modular block.
14.10.1 Hydraulic Properties of Selected Dynamic Wave Sewer Models Wellknown models, in which the highest hydraulic level, the dynamic wave simulation is employed, are listed in Table 14.21. In the table, the subscript o denotes the sewer receiving outflow from the junction. All these models were developed for flow simulation rather than for design of sewers in a network. CAREDAS, UNSTDY, HYDROWORKS, and MOUSE are proprietary models. Among the nonproprietary models, only two [ISS and Stormwater Management Models Extended Transportation (SWMM-EXTRAN)] have user’s manuals published and available to the public. For dynamic wave and noninertia models, the junction conditions and surcharge transition conditions—if surcharge is allowed—are important for reliable and realistic simulation of the flow. However, for most of the models listed in Table 14.21, information about the details and assumption on the surcharge transition and on junctions is inadequately given. Also, except ISS which cannot handle flow having a Froude number greater than 1.6, it is not known whether the other models can handle supercritical flow with roll waves, and if so, what assumptions are involved. In the following, dynamic wave models listed in Table 14.21 are briefly discussed in three groups, namely, the explicit scheme model (SWMM-EXTRAN), the models that handle only open-channel sewer flows, and the models that handle both open-channel and surcharge sewer flows. The allowed network size given in Table 14.21 is that indicated in the quoted literature. For most models, this number has been increased with later developments. 14.10.1.1 Explicit scheme model: SWMM-EXTRAN. The Storm Water Management Model (SWMM) developed under continuous support of the U.S. Environmental Protection Agency is one of the best known among all the sewer models. The Extended Transport block (EXTRAN) (Roesner and Shubinski, 1982; Roesner et al., 1984) was added to the SWMM Version III to provide the model with dynamic wave simulation capability. The entire sewer length is considered as a single computational reach, and the dynamic wave equation is written in backward time difference between the time levels n 1 and n for the sewer, and expressed explicitly as 1 gn2∆t Au,n Ad,n Qn 1 1 |V | Qn 2V n ∆A V n ∆t n 4/3 2. 2 1 R L n
hu,n hd,n gA n ∆t L
(14.98)
where all the symbols have been defined previously, the subscript u the upstream end of the sewer (that is, entrance) and d the downstream end of the sewer (that is, exit), the bar indicates the average of values at the entrance and exit, and presumably ∆A An1 An is also the average of the values at the sewer ends. The junction condition used is the continuity equation, Eq. (14.47), expressed explicitly in terms of the depth and discharge values at the time n∆t as
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Sewer Downstream Condition
Solution Scheme
Interior Junction
All pipes entering a junction are full or the highest entering pipe is submerged
ΣQ ds/dt Q Qf and ho H (KV2/2g) or ΣQ 0 and hi ho
DAGVL-A Implicit, h, Q Manning Junction water Simultaneous Yes continuity surface or (double sweep) six-point, critical depth momentum four-point, w 0.55
h/d 0.91
ΣQ ds/dt and NA ho H (V2/2g) or ΣQ O and hi ho
ΣQ ds/dt and hi ho
Transition Condition
ΣQ ds/dt and hi ho
Characte- h, V DarcyJunction water Simultaneous Yes ristic Weisbach surface or on overlapping critical depth segments; pipe by pipe
Yes
Detention Equations Storage
CAREDAS Four-point h, Q Chezy or Junction water Simultaneous Yes implicit Manning surface (double sweep)
ISS
Sf
h, Q Manning Junction water One sweep, surface or pipe by pipe assumed condition
Numerical Para Scheme meters
Open-Channel Flow
NA
—
Preissmann slot
Preissmann slot
—
Sevuk et al. (1973); Sevuk and Yen (1982)
26
Sjöberg (1976, 1982)
28 (for Chevereau dynamic et al. (1978); wave) Cunge and Mazaudou (1984)
54
Roesner and Shubinski (1982); Roesner et al. (1984)
Reported or Programmed
Network References size
One sweep, 200 pipe by pipe
Numerical Solution Scheme Scheme
Surcharge Flow Surcharge Hydraulics Employ open- Explicit channel equations with junction head computed using assumed adjustment factors, excess water lost
Summary of Hydraulic Properties of Selected Dynamic Wave Sewer Network Models
SWMM- Explicit EXTRAN
Model
TABLE 14.21
HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
Hydraulic Design of Urban Drainage Systems 14.95
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Interior Junction Detention Equations Storage
Transition Condition
Four-point h, Q Manning Junction water Simultaneous implicit, surface w 0.55, 0.6, 0.75, or 1.0 Yes ΣQ ds/dt and Q Qf or ho H (KV2/2g) submerged or ΣQ 0 exit and hi ho
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Four-point h, Q Colebrook-Junction water Simultaneous Yes ΣQ ds/dt implicit White or surface or (double sweep) and Manning critical depth ho H (KV2/2g)
SPIDA/ HYDROWORKS
.
Six-point implicit, w 0.5
MOUSE
h, Q Manning Junction water Simultaneous Yes ΣQ ds/dt Not given surface (double sweep) and ho H (KV2/2g) or ΣQ 0 and hi ho
SURDYN Four-point h, Q Manning Junction water Simultaneous implicit, surface or w 0.55 critical depth
Joliffe
Not given
Solution Scheme
No ΣQ 0 and hi ho
Sewer Downstream Condition Not given
Sf
UNSTDY Four-point h, Q Manning Junction water Simultaneous Yes ΣQ ds/dt implicit surface or (double sweep) and hi ho sluice gate
Numerical Para Scheme meters
Open-Channel Flow
40
300
size
Preissmann slot
Preissmann slot
Abbott et al. (1982); DHI (1994); HoffClausen et al. (1982) 5000 Wallingford Software (1991, 1997)
(87)
Pansic (1980)
Joliffe (1984a, b)
Book et al. (1982); Chen and Chai (1991); Labadie et al. (1978)
Reported or Programmed
Network References
Simultaneous 10
Preissmann slot
Preissmann slot
Numerical Solution Scheme Scheme
Quasi-steady Implicit dynamic, junction head losses considered
Surcharge Hydraulics
Surcharge Flow
(Continued) Summary of Hydraulic Properties of Selected Dynamic Wave Sewer Network Models
14.96
Model
TABLE 14.21
HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
Chapter Fourteen
HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
Hydraulic Design of Urban Drainage Systems 14.97
Hn 1 Hn ∆t (∑Qi,n Qj,n) Aj
(14.99)
where the subscript j indicates junotion. The junction dynamic relation is simplified as a common water surface [Eq. (14.51)]. Equations (14.98) and (14.99) are solved explicitly by using a modified Euler method and half-step and full-step calculations. Courant’s stability criterion is adopted to select the computational ∆t. In EXTRAN, when a junction is surcharged, instead of properly applying the continuity equation (Eq. 14.53), it assumes the point-type junction continuity relationship (Eq. 14.50) applies. On the basis of this point-type junction continuity equation, an expression of the junction water head is derived through an improper application of the chain rule of differentiation, for which a Taylor expansion would have been more appropriate. The unsatisfactory result was apparently recognized, and remedies were attempted through the introduction of an adjustment factor and the assumption on the numerical iterations to either reach a maximum number set by the user or the algebraic sum of the inflows and outflows of a junction being less than a tolerance. In an earlier version of EXTRAN that was applied to a project in San Francisco, California, an attempt was made to artificially modify the geometry of the junction so that numerical solution could be obtained. The SWMM-EXTRAN, with its explicit difference formulation, solves the flow sewer by sewer by using the one-sweep explicit solution method with no need for simultaneous solution of the sewers of the network. Therefore, it is relatively easy to program. Nonetheless, because of the assumptions on the surcharge condition, and also the stability and convergence (accuracy) problems of the explicit scheme for the open-channel condition, on a theoretical basis EXTRAN is inferior to other dynamic wave models listed in Table 14.21. The other models, of course, have their share of problems concerning the assumptions on the transitions between open–channel and surcharge flows, between supercritical and subcritical flows, and on roll waves. 14.10.1.2 Dynamic wave model handling only open-channel flow: ISS. The Illinois Storm Sewer System Simulation (ISS) model (Sevuk et al. 19973) solves the dynamic wave equation using the first-order scheme of the method characteristics. The SaintVenant equations [Eqs. (14.2) and (14.5)] or similar type partial differential equations are transformed mathematically into two sets of characteristic equations, each set consisting of a pair of ordinary differential equations which are solved numerically using a semiimplicit scheme. The formulation can be found in Sevuk and Yen (1982). The junction conditions used for a storage junction are Eq. (14.47), together with the equations in Table 14.15, for sewer exits, and for sewer entrances Eq. (14.48) with Ki 0, that is, H (V2/2g) h Z (14.100) For a point junction, the equations are Eq. (14.50), together with Eqs. (14.51) or (14.52). The ISS model program considers direct backwater effects for up to three sewers in a junction. For junction with more than three joining sewers, the excess sewers (preferably those with small backwater effects from the junction) are treated as direct inflow, that is, Qj in Eqs. (14.47) or (14.50). The flow in the network is solved by using the overlapping segment method. The outlet of the network can be any one of the following: (1) a free fall, (2) flow continuing to approach normal flow, (3) a stage hydrograph h f(t), (4) a rating curve Q f(h), (5) a velocity-depth relationship v f(h), and (5) a discharge-time relationship Q f(t).
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HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
14.98
Chapter Fourteen
When used to compute the required pipe diameter of a sewer, ISS is the only model among those listed in Table 14.21 that uses a maximum depth criterion to ensure gravity flow in the sewer for the design situation. Other models compute the required pipe diameter on the basis of the peak discharge that does not guarantee gravity flow because, for unsteady flow, maximum depth usually does not occur at the same time as the maximum discharge in a sewer. The ISS model can easily be modified to account also for surcharge flow by adding the Preissmann hypothetical slot. 14.10.1.3 Dynamic wave models handling both open-channel and surcharge flows. Among the seven models belonging to this group listed in Table 14.21, four of them— CAREDAS, UNSTDY, Joliffe, and HYDROWORKS—are numerically similar, using a four-point implicit scheme and adopting the Preissmann fictitious open slot to simulate surcharge flow. Details of the four-point implicit scheme can be found in Liggett and Cunge (1975) and Lai (1986). In fact, the same four-point numerical scheme is also used in SURDYN (Pansic, 1980). SURDYN is the only model in this group of seven that simulates the surcharge flow by using the standard pressurized conduit approach and solving it simultaneously with the open–channel flow. The surcharge equation used in this model is a quasi-steady dynamic equation obtained by dropping the local acceleration (∂V/∂t) term in Eq. (14.2). For the rising transition from open-channel flow, surcharge is assumed to occur when the discharge exceeds Qf or when the pipe exit is submerged. Falling transition from surcharge to open–channel flow is assumed to occur when the pipe entrance is not submerged, when the discharge falls less than Qf or when the pipe exit is not submerged. Pansic (1980) reported that the model simulates the unsteady flow reasonably well. But oscillations often occur at transitions between open-channel and surcharge conditions. This oscillation problem is partly numerical, partly hydraulic, and partly due to assumptions. Among these models, HYDROWORKS, MOUSE, UNSTDY, and CAREDAS are proprietary. They are briefly introduced in the following: 1. HYDROWORKS. The dynamic wave sewer flow routing option of HYDROWORKS is based on an earlier model SPIDA from the same company, Hydraulics Research, in England. HYDROWORKS also contains noninertial (WALLRUS) and nonlinear kinematic wave (WASSP-SIM) sewer routing options. The model can handle a looped-type network as well as a dendritic type. For dynamic wave routing, the inertia terms are linearly phased out from a Froude number equal to 0.8-1.1. Essentially, for supercritical flow, the noninertia approximation is used. For pressurized flow, the hypothetical slot width is assumed one-twentieth of the maximum pipe diameter. 2. MOUSE. This model was upgraded from Danish Hydraulic Institute’s (DHI) System 11-sewer (S11-S) model. It was first released in 1985 and subsequently updated with personal computer PC technology advancements. It uses the Abbott-Ionescu six-point implicit scheme (Abbott and Basco, 1990) which is relatively stable and consistent but costly in computation. The model allows loop network. In addition to dynamic wave routing, it also has noninertia (identified in the model as diffusion wave) and kinematic wave routing options for sewers. 3. UNSTDY. The UNSTDY model uses four-point noncentral implicit schemes to solve the Saint-Venant equations for subcritical flow. Supercritical flow is simulated by using the kinematic wave approximation. The model can solve a looped network in the system.
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Four-point implicit
NISN
h, Q
Manning
Manning
Six-point implicit w 0.5
h, Q
Sewer Downstream Condition
Junction water surface or critical depth
Junction water surface
Junction water surface or critical depth
Colebrook- Unspecified or White rating curve
Sf
MOUSE
h, Q
Parameters
Open-Channel Flow
Manning
Numerical Scheme
Simultaneous, over-lapping segment
Simultaneous (double sweep)
Simultaneous (double sweep)
Pipe by pipe
Solution Scheme
Yes
Yes
Yes
No
Standard pressurized pipe flow
Preissmann slot
Preissmann slot Preissmann slot
ΣQ ds/dt and ho H (KV2/2g) or ΣQ 0 and hi h0 ΣQ 0 and hi h0ΣQ 0 and hi h0 or ΣQ ds/dt and ho H (KV2/2g)
Flow
Surcharge
ΣQ 0 and hi ho
Junction Equation
Interior Junctions Detention Storage
Summary of Hydraulic Properties of Noninertia Sewer Network Models
DAGVL- Implicit, six-point h, Q DIFF continuity four–point momentum w 0.55
HVM
Model
TABLE 14.22
Pagliara and Yen (1997)
DHI (1994)
Sjöberg (1982)
Geiger and Dorsch (1980); Klym et al. (1972); Vogel and Klym (1973)
References
HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
Hydraulic Design of Urban Drainage Systems 14.99
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HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
14.100
Chapter Fourteen
4. CAREDAS. This is one of the earliest full dynamic wave sewer flow routing models developed by SOGREAH at Grenoble, France. This is the first model to incorporate the Preissmann slot to simulate surcharge flow. In applying CAREDAS, a sewer network is first checked for the sewers with sufficiently steep slope for which the kinematic wave equation can be applied as an adequate approximation. The dynamic wave model is applied to each group of the connected, gently sloped sewers.
14.10.2 Hydraulic Properties of Noninertia Sewer Models The noninertia approximation of the unsteady flow momentum equation [Eq. (14.1)] is probably the most efficient option among the dynamic-wave momentum equation options to solve unsteady open-channel sewer flow problems. It accounts for downstream backwater effect, and it allows reversal flow. Computationally, it is much simpler than the full dynamic wave option. It is only for rare highly unsteady cases that the noninertia option is inadequate and the full dynamic wave or the exact momentum options are required. However, only a few noninertia sewer models have been developed; only four are reported in the literature and they are summarized in Table 14.22. The proprietary HVM-QQS model was developed by Dorsch Consult (Klym et al., 1972; Vogel and Klym, 1973) at Munich, Germany. It has been misquoted as a dynamic wave model (Brandstetter, 1976). Examination of the equations [Eqs. (3) and (4) in Vogel and Klym, 1973] reveals that, in fact, it is a noninertia model. It was stated that to avoid simultaneous solution of all the sewers in the network, further assumptions were made. One assumption is to let Sf So(Q/Qo)2, where Qo is defined as a normal flow discharge corresponding to So, but it is not clear what depth is used in computing Qo. Another issue that the sewer downstream boundary condition at the exit is either unspecified or a rating curve h h(Q), or the exit depth hydrograph h(t) is known. In fact, with unspecified downstream boundary condition, this model does not really account for the backwater effect, and thus, it omits one of the important advantageous properties of the noninertia model. No information is given on whether the flow equations are solved implicitly or explicitly. The DAGVL-DIFF model was developed at the Chalmers University of Technology (Sjöberg, 1982) at Göteborg, Sweden. The equations in the model are solved in a manner similar to the dynamic wave model DAGVL-A and were found generally satisfactory. No further development or support of the DAGVL models has been provided since the development of S11S/MOUSE. The proprietary DHI (1994) model MOUSE contains noninertia and kinematic wave sewer routing options in addition to dynamic wave routing. The noninertia option simulates the flow the same way as the dynamic wave option; thus, it does not take full advantage of the simplicity and computational efficiency of the noninertia modeling. The NISN model (Pagliara and Yen, 1997) utilizes the overlapping segment method to solve for the flow in a network. For each segment, the flow equations are solved simultaneously using the Preissmann four-point implicit scheme. Junction storage and headloss are allowed. There is no network size limit for this model.
14.10.3 Nonlinear Kinematic Wave Models Unlike the dynamic wave and noninertia sewer models, there exist many kinematic wave models. Only a few nonlinear kinematic wave models are listed in Table 14.23 for dis-
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Nonlinear kinematic wave
Nonlinear kinematic wave
MuskingumCunge
MuskingumCunge
Improved nonlinear kinematic wave
ILSD-B2
ILSD-B3
WASSPSIM
SWMMTRANSPORT
A, Q
h, Q
h, Q
h, Q
A, Q
Manning
Manning
Manning
Sf
Manning
Cascade
NA
ΣQ 0 NA
NA
Q Qf
Four-point One sweep, ΣQ 0 Q Qf implicit, pipe by unless storage w 0.55 pipe block is used
Q Qf
Solution Scheme
Surcharge Flow References
NA
NA
Yen et al. (1976)
Yen and Sevuk (1975), Yen et al. (1976)
Store excess Pipe by pipe Huber and water, release Heaney later (1982), Huber et al. (1984); Metcalf & Eddy Inc. et al (1971)
H calculated, Implicit Price headlosses simultaneous (1982a, b) considered relaxation
NA
NA
Store excess Pipe by pipe Dawdy et al. water, release (1978) later
Surcharge Interior Hydraulics Juntion
Q Qf Unsteady or assumed dynamic submergence equation conditions
NA
ΣQ 0
Q Qf
ΣQ ds/dt
Transition Condition
Junction
Interior
One sweep, ΣQ ds/dt pipe by pipe
Four-point Cascade implicit
Four-point Cascade implicit
Explicit
Numerical Solution Scheme Scheme
Open-channel flow
DarcyQuasi Weisbach explicit and ColebrookWhite
Sewer Parameters Hydraulics
Sewer Hydraulic Properties of Selected Nonlinear Kinematic Wave Models
USGS
Model
TABLE 14.23
HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
Hydraulic Design of Urban Drainage Systems 14.101
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HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
14.102
Chapter Fourteen
cussion. All of the models listed in these tables, except the USGS model, have provision to compute the required diameter for a specified discharge using the Manning or DarcyWeisbach formula. All the nonlinear kinematic wave models listed in Table 14.23 consider the backwater effect from upstream (entrance) of the sewer within the realm of a single sewer and not beyond, and not the backwater effect from downstream (sewer exit). The kinematic wave models, unable to compute reliably the sewer flow cross-section area A, depth h, and velocity V, are of questionable usefulness in coupling with a waterquality equation for water-quality evaluation. Unless the downstream backwater effect is always insignificant, otherwise a water-quantity model having a hydraulic level of noninertia approximation or higher should be used. Nonlinear kinematic wave models may be classified further according to the manner the flow equations are formulated for solution. The first group includes the models solving directly the nonlinear kinematic wave equations. The first two models in Table 14.23, [U.S. Geological Survey’s Distributed Routing Rainfall-Runoff Model (USGS) (Dawdy et al., 1978)] and [Illinois Least-Cost Sewer System Design Model, option B2 (ILSD-B2) (Yen et al., 1976)] belong to this group. The second group includes the models that solve an explicit linear algebraic equation of the Muskingum equation form. The Illinois Least-Cost Sewer System Design Model, option B3 (ILSD-B3) (Yen et al., 1976) and the British Hydraulics Research’s Wallingford Storm Sewer Design and Analysis Package Simulation Method (WASSP-SIM) (Price, 1982a,b) belong to this group. The third group consists of the models using other modified nonlinear kinematic wave equations for solution such as the TRANSPORT Block in SWMM (Metcalf and Eddy. et al., 1971). 1. ILSD-B2 and USGS models. In the first group, the continuity equation is written as a finite difference algebraic equation of one variable (usually h or Q) or two variables (e.g., h, Q or A, Q) and solved iteratively with the aid of the simplified momentum equation, So Sf, where Sf is approximated by Manning’s or similar formulas to relate the depth or area to discharge. A formulation used in Yen and Sevuk (1975) and adopted in ILSD-B3 is given in the following as an example. Noting that B(h) ∂A/∂h and G(h) ∂Q/∂h, Eq. (14.4) can be rewritten as ∂h ∂h B(h) G(h) 0 ∂t ∂x For partially filled circular pipes (Fig. 14.3), φ B(h) D sin 2 and by using Manning’s formula
(14.101)
(14.102)
sin φ K 1 G(h) n So1/2R2/3 B 5 2 1 n 3 sin (φ/2) φ
0.132K sin φ n So1/2D5/3 1 n φ
2/3
5
(14.103)
sinφ 1 sin φ 1 sin (φ/2) φ 2
where the central angle φ in radians is (Fig. 14.3): (14.104) φ 2 cos 1 [1 (2h/D)] Consider the four computational grid points boxed by the time levels n and n 1 and space levels i and i 1, Eq. (14.101) can be transformed into the following implicit four-point forward–difference equation:
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HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
Hydraulic Design of Urban Drainage Systems 14.103
1 (Bi,n 1 Bi 1,n 1)(hi,n 1 hi 1,n 1 hi,n hi 1,n) 2∆t
[
]
1 (Gi,n 1 Gi 1,n 1)(hi 1,n 1 hi,n 1) 0 (14.105) ∆x This equation is nonlinear only with respect to the unknown flow depth hi 1, n 1 since Bi 1, n 1 and Gi 1, n 1 are both expressed in terms of the depth [Eqs. (14.102) and (14.103)], and hence it can readily be solved by using Newton’s iteration method. The solution proceeds sewer by sewer from upstream toward downstream. Within each sewer, the flows for all the reaches are determined for a given time before proceeding to the next time step. In ILSD, there are actually several sewer flow routing schemes of different hydraulic levels, including options B2 and B3 listed in Table 14.23 and the option of hydrograph time lag adopted in ILSD-1 and 2. The objective of ILSD is to develop an efficient and practical optimization model for the least cost system design of sewer networks. Therefore, the sensitivity and significance of the sophistication of hydraulics on optimal design of sewer systems were investigated. It was found that for the purpose of designing sewers, because of the discrete sizes of commercially available pipes, unsophisticated hydraulic schemes often suffice, and hence the hydrograph time lag method, instead of options B2 and B3, is adopted in ILSD-1 and 2. Yen and Sevuk (1975) also arrived at a similar conclusion that for design, a low hydraulic level routing method is often acceptable, whereas for evaluation and simulation of flow in sewers, a high hydraulic level routing is usually required. In the USGS model, the finite difference equation is formulated from Eq. (14.47) similar to ILSD-B2. However, the nonlinear relation between Q/Qf and A/Af is approximated by a straight line, and the flow area A is expressed explicitly as
[
]
Ai 1,n 1 f(Ai,n 1, Ai 1,n)
(14.106)
Hence, solution for all the reaches within a sewer must be obtained at each time for the time increments. However, for the sewers in a network, the solution technique can be either the cascade method or the one-sweep method. No information on which one is used in the model is given in the literature. 2. SWMM-TRANSPORT. Only one model in the third group of modified nonlinear kinematic wave models is listed in Table 14.23. The SWMM is a comprehensive urban storm water runoff quality and quantity simulation model for evaluation and management. A good summary of the model is given in Huber and Dickinson (1988), Huber and Heaney (1982), and Metcalf and Eddy et al. (1971). It has two sewer flow routing options, TRANSPORT and EXTRAN, not counting the crude gutter-type routing in the RUNOFF block. EXTRAN was discussed above. TRANSPORT is the original sewer-routing submodel built in the progam. In TRANSPORT, the continuity equation is first normalized using the just-full steady uniform flow discharge Qf and area Af, then the equation is written in finite differences and expressed as a linear function of the normalized unknowns A/Af and Q/Qf at the grid point x (i 1)∆x and t (n 1)∆t: (Q/Qf)i 1,n 1 C1(A/Af)i 1,n 1 C2 0
(14.107)
where C1 and C2 functions of known quantities. From the simplified dynamic equation Sf So and Manning’s formula, we have
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HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
14.104
Chapter Fourteen
(Q/Qf) AR2/3/AfRf2/3 f(A/Af)
(14.108)
Accordingly, curves of normalized discharge-area relationship Q/Qf versus A/Af for steady uniform flow in pipes of different cross-sectional geometries are established and solved together with Eq. (14.107) for Q/Qf and A/Af. In the kinematic wave method of solving Eqs. (14.107) and (14.108), in addition to the initial condition, only one boundary condition is needed, which is usually the inflow hydrograph at the sewer entrance. No downstream boundary condition is required, and hence, no backwater effect from the downstream can be accounted for if the flow is subcritical. However, in TRANSPORT through a formulation of friction slope calculation using the previous time values at the spatially forward point, the downstream backwater effect is partially accounted for at one time step behind. In routing the unsteady nonuniform flow by using Eqs. (14.107) and (14.108), the value of Qf is not calculated as the steady uniform full-pipe discharge. Instead, it is adjusted by assuming that ∂h V ∂V hi 1,n hi,n Vi2 1,n Vi2,n Sf So So (14.109) ∂x g ∂x ∆x 2g∆x To improve computational stability, it is further assumed in TRANSPORT that at any iteration k, Qfk is taken as the average of previous and current values: that is, 1 Kn Qfk Qf(k 1) AfRf2/3 2 (14.110)
Vi2,(k 1) Vi2 1,(k 1) So∆x hi,(k 1) hi 1,(k 1) 2g 2g
1/2
where all the values of h and V are those at the previous time n∆t that are known if the one–sweep or implicit solution method is used to solve for the flow in individual sewers at incremental times. Incorporating Eq. (14.109) for Sf in Manning’s formula yields a quasi-steady dynamic wave approximation instead of the kinematic wave. Thus, use of Eq. (14.110) to compute Qf indirectly gives a partial consideration of the downstream backwater effect with a time lag. This improvement of the kinematic wave approximation makes SWMM TRANSPORT hydraulically more attractive than the standard nonlinear kinematic wave models. Presumably, the partial accounting of the downstream backwater effect is effective as long as the flow does not change rapidly with time, and no hydraulic jump or hydraulic drop is allowed. A hydraulic comparison of EXTRAN to improvement and advantages over TRANSPORT has not been reported and would be interesting. Nonetheless, since the downstream boundary condition is not truly accounted for, it is recommended in SWMMTRANSPORT that for a sewer with a large downstream storage element from which the backwater effect is severe, the water surface is assumed as horizontal from the storage element going backward until it intercepts the sewer invert. Moreover, when the sewer slope is steep, presumably implying high-velocity supercritical flow, the flood may simply be translated through the sewer without routing, that is, shifting of the hydrograph without time lag. Also, if the backwater effect is expected to be small and the sewer is circular in cross section, the gutter flow routing method in the RUNOFF Block may be applied to the sewer as an approximation. In SWMM, large junctions with significant storage capacity and storage facilities are called storage elements, equivalent to the case of storage junction (that is, ds/dt 0), which was discussed above. Only the continuity equation, Eq. (14.35), is used in storage element routing. No dynamic equation is considered except for the cases with weir or orifice outlets. Small junctions are treated as point-type junctions with ds/dt 0.
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HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
Hydraulic Design of Urban Drainage Systems 14.105
3. ILSD-B3 and WASSP-SIM. In the second group of nonlinear kinematic wave models, both ILSD-B3 and WASSP-SIM adopt the Muskingum-Cunge method. The Muskingum routing formula can be written for discharge at x (i 1)∆x and t (n 1)∆t as Qi 1,n 1 C1Qi,n C2Qi,n 1 C3Qi 1,n
(14.111)
in which KX 0.5∆t C1 K(1 X) 0.5t
(14.112a)
0.5∆t KX C2 K(1 X) 0.5∆t
(14.112b)
K(1 K) 0.5∆t (14.112c) C3 K(1 X) 0.5∆t where K is known as the storage constant having a dimension of time and X a factor expressing the relative importance of inflow. Cunge (1969) showed that by taking K and ∆t as constants, Eq. (14.111) is an approximate solution of the nonlinear kinematic wave equation [Eqs. (14.4) and (14.102) or Eq. (14.104)]. He further demonstrated that Eq. (14.111) can be considered as an approximate solution of Eq. (14.104) if and
K ∆x/c
(14.113)
and
1 X (ε/c∆x) (14.114) 2 where ε is the “diffusion” coefficient and c is the celerity of the flood peak that can be approximated as the length of the reach divided by the flood peak travel time through the reach. Assuming K ∆t and denoting α 1 2X, Eq. (14.111) can be rewritten as
2α Q α Q Qi 1,n 1 α Q (14.115) 2 α i, n 2 α i, n 1 2 α i 1, n In the traditional Muskingum method, X and, consequently, α are regarded as constant. In the Muskingum method as modified by Cunge, α is allowed to vary according to the channel geometry and is computed as α KQ/So(∆x)2B
(14.116)
in which B is the surface width of the flow and So the sewer slope. The values of α are restricted to being between 0 and 1 so that C1, C2, and C3 in Eq. (14.112) will not be negative. It is the variation of α, and hence C1, C2, and C3, that classifies the MuskingumCunge method as a nonlinear kinematic wave approximation. The Muskingum-Cunge method offers two advantages over the standard nonlinear kinematic wave methods. First, the solution is obtained through a linear algebraic equation [Eq. (14.111) or Eqs. (14.115) and 14.116)] instead of a partial differential equation, permitting the entire hydrograph to be obtained at successive cross sections instead of solving for the flow over the entire length of the sewer pipe for each time step as for the standard nonlinear kinematic wave method. Second, because of the use of Eq. (14.116), a limited degree of wave attenuation is included, permitting a more flexible choice of the time and space increments for the computations as compared to the standard nonlinear kinematic wave method.
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HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
14.106
Chapter Fourteen
In ILSD-B3, the coefficient α in Eq. (14.115) is computed at each grid point by using Eq. (14.116), while B and K both change with respect to time and space. The values of K are computed by using Eq. (14.113) with the celerity c evaluated by c ∂Q/∂A
(14.117)
or for a partially filled pipe using Manning’s formula 0.132K sin φ c n So1/2 D2/3 1 n φ
φ
sin φ
5 sin 2 φ 1 2/3
2
(14.117)
The initial flow condition is the specified base flow as in ILSD-B2. The upstream boundary condition of the sewer is the given inflow hydrograph. The flow depth and other geometric parameters at the sewer entrance can be computed from the geometric equations given in Fig. 14.3. The junction condition used is the continuity relationship, Eq. (14.53). The solution is obtained over the entire time period at a flow cross section before proceeding to the next cross section. The solution then proceeds downstream section by section and then sewer by sewer in a cascading sequence. More details on the computational procedure of ILSD-B3 can be found in Yen et al. (1976). The British model WASSP is a sewer design and analysis package consisting of four submodels (Price, 1982b): A modified rational method for design of sewers, a hydrograph method for design of sewers using the Muskingum-Cunge routing, an optimal design method, and a simulation method using the fixed parameter Muskingum-Cunge technique for open-channel routing in sewers and the unsteady dynamic equation for surcharge flow computations. Open-channel flow is routed using Eq. (14.111) with the coefficients C1, C2, and C3 expressed as functions of c and µ Q/2BSo. In computation, c is taken as the full-pipe velocity and µ is evaluated at h/D 0.6. Sewers under open-channel flow are solved pipe by pipe, using a directionally explicit algorithm to calculate the discharge at the sewer exit. The space increment ∆x along the sewer is selected automatically in terms of ∆t to enhance computational accuracy. Connected surcharged sewers are solved simultaneously. For surcharge flow, a time increment as small as a few seconds may be necessary if surges occur. The transition between open-channel flow and surcharge flow is assumed to occur when the discharge exceeds Qf, when the sewer entrance and exit are submerged, or when the water depth in the junction is higher than the sewer flow depth plus the entrance or exit headloss (Bettess et al num., 1978). At a junction, only the continuity equation is considered for open-channel flow. For surcharge flow, in addition to the continuity equation, junction headloss is considered and incorporated into the surcharge unsteady dynamic wave equation. The headloss coefficient is assumed to be 0.15 for a junction with straight pipes, 0.50 for 30º bend pipes, and 0.90 for 60º bend pipes. Some details of WASSP-SIM are reported in Price (1982b).
REFERENCES Abbott, M. B., and Basco, D. R., Computational Fluid Dynamics, John Wiley & Sons, New York, 1990. Abbott, M. B., K. Havnø, N. E. Hoff-Clausen, and A. Kej, “A Modelling System for the Design and Operation of Storm-Sewer Networks,” in M. B. Abbott and J.A. Cunge, eds., Engineering Applications of Computational Hydraulics, Vol. 1, Pitman, London, 1982, pp. 11—36. Abt, S. R., and N. S. Grigg, “An Approximate Method for Sizing Detention Reservoirs,” Water Resources Bulletin, 14(4):956–965, 1978. Ackers, P., “An Investigation of Head Losses at Sewer Manholes,” Civil Engineering Public Works Review, 54:882—884 and 1033–1036, 1959. Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
Hydraulic Design of Urban Drainage Systems 14.107 Akan, A. O., “Similarity Solution of Overland Flow on Pervious Surface,” Journal of Hydraulic Engineering, ASCE, 111(7): 1057–1067, 1985a. Akan, A. O., “Overland Flow Hydrographs for SCS Type II Rainfall,” Journal of Irrigation & Drainage Engineering, ASCE, 111(3):276–286, 1985b. Akan, A. O., “Kinematic-Wave Method for Peak Runoff Estimation,” Journal of Transportation Engineering, ASCE, 111(4):419–425, 1985c. Akan, A. O., “Overland Flow on Pervious Converging Surface,” Nordic Hydrology, 19:153–164, 1988. Akan, A. O., “Detention Pond Sizing for Multiple Return Periods,” Journal of Hydraulic Engineering, 115(5):650–665, 1989a. Akan, A. O., “Time of Concentration Formula for Overland Flow,” Journal of Irrigation & Drainage Engineering, ASCE, 115(4):733–735, 1989b. Akan, A. O., “Single-Outlet Detention-Pond Analysis and Design,” Journal of Irrigation & Drainage Engineering, 116(4):527–536,1990. Akan, A. O., Urban Stormwater Hydrology, Technomic, Lancaster, PA, 1993. Akan, A. O., and B. C. Yen, “Unsteady Gutter Flow into Grate Inlets,” Civil Engineering Studies Hydraulic Engineering Series, 36, University of Illinois at Urbana-Champaign, Urbana, IL, 1980. Akan, A. O., and B. C. Yen, “Mathematical Model of Shallow Water Flow Over Porous Media,” Journal of Hydraulics Division, ASCE, 107(HY4):479–494, 1981a. Akan, A. O., and B. C. Yen, “Diffusion Wave Flood Routing in Channel Networks,” Journal of Hydraulics Division, ASCE, 107(HY6):719–732, 1981b. American Society of Civil Engineers, Hydrology Handbook, 2d ed., Manuals and Reports on Engineering Practice, No. 28, ASCE. Reston, Virginia,1996. American Society of Civil Engineers, Urban Runoff Quality Management, Manual and Report on Engineering Practice, No. 87, ASCE Reston, Virginia, 1998. Archer, B., F. Bettess, and P. J. Colyer, “Head Losses and Air Entrainment at Surcharged Manhole,” Report. IT185, Hydraulics Research Station, Wallingford, UK 1978. Arnell, V., “Description and Validation of the CTH-Urban Runoff Model,” Report A5, Department of Hydraulics, Chalmers University of Technology, Göteborg, Sweden, 1980. Aron, G., and D. F. Kibler, “Pond Sizing for Rational Formula Hydrographs,” Water Resources Bulletin, 26(2):255–258, 1990. ASCE Task Committee on Definition of Criteria for Evaluation of Watershed Models of the Watershed Management Committee, “Criteria for Evaluation of Watershed Models,” Journal of Irrigation & Drainage Engineering, ASCE, 119(3):429–443, 1993. Baker, W. R., “Storm-Water Detention Basin Design for Small Drainage Areas,” Public Works, 108(3):75–79, 1979. Behlke, C. E., and H. D. Pritchett, “The Design of Supercritical Flow Channel Junctions,” Highway Research Record No. 123:17–35, National Research Council Highway Research Board, Washington, DC, 1966. Bermeuleu, L. R., and J. T. Ryan, “Two-Phase Slug Flow in Horizontal and Inclined Tubes,” Canadian Journal of Chemical Engineering, 49:195–201, 1971. Best, J. L., and I. Reid, “Separation Zone at Open-Channel Junctions,” Journal of Hydraulic Engineering, ASCE, 110(11):1588–1594, 1984. Bettess, R., R. A. Pitfield, and R. K. Price, “A Surcharging Model for Storm Sewer Systems,” in P. R. Helliwell, ed., Urban Storm Drainage, Procedings 1st International Conference, pp. 306–316 Pentech Press, London and Wiley-Interscience, New York, 1978. Blaisdell, F. W., and P. W. Mason, “Energy Loss at Pipe Junction,” Journal of Irrigation & Drainage Division, ASCE, 93(IR3):59–78, 1967; Discussions, 94(IR2):280–282, 1968. Bo Pedersen, F., and O. Mark, “Head Losses in Storm Sewer Manholes: Submerged Jet Theory,” Journal of Hydraulic Engineering, ASCE, 116(11):1317–1328, 1990. Bodhaine, G. L., “Measurement of Peak Discharge at Culvert by Indirect Methods,” Techniques of Water Resources Investigations, Book 3, Chapter A3, U.S. Geological Survey, Washington, DC, 1968.
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14.108
Chapter Fourteen
Book, D. E., J. W. Labadie, and D. M. Morrow, “Dynamic vs. Kinematic Routing in Modeling Urban Storm Drainage,” in B. C. Yen, ed., Urban Stormwater Hydraulics and Hydrology, pp. 154–163, Water Resources Publications, Highlands Ranch, CO, 1982. Bowers, C. E., “Studies of Open-Channel Junctions,” Technical Paper, No. 6, Series B, St. Anthony Falls Hydraulic Laboratory, University of Minnesota, Minneapolis, MN, 1950. Brandstetter, A., “Assessment of Mathematical Models for Urban Storm and Combined Sewer Management,” Environmental Protection Technology Series, EPA-600/2-76-175a, Municipal Environmental Research Laboratory, U.S. Environmental Protection Agency, Cincinnati, OH, 1976. Chaudhry, M. H., Applied Hydraulic Transients, Van Nostrand-Reinhold, Princeton, NJ, 1979. Chen, Y. H., and S.-Y. Chai, “UNSTDY Combined Storm Sewer Model User’s Manual,” Report, Chen Engineering Technology, Fort Collins, CO, 1991. Chevereau, G., F. Holly, and A. Preissmann, “Can Detailed Hydraulic Modeling be Worthwhile when Hydrologic Data is Incomplete?” in P. R. Helliwell, ed., Urban Storm Drainage, Proc. 1st International Conference, pp. 317–326, Pentech Press, London and Wiley-Interscience, New York, 1978. Chow, V. T., Open-Channel Hydraulics, McGraw-Hill, New York, 1959. Chow, V. T., ed., Handbook of Applied Hydrology, McGraw-Hill, New York, 1964. Chow, V. T., and B. C. Yen, “Urban Stormwater Runoff—Determination of Volumes and Flowrates,” Environmental Protection Technology Series EPA–600/2-76-116, Municipal Environmental Research Lab., U.S. Environmental Protection Agency, Cincinnati, OH, 1976. Colyer, P. J., and R. W. Pethick, “Storm Drainage Design Methods: A Literature Review,” Report No. INT 154, Hydraulics Research Station, Wallingford, UK, 1976. Cunge, J. A., “On the Subject of a Flood Propagation Computation Method,” Journal of Hydraulic Research, 7: 205–230, 1969. Cunge, J. A., and B. Mazaudou, “Mathematical Modelling of Complex Surcharge Systems: Difficulties in Computation and Simulation of Physical Situations,” in P. Balmer, P. A. Malmqvist, and A. Sjöberg, eds., Proc. 3rd International Conference Urban Storm Drainage, 1:363–373, Chalmers University of Technology, Göteborg, Sweden, 1984. Cunge, J. A., and M. Wegner, “Intégration Numérique des Équations d’Écoulement de Barré de Saint-Venant par un Schéma Implicite de Différences Finies: Application au Cas d’Une Galerie Tantôt en Charge, Tantôt a Surface Libre,” La Houille Blanche, 1:33-39, 1964. Cunge, J. A., F. M. Holly, and A. Verwey, Practical Aspects of Computational River Hydraulics, Pitman, London, 1980. Currey, D. L., “A Two-Dimensional Distributed Hydrologic Model for Infiltrating Watersheds with Channel Networks,” M.S. thesis, Old Dominion University, Departament of Civil and Environmental Engineering, Norfolk, VA, 1998. Currey, D. L., and A. O. Akan, “Single Outlet Detention Pond Design for Differing Hydrograph Shapes” (in press). Dawdy, D. R., J. C. Schaake, Jr., and W. M. Alley, “User’s Guide for Distributed Routing RainfallRunoff Model,” Water Resources Investigation, U.S. Geological Survey, pp. 78-90, 1978. DeGroot, C. F., and M. J. Boyd, “Experimental Determination of Head Losses in Stormwater Systems,” Proc. 2d National Conference on Local Government Engineering, pp. 19–22, Brisbane, Australia, September 1983. DHI, “MOUSE: Reference Manual Version 3.2,” Danish Hydraulic Institute, Copenhagen, Denmark, 1994. Federal Highway Administration, Urban Drainage Design Manual, Hydraulic Engineering Circular, No. 22, Washingtun, DC, 1996. Fried, E., and I. E. Idelchik, Flow Resistance, Hemisphere, New York, 1989. Fujita, S., “Experimental Sewer System: Its Application and Effects,” in W. Gujer and V. Krejci, eds., Urban Stormwater Quality Planning and Management, pp. 357–362, Water Resources Publications, Highlands Ranch, CO, 1987.
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Hydraulic Design of Urban Drainage Systems 14.109 Geiger, W. F., and H. R. Dorsch, Quantity-Quality Simulation (QQS): A Detailed Continuous Planning Model for Urban Runoff Control,” Report EPA-600/2-80–011, U.S. EPA, 1980. Hager, W.H., Abwasser-Hydraulik: Theorie und Praxis, Springer-Verlag KG, Berlin, Germany, 1994. Hampton Roads Planning District Commission, Best Management Practices Design Guidance Manual for Hampton Roads, 1991. Harms, R. W., “Application of Standard Unit Hydrograph in Storm Sewer Design,” in B. C. Yen, ed., Urban Stormwater Hydraulics and Hydrology, pp. 257–265,Water Resources Publications, Highlands Ranch, CO,1982. Harris, G. S., “Development of a Computer Program to Route Runoff in The Minneapolis-St. Paul Interceptor Sewers,” Memo M121, St. Anthony Falls Hydraulic Laboratory, University of Minnesota, Minneapolis, MN, 1968. Hartigan, J. P., “Basis for Design of Wet Detention Basin BMPs,” in L. A. Roesner et al., eds., Design of Urban Runoff Quality Controls, American Society of Civil Engineers, Reston, VA, 1989. Hoff-Clausen, N. E., K. Havnø, and A. Kej, “System 11 Sewer—A Storm Sewer Model,” in B. C. Yen, ed., Urban Stormwater Hydraulics and Hydrology, pp. 137–146, Water Resources Publications, Highlands Ranch, CO, 1982. Howarth, D. A., and A. J. Saul, “Energy Loss Coefficients at Manholes,” in P. Balmer, et al., eds. Procedings 3rd International Conference on Urban Storm Drainage, 1:127–136, Chalmers University of Technology, Göteborg, Sweden, 1984. Huber, W. C., and R. E. Dickinson, “Storm Water Management Model, Version 4: User’s Manual,” Department of Environmental Engineering Sciences, University of Florida, Gainesville, FL, 1988. Huber, W. C., and J. P. Heaney, “The USEPA Storm Water Management Model, SWMM: A Ten Year Perspective,” in B. C. Yen, ed., Urban Stormwater Hydraulics and Hydrology, pp. 247–256, Water Resources Publications, Highlands Ranch, CO, 1982. Huber, W. C., J. P. Heaney, S. J. Nix, R. E. Dickinson, and D. J. Polmann, “Storm Water Management Model Users Manual Version III,” Environmental Protection Technology Series EPA–600/2–84–109a, Municipal Environmental Research Laboratory, U.S. Environmental Protection Agency, Cincinnati, OH, 1984. Hydrologic Engineering Center, “Urban Runoff: Storage, Treatment and Over Flow Model— STORM,” U.S. Army Corps of Engineers Hydrologic Engineering Center Computer Program 723-58-L2520, Davis, CA, 1974. Hydrologic Engineering Center, “HEC-1 Flood Hydrograph Package, User’s Manual,” U.S. Army Corps of Engineers, Davis, CA, 1990. Izzard, C.F., “Hydraulics of Runoff from Developed Surfaces”, Proceedings Highway Research Board 6 Vol. 26,129–146, 1946. Johnston, A. J., and R. E. Volker, “Head Losses at Junction Boxes,” Journal of Hydraulic Engineering, ASCE, 116(3): 326–341, 1990. Joliffe, I. B., “Accurate Pipe Junction Model for Steady and Unsteady Flows,” in B. C. Yen, ed., Urban Stormwater Hydraulics and Hydrology, pp. 92–100, Water Resources Publications, Highlands Ranch, CO, 1982. Joliffe, I. B., “Computation of Dynamic Waves in Channel Networks,” Journal of Hydraulic Engineering, ASCE, 110(10):1358–1370, 1984a. Joliffe, I. B., “Free Surface and Pressurized Pipe Flow Computations,” in P. Balmer et al., eds. Proceding 3rd International Conference on Urban Storm Drainage, 1:397–405, Chalmers University of Technology, Göteborg, Sweden, 1984b. Jun, B. H., and B. C. Yen, “Dynamic Wave Simulation of Unsteady Open Channel and Surcharge Flows in Sewer Network,” Civil Engineering Studies Hydraulic Engineering Series No. 40, University of Illinois at Urbana-Champaign, Urbana, IL, 1985. Kanda, T., and T. Kitada, “An Implicit Method for Unsteady Flows with Lateral Inflows in Urban Rivers,” Proceedings 17th Congress Internatinal Association for Hydraulic Research, BadenBaden, Germany, 2:213–220, 1977.
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14.110
Chapter Fourteen
Keifer, C. J, C. Y. Hung, and K. Wolka, “Modified Chicago Hydrograph Method,” in B. C. Yen, ed., Storm Sewer Design, pp. 62–81, Department of Civil Engineering, University of Illinois at Urbana–Champaign, Urbana, IL, 1978. Kessler, A., and M. H. Diskin, “The Efficiency Function of Detention Reservoirs in Urban Drainage Systems,” Water Resources Research, 27(3):253–258, 1991. Kibler, D.F., ed., Urban Stormwater Hydrology, Water Resources Monograph No. 7, American Geophysical Union, Washington, DC, 1982. Klym, H., W. Königer, F. Mevius, and G. Vogel, “Urban Hydrological Processes,” paper presented in the Seminar on Computer Methods in Hydraulics, Swiss Federal Institute of Technology, Zurich, Switzerland, 1972. Labadie, J. W., D. M. Morrow, and R. C. Lazaro, “Urban Stormwater Control Package for Automated Real-Time Systems,” Project Report No. C6179, Department of Civil Engineering, Colorado State Univ,. Fort Collins, CO, 1978. Lai, C., “Numerical Modeling of Unsteady Open-Channel Flow,” in B. C. Yen, ed., Advances in Hydroscience, Vol. 14, pp. 162–333, Academic Press, Orlando, FL, 1986. Larson, C. L., T. C. Wei, and C. E. Bowers, “Numerical Routing of Flood Hydrographs Through Open Channel Junctions,” Water Resources Research Center Bulletin, No. 40, University of Minnesota, Minneapolis, MN, 1971. Lee, K. T., and B. C. Yen, “Geomorphology and Kinematic-Wave-Based Hydrograph Derivation,” Journal of Hydraulic Engineering, ASCE, 123(1):73–80, 1997. Liggett, J. A., and J. A. Cunge, “Numerical Methods of Solution of the Unsteady Flow Equation,” in K. Mahmood and V. Yevjevich, eds., Unsteady Flow in Open Channels, Vol. 1, Water Resources Publications, Highlands Ranch, CO, 1975. Lin, J. D., and H. K. Soong, “Junction Losses in Open Channel Flows,” Water Resources Research, 15:414–418, 1979. Lindvall, G., “Head Losses at Surcharged Manholes with a Main Pipe and a 90º Lateral,” in P. Balmer, P.A. Malmqvist, and A. Sjöberg, eds. Proceedings 3rd International Conference on Urban Storm Drainage, 1:137–146, Chalmers University of Technology, Göteborg, Sweden, 1984. Lindvall, G., “Head Losses at Surcharged Manholes,” in B. C. Yen, ed., Urban Stormwater Hydraulics and Hydrology (Joint Proceedings 4th International Conference on Urban Storm Drainage and IAHR 22nd Congress, Lausanne, Switzerland), pp. 140–141, Water Resources Publications, Highlands Ranch, CO, 1987. Loganathan, G. V., D. F. Kibler, and T. J. Grizzard, “Urban Stormwater Management,” in L. W. Mays, ed., Handbook of Water Resources Engineering, McGraw-Hill, New York, 1996. Maidment, D., Handbook of Hydrology, McGraw-Hill, New York, 1993. Marsalek, J., “Head Losses at Sewer Junction Manholes,” Journal of Hydraulic Engineering, ASCE, 110(8):1150–1154, 1984. Marsalek, J., “Head Losses at Selected Sewer Manholes,” Special Report No. 52, American Public Works Association, Chicago, IL, 1985. Mays, L. W., “Sewer Network Scheme For Digital Computations,” Journal of Environmental Engineering Division, ASCE, 104(EE3):535–539, 1978. McEnroe, B. M., “Preliminary Sizing of Detention Reservoirs to Reduce Peak Discharges,” Journal of Hydraulic Engineering, ASCE, 118(11):1540–1549, 1992. Metcalf & Eddy, Inc., University of Florida, and Water Resources Engineers, Inc., “Storm Water Management Model,” Water Pollution Control Research Series, 11024 DOC, Vol. 1–4, U.S. Environmental Protection Agency, 1971. Miller, D.S., Internal Flow Systems, 2d ed., Gulf Publishing Co., Houston, TX, 1990. Montgomery County Department of Environmental Protection, “Stormwater Management Pond Design Review Checklist,” Stormwater Management Division, Rockville, MD, 1984. Morgali, J., and R. K. Linsley, “Computer Analysis of Overland Flow,” Journaly of Hydraulics Division, ASCE, 91(HY3):81–100, 1965.
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HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS
Hydraulic Design of Urban Drainage Systems 14.111 Morita, M., R. Nishikawa, and B.C. Yen, “Application of Conjunctive Surface-Subsurface Flow Model to Infiltration Trench,” Proceedings 7th IAHR/IAWQ International Conference on Urban Storm Drainage, pp. 527–532, Hannover, Germany, 1996. Northern Virginia Planning District Commission, BMP Handbook for the Occoquan Watershed, Annandale, VA, 1987. Northern Virginia Planning District Commission, Evaluation of Regional BMPs in the Occoquan Watershed, Annandale, VA, 1990. Pagliara, S., and B. C. Yen, “Sewer Network Hydraulic Model”: NISN, Civil Engineering Studies Hydraulic Engineering Series No. 53, University of Illinois at Urbana-Champaign, Urbana, IL, 1997. Pansic, N., “Dynamic-Wave Modeling of Storm Sewers with Surcharge,” M. S. thesis, Department of Civil Engineering, University of Illinois at Urbana-Champaign, Urbana, IL, 1980. Papadakis, C. N., and H. C. Preul, “University of Cincinnati Urban Runoff Model,” Journal of Hydraulics Division, ASCE, 98(HY10):1789–1804, 1972. Price, R. K., “A Simulation Model for Storm Sewers,” in B. C. Yen, ed., Urban Stormwater Hydraulics and Hydrology, pp. 184–192 Water Resources Publications, Highlands Ranch, CO, 1982a. Price, R. K., “The Wallingford Storm Sewer Design and Analysis Package,” in B. C. Yen, ed., Urban Stormwater Hydraulics and Hydrology, pp. 213–220 Water Resources Publications, Highlands Ranch, CO, 1982b. Prince George’s County Department of Environmental Resources, “Stormwater Management Design Manual,” MD, 1984. Radojkovic, M., and C. Maksimovic, “Internal Boundary Conditions for Free Surface Unsteady Flow in Expansions and Junctions,” Proceedings 17th Congress International Associatin for Hydraulic Research, Baden–Baden, Germany, 2:367–372, 1977. Radojkovic, M., and C. Maksimovic, “Development, Testing, and Application of Belgrade Urban Drainage Model,” Proceedings 3rd IAHR/IAWQ International Conference on Urban Storm Drainage, Chalmers University of Technology, Göteborg, Sweden, 4:1431–1443, 1984. Ramamurthy, A. S., and W. Zhu, “Combining Flows in 90º Junctions of Rectangular Closed Conduits,” Journal of Hydraulic Engineering, ASCE, 123(11):1012–1019, 1997. Rhodes, E., and D. S. Scott, “Cocurrent Gas-Liquid Flow,” Procedings International Symposium on Research in Concurrent Gas-Liquid Flow, pp. 1–17 University of Waterloo, Ontario, Canada, 1968. Roesner, L. A., and R. P. Shubinski, “Improved Dynamic Routing Model for Storm Drainage Systems,” in B. C. Yen, ed., Urban Stormwater Hydraulics and Hydrology, pp. 164–173, Water Resources Publications, Highlands Ranch, CO, 1982. Roesner, L. A., R. P. Shubisnki, and J. A. Aldrich, “Stormwater Management Model User’s Manual Version III, Addendum I EXTRAN,” Environmental Protection Technology Series EPA–600/2–84–109b, Municipal Environmental Research Laboratory, U.S. Environmental Protection Agency, Cincinnati, OH, 1984. Sangster, W. M., H. W. Wood, E. T. Smerdon, and H. G. Bossy, “Pressure Changes at Storm Drain Junctions,” Bulletin No. 41, Engineering Experiment Station, University of Missouri, Columbia, MO, 1958. Sangster, W. M., H. W. Wood, E. T. Smerdon, and H. G. Bossy, “Pressure Changes at Open Junctions in Conduit,” Transactions, ASCE, 126, Part I:364–396, 1961. Schueler, T. B., “Controlling Urban Runoff: A Practical Manual for Planning and Designing Urban BMPs,” Washington Metropolitan Water Resources Planning Board, 1987. Serre, M., A. J. Odgaard, and R. A. Elder, “Energy Loss at Combining Pipe Junction,” Journal of Hydraulic Engineering, ASCE, 120(7):808–830, 1994. Sevuk, A. S., “Unsteady Flow in Sewer Networks,” Ph.D. thesis, Department of Civil Engineering, University of Illinois at Urbana-Champaign, Urbana, IL, 1973. Sevuk, A. S., and B. C. Yen, “Comparison of Four Approaches in Routing Flood Wave Through Junctions,” Proceedings 15th Congress International Association for Hydraulic Research, Istanbul, Turkey, 5:169–172, 1973.
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14.112
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Sevuk, A. S., and B. C. Yen, “Sewer Network Routing by Dynamic Wave Characteristics,” Journal of Hydraulics Division, ASCE, 108(HY3):379–398, 1982. Sevuk, A. S., B. C. Yen, and G. E. Peterson, II, “Illinois Storm Sewer System Simulation Model: User’s Manual,” Research Report No. 73, Water Resources Center, University of Illinois at Urbana-Champaign, Urbana, IL, 1973. Shen, H. W., and R. M. Li, “Rainfall Effects on Sheet Flow over Smooth Surface,” Journal of Hydraulics Division, ASCE, 99(HY5):771–792, 1973. Singh, V. P., Kinematic Wave Modeling in Water Resources: Surface Water Hydrology, John Wiley & Sons, New York, 1996. Sjöberg, A., “Calculation of Unsteady Flows in Regulated Rivers and Storm Sewer Systems,” Report, Division of Hydraulics, Chalmers University of Technology, Göteborg, Sweden, 1976. Sjöberg, A., “Sewer Network Models DAGVL-A and DAGVL-DIFF,” in B. C. Yen, ed., Urban Stormwater Hydraulics and Hydrology, pp. 127–136,. Water Resources Publications, Highlands Ranch, CO, 1982. Soil Conservation Service, “Urban Hydrology for Small Watersheds,” Technical Relase 55, U.S. Department of Agriculture, Washington, DC, 1986. Stahre, P., and Urbonas, B., Stormwater Detention, Prentice–Hall, Engelwood Cliffs, NJ, 1990. Stephenson, D., Pipeflow Analysis, Elsevier, Amsterdam, 1984. Taitel, Y., N. Lee, and A. E. Dukler, “Transient Gas-Liquid Flow in Horizontal Pipes: Modeling the Flow Pattern Transitions,” J. AICHE, 24:920–934, 1978. Takaaki, K., and S. Fujita, Planning and Dimensioning of a New Sewer System—Experimental Sewer System for Reduction of Urban Storm Runoff, Kajima Institute Corporation, Tokyo, Japan, 1984. Taylor, E. H., “Flow Characteristics at Rectangular Open Channel Junctions,” Transactions, ASCE, 109:893–902, 1944. Terstreip, M.l., and J.B. Stall, “The Illinois Urban Drainage Area Simulator, ILLUDAS, Bulletin 58, Illinois State wWater Survey, Champaign, IL, 1974. Tholin, A. L., and C. J. Keifer, “Hydrology of Urban Runoff,” Transactions, ASCE, 125:1308–1379, runbach 1960. Townsend, R. D., and J. R. Prins, “Performance of Model Storm Sewer Junctions,” Journal of Hydraulics Division, ASCE, 104(HY1):99–104, 1978. Toyokuni, E., and M. Watanabe, “Application of Stormwater Runoff Simulation Model to Matsuyama City Drainage Basin,” Proceedings 3rd IAHR/IAWQ International Conference on Urban Storm Drainage, pp. 555–564 Chalmers University of Technology, Göteborg, Sweden, 1984. University of Cincinnati, “Urban Runoff Characteristics,” Water Pollution Control Research Series, 11024DQU, U.S. Environmental Protection Agency, 1970. Urbonas, B., and P. Stahre, Stormwater, Prentice–Hall, Engelwood Cliffs, NJ, 1993. U.S. Environmental Protection Agency, “Methodology for Analysis of Detention Basins for Control of Urban Runoff Quality,” EPA 440/5–97-001, U. S. Environmental Protection Agency, Washington, DC, 1986. Vermeuleu, L. R., and J. T. Ryan, “Two-Phase Slug Flow in Horizontal and Inclined Tubes,” Journal Chemical Engineering, 49:195–201, 1971. Vogel, G., and H. Klym, “Die Ganglinien-Volumen-Methode,” paper presented at the Workshop on Methods of Sewer Network Calculation, Dortmund, Germany, 1973. Walker, W. W., “Phosphorus Removal by Urban Runoff Detention Basins”,. Lake and Reservoir Management Vol. III, North American Lake Management Society, Washington, DC, 1987. Wallingford Software, “SPIDA User Manual—Version Alpha-3,” Hydraulic Research, Ltd., Wallingford, UK, 1991. Wallingford Software, “HYDROWORKS User Manual”, Hydraulic Research Ltd., Wallingford, UK, 1997. Webber, N. B., and C. A. Greated, “An Investigation of Flow Behaviour at the Junction of Rectangular Channels,” Proceedings Institution of Civil Engineering, (London), 34:321–334, 1966.
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Hydraulic Design of Urban Drainage Systems 14.113 Wood, D. J., “The Analysis of Flow in Surcharged Storm Sewer Systems,” in Proceedings International Symposium on Urban Storm Runoff, pp. 29–35, University of Kentucky, Lexington, KY, 1980. Wycoff, R. L., and U. P. Singhm, Preliminary Hydrologic Design of Small Flood Detention Reservoirs,” Water Resources Bulletin, 12(2):337–349, 1976. Wylie, E. B., and V. L. Streeter, Fluid Transients, FEB Press, Ann Arbor, MI, 1983. Yen, B. C., “Methodologies for Flow Prediction in Urban Storm Drainage Systems,” Research Report No. 72, Water Resources Center, University of Illinois at Urbana-Champaign, Urbana, IL, 1973a. Yen, B. C., “Open-Channel Flow Equations Revisited,” Journal of Engineering Mechanics Division, ASCE, 99(EM5):979–1009, 1973b. Yen, B. C., “Further Study on Open-Channel Flow Equations,” Sonderforschungsbereich 80, Report No. SFB80/T/49, University of Karlsruhe, Karlsruhe, Germany, 1975. Yen, B. C., “Hydraulic Instabilities of Storm Sewer Flows,” in PR. Helliwell, ed., Urban Storm Drainage (Proceedings 1st International Conference), pp. 282–293, Pentech Press, London and Wiley-Interscience, New York, 1978a. Yen, B. C., ed., Storm Sewer System Design, Department. of Civil Engineering, University of Illinois at Urbana–Champaign, Urbana, IL, 1978b. Yen, B. C., “Some Measures for Evaluation and Comparison of Simulation Models” in B. C. Yen, ed., Urban Stormwater Hydraulics and Hydrology, pp. 341–349, Water Resources, Publications, Highlands Ranch, CO, 1982. Yen, B. C., “Hydraulics of Sewers,” in B. C. Yen, ed., Advances in Hydroscience, Vol. 14, pp. 1–122, Academic Press, Orlando, FL, 1986a. Yen, B. C., “Rainfall-Runoff Process on Urban Catchments and Its Modelings,” in C. Maksimovic and M. Radojkovic, eds., Urban Drainage Modelling, pp. 3-26, Pergamon Press, Oxford, UK, 1986b. Yen, B. C., “Urban Drainage Hydraulics and Hydrology: From Art to Science,” (Joint Keynote at 22nd IAHR Congress and 4th International Conference on Urban Storm Drainage, EPF–Lausanne, Switzerland), Urban Drainage Hydraulics and Hydrology, pp. 1–24, Water Resources Publications, Highlands Ranch, CO, 1987. Yen B. C., “Hydraulic Resistance in Open Channels,” in B.C. Yen, ed., Channel Flow Resistance: Centennial of Manning's Formula, pp. 1–135, Water Resources Publications, Highlands Ranch, CO, 1991. Yen, B. C., “Hydraulics for Excess Water Management,” in L. W. Mays, ed., Handbook of Water Resources, pp. 25-1 – 25-55, McGraw-Hill, New York, 1996. Yen, B. C., and A. O. Akan, “Flood Routing Through River Junctions,” Rivers '76, 1:212–231, ASCE, New York, 1976. Yen, B. C., and A. O. Akan, “Effects of Soil Properties on Overland Flow Infiltration,” Journal of Hydraulic Research, IAHR, 21(2):153–173, 1983. Yen, B. C., and V. T. Chow, “Local Design Storms, Vol. I to III,” Report No. FHWA-RD-82-063 to 065, U.S. Dept. of Transportation Federal Highway Administration, Washington, DC, 1983. Yen, B. C., and J. A. Gonzalez, “Determination of Boneyard Creek Flow Capacity by Hydraulic Performance Graph,” Research Report 219, Water Resources Center, University of Illinois at Urbana-Champaign, Urbana, IL, 1994. Yen, B. C., and A. S. Sevuk, “Design of Storm Sewer Networks,” Journal of Environmental Engineering Division, ASCE, 101(EE4):535–553, 1975. Yen, B. C., V. T. Chow, and A. O. Akan, “Stormwater Runoff on Urban Areas of Steep Slopes,” Environmental Protection Technical Series, EPA–600/2–77–168, U.S. Environmental Protection Agency, Cincinnati, OH, 1977. Yen, B. C., H. G. Wenzel, Jr., L. W. Mays,, and W. H. Tang, “Advanced Methodologies for Design of Storm Sewer Systems,” Research Report No. 112, Water Resources Center, University of Illinois at Urbana-Champaign, Urbana, IL, 1976.
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14.114
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Yen, B. C., S. T. Cheng, B.-H. Jun, M. L. Voorhees, H. G. Wenzel, Jr., and L. W. Mays, “Illinois Least-Cost Sewer System Design Model: ILSD-1&2 User’s Guide,” Research Report No. 188, Water Resources Center, University of Illinois at Urbana-Champaign, Urbana, IL, 1984. Yevjevich, V., and A. H. Barnes, “Flood Routing Through Storm Drains, Parts I-IV,” Hydrologic Papers No. 43–46, Colorado State University, Fort Collins, CO, 1970. Yu, S. L., and R. J. Kaighn, Jr., “VDOT Manual of Practice for Planning Stormwater Management,” Virginia Transportation Research Council, Charlottesville, VA, 1992.
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Source: HYDRAULIC DESIGN HANDBOOK
CHAPTER 15
HYDRAULIC DESIGN OF CULVERTS AND HIGHWAY STRUCTURES I. Kaan Tuncok Stanley Consultants, Inc. Phoenix, Arizona
Larry W. Mays Department of Civil and Environmental Engineering Arizona State University Tempe, Arizona
15.1 INTRODUCTION The aim in highway drainage is to prevent on-site water standing on the surface and convey the off-site storm runoff from one side of the roadway to the other. To accomplish the off-site drainage either a culvert or a bridge can be used. Culverts are closed conduits in which the top of the structure does not form part of the roadway. Bridges are mainly provided for large streams and the road is practically a part of the span or drainage structure. National Bridge Inspection Standards (NBIS) define bridges as those structures that have at least 20 ft of length along the roadway centerline. The main operational differences between culverts and bridges may be described in terms of: • economics, • hydraulics, • structural aspects, • maintenance attention requirements. Economically, the initial and operating costs of culverts are considerably less than that of bridges. Thus, the total investment of public funds for culverts constitutes a substantial share of the highway budget in relation to the investment for bridges. The hydraulic properties for both the culverts and bridges can be computed by using the conservation of mass, energy, and momentum principles defined in Chap. 3. In the case of bridges the magnitude of energy losses must be carefully computed. One part of this loss is due to the contraction and expansion that occur in reaches immediately upstream and downstream from the bridge. The other part is due to the losses at the structure itself and is calculated with either the momentum or energy principles (U.S. Army Corps of Engineers, 1990). To compute the structural losses, the flow must be identified 15.1 Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
HYDRAULIC DESIGN OF CULVERTS AND HIGHWAY STRUCTURES
15.2
Chapter Fifteen
as low flow, pressure flow, or weir flow and equations based on the momentum principle, orifice flow, and a standard weir equation can be used, respectively, to separate the associated losses. The hydraulics of culverts is complicated and the concept of inlet control and outlet control is commonly used to simplify the analysis. Inlet and outlet control culverts have a barrel capacity larger and smaller than the capacity of the culvert entrance, respectively. This chapter will focus on the hydraulic design of culverts and stream stability at highway structures. Structurally, culverts are designed with a heavy dependence on good and proper backfill around the perimeter of the culvert conduit. Often culverts must accommodate significant dead loads (i.e., embankment loads) in addition to the live loads (i.e., vehicles and pedestrians). The earth load transmitted to a culvert is largely dependent on the type of installation, and the four common types are: • trench, used in the construction of sewers, drains and water mains, • positive projecting embankment, used in relatively flat streambed or drainage path • negative projecting embankment, used in relatively narrow and deep streambed or drainage path, • induced trench, used in the construction of culverts placed under high embankment. For the details of these different installation types the reader should refer to the American Concrete Pipe Association (1992). Bridges barely have significant embankment loads and therefore the live load on bridges is the predominant structural consideration. Maintenance requirements for culverts are considerable. Constant efforts must be made to ensure: • clear and open conduits, • protection or repair against corrosion and abrasion, • repair and protection against local and general scour, • structural distress repair, • maintenance of traffic safety devices pertinent to the culvert. Bridges require some of the same items of maintenance attention, but not to the extent required by culverts.
15.2 DESIGN PARAMETERS To better understand the design of the culverts, the basic design parameters must be carefully studied. The headwater, outlet velocity, and tailwater are factors of significant importance.
15.2.1 Headwater and Tailwater Headwater is the vertical distance from the culvert invert at the entrance, to the energy line (depth velocity head) of the headwater. Because of the low velocities in most entrance pools and the difficulty in determining the velocity head for all flows, the water surface and the energy line at the entrance are assumed to be coincident (American Iron and Steel Institute, 1983). Therefore, headwater is the depth of water at the upstream face of the culvert. The main reason for the accumulation of water is to build up the energy required to pass the water through the culvert opening and to overcome the friction,
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HYDRAULIC DESIGN OF CULVERTS AND HIGHWAY STRUCTURES
Hydraulic Design of Culverts and Highway Structures 15.3
FIGURE 15.1 Typical inlet control flow condition. dc critical depth; HW headwater; TW tailwater; WS water surface. (From Normann et al., 1985)
entrance, and exit losses along the culvert. By this process, the potential energy accumulated in water is transformed into kinetic energy through the culvert. Figure 15.1 shows a typical inlet control flow condition with the headwater depicted on it. The designer should not ignore the design limitations on the maximum or allowable headwater (AHW). AHW is the level to which the culvert headwater may rise before causing an unwanted inundation or damage under the circumstances of the design flood (Reagan, 1993). The level of AHW may be based on the fill heights, elevation level of property, or features which would be affected by ponded water due to culvert headwater levels. In cases with a heavy load of debris in the runoff, provision for its passage (or retention) must be made. The designer should avoid the use of very high allowable headwater that may result in unacceptable turbulence and objectionable velocities of flow into and through the culvert. The criteria in establishing the AHW can be summarized as: (1) AHW should not give damage to upstream properties, (2) AHW should be below the traffic lanes of interest or lower than the shoulder, (3) AHW should be lower than the low point in the road grade, (4) AHW should not be equal to the elevation where flow diverts around the culvert. The ponding affect at the upstream face of the culvert may cause an attenuation in the peak discharges, therefore resulting in smaller culvert size requirements for the design. The tailwater is the depth of water downstream of the culvert measured from the outlet invert. It can be a very significant factor in culvert hydraulic operation. The depth of the tailwater can affect the outlet velocity, and depending on the character of the culvert flow it can also affect the operating headwater on the culvert.
15.2.2 Outlet Velocity For design purposes the outlet velocity should be similar in magnitude to the velocity in the channel to provide appropriate protection at the downstream end (American Concrete Pipe Association, 1992). But generally due to the constriction inside the culvert opening, the outlet velocity is higher than the channel velocity. High channel velocities can be mitigated in several different ways such as the following. (1) Stabilization of the channel. (2) Construction of energy dissipation structures, such as hydraulic jump basin, drop structure, stilling basin, riprap, and sill. For a detailed description of the individual methods, the designer should refer to Corry et., al. (1983). (3) Changing the size or the roughness of the culvert.
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HYDRAULIC DESIGN OF CULVERTS AND HIGHWAY STRUCTURES
15.4
Chapter Fifteen
A more detailed discussion of outlet velocity will be given in Sec. 15.3.3. Another important factor is the minimum velocity that must be maintained within the culvert for an efficient operation. The ideal minimum velocity should be adjusted such that the sediment particles transported through the culvert should not be allowed to settle. The culvert barrel should result in a tractive force, τ, greater than the critical τ of the transported stream bed material at low flow rates. In case of unknown stream bed material size a culvert velocity of 2.5 ft/s should be maintained within the culvert. If clogging is probable, installation of a sediment trap or sizing the culvert to facilitate the cleaning should be considered.
15.3 CHARACTERISTICS OF FLOW A culvert is a type of structure that can transmit water as full or partly full. Full flow is not common for culverts unless governed by a high downstream or upstream water surface elevation. Full flow can be described by the fundamentals of pipe flow. Partly full flow culverts follow the rules of open-channel flow and need to be classified as either subcritical or supercritical flow to accomplish the design procedure. The reader should refer to Chap. 3 for the details of open-channel flow and the related terminology that will be used in this chapter. An exact theoretical analysis of culvert flow is difficult due to the following factors: (1) the culvert may have both gradually and rapidly varied flow zones and non-uniform flow conditions must be considered to analyze the flow, (2) the location of hydraulic jump, if it occurs, must be identified, (3) the results of hydraulic model studies must be applied, (4) the change in the flow type as a function of discharge and hydraulic characteristics of the culvert must be studied. In the culvert design procedures of this chapter, the concept of control section, as related to the relationship between the flow rate and the upstream water surface elevation will be used. A culvert can operate either under inlet or outlet control depending on whether the barrel or the inlet has a greater hydraulic capacity. Under inlet control, the cross sectional area of the culvert barrel, the inlet geometry, and the amount of headwater at the entrance are of primary importance. Outlet control involves the additional consideration of the tailwater elevation in the outlet channel, the slope, the roughness and the length of the culvert barrel. When a culvert operates under inlet control, the barrel will flow partly full and depending on the headwater, the flow may be divided either by using the weir or orifice equations. For outlet control, the culvert barrel is intended to flow full for design conditions. A more detailed description of both the inlet and outlet control culverts will be described in the following sections.
15.3.1 Inlet Control A culvert operates under inlet control when the barrel hydraulic capacity is higher than that of the inlet. A typical flow condition is critical depth near the inlet and supercritical flow in the culvert barrel. Depending on the tailwater, a hydraulic jump may occur downstream of the inlet. The barrel geometry and roughness have no direct influence on the hydraulic characteristics of the culvert.
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HYDRAULIC DESIGN OF CULVERTS AND HIGHWAY STRUCTURES
Hydraulic Design of Culverts and Highway Structures 15.5
Due to the severe constriction of the flow at the culvert entrance, the inlet configuration have a significant effect on the hydraulic performance. To increase the capacity of the culvert the designer can use beveled inlet edges, side-tapered inlet, or slope-tapered inlet. The details of different inlet conditions can be found in Sec. 15.7.4. Typically both the inlet and outlet control culverts can be studied in four different types and can be summarized as: (1)
Inlet and outlet unsubmerged,
(2)
Inlet unsubmerged, outlet submerged,
(3)
Inlet submerged, outlet unsubmerged,
(4)
Inlet and outlet submerged.
Culverts operating under steep slope regimes usually operate with inlet control of the headwater, and as a result supercritical flow is a very common flow condition. In cases where the specific energy within the culvert barrel is in the vicinity of minimum specific energy, the flow depth is sensitive to the changes (i.e., roughness) in the culvert. In application, if the normal depth and critical depth are within 5 percent of each other, hydraulic computations should be performed separately to identify the worst case for the design. In the case of inlet control, these four cases are depicted in Fig. 15.2. The flow will be governed by weir flow when the entrance is unsubmerged, and by orifice flow in the case of a submerged entrance. Laboratory investigations showed that for an entrance, either inlet or outlet control, to be considered as unsubmerged, the headwater must be less than a critical value Hcr and can be approximated as (1.2–1.5) times the height of the barrel. Another important factor is the identification of the culvert as hydraulically short or long (Chow, 1959), which depends on the length, slope, size, entrance geometry, headwater, entrance, and outlet conditions. The most typical case of inlet submerged and outlet unsubmerged condition is shown in Fig. 15.2c. As soon as the flow enters the culvert, critical depth is observed and the freeflow conditions at the downstream end results in supercritical normal depth. Supercritical flow is maintained along the culvert barrel. A very similar condition with both inlet and outlet unsubmerged is shown in Fig. 15.2a. The difference from the previous condition is the low headwater at the culvert entrance, but again the flow passes through critical depth at the culvert entrance. In the case of inlet unsubmerged and outlet submerged (Fig. 15.2b), even though the high downstream water surface depth cannot enforce outlet control conditions, a hydraulic jump is observed within the culvert. Similar to the previous cases the flow passes through critical depth at the culvert entrance, but the partly full flow is maintained only upstream of the hydraulic jump (ARMCO, 1950). An uncommon condition shown in Fig. 15.2d depicts submerged inlet and outlet with partly full flow. Similar to the previous case, a hydraulic jump will be observed within the culvert.
15.3.2 Outlet Control The culvert will operate under outlet control conditions if the culvert barrel has a smaller hydraulic capacity than the inlet does. The flow regime is always subcritical; as a result, the control of the flow is either at the downstream end of the culvert or further downstream of the culvert outlet, depending on the depth of the tailwater. (Portland Cement Association, 1964). Typical flow conditions include a full or partially full culvert barrel for all or part of its length. Similar to the inlet control culverts, four types of control are available for the outlet control culverts (Fig. 15.3). In the first flow condition in which both the inlet and the out-
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HYDRAULIC DESIGN OF CULVERTS AND HIGHWAY STRUCTURES
15.6
Chapter Fifteen
FIGURE 15.2 Types of inlet control: (A) outlet unsubmerged, (B) outlet submerged, inlet unsubmerged, (C) inlet submerged (D) outlet submerged. (From Normann et al., 1985)
let are unsubmerged, the flow is subcritical and the culvert barrel is partly full over its entire length. The unsubmerged flow condition results in critical depth at the culvert outlet. For the second flow condition, in which the inlet is unsubmerged and outlet is submerged, headwater is an important factor. Typically, the water surface will drop at the culvert’s entrance and experience a contraction through the culvert’s opening. This is depicted in Fig. 15.3B and is a result of shallow headwater. The unusual case shown in Fig. 15.3C can be established with high submergence at the culvert inlet. Although there is no tailwater, the submergence at the inlet maintains the pressure flow along the culvert barrel. The condition where the inlet is submerged and outlet is unsubmerged, (Fig. 15.3D) is a result of tailwater that does not submerge the culvert depth but is greater than the critical depth at the outlet. Finally, the common condition where both the inlet and the outlet are submerged (Fig. 15.3A) results in pressure flow through the culvert opening. In all of these flow types, in addition to the factors that affect the inlet control culverts, the barrel characteristics such as roughness, area, length, and slope also play an important role. Hydraulics To evaluate the outlet control hydraulics the condition of full flow in the culvert barrel will be used. The energy equation, Eq. (15.1), must incorporate the losses
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HYDRAULIC DESIGN OF CULVERTS AND HIGHWAY STRUCTURES
Hydraulic Design of Culverts and Highway Structures 15.7
FIGURE 15.3 Types of outlet control. (From Normann et al., 1985)
due to entrance (he), friction (hhf), exit (hex), bend (hb), junctions (hhj), and grates (hg) and can be written as HL he hf hex hb hj hg
(15.1)
where HL total energy loss (ft). The velocity head, (hv), can be expressed as (V V2 V2a) hv 2g
(15.2)
where V velocity of flow in the culvert barrel (ft/s), and can be computed as
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HYDRAULIC DESIGN OF CULVERTS AND HIGHWAY STRUCTURES
15.8
Chapter Fifteen
Q V b A
(15.3)
where Qb discharge per barrel (ft3/s), A cross-sectional area of flow with the barrel full (ft2), Va approach velocity of flow (ft/s), and g acceleration due to gravity (32.2 ft/s2). For design purposes, the approach velocity usually is ignored and the velocity head can be expressed as V2 hv 2g
(15.4)
The entrance loss can be computed in terms of velocity head as V2 he ke 2g
(15.5)
where ke entrance loss coefficient (Table 15.1). Similar to the entrance loss, the exit loss can be computed as V2 hex 2g
TABLE 15.1
(15.6)
Entrance Loss Coefficients–Outlet Control, Full or Partly Full V2 He ke 2g
Type of Structure and Design of Entrance Pipe, concrete Mitered to conform to fill slope End-section conforming to fill slope* Projecting from fill, sq. cut end Headwall or headwall and wingwalls Square-edge Rounded (radius 1/12D) Socket end of pipe (groove-end) Projecting from fill, socket end (groove-end) Beveled edges, 33.7º or 45º bevels Side- or slope-tapered inlet Pipe, or pipe-arch, corrugated metal Projecting from fill (no metal) Mitered to conform to fill slope, paved or unpaved slope Headwall or headwall and wingwalls aquare-edge
Coefficient ke
0.7 0.5 0.5 0.5 0.2 0.2 0.2 0.2 0.2 0.9 0.7 0.5
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HYDRAULIC DESIGN OF CULVERTS AND HIGHWAY STRUCTURES
Hydraulic Design of Culverts and Highway Structures 15.9 TABLE 15.1
Entrance Loss Coefficients–Outlet Control, Full or Partly Full V2 He ke 2g
Type of Structure and Design of Entrance
Coefficient ke
End-section conforming to fill slope* Beveled edges, 33.7º or 45º bevels Side-or slope-tapered inlet
0.5 0.2 0.2
Box, reinforced concrete Wingwalls parallel (extension of sides) Square-edges at crown Wingwalls at 10º–25º or 30º–75º to barrel Square-edged at crown Headwall parallel to embankment (no wingwalls) Square-edged on three edges Rounded on three edges to radius of 1/12 barrel dimension, or beveled edges on 3 sides Wingwalls at 30º–75º to barrel Crown edge rounded to radius of 1/12 barrel dimension, or beveled top edges Side- or slope-tapered inlet
0.7 0.5 0.5 0.2
0.2 0.2
Source : From Normann et al., (1995). *Note: “End section conforming to fill slope,” made of either metal or concrete, are the sections commonly available from manufacturers. From limited hydraulic tests they are equivalent in operation to a headwall in both inlet and outlet control. Some end sections, incorporating a closed taper in their design have a superior hydraulic performance. These latter sections can be designed using the information given for the beveled inlet.
The friction loss can be expressed as hf Sf L (15.7) where Sf friction slope, and can be computed by a manipulation of Manning's equation (Brater, et al 1996) and L length of the culvert barrel (ft).
冤
Qn Sf 1.486 A R0.67
冥
2
in American customary units where, n Manning’s roughness coefficient, A cross-sectional area (ft2), R hydraulic radius (ft), A and R are based on full-flow conditions. When the expression for friction slope is inserted into Eq. (15.7), the expression for friction loss can be written as 29 n2 L V2 hf 1.3 (15.8) 3 2g R The bend losses are also a function of velocity head and can be computed as V2 hb Kb 2g
(15.9)
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HYDRAULIC DESIGN OF CULVERTS AND HIGHWAY STRUCTURES
15.10
Chapter Fifteen TABLE 15.2
Loss Coefficients for Bends
Radius of Bend Equivalent Diameter (ft)
Angle
of
Bend,º
90
45
22.5
1
0.50
0.37
0.25
2
0.30
0.22
0.15
4
0.25
0.19
0.12
6
0.15
0.11
0.08
8
0.15
0.11
0.08
Source: From Normann, et al. (1985).
where Kb bend loss coefficient (Table 15.2). Junction losses, hj, are significant whenever more than two culverts flow into a single culvert downstream, as shown in Fig. 15.4. In such cases the energy and momentum principles can be used to calculate hj. The energy equation for the junction can be written as hj y hvu hvd
(15.10)
where hj junction loss in the main culvert (ft) run in hvu = velocity head in the upstream culvert (ft), hvd = velocity head in the downstream culvert (ft),and y = change in hydraulic grade line through the junction (ft). The momentum principle can be used to compute y as
FIGURE 15.4 Culvert junction. (From Normann et al., 1985)
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HYDRAULIC DESIGN OF CULVERTS AND HIGHWAY STRUCTURES
Hydraulic Design of Culverts and Highway Structures 15.11
y
Q2V2 Q1V1 Q3 V3 Cos θ 0.5 ((A1 A2)g
(15.11)
where A1 and A2 cross-sectional areas for culverts 1 and 2, respectively; V1 and V2 velocity for culverts 1 and 2, respectively; and θ angle between culverts 1 and 3. If only the entrance, friction, and exit losses are considered, and the respective equations are inserted into Eq. (15.1), the total headloss can be expressed in American customary units as 29n2L V2 H 1 ke 1.3 hex he hf 3 2g R
(15.12)
By using the schematic in Fig. 15.5 that depicts the hydraulic and energy grade lines, the energy equation with the entrance, exit, and friction loss terms for a full flow, can be written between the upstream and downstream ends of a culvert system as V2 V2 HW W0 u TW d H 2g 2g
(15.13)
where HW0 headwater depth above the outlet invert, TW tailwater depth above the outlet invert, and the subscripts u and d denote the upstream and downstream conditions, respectively. Because of the smaller values of upstream and downstream velocities, compared to the culvert velocity, they can be neglected. Then the energy equation can be expressed as HW W0 TW H S0L
(15.14)
where So L drop in elevation from the inlet invert to the outlet invert A practical way to compute the headloss H is to use the outlet control nomographs (Figs. 15.6 and 15.7). Outlet control nomographs presented for full flow can also be used
he hf
FIGURE 15.5 Full–flow energy and hydraulic grade lines. (From Normann et al., 1985)
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HYDRAULIC DESIGN OF CULVERTS AND HIGHWAY STRUCTURES
15.12
Chapter Fifteen
for the partly full flow for the computation of H. Details on the use of outlet control nomographs will be given in Sec. 15.4.2.2. Equations (15.1–15.14) were developed for full barrel flow in which TW is greater than or equal to the culvert diameter, D. However in the case of partly full flow, backwater calculations may be required starting at the downstream end of the culvert and proceeding upstream. During the backwater computations if the hydraulic grade line cuts the top of the barrel, full flow will be observed upstream of this intersection point. FHWA has also developed an approximate method to overcome the tedious backwater computations for partly full flow. They found out that the hydraulic grade line at the culvert outlet is at a point between the critical depth and the culvert diameter, and can be computed as (ddc D)/2. It was oncluded that the observed TW should only be used if it is higher in magnitude than (ddc D)/2. In such a case the following equation should be used: HW Wo ho H SoL where ho max [TW, W (dc D) / 2].
(15.15)
The use of these two equations reasonable results for HW depths greater than 0.75D.
FIGURE 15.6 Head for concrete pipe culverts. (n 0.012) (From Normann et al., 1985)
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HYDRAULIC DESIGN OF CULVERTS AND HIGHWAY STRUCTURES
Hydraulic Design of Culverts and Highway Structures 15.13
FIGURE 15.7 Head for concrete box culverts flowing full (n 0.012) (From Normann, et al., 1985)
15.3.3 Outlet Velocity Outlet velocity is an important factor to define the type of outlet protection. Because of the decreased area of flow in the culvert, the velocity increases through the culvert barrel and results in higher velocities than that of the natural stream. Because of the high velocities, outlets must be protected by using riprap or an energy dissipator. To accurately compute the outlet velocity, the type of control must be defined. For inlet control culverts, either an exact or approximate method can be used. In the exact method, water surface profile computations start at the upstream and proceed downstream and the velocity is computed by using the cross-sectional area at the exit. For the approximate method, normal depth is assumed to occur at the culvert outlet as depicted in Fig. 15.8. This method is more commonly used and results in more conservative estimates of the outlet velocity. Normal depth can be computed by using Manning's equation. For outlet control culverts, the outlet velocity depends on the outlet geometry and the magnitude of tailwater depth with respect to either the critical depth or the barrel diameter. The following procedure can be used to define the tailwater depth and to compute the outlet velocity with the given geometry. Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
HYDRAULIC DESIGN OF CULVERTS AND HIGHWAY STRUCTURES
15.14
Chapter Fifteen
If
TW dc
Depth at culvert outlet dc.
If
dc TW D
Depth at culvert outlet TW. W
If
TW D
Depth at culvert outlet D.
The schematic in Fig. 15.9 also summarizes these three flow conditions.
15.3.4 Roadway Overtopping Roadway overtopping starts soon after the headwater elevation reaches the top of roadway elevation. To compute the magnitude of the overtopping flow, the following broad-crested equation can be used: Wr)1.5 Qr Kt Cd L (HW
(15.16)
where Qr overtopping flow rate (ft3/s), Cd overtopping discharge coefficient kt Cr , the discharge coefficient Cr , and the submergence factor kt, are presented in Fig. 15.10; L length of roadway crest, (ft) , HW Wr the upstream depth, measured above the roadway crest, (ft)
FIGURE 15.8 Inlet control outlet velocity. (From Normann et al., 1985)
FIGURE 15.9 Outlet control outlet velocity. (From Normann et al., 1985)
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HYDRAULIC DESIGN OF CULVERTS AND HIGHWAY STRUCTURES
Hydraulic Design of Culverts and Highway Structures 15.15
The main reason for this type of equation is the similarity of this flow to the broad-crested weir flow. One of the most important factors in the computation of Qr is a good estimate of the length of roadway crest. The length can be established in two different ways. The first method, although time consuming, results in more accurate estimates of the length, especially if the elevation of the roadway crest varies. It is based on an approximation of the vertical curve as a series of horizontal segments and can be applied to cases where the crest is defined by a roadway sag vertical curve (for definition see Sec. 15.7.3) and is depicted in Fig. 15.11. Then the flow over each segment is calculated for a given headwater and finally the flows for each segment are added together to determine the total flow. In the second method, which is more practical, the length can be represented by a single horizontal line at a constant roadway elevation. For correct estimates of Qr, this horizontal line must span the roadway profile accurately. The depth of flow used in this method is the average depth of upstream pool above the roadway. Given the headwater, length and the coefficient, Qr can be computed by using Eq.
FIGURE 15.10 Discharge coefficient and submergence factor for roadway overtopping. (From Flood Control District of Maricopa County, 1996)
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HYDRAULIC DESIGN OF CULVERTS AND HIGHWAY STRUCTURES
15.16
Chapter Fifteen
FIGURE 15.11 Weir crest length determinations for roadway overtopping. (A) Method 1— subdivision in to to segments. (B) Method 2—use of a single segment. (From Normann et al., 1985)
(15.16). The difficult task is to determine the magnitude of total flow which equals roadway overflow plus the culvert flow and can be determined by using a trial and error procedure. Performance curves, as defined in Sec. 15.5, can also be used to compute the total flow.
15.4 METHOD OF CULVERT DESIGN The culvert design procedures for both the inlet and outlet control culverts must consider some important factors (AASHTO, 1990) namely, • Establishment of hydrology • Design of downstream channel • Assumption of a trial configuration • Computation of inlet control headwater • Computation of outlet control headwater at inlet • Evaluation of the controlling headwater • Computation of discharge over the roadway and then the total discharge • Computation of outlet velocity and normal depth • Comparison of headwater and velocity to limiting values • Adjustment of configuration (if necessary) • Recomputation of hydraulic characteristics (if necessary)
15.4.1 Inlet Control The computation of headwater for inlet control culverts is based on either the design equations or the nomographs.
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HYDRAULIC DESIGN OF CULVERTS AND HIGHWAY STRUCTURES
Hydraulic Design of Culverts and Highway Structures 15.17
15.4.1.1 Design equations. The design equations to be used depend on the condition of the inlet control. A culvert performs as an orifice when the inlet is submerged and as a weir when it is unsubmerged. The submerged (weir) equation can written as HWi D
Q 2 Q c for 0.5 Y Z 0.5 4.0 AD AD
(15.17)
where HW Wi headwater depth above the inlet control section invert (ft); D interior height of the culvert barrel (ft), Q discharge (ft3/s), A full cross-sectional area of the culvert barrel (ft2), c, Y constants from Table 15.3, and Z term for culvert barrel slope (ft/ft). For mitered inlets, Z 0.7S. For all other conditions, Z 0.5S. The unsubmerged (orifice) condition can be presented in two forms. The first one is based on the specific head at critical depth, and can be written as HWi D
H Q M c K 0.5 Z D AD
for
Q 0.5 AD
3.5
(15.18)
where Hc specific head at critical depth (H Hc dc V2c / 2g) (ft), and K, M constants from Table 15.3. The second form is similar to a weir equation and has a simpler form: HWi
Q M 0.5 AD
for
Q 0.5 AD
3.5 (15.19) D The latter form is the only documented form for some of the design inlet control nomographs in Normann, et al., (1985). Nomographs As an alternative procedure, the inlet control nomographs presented in Figs. 15.12 and 15.13 can be used as described in the following: (1) Identify the culvert size and flow rate to be used. It is important to note that for box culverts, the flow rate per foot of barrel width is used. (2) Connect the culvert size and discharge and extend a straight line to cut the HW/ W/D axis. In case HW/ W/D is required in another scale, extend the value at the original intersection horizontally to cut the appropriate scale. W/D ratio is computed either by the design equations or the nomograph, (3) Once the HW/ compute the inlet control headwater depth, HW Wi, by multiplying the barrel diameter by the ratio HW/ W/D.
15.4.2 Outlet Control Similar to the inlet control culverts, the headwater can be determined either by the design equations or the outlet control nomographs. 15.4.2.1 Design equations. Unlike the inlet control culverts, the headwater required for an outlet culvert cannot be computed by using a single equation, but Eq. (15.12) can be used to compute the energy losses. This concept is illustrated in the xample in Sec. 15.4.2.2. Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
HYDRAULIC DESIGN OF CULVERTS AND HIGHWAY STRUCTURES
15.18
Chapter Fifteen
15.4.2.2 Nomographs. The nomographs can be used in the following procedure: (1) Identify the culvert size, D, and length, L, (2) Use the appropriate scale for ke to connect D and L with a straight line and identify the intersection of this line with the turning line (i.e., point X),
TABLE 15.3
Constants for Inlet Control Design Equations
Shape and Material
Inlet Edge Description
Unsubmerged Equation No. K
Submerged M
c
Y
Circular concrete
Square edge w/headwall Groove end w/headwall Groove end projecting
15.18
0.0098 0.0078 0.0045
2.000 2.000 2.000
0.0398 0.0292 0.0317
0.67 0.74 0.69
Circular CMP
Headwall Mitered to slope Projecting
15.18
0.0078 0.0210 0.0340
2.000 1.330 1.500
0.0379 0.0463 0.0553
0.69 0.75 0.54
Circular ring
Beveled ring, 45º bevels Beveled ring 33.7º bevels
15.18
0.0018
2.500
0.0300
0.74
0.0018
2.500
0.0243
0.83
30º–75º wingwall flares 90º and 15º winwall flares 0º wingwall flares
15.18
0.0260
1.000
0.0385
0.81
0.0610
0.750
0.0400
0.80
0.0610
0.750
0.0423
0.82
Rectangular box
45º wingwall flare 18º–33.7º wingwall flare
15.19
0.5100 0.4860
0.667 0.667
0.0309 0.0249
0.80 0.83
Rectangular box
90º headwall w/ 3/4 in chamfers 90º headwall w/ 45º bevels 90º headwall w/ 33.7º bevels
15.19
0.5150
0.667
0.0375
0.79
0.7950
0.667
0.0314
0.82
0.4860
0.667
0.0252
0.87
3/4 in chamfers, 45º skewed headwall 3/4 in chamfers, 30º skewed headwall 3/4 in chamfers, 15º skewed headwall 45º bevels,10– 45º skewed wall
15.19
0.5220
0.667
0.0402
0.73
0.5330
0.667
0.0425
0.71
0.5450
0.667
0.0451
0.68
0.4980
0.667
0.0327
0.75
0.4970
0.667
0.0339
0.80
0.4930
0.667
0.0361
0.81
0.4930
0.667
0.0386
0.71
Rectangular box
Rectangular box
Rectangular box, 3/4 in. chamfers
45º non offset 15.19 wingwall flares 18.4º non offset wingwall flares 18.4º non offset wingwall flares, 30º skewed barrel
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HYDRAULIC DESIGN OF CULVERTS AND HIGHWAY STRUCTURES
Hydraulic Design of Culverts and Highway Structures 15.19 TABLE 15.3
(Continue)
Shape and Material Rectangular box, top bevels Corrugated metal boxes
Inlet Edge Descrition
Unsubmerged Equation No. K M
45º wingwall flares—offset 15.19 33.7º wingwall flares—offset 18.4º wingwall flares—offset 90º headwall 15.18 Thick wall projecting Thin wall projecting
0.4970 0.4950 0.4930 0.0083 0.0145 0.0340
0.667 0.667 0.667 2.000 1.750 1.500
Submerged c
Y
0.0302 0.0252 0.0227 0.0379 0.0419 0.0496
0.84 0.88 0.89 0.69 0.64 0.57
Source: From Normann et al. (1985).
FIGURE 15.12 Headwater depth for box culverts with inlet control. (From Normann et al., 1985)
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HYDRAULIC DESIGN OF CULVERTS AND HIGHWAY STRUCTURES
15.20
Chapter Fifteen
FIGURE 15.13 Headwater depth for inlet control rectangular box culverts, 90º headwall chamfered or beveled inlet edges. (From Normann et al., 1985)
(3) Draw a new line that connects the flow rate Q and headloss H and goes through point X, X (4) The value read from this nomograph for H, includes the entrance, friction, and exit losses. Some of the important steps for design of outlet control culverts can be summarized as (1) Compute the tailwater depth (2) Compute the critical depth, and remember the following restrictions; • dc cannot exceed the diameter of the culvert, D. • if dc 0.9D, use Fig. 15.14 • if dc 0.9D, (use the guidelines in Chapter 3 for a more accurate estimate) (3) Compute (ddc D) / 2 (4) Compute h0
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HYDRAULIC DESIGN OF CULVERTS AND HIGHWAY STRUCTURES
Hydraulic Design of Culverts and Highway Structures 15.21
ho max [TW, W (dc D) /2]
(15.20)
(5) Determine the entrance loss coefficient, ke (6) Determine h either by using the design equations or the nomograph For Maning's n value different from that of the outlet nomograph, a modified length L1 is used as the length scale, given by
FIGURE 15.14 Critical depth rectangular section. (From Normann et al., 1985)
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HYDRAULIC DESIGN OF CULVERTS AND HIGHWAY STRUCTURES
15.22
Chapter Fifteen n 2 L1 L 1 n
(15.21)
where L1 = adjusted culvert length (ft), L = actual culvert length (ft), n1 = desired Manning n value, and n = Manning n value from the chart, (8) Compute the outlet control headwater by using the following equation HW Wo H ho SoL
(15.22)
Once the inlet control headwater, HW Wi and the outlet control headwater, HWo are computed, the controlling headwater is determined by comparing HW Wi and HWo, if HWo HW Wi, the culvert is inlet control; Wi, the culvert is outlet control. if HWo HW Then the discharge over the roadway is computed as given in Sec. 15.3.4. After computing the total discharge, Qt, the outlet velocity, Vo' and the normal depth, dn, the results are evaluated so that the barrels have adequate cover, the headwalls and wingwalls fit to the site conditions, the allowable headwater is not exceeded, the allowable overtopping flood frequency is not exceeded. Example. Design a reinforced concrete box culvert for a roadway crossing to pass a 50-year discharge of 400 ft3/s. The site conditions are as follows: Shoulder elevation 155 ft Streambed elevation at culvert face 140 ft Natural stream slope 1.5% Tailwater depth 3.0 ft Approximate culvert length 200 ft Downstream channel is a 10 10 ft rectangular channel The inlet is not depressed. Solution. The design will be performed by using the procedures given in Sec. 15.4 Step 1. The 50-year design discharge is given as 400 ft3/s. Step 2. The geometry for the downstream channel is given (10 ft 10 ft rectangular). o
Step 3. Select a 7 ft 5 ft reinforced concrete box culvert with 45 beveled edges in a headwall. Step 4. Determine inlet control headwater HW Wi 1. Either by using the inlet control nomograph (Fig. 15.13) a) D 5 ft b) Q/B / 400/7 57 ft3 / s / ft c) HW/ W/D (1) 1.80 for 3/4 in chamfer (2) 1.64 for 45o bevel W/D) D 1.64 5 8.2 ft d) HW Wi (HW/ Neglect the approach velocity.
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HYDRAULIC DESIGN OF CULVERTS AND HIGHWAY STRUCTURES
Hydraulic Design of Culverts and Highway Structures 15.23
2. Or the design equations (Eqs. 15.17, 15.18, or 15.19). To identify whether the inlet is submerged or unsubmerged, Q 400 Compute 5.11 4.0 0.5 AD (35) (5)0.5
Therefore the inlet is submerged; use Eq. 15.17 HWi D HWi 5.0
Q 2 c 0.5 Y 0.5S AD
400 2 0.0314 0.5 0.82 0.5(0.015) (35 ) (5)
HW Wi 8.20 ft Step 5. Determine the outlet control headwater depth at inlet, 1. The tailwater depth is specified as 3.0 ft which is obtained either from a backwater computation or normal depth calculation. 2. Compute critical depth, dc a. Either by using the following equation (Chaurhy, 1993) dc
冪莦qg莦 2
total discharge where q [(ft3. / Ps)/ft] unit discharge , and g gravitaculv ert width tional acceleration, 32.2 ft/s2; then dc
/ 7) 4.7 ft 冪 莦(40莦30莦2莦 .2莦莦 2
b. Or by using Fig. 15.14, dc 4.7 ft 3. (ddc D) / 2 (4.7 5.0) / 2 4.85 ft 4. ho max [TW, W (dc D) / 2 max [3.0, 4.85] 4.85 ft 5. ke 0.2 from Table 15.1 6. Determine H a. Either by using Eq. (15.12) 29n2L V2 H 1 ke 1.3 3 2g R
where A 7 5 35 ft2, V 400/35 11.4 ft/s, and R A/P / 30 / (7 7 5 5) 1.25 ft, then 29(0.012)2 (200) (11.4)2 3.7 ft H 1 0.2 (1.25)1.33 2(32.2)
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HYDRAULIC DESIGN OF CULVERTS AND HIGHWAY STRUCTURES
15.24
Chapter Fifteen
b. Or by using Fig. 15.7 where ke scale 0.2, culvert length, L 200 ft (n 0.012, same as on chart), area 35ft2, H 3.7 ft. 7. Compute the outlet control headwater by using Eq. (15.22) HW Wo H ho SoL 3.7 5.0 (0.015) (200) 5.7 ft Step 6. Determine the controlling headwater, HW Wc: HW Wi 8.2 ft . 5.7 ft. therefore HW Wc 8.2 ft and the culvert is in inlet control. Step 7. Compute the discharge over the roadway, Qr 1. Calculate the depth over the roadway, HW Wr: HW Wr HW Wc hc where hc 155 140 15 ft, then HW Wr 8.2 15 6.8 ft 2. As HW Wr 0, Qr 0 Step 8. Compute the total discharge. Qt Qd Qr 400 0 400 ft3 / s Step 9. Compute the outlet velocity, Vo and the normal depth, dn Normal depth for the culvert can be computed using Manning's equation A Q 1.49 R2/3 S1/2 n (7dn) (7ddn) 2/3 400 1.49 (0.015)1/2 0.012 (7 2dn) By trial and error, dn 2.8 ft. Normal depth in the culvert is less than the critical depth, therefore, the type of flow in the culvert is supercritical, a typical case for inlet control culverts. 400 Compute the velocity at the culvert outlet, Vo 20.4 ft/s (7) (2.8) 400 Compute the velocity at the downstream channel, V 13.3 ft/s (7) (3) Since V 1.5 Vo, an armoring riprap as defined in Sec. 15.8.1 can be appropriate for erosion protection at culvert outlet. Step 10.Review the results: the culvert barrel has 5 ft of cover, which is adequate and The allowable headwater (10 ft) is greater than HW Wi 8.2 ft, which is appro priate for design.
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HYDRAULIC DESIGN OF CULVERTS AND HIGHWAY STRUCTURES
Hydraulic Design of Culverts and Highway Structures 15.25
15.5 PERFORMANCE CURVES A performance curve is a plot of flow rate versus headwater depth or elevation for a culvert. Because a culvert has several possible control sections (inlet, outlet, throat), a given installation will have a performance curve for each control section and one for roadway overtopping, as shown in Fig. 15.15. In addition to these individual performance curves, an overall culvert performance curve can be constructed by the controlling portions of the individual performance curves for inlet, outlet, and overtopping. Inlet control. The curves for the inlet control culverts can be plotted either by using the design equations or the inlet control nomographs described in Sec. 15.4.1. Outlet control. For the outlet control culverts there are a number of ways to plot the performance curves; the use of Eqs. (15.1–15.14), outlet control nomographs, or backwater calculations. In the first two cases, the flow rates that will be used for the design of the culvert are chosen, then the corresponding total losses are computed and are added to the elevation of the hydraulic grade line at the culvert outlet to obtain the headwater. In the backwater method the headwater elevations are computed for the corresponding flow rates by adding the inlet losses to the energy equation. The energy grade line at the culvert inlet should reflect this adjustment. Roadway overtopping. An overall performance curve is a useful tool to separate the culvert flow from the roadway overtopping flow and to compute their respective magnitudes. In order to develop a typical overall performance curve as shown in Fig. 15.15, the step-by-step procedure as defined in the following can be used: 1. A set of discharge values that will be used in the design of the culvert must be selected. Then the corresponding inlet and outlet headwater values must be computed. 2. The inlet and outlet control performance curves must be combined to develop a single performance curve for the culvert. 3. To compute the flow rates of the overtopping flow; calculate the depth of upstream water surface above the roadway for each one of the flow rates selected in Step 1. and use Eq. (15.16) and the depth calculated in Step 3i. 4. Then the overall performance curve will be generated by adding the overtopping and culvert flow at the corresponding backwater elevations. Example. Develop a performance curve for two 48-in corrugated metal pipe culverts (Manning's n 0.024) with metal end sections. The culvert is 150 ft long and on a 0.08 percent slope. The roadway is a 30-ft-wide paved crossing that can be approximated as a broadcrested weir 130 ft long. The roadway centerline elevation is 109 ft. The culvert invert elevations are 100 ft. at the inlet, and 99.98 ft at the outlet. Tailwater discharges and depths are listed below: Q (ft3 /s)
80
TW (ft)
102.4
120
160
200
240
280
320
360
103.0
103.3
103.5
103.7
104.1
104.3
104.5
Step 1. Compute inlet and outlet control headwater elevations as tabulated below. Step 2. Develop the rating curve for the overtopping flow by using Eq. (15.16) Qr kt Cd L (HW Wr)1.5 Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
HYDRAULIC DESIGN OF CULVERTS AND HIGHWAY STRUCTURES
15.26
Chapter Fifteen
FIGURE 15.15 Culvert performance curve with roadway overtopping. (From Normann et al., 1985)
HWr
Cd
kt
L
Qr (Overtopping Flow)
Qc (Culvert Flow)
Qt (Total Flow)
0.30
3.00
1
130
64
235
299
0.50
3.05
1
130
184
240
424
Step 3. Draw the performance curve for the inlet control, outlet control, and overtopping flow, shown in Fig. 15.17, by using the headwater calculations and Step 2.
Inlet Control Total flow Flow per barrel Q/N HWi/D HWi ELhi (1) (2) (3) 40 102.72 2.72 80 0.68
Tw
de (4)
Outlet Control ke (dc+D)/2 ho (6) (5)
h (7)
ELho (8)
ELh (9)
2.40 1.84
2.92
2.92 0.50 0.65 103,55 103,55
120
60
0.88
3.52
103.52 3.00 2.31
3.16
3.16 0.50 1.50 104,64 104,64
160
80
1.08
4.32
104.32 3.30 2.73
3.37
3.37 0.50 2.60 105,95 105,95
200
100
1.30
5.20
105.20 3.50 3.11
3.56
3.56 0.50 4.20 107,74 107,74
240
120
1.58
6.32
106.32 3.70 3.47
3.74
3.74 0.50 5.80 109,52 109,52
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HYDRAULIC DESIGN OF CULVERTS AND HIGHWAY STRUCTURES
Hydraulic Design of Culverts and Highway Structures 15.27
(1) (2) (3) (4) (5) (6) (7) (8) (9)
Use Q/NB for box culverts Use Figure 15.16, Scale 1 ELhi HWi 100 dc 0.325 (Q/ND) ^ 0.66 0.083 Use Equation 15.1 Use Table 15.1 Use Figure 15.17 ELho 99.98 h ho Elh max (Elhi, ELho)
15.6 MATERIALS AND CULVERT GEOMETRY Culverts are required to serve under various conditions including wide temperature variations, heavy abrasion, erosion and sedimentation within the culvert barrel. Because of these factors there are a variety of culvert materials used to maintain the long life and durability of culverts. The three main types used in application are the precast concrete, monolithic or cast-in-place concrete, and corrugated aluminum and steel. Some other options are listed in Table 15.4. The culvert must be able to carry loads without large deflection and efficiently convey the water through the culvert opening. Therefore, the selection of culvert material should consider the desired hydraulic efficiency, structural, and operational characteristics. One of the factors in the physical strength of the culverts is the type of the material used. The two main applications are with rigid and flexible culverts. Rigid culverts are usually of reinforced concrete pipe and their primary strength is built into the pipe itself. Special designs are possible that permit the pipe to support any magnitude of overload. Their durability against freezing, thawing and erosion are major advantages, combined with good hydraulic efficiency. In design with smaller culvert size requirements and lighter loads, nonreinforced concrete pipe can be used. Flexible culverts, generally corrugated metal pipe or plastic pipe, provide their strength by the interaction of the pipe and the surrounding backfill material. Corrosion is a concern with the corrugated metal pipes. In many applications the operational characteristics of a culvert are as important as the hydraulic and structural elements. The culvert designer must select the material with the consideration of hydraulics, structures, maintenance, and geotechnical characteristics of the culvert site. The proper culvert design should optimize the hydraulic and structural characteristics of the culvert while minimizing the total cost. The culvert material which directly affects the service life, is a key parameter in the total cost of the culvert. Another important factor for the efficient design is the shape of the culvert conduit. The shape has an impact on the hydraulics and structural characteristics. The type of the maintenance and accessibility for deterioration and deformation repairs are also affected by the shape. The most common culvert conduit shapes are circular, rectangular, pipearch, and arch.
15.7 LOCATION AND ALIGNMENT OF CULVERTS The location and alignment has an important affect on the economics of culvert design. As a general criteria, culvert location should not alter the natural drainage, and be placed at the
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HYDRAULIC DESIGN OF CULVERTS AND HIGHWAY STRUCTURES
15.28
Chapter Fifteen
FIGURE 15.16 Headwater depth for CM pipe culverts with inlet control. (From Normann et al., 1985)
bottom of the ravine. It is almost impossible to find a unique rule that could be applied to every culvert. The final location and alignment of the culverts depend on the designer's own experience in hydrology, hydraulics and structural aspects of culvert design. This will help the designer to achieve maximum economy, utility, and safety (Hendrickson, 1957). There are different options to locate the culvert; namely, bottom location placement, and top location placement.
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HYDRAULIC DESIGN OF CULVERTS AND HIGHWAY STRUCTURES
Hydraulic Design of Culverts and Highway Structures 15.29 TABLE 15.4
Culvert Materials
Shape
Material
Protective Coating
Circular
Steel
Galvanizing
Rectangular
Aluminium
Aluminized
Arch
Al/Fe Alloy
Bituminous
Pipe-arch
Concrete
Polymer
Elliptical
Plastic
Concrete
Source: From Reagan (1993).
FIGURE 15.17 Head for standard CM pipe culverts flowing full. (n 0.024). (From Normann et al., 1985)
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HYDRAULIC DESIGN OF CULVERTS AND HIGHWAY STRUCTURES
15.30
Chapter Fifteen
FIGURE 15.18 Example performance curve.
15.7.1 Bottom Location Placement The invert of the culvert with bottom location placement follows the line and grade of the natural stream channel as closely as possible (Fig. 15.19). Although this type of installation may result in some curvature and possible changes in grade to obtain satisfactory bedding, it has a good hydraulic performance. However, there are several disadvantages to this type of location. In applications with steep natural gradients, the velocity at the culvert outlet can be high and may require erosion and scour protection. In cases where the bottom of the stream channel lies in a ravine, the culverts can be under high fills resulting in long culverts and heavy loads. Cost comparison with other locations must include consideration of headwalls, spillways, energy dissipation structures, channel changes, and extra maintenance. In cases of steep channels, it is possible to reduce the length and required strength of culvert by modifying the natural slope and alignment. This can be done by keeping the culvert inlet at the bottom of the channel, but shifting the culvert outlet horizontally and vertically and placing it in the downstream face of the embankment as shown in Fig. 15.20. Although this configuration can result in a more economical solution, an appropriate design must be provided to return the flow at the culvert outlet to the natural drainage channel. It may be necessary to protect the downstream slope against erosion, depending on the type of the material in the fill. In some instances, a form of spillway may be required, particularly if the return channel is steep. In other cases it may be feasible to project the culvert beyond the toe of the embankment and allow the water to drop into a pool.
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HYDRAULIC DESIGN OF CULVERTS AND HIGHWAY STRUCTURES
Hydraulic Design of Culverts and Highway Structures 15.31
FIGURE 15.19 Bottom location culverts. (From Hendrickson, 1957)
15.7.2 Top Location Placement In some applications the culvert can be placed near the top of the embankment fill just beneath the roadway grade. The main advantage over the bottom installation is its shorter length and considerably reduced load that the culvert has to carry. This type of installation forces the embankment to serve as a dam and results in a rise in the level of the upstream pool until water flows through the culvert. Therefore the embankment material must ensure stability during flood events. One other concern is the possible accumulation of sediment and debris due to stagnant ponding on the upstream side when the water level drops below the level of the culvert invert. Similar to the second solution of the bottom location culverts, spillways or auxiliary channels might be required to carry water from the outlet without undercutting or eroding the embankment.
15.7.3 Siphons The site conditions might require the modification in the horizontal and vertical orientation, and under certain conditions a culvert may become a siphon. A siphon is defined as a tube by which liquid can be transferred by means of atmospheric pressure from a higher to a lower level over an intermediate elevation. This is the case for a culvert laid on a broken grade line as shown in Fig. 15.21A. Due to the change in grade, the hydraulic grade line drops below the crown of the culvert when the inlet and outlet are submerged. The negative pressure or vacuum in the center section of the culvert results in a siphon. In certain cases the designer may have to use the inverted siphon alternative, which is also named as sag culvert. The inverted siphon is a consequence of a hydraulic grade line above the crown of the culvert, thus causing the culvert to flow under pressure. These culverts can be constructed under low fills where it is necessary to excavate down to get the
FIGURE 15.20 Modified bottom location culverts. (From Hendrickson, 1957)
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HYDRAULIC DESIGN OF CULVERTS AND HIGHWAY STRUCTURES
15.32
Chapter Fifteen
required cross-sectional area for hydraulic capacity (Fig. 15.21B), or they can result from the gradual accumulation of sediment and debris at the upstream and downstream ends. In irrigated areas, inverted siphons are used to carry the water from open ditches and canals below highways, railroads, and other obstructions. This type of culvert must be periodically maintained as a protection against sediment and debris accumulation. Also they are not suggested for intermittent streams, because stagnant water can lay in them for a long period of time and can result significant health hazards.
15.7.4 End treatments End treatments are applied to culverts for many purposes, including; retention of roadway embankment, improvement of hydraulic efficiency, protection of highway embankment from discharge momentum, protection against piping, debris control, traffic safety. Headwall end treatments are very common structures standardized with construction details and are economical and practical. In some instances, hydraulic efficiency can be improved by minor details such as flared wingwalls on the headwall. Prefabricated end sections are very popular due to their convenience of installation and economy. Their application is limited to smaller culverts. The flowline length of a mitered end culvert (Fig. 15.22) corresponds to the distance between the upstream and downstream roadway toe-of-slope locations. The top length is measured between the points of interception with the roadway side slopes and the barrel top. Mitered end treatments are economical and relatively simple in construction. Projecting end treatments are similar to mitered end treatments except that the upper portion of the culvert barrel protrudes from the side slopes of the roadway embankment as shown in Fig. 15.22. Projecting end treatments are convenient and economical. However, they have serious problems regarding, hydraulics (greater entrance losses), structural (inadequate perimeter containment), operational (potential for collision with maintenance operations). Another alternative is the use of different improved inlet conditions. Since a culvert usually represents a severe constriction to the stream flow, the physical characteristics of
A
B FIGURE 15.21 A. Culvert siphons: culvert siphon, B. inverted siphon culvert. (From Hendrickson, 1957)
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HYDRAULIC DESIGN OF CULVERTS AND HIGHWAY STRUCTURES
Hydraulic Design of Culverts and Highway Structures 15.33
the entrance to the culvert have a significant effect on the operational capacity. This effect is reflected by the headloss coefficient, ke, associated with the entrance type used. With reference to the table of entrance loss coefficients in Table 15.1, note that some ke are much lower than others depending on the type of entrance. Often, improvement in the hydraulic capacity of the culvert can be realized by simple and inexpensive expedients such as, beveling or rounding of the inside perimeter of the entrance to the conduit and ensuring installation of reinforced concrete pipe with the grooved end upstream. Beveled edges are commonly used with box culverts and headwall structures. As given in Normann, et al., (1985), design charts are available for two bevel angles, 45º and 33.7º (Fig. 15.23). Although the 33.7º bevels result in a better inlet performance than the 45º bevel, it requires additional structural modifications. As a result, the 45o bevel is the preferred alternative. Groove ends provide similar hydraulic performance as the bevels. Figs. 15.24 and 15.25 demonstrate bevels with other inlet improvements. Side-tapered inlets provide enlarged culvert entrance with a transition to the original barrel dimensions. The
FIGURE 15.22 Four standard inlet types. (From Norman et al., 1985)
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HYDRAULIC DESIGN OF CULVERTS AND HIGHWAY STRUCTURES
15.34
Chapter Fifteen
test results of FHWA (Harrison et al., 1972) resulted in a section geometry as shown in Fig. 15.24. The throat section as identified in this figure is used to control the capacity and maximize the efficiency of side-tapered inlets. Slope-tapered inlets have steeper slopes at the entrance as compared to the culvert barrel. The steeper slope is provided to increase the head on the throat section and create additional fall. Depending on the magnitude of the available fall, the inlet capacities can be significantly higher than the conventional culvert with square edges. Culvert inlet performance also can be improved cost effectively, by the incorporation of special entrance geometries. Effectively, such improvements are used to reduce the headlosses due to flow contractions and to increase capacity of culverts operating under steep slope regimes. Expanded entrances have a “funneling” effect on the flow. These flared entrances allow more discharge into the conduit, thus significantly improving the efficiency of the conduit. Their main advantage is in the reduction of constriction loss coefficients. While there may be a minor loss reduction in outlet control culvert operations, they are mostly effective in situations of steep slope regime. Culverts operating in a supercritical flow regime often exhibit excessive outlet velocities. The usual effect of an improved entrance on such a configuration results in more efficient use of the culvert barrel with the further result of a lower outlet velocity. Transitions of flow from the relatively wide channel approach to the constricted culvert opening may be useful in some instances. Both expanded entrances and transitions have the potential of clogging due to debris. Larger debris is allowed to enter the culvert partially, and subsequently becomes lodged in the smaller opening provided by the main conduit. Improved inlets are useful in some situations but the following considerations should be made in design:
FIGURE 15.23 Beveled edges. (From Normann et al., 1985)
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HYDRAULIC DESIGN OF CULVERTS AND HIGHWAY STRUCTURES
Hydraulic Design of Culverts and Highway Structures 15.35
FIGURE 15.24 Typical side-tapered inlet detail. (From State of Florida Department of Transportation, 1987)
• The improved inlet often is a relatively expensive item, sometimes costing more than the rest of the culvert. • Improved inlets are effective only in supercritical flow situations. • Debris, erosion, and other maintenance problems can represent significant operational costs when improved inlets are used. • The design charts and methods for improved inlet design are applicable to one or two barrels only of reinforced concrete box culvert configurations. • The design charts require a culvert face which is perpendicular to the stream flow. This can be a construction and maintenance problem when the culvert is skewed to the roadway centerline.
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HYDRAULIC DESIGN OF CULVERTS AND HIGHWAY STRUCTURES
15.36
Chapter Fifteen
FIGURE 15.25 Typical slope-tapered inlet detail. (From State of Florida Department of Transportation, 1987)
15.8 SPECIAL CONSIDERATIONS 15.8.1 Erosion The high outlet velocities observed at the culvert outlets may result in excessive scour of the channel in the vicinity of the outlet. The variety in the soil type of natural channels and varying flow characteristics at the culvert outlet enforces the use of different methods to
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HYDRAULIC DESIGN OF CULVERTS AND HIGHWAY STRUCTURES
Hydraulic Design of Culverts and Highway Structures 15.37
control or protect the channel against potential damaging effects. A method to predict and analyze scour conditions is given by Reagan (1993). Some of the commonly used techniques to provide protection against scour are: minor structural elements, velocity protection devices, velocity control device Minor structural elements. In the case of low outlet flow velocities (outlet velocity less than 1.3 times the average natural stream velocity) cutoff walls are commonly used. They are provided in streams with adequate downstream protection or where the outlet hydraulics and soil conditions do not result in structural instability. Cutoff walls should be constructed such that its depth equals the depth of scour, and its width covers at least onethird the scour width. In cases where the cutoff wall depth exceeds 6.0 ft, either a different form of protection must be provided or the structural stability of the walls must be maintained (Flood Control District of Maricopa County, 1996). Velocity protection device. For culvert outlet velocities greater than 1.3 times the natural stream velocity, but less than 2.5 times, armoring riprap is used as a protection against the damaging potential. Different types of armoring material can be used including concrete riprap, rock riprap, vegetation and synthetic sodding methods. The details on the size and placement of riprap for various types of design conditions can be found in Corry et al. (1983), Maccaferri Gabions, Inc. (1997), and Simons and Senturk.(1977). Velocity control device. Energy dissipators are appurtenances that may be used at the downstream end of a culvert for the purpose of reducing the damaging effects of outlet velocities greater than 2.5 times the natural stream velocity. The control is accomplished by the use of an energy dissipator. The turbulent outlet flow is contained within this dissipator, resulting in a more tranquil flow downstream of it. Due to the high cost of an energy dissipator structure, it must be properly designed and constructed to control the excessive outlet velocity efficiently. For more detailed information on the design of energy dissipators the reader is referred to Corry et al. (1983) and Simons and Senturk. (1977). A consequence of high outlet velocities is the local scour in the vicinity of the culvert exit. A detailed discussion of the local scour at culvert outlets is presented by Breusers and Raudkivi (1991). Also some of the research on scour at culvert outlets by Ruff (1982) can be summarized as follows: • The dimension of the scour hole can not be predicted as a unique function of outlet velocity or exit Froude number, because the variable tailwater has a significant effect on the scouring process. • The scour depth is dependent on the dimensionless ratio (outlet velocity/outlet critical shear velocity), whereas scour length and width are correlated to the exit Froude number. • The effect of gradation on scour depth is significant at low exit velocities only, therefore it is appropriate to use the median grain diameter, d50 , in computations. 15.8.2 Sedimentation Sedimentation is a concern in the design of culverts and may result in excessive loss in efficiency at the inlet, outlet, or within the culvert barrel. The two major factors that influ-
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HYDRAULIC DESIGN OF CULVERTS AND HIGHWAY STRUCTURES
15.38
Chapter Fifteen
ence the sedimentation characteristics of a culvert are the barrel roughness and the slope. Sediment depositing within the culvert will change it's roughness (i.e., Manning's coefficient, n) and will adversely affect the flow conditions within the culvert. In terms of the stability of the culvert system, it is desirable to align the culvert at the natural stream slope. In cases with milder culvert gradients, low-flow velocities within the culvert will force the sediment particles to deposit. If the culvert is set at a steeper slope than that of the natural channel, the increased velocities within the culvert may result in excessive erosion along the culvert. To maintain the stability and erosion/sedimentation characteristics of a culvert, the sediment characteristics of the channel system and the specific site conditions must be carefully studied. A channel system with a degrading pattern may not result in a sedimentation problem at the outlet but will have the potential to provide excessive sediment to the culvert inlet. For aggrading systems, sediment accumulation in the channel will reach to the culvert outlet and will gradually move upstream within the culvert. The site-specific conditions may result in the use of multibarrel installation or culverts with depression. Multibarrel culverts are used with wide and shallow channel systems or low fills and when used in a curved alignment, sedimentation is a natural consequence. In such cases the recommended installation is straight culverts aligned with the channel upstream. Culverts with depression may experience sediment due to a milder culvert slope than that of the upstream channel. Storm events with increased culvert velocities result in a self-cleansing effect. Details of the erosion and sedimentation characteristics of the channel and the culvert system can be found in Raudkivi (1990, 1993), Richardson et al. (1990), Simons and Senturk (1977), Simons, Li & Associates (1982), Vanoni (1977), and Yang (1996).
15.8.3 Control of Debris To control the debris that flows through culverts, certain aspects of the site such as stream velocity, natural channel slope and alignment, existing and future vegetation and land use patterns, and frequency of flood events must be carefully analyzed. Debris can either accumulate at the culvert entrance or within the culvert barrel, resulting in upstream flooding and roadway overtopping. For small volumes of debris accumulation, regular maintenance and inlet conditions will affect the hydraulic performance of culverts. Smooth inlets with single barrel configuration, aligned with the natural stream will enhance the hydraulic efficiency of culverts. Typically three debris control methods are used for sites with higher volumes of debris accumulation: 1) debris is intercepted in the vicinity of the culvert inlet and can be accomplished by the use of debris racks, floating drift booms, and debris basins, 2) debris is deflected away from the entrance and collected in a basin and generaly debris risers and cribs are used, 3) debris is transported through the barrel by using an oversized culvert. The guidance for considerations and design of debris control structures can be found in FHWA HECNo. 9 (Reihsen and Harrison., 1971).
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HYDRAULIC DESIGN OF CULVERTS AND HIGHWAY STRUCTURES
Hydraulic Design of Culverts and Highway Structures 15.39
15.9 STREAM STABILITY AT HIGHWAY STRUCTURES 15.9.1 Basic Engineering Analysis The factors that affect stream stability and potentially bridge stability at highway stream crossings can be classified as geomorphic factors (Fig. 15.26) and hydraulic factors (Fig. 15.27). The FHWA Hydraulic Engineering Circular (HEC) No. 20 (Lagasse et al., 1995) presents detailed descriptions of the various factors and how they affect stream stability. Much of the material presented in this section is adapted from the FHWA HEC No. 20 which recommends a three-level procedure for analyzing stream stability: Level 1. Steps in qualitative and other geomorphic analysis, (1) stream characteristics, (2) land use changes, (3) overall stability, (4) lateral stability , (5) vertical stability, and (6) stream response Level 2. Steps in basic engineering analysis (see Fig. 15.28). Level 3. Mathematical and physical model studies. The remaining discussion will be on the steps in the basic engineering analysis (Fig. 15.28). Evaluation of flood history (Step 1) is an integral step in characterizing watershed response and morphologic change, particularly in arid regions. It is important to study flood records and corresponding stream responses using aerial photography or other physical information. Development of the rainfall-runoff relation is important to understand watershed conditions and historical changes in the watershed. To evaluate the hydraulic conditions (Step 2), it is common practice to compute water surface profiles and hydraulic conditions using a computer code. For the analysis and design of bridge crossings, the Federal Highway Administration's WSPRO (see Shearman, 1987) computer program is recommended because the computational procedure for evaluating bridge hydraulics is superior to other models such as the U.S. Army Corps of Engineers’ HEC-2 (1991) and The U.S. Army Corps of Engineers’ (1995) HEC– RAS. (River Analysis System). The input structure for WSPRO was specifically developed to facilitate bridge design. Step 3 involves performing an analysis of the bed and bank material to determine particle size gradation and other properties such as shape, fall velocity. Step 4 involves evaluating the watershed sediment yield using a method such as the Universal Soil Loss Equation (USLE). Step 5 involves performing an incipient motion analysis to evaluate the relative channel stability. For most river conditions the following equation derived from the Shields diagram (see Chap. 6) can be used (Lagasse et al., 1995). τ Dc 0.047(γs γ) γ
(15.23)
where Dc is the diameter of the sediment particle at incipient motion condition in (m), τ is the boundary shear stress in N/m2 (pa); γs and γ are the specific weights of the sediment and water, respectively, in N/m3, and 0.047 (for sand-bed channels) is the dimensionless coefficient referred to as the Shields parameter. Step 6 is to evaluate armoring potential. Armoring is the natural process where by an erosion-resistant layer of relatively large particles is formed on a streambed due to the removal of finer particles by the stream flow. An armoring layer which is sufficient to protect the bed against moderate discharges can be disrupted during high discharges and then may be restored as flows diminish. By determining the percentage of bed material equal
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15.40
Chapter Fifteen
FIGURE 15.26 Geomorphic factors that affect stream stability. (Fom Brice and Blodgett, 1978 as presentel in Laggasse, et al., 1995)
to or larger than the armor size, Dc, computed using Eq. (15.23) for incipient motion, the depth of scour ys necessary to establish an armor layer can be determined using (Pemberton and Lara, 1984) the following equation as 1 ys ya[ 1] Pc
(15.24)
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HYDRAULIC DESIGN OF CULVERTS AND HIGHWAY STRUCTURES
Hydraulic Design of Culverts and Highway Structures 15.41
FIGURE 15.27 Hydraulic and location factors that affect stream stability. (From Laggasse et al., 1995)
where ya is the thickness of the armor layer and Pc is the decimal fraction of material coarser than the armoring size, ya ranges from one to three times the armor size, Pc ' depending on the value of Dc (Lagasse et al., 1995). Step 7 is the evaluation of stage-discharge rating curve shifts. Scour and deposition is the most common cause of rating curve shifts. Rating curves that continually shift indicates channel instability. Step 8 is to evaluate the scour condition at bridge crossings.
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15.42
Chapter Fifteen
FIGURE 15.28 Flowchart for level 2: Basic Engineering Analyses. (From Lagasse et al., 1995)
Section 15.10 provides a detailed description of procedures based on the FHWA-HEC No. 18.(Richardson and Davis, 1995). Calculation of the three components of scour (local scour, contraction scour, and regional aggradation /degradation) quantifies the potential instability at a bridge crossing. Once the above steps have been completed, if a more detailed analysis is needed, then the level 3 analysis is performed in which mathematical and/or physical models are used to evaluate and assess stream stability. If a more detailed analysis is not needed then the counter measures are selected and designed. Historically the need for a level 3 type analysis has been for highrisk locations and for extraordinarily complex problems. However,
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HYDRAULIC DESIGN OF CULVERTS AND HIGHWAY STRUCTURES
Hydraulic Design of Culverts and Highway Structures 15.43
because of the importance of stream stability to the safety and integrity of bridges, the level 3 analysis should be performed routinely according to Lagasse et al. (1995).
15.9.2 Countermeasures (Flow Control Structure) for Stream Instability Countermeasures are measures incorporated into a highway-stream crossing system to monitor, control, inhibit, change, delay, or minimize stream and bridge stability problems or action plans for monitoring structures during and/or after floods events (Lagasse et al., 1995). Selection of a countermeasure for a bank erosion problem is dependent on the erosion mechanism, stream characteristics, construction and maintenance requirements, potential for vandalism and costs. Of particular interest are flow control structures which are defined as structures, either within or outside a channel, that acts as a countermeasure by controlling the direction, velocity, or depth of flowing water. (Richardson et al., 1990). These types of structures are also referred to as river training works. Various types of flow control structures are shown in Fig. 15.29. 15.9.2.1 Spurs. Spurs are structures or embankments that are projected into streams at the same angle in order to deflect flowing water away from critical zones to prevent erosion of the banks, and to establish a more desirable channel alignment or width (Richardson et al., 1990). Spurs have several functions: (1) to protect highway embankments that form approaches to bridge crossings; (2) to channelize wide, poorly defined streams into well-defined channels;
FIGURE 15.29 Placement of flow control structures relative to channel banks, crossing, and floodplain. Spurs, retards, dikes, and jack fields may be either upstream or downstream from the bridge. (From Brice and Blodgett, 1978)
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15.44
Chapter Fifteen
(3) to establish and maintain new alignments through deposition of sediments; and (4) to halt meander migration at banks. Spurs are classified as retarder spurs, retarder/deflector spurs, and deflector spurs based on their permeability, which is defined as the percentage of the spur surface area facing the streamflow. Accordingly, retarder spurs are permeable and function by retarding flow velocities near stream banks. Retarder/deflector spurs are permeable and function by retarding flow velocities at stream banks and direct flow away from the banks. Deflector spurs are impermeable, and deflect flow currents away from banks. Guide banks (spur dikes). are placed at or near the ends of approach embankments to guide the streamflow through a bridge opening. They are used to prevent erosion of the approach embankments by cutting off the flow adjacent to the embankment, and by guiding flow through the opening. Guide banks reduce separation of flow at the upstream abutment face and reduce abutment scoar as a result of less turbulent flow at the abutment face. Figure 15.30 illustrates a typical guide bank plan view. Figure 15.31 can be used to determine guide bank length, Ls, for designs greater than 15 m and less than 75 m. Lagasse et al. (1995) recommend that 15 m is the minimum length of guide banks and if the figure indicates a length larger than 75 m the design should be set at 75 m. FHWA practice has shown that a standardized length of 46 m has performed well (Lagasse et al., 1995). Other guidelines for design include the following (refer to Lagasse et al. (1995) for more detailed design concerns):
FIGURE 15.30 Typical guide bank. Guide banks should start at and be set parallel to the abutment and extend upstream from the bridge opening. The distance between the guide banks at the bridge opening should be equal to the distance between bridge abutments. Best results are obtained by using the guide banks with a plan form shape in the form of a quarter of an ellipse, with the ratio of the major axis (length Ls) to the minor axis (offset) of 2.5:1. This allows for a gradual constriction of the flow. If the length of the guide bank measured perpendicularly from the approach embankment to the upstream nose of the guide bank is denoted as Ls, the amount of expansion of each guide bank (offset), measured from the abutment parallel to the approach roadway, should be 0.4Ls. (Fom Lagasse et al. 1995 as modified from Bradley 1978)
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HYDRAULIC DESIGN OF CULVERTS AND HIGHWAY STRUCTURES
Hydraulic Design of Culverts and Highway Structures 15.45
Ls – LENGTH OF GUIDE BANKS (m) FIGURE 15.31 Nomograph to determine guide bank length. (From Lagasse et al., 1995)
• A minimum freeboard of 0.6 m above the design water surface elevation should be used. • Generally top widths of 3–4 m are used. • Upstream end of the guide bank should be round nosed. • Side slopes should be 1V:2H or smaller. • Rock riprap should be placed on the stream side face and around the end of the guide bank. • A gravel, sand, or fabric filter may be required to protect the underlying embankment material. To use Fig.15.31 the following parameters are needed: Qf is the lateral or floodplain discharge of either floodplain intercepted by the embankment in m3/s; Q30 m is the dis-
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HYDRAULIC DESIGN OF CULVERTS AND HIGHWAY STRUCTURES
15.46
Chapter Fifteen
charge in 30 m of stream adjacent to the abutment in m3/s; Q is the total discharge of the stream in m3/s; An2 is the cross-sectional flow area at the bridge opening at normal stage in m2; Vn2 Q/A / n2 is the average velocity through the bridge opening in m/s, and Qf /Q30m is the guide bank discharge ratio. To design the guide bank for the situation shown in Fig. 15.32 the following steps are followed: 1. Determine hydraulic design parameters (depths and velocities) using computer program WSPRO, HEC –2 or HEC–RAS. 2. Determine Qf in the left overbank and right overbank. 3. Determine Q30 m and Qf /Q30 m for the left and right overbank. 4. Determine length of the guide bank, Ls. 5. Select crest width, crest elevation, and riprap design. 15.9.2.2 Check dams (channel drop structures). Check dams or channel drop structures are countermeasures placed downstream of highway crossings to arrest head cutting and to maintain stable streambed elevation in the vicinity of the bridge. They are typically constructed of rock riprap, concrete, sheet piles, gabions, or treated timber piles. A typical vertical drop structure with a free overfall is shown in Fig. 15.33A. Pemberton and Lara (1984) recommended the following equation to estimate the depth of scour downstream of a vertical drop
FIGURE 15.32 Example guide bank design. (From Lagasse et al., 1995)
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HYDRAULIC DESIGN OF CULVERTS AND HIGHWAY STRUCTURES
Hydraulic Design of Culverts and Highway Structures 15.47
ds KH Ht0.225 q0.54 dm
(15.25)
where ds is the local scour depth for a free overfall, measured from the streambed downstream of the drop in m; q is the discharge per unit width in m3/s/m; Ht is the total drop in head, measured from the upstream to the downstream energy grade line in m; dm is the tailwater depth in m; and K 1.90. Equation (15.25) is independent of grain size of the bed material and will acknowledge that the bed will scour regardless of the type of material. Based on the energy equation V2 V2 Ht Yu u Zu Yd d Zd 2g 2g
(15.26)
where Y is the flow depth in m; V is the velocity in m/s; Z is the bed elevation in m, and g is the acceleration due to gravity (9.81 m/s2). Subscripts u and d refer to upstream and downstream of the channel drop as shown in Fig. 15.33. Example.(adapted from Lagas, et al.,1995) For the following hydraulic parameters, determine the depth of local scour. Design discharge 167 m3/s, channel width 32 m; Yu 3.22 m; dm 2.9 m; Yd 2.9 m; unit discharge (q) 5.22 m3/s/m; Vu 1.62 m/s; Vd 1.80 m/s; drop height (h) 1.4 m. The results are presented in Fig. 15.33B. Solution: Compute Ht using Eq. (15.26) 1.622 1.802 Ht 3.22 1.4 2.9 0 2 (9 .8 1 ) 2 (9 .8 1 )
1.69 m Compute ds using Eq. (15.25)
Ht
Yu
yd Zu Zd
FIGURE 15.33. Schematic of a vertical drop caused by a check dam (From Laggasse et al., 1995)
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15.48
Chapter Fifteen
ds 1.90(1.69)0.225(5.22)0.54 2.9 2.3 m The unsupported height of the structure is h ds 3.7 m.
15.10 BRIDGE SCOUR This section presents methods and equations for determining total scour at a bridge, that is, long-term aggradation or degradation, contraction scour, and local scour. Bridge scour is the erosion or removal of streambed or bank material from bridge foundations due to flowing water, usually considered as long-term bed degradation, contraction, and local scour. Much of the material in this section has been adapted from the FHWA, Hydraulic Engineering Circular No. 18 by (Richardson and Davis 1995).
15.10.1 Design Approach Before applying the various methods for estimating scour, it is necessary to (1) determine the fixed-bed channel hydraulics, (2) determine the long-term impact of degradation or aggradation on the bed profile, (3) if degradation occurs, adjust the fixed-bed hydraulics to reflect this change, and (4) compute the bridge hydraulics are 1995. The seven steps recommended in “Evaluating Scour at Bridges” (Richardson and Davis, 1995) are. (1) determine the scour analysis variables (2) analyze long-term bed elevation change (3) evaluate the scour analysis method (4) compute the magnitude of contraction scour (5) compute the magnitude of local scour at piers (6) compute the magnitude of local scour at abutments (7) plot and evaluate the total scour depths. Many of the hydraulic variables used in estimating scour can be obtained from computer models for water profile computation such as WSPRO developed by the U.S. Department of Transportation (1990). The following definitions are from the FHWA Hydraulic Engineering Circular No. 18 by Richardson and Davis (1995). Aggradation and degradation are long-term streambed elevation changes due to natural or human-induced causes which can affect the reach of the river on which the bridge is located. Aggradation involves the deposition of material eroded from the channel or watershed upstream of the bridge; whereas, degradation involves the lowering or scouring of the streambed due to a deficit in sediment supply from upstream. Contraction scour in a natural channel or at a bridge crossing, involves the removal of material from the bed and banks across all or most of the channel width. This component of scour can result from a contraction of the flow area, an increase in discharge at the bridge, or both. It can also result from a change in downstream control of the water surface elevation. The scour is the result of increased velocities and shear stress on the channel bed. Contraction of the flow by bridge approach embankments encroaching onto the floodplain and/or into the main channel is the most common cause of contraction scour. Contraction scour can be either clear-water or live-bed. Live-bed contraction scour occurs when there is transport of bed material in the approach reach, whereas clear-water contraction scour occurs when there is no bed mate-
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HYDRAULIC DESIGN OF CULVERTS AND HIGHWAY STRUCTURES
Hydraulic Design of Culverts and Highway Structures 15.49
rial transport in the approach reach or the bed material being transported in the upstream reach is so fine that it washes through the contracted section. Live-bed contraction scour typically occurs during the rising stage of a runoff event, while refilling of the scour hole occurs during the falling stage. Also, clear-water scour at low or moderate flows can change to live-bed scour at high flows. This cyclic nature creates difficulties in measuring contraction scour after a flood event. Local scour involves removal of material from around piers, abutments, spurs, and embankments. It is caused by an acceleration of flow and resulting vortices induced by the flow obstructions. Local scour can also be either clear-water or live-bed scour. Live-bed local scour is cyclic in nature; that is, the scour hole that develops during the rising stage refills during the falling stage. Lateral stream migration of the main channel of a stream within a floodplain may increase pier scour, erode abutments or the approach roadway, or change the total scour by changing the flow angle of attack at piers. Factors that affect lateral stream movement also affect the stability of a bridge. These factors are the geomorphology of the stream, location of the crossing on the stream, flood characteristics, and the characteristics of the bed and bank materials.
15.10.2 Contraction Scour 15.10.2.1 Live-bed contraction scour. Live-bed contraction scour occurs at a bridge when there is transport of bed material in the upstream reach into the bridge cross section,so that the area of the contracted section increases until, the transport of sediment out of the contracted section equals the sediment transported into the section (Richardson and Davis, 1995). The width of the contracted section is constrained and depth increases until the limiting conditions are reached. Laursen (1960) derived the following live-bed contraction scour equation based on a simplified transport function, transport of sediment in a long contraction, and other simplifying assumptions. Q2 6/7 W1 k1 n2 k2 y y2 1 Q1 W2 n1
(15.27)
The average scour depth is ys y2 y0, where y1 average flow depth in the upstream main channel (m), y2 average flow depth in the contracted section (m), y0 existing flow depth in the contracted section before scour (m), Q1 flow in the upstream channel transporting sediment (m3/s), Q2 flow in the contracted channel (m3/s), which is often equal to the total discharge unless the total flood flow is reduced by relief bridges, water overtopping the approach roadway, or in the setback area, W1 bottom width of the upstream main channel (m), W2 bottom width of main channel in the contracted section (m), n2 Manning's n for contracted section, n1 Manning's n for upstream main channel, k1 and k2 exponents depending on the mode of bed material transport (Table 15.5), V* (gyS1)1/2 shear velocity in the upstream section (m/s), ω median fall velocity of the bed material based on the D50, (m/s) (see Fig. (15.34), g acceleration of gravity (9.81 m/s2), S1 slope of energy grade line of main channel (m/m) and D50 median diameter of the bed material (m). The location of the upstream section needs to be located with engineering judgment. If WSPRO is used to obtain the values of the quantities, ( y1, Q1, W1, and n1) the upstream channel section is located a distance equal to one bridge opening from the upstream face of the bridge.
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HYDRAULIC DESIGN OF CULVERTS AND HIGHWAY STRUCTURES
15.50
Chapter Fifteen
A modified version of Laursen's (1960) equation for live-bed scour at a long contraction is recommended to predict the depth of scour in a contracted section (Richardson and Davis, 1995). The original equation given as Eq. (15.27) is modified by eliminating the ratio of Manning's n for the contracted and upstream sections. The equation assumes that bed material is being transported in the upstream section, and is given as Q2 6/7 W1 k1 y y2 1 Q1 W2
(15.28)
and the average scour depth (m) is ys y2 yo and k1 is obtained from Table 15.5(b). The Manning's n ratio in Eq. (15.27) can be significant for the condition of a dune bed in the main channel and a corresponding plane bed, washedout dunes or antidunes in the contracted channel (Richardson and Davis, 1995). However, Eq. (15.27) does not adequately account for the increase in transport which occurs as a result of the bed planing out (which decreases resistance to flow, increases the velocity and the transport of bed material at the bridge). Laursen's Eq. (15.27) results in a decrease in scour for this case, whereas in reality, there would be an increase in scour depth. In addition, at flood flows, a plain bedform will usually exist upstream and through the bridge waterway, and the values of Manning's n will be equal. Consequently, the n value ratio is not recommended by Richardson and Davis (1995) as presented in the recommended Eq. (15.28). Scour depths with live-bed contraction scour may be limited by coarse sediments in the bed material armoring the bed. Where coarse sediments are present, Richardson and Davis (1995) recommend that scour depths be calculated for live-bed scour conditions using the clearwater scour equation (given in Sec. 15.10.2.2) in addition to the live-bed equation, and that the smaller calculated scour depth be used. 15.10.2.2 Clear-water contraction scour. Clear-water contraction scour occurs in a long contraction when (1) there is no bed material transport from the upstream reach into the downstream reach or (2) the material being transported in the upstream reach is transported through the downstream reach mostly in suspension and at less than capacity of the flow (Richardson and Davis, 1995). The area of the contracted section increases until, the velocity of the flow (V) V or the shear stress (τo) on the bed is equal to the critical velocity (V Vc) or the critical shear stress (τc) of a certain particle size (D) in the bed material. Widths (W) W of contracted section usually are constrained and the depth (y) increases until the limiting conditions are reached. Following the development given by Laursen (1963), equations for determining the clear-water contraction scour in a long contraction were developed in SI units. For equilibrium in the contracted reach, τo τc where τo is the average bed shear stress, contracted section, Pa (N/m2) and τc is the critical bed shear stress at incipient motion, Pa (N/m2). The average bed shear stress based on the hydraulic radius (R y) and Manning's equation to determine the slope (SSf) is expressed as
gn2V2 τo γyS γ Sf 1/ (15.29) y 3 For noncohesive bed materials and fully developed clear-water contraction scour, the critical shear stress can be determined using Shields relation (Laursen, 1963) τc KS(ρs ρ)gD
(15.30)
The bed in a long contraction scours until τo τc so that using Eqs. (15.29) and (15.30) implies
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HYDRAULIC DESIGN OF CULVERTS AND HIGHWAY STRUCTURES
Hydraulic Design of Culverts and Highway Structures 15.51
gn2V2 K (ρ ρ)gD 1/ s s y 3
(15.31)
so that the depth (y) in the contracted section is 3 n2V2 y Ks(Ss 1)D
(15.32)
and in terms of discharge (Q), the depth is
Table 15.5
Exponents for Live-Bed Contraction Scour Equation
(a) For Long Section V•/w
k1
k2
Mode of Bed Material Transport
0.50
0.59
0.066
Mostly contact bed material discharge
0.50–2.0
0.64
0.21
Some suspended bed material discharge
2.0
0.69
0.37
Mostly suspended bed material discharge
(b) For Contracted Section V•/ω
k1
Mode of Bed Material Transport
0.50
0.59
Mostly contact bed material discharge
0.50–2.0
0.64
Some suspended bed material discharge
2.0
0.69
Mostly suspended bed material discharge
Source: From Laursen (1960).
FIGURE 15.34 Fall velocity of sand-sized particles. (From Richardson and Davis, 1995)
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HYDRAULIC DESIGN OF CULVERTS AND HIGHWAY STRUCTURES
15.52
Chapter Fifteen 3/7 n2Q2 y 2 K (S 1 )DW W s s
(15.33)
where y average flow depth in the contracted section after contraction scour (m), Sf slope of the energy grade line (m/m), V average velocity in the contracted section (m/s), D diameter of smallest nontransportable particle in the bed material (m), Q discharge (m3/s), W bottom width of contracted section (m), g acceleration of gravity (9.81 m/s2), n Manning's roughness coefficient, Ks Shield's coefficient Ss specific gravity (2.65 for quartz), γ unit weight of water (9800 N/m3), ρ density of water (1000 kg/m3) and ρs density of sediment (quartz, 2647 kg/m3) Equations (15.32) and (15.33) are the basic equations for the clear-water scour depth (y) in a long contraction. Laursen used a value of 4 (in American customary units) for Ks(ρs - ρ)g; D50 for the size (D) of the smallest nonmoving particle in the bed material and Strickler's (1923) approximation for Manning's n (n 0.034 D501/6). Laursen’s assumption that τc 4D50 with Ss 2.65 is equivalent to assuming a Shield's parameter Ks 0.039. Shield’s coefficient (K Ks) to initiate motion ranges from 0.01 to 0.25 and is a function of particle size, Froude number, and size distribution. Some typical values for Ks for Fr 0.8 and as a function of bed material size are (1) Ks 0.047 for sand (0.065 mm D50 2 mm); (2) Ks 0.03 for medium coarse bed material (2 mm D50 40 mm) and (3) Ks 0.02 for coarse-bed material (D50 40 mm). In SI units, Strickler's equation for n as given by Laursen is n 0.041 D501/6, where D50 is in meters. Richardson et al. (1990) recommend the use of the effective mean bed material size (Dm) in place of the D50 size for the beginning of motion (D50 1.25Dm). Changing D50 to Dm in the Strickler's equation gives n 0.040 Dm1/6. Substituting Ks 0.039 into Eq.(15.32) and (15.33) gives the following equation for y: V2 3 y [ ] 40Dm2/3
(15.34)
Q2 3/7 y [ 2/] 40Dm3 W2
(15.35)
ys y y0 (average scour depth, m) where Q discharge through contraction (m3/s), Dm diameter of the bed material (D50/1.25) in the contracted section (m), W bottom width in contraction (m) and y0 existing depth in the contracted section before scour (m) The above clear-water contraction scour equations (15.32 – 15.35) assume homogeneous bed materials. For clear-water scour in stratified materials, using the layer with the finest D50 would result in the most conservative estimate of contraction scour. The clear-water contraction scour equations can be used sequentially for stratified bed materials. The distribution of the contraction scour in the cross section can not be determined from the above scour equations. Assuming a uniform contraction scour depth across the opening may not be in error (e.g., short bridges, relief bridges, bridges with simple cross sections, and on straight reaches). For wide bridges, bridges on bends, bridges with large overbank flow, or crossings with a large variation in bed material size distribution, the contraction scour depths will not be uniformly distributed across the bridge opening (Richardson and Davis, 1995). In these cases, Eq. (15.32) and (15.34) can be used if the distribution of the velocity and/or the bed material is known. WSPRO uses the concept of stream tubes to determine the discharge and velocity distribution in cross sections (Figs. 15.35). Eqs. (15.32) and (15.34) can be used to estimate the distribution of the
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HYDRAULIC DESIGN OF CULVERTS AND HIGHWAY STRUCTURES
Hydraulic Design of Culverts and Highway Structures 15.53
contraction scour depths with the WSPRO produced discharge and velocity distribution. Then Equations (15.33) and (15.35) can be used to determine the average contraction scour depth in the section. The velocity and depth given in Eq. (15.32) are associated with initiation of motion of the indicated particle size (D). Eq. (15.32) can be rearranged to give the critical velocity (V Vc) for beginning of motion of bed material of size D results in Ks1/2 (Ss 1)1/2D1/22y1/6 Vc n
(15.36)
Equation 15.16 can be simplified using Ks 0.039, Ss 2.65, and n 0.041 D1/6, to obtain Vc 6.19y1/6D1/3
(15.37)
where Vc critical velocity above which bed material of size D and smaller will be transported (m/s), Ks Shields parameter, Ss specific gravity of the bed material, D size of bed material (m), y depth of flow (m) and n Manning's roughness coefficient. Example. 15.10.1 This example problem and the succeeding examples in Sec. 15.10 are based on the examples taken from Arneson et al. (1991) and also used by Richardson and Davis (1995) in FHWA Hydraulic Engineering Circular No. 18. WSPRO was used to obtain the hydraulic variables listed in Tables 15.6–15.10. A 198.12-m long bridge (Fig. 15.36) is to be constructed over a channel with spillthrough abutments (slope of 1V:2H). H The left abutment is set approximately 60.5 m back from the channel bank. The right abutment is set at the channel bank. The bridge deck is set at elevation 6.71 m and has a girder depth of 1.22 m. Six round-nose piers are evenly spaced in the bridge opening. The piers are 1.52 m thick, 12.19 m long, and are aligned with the flow. The 100-year design discharge is 849.51 m3/s. The 500-year flow of
FIGURE 15.35 Cross section of proposed bridge. (From Richardson and Davis, 1995)
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HYDRAULIC DESIGN OF CULVERTS AND HIGHWAY STRUCTURES
15.54
Chapter Fifteen
TABLE 15.6
Hydraulic Variables from WSPRO for Estimation of Live-Bed Contraction Scour Remarks
Q (m3/s)
849.51
Total discharge Conveyance of main channel of approach
K1 (approach)
19,000
Ktotal (approach)
39,150
W1 or TOPW (approach) (m)
121.9
Ac (approach) (m2) WETP (approach) (m)
320 122.0
Kc (bridge)
11,330
Ktotal (bridge)
12,540
Total conveyance of approach section Top width of flow (TOPW). Assumed to represent active live bed width of approach Area of main channel approach section Wetted perimeter of main channel approach section Conveyance through bridge Total conveyance through bridge
Ac (bridge) (m2)
236
Area of the main channel, bridge section
Wc (bridge) (m)
122
Channel width at the bridge. Difference between subarea break-points defining banks at bridge
W2 (bridge) (m)
115.9
Channel width at the bridge. less four channel pier widths (6.08 m)
Sf (m/m)
0.002
Average unconstrictected energy slope (SF)
Source: From Richardson et al. (1995).
TABLE 15.7 Hydraulic Variables from WSPRO for Estimation of Clear-Water Contraction Scour on Left Overbank Remarks Q (m /s)
849.51
Total discharge, (see Table 16.10.2).
Qchan (bridge)
767.54
Flow in main channel at bridge. Determined in live-bed computation of Step 5A.
3
Q2 (bridge)
Dm (bridge overbank) (m)
81.97
0.0025
Flow in left overbank through bridge. Determined by subtracting Qchan (listed above) from total discharge through bridge. Grain size of left overbank area. Dm 1.25 D50.
Wsetback (bridge) (m)
68.8
Top width of left overbank area. (SA #1) at bridge
Wcontracted (bridge) (m)
65.8
Setback width less two pier widths (3.04 m).
Aleft (bridge) (m2)
57
Area of left overbank at the bridge
Source: From Richardson et al. (1995).
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HYDRAULIC DESIGN OF CULVERTS AND HIGHWAY STRUCTURES
Hydraulic Design of Culverts and Highway Structures 15.55 TABLE 15.8 Number 12)
Hydraulic Variables from WSPRO for Estimation of Pier Scour (Conveyance Tube Remarks
V1 (m)
3.73
Velocity in conveyance tube #12
V1 (m)
2.84
Mean depth of tube #12
TABLE 15.9 Hydraulic Variables from WSPRO for Estimation of Abutment Scour Using Froehlich’s Equation for Left Abutment Remarks Q (m3/s) qtube (m /s) 3
#Tubes
849.51 42.48 3.5
Total discharge (see Table 16.10.2). Discharge per equal conveyance tube, defined as total discharge divided by 20. Number of approach section conveyance tubes which are obstructed by left abutment. Determined by superimposing abutment geometry onto the approach section.
Qe (m3/s)
148.68
Flow in left overbank obstructed by left abutment. Determined by multiplying # tubes and qtube.
Ae (left abut)
264.65 (m2)
Area of approach section conveyance tubes number 1, 2, 3, and half of tube number 4.
L' (m)
232.80
Length of abutment projected into flow, determined by adding top widths of approach section conveyance tubes number 1, 2, and 3, and half of tube number 4.
Source: From Richardson et al. (1995).
TABLE 15.10 Hydraulic Variables from WSPRO for Estimation of Abutment Scour Using HIRE equation for left abutment Remarks Vtube (m/s) (bridge section)
1.29
Mean velocity of conveyance tube #1, adjacent to left abutment
y1 (m) (bridge section)
0.83
Average depth of conveyance tube #1
Source: From Richardson et al. (1995).
1444.16 m3/s was estimated by multiplying the Q100 by 1.7 since no hydrologic records were available to predict the 500-year flow. D50 is 0.002 m. Determine whether the flow condition in the main channel is a live-bed or a clear-water condition. Solution. Determine whether the flow condition in the main channel is live-bed or clearwater by comparing the critical velocity for sediment movement and the average channel velocity at the approach section. The discharge in the main channel at the approach section, Q1, is computed as Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
HYDRAULIC DESIGN OF CULVERTS AND HIGHWAY STRUCTURES
15.56
Chapter Fifteen
A
B
C FIGURE 15.36 A. Equal conveyance tubes of approach section. B. Equal conveyance tubes of bridge section C. Velocity distribution at bridge crossing (From Richardson and Davis, 1995)
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HYDRAULIC DESIGN OF CULVERTS AND HIGHWAY STRUCTURES
Hydraulic Design of Culverts and Highway Structures 15.57 K1 m3 19,000 Q1 Q 849.51 412.28 m3/s Ktotal s 39,150
where K1 and Ktotal are the conveyance of the main channel of approach and the total conveyance of the approach section, respectively, from Table 15.6. The average velocity in the main channel of the approach section is V1 Q1/A / c 412.28/320 1.29 m/s where Ac is the area of the main channel approach section from Table 15.6. The average depth of flow y1 in the approach section is y1 A1/W W1 320/121.9 2.63 m, where W1 is the top width of flow. The critical velocity for D50 size sediment is computed using Eq. (15.37) thus: Vc 6.19y11/6D501/3 6.19(2.63 m)1/6(0.002 m)1/3 0.92 m/s Comparing V1 and Vc, V1 Vc indicating that the flow condition is live-bed Example.15.10.2 Compute the live-bed contraction scour for the main channel for the above example problem. Solution. Equation 15.28 is used to compute y2/y1, and the scour depth is computed using ys y2 y0 so that
冢 冣
y ys 2 y1 yo y1 Q 6/7 W k1 2 1 y1 yo Q1 W2
The first step is to compute all the parameters in this equation to evaluate ys. Q1 was evaluated as 412.28 m3/s in the above Example, and Q2, the discharge in the main channel at the bridge, is K2 m3 11,330 849.51 767.54 m3/s Q2 Q Ktotal s 12,540
The channel widths are W1 121.9 m and W2 115.9 m in Table 15.6, y1 2.63 m was computed in Example 15.10.1. The bridge channel flow depth is the area divided by the top width y0 236 m2/122 m 1.93 m. The only remaining unknown is K1 which can be obtained from Table 15.5.B for V*/ . To compute V*/ω, first compute the shear velocity V* using V* 兹 兹τ苶0/ρ 苶 where τ0 γRS γ 9810 (N/m3)(2.62 m) (0.002 m/m) 51.4 N/m2 51.4 Pa. The shear velocity is then V* 兹5苶苶 兹 1苶 .4苶 /1苶0苶00苶 0.227 m/s. The bed material is sand with D50 0.002 m (2 mm). Fall velocity ω 0.21 m/s from Fig. 15.34. Then V*/ω 0.227/0.21 1.08. From Table 15.5B K1 0.64 which indicates that the mode of bed transport is a mixture of suspended and contact bed material discharge.
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HYDRAULIC DESIGN OF CULVERTS AND HIGHWAY STRUCTURES
15.58
Chapter Fifteen
The live-bed contraction scour of the channel is computed using Table 15.5A as ys (767.54/412.28)6/7(121.9/115.9)0.64(2.63) 1.93 2.7 m This scour is large and could be minimized through a larger bridge opening, by putting relief bridges in the overbank, or possibly in some cases providing for highway approach overtopping (Richardson and Davis, 1995). Example 15.10.3. Clear-water contraction scour will occur in the overbank area between the left abutment and the left bank of the bridge opening Example 15.10.1. Compute the clear-water contraction scour in the left bank based on the discharge and depth of flow passing under the bridge and use the hydraulic variables from the example in Table 15.7. Solution. Using Eq. (15.35), 3/7 Q2 y 2/3 W 2 40D m contacted 0.025(81.97 m3/s)2 3/7 (0.0025m)2/3 (65.8m)2
1.38 m Average flow depth (y0) in the left overbank bridge section is y0 a/W Wsetback (57.0 m2)/(68.8 m) 0.83 m. The clear-water contraction scour in the left overbank of the bridge opening is ys y2 y0 1.38 m 0.83 m 0.55 m. 15.10.3 Local Scour at Piers Local scour at piers or abutments is caused by the formation of horseshoe vortices at their base (Fig. 15.37). The horseshoe vortex results from the increased head water on the upstream surface of the obstruction and subsequent acceleration of the flow around the nose of the pier or abutment. Vortex action removes bed material from around the base of the obstruction, where the transport rate of sediment away from the base region is greater than the transport rate into the region, and consequently, a scour hole develops. (Richardson and Davis, 1995). There are also vertical vortices called wake vortices downstream of the pier (Fig. 15.37). which also remove material from the pier base region. The intensity of wake vortices diminishes rapidly as the distance downstream of the pier increases, so that there is often deposition of material, immediately downstream of a long pier. (Richardson and Davis, 1995) Factors that affect the magnitude of local scour depth at piers and abutments are (1) velocity of the approach flow, (2) depth of flow, (3) width of the pier, (4) discharge intercepted by the abutment and returned to the main channel at the abutment, (5) length of the pier if skewed to flow, (6) size and gradation of bed material, (7) angle of attack of the approach flow to a pier or abutment, (8) shape of a pier or abutment, (9) bed configuration, and (10) ice formation or jams and debris (Richardson and Davis, 1995). There have been many scour equations reported in the literature for live-bed scour in cohesionless sand-bed streams. The equation presented here is recommended for both live-bed and clear-water pier scour, given as (Richardson et al., 1990)
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HYDRAULIC DESIGN OF CULVERTS AND HIGHWAY STRUCTURES
Hydraulic Design of Culverts and Highway Structures 15.59
FIGURE 15.37 Schematic representation of scour at a cylindrical pier. (From Richardson and Davis, 1995)
a 0.65 y ys 2.0K1 K2 K3 K4 y Fr10.43 1 1
(15.38)
In terms of ys /a, where a is the pier width, Eq. (15.38) is y 0.35 y as 2.0K1 K2 K3 K4 a1 Fr10.43
(15.39)
where ys scour depth (m), y1 flow depth directly upstream of the pier (m), K1 correction factor for pier nose shape (Fig. 15.38 and Table 15.11), K2 correction factor for angle of attack of flow from Table 15.12, K3 correction factor for bed condition from Table 15.13, K4 correction factor for armoring by bed material size from the list inmediately below, a pier width (m), L length of pier (m), Fr1 Froude number directly upstream of the pier V1/(gy1)1/2, V1 mean velocity of flow directly upstream of the pier (m/s) and g acceleration of gravity (9.81 m/s2) The limits for bed material size and K4 value are (Richardson and Davis, 1995). Minimum Bed Material Size
Minimum K4 Value
VR 1.0
D50 0.06
0.7
1
Example 15.10.4. Compute the magnitude of the local scour at the pier. Anticipating that any pier under the bridge could be subject to the maximum flow depth and velocities, only one computation of pier scour of the side pier is needed. This assumption is acceptable because the thalweg is prone to shift and there is a possibility of lateral channel migration. Solution. This solution requires the parameters in Eq. (15.38) a 0.65 y ys 2.0K1 K2 K3 K4 y Fr10.43 1 1
The Froude number, Fr1, for the pier scour computation is based on the hydraulic characteristics of conveyance tube number 12. The velocity V1 3.73 m/s, and y1 2.89 m are from Table 15.8, so Fr1 is:
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HYDRAULIC DESIGN OF CULVERTS AND HIGHWAY STRUCTURES
15.60
Chapter Fifteen
FIGURE 15.38 Common pier shapes (A) square nose, (B) round nose, (C) cylinder, (D) sharp nose, (E) group of cylinder (see Multiple Columns). (From Richardson and Davis, 1995)
Table 15.11 Correction Factor, K1, for Pier Nose Shape* Shape of pier Nose
K1
(a) (b) (c) (d) (e)
1.1 1.0 1.0 1.0 0.9
Square nose Round nose Circular cylinder Group of cylinders Sharp nose
Source: From Richardson et al. (1995). *The correction factor K for pier nose should be determined using 1 Table 15.11 for angles of attack up to 5º. For greater angles. K2 domL a is larger than 12, inates and K1 should be considered as 1.0. If L/ use the values for L/ L a 12 as a maximum in Table 15.12
3.73 m / s V1 Fr1 [(9.81 m / s2)(2.84 m)]0.5 0.71 (gyy1)0.5 For a round-nose pier, aligned with the flow and sand-bed material (see Tables 15.11 and 15.12), K1 K2 K4 1.0. For plane-bed condition (Table 15.13), K3 1.1. Using Eq. (15.39) 1.52 m 0.65 ys 2.0(1) (1) (1.1) (1) (0.71)0.43 2.84 2.84 m
and ys 3.3 m The maximum local pier scour depth will be 3.3 m.
15.10.4 Live-Bed Scour at Abutments When abutments obstruct flow, local scour occurs as a result of a horizontal vortex that starts at the upstream end of the abutment and runs along the toe of the abutment, similar
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HYDRAULIC DESIGN OF CULVERTS AND HIGHWAY STRUCTURES
Hydraulic Design of Culverts and Highway Structures 15.61 TABLE 15.12
Correction Factor, K2, for Angle of Attack, θ, of the Flow*
Angle
L/a 4
L/a 8
L/a 12
0 15 30 45 90
1.0 1.5 2.0 2.3 2.5
1.0 2.0 2.75 3.3 3.9
1.0 2.5 3.5 4.3 5.0
Source: From Richardson and Davis. (1995). The value of the correction factor K2 should be applied only when the field conditions are such that the entire length of the pier is subjected to the angle of attack of the flow. Use of this factor directly from the table will result in a significant overprediction of scour if (1) a portion of the pier is shielded from the direct impingement of the flow by an abutment or another pier, or (2) an abutment or another pier redirects the flow ia a direction parallel to the pier. For such cases, judgment must be exercised to reduce the value of the K2 factor by selecting the effective length of the pier actually subjected to the angle of attack of the flow. (Richardson and Davis, 1995). *
TABLE 15.13
Increase in Equilibrium Pier Scour Depths, K3, for Bed Condition*
Bed Condition Clear – water scour Plane bed and antidune flow Small dunes Medium dunes Large dunes
Dune Height m
K3
NA NA 3 H 0.6 9H3 H9
1.1 1.1 1.1 1.2–1.1 1.3
Source: From Richardson and Davis (1995). Abbreviations: NA, not aplicable. *The correction factor K results from the fact that for plane-bed conditions, which is typical of most 3 bridge sites for the flood frequencies employed in scour design, the maximum scour may be 10 percent greater than computed with Eq. (15.38). In the unusual situation large where a dune bed configuration with large dunes exists at a site during flood flow, the maximum pier scour may be 30 percent greater than the predicted equation value. This may occur on very large rivers, such as the Mississippi. For smaller streams that have a median dune configuration at flood flow, the dunes will be smaller and the maximum scour may be only 10 to 20 percent larger than equilibrium scour. For antidune bed configuration the maximum scour depth may be 10 percent greater than the computed equilibrium pier scour depth. (Richardson and Davis, 1995)
to the horizontal vortex that forms at piers. A vertical wake vortex forms at the downstream end of an abutment similar to that which forms downstream of a pier or downstream of any flow separation. Figure 15.39 is a definition sketch for abutment scour. There are three general abutment shapes as indicated in Fig. 15.40. The potential for lateral channel migration, long-term degradation, and contraction scour should be considered in determining abutment foundation depths near the main channel. The FHWA, HEC-18 recommends that abutment scour equations be used to develop insight as to the scour potential of an abutment. Then, the abutment may be designed to resist the computed scour or as an alternative, riprap and guide banks can be used to protect the abutment from scour and erosion (Richardson and Davis, 1995). Two approaches, Froelich's live-bed scour equation and a U.S. Army Corps of Engineers’ equa-
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HYDRAULIC DESIGN OF CULVERTS AND HIGHWAY STRUCTURES
15.62
Chapter Fifteen
FIGURE 15.39 Definition sketch for abutment scour. (From Richardson et al., 1990)
FIGURE 15.40 Abutment shapes (A) spill through, (B) vertical wall, (C) vertical wall with flared wingwalls. (From Richardson and Davis, 1995)
tion, are discussed in this section. To check the potential depth of scour for the design of the foundation and placement of rock riprap or guide banks, Froelich’s live-bed scour equation can be used. Froelich (1989) analyzed 170 live-bed scour measurements in laboratory flumes and developed the following regression equation:
where: K1 K2
L' Ae
L' 0.43 y ys 2.27 K1 K2 y Fr10.61 1 a a
(15.40)
coefficient for abutment shape (see the list immediately following) coefficient for angle of embankment to flow (θ/90)0.13 (see Fig. 15.41 for definition of θ θ 90º if embankment points downstream θ 90º if embankment points upstream length of abutment (embankment) projected normal to flow (m) flow area of the approach cross section obstructed by the embankment (m2)
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HYDRAULIC DESIGN OF CULVERTS AND HIGHWAY STRUCTURES
Hydraulic Design of Culverts and Highway Structures 15.63
Fr1 Froude number of approach flow upstream of the abutment Ve /(gya )1/2 Ve Qe /A / e , m/s Qe
flow obstructed by the abutment and approach embankment (m3/s)
ya
average depth of flow on the floodplain (m)
ys
scour depth (m)
The abutment shape coefficients are (Richarson and Davis, 1995): Description
K1
Vertical-wall abutment
1.00
Vertical-wall abutment with wing walls
0.82
Spill-through abutment
0.55
Equation (15.40) is not consistent because as L' tends to 0, ys also tends to 0. The 1 was added to the equation so as to encompas 98 percent of the data. Field data of scour at the end of spurs in the Mississippi River (obtained by the U.S. Army Corps of Engineers) can also be used for estimating abutment scour. This field situation closely resembles the laboratory experiments for abutment scour in that the discharge intercepted by the spurs was a function of the spur length. Equation (15.41), referred to as the Highways in the River Environment (HIRE) equation (Richardson et al., 1990), is applicable when the ratio of projected abutment length (L') to the flow depth (y1) is greater than 25. This equation can be used to estimate scour depth (ys) at an abutment where conditions are similar to the field conditions from which the equation was derived (Richardson and Davis, 1995): y K1 ys 4 Fr10.33 0.5 5 1
(15.41)
where: y1 depth of flow at the abutment on the overbank or in the main channel (m), Fr1 Froude number based on the velocity and depth adjacent to and upstream of the abutment and K1 abutment shape coefficient (see list detailing abutment shape coefficients, above). To correct Eq. (15.41) for abutments skewed to the stream, use Figure 15.41. For clear-water scour at an abutment, use either Eq (15.40) or (15.41) because clearwater scour equations potentially decrease scour at abutments due to the presence of coarser material, which is unsubstantiated by field data (Richardson and Davis, 1995). Froelich's equation will generally result in deeper scour predictions than experienced in the field, according to Richardson and Davis (1995). These scour depths could occur if the abutments protruded into the main channel flow, or when a uniform velocity field is cut off by the abutment in a manner that most of the returning overbank flow is forced to return to the main channel at the abutment end. All of the abutment scour computations (left and right abutments) assumed that the abutments were set perpendicular to the flow. If the abutments were angled to the flow, a correction utilizing K2 would be applied to Froelich's equation or, using Fig. 15.41 would be applied to Eq. (15.41). However, the adjustment for skewed abutments is minor when compared to the magnitude of the computed scour depths.
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HYDRAULIC DESIGN OF CULVERTS AND HIGHWAY STRUCTURES
15.64
Chapter Fifteen
FIGURE 15.41 Adjustment of abutment scour estimate for skew. (From Richardson et al., 1990 as presented in Richardson and Davis, 1995)
Example 15.10.5 Compute the magnitude of the local scour at the left abutment for the previous example in this section. Use Froelich's live-bed scour equation for the computation. Solution. First determine all the parameters for Froelich's equation L' 0.43 y ys 2.27 K1 K2 y Fr10.61 1 a a
For spill-through abutments, K1 0.55, and for abutments perpendicular to the flow, K2 1.0. Abutment scour can be estimated using Froelich's equation with data derived from the WSPRO output (Table 15.9). ya at the abutment is assumed to be the average flow depth in the overbank area, computed as the cross-sectional area of the left overbank cut off by the left abutment divided by the distance the left abutment protrudes into the overbank flow A 264.6 5 m2 1.14 m ya e 2 3 2 . 8 0m L' The average velocity of the flow in the left overbank which is cut off by the left abutment is computed as the discharge cutoff by the abutment divided by the area of the left overbank cut off by the left abutment. Q 148.68 m3 / s Ve e 0.56 m /s Ae 264.65 m2 The Froude number of the overbank flow is 0.56 m / s Ve Fr [(9.81 m / s2)(1.14 m)]0.5 0.17 (gyya)1/2 232.8 0.43 ys 2.27 (0.55) (1.0) (0.17)0.61 1 1.1 4 1.14
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HYDRAULIC DESIGN OF CULVERTS AND HIGHWAY STRUCTURES
Hydraulic Design of Culverts and Highway Structures 15.65
ys 5.17 1.1 4 ys 5.9 m The abutment scour at the left abutment based on Froelich's equation is 5.9 m. Example 15.10.6. Compute the magnitude of the local scour at the left abutment for the first example in this section using the HIRE Equation (Eq. 15.41). Solution. The HIRE Equation (Eq. 15.41) is y K1 ys 4 Fr10.33 0.5 5 1 which is based on the velocity and depth of flow passing through the bridge opening adjacent to the abutment listed in Table 15.10. The Froude number is 1.29 m / s V ube Fr1 t [(9.81 m / s2)(0.83 m)]0.5 3.07 (ggy1)0.5 and from the list abutment shape coefficients, above, K1 0.55, using Equation (15.41): 0.55 ys 4Fr10.33 4(0.45)0.33 3.07 0.83 m 0.55
ys 2.6 m The depth of scour at the left abutment, as computed using the HIRE equation, is 2.6 m. Example 15.10.7. Use the HIRE equation (15.41) and the data below to compute the magnitude of the local scour at the right abutment using Vtube 2.19 m/s and y1 1.22 m. Solution. The HIRE equation is also applicable to the right abutment since L/y1 is greater than 25. The HIRE equation is based on the velocity and depth of the flow passing through the bridge opening adjacent to the end of the right abutment listed above. The Froude number is Fr1
2.19 m / s 0.63 [(9.81 m / s2)(1.22 m)] 0.5
and K1 0.55 (see list detailing abutment shape coefficients), using the HIRE equation, is 0.55 ys 4(0.63)0.33 3.43 4 Fr10.33 1.22 m 0.55
ys 4.2 m The depth of scour at the right abutment is 4.2 m. Example15.10.8 For the preyton examples in this section 15.10 the final step is to plot the results of the scour computation. Solution. Figure 15.42 is a plot of the total scour on the bridge cross section. Only the computation for pier scour with piers aligned with the flow was used, and only the abutDownloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
HYDRAULIC DESIGN OF CULVERTS AND HIGHWAY STRUCTURES
15.66
Chapter Fifteen
ment scour computation for Eq. (15.41) was used. The top width of the local scour holes suggested is 2.0 ys (Richardson and Davis, 1995).
15.11
COMPUTER MODELS FOR CULVERTS AND SEDIMENTATION
Numerous computer programs are available to aid in the design and analysis of highway culverts and sedimentation issues. Familiarity with culvert hydraulics and sediment transport is necessary to provide a solid basis for designers to take advantage of the speed, accuracy, and increased capabilities of hydraulic design computer programs.
15.11.1 Computer Models for Culverts 15.11.1.1 Integrated Drainage Design Computer System. The computer programs for the hydraulic design of culverts available from the FHWA are more fully described in the HDS No. 5 (Normann et al., 1985). More detailed information on the FHWA publications can be obtained from http://www.fhwa.dot.gov. The recommended personal computer system is the HYDRAIN-Integrated Drainage Design Computer System. There are currently two culvert design and analysis programs within the HYDRAIN system. They are the Culvert Design System (CDS) and the HY8-Culvert Analysis program. HYDRAIN Version 6.0 was developed by GKY and Associates, Inc. (1996), under contract with the FHWA. HYDRAIN software and user support is available through, McTrans at the University of Florida (
[email protected]), and PCTrans at the University of Kansas ( (
[email protected] ). 15.11.1.2 Culvert Design System (CDS) CDS is a program that can be used for the hydraulic design or the analysis of an existing or proposed culvert. The model can accommodate a variety of hydrograph relationships, culvert shapes, materials, and inlet types
FIGURE 15.42 Plot of total scour for example problem. (From Richardson and Davis, 1995)
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HYDRAULIC DESIGN OF CULVERTS AND HIGHWAY STRUCTURES
Hydraulic Design of Culverts and Highway Structures 15.67
(Brater et al., 1996). Any of the six culvert types defined in Section 15.6 can be used. The model starts its design process by selecting a culvert size and number of barrels that are compatible with geometric, environmental and construction constraints. The culvert size is based on the design headwater, headwater-to-diameter ratio, inundation, outlet velocity or cover limitations. 15.11.1.3 HY8. HY8 is a program that uses FHWA procedures as defined in HDS No. 5 for the hydraulic design of culverts, and incorporates factors such as hydrological inputs, storage and routing considerations, and energy dissipation devices (Norman et al.,) 1985. The program can accommodate various culvert shapes including circular, rectangular, elliptical, arch, and user-defined geometry. Improved inlets can be specified and various design conditions can be analyzed including inlet and outlet control for full and partially full culverts, flow over the roadway embankment, and balance of flows through multiple parallel culverts. The program features include the development of performance curves, generation of rating curves for uniform flow, velocity, and maximum shear for the downstream channel. 15.11.1.4 CULVERT2 (English units), CULVERT3 (metric units). CULVERT2 and 3 are provided by Caltrans [California Department of Transportation (DOT)], have corrosion criteria for culverts and present alternative culvert materials and material thickness acceptable for 50 years of service using sitespecific test data. The minimum resistivity test data (ohm-cm) and pH of the site soils and/or water are required for all analysis. Water-soluble sulfate and chloride concentrations (ppm) are required for aggressive sites when the minimum resistivity is less than 100 ohm-cm. Culvert materials addressed include corrugated steel pipe, (CSP), corrugated aluminized steel pipe, (CASP); corrugated aluminum pipe, (CAP); and reinforced concrete pipe, (RCP). Documentation and software can be ordered from http://www-mctrans.ce.ufl.edu/info-cen. 15.11.1.5 Culvert Analysis Program (CAP). The Culvert Analysis Program (CAP) is a program (Fulford, 1995) used to compute discharges through a culvert and develop stagedischarge relationships for a culvert from the measurements of upstream and downstream water surface elevations. Types of culverts that can be studied include rectangular, circular, pipe arch, and nonstandard shaped culverts. CAP will not analyze culverts that vary in cross section or material, culverts that have a nonuniform slope or break, or culverts that have a severe adverse slope. The computation procedure that CAP uses is based upon the U.S. Geological Survey (USGS) method described by Bodaine (1968) entitled Measurement of Peak Discharge of Culverts by Indirect Methods found in the USGS document Techniques of WaterResources Investigations, Book 3, Chapter A3. The method is based upon USGS field investigations and laboratory investigations conduced by thde USGS, Bureau of Public Roads, and various universities. CAP Version 97-01 is written in FORTRAN 77. The source program has been compiled for IBM personal computers, Macintosh personal computers, and mainframe computers, such as UNIX. To execute CAP, preparation of a input file utilizing a text editor that produces American Standard Code for Information Interchange (ASCII) files is required. A computer with 386 MHz and 2 Megabytes ramdon access memory (RAM) is also required.
15.11.2 Computer Models for Sedimentation 15.11.2.1 HEC-6. HEC-6 (U.S. Army Corps of Engineers, 1993) is a one-dimensional, moveable-boundary, steady-state, open-channel flow model used to compute degradation Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
HYDRAULIC DESIGN OF CULVERTS AND HIGHWAY STRUCTURES
15.68
Chapter Fifteen
(scour) and aggradation (deposition) in river systems by simulating the interaction between hydraulics of the flow and the rate of sediment transport (Morris et al., 1996). A river system consisting of a main stem, tributaries, and local inflow/outflow points can be simulated. The program processes a water discharge hydrograph as a sequence of steady flows of variable durations. Using continuity of sediment, changes are calculated with respect to time and distance along the study reach for the total sediment load, volume and gradation of sediment that is scoured or deposited, armoring of the bed surface, and the cross-sectional elevations. HEC-6 is designed to analyze long-term scour and/or deposition. The bed-material transport algorithms assume that equilibrium conditions are reached within each time step. HEC-6 raises or lowers cross-sectional elevations to reflect deposition and scour. The horizontal locations of the channel banks are considered fixed and the floodplains on each side of the channel are considered as having fixed ground elevations. The hydraulic parameters needed to calculate sediment transport potential are obtained by using the standard step method. The hydraulic parameters are calculated at each cross-section for each successive discharge. Manning's equation and n values for overbank and channel areas may be specified by discharge or elevation. Sediment transport rates are calculated for each flow in the hydrograph for each grain size (upper limit for grain size is 2048 mm). The transport potential is calculated for each grain size in the bed as though that size comprised 100 percent of the bed material. Transport potential is then multiplied by the fraction of each size class present in the bed at that time to yield the transport capacity for that size class. For deposition and erosion of clay and silt sizes up to 0.0625 mm, Krone’s (1962) method is used for deposition and Ariathurai and Krone’s (1976) adaption of Parthenaides’s (1965) method is used for scour. The model and the user's manual can be downloaded from http://www.hec.usace.army.mil/software. 15.11.2.2 Generalized stream tube model alluvial river simulation. (GSTARS 2.0). Most of the sediment and water routing models, such as the HEC-6 model were developed for solving one-dimensional alluvial river problems. Although there are truly two-dimensional and three-dimensional models for alluvial river simulation, they are too computationally intensive for engineering applications. The Generalized Stream Tube model for Alluvial River Simulation (GSTARS) was developed by Molinas and Yang (1986) to simulate the flow conditions in a semi-two-dimensional manner and the change of channel geometry in a semi-three-dimensional manner. GSTARS, Version 2.0, is an enhanced personal computer version of the original GSTARS program. The stream tubes are imaginary channels that are bounded by streamlines and convey the same discharge. They are used to compute the lateral variation of hydraulic and sediment parameters within the cross-section. The sediment transport results for stream tubes may vary and one stream tube may be aggrading and the other degrading. Stream boundaries are updated for each cross section and time step to result in equal hydraulic conveyance. The model can also solve for an unknown channel width or compute the changes in the channel width based on minimum stream power theory. The GSTARS 2.0 program and the user's manual can be requested from http://www.usbr.gov/srhg/gstars/2.0. 15.11.2.3 Surface water modeling system. Surfacewater Modeling System (SMS) is a powerful, and comprehensive two-dimensional surfacewater modeling package. The software models the water surface elevation, flow velocity, contaminant transport and dispersion, and sediment transport and deposition for complex two-dimensional horizontal flow problems. SMS provides complete support for the U.S. Army Corps of Engineers RMA two-dimensional hydrodynamic and contaminant transport, SED-2D two-dimensional sediment transport and deposition, HIVEL-2D two dimensional hydrodynamic supercritical and subcritical flow, and U.S Federal Highway Administration FESWMS two-dimenDownloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
HYDRAULIC DESIGN OF CULVERTS AND HIGHWAY STRUCTURES
Hydraulic Design of Culverts and Highway Structures 15.69
sional hydrodynamic and bridge scour finite element models. Single- and multiple-opening bridge and culvert roadway crossings, channel networks, sinuous rivers, harbors, bays, estuaries, wetlands, irregular floodplains, split flows, and other complex simulations can be modeled. The analysis results can be output or displayed graphically using a variety of plots, including vector plots, contour plots, color-shaded contour plots, and time-history plots. Time-history plots can be requested at any location to illustrate fluctuations in water surface elevation, velocity, discharge, contaminant concentration, and bed elevation. Along the flow boundaries and at each node in the finite element mesh, the water surface elevation, flow velocity, pollution contaminant concentration, and bed scour and deposition are computed. Flow separations and eddy currents are accurately modeled. The software can find solutions for a single instance in time (steady-state solution), or during a series of time-steps (transient solution). Transient solutions can be used to model flow fluctuations caused by inflow hydrographs, tidal cycles, and storm surges. For subcritical-supercritical mixed flow regimes, hydraulic jumps are automatically located. The software and the user's manual can be requested from http://www.bossintl.com.hk/html/products.html. 15.11.2.4 Bridge stream tute model for alluvial river simulation (BRI-STARS). Bridge Stream Tube model for Alluvial River Simulation (BRI-STARS, Version 3.3) is a program developed for the National Cooperative Highway Research Program Project 15-11, Computer-Aided Analysis of Highway Encroachments on Mobile Boundary Systems. It is a semi-two-dimensional model capable of computing alluvial scour/deposition through subcritical, supercritical, and a combination of both flow conditions involving hydraulic jump. This model, unlike conventional water and sediment routing computer models, is capable of simulating channel widening/narrowing phenomenon as well as local scour due to highway encroachments. It couples a fixed-width stream tube computer model, which simulates the scour/deposition process taking place in the vertical direction across the channel, with a total stream power minimization algorithm. The decision-making algorithm, using rate of energy dissipation or total stream power minimization, determines whether the simulated sediment erosion satisfying the sediment continuity equation should take place in the lateral or vertical direction. It is this second component that allows the lateral changes in channel geometries. Finally, the bridge component allows computation of the hydraulic flow variables and the resulting scour due to highway encroachments. The model also contains a rule-based expert system program for classifying streams by size, bed and bank material stability, platform geometry, and other hydraulic and morphological features. Documentation and software can be ordered from http://wwwmctrans.ce.ufl.edu/info-cen. 15.11.2.5 HY9. HY9 is based on the FHWA Publication Interim Procedures for Evaluating Scour at Bridges. It computes contraction scour, pier scour, and abutment scour using the equations presented in this chapter. Scour at Bridges (HY-9), Version 4.0 and the accompanying reports HEC-18, Evaluating Scour at Bridges and HEC-20, Stream Stability at Highway Structures, are available from http://www-mctrans.ce.ufl.edu/info-cen.
REFERENCES American Association of State Highway and Transportation Officials (AASHTO) Drainage Manual. AASHTO Task Force on Hydraulics and Hydrology, 1990. American Iron and Steel Institute, Handbook of Steel Drainage and Highway Construction Products, W. P. Reyman Associates, New York, 1983. American Concrete Pipe Association, Concrete Pipe Design Manual, Vienna, VA, 1992. Ariathurai R., and R. B. Krone, "Finite Element Model for Cohesive Sediment Transport," Journal of the Hydraulics Division, American Society of Civil Enqineers, 323–338, March 1976.
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HYDRAULIC DESIGN OF CULVERTS AND HIGHWAY STRUCTURES
15.70
Chapter Fifteen
ARMCO Drainage and Metal Products, Handbook of Culvert and Drainage Practice, R. R. Donnelley & Sons, Chicago, IL, 1950. Arneson, L., Shearman, J. O., and Jones, J. S., "Evaluating Scour at Bridges Using WSPRO," unpublished paper presented at the 71st Annual Transportation Research Board meeting, Washington, DC, January, 1991. Bodaine, G. L., “Measurement of Peak Discharge at Culverts by Indirect Methods,” Techniques of Water—Resources Investigations, U.S. Geological Survey, 1968. A3. Brater E. F., H. W. King, J. E. Lindell, and C. Y. Wei, Handbook of Hydraulics, Mc Graw Hill, New York, 1996. Brice, J. C., and J. C. Blodgett, Countermeasures for Hydraulic Problems at Bridges, Vol. 1, Analysis and Assessment, FHWA/RD-78-162, Federal Highway Administration, Washington, DC, 1978. Bradley, J.N. Hydraulics of bride Waterways, Hydraulic Design Series No. 1 U.S. Department of Trasnportation, FHWA, 1978. Breusers, H. N. C., and A. J. Raudkivi, Scouring, A. A. Balkema, Rotterdam, the Netherlands, 1991. Brice, J. C., and J. C. Blodgett, Countermeasures for Hydraulic Problems at Bridges, Vol. 2, Case Histories for Sites I-283, FHWA/RD-78-163, Federal Highway Administration, Washington, DC, 1978b. Chaudry, M. H., Open-Channel Flow, Prentice-Hall, NJ, 1993. Chow, V. T., Open-Channel Hydraulics, McGraw-Hill, New York, 1959. Corry, M. L., P. L. Thompson, F. J. Watts, J. S. Jones, and D. L. Richards, Hydraulic Design of Energy Dissipators for Culverts and Channels, Hydraulic Engineer Circular No. 14, Hydraulics Branch, Bridge Division, Office of Engineering, Federal Highway Administration, Washington, DC, 1983. Flood Control District of Maricopa County, Drainage Design Manual for Maricopa County, Vol. II, Hydraulics, 1996. Froelich, D. C., Abutment Scour Prediction, Presentation, Transportation Research Board, Washington, DC, 1989. Fulford, J. M., User’s Guide to the Culvert Analysis Program, U.S. Geological Survey, Open file report 95-137, 1995. GKY and Associates, Inc., Culvert Analysis Program HY8 Version 6.0, Springfield, VA, 1996. Harrison, L. J., J. L. Morris, J. M. Normann, and F. L. Johnson, Hydraulic Design of Improved Inlets for Culverts, Hydraulic Engineering Circular No. 13, Federal Highway Administration, U.S. Department of Transportation, Washington, DC, 1972. Hendrickson, J. G., Hydraulics of Culverts, American Concrete Pipe Association, Chicago, IL, 1957. Krone, R. B., “Flume Studies of the Transport of Sediment in Estuarial Shoaling Processes,” Hydraulic Engineering Laboratory, University of California, Berkeley, CA, 1962. Lagasse, P. F., Schall, F., Johnson, F., Richardson, EV. and Chang, F., Stream Stability at Highway Structure, Department of Transportation, Federal Highway Administration, Hydraulic Engineering Circular No. 20, Washington, DC, 1995. Laursen, E. M., "Scour at Bridge Crossings, "Journal of the Hydraulics Division, American Society of civil Enqineers, 86(1142) 1960. Maccaferri Gabions Inc., "Gabion and Reno Mattress Short Course," West Sacramento, CA, 1997. Molinas, A., and Yang, C. T., “Computer Program User's Manual for GSTARS,” U.S. Bureau of Reclamation, Denver, Col, 1986. Morris G. L., and J. Fan, Reservoir Sedimentation Handbook, McGraw-Hill, New York, 1997. Normann, J. M., R. J. Houghtalen, and W. J. Johnston, Hydraulic Design of Highway Culverts, HDS No. 5, Federal Highway Administration (FHWA), U. S. Department of Transportation, Norfolk, VA, 1985. Parthenaides, E., "Erosion and Deposition of Cohesive Soils," Journal of the Hydraulics Division, American Society of Civil Enqineers, 755–771, 1965.
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HYDRAULIC DESIGN OF CULVERTS AND HIGHWAY STRUCTURES
Hydraulic Design of Culverts and Highway Structures 15.71 Pemberton, E. L., and J. M. Lara, Computing Degradation and Local Scour, Technical Guideline for Bureau of Reclamation, Engineering Research Center, Denver, C, 1984. Portland Cement Association, Handbook of Concrete Culvert Pipe Hydraulics, Skokie, IL, 1964. Raudkivi, A. J., Loose Boundary Hydraulics, Pergamon Press, New York, 1990. Raudkivi, A. J., Sedimentation: Exclusion and Removal of Sediment from Diverted Water, A. A. Balkema, Rotterdam, the Netherlands, 1993. Reagan, D., “ Highway Drainage Design,” C-TAP Course, 1993. Reihsen, G., and L. J. Harrison, Debris Control Structures, Hydraulic Engineering Circular No. 9, Federal Highway Administration, U.S. Department of Transportation, Washington, DC, 1971. Richardson, E. V., and S. R. Davis, Evaluating Scour at Bridges, 3rd ed., Hydraulic Engineering Circular No. 18, Publication No. FHWA-IP-90-017, Federal Highway Administration, U.S. Department of Transportation, Washington, DC, 1995. Richardson, E. V., Simons, D. B., and P. Y. Julien, Highways in the River Environment, FHWAHI90-016, Federal Highway Administration, U.S. Department of Transportation, Washington, DC, 1990. Ruff, J. F. , Scour at Culvert Outlets in Mixed Bed Materials, Federal Highway Administration/RD82/011, 1982. Shearman, J.O. Bridge Waterways Analysis Model for Mainframe and Micro computers, WSPRO / HY-7, Federal Higway Administration, U.S. Department of Transportation, Washington, D.C., 1987. Simons, D. B., and F. Senturk, Sediment Transport Technology, Water Resources Publications, Fort Collins, Co, 1977. Simons, Li & Associates, Engineering Analysis of Fluvial Systems, Fort Collins, Co, 1982. State of Florida Department of Transportation, Drainage Manual, Vol. 3, Theory, Drainage Design Office, FL, 1987. Strickler, A., “Beiträge zur Frage der Geschwindigkeitsformel und der Rauhigkeitszahlen für ströme, Kanäle und geschlossene Leitungen,” Mitteilungen des Eidgenössischen Amtes für Wasserwirtschaft 16, Bern, Switzerland, 1923 (Translated as “Contributions to the Question of a Velocity Formula and Roughness Data for Streams, Channel, and Closed Pipelines,” by T. Roesgan and W. R. Brownie, Translation T-10, W. M. Keck Laboratory of Hydraulics and Water Resources, California Institute of Technology, Pasadena, CA, January 1981). U.S. Army Corps of Engineers CA, HEC-2, Water Surface Profiles, Hydraulic Engineering Center, Davis, CA, 1991. U.S. Army Corps of Engineers, HEC-6, Scour and Deposition in River and Reservoirs, Hydrologic Engineering Center, Davis, CA, 1993. U.S. Army Corps of Engineers, River Analysis System, HEC–RAS, User’s Manual Version 1.0, Hydrologic Engineering Center, Davis, CA, 1995 U.S. Department of Transportation, User's Manual for WSPRO–A Computer Model for Water Surface Profile Computation, Report No. FHWA-IP-89-027, Federal Highway Administration, Washington, DC, 1990. Vanoni, V. A., ed., Sedimentation Engineering, ASCE Manual and Reports on Engineering Practice, 54, American Society of Civil Engineers, New York, 1975. Yang, C. T., Sediment Transport: Theory of Practice, McGraw-Hill, New York, 1996.
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Source: HYDRAULIC DESIGN HANDBOOK
CHAPTER 16
HYDRAULIC DESIGN OF FLOOD CONTROL CHANNELS George K. Cotton Simons & Associates, Inc. Fort Collins Colorado
16.1 INTRODUCTION Flood protection is essential to the economic and environmental integrity of most civil engineering projects. The reliability of transportation systems, the livability of urban and suburban developments, the long-term stability of landfills, the operation of mining excavations, the reclamation of disturbed lands, and the management of forest and agricultural lands require careful planning for flood hazards. In nearly all these cases, design of a flood-control system will include a variety of conveyance channels referred to as floodcontrol channels. To perform reliably, flood-control channels must behave in a stable, predictable manner. This ensures that a known flow capacity will be available for a planned flood event. In most cases, the design goal is a noneroding channel boundary, although, in certain cases, a dynamic channel is sought. Since most soils erode under a concentrated flow, channel linings are needed either temporarily or permanently to achieve channel stability. Channel linings can be classified in two broad categories: rigid or flexible. Rigid lining include channel pavements of concrete or asphaltic concrete and a variety of precast interlocking blocks and articulated mats. Flexible linings include such materials as loose stone (riprap), vegetation, manufactured mats of light-weight materials, fabrics, or combinations of these materials. The selection of a particular lining is a function of the design context, involving issues related to the consequences of flooding, the availability of land, and environmental needs. A rigid lining is capable of high conveyance and high-velocity flow. Flood-control channels with rigid linings are often used to reduce the amount of land required for a surface drainage system. When land is costly or unavailable because of restrictions, use of rigid channel linings is preferred. Flexible channel linings are distinguished from rigid linings because they can respond to a change in channel shape. They can sustain some adjustment of the channel’s shape and still maintain their integrity. This would not be the case for a rigid lining, where local damage to the lining may lead to a general unraveling. Damage to a rigid lining can result from secondary forces, such as frost heave, piping, or slumping, although uplift and shear forces are often causes of failure. Flexible linings are used as temporary channel linings for control of erosion during construction or reclamation of disturbed areas. Also, when environmental requirements 16.1 Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
HYDRAULIC DESIGN OF FLOOD CONTROL CHANNELS
16.2
Chapter Sixteen
are part of the design, flexible lining materials have several advantages. Flexible linings are inexpensive, permit infiltration and exfiltration, and allow growth of vegetation. Hydraulically, flow conditions in the channel lined with flexible materials generally can be made to conform to conditions found in a natural channel. This provides better habitat opportunities for local flora and fauna. By permitting the growth of vegetation in the channel, flexible linings can provide a buffering effect for runoff contaminant’s and sediment. We are fortunate that a large number of innovative and traditional products are currently available for channel linings. In most cases, these products have received extensive testing, including laboratory flume studies, large-scale prototype modeling, and documentation of field case histories. From this effort has come a better understanding of complicated flow conditions associated with each type of lining and a better understanding of product performance and of methods needed for design. Construction experience has resulted in better specifications for channel lining materials and their installation. The presentation in this chapter covering flexible lining materials is based on work in preparing Design of Roadside Channels with Flexible Linings (Hydraulic Engineering Circular No. 15) for the Federal Highway Administration. Chen and Cotton (1988), Since 1988, when that manual was published, the commercial market for channel-lining products has expanded. Because product testing and performance monitoring also have increased, this chapter also serves to update the reader on the status of current practice. The so-called tractive force (or boundary shear stress) that acts on the channel’s perimeter describes the basic mechanics of channel stability. Design and field methods based on tractive force are well suited for the evaluation of the stability of smaller channels where the grade of the channel dominates. In these cases, the calculation of shear stress is simple, the determination of channel slope is the most difficult estimation. For field observations on a channel of known grade, depth alone needs to be measured to estimate the maximum shear stress. Field estimates of flow velocity require some type of current meter. In design, the performance criteria are simple to recall for a specific type of lining because it is represented by a single permissible shear stress value. This permissible shear stress value is applicable over a wide range of channel slopes and shapes. Permissible velocity criteria, on the other hand, are a function of channel slope, lining roughness, and channel shape.
16.2 DESIGN CONCEPTS The design methods are based on the concept of maximum permissible tractive force, coupled with the hydraulic resistance of the particular lining material. The method includes two parts: computation of the flow conditions for a given design discharge and determination of the degree of erosion protection required. The flow conditions are a function of the channel geometry, design discharge, channel roughness, and channel slope. The erosion protection required can be determined by computing the shear stress on the channel at the design discharge and by comparing the calculated shear stress to the permissible value for the type of channel lining used.
16.2.1 Flood-Control Channel Design The methods given in this chapter are for small (less than 1 m3/s) to medium-sized floodcontrol channels (less than 10 m3/s). The design of larger flood-control channels is increasingly complex in terms of the physical environment, the nature of the flood risk, and the number of issues and constraints that a design team must address. Small channels
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HYDRAULIC DESIGN OF FLOOD CONTROL CHANNELS
Hydraulic Design of Flood Control Channels 16.3
can often be designed on the basis of simplified hydrologic analysis (i.e., the rational method), without detailed geotechnical data, and by assuming uniform flow. A mediumsized flood-control channel typically requires hydrologic routing, information on soil properties, and a hydraulic analysis based on nonuniform flow. The level of effort for design increases steadily as the size of the channel increases. It is expected that it may require approximately four times the hours for technical staff to accomplish a medium-sized flood channel compared with a small channel. Since costs will increase by a factor of about 10, there is a definite economy of scale in the design of larger channels. However, the larger a channel becomes, the more tributary and appurtenant features will be needed for a complete design. Therefore, the design of a medium-sized channel often includes the design of several small channels and other features.
16.2.2 Open-Channel Flow 16.2.2.1 Types of flow. Open-channel flow can be classified according to three general conditions: (1) uniform or nonuniform flow, (2) steady or unsteady flow, and (3) subcritical or supercritical flow. In uniform flow, the depth and discharge remain constant along the channel. In steady flow, no change in discharge occurs over time. Most natural flows are unsteady and are described by a runoff hydrography. One can assume, in most cases, that the flow will vary gradually and can be described as steady, uniform flow for short periods. Subcritical flow is distinguished from supercritical flow by a dimensionless number called the Froude number (Fr), which is defined as the ratio of inertial forces to gravitational forces in the flow. Subcritical flow (Fr 1.0) is characterized as tranquil and has deep, slower velocity flow. Supercritical flow (Fr 1.0) is characterized as rapid and has shallow, high-velocity flow. For design purposes, uniform flow conditions also are considered to be steady. The channel grade So, water-surface grade Sw, and the energy grade Sf are assumed to be equal. This allows the development of a flow equation based on a friction loss formula (such as Manning’s equation) and channel grade. Computation of uniform flow is suitable for small channels and a useful approximation for medium-sized channels. In medium channels, steady uniform flow is rare, and a more complete analysis using gradually varied flow methods is required. In solving gradually varied flow, the designer should rely on a current computer program, the accuracy of which is well documented. Design of steep channels with supercritical flow presents a number of special concerns. Waves can form in a channel bed with a flexible lining (beginning near a Fr of 0.8) that ultimately approach the depth of flow. In extremely steep channels, the flow may splash and surge in a violent manner and additional freeboard is required. Ultimately, at Fr’s near 2.0, the flow becomes unstable. Channel designs in this range should be avoided. 16.2.2.2 Resistance to flow and boundary shear stress. Flow resistance is the result of the drag of moving water against the channel boundary. For practical purposes, the flow in most channels will be fully turbulent and the velocity, V, will be proportional to the square root of the shear stress, , on the channel boundary. This gives the following simple formula for estimating channel velocity: 兹苶 苶 / 苶 V Cf 兹苶
(16.1)
where Cf the conveyance factor and the density of water.
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HYDRAULIC DESIGN OF FLOOD CONTROL CHANNELS
16.4
Chapter Sixteen
FIGURE 16.1 Definition sketch for boundary shear stress
For uniform flow, the average shear stress on the channel boundary acts opposite to the weight component of the flow, as shown in Fig. 16.1. For channel grades of less than 10°, the sine of the grade can approximated by the tangent and, hence, the slope of the bed. This leads to the equation for mean boundary shear: τ∆ ∆L P γ A ∆ ∆L Sf τ = γ R Sf
(16.2)
where ∆L ∆ the incremental reach length (Fig. 16.1), P the wetted perimeter, γ the specific weight of water, r A the cross sectional area of flow, and Sf the friction slope (energy grade). The conveyance factor in Eq. 16.1 is not constant but is a function of both boundary and flow conditions. If this were not the case, there would be little difference between a velocity-based approach and a tractive shear-based approach to channel design. In Manning’s formulation, the conveyance factor varies as the one-sixth power of the hydraulic radius, R, giving 1\6 Cf 1 R n
(16.3)
Note: Because Cf is dimensionless, Manning’s coefficient n has the dimension of length to the one-sixth power. Combining Eqs. 16.1, 16.2, and 16.3 gives the standard form of the Manning’s resistance equation: 2\3 1\2 V 1 R Sf n
(16.4)
Not all channel linings behave according to the Manning’s formulation for the conveyance factor. Most rigid channel linings have a Manning’s n that is approximately constant. For shallow flows, the n value increases in rigid channels, but this effect is often
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HYDRAULIC DESIGN OF FLOOD CONTROL CHANNELS
Hydraulic Design of Flood Control Channels 16.5
neglected even for small flood-control channels. However, for flexible types of channel linings, the conveyance factor is difficult to describe using Eq. 16.3. A channel lined with a good stand of vegetation cannot be described by a single n value. The flow resistance is complicated by the mechanics of the vegetation since the stem of the plant will bend because of the drag force, changing the height of the plant relative to the flow depth. The Soil Conservation Service (SCS) (USDA, 1954), through the work of Ree and Palmer (1949), developed standard classifications for vegetative flow retardance. Grasses are classified into five broad categories of flow retardance: Class A identifies grasses with the highest flow retardance and Class E identifies the grasses with the lowest flow retardance. In general, high flow-retardance species of grass form a dense cover, are tall, and have stiff stems. Sparse cover or short, flexible grasses have lower flow retardance. Temple (1980; Temple et al., 1987) and Kouwen, (1988; Kouwen and Li, 1980; Kouwen and Unny, 1969) have developed modern approaches to the flow resistance of vegetation. Temples approach is based on a mathematical fit to the SCS retardance charts and a generalized retardance index that is a function of two properties of gress: stem length and cover density. Temple compiled standard values of these properties for selected species for reference stem densities representing “good” cover conditions. Adjusting these reference properties for local conditions and expected growth allows the designer to determine an appropriate index for specific design conditions. Kouwen analyzed the biomechanics of vegetation and developed conveyance factors that are a function of the density, height, and stiffness of the grass. He showed that the resulting method gives results that are consistent with the SCS retardance curves. Since Kouwen’s method is physically based, it is a useful tool for extending retardance curves to channel conditions that were not studied by the SCS, including stiff vegetation and mild channel grades.
16.2.3 Flood-Control Channel Components A basic flood-control channel consists of a minimum of five components. There are specific design issues associated with the local flow conditions at each component. More complex flood-control systems also may include additional components for flood storage (detention basins) and grade control (low dams). Channel inlet. At the point where water enters a flood-control channel, local shear stresses can develop as a result of local acceleration of the flow. For small flood-control channels, the inlet may consist of small berms or dikes that collect runoff and direct this diffuse flow to the channel. A medium-sized flood-control channel will typically be the result of the combination of several tributaries or the extension of an existing channel. Collection and control of incoming flow is essential to the proper operation of a larger channel. In most cases, a flood-control channel will discharge as a tributary to a larger channel. Reach. The reach component is the length of channel, with only minor variation in the properties of grade, discharge, cross section, and lining material. A reach may have a straight or curving alignment. In larger flood-control channels, a reach also may include a bridge or culvert. Flow around a bend in an open channel induces centrifugal forces created by the change in the direction of flow. This results in a superelevation of the water surface, with its surface being higher at the outside of the bend than at the inside of the bend. Flow around a channel bend also imposes higher shear stress on the channel bottom and banks.
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HYDRAULIC DESIGN OF FLOOD CONTROL CHANNELS
16.6
Chapter Sixteen
The increased shear stress requires additional design considerations within and downstream of the bend. Confluence. A confluence is the site where two or more flows merge without a significant grade difference between each channel. The process of merging flow involve turbulent mixing and a local increase in energy loss and boundary shear stress. Flow separation, wave formation, and bank impingement can occur in a confluence that creates much higher local boundary shear stresses and a potential for erosion. Control of the velocity and direction of tributary inflows often is required to prevent damage to the channel within a confluence. Where a large tributary joins the main channel, performance of the confluence can be improved if the direction of the two flows is as nearly parallel as possible. Side inlet. A side inlet allows flow to enter the channel over the bank. Example of the need for a side inlet to a channel are flow from a field or a street. Often, the flow enters the channel from a chute constructed on the channel bank. When the bank channel is at a mild grade, vegetation may be a sufficient lining for the chute. For steeper bank slopes, however, riprap or gabion linings are common. Channel crossing. A channel crossing is required where a private or public road passes over the channel. Structures used for this purpose are stream fords, culverts, and bridges. Stream fords are best suited for low-traffic-volume roads crossing channels with an intermittent type of flow. Fords can be hazardous if flow depths of more than 0.30 to 0.45 m. are expected. Culverts of concrete or metal pipe provide an economical crossing for small channels. Since the shape of the culvert is typically circular, elliptical, or arched, the culvert will obstruct a portion of the channel section. Bridges of concrete, steel, or timber are used when the channel must remain largely unobstructed. Transition. A transition is a gradual expansion or contraction between two channel sections. Transitions can occur between one reach and another or between a reach and the channel inlet or outlet. Channel Outlet. At the point where water exits a flood control channel, local shear stresses can develop as aresult of deceleration of the flow. At the outlet, the flow changes to match the local velocity and depth of the receiving channel.
16.2.4 Stable Channels 16.2.4.1 Stable channel modes. A stable channel can be either static or dynamic, but over time the net effect of scour and deposition must be zero. In a static stable channel, scour and deposition occur within the limits of a channel boundary that effectively resists the erosive force of the flow. Although the transport of sediment can be significant in a static channel, the effective change in channel section at any given time is small. This implies equilibrium between the incoming supply of sediment and the transport of sediment within a channel reach. When the supply of sediment is small, the principles of rigid boundary hydraulics can be applied to evaluate the channel’s capacity and the lining’s stability. When the supply of sediment is large, then the determination of the channel’s capacity must consider the transport of both water and sediment. The transport of sediment affects the flow resistance and therefore the shear forces and the stability of the channel lining. In a dynamic stable channel, some change in the channel bed and banks are expected. The channel is considered to be stable if the morphology (i.e., shape) of the channel remains unchanged over time. Important measures of channel shape include the channel width, depth, slope, pattern (straight, meandering, or braided), hydraulic variables (flow
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HYDRAULIC DESIGN OF FLOOD CONTROL CHANNELS
Hydraulic Design of Flood Control Channels 16.7
velocity and discharge), and sediment variables (particle size, suspended concentration of sediment). A dynamic channel can be regarded as stable as long as local changes in channel shape do not have adverse consequences. For most small- or moderate-capacity floodcontrol systems, design of a dynamic channel is often too complex or it presents too many uncertainties. Consequently, development of static stable channels is usually preferable to using dynamic approaches to the design. Several empirical methods are commonly applied to determine whether a channel is stable. These methods are defined as permissible velocity methods. Permissible velocity approaches were first developed in the early 1920s (Lacey, 1920, Lindley, 1919) and are related to the successful development of regime theory for large irrigation canals beginning in 1885 with Kennedy’s formula. Regime theory continues to be a basic tool for evaluating large rivers and irrigation canals (Blench, 1969; Simons and Albertson, 1963); however, its empirical nature limits its application in the case of flood control channels. In the 1950s, under the direction of Lane at the U.S. Bureau of Reclamation, research was conducted that refined the permissible tractive force method (Glover and Florey, 1951; Lane, 1955). This work and subsequent research clarified the actual physical processes occurring for flow in stable channels. This method extended the range of stable channel design beyond the empirical limits of regime theory. In most cases, a more realistic model of channel stability is based on permissible tractive force. 16.2.4.2 Tractive force. The flow resistance of water moving against a channel boundary results in shear force along the channel boundary referred to as the tractive force. As was discussed in Sec. 16.1.2, there is a close relationship between the tractive force and the flow velocity. In a uniform flow with a normal (semilog) velocity distribution, the tractive force is equal to the effective component of the gravitational force acting on the body of water parallel to the channel bottom (see derivation of Eq. 16.2). However, shear stress is not uniformly distributed along the wetted perimeter in a channel, and Eq.16.2 describes only the average shear force on the channel. For design, it is necessary to determine the maximum shear along the channel perimeter so that the lining material is suitable for resisting the shear. At the same time, the stability of a soil particle decreases as the lateral slope of the channel increases. The relative combination of particle stability and boundary shear is needed to attain a stable channel. Simply stated, the principle of stable channel design using tractive force is to have the shear strength of the lining material τp , exceed the boundary shear force τb at every point in the channel’s wetted perimeter: τp τb
(16.5)
The permissible shear stress of the lining material τp is the maximum shear stress that the lining can safely withstand. Using the critical shear stress τc for a soil particle as a reference value, the permissible shear stress can be defined as τp Ca τc,
(16.6)
where Ca is the critical shear-stress adjustment factor. The design shear stress is affected by a number of factors, including: the density of the soil particles, the submerged soil friction coefficient (µ tan φ, where φ is the angle of repose), the lateral (side slope) and down-slope components of soil particle weight, and the direction of flow. The boundary shear stress τb varies within a channel section primarily as a function of depth and also in response to the lateral diffusion of momentum in the channel section. Using the mean shear stress τo as a reference value, the local boundary shear stress can be defined as
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HYDRAULIC DESIGN OF FLOOD CONTROL CHANNELS
16.8
Chapter Sixteen
FIGURE 16.2 Boundary shear distribution in channel section
FIGURE 16.3 Boundary shear distribution in a bend
τb Ka τo
(16.7)
where Ka is the boundary shear-stress adjustment factor. Channel alignment, the relative channel width and depth (aspect ratio), and the channel side slope affect the boundary shear stress. Figure 16.2 shows a typical distribution in a parabolic channel. Boundary shear is greatest at the middle of the channel (maximum flow depth) and tends toward zero along the channel bank. The distribution shear of in a channel bend is shown in Fig. 16.3.
16.2.5 Design Parameters Flood frequency. The most important question facing the designer of a flood-control channel is - What is the probability of failure? Because flood-control channels are built to protect valuable facilities, damage should occur only rarely. The probability of flood damage, although never zero, can be low enough to make the risk to the facility acceptable. Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
HYDRAULIC DESIGN OF FLOOD CONTROL CHANNELS
Hydraulic Design of Flood Control Channels 16.9 TABLE 16.1
Commonly Used Flood Frequencies
Type of Facility
Potential Flood Damage
Flood Frequency (Return Period, years)
Collector road Roadside or median ditch
Delay costs Impeded traffic
2 to 5 years
Urban collector streets
Delay costs Impeded emergency access Detour costs
10 to 25 years
Rural flood control
Crop damage Road damage
25 to 50 years
Urban flood control
Property damage Infrastructure damage
100 years
When flows are small and potential damages are small, such as in the case of a roadway drainage system, more frequent flooding is allowed. For larger flood-control channels, where flood damage is severe, a low frequency of flooding is desired. Although economic analysis can be used to determine the optimal range of flood frequency for a floodcontrol facility, only larger projects warrant such an effort. In most cases, standard flood frequencies are used. Table 16.1 provides a brief summary of common flood frequencies and their use in the design of various types of facilities. Duration of flooding. Small flood-control channels with intermittent flooding tend to have short times of flooding, typically on the order of several hours. Larger flood-control channels, or channels with sustained flows, require a design that considers the persistence of boundary stresses. Channels that sustain only short periods of boundary stress often can sustain higher stresses without damage. Channel profile. Major project features, such as roads generally dictate the slope of smaller channels. If channel stability cannot be maintained for these conditions, it may be feasible to reduce the channel gradient slightly relative to other grading. This typically results in short, steep reaches that are constructed as rigid chutes or low drops. Channel slope is among the major parameters affecting boundary shear stress. Thus at a given discharge, the shear stress is less for a mild (subcritical regime) gradient compared with a steep (supercritical regime) gradient. It is not uncommon for smaller flood-control channels to operate mainly in supercritical regime. However, extremely steep channels (above 10 percent) and larger channels require additional design considerations. Channel section. The most common shapes of drainage channel are trapezoidal or triangular. These shapes are a close approximation of the most efficient hydraulic section, a semicircle. For smaller channels with flexible linings, these shapes tend to be constructed with slightly rounded corners, which gives the as-built section a slightly parabolic shape. The banks are the least stable portions of the channel section. Bank stability increases as the side slope flattens. For smaller channels, a 3:1 side slope is sufficient to prevent erosion in excess of the channel bed. For larger channels, or for small channels that require steep side slopes, the stability of the side slope should be checked. In larger flood-control channels with persistent low flows, a low-flow channel often is included within the channel section. It is advisable to situate the low-flow channel near
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HYDRAULIC DESIGN OF FLOOD CONTROL CHANNELS
16.10
Chapter Sixteen
the center of the main channel to avoid creating increased shear near the channel bank. Low-flow channels are often designed to meander within the larger section. The meander reduces the gradient of the channel, allowing the enhancement of the channel section for habitat or appearance. The potential effect at meander bends on the main channel bank line should be considered. Although a channel section can be widened to reduce boundary shear, channels with large width depth to ratios typically are not stable. Over time, a high stress area will develop within the section and cause erosion that will form a narrower, deeper channel, the channel eventually fails.
16.3 CHANNEL LININGS Considerable research and development has produced a variety of rigid and flexible lining materials. Before the late 1960s, channel linings were constructed primarily of natural materials, such as rock riprap, stone masonry, concrete and vegetation. Material manufactured or fabricated into rolls offered several advantages for erosion control, particularly during construction and during the establishment of vegetation. Use of rolled erosioncontrol products for temporary erosion control assured improved long-term performance of vegetative linings.
16.3.1 Lining Types Because of the large number of channel stabilization materials currently available, it is useful to classify materials according to their performance characteristics. Lining types are classified as either rigid or flexible. Flexible linings are grouped further as either permanent or temporary. Rigid linings Cast-in-place concrete Cast-in-place asphaltic concrete Stone masonry Soil cement Fabric formed concrete Grouted riprap Flexible linings (Long-term nondegradable) Riprap Wire-enclosed stone Vegetation Gravel Synthetic mat
16.3.2
Flexible linings (temporary degradable) Straw with net Curled wood mat Jute net Woven paper net Fiberglass roving
Performance Data
16.3.2.1 Rigid linings. Rigid linings are useful in applications where high shear-stress or nonuniform flow conditions exist, such as at transitions in channel shape or at an energy dissipation structure. In areas where loss of water or seepage from the channel is undesir-
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HYDRAULIC DESIGN OF FLOOD CONTROL CHANNELS
Hydraulic Design of Flood Control Channels 16.11
able, a rigid lining can provide an impermeable barrier. Because rigid linings are nonerodible, the designer can use any channel shape that provides adequate conveyance. Rigid linings may be the best option if right-of-way limitations restrict the channel size. Despite the nonerodible nature of rigid linings, they are highly susceptible to failure from structural instability. For example, cast-in-place or masonry linings often break up and deteriorate if the foundation is poor. Once a rigid lining deteriorates, it is highly susceptible to erosion because it forms large, flat, broken slabs that are easily moved by flow. The major causes of structural instability and failure of rigid linings are freeze-thaw, soil swelling, and excessive soil pore-water pressure. Freeze-thaw and swelling soils exert upward forces against the lining, and the cyclic nature of these conditions eventually causes failure. Excessive soil-pore pressure may occur when the flow levels in the channel drop quickly but the soil behind the lining remains saturated. This can result in instability of the bank’s slope caused by the high water-table gradients within the channel bank. Construction of rigid linings requires specialized equipment using relatively costly materials. As a result, the cost of rigid channel linings is high. Prefabricated linings can be a less expensive alternative if shipping distances are not excessive. There often are significant environmental issues associated with rigid channel linings. In environmentally sensitive areas, replacement of concrete linings with articulated block mats is a partial solution, because openings between the blocks allows vegetation to take hold. 16.3.2.2 Flexible linings. Riprap and gabion are suitable linings for hydraulic conditions similar to those requiring rigid linings. Because flexible linings are permeable, they may require protection of the underlying soil to prevent washout. For example, filter cloth is often used with riprap to inhibit soil piping. Vegetative and temporary linings are suited to hydraulic conditions where uniform flow exists and shear stresses are moderate. Most grass linings cannot survive sustained flow conditions or long periods of submergence. Grass-lined channels with sustained low flow and intermittent high flows often are designed with a composite lining of a riprap or concrete low-flow section. The primary use of temporary linings is to provide protection from erosion until vegetation is established. In most cases, the lining will deteriorate over the period of one growing season, which means that successful revegetation is essential to the overall channel stabilization effort. Temporary channel linings can be used without vegetation to control erosion on construction sites temporarily. 16.3.3 Information About Flexible Linings The erosion control industry has grown in response to continued infrastructure development and increased awareness of water-quality problems. Traditional methods of erosion control are well documented, and a variety of specifications exist for these types of linings. In 1997, the Erosion Control Technology Council (ECTC) established standard terminology and testing methods for the variety of manufactured products used for erosion control. The ECTC terminology for rolled erosion control products (RECPs) is used here for clarity. 16.3.3.1 Long-term, Nondegradable flexible linings. Vegetation. Vegetative linings consist of planted or sodded grasses placed in and along the drainage (Fig. 16.4). If planted, grasses are seeded and fertilized according to the requirements for the particular variety or mixture and to soil conditions. Sod is laid parallel to the direction of flow and can be secured with staples or stakes.
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HYDRAULIC DESIGN OF FLOOD CONTROL CHANNELS
16.12
Chapter Sixteen
FIGURE 16.4 Vegetative channel lining
FIGURE 16.5 Rock channel lining
Rock riprap. Rock riprap is dumped or hand placed on prepared ground with a filter blanket or a prepared bedding material interface (Fig. 16.5). The stone layer is placed to form a well-graded mass with a minimum of voids. Stones should be hard, durable, prefer-
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HYDRAULIC DESIGN OF FLOOD CONTROL CHANNELS
Hydraulic Design of Flood Control Channels 16.13
FIGURE 16.6 Wire-enclosed stone channel lining
ably angular in shape, and free from overburden, shale, and organic material. Resistance to disintegration from channel erosion should be determined from service records or from specified field and laboratory tests.
FIGURE 16.7 Gravel channel lining
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HYDRAULIC DESIGN OF FLOOD CONTROL CHANNELS
16.14
Chapter Sixteen
Wire-enclosed stone. Wire-enclosed stone consists of a wire basket or tube filled with stone (Fig. 16.6). The wire basket is made of steel wire woven in a uniform pattern and reinforced on corners and edges with heavier wire. Common forms of wireenclosed stone include boxlike baskets, thin mattresses, and tubes. The containers are filled with stone, connected together, and anchored to the channel side slope. Stones are graded fairly uniformly, with the smallest size larger than the wire mesh opening. The stones should be hard, durable, and free from overburden, shale, and organic material. Wire-enclosed stone typically is used when rock riprap is either not available or not large enough to be stable
FIGURE 16.8 Turf-reinforcement mat channel lining
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HYDRAULIC DESIGN OF FLOOD CONTROL CHANNELS
Hydraulic Design of Flood Control Channels 16.15
Gravel. Gravel consists of coarse gravel or crushed stone placed on filter fabric or prepared bedding material to form a well-graded lining with a minimum of voids (Fig. 16.7). The stones should be hard, durable, and free from overburden, shale, and organic material.
FIGURE 16.9 Woven-paper net channel lining
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HYDRAULIC DESIGN OF FLOOD CONTROL CHANNELS
16.16
Chapter Sixteen
Turf reinforcement mat. Turf reinforcement mat is composed of ultraviolet-stabilized, nondegradable, synthetic fibers, netting, or filaments processed into a three-dimensional reinforcement matrix ranging in thickness from 6 to 20 mm (Fig. 16.8). The mat provides sufficient thickness, strength, and void space to permit soil filling and development of vegetation within the matrix. The mat is laid parallel to the direction of flow is stapled or staked to the channel surface, and is anchored into cutoff trenches at regular intervals along the channel. The mat is top-dressed with fine soil at a depth equal to the mat’s thickness and is seeded and fertilized. Turf reinforcement occurs as root growth penetrates and entangles with the mat.
FIGURE 16.10 Jute net channel lining
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HYDRAULIC DESIGN OF FLOOD CONTROL CHANNELS
Hydraulic Design of Flood Control Channels 16.17
16.3.3.2 Temporary degradable flexible linings. Woven paper net. Woven paper net is erosion-control net that consists of knotted plastic netting interwoven with paper strips (Fig. 16.9). The net is applied evenly on the channel slopes, with the fabric running parallel to the channel’s direction of flow. The net is stapled to the ground and placed into cutoff trenches at regular intervals along the channel. Woven paper net is usually installed immediately after seeding operations.
FIGURE 16.11 Curled-wood mat channel lining
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HYDRAULIC DESIGN OF FLOOD CONTROL CHANNELS
16.18
Chapter Sixteen
FIGURE 16.12 Straw with net channel lining
Jute net. Jute net is an erosion control mat that consists of jute yarn approximately 6 mm in diameter that is woven into a net with openings approximately 10 to 20 mm (Fig. 16.10). The jute net is loosely laid in the channel parallel to the direction of flow. The net is secured with staples and is placed int cutoff trenches at regular intervals along the channel. Jute net is usually installed immediately after seeding operations. Curled wood mat. Curled wood mat is an erosion control blanket that consists of wood fibers, 80 percent of which are 150 mm or longer, with a consistent thickness and an even distribution of fiber over the entire mat (Fig. 16.11). The topside of the mat is covered with biodegradable plastic mesh. The mat is placed in the channel parallel to the direction of flow and is secured with staples and cutoff trenches. Straw with net. Straw with net can be manufactured as an erosion control blanket or can be constructed from straw mulch secured with erosion control net. The constructed version consists of plastic mesh with 20-mm2 square openings overlaying straw mulch (Fig. 16.12). Straw is spread uniformly over the channel surface at a rate of approximately 4.5 metric tons/hectare and can be crimped into the soil. Plastic mesh is placed over the straw and stapled to the soil. The manufactured version consists of straw, either sewn to a layer of erosion control net or sandwiched between to two layers of erosion control net. Straw weight is approximately 2.5 metric tons/hectare. The mat is placed parallel to the direction of flow and is secured with staples and cutoff trenches. Fiberglass roving. Fiberglass roving consists of continuous fibers drawn from molten glass that is coated and lightly bound together into roving (Fig. 16.13). The roving is ejected by compressed air, forming a random mat of continuos glass fibers. The material is spread uniformly over the channel and anchored with asphaltic materials.
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HYDRAULIC DESIGN OF FLOOD CONTROL CHANNELS
Hydraulic Design of Flood Control Channels 16.19
FIGURE 16.13 Fiberglass roving channel lining
16.4 MILD-GRADIENT CHANNEL DESIGN (SYMMETRIC SECTION) This section outlines a method of stable channel design for channel gradients of less than 10 percent. Most of the lining types presented will perform acceptably up to grades of
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HYDRAULIC DESIGN OF FLOOD CONTROL CHANNELS
16.20
Chapter Sixteen
TABLE 16.2
Mannings Roughness Coefficients–Mild Gradients (< 10%) ks Depth Ranges Lining Category Lining Type mm 0.0–0.15 m 0.15–0.60 m Rigid
Unlined
> 0.60 m
Concrete
–––
0.015
0.013
0.013
Grouted riprap
–––
0.040
0.030
0.030
Stone masonry
–––
0.042
0.032
0.030
Soil cement
–––
0.025
0.022
0.020
Asphalt
–––
0.018
0.016
0.016
Bare soil (compacted)
–––
0.023
0.020
0.020
Rock cut
–––
0.045
0.035
0.025
Temporary
Woven paper net
1.2
0.016
0.015
0.015
(degradable)
Jute net
11.6
0.028
0.022
0.019
Flexible Linings
Long-term Gravel Large Stone
Straw with net
36.6
0.065
0.033
0.025
Curled wood mat
33.5
0.066
0.035
0.028
Fiberglass roving
10.7
0.028
0.021
0.019
Synthetic mat (unvegetated)
19.8
0.036
0.025
0.021
25 mm D50
25.0
0.044
0.033
0.030
50 mm D50
50.0
0.066
0.041
0.034
150 mm D50
150.
0.104
0.069
0.035
300 mm D50
300.
–––
0.078
0.040
about 4 percent, whereas fewer are effective as the grade approaches 10 percent. On steeper grades, stone lining (riprap and wire-tied rock) is the most common lining type.
16.4.1 Resistance to Flow 16.4.1.1 Rigid and flexible lining materials. Channel roughness is affected by the relative height of the roughness compared to the flow depth. As a result, channel roughness increases for shallow flow depths and decreases as flow-depth increases. For flow depths between 0.15 and 0.60 m (a typical range for small drainage channels), roughness is higher compared to deeper flows. Recommended values of Manning’s roughness coefficients are summarized in Table 16.2. A more detailed table is presented in Chapter 3, (Appendix 3.B) These values are means of the range; the maximum and minimum of the range are about 10% of the mean value. 16.4.1.2 Vegetative linings. For vegetative linings, the channel roughness varies significantly, depending on the amount of submergence of the vegetation. Because vegetation is flexible, the amount of submergence will increase as the drag force bends the plant stems toward the channel bed. For shallow flow, plants are erect and may protrude from the flow. As flow depth and velocity increase, plants bend and are submerged by the flow. At higher velocities, the plants are bent flat to the channel bed. Kouwen’s method (1990) provides
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HYDRAULIC DESIGN OF FLOOD CONTROL CHANNELS
Hydraulic Design of Flood Control Channels 16.21
a practical means of determining Manning’s roughness value for a wide range of flow conditions and vegetation types. The mechanical properties of individual plants (height and stiffness) and the density of plant cover affect flow resistance. Kouwen and Li (1980) found that the effective roughness height ks of vegetation varies as a function of shear stress exerted on the lining and density-stiffness properties: MEI 0.25 1.59 o h ks 0.14 h (16.8)
Values of the regetation height h and the density stiffness parameter MEI that are equivalent to the five standard retardance classifications (Class A representing vegetation types with the highest flow resistance and Class E, with the lowest flow resistance) are listed below: Retardance class
Average Height, h, mm
Stiffness, MEI, N m2
A
900
300.0
B
600
20.0
C
200
0.5
D
100
0.05
E
40
0.005
With grasses, there is a good correlation between the length of vegetation and the density-stiffness parameter MEI. Empirical relationships (Kouwen, 1988) are useful in the field estimation and determination of seasonal effects (note that h is in meters): Green grass: MEI 319 h3.30
(16.9)
Dormant grass: MEI 24.5 h
2.26
(16.10)
The Manning’s roughness coefficient for vegetation is a function of the relative roughness (ratio of flow depth to roughness height). The variation of Manning’s roughness with relative roughness follows the well-known semilog relationship: 1/6
R n 兹 (a b log (R/kks)) 兹苶 兹g
(16.11)
The a and b coefficients are based on a classification of the three types of flow conditions with vegetation: erect, submerged (bent), and flattened. The initial shear stress that bends the vegetation from an erect position is referred to as the vegetative critical shear stress. τcv minimum of (0.78 354 MEII2 40100 MEII4, 53 MEII0.212)
(16.12)
As a practical matter, the first term of Eq. (16.12) controls for vegetative stiffness of less than 0.16 N-m2 (D retardance). Stiffer vegetation (A, B, and C retardances) has critical shear stresses governed by the second term of Eq. (16.12). The minimum vegetative
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HYDRAULIC DESIGN OF FLOOD CONTROL CHANNELS
16.22
Chapter Sixteen
critical shear stress is about 0.80 N/m2, which occurs for stiffness values lower than approximately 0.010 N–m2 (E retardance). The values of a and b as a function of square root of the ratio of the mean boundary shear stress to the vegetative critical shear stress are listed in Table 16.3. For design, it is acceptable to assume that the grass would be bent completely flat. As long as the underlying soil and plant roots are not eroded, damage to the plant stems is not a critical factor in the lining’s performance. To simplify the estimation of flow resistance for vegetated linings, HEC-15 (Chen and Cotton, 1988) uses simplified formulas for Class A through E vegetation, as shown below: R1/6 nA 15.8 19.97 log(R1.4S0.4)
R1/6 nB 23.0 19.97 log(R1.4S0.4)
R1/6 nC 30.2 19.97 log(R1.4S0.4)
R1/6 nD 34.6 19.97 log(R1.4S0.4)
R1/6 nE 37.7 19.97 log(R1.4S0.4) 16.4.1.3 Flexible linings. Flexible linings have more variation in resistance compared with rigid linings (see Table 16.2). This is particularly true of linings with a large relative roughness, such as mats. HEC-15 (Chen and Cotton, 1988) gives the coefficients in Table 16.4 for estimation of Manning’s n using Eq. (16.11).
TABLE 16.3 Classification
Values of “a” and “b” Criteria o
冪莦 莦 1.0
Erect
“a”
Parameter “b”
0.42
5.23
cv
Submerged (Bent)
1.0
Flat
2.5
冪莦莦 2.5 o
Linearly interpolate between erect and prone parameter values
cv
冪莦莦 o
0.82
9.90
cv
Source: (Kouwen and Li, 1980)
TABLE 16.4
Flexible Lining Coefficients for the Resistance Equation (kks in meters) Woven Paper Net
a
0.73
Jute Net 0.74
Straw with Net 0.72
Curl Wood Mat 0.65
Fiberglass Roving 0.73
Synthetic Mat (unveg.) 0.96
b
8.00
8.04
7.83
7.10
8.00
8.13
ks
0.0012
0.0116
0.0366
0.0335
0.0107
0.0198
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HYDRAULIC DESIGN OF FLOOD CONTROL CHANNELS
Hydraulic Design of Flood Control Channels 16.23
Example #1: Flow resistance. The following example illustrate the variation in Manning’s n for increasing shear stress on a grass-lined channel.determine Manning’s n for a range of hydraulic radius 0.1 to 0.6. Inputs:
So = 0.015 m/m R = 0.1 to 0.6 h = 200 Condition: Green grass
Outputs: a. Estimate stem stiffness using Eq. (16.9): MEI = 1.57 Nm2. b. Compute relative roughness height using Eq. (16.8): R
τo M/m2
0.1 0.2 0.3 0.4 0.5 0.6
14.7 29.4 44.1 58.8 73.5 88.2
ks/ h 0.74 0.57 0.48 0.43 0.39 0.37
ks m 0.149 0.113 0.096 0.086 0.079 0.073
c. Compute vegetative critical shear stress using Eq. (16.12):
τcv =
58.4
N/m2
d. Compute semilog coefficients (Table 16.3): τo/τcv
a
b
0.252 0.504 0.756 1.008 1.260 1.511
0.42 0.42 0.42 0.42 0.49 0.56
5.23 5.23 5.23 5.25 6.04 6.82
e. Compute Manning’s n Eq. (16.11) R
R/kks
n
0.1 0.2 0.3 0.4 0.5 0.6
1.00 1.77 3.12 4.66 6.37 8.21
0.518 0.142 0.087 0.070 0.053 0.043
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HYDRAULIC DESIGN OF FLOOD CONTROL CHANNELS
16.24
Chapter Sixteen
1 lb/ft2 48 n/m2
FIGURE 16.14 Permissible shear stress for non-cohesive soils(after Lane, 1995)
16.4.2 Tractive Force Design To achieve a stable channel lining, an anchoring force must counter the hydraulic force that acts to detach the lining. The unit force acting to detach the lining is referred to as the tractive force, or boundary shear stress τb. The unit force acting to prevent movement of
Table 16.5
Permissible Shear Stresses for Lining Materials
Lining Category Temporary degradable
Long-term nondegradable
Lining Type Erosion control net Erosion control mat Erosion control blanket Single net Double net Turf reinforcement mat Unvegetated Vegetated
Permissible Shear (N/m2) 4.5 to 9.5 20 to 140 70 to 95 95 to 140 140 to 280 240 to 380
Source: ECTC (1997) S
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HYDRAULIC DESIGN OF FLOOD CONTROL CHANNELS
Hydraulic Design of Flood Control Channels 16.25
FIGURE 16.15 Permissible shear stress for cohesive soils (after Smerdon and Beaseley, 1959)
the lining is the component of the anchoring force perpendicular to the slope and is referred to as the permissible shear stress τp. To provide a stable lining, the permissible shear stress must exceed the boundary shear stress: τp τb
(16.6)
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HYDRAULIC DESIGN OF FLOOD CONTROL CHANNELS
16.26
Chapter Sixteen
16.4.3 Permissible Shear Stress The critical shear stress τc is the force required to initiate movement of the lining material or the underlying soil. In most cases, before movement of the lining, the underlying soil is relatively protected from erosion. However, for vegetative linings and other types of flexible lining, soil movement may occur before movement of the lining. Once the lining material is lost, the underlying soil is exposed to the full erosive force of the flow. The consequence of lining failure on highly erodible soils is great because the erosion rate after failure is high compared with the rate for soils of low erodibility. Values for critical shear stress for soils and lining materials are based on research conducted in laboratory flumes and in the field. The values presented here are judged to be conservative and appropriate for design use. The critical shear stress for noncohesive soils is a function of the mean diameter of the particle gradation, as shown in Fig. 16.14. For larger stones, the critical shear stress (assuming a stone density of 2.65 and a Shields parameter of 0.047) can be determined by the following equation: τc 0.76 D50
(16.13)
For cohesive material, the variation in critical shear stress depends on the concentration of the clay particles within the soil. The plasticity index of cohesive soil provides a simple guide to the permissible shear stress (Fig. 16.15). Table 16.5 presents the Erosion Control Technology council (ECTC, 1997) guidelines for maximum shear stress values for various rolled erosion control products. Since the above guidelines present a large range and are not product specific, it is important for product information to be carefully reviewed. In general, the upper end of the ECTC range is based on linings with good vegetative cover, short-duration flow, and excellent installation. The lower end of the ECTC range is typical of linings with little or no vegetation. In arid climates or semiarid climates, it often is difficult to achieve good vegetative cover in a channel. A lining with a good vegetative cover density in excess of 70 percent is typically required to resist shear stresses in excess of approximately 100 N/m2. The critical shear stress is the maximum shear force that the lining system can sustain. For thin lining systems, such as rolled erosion control products, the maximum shear force
TABLE 16.6
Values of Factor Cr
Side Slope, Z
Very Rounded Small Large Ar 36° Ar 39°
Crushed Large Ar 42°
0.5
2.75
2.47
2.18
1.0
1.38
1.23
1.09
1.5
0.92
0.82
0.73
2.0
0.69
0.62
0.55
3.0
0.46
0.41
0.36
4.0
0.34
0.31
0.27
5.0
0.28
0.25
0.22
Horizontal
0.00
0.00
0.00
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HYDRAULIC DESIGN OF FLOOD CONTROL CHANNELS
Hydraulic Design of Flood Control Channels 16.27 Values of Factor Cl (assumes Ar 39°)
TABLE 16.7
Side Slope, Z
1/2
τo/τc 2/3
1
0.5 1.0 1.5 2.0 3.0 4.0 5.0 horizontal
1.640 1.568 1.487 1.416 1.299 1.220 1.167 1.000
1.554 1.474 1.389 1.318 1.212 1.148 1.108 1.000
1.418 1.336 1.256 1.197 1.118 1.077 1.054 1.000
TABLE 16.8
Values of Factor Cb (assumes Ar = 39°)
Side Slope, Z
1/2
τo/τc 2/3
5 1.0 1.5 2.0 horizontal
0.976 0.961 0.939 592 0.000
0.958 0.934 0.898 0.440 0.000
1 0.912 0.868 0.806 0.000
is typically controlled by the erodibility of the underlying soil. Often the shear strength of a manufactured product will exceed the erodibility of the underlying soil. For noncohesive soils, several factors reduce the critical shear stress. The permissible shear stress is the maximum shear stress that a lining can safely withstand given local conditions. The permissible shear stress, τp, is given by: τp Ca τc
(16.6)
Factors that are known to affect the permissible shear stress include the density of the soil particles, the angle of repose, the channel side slope, the flow angle, and the required factor of safety. On channel side slopes, the soil particle has a reduced weight component with which to anchor the particle and resist detachment. The angle of repose and the required safety factor also modify the allowable shear stress. The critical shear adjustment factor is defined as follows (see appendix for complete derivation): Ca 1Cl Cz Cw (1 Cr Cb SF) F SF
(16.14)
where Cz cos(Az ( ) Cw (ρs 1) / 1.65
Az tan 1(1/z /)
Cr tan(Az ( ) / tan(Ar) (
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HYDRAULIC DESIGN OF FLOOD CONTROL CHANNELS
16.28
Chapter Sixteen
Cl 2/[1 sin(Al ( Ab)]
Ab tan 1{cos[Al [ ] / [E + sin(Al ( )]}
Cb cos(Ab ( ) E 2 (τc / τb) Cz Cr where Az angle of the channel side slope, Ar angle of repose of the soil, Ab angle of particle movement, Al oblique flow angle, SF design safety factor, ρs soil particle density, τc critical shear, and τb local boundary shear. The factor Cz accounts for the effect of the channel side slope on particle stability. As is summarized below, the factor Cz decreases, as the side slope becomes steeper. The value Z is defined as the ratio of the side slope’s horizontal to vertical distance. Z
0.5
1.0
1.5
2.0
3.0
4.0
5.0
horizontal
Cz
0.45
0.71
0.83
0.89
0.95
0.97
0.98
1.00
The factor Cw is an adjustment for the effect of stone density on particle stability. The density of a sediment particle is a function of the mineral composition. The specific gravity of sediments ranges from 2.3 for coal to 7.5 for galena. Alluvial sediments consist mainly of common minerals of quartz and feldspar with a specific gravity between 2.6 and 2.8. The default stone density for computation of critical shear stress is 2.65. The variation of factor Cw with specific gravity, ρs. ρs
2.3
2.5
2.65
2.8
3.0
4.0
Cw
0.79
0.91
1.00
1.09
1.21
1.82
The factor Cr accounts for the effect of the size and shape of soil particles on a slope. The flatter the channel side slope, or the larger and more angular the particle size, the more stable the lining. The size and angularity of the stone effect the slope at which particles will stabilize (the angle of repose). Angle of repose for noncohesive sediments and stones ranges from a low of 30° to a maximum of about 42.5° for large crushed rock (see Fig. 16.14). Table 16.6 provides values of Cr for a range of side slopes and angle of repose conditions. The factors Cl and Cb account for the hydraulic forces resulting from flow past soil layer particles. Because the direction of flow at a particle need not be parallel to the stream bank, the angle at which forces resolve must be calculated before Cl and Cb can be determined. The angle Al is the oblique flow angle. Oblique flow typically occurs in contraction and expansion reaches, such as at bridge or culvert openings. The angle Ab is the direction at which the combined forces of lift and drag on the particle and the down-slope weight of the particle push a particle. The angle depends on the angle of the side slope, the particle’s size and shape, and the ratio of the local boundary shear stress to the critical shear stress. For a low boundary shear stress, the direction of particle movement tends to follow the bank slope, whereas at higher shear, the direction of the particle movement tends to be more in the direction, of flow. Tables 16.7 and 16.8 summarize the values of Cl and Cb coefficients for a range of shear and bank slope conditions without oblique flow ((Al = 0°). In general, values of Cl and Cb decrease with decreasing side slope and increasing shear stress. Cl approaches a value of 1.0, and Cb approaches zero. For cohesive soils, particle stability is affected by the same factors that effect non cohesive soils, but chemical forces tend to hold the soil together as a larger mass. For wellcompacted homogeneous soils forming small channels, the critical shear adjustment factor is equal to the inverse of the safety factor (1/SF). F For larger channels, the various components of Eq. (16.11) should be considered in detail.
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HYDRAULIC DESIGN OF FLOOD CONTROL CHANNELS
Hydraulic Design of Flood Control Channels 16.29
kbs
FIGURE 16.16 Boundary shear stress adjustment factor for channel bed
16.4.4 Boundary Shear Stress The mean boundary shear stress is the tractive force acting parallel to the energy gradient for the flow per unit of wetted area: τo γ R Sf
(16.2)
The average boundary shear stress also can be calculated from the Manning’s Eq. (16.4): 2 τo n1/6 V2 γV R
(16.15)
The local boundary shear stress varies within a river reach as a consequence of the nonuniform distribution of velocity in the cross section or because of other channel features. For design, it is important to assess the maximum shear stress that can occur at specific locations in the reach. The local boundary shear stress is determined by multiplying the mean boundary shear stress by an associated adjustment factor that accounts for local hydraulic conditions. The maximum boundary shear stress is then calculated using Eq. (16.7). For a simple channel section on a mild slope, there are four basic types of local boundary shear-stress adjustments, which are summarized as follows:
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HYDRAULIC DESIGN OF FLOOD CONTROL CHANNELS
16.30
Chapter Sixteen
TABLE 16.9
Properties of Grass Channel Linings (good, uniform stands of each cover)
Cover Factor (C CF)
Grass Cover
Growth Form*
Stem Density M (stems/m2)†
0.90
Bermudagrass
S
5380
Centipedegrass
S
5380
0.87
Buffalograss
S
4300
Kentucky bluegrass
S
3770
Blue grama
S
3770
0.75
Grass mixture
B-S
2150
0.50
Weeping lovegrass
B
3770
Yellow bluestem
F
2690
Alfalfa
F
5380
Lespedeza sericea
B
3230
Common lespedeza
B
1620
Sudangrass
B
540
Multiply the stem densities given by 1/3, 2/3, 1, 4/3, and 5/3 for poor, fair, good, very good, and excellent covers, respectively. Legumes, large-stemmed. *B = Bunchgrass; S = sod-former; F = forb. †
Ka
Description
Kbs
Channel bed in a straight, symmetric reach
Kss
Channel bank in a straight, symmetric reach
Kf
Flexible lining
Kr
Channel bed or bank in a bendway, symmetric reach
Channel-bed adjustment factor in a straight, symmetric reach. In a straight reach of channel, the shear stress on the channel bed varies across the channel section. The maximum boundary shear stress is a function of the bank slope and the cross-sectional aspect ratio B/R (the ratio of the channel bottom width to the hydraulic radius). Figure 16.17 provides the boundary shear-stress adjustment factor for the channel bed for a trapezoidal section. ((Note: For aspect ratios greater than 5, the average flow depth can approximate the hydraulic radius.) Channel-bank adjustment factor in a straight, symmetric reach. The maximum boundary shear stress on the bank of a channel also is a function of the bank slope and the cross-sectional aspect ratio B/R. Figure 16.16 provides the boundary shear-stress adjustment factor for the channel bank for a trapezoidal section. Flexible lining adjustment factor. Flexible linings tend to deflect shear away from the channel surface. This effect creates a condition where part of the shear is absorbed by the movement of the lining material, resulting in a lower shear stress on the soil surface. This is the primary mechanism by which-flexible lining reduces erosion in a channel. The lining should be anchored well to the soil to provide the shear reduction either by mechanical means in the case of synthetic materials or in the case of well-established roots in the case of vegetation. Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
HYDRAULIC DESIGN OF FLOOD CONTROL CHANNELS
Hydraulic Design of Flood Control Channels 16.31
Kss
FIGURE 16.17 Boundary shear stress adjustment factor for channel side slope (after Anderson, et al., 1970)
Temple (1980) developed a method to estimate the shear stress acting on the soil for a vegetative channel lining. The adjustment factor is n Kf (1 C CF) s n1
(16.16)
Temple (1980) estimated CF values for a range of vegetative densities and grass covers. CF values for untested covers can be estimated by recognizing that density and uniformity of cover near the soil surface dominate the cover factor. Thus, the sod-forming grasses (top of Table 16.9) exhibit higher CF values than do the bunch grasses and annuals (near the bottom of Table 16.9). For large-stemmed plants, such as forbs and shrubs, the effective stem density should be increased. In the case of the legumes, the effective stem density is about five times the actual stem density. For turf-reinforcing mats, increasing the stem density by 4/3 is recommened.
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HYDRAULIC DESIGN OF FLOOD CONTROL CHANNELS
16.32
Chapter Sixteen
Bend adjustment factor. A channel bend causes an increase in local boundary shear on the bank of a channel from the beginning of the bendway, through the bend for a significant distance downstream of the bend. The boundary shear-stress adjustment factor for the channel bank in the bend is a function of the radius of the curvature of the bend and the width of flow: Kr and
3.16 Rc Rc/T 10.0(16.17) T
冪莦莦
Kr 1.0
Rc/T 10.0.
The extent of the downstream effect of a bend depends on the channel roughness in the bend, and on the average depth of flow, as described by the hydraulic radius. R Lp 0.604 R nb 1/6
(6.18)
Example 2: Channel lining evaluation. This example illustrates the hydraulic analysis of a grass-lined channel. A rating is performed in order to illustrate the effect of increasing depths of flow on channel performance. Inputs:
Channel attributes So = 0.020 m/m
Grade
Straight Compact, Cohesive Soil PI = 20 B = 1.5 m z = 3:1 Lining attributes Green grass h = 0.250 m Cover factor = 0.75
Alignment Soil type Bottom width Side slope Condition Grass height
Outputs: Range of rating d = 0.1 to 0.9 m a. Estimate the stem stiffness Eq. (16.9): MEI = 3.29 Nm2 b. Compute geometry rating for the channel d
Area m2
Perimeter m
Hyd. Rad. R, m
Mean Depth da, m
0.1
0.18
2.13
0.08
0.09
0.2
0.42
2.76
0.15
0.16
0.3
0.72
3.40
0.21
0.22
0.4
1.08
4.03
0.27
0.28
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HYDRAULIC DESIGN OF FLOOD CONTROL CHANNELS
Hydraulic Design of Flood Control Channels 16.33
d
Area m2
Perimeter m
Hyd. Rad. R, m
0.5 0.6 0.7 0.8 0.9
1.50 1.98 2.52 3.12 3.78
4.66 5.29 5.93 6.56 7.19
0.32 0.37 0.43 0.48 0.53
Mean Depth da, m 0.33 0.39 0.44 0.50 0.55
c. Compute relative roughness height using Eq. (16.8): R
τ0
ks/h
ks
0.08 0.15 0.21 0.27 0.32 0.37 0.43 0.48 0.53
16.5 29.8 41.5 52.5 63.1 73.3 83.3 93.2 103
0.67 0.53 0.46 0.42 0.39 0.37 0.35 0.34 0.32
0.17 0.13 0.12 0.11 0.10 0.09 0.09 0.08 0.08
d. Compute vegetative critical shear stress using Eq. (16.12)
τcv = 68.2 N/m2 e. Compute semi-log coefficients using Table 16.3: d
τ0/τcv
a
b
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.243 0.436 0.609 0.770 0.924 1.074 1.222 1.367 1.510
0.42 0.42 0.42 0.42 0.42 0.44 0.48 0.52 0.56
5.23 5.23 5.23 5.23 5.23 5.46 5.92 6.37 6.82
f. Compute Manning’s n using Eq. (16.11) d
R/ks
0.10 0.20 0.30 0.40 0.50 0.60 0.70
1.00 1.15 1.83 2.54 3.28 4.05 4.84
n 0.504 0.317 0.138 0.101 0.085 0.072 0.061
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HYDRAULIC DESIGN OF FLOOD CONTROL CHANNELS
16.34
Chapter Sixteen d
R/ks
0.80 0.90
5.67 6.51
n 0.053 0.047
g. Compute channel velocity (Eq. 16.4) and discharge: d
R2/3
V, m/s V
Q, m3/s
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.215 0.342 0.448 0.543 0.630 0.711 0.788 0.862 0.932
0.060 0.15 0.46 0.76 1.05 1.39 1.83 2.30 2.80
0.01 0.06 0.33 0.82 1.58 2.76 4.60 7.16 10.60
h. Maximum boundary shear (Figs. 16.16 and 16.17): d 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
B/da 17.5 9.6 6.9 5.4 4.5 3.9 3.4 3.0 2.7
kbs 1.16 1.26 1.34 1.40 1.45 1.47 1.49 1.49 1.49
τbed N/m2 19.2 37.6 55.5 73.5 91.4 108 124 139 153
kss 0.99 1.07 1.16 1.21 1.25 1.27 1.30 1.33 1.35
τside N/m2 16.3 32.0 48.2 63.6 78.8 93.1 108 124 139
i. Shear stress below lining (Eq. 16.16):
Depth m 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
ns 0.023 0.020 0.020 0.020 0.020 0.020 0.020 0.020 0.020
Kf 0.011 0.016 0.036 0.049 0.059 0.069 0.082 0.094 0.106
τbed N/m2 0.22 0.59 2.0 3.6 5.4 7.5 10.2 13.1 16.3
τside N/m2 0.19 0.50 1.8 3.1 4.6 6.4 8.9 11.7 14.8
exceeds permissibleshear exceeds permissibleshear
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HYDRAULIC DESIGN OF FLOOD CONTROL CHANNELS
Hydraulic Design of Flood Control Channels 16.35
j. Permissible shear stress (Fig. 16.15): 11.0 N/m2 Note: The permissible shear is first exceeded on the channel bed at a depth of approximately 0.73 meters.
16.5 STEEP-GRADIENT CHANNEL DESIGN The maximum permissible shear stress is less for channels on steep slopes. There are a couple of reasons for this effect on the channel lining. One reason is that as velocity increases and flow depth decreases the exchange of momentum between portions of the channel becomes more efficient. In low-gradient channels, the maximum shear stress is concentrated over a small portion of the channel boundary. For a steep slope, the channel boundary away from the zone of maximum shear stress receives increased shear stress that approach the maximum. A second reason is the creation of localized shear zones near irregularities in the lining. Steep slopes typically require linings with larger particle sizes, that make construction of a uniform lining difficult. Use of large stones to construct a relatively small channel section should be avoided. Design flow depths should exceed the largest stone size by several times. Protrusions of large stones from the lining are generally a problem. If large stones are desired, then the stability of these stones should be assessed individually.
16.5.1 Resistance to Flow in Steep-Gradient Channels Most of the flow resistance in channels with large-scale roughness is derived from the form drag of the roughness elements and the distortion of the flow as it passes around roughness elements. Consequently, a flow-resistance equation for these conditions must account for skin friction and form drag. Because of the shallow depths of flow and the large size of the roughness elements, the flow resistance will vary with relative roughness area and roughness geometry, the Fr (the ratio of inertial forces to gravitational forces), and the Reynolds number (Re) (the ratio of inertial forces to viscous forces). Bathhurst et al., (1979) quantified these relationships experimentally. This work showed that for Re’s in the range of 4 x 104 to 2 x 105, the effect on resistance remains constant. When roughness elements protrude through the free surface, resistance increases significantly because of the effects of the Fr (standing waves, hydraulic jumps, and free-surface drag). For the channel as a whole, free-surface drag decreases as the Fr and the relative submergence increases. Once the elements are submerged, ther effect of Fr related to free-surface drag is small, but those related to standing waves are important. The Manning’s form of the Bathurst equation is 1/6
R n 兹 f( 兹苶 兹g f Fr) f( f REG) f( f CG)
(16.19)
The function f (Fr) accounts for the free surface drag of the elements: 0.28 c1 f (Fr) Fr , b
1.0
0.755 where c1 log b
for b 0.755
(16.20)
b 0.755.
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HYDRAULIC DESIGN OF FLOOD CONTROL CHANNELS
16.36
Chapter Sixteen
The parameter b describes the relationship between the effective roughness concentration and the relative submergence of the roughness bed: D50 0.453 R 0.814 b 1.14 T D50
The function f( f REG) accounts for the roughness geometry: T 0.492 f REG) c2 f( D50
(16.21)
0.118
where c2 13.434 b 1.025(T/D50)
The function ff(CG) accounts for the relative roughness area:
b
F(CG) T R
(16.22)
16.5.2 Permissible Shear Stress in Steep Gradient-Channels Flow in steep channels is generally a shallow high-velocity flow. The main force acting on a lining composed of large particles, such as stone, is the drag force. The lift force will be absent until there is a sufficient depth of flow over the particle. The effect is to increase the shear parameter to approximately 0.094, or roughly twice the value for fully submerged flow. The critical shear stress for shallow flow is determined by the following equation: τc 1.5 D50
for R/D / 50 2.0
(16.23)
16.5.3 Boundary Shear Stress in Steep Gradient Channels Correction for slope. For a steep channel (10%), the approximation of the energy gradient by the tangent of the grade is not accurate. The mean boundary shear-stress adjustment for steep slopes is: Kθ cos(θ)
(16.24)
Correction for high Froude number. For Fr’s than 0.6, the distribution of shear stress in the channel is affected by increased flow turbulence (Davidian and Cahal, 1963). An increase in the Fr will cause • the average wall shear to approach the maximum wall shear, thereby decreasing the ratio of the maximum wall shear to the average wall shear toward unity. • the average total cross section shear to approach the average bottom shear, thereby decreasing the ratio of the average floor shear to the average total cross-sectional shear toward unity. • the average wall shear to approach the average bottom shear, thereby increasing the ratio of the average wall shear to the average bottom shear toward unity. The result is a more uniform distribution of shear on the boundary. The channel banks are adjusted for the high Fr by using a secondary factor that adjusts the bed shear
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HYDRAULIC DESIGN OF FLOOD CONTROL CHANNELS
Hydraulic Design of Flood Control Channels 16.37
stress. The resulting channel-bank shear-adjustment factor is then compared to the bank adjustment factor that is determined without consideration of the Froude effect. The higher of the two factors is then used. The boundary stress adjustment factor for the channel bank is Ksf maximum of (Fsf * Kbs, Kss) where Fsf 0.23 log (Fr) 0.91 1.0
(16.25) for Fr 2.5 for Fr 2.5.
Example 3: Steep channel lining evaluation. This example illustrates hydraulic analysis of a steep riprap channel. A rating is performed in order to illustrate the effect of increasing depths of flow on channel performance. Inputs:
Channel attributes So = 0.200 m/m Sine = 0.196 Straight Alignment B = 1.2 m z = 3:1
Bottom width Side slope
Lining attributes Angular riprap
Type
D50 = 0.450 m
Size
Grade
Outputs: Range of rating d = 0.1 to 0.9 m a. Compute the geometry rating for the channel: d
Area m2
Perimeter m
Top width m
Hyd.Rad. R, m
0.1 0.2 0.3 0.4 0.5 0.6
0.15 0.36 0.63 0.96 1.35 1.80
1.83 2.46 3.10 3.73 4.36 4.99
1.80 2.40 3.00 3.60 4.20 4.80
0.082 0.15 .20 0.26 0.31 0.36
b. Bathhurst resistance equation coefficients: d 0.1 0.2 0.3 0.4 0.5 06
b 0.152 0.214 0.253 0.282 0.306 0.326
c1 0.696 0.548 0.475 0.428 0.393 0.365
f f(Fr) 1.653vˆc1 1.051vˆc1 0.891vˆc1 0.818vˆc1 0.777vˆc1 0.752vˆc1
c2 1.382 1.955 2.305 2.559 2.763 2.936
f f(REG)
f f(CG)
2.733 4.455 5.862 7.119 8.290 9.409
0.625 0.550 0.506 0.475 0.451 0.430
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HYDRAULIC DESIGN OF FLOOD CONTROL CHANNELS
16.38
Chapter Sixteen
c. Manning n function -- 1/n = a Vb and hydraulics: d
a
b
0.1 0.2 0.3 0.4 0.5 0.6
13.4 11.1 10.8 10.9 11.1 11.3
0.696 0.548 0.475 0.428 0.393 0.365
n 0.075 0.090 0.093 0.092 0.090 0.088
V m/s
Q m3/s
τ0 N/m2
Fr
1.5 2.0 2.6 3.2 3.8 4.3
0.22 0.72 1.64 3.07 5.09 7.8
157 281 391 495 595 693
1.62 1.66 1.85 2.02 2.17 2.30
d. Maximum boundary shear using Figs. 16.16 and 16.17 and Eq. (16.25): τbed
τbank 2
d
B/da
Kbs
Kss
Fsf
N/m
M/m2
0.1 0.2 0.3 0.4 0.5 0.6
14.7 8.2 5.9 4.7 3.9 3.3
1.19 1.30 1.34 1.40 1.44 1.47
1.01 1.10 1.13 1.18 1.22 1.25
0.958 0.961 0.971 0.980 0.987 0.993
187 364 524 693 856 1018
179 350 509 679 846 1011
OK OK
e. Permissible shear stress (Eqs. 16.13, 16.14, and 16.21); D50 450mm
τc 675 N/m2
SF
Cz
Cw
Cr
1.3
0.95
1.00
0.36
τpbed 519 N/m2
τp_bank d
E
Ab
Cl
Cb
Ca
N/m2
0.1 0.2 0.3 0.4 0.5 0.6
2.58 1.32 0.91 0.68 0.55 0.46
0.370 0.648 0.834 0.973 1.071 1.143
1.47 1.25 1.15 1.09 1.07 1.05
0.93 0.80 0.67 0.56 0.48 0.42
0.61 0.57 0.58 0.59 0.60 0.62
408 OK 386 OK 389 398 408 416
Note: The riprap lining is adequate to a depth of slightly over 0.2 m. The channel bank is weaker than the bed and is the limiting condition.
16.6 COMPOSITE-SECTION CHANNEL DESIGN Because the highest shear stresses occur on the bed of the channel or on the channel bank near the bed, it is often desirable to use a combination of linings in a channel section. A composite lining consists of two or more lining materials with a higher strength lining material used selectively in high shear areas. Other requirements may dictate the design of a composite lining. Many times, low-flow channels are included within the main channel or channel banks are vegetated or allowed to remain in a natural condition as part of a bioengineering approach.
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HYDRAULIC DESIGN OF FLOOD CONTROL CHANNELS
Hydraulic Design of Flood Control Channels 16.39
16.6.1 Resistance to Flow Computation of flow conditions in a composite channel requires the use of an equivalent Manning’s n-value for the entire perimeter of the channel. For determination of equivalent roughness, the channel perimeter is divided into similar parts. For channels consisting of a bed of one lining type and banks of a second lining type, the Manning’s n-value for the entire section is n Kc ns
(16.26)
P 0.667 Ps nr 1.5 s Kc 1 P ns P
(16.27)
16.6.2 Boundary Shear The maximum boundary shear acting on the bed and the bank of the channel is determined using Eq. 16.7, and the permissible shear stress for each lining is determined using Eq. 16.6.
16.6.3 Special Considerations When two lining materials have significantly different roughnesses and are adjacent to each other in the flow, erosion may occur near the boundary of the two linings and erosion of the weaker lining material may damage the lining as a whole. In the case of a composite channel lining with bank vegetation, this problem can occur in the early stages of establishing the vegetation. Thus a temporary lining should be used adjacent to the channel bed to provide temporary erosion protection until the lining vegetation is well established. Low-flow channels are needed when a flow of long duration is expected, such as the outlet from a detention pond or from areas with sustained groundwater inflow to the channel. Establishing vegetative linings under these conditions can be difficult, particularly if the soil is highly erodible without the lining material. In erodible soils, gullies form in the channel invert that weakens the lining during higher flows. One solution is to provide a nonvegetative channel lining, such as concrete or riprap, for the channel invert. The dimensions of the low-flow channel are determined in the basis of the low flow only, with the remainder of the channel covered in vegetation.
16.7 CHANNELS WITH SEDIMENT TRANSPORT All flood-control channels will transport some sediment and will require routine maintenance as necessary to prevent excessive accumulation. However, when the supply of sediment to a flood-control channel increases, the channel must be designed with a sufficient capacity to transport the load of both the water and the sediment. The purpose of this section is to present general concepts for approaching sediment transport analysis for flood-control channels.
16.7.1 Sediment Supply For a flood-control channel to operate effectively, it must be able to transport the supply of sediment that it receives from the upper watershed. Ideally, the transport capacity of the Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
HYDRAULIC DESIGN OF FLOOD CONTROL CHANNELS
16.40
Chapter Sixteen
channel will equal the supply over the duration of a flood event. Watershed sediments can be derived from upstream erosion of the stream network (erosion of stream banks and bed) or from erosion of catchment areas (typically a combination of soil detachment by the impact of rain drops and small-scale rilling of the soil surface). For undisturbed watersheds, the supply of sediment can be estimated based on the sediment transport capacity of the existing stream network, assuming that the entire system is near equilibrium. The existing drainage network often is a good guide to the type of channel sections and grades required to achieve adequate sediment transport in a flood-control channel. The development of land can greatly alter the supply of watershed sediment. Construction can result in a large temporary increase in the potential supply of sediment. Long-term change in land use, such as agricultural development, urbanization, and deforestation, can greatly increase or decrease the yield of watershed sediment. This can result in a complex adjustment within the stream network that may require many years for equilibrium to be reestablished. Estimation of sediment yield for disturbed watersheds should consider potential sources of sediment from both catchment erosion and erosion within the stream network. Both are typically altered in the course of development. The amount of erosion can be affected by installation of erosion control systems. Methods of analyzing the supply of catchment-level sediment have been developed and tested primarily for agricultural lands. These methods have been adapted successfully to a variety of other land alterations. The widely used Universal Soil Loss Equation, USLE (Wischmeier and Smith, 1978) has recently been updated as the Revised Universal Soil Loss Equation, RUSLE (Renard etal., 1992). Methods of watershed stream-network sediment supply depend on an evaluation of stream transport capacity relative to catchment sediment supply. A qualitative analysis can be conducted for channels in non-cohesive sediments using Lane’s relationship (Lane, 1955). Q S Gs ds
(16.28)
The Lane relationship is used by assuming that two of the variables are either constant or has a known behavior. For example, an increase in bed load Gs, with no change in sediment size ds or stream discharge Q, will result in an increase in stream slope S. This is interpreted as a potential for sediment deposition in the existing stream channel. In another case, an increase in stream discharge Q, with no change in sediment supply Gs, or ds, will result in a decrease in stream slope S. This is interpreted as a potential for scour within the existing stream channel.
16.7.2
Sediment Transport
Sediment is transported in a stream channel as a combination of bed load (sediment that is in frequent contact with the bed of the channel) and suspended bed load (sediment from the bed that is mixed with the water flow by turbulence). The combined transport is referred to as the total sediment load. Incidental to the bed load is wash load, which consists of fine particles that stay in suspension and are derived from the erosion of watershed soil. Sediment transport begins when the critical shear stress of the bed sediment is exceeded. As the mean shear stress and flow velocity increase, the total rate of sediment-transport increases. Sediment is continuously exchanged vertically in the water column and, to a
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HYDRAULIC DESIGN OF FLOOD CONTROL CHANNELS
Hydraulic Design of Flood Control Channels 16.41
lesser extent, laterally. Areas of the channel where critical shear stress has been exceeded are referred to as active channel areas. The sediment discharge for the channel section is Gs gs W W’b
(16.29)
Numerous formulas for estimating sediment transport exist (Yang, 1996). These formulas need to be evaluated carefully before use; because many are designed for specific sizes of sediment or a specific range of hydraulic conditions. Total transport often is computed by computing portions of the transport process in separate steps (bed load first, then suspended load). Many formulas require careful calibration to obtain a valid result. Similarly, it is possible to find formulas developed for use in similar conditions that differ significantly in estimates sediment transport. Therefore, once a method of analysis is selected, it is important to maintain consistency within that analysis. It is extremely difficult to generalize sediment-transport formulas. However, it is important to have an understanding of the relation between the dominant hydraulic variables (velocity and flow depth) and the total transport of bed material. From Eq. (16.28), we can state the dominant variables for sediment transport are stated qualitatively as qS Vτ gs ∝ f f o ds ds
(16.30)
The product of velocity and shear stress is the stream power per unit of bed area. The functional form of Eq. (16.30) can be expressed as a basic power function: gs a1(Vτ V o Vcτc)b1
(16.31)
For most channels, the exponent b1 typically ranges from 1.5 to 2.0. Since boundary shear stress is proportional to the square of the velocity, sediment transport is roughly proportional to velocity to the 4.5 to 6.0 power. In other words, small increases or decreases in velocity can greatly increase or decrease capacity for the sediment transport. It is often useful to develop sediment-rating functions for selected channel reaches using a sediment transport formula, such as Eq. (16.31) or actual measurements at gaging stations. b
Gs a2Q
2
(16.32)
Comparison of successive rating relationships can be used to give a preliminary estimate of aggradation and degradation trends in the channel. The drawback to this type of analysis is that, in reality, the rating will shift with changes in the channel bed and adjustment of hydraulic conditions. However, this approach, although essentially qualitative, can be used as a refinement of the Lane relationship Eq. (16.29).
16.6.3 Aggradation-Degradation Sediment is transported from the watershed through stream channels to a place of deposition. If we observe sediment movement at a cross section in a stream we can note two conditions (Einstein, 1964): (1) every sediment particle must have been eroded somewhere in the watershed above the cross section and (2) the sediment particle must be transported by the flow to the cross section. If the supply of sediment from the watershed exceeds the sediment transport at a cross-section, then sediment will accumulate: i.e., aggrade. If the transport at the cross section exceeds the supply of sediment, then sediment will be removed from the channel: i.e., degrade.
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HYDRAULIC DESIGN OF FLOOD CONTROL CHANNELS
16.42
Chapter Sixteen
Degradation will generally occur within the active area of the channel bed, whereas aggradation will be distributed fairly uniformly over the entire channel bed. The process of aggradation-degradation involves a mass balance of sediment within a channel reach: (W’b ∆L ∆z)s Gsup Gsxs ∆ 0 t
(16.33)
Accurate solution of Eq. (16.32) involves a repetitive solution of stream hydraulics and sediment transport for a sequence of many time intervals. This is accomplished through computer programs (Fan, 1988).
16.7.4 Resistance to Flow Among the most important aspects of flood-control channels that transport sediment is the effect of that transport on flow resistance. The transport of sediment creates features in the channel bed referred to as bed forms. As shear increases on the channel, bed forms increase in size and have an increasing effect on flow resistance in the channel. The general progression of bed forms is from lower regime (ripples to dunes) through a transition (washed out dunes to plane bed) to upper regime (antidunes). Allen (1978) determined the height of lower regime and transition bed forms relative to flow depth experimentally as follows: 2
3
4
θ θ θ θ H 0.080 2.24 18.13 70.90 88.3 d 3 3 3 3
H 0 d
θ 1.5
θ 1.5
(16.34)
τ θ o D50 (16.35) γ’s (Note: θ 1.1 lower regime, 1.1 θ 1.5 transition, and θ 1.5 upper regime.) For sand, Karim (1990) developed a simple equation for estimating the Manning’s roughness coefficient based on Allen’s bed-form height Eq. (16.34): f 0.465 n 0.037 D50 0.126 fo
(D50 in m)
(16.36)
where f H 1.20 8.92 fo d For upper regime flow, a typical range of Manning’s resistance coefficients is 0.015 to 0.030. Flow resistance increases with increasing Fr as antidunes become unstable and form breaking waves and eventually form chutes and pools.
REFERENCES Allen, J. R. L., “Computational methods for dune. Calculations using Stein’s rule for dune height, “ Sedimentary Geol., 20(3), 165–216, 1978) Anderson, A. G., A. S. Paintal, and J. T. Davenport, “Tentative Design Procedure for Riprap–Lined Channel,” Highway Research Board, National Cooperative Highway Research Program, Report 108, Washington, DC., 1970. Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
HYDRAULIC DESIGN OF FLOOD CONTROL CHANNELS
Hydraulic Design of Flood Control Channels 16.43 Apperly, L. W., Effect of Turbulence on Sediment Entrainment, doctoral dissertation, University of Auckland, New Zealand, 1968. Austin, D. N., and L. E. Ward, ECTC Provides Guidelines for Rolled Erosion-Control Products, Geotechnical Fabrics Report, Industrial Fabrics Association International, Jan/Feb 1996. Bathurst, J. C., R. M. Li, and D. B. Simons, Hydraulics of Mountain Rivers, Colorado State University Experiment Station, CER78–79JCB–RML–DBS55, May 1979. Benedict, B. A., and B. A. Christensen, “Hydrodynamic Lift on a Stream Bed,” Chap. 5, Sedimentation, H W. Shen, ed. and Pub., 1972. Benedict, B. A.,“Hydrodynamic Lift in Sediment Transport,” doctoral Dissertation, University of Florida, Gainesville, Florida, 1968. Blench, T., Mobile-Bed Fluviology: A Regime Treatment of Canals and Rivers, University of Alberta Press, Edmonton, Canada, 168, 1969. Blodgett, J. C., and C. E. McConaughy, Rock Riprap Design for Protection of Stream Channels near Highway Structures, Vols. 1 and 2, Water-Resources Investigations Report No. 86–4128, U.S. Geological Survey, 1986. Chen, Y. H., and G. K. Cotton, “Design of Roadside Channels with Flexible Linings,” Hydraulic Engineering Circular No. 15 (HEC-15), FHWA, Publication No. FHWA–IP–87–7, U.S. Department of Transportation, McLean, VA., 1988. Davidian, J., and D. I. Cahal, “Distribution of Shear in Rectangular Channels,” Article 113, U.S. Geological Survey Professional Paper No. 475–C, pp. C206–C208, Washington, DC, 1963. Chepil, W. S. “ The use of Evenly Spaced Hemispheres to Evaluate Aerodynamic Forces on a Soil Surface” Transactions American Geophysical Union, 39(3), 397–404, 1958. Einstein, H. A., “Sedimentation, Part II. River Sedimentation,” Handbook of Applied Hydrology, V. T. Chow, ed., Sec 17, McGraw-Hill, New York, 1964. El–Samni, E. A., “Hydrodynamic Forces Acting on the Surface Particles of a Stream bed,” doctoral dissertation, University of California, Berkley, California, 1949. Erosion Control Technology Council (ECTC), Technical Guidance Manual: Terminology and Index Testing Procedures for Rolled Erosion Control Products, ECTC Testing and Standards Committee, January 1997. Fan, S. S., ed., “Twelve Selected Computer Stream Sedimentation Models Developed in the United States,” Interagency Sedimentation Work Group, Washington, DC, 1988. Glover, R. E., and Q. L. Florey, Stable Channel Profiles, Vol. 235, U.S. Bureau of Reclamation, Denver, CO., 1951. Karim, M. F., “Menu of Coupled Velocity and Sediment Discharge Relations for Rivers,” Journal of Hydraulic Engineering, American Society of Civil Engineers. 116: 987–996, 1990. Kennedy, R. G., “The Prevention of Silting in Irrigation Canals,” Proceedings of the Institute of Civil Engineers, 1885. Kouwen, N., “Field Estimation of the Biomechanical Properties of Grass,” Journal of Hydraulic Research, 26: 559–568, 1988. Kouwen, N., and R. M. Li, “Biomechanics of Vegetated Channel Linings,” Journal of the Hydraulics Division, American Society of Civil Enginneers, 106, 1085–1103, 1980. Kouwen, N., and T. E. Unny, “Flexible Roughness in Open Channels,” Journal of the Hydraulics Division, American Society of Civil Enginneers, 99: 713-728, 1969. Lacey, G., “Stable Channels in Alluvium,” Proceedings of the Institute of Civil Engineers, 229:259–292, 1920. Lane, E. W., “Design of Stable Channels,”Transactions”, ASCE 120: 1234–1279, 1995 Lane, E. W., “Design of Stable Channels,”Transactions of the American Society Of Civil Engineers, 120:1234–1260, 1955. Lane, E.W., “The Importance of Fluvial Morphology in Hydraulic Engineering,” Proceedings of the American Society of Civil Engineers, 81:1–17, 1920. Lane, E. W., and E. J. Carlson, “Some factors affecting the stability of canals constructed in coarse granular material,” Proceedings of the Minnesota International Hydraulics Convention, September 1953.
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HYDRAULIC DESIGN OF FLOOD CONTROL CHANNELS
16.44
Chapter Sixteen
Lindley, E. S., “Regime channels,” Proceeding of the Punjab Engineers Congress, Vol. 7, Punjab, India, 1919. Ree, W. O., and V. J. Palmer, Flow of Water in Channels Protected by Vegetative Linings, U.S. Soil Conservation Service Bulletin No. 967, 1949. Renard, K. G., G. R. Foster, G. A. Weesies, D. K. McCool, and D. C. Yoder, “Predicting Soil Erosion by Water: A Guide to Conservations Planning with the Revised Universal Soil Loss Equation (RUSLE),” USDA\ARS Agricultural Handbook, No. 537, 1992. Simons, D. B., and M. Albertson, “Uniform Water Conveyance Channels in Alluvial Material,” Transactions of the American Society of Civil Engineers, 128(1): 65–167, 1963. Simons, D. B., Y. H. Chen, and L. J. Swenson, “Hydraulic Test to Develop Design Criteria for the Use of Reno Mattress,” Maccafferri Steel Wire Products, March 1984. Simons, Li & Associates, Inc., “Designing Stable Channels with Armorflex Articulated Concrete Block Revetment Systems,” Nicolon Corporation, Feb 1990. Smerdon, E. T. and Beaseley, R. P., “The Tractive Force Theory Applied to stability of Open Channels in Cohesive soils,” Agricultural Experiment Station, Research Bulletin No. 715, University of Missouri, Columbia, Missouri, October 1959. Temple, D. M., “Tractive force design of vegetated channels,” Transactions of the American Society for Agricultural Engineers, 23: 884–890, 1980. Temple, D. M., K. M. Robinson, R. M. Ahring, and G.S. Davis, “Stability design of grass-lined channels,” Agricultural Handbook No. 667, Agricultural Research Service, U.S.Department of Agriculture, Washington, D.C., 1987. USDA, Handbook of Channel Design for Soil and Water Conservation, SCS–TP–61, U.S. Department of Agriculture, Washington, D.C, Soil Conservation Service, as revised 1954. Yang, C. T., Sediment Transport, Theory and Practice, McGraw-Hill, New York, 1996. Wischmeier, W. H. and D. D. Smith, “Predicting Rainfall Erosion Losses: A Guide to Conservation Planning,” USDA Agricultural Handbook 537, 1978.
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Source: HYDRAULIC DESIGN HANDBOOK
CHAPTER 17
HYDRAULIC DESIGN OF SPILLWAYS H. Wayne Coleman C. Y. Wei James E. Lindell Harza Engineering Company Chicago, Illinois
17.1 INTRODUCTION The spillway is among the most important structures of a dam project. It provides the project with the ability to release excess or flood water in a controlled or uncontrolled manner to ensure the safety of the project. It is of paramount importance for the spillway facilities to be designed with sufficient capacity to avoid overtopping of the dam, especially when an earthfill or rockfill type of dam is selected for the project. In cases where safety of the inhabitants downstream is a key consideration during development of the project, the spillway should be designed to accommodate the probable maximum flood. Many types of spillways can be considered with respect to cost, topographic conditions, dam height, foundation geology, and hydrology. The spillways discussed in this chapter include overflow, overfall, side-channel, orifice, morning-glory, labyrinth, siphon, tunnel, and chute spillways A section on design of spillways that considers cavitation and aeration also is included.
17.2 OVERFLOW SPILLWAY An overflow spillway can be gated or ungated, and it normally provides for flow over a gravity dam section. The flow remains in contact with the spillways surface (except for possible aeration ramps) from the crest of the dam to the vicinity of its base. The hydraulic characteristics are defined as follows: 1.
Determine design head Ho. Normally, Ho is 75 to 80% of maximum head Hmax.
2.
Use the depth from the crest to ground surface P to find the basic discharge coefficient Co from Fig. 17.1.
3.
Find discharge coefficient C for the full range of heads from Fig. 17.2.
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HYDRAULIC DESIGN OF SPILLWAYS
17.2
Chapter Seventeen
4.
Correct discharge coefficient C for the sloping upstream face from Fig. 17.3. The sloping upstream face is normally for structural stability, not hydraulic efficiency.
5.
Correct discharge coefficient C for the downstream apron from Fig. 17.4.
6.
Correct discharge coefficient C for tailwater submergence from Fig. 17.5
7.
Define the shape of the pier nose. Normally, use Type 3 or 3A from Fig. 17.6.
8.
Check the minimum crest pressure from Fig. 17.7 or 17.8. If the minimum pressure is below 1/2 atmosphere, increase Ho and start over.
9.
Define the crest shape from Fig. 17.9(a) and (b).
10.
Determine the effective crest length for the full range of heads from L L' 2(NK Kp Ka) He
(17.1)
where: L effective length of crest, L' net length of crest, N number of piers, K pier contraction coefficient, Ka abutment contraction coefficient, and He total head on crest. The following are values of the pier contraction coefficient
For square-nosed piers with corners rounded on a radius equal to approximately 0.1 of the pier thickness for round-nosed piers for pointed-nose piers
Kp 0.02
0.01 0.00
The following are values of the abutment contraction coefficient Ka for square abutments with headwall 0.20 at 90° to the direction of flown for rounded abutments with headwall 0.10 at 90° to the direction of flow Ho when 0.5H Ho r 0.15H for rounded abutments, where 0.00 r 0.5H Ho, and the headwall is placed no more than 45° to the direction of flow, where r radius of the abutment rounding. 11.
Determine the discharge rating curve from Q C L He3/2
(17.2)
Exhibits 17.1 and 17.2 illustrate overflow spillings for hydroelectric projects.
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HYDRAULIC DESIGN OF SPILLWAYS
Hydraulic Design of Spillways 17.3
FIGURE 17.1 Discharge coefficients for a vertical-faced ogee crest. (USBR, 1987).
FIGURE 17.2 Coefficient of discharge for other than the design head. (USBR, 1987).
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HYDRAULIC DESIGN OF SPILLWAYS
17.4
Chapter Seventeen
FIGURE 17.3 Coefficient of discharge for an ogee-shaped crest with a sloping upstream face. (USBR, 1987)
FIGURE 17.4 Ratio of discharge coefficients associated with the apron effect. (USBR, 1987).
FIGURE 17.5 Ratio of discharge coefficients associated with the tailwater effect. (USBR, 1987).
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HYDRAULIC DESIGN OF SPILLWAYS
FIGURE 17.6 Pier-contraction coefficients for high-gated overflow crests. (USACE, 1988)
Hydraulic Design of Spillways 17.5
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HYDRAULIC DESIGN OF SPILLWAYS
17.6
Chapter Seventeen
FIGURE 17.7 Crest pressures along piers (Type 3A) for high-overflow dams. (USACE, 1988)
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HYDRAULIC DESIGN OF SPILLWAYS
Hydraulic Design of Spillways 17.7
FIGURE 17.8 Crest pressures along the centerline of a pier (Type 3A) bay. (USACE, 1988)
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HYDRAULIC DESIGN OF SPILLWAYS
17.8
Chapter Seventeen
FIGURE 17.9 (a) Factors for the definition of nappe-shaped crest profiles (USBR, 1987)
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HYDRAULIC DESIGN OF SPILLWAYS
Hydraulic Design of Spillways 17.9
FIGURE 17.9(b) Factors (xc, yc, R1, and R2) for definition of nappe-shaped crest profiles.(USBR, 1987)
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HYDRAULIC DESIGN OF SPILLWAYS
17.10
Chapter Seventeen
(a)
(b)
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HYDRAULIC DESIGN OF SPILLWAYS
Hydraulic Design of Spillways 17.11
(c)
(d) Exhibit 17.1 Macagua hydroelectric project, Venezuela (Courtesy CVG–EDELCA, Caracas.) (a) General view of the spillway in operation. (b) Free flow condition at the ogee crest. (c) Flow condition at the ogee crest with gate partially open. (d) Layout of the spillway and stilling basin.
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HYDRAULIC DESIGN OF SPILLWAYS
17.12
Chapter Seventeen
(a)
(b) Exhibit 17.2 Tarbela hydroelectric project, Pakistan (Courtesy water and power development authority, Pakistan) (a) General view of the spillway in operation. (b) Layout of the spillway including the flip bucket.
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HYDRAULIC DESIGN OF SPILLWAYS
Hydraulic Design of Spillways 17.13
17.3 OVERFALL SPILLWAY An overfall spillway can be gated or ungated and provide for flow over an arch or archbuttress dam, wherein the flow free-falls some distance before entering a plunge-pool energy dissipator in the tailrace. The hydraulic characteristics are defined as follows: 1.
The crest structure and discharge rating are similar to those for the overflow spillway.
2.
The flow normally leaves this structure shortly below the crest. The exit structure is normally some variation of a flip-bucket.
3.
The flip-bucket radius for an overfall spillway is normally smaller than the ideal, which is at least 5d, where d is the flow depth at the bottom of the bucket. The radius is usually undersized to minimize the size of the overhang, which can destabilize the top of a thin-arch dam. However, the radius should be sufficient to fully— deflect a significant flood say, the 100-year event.
4.
The bucket exit angle is selected to throw the jet to a suitable location in the tailrace. The trajectory can be estimated by 2 y x tan x 3.6H cos2
(17.3)
where: y vertical distance from the bucket lip x horizontal distance from the buck et lip bucket exit angle, and H depth velocity head at the bucket lip. 5.
The trajectory can be estimated from Step 4 as long as the bucket radius exceeds 5d. For larger depths, the flow overrides the bucket and will fall short of the maximum trajectory. The trajectory in this range is determined best by a physical model.
6.
Pressure load on the bucket can be estimated from Fig. 17.10. For larger floods, where d R/5, this load is determined best by a physical model.
7.
The energy from an overfall spillway is normally dissipated by a plunge pool, which can be lined or unlined. If unlined, the scour and the scour rate will be based on both flow and geology. The scour hole development is usually indeterminate. However, the terminal scour depth for a uniformly erodible material can be esti-mated from the following empirical formula and from Fig. 17.11 (Coleman, 1982; USBR, 1987) ys ds sinα terminal vertical scour depth
(17.4)
ds Cs H0.225 q0.54
(17.5)
and
where Cs 1.32 (for units in ft and cfs/ft) 1.90 (for units in m and cms/m), H effective head at tailwater level, q unit discharge Q/B / , B width of the bucket, and α average jet entry angle. The extent of the scour hole is based on judgment of stable slope of material surrounding the deepest hole. A physical model is normally used where topography is complex and where scour can endanger project structures.
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HYDRAULIC DESIGN OF SPILLWAYS
17.14
Chapter Seventeen
8.
As the jet plunges into the pool, it diffuses almost linearly and entrains air at the surface of the pool and the water from the pool at the boundary of the jet. The behavior of the plunging jet, including dynamic pressures, can be approximated (Hinchliff and Houston, 1984) using Fig. 17.12 and Table 17.1 for both rectangular and circular jets.
9.
If the plunge pool is lined because the scour would be unpredictable and unacceptable, the lining must be designed for pressure pulsations from the jet’s impact. The design pressures can be determined best from physical model studies.
FIGURE 17.10 Pressures for flip-buckets and toe curves of an overflow spillway. (USACE, 1988).
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HYDRAULIC DESIGN OF SPILLWAYS
Hydraulic Design of Spillways 17.15
FIGURE 17.11 Definition sketch of free-jet trajectory and scour depth of an overflow spillway. (Coleman, 1982)
FIGURE 17.12 Schematic diagram of a diffusing plunging jet. (Vischer and Hager, 1995).
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HYDRAULIC DESIGN OF SPILLWAYS
17.16
Chapter Seventeen TABLE 17.1
Plunging Jet Characteristics (Whittaker and Schleiss, 1984) Rectangular jet
y yk
y ≥ yk
vz vu pz pu Q Qu E Eu v vz p pz vz vu pz pu Q Qu e Eu v vz p pz
Circular jet
1
1
1
1
1 0.414 y/y / k
1 0.507 y/y / k 0.500 (y/y / k)2
1 0.184 y/y / k
1 0.550 y/y / k 0.217(y/y / k)2
e-π/8(1 x/x/Buⴢy/y/ /y/ k–yk/y)2 e-1/2(1 r/r/RuⴢYYk/y-yk/y)2 e-π/16(x/x/Bu)2
e-1/2(r/r/Ru)2
yk /y /
yk/y
yk/y
(yk/y)2
1.414 y/yk 0.816 y/k y
2y/y / k 0.667 yk/y
-π/8(x\B x uⴢy /y)2 k
e
e-1/2(r/r/Ruⴢyk/y)2
x uⴢyk/y)2 e-π16(x/B
e-1/4(r/r/Ruⴢyk/y)2
Source: Whittaker and Schleiss (1984).
(a)
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HYDRAULIC DESIGN OF SPILLWAYS
Hydraulic Design of Spillways 17.17
(b)
(c) Exhibit 17.3 Boyd’s corner dam project, New York. (a) General view of the spillway (b) A view of the overfall spillway in operation. (c) Layout of the spillway including the flip bucket.
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HYDRAULIC DESIGN OF SPILLWAYS
17.18
Chapter Seventeen
Exhibit 17.3 illustrates an overfall spilling in operation.
17.4 SIDE-CHANNEL SPILLWAY A side-channel spillway can be gated or ungated and provides for flow into a chute or tunnel at right angles because the abutment topography is not favorable for a normal crest alignment. Figure 17.13 shows a typical arrangement. Exhibit 17.4 illustrates a side-channel spillway in operation The hydraulic features are defined as follows (USBR, 1987): 1. The crest is often ungated since a long crest may be suitable because of favorable topography. 2. The crest shape is based on the same criteria as the criteria for an overflow spillway. 3. The trough is sized by trial and error to prevent the maximum discharge water surface from encroaching on the crest’s free-discharge capacity. The trough should be as nearly V-shaped as possible to maximize efficient dissipation of energy. 4. The chute crest is proportioned to produce subcritical flow in the trough for all discharges to dissipate the overflow energy and produce uniform flow into the chute. 5. The trough geometry for the first trial is proportioned to produce an approximately uniform reduction in trough velocity from downstream to upstream. This will usually minimize the trough size. 6. The water surface profile in the trough is estimated by the following: Q2 (V V1 V2) (Q2 – Q1) ∆Y V V1) V1 (V g (Q1 Q2) 2 Q2
(17.6)
where ∆Y = change in water level between two sections ∆X ∆ apart, Q2,V V2 = discharge and velocity at the downstream section, and Q1,V V1 = discharge and velocity at the upstream section. The computation begins with a known water level at the downstream end produced by critical depth control at the chute crest, and proceeds upstream in increments of ∆X ∆ by trial and error, calculating the water level along the length of the side channel. If the calculated water profile encroaches on the side channel’s capacity, the geometry of the trough or chute crest or both should be adjusted.
FIGURE 17.13 Typical arrangement of a side-channel spillway. (USBR, 1987).
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HYDRAULIC DESIGN OF SPILLWAYS
Hydraulic Design of Spillways 17.19
(a)
(b) Exhibit 17.4 Ullum dam project, Argentina. (a) General view of the side–channel spillway in operation (b) Layout of the spillway including the stiling basin.
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HYDRAULIC DESIGN OF SPILLWAYS
17.20
Chapter Seventeen
7. Once an acceptable geometry for the trough and chute crest has been determined, the chute should be sloped to provide supercritical flow away from the chute crest. 8. Large side-channel spillways are almost always model-tested because of the complex flow conditions in the trough area.
17.5 ORIFICE SPILLWAY An orifice spillway is normally gated and is used when substantial discharge capacity is needed at low reservoir levels, as illustrated in Exhibit 17.5. For instance, it is useful when sediment sluicing is required. It also is useful for diverting flow during construction. Gate sizes are normally smaller for these spillways, but higher head and sealing details can make them expensive. Orifice spillways can be found in gravity dams, adjacent to embankment dams, and in arch dams. Figs. 17.14 and 17.15 show a typical layout. The proportions of the orifice are defined as follows: 1. Gates are sized on the basis of discharge requirements at maximum and minimum headwater levels. The discharge coefficient can be assumed to be 0.90. 2. The roof curve must be shaped carefully to prevent cavitation if the head is high. For instance, the design in Fig. 17.15 has a roof curve composed of a circular curve with a large radius, followed by a 1:15 roof slope to the gates top seal. This shape was based on a physical model test (Fig. 17.14).
(a) Exhibit 17.5 Mangla hidroelectric project, Paskistan A three–dimensional rendition of the spillway control structure.(Courtesy Water and Power Development Authority, Pakistan). Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
HYDRAULIC DESIGN OF SPILLWAYS
FIGURE 17.14 Typical arrangement of an orifice spillway. (Institute of Civil Engineers, (1968).
Hydraulic Design of Spillways 17.21
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HYDRAULIC DESIGN OF SPILLWAYS
17.22
Chapter Seventeen
FIGURE 17.15 Typical arrangement of an orifice spillway control structure. (Institute of Civil Engineers, 1968)
3. The floor curve is less critical; however, for the design in Figs. 17.14 and 17.15 the curve radius was 150 ft. 4. Potential formation of vortex is a special concern for orifice spillways. A vortex suppressor will generally be required and can be determined by a physical model. 5. The spillway downstream of the gate structure changes to a chute that may lead to a stilling basin (see Figs. 17.14 and 17.15) or a flip-bucket.
17.6 MORNING GLORY SPILLWAY The morning glory spillway is normally used in conjunction with a tunnel spillway when the intake is a vertical shaft. Also, because of flow entry from the entire periphery, the crest capacity is relatively high. Crest gates are not normally used because of access and cost considerations as well as poor hydraulic conditions in the shaft with partially open gates Exhibit 17.6 illustrates a morning glory spillway. Figures 17.16 and 17.17 show the range of possible flow conditions in the crest area for a morning-glory spillway (USBR,1987). The design procedure is as follows: 1. By trial and error, determine the required design head Ho and the crest radius from Fig. 17.18; Ho/Rs / = 0.3 is recommended. Note that in Fig. 17.18, the discharge coefficient Co is for English units. For metric units, the coefficient should be multiplied by a conversion factor of 0.552. 2. Determine the discharge rating curve for the full range of heads from Fig. 17.19. 3. Determine the lower nappe profile from Figs. 17.17 and 17.20 and Tables 17.2, 17.3, and 17.4.
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HYDRAULIC DESIGN OF SPILLWAYS
Hydraulic Design of Spillways 17.23
4. Check for throat control in the shaft using the following: Q1/2 R CR (17.7) Ha1/4 3 3 where: CR 0.275 for units in m and m /sec, 0.204 for units in ft and ft /sec, R
FIGURE 17.16 Possible flow conditions for a morning-glory spillway. (USBR, 1987).
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HYDRAULIC DESIGN OF SPILLWAYS
17.24
Chapter Seventeen
FIGURE 17.17 Elements of nappe-shaped profile for a morning-glory spillway. (USBR, 1987)
FIGURE 17.18 Circular crest coefficients for a morning-glory spillway with aerated nappe. (USBR, 1987)
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HYDRAULIC DESIGN OF SPILLWAYS
FIGURE GURE 17 17.19 19 Ci Circular l crest coefficients ffi i off discharge di h for f other h than h design d i head. h d (USBR, (USBR 1987)
Hydraulic Design of Spillways 17.25
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HYDRAULIC DESIGN OF SPILLWAYS
17.26
Chapter Seventeen
TABLE 17.2 Coordinates of Lower Nappe Surface for Different Values of Hs/R / When P/ P/R 0.15 H s R
0.20
0.2
0.30
X Hs
0.35
0.40
0.45
0.50
0.60
0.80
Y For portion of the profile above the weir crest Hs
0.000 .010 .020 .030 .040
0.0000 .0120 .0210 .0285 .0345
0.0000 .0120 .0200 .0270 .0335
0.0000 .0115 .0195 .0265 .0325
0.0000 .0115 .0190 .0260 .0310
0.0000 .0110 .0185 .0250 .0300
0.0000 .0110 .0180 .0235 .0285
0.0000 .0105 .0170 .0225 .0265
0.0000 .0100 .0160 .0200 .0230
0.0000 .0090 .0140 .0165 .0170
.050 .060 .070 .080 .090
.0405 .0450 .0495 .0525 .0560
.0385 .0430 .0470 .0500 .0530
.0375 .0420 .0455 .0485 .0510
.0360 .0400 .0430 .0460 .0480
.0345 .0380 .0410 .0435 .0455
.0320 .0355 .0380 .0400 .0420
.0300 .0330 .0350 .0365 .0370
.0250 .0365 .0270 .0270 .0265
.0170 .0165 .0150 .0130 .0100
.100 .120 .140 .160 .180
.0590 .0630 .0660 .0670 .0675
.0560 .0600 .0620 .0635 .0635
.0535 .0570 .0585 .0590 .0580
.0500 .0520 .0525 .0520 .0500
.0465 .0480 .0475 .0460 .0435
.0425 .0435 .0425 .0400 .0365
.0375 .0365 .0345 .0305 .0260
.0255 .0220 .0175 .0110 .0040
.0065
.200 .250 .300 .350 .400
.0670 .0615 .0520 .0380 .0210
.0625 .0560 .0440 .0285 .0090
.0560 .0470 .0330 .0165
.0465 .0360 .0210 .0030
.0395 .0265 .0100
.0320 .0160
.0200 .0015
.450 .500 .550
.0015
Y Hs
X For portion of the profile below the weir crest Hs
—0.000 —.020 —.040 —.060 —.080
0.454 .499 .540 .579 .615
0.422 .467 .509 .547 .583
0.392 .437 .478 .516 .550
0.358 .404 .444 .482 .516
0.325 .369 .407 .443 .476
0.288 .330 .368 .402 .434
0.253 .292 .328 .358 .386
0.189 .228 .259 .286 .310
0.116 .149 .174 .195 .213
—.100 —.150 —.200 —.250 —.300
.650 .726 .795 .862 .922
.616 .691 .760 .827 .883
.584 .660 .729 .790 .843
.547 .620 .685 .743 .797
.506 .577 .639 .692 .741
.462 .526 .580 .627 .671
.412 .468 .516 .557 .594
.331 .376 .413 .445 .474
.228 .263 .293 .319 .342
—.400 —.500 —.600 —.800 —1.000
1.029 1.128 1.220 1.380 1.525
.988 1.086 1.177 1.337 1.481
.947 1.040 1.129 1.285 1.420
.893 .980 1.061 1.202 1.317
.828 .902 .967 1.080 1.164
.749 .816 .869 .953 1.014
.656 .710 .753 .827 .878
.523 .567 .601 .655 .696
.381 .413 .439 .473 .498
—1.200 —1.400 —1.600 —1.800 —2.000
1.659 1.780 1.897 2.003 2.104
1.610 1.731 1.843 1.947 2.042
1.537 1.639 1.729 1.809 1.879
1.411 1.480 1.533 1.580 1.619
1.228 1.276 1.316 1.347 1.372
1.059 1.096 1.123 1.147 1.167
.917 .949 .973 .997 1.013
.725 .750 .770 .787 .801
.517 .531 .544 .553 .560
—2.500 —3.000 —3.500 —4.000 —4.500
2.340 2.550 2.740 2.904 3.048
2.251 2.414 2.530 2.609 2.671
2.017 2.105 2.153 2.180 2.198
1.690 1.738 1.768 1.780 1.790
1.423 1.457 1.475 1.487 1.491
1.210 1.240 1.252 1.263
1.049 1.073 1.088
.827 .840
—5.000 —5.500 —6.000
3.169 3.286 3.396
2.727 2.769 2.800
2.207 2.210
1.793
Hs R
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.60
0.80
Source: USBR (1987).
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HYDRAULIC DESIGN OF SPILLWAYS
Hydraulic Design of Spillways 17.27 TABLE 17.3 Coordinates of Lower Nappe Surface for Different Values of Hs/R / When P/ P/R 0.30 H s R
0.20
0.2
0.30
X Hs
0.35
0.40
0.45
0.50
0.60
0.80
Y For portion of the profile above the weir crest Hs
0.000 .010 .020 .030 .040
0.0000 .0130 .0245 .0340 .0415
0.0000 .0130 .0242 .0335 .0411
0.0000 .0130 .0240 .0330 .0390
0.0000 .0125 .0235 .0320 .0380
0.0000 .0120 .0225 .0300 .0365
0.0000 .0120 .0210 .0290 .0350
0.0000 .0115 .0195 .0270 .0320
0.0000 .0110 .0180 .0240 .0285
0.0000 .0100 .0170 .0210 .0240
.050 .060 .070 .080 .090
.0495 .0560 .0610 .0660 .0705
.0470 .0530 .0575 .0620 .0660
.0455 .0505 .0550 .0590 .0625
.0440 .0490 .0530 .0565 .0595
.0420 .0460 .0500 .0530 .0550
.0395 .0440 .0470 .0500 .0520
.0370 .0405 .0440 .0460 .0480
.0325 .0350 .0370 .0385 .0390
.0245 .0250 .0245 .0235 .0215
.100 .120 .140 .160 .180
.0740 .0800 .0840 .0870 .0885
.0690 .0750 .0790 .0810 .0820
.0660 .0705 .0735 .0750 .0755
.0620 .0650 .0670 .0675 .0675
.0575 .0600 .0615 .0610 .0600
.0540 .0560 .0560 .0550 .0535
.0500 .0510 .0515 .0500 .0475
.0395 .0380 .0355 .0310 .0250
.0190 0.120 .0020
.200 .250 .300 .350 .400
.0885 .0855 .0780 .0660 .0495
.0820 .0765 .0670 .0540 .0370
.0745 .0685 .0580 .0425 .0240
.0660 .0590 .0460 .0295 .0100
.0575 .0480 .0340 .0150
.0505 .0390 0.220
.0435 .0270 .0050
.0.180
.450 .500 .550
.0300 .0090
.0170 —.0060
.0025
Y Hs
X For portion of the profile below the weir crest Hs
—0.000 —.020 —.040 —.060 —.080
0.519 .560 .598 .632 .664
0.488 .528 .566 .601 .634
0.455 .495 .532 .567 .600
0.422 .462 .498 .532 .564
0.384 .423 .458 .491 .522
0.349 .387 .420 .451 .480
0.310 .345 .376 .406 .432
0.238 .272 .300 .324 .348
0.144 .174 .198 .220 .238
—.100 —.150 —.200 —.250 —.300
.693 .760 .831 .893 .953
.664 .734 .799 .860 .918
.631 .701 .763 .826 .880
.594 .661 .723 .781 .832
.552 .618 .677 .729 .779
.508 .569 .622 .667 .708
.456 .510 .558 .599 .634
.368 .412 .451 .483 .510
.254 .290 .317 .341 .362
—.400 —.500 —.600 —.800 —1.000
1.060 1.156 1.242 1.403 1.549
1.024 1.119 1.203 1.359 1.498
.981 1.072 1.153 1.301 1.430
.932 1.020 1.098 1.227 1.333
.867 .938 1.000 1.101 1.180
.780 .841 .891 .970 1.028
.692 .745 .780 .845 .892
.556 .595 .627 .672 .707
.396 .424 .446 .478 .504
—1.200 —1.400 —1.600 —1.800 —2.000
1.680 1.800 1.912 2.018 2.120
1.622 1.739 1.849 1.951 2.049
1.543 1.647 1.740 1.821 1.892
1.419 1.489 1.546 1.590 1.627
1.240 1.287 1.323 1.353 1.380
1.070 1.106 1.131 1.155 1.175
.930 .959 .983 1.005 1.022
.733 .757 .778 .797 .810
.521 .540 .551 .560 .569
—2.500 —3.000 —3.500 —4.000 —4.500
2.351 2.557 2.748 2.911 3.052
2.261 2.423 2.536 2.617 2.677
2.027 2.113 2.167 2.200 2.217
1.697 1.747 1.778 1.796 1.805
1.428 1.464 1.489 1.499 1.507
1.218 1.247 1.263 1.274
1.059 1.081 1.099
.837 .852
—5.000 —5.500 —6.000
3.173 3.290 3.400
2.731 2.773 2.808
2.223 2.228
1.810
Hs R
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.60
0.80
Source: USBR (1987).
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HYDRAULIC DESIGN OF SPILLWAYS
17.28
Chapter Seventeen
TABLE 17.4 P/R 2.00 P/ H s R X Hs
0.00
0.000 .010 .020 .030 .040 .050 .060 .070 .080 .090 .100 .120 .140 .160 .180 .200 .250 .300 .350 .400 .450 .500 .550 .600 .650
0.0000 .0150 .0280 .0395 .0490 .0575 .0650 .0710 .0765 .0820 .0860 .0940 .1000 .1045 .1080 .1105 .1120 .1105 .1060 .0970 .0845 .0700 .0520 .0320 .0000
Coordinates of Lower Nappe Surface for Different Values of Hs/R / When 0.10*
0.20
H R
0.30
0.35
0.40
0.45
0.50
0.60
0.80
1.00
1.20
1.50
2.00
0.0000 .0104 .0180 .0231 .0268 .0292 .0305 .0308 .0301 .0287 .0264 .0195 .0101
0.0000 .0095 .0159 .0198 .0220 .0226 .0220 .0201 .0172 .0135 .0089
0.0000 .0086 .0140 .0168 .0176 .0168 .0147 .0114 .0070 .0018
0.0000 .0077 .0115 .0126 .0117 .0092 .0053 .0001
0.0000 .0070 .0090 .0085 .0050
Y For portion of the profile above the weir crest Hs 0.0000 .0145 .0265 .0365 .0460 .0535 .0605 .0665 .0710 .0765 .0180 .0880 .0935 .0980 .1010 .1025 .1035 .1000 .0930 .0830 .0700 .0520 .0320 .0080
0.0000 .0133 .0250 .0350 .0435 .0506 .0570 .0627 .0677 .0722 0.762 .0826 .0872 .0905 .0927 .0938 .0926 .0850 .0750 .0620 .0450 .0250 .0020
X Hs 0.000 –.020 –.040 –.0.60 –.080 –.100 –.150 –.200 –.250 –.300 –.400 –.500 –.600 –.800 –.1.000 –.1,200 –.1.400 –.1.600 –.1.800 –.2.000 –.2.500 –.3.000 –.3.500 –.4.000 –.4.500 –.5.000 –.5.500 –.6.000
0.25
0.0000 .0130 .0243 .0337 .0417 .0487 .0550 .0605 .0655 .0696 .0734 .0790 .0829 .0855 .0872 .0877 .0850 .0764 .0650 .0500 .0310 .0100
0.0000 .0128 .0236 .0327 .0403 .0471 .0531 .0584 .0630 .0670 .0705 .0758 .0792 .0812 .0820 .0819 .0773 .0668 .0540 .0365 .0170
0.0000 .0125 .0231 .0317 .0389 .0454 .0510 .0560 .0603 .0640 .0672 .0720 .0750 .0760 .0766 .0756 .0683 .0559 .0410 .0220 .000
0.0000 .0122 .0225 .0308 .0377 .0436 .0489 .0537 .0578 .0613 .0642 .0683 .0005 .0710 .0705 .0688 .0596 .0446 .0280 .0060
0.0000 .0119 .0220 .0299 0.363 .0420 .0470 .0514 .0550 .0581 .0606 .0640 .0654 .0651 .0637 .0611 .0495 .0327 .0125
0.0000 .0116 .0213 .0289 .0351 .0402 .0448 .0487 .0521 .0549 .0570 .0596 .0599 .0585 .0559 .0521 .0380 .0174
0.0000 .0112 .0202 .0270 .0324 .0368 .0404 .0432 .0455 .0471 .0482 .0483 .0460 .0418 .0361 .0292 .0068
Y For portion of the profile below the weir crest Hs 0.668 .705 .742 .777 .808 .838 .913 .978 1.040 1.100 1.207 1.308 1.397 1.563 1.713 1.846 1.970 2.085 2.196 2.302 2.557 2.778
0.615 .652 .688 .720 .752 .784 .857 .925 .985 1.043 1.150 1.246 1.335 1.500 1.646 1.780 1.903 2.020 2.130 2.234 2.475 2.700 2.916 3.114 3.306 3.488 3.653 3.820
0.554 .592 .627 .660 .692 .722 .793 .860 .919 .976 1.079 1.172 1.260 1.422 1.564 1.691 1.808 1.918 2.024 2.126 2.354 2.559 2.749 2.914 3.053 3.178 3.294 3.405
0.520 .560 .596 .630 .662 .692 .762 .826 .883 .941 1.041 1.131 1.215 1.369 1.508 1.635 1.748 1.855 1.957 2.053 2.266 2.428 2.541 2.620 2.682 2.734 2.779 2.812
0.487 .526 .563 .596 .628 .657 .725 .790 .847 .900 1.000 1.087 1.167 1.312 1.440 1.553 1.653 1.742 1.821 1.891 2.027 2.119 2.171 2.201 2.220 2.227 2.229 2.232
0.450 .488 .524 .557 .589 .618 .686 .745 .801 .852 .944 1.027 1.102 1.231 1.337 1.422 1.492 1.548 1.591 1.630 1.701 1.748 1.777 1.796 1.806 1.811
0.413 .452 .487 .519 .549 .577 .641 .698 .750 .797 .880 .951 1.012 1.112 1.189 1.248 1.293 1.330 1.358 1.381 1.430 1.468 1.489 1.500 1.509
0.376 .414 .448 .478 .506 .532 .589 .640 .683 .722 .791 .849 .898 .974 1.030 1.074 1.108 1.133 1.158 1.180 1.221 1.252 1.267 1.280
0.334 .369 .400 .428 .454 .478 .531 .575 .613 .648 .706 .753 .793 .854 .899 .933 .963 .988 1.008 1.025 1.059 1.086 1.102
0.262 .293 .320 .342 .363 .381 .423 .459 .490 .518 .562 .598 .627 .673 .710 .739 .760 .780 .797 .810 .838 .853
0.158 .185 .212 .232 .250 .266 .299 .326 .348 .368 .400 .427 .449 .482 .508 .528 .542 .553 .563 .572 .588
0.116 .145 .165 .182 .197 .210 .238 .260 .280 .296 .322 .342 .359 .384 .402 .417 .423 .430 .433
0.093 .120 .140 .155 .169 .180 .204 .224 .239 .251 .271 .287 .300 .320 .332 .340 .344
0.070 .096 .115 .129 .140 .150 .170 .184 .196 .206 .220 .232 .240 .253 .260 .266
0.048 .074 .088 .100 .110 .188 .132 .144 .153 .160 .168 .173 .179 .184 .188
0.00
0.10
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.60
0.80
1.00
1.20
1.50
2.00
H H •The tabulation for s = 0.10 was obtained by interpolation between s = 0 and 0.20. R R Source: USBR (1987).
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HYDRAULIC DESIGN OF SPILLWAYS
Hydraulic Design of Spillways 17.29
(a)
(b) Exhibit 17.6 Big Dalton Dam, California (a) General view of the modified morning–glory spillway. (b) Layout of the spillway. Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
HYDRAULIC DESIGN OF SPILLWAYS
17.30
Chapter Seventeen
required shaft radius for throat control (m or ft), Ha head from headwater level to throat location (m or ft), and Q discharge under consideration (m3/sec or ft3/sec). If R exceeds the radius of the shaft for a given flow, then throat control exists and the discharge is based on the shaft radius. 5. The downstream tunnel is ordinarily sized to flow no more than three-fourths full with the maximum value of Manning’s roughness coefficient n 0.016 to avoid potentially unstable flow conditions. If the tunnel flows full, with throat control already developed, the capacity is dictated by full flow throughout. 6. An ideal design has crest control throughout the range of discharge.
17.7 LABYRINTH SPILLWAY The labyrinth spillway is used to concentrate discharge into a narrow chute, where space does not permit a linear ungated crest. It generally minimizes approach excavation, whereas the concrete weir is more complicated to construct. Labyrinth spillways have been built with a wide range of sizes and discharge capacities and are well suited for rehabilitation of existing spillway structures when increased spillway capacity is needed (Hinchliff and Houston, 1984; Tacail et al., 1990; Tullis el al., 1995). Labyrinth structures can be built economically if an adequate foundation is available. Figure 17.21 is a typical layout (Tacail et al., 1990). The most efficient spillway entrance for most reservoir applications is a curved approach adjacent to each end cycle of the spillway, with the approach flow parallel to the centerline of the spillway cycles for more uniform approach flow. The spillway entrance should be placed as far upstream in the reservoir as possible to reduce localized upstream head losses. To avoid the submergence effect, the supercritical flow condition should be maintained at the chute crest downstream from the labyrinth. The number of spillway cycles should be determined on the basis of the magnitude of the upstream head, the effect of nappe interference, and the economics of the design. With normal operating conditions, the vertical aspect ratio for each labyrinth cycle, w/P (Fig. 17.22), should be 2.5 or greater (Tullis T et al., 1995). Subatmospheric pressures under the nappe will cause nappe oscillation and noise and should be avoided by venting for structural reasons. Splitter piers can be placed along the spillway’s side walls, or crushed stone can be placed along the downstream edge of the crest. The piers should be located at a distance equal to 8 to 10 percent of the wall length upstream of the downstream apexes (Tullis et al., 1995). Although the use of crushed stone may reduce the spillway’s capacity, it can be cost effective. Discharge capacity is a function of the head over the crest, the height of the crest wall, the shape of the crest, the angle of the labyrinth, the number of cycles, and the length of the side leg (Tullis et al., 1995). An efficient design will result from the following procedure: 1. Knowing the design discharge and the maximum head, determine the required effective length from Q 2Cd L 2gHt3/2 3 where Cd 0.30 is derived from Fig. 17.23, with 8°, and Ht/p 0.8.
(17.8)
2. Fixing the apex A, determine the number of cycles in the labyrinth N from
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HYDRAULIC DESIGN OF SPILLWAYS
Hydraulic Design of Spillways 17.31
FIGURE 17.21
Layout of a typical labyrinth spillway. (Tacail et al., 1990).
FIGURE 17.22 Layout and details of a labyrinth weir. (Tullis et al., 1995).
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HYDRAULIC DESIGN OF SPILLWAYS
17.32
Chapter Seventeen
FIGURE 17.23 Crest discharge coefficient for labyrinth spillways. (Tullis et al., 1995).
L 2N (A ( L2)
(17.9)
where L2 is sized to produce an integral number N. 3. If the economics of labyrinth size must be confirmed, other trial sizes can be determined by assuming angles other than 8° or by varying Httt/p ( 0.9) and repeating Steps 1 and 2. The costs of the approach channel and spillway chute, including the dissipator, may influence the ultimate selection of labyrinth geometry. 4. After selecting the labyrinth size, detail as follows: t P/6, P RP P/12, and A 1 to 2 t. The crest shape is quarter-round. 5. The width of the spillway chute downstream is equal to W W, the width of the labyrinth. 6. The chute slope must be supercritical for the entire flow range.
17.8 SIPHON SPILLWAY 17.8.1 Standard Siphon Spillway A standard siphon spillway is used when a large discharge capacity is required in an extremely narrow head range without the use of operating gates. It is ideal for emergency overflows in remote locations. The type used most commonly is the standard or blackwater siphon, where there is a definite priming point, after which there is no air in the flow. Figures 17.24 and 17.25 show the design details for this spillway (USBR, 1987). To minimize losses, the upper leg transition should be well proportioned to provide gradually contracting area, and the inlet area should be about two or three times the area of the throat. The recommended design procedure is as follows:
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HYDRAULIC DESIGN OF SPILLWAYS
Hydraulic Design of Spillways 17.33
1.
Set the crest elevation.
2.
Determine head available (H Ht local atmospheric head).
3.
Estimate number of siphon barrels required.
4.
Determine d, B, D, and Rc, knowing the Q/barrel required, so that Q/barrel CBD2 g Ht
(17.10)
where d depth of water at outlet, D throat width, B width of the rectangular siphon 2D (recommended), C discharge coefficient based on d/D and RCL (Fig. 17.25), RCL radius of center-line of throat, and Rc radius of crest of throat RCL-D/2. Recommended ratio of RCL to D is 2.5 for the throat at the upper and lower bends. 5.
Check the theoretical limiting Q/barrel from Q/barrel BRc 0 .7 (2gh / c) at) ln(Rs/R
(17.11)
where Rs the radius of curvature at the summit of the throat and hat local atmospheric head. 6.
Adjust Rc (and D, if necessary) if the Q/barrel from Step 5 is less than from Step 4, until the result from Step 5 is than the result from Step 4.
7.
Select the priming head on the basis of the desired operating level of the reservoir. If the priming head is less than D/2, locate P.C. of the lower bend on the trajectory of the nappe.
8.
Determine the size and the inlet elevation of the siphon breaker (area of breaker area of throat/24).
9.
Select and design the priming aids to be used (slot to aerate the nappe, vent to equalize pressure above and below the nappe, and vent at the lower bend).
FIGURE 17.24 Typical standard siphon spillway. (USBR, 1987).
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HYDRAULIC DESIGN OF SPILLWAYS
17.34
Chapter Seventeen
FIGURE 17.25 Discharge coefficients for standard siphon spillways. (USBR, 1987).
10. Set the upper bend-aeration slot at 60° from the vertical axis of the crest. Set the pipe vent in the sidewalls and connect the slot to the summit area. 11. Set the lower bend P.C. on the trajectory of nappe according to the priming head at the crest. Set the top of the vertical sections of the siphons lower bend at the elevation of the outlet bottom. 12. Set the elevation of the lower bend air trap at the elevation of the outlet bottom plus the priming head. Set the vent area throat area / 48 (approximate). 13. Set the length of the diverging tube using an 8°30' angle of divergence, and set the vent outlet at two-thirds the depth of the outlet at maximum capacity. 14. Set the pipe vent of the siphon breaker at the summit, with the inlet end set at or slightly below the normal level of the reservoir. Set the minimum pipe area throat area/24. 15. Set the submerged lip of the upper leg deep enough to provide good seal and to prevent excessive drawdown.
17.8.2 Air-Regulated Siphon Spillway A disadvantage of the standard (nonaerated blackwater) siphon is the sudden increase in discharge on priming and the potential for “hunting” when the inflow is between priming flow
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HYDRAULIC DESIGN OF SPILLWAYS
Hydraulic Design of Spillways 17.35
FIGURE 17.26 Typical air-regulated siphon spillway. (Ackers and Thomas, 1975).
FIGURE 17.27 Typical discharge rating curve for air-regulated siphon spillways. (Ackers and Thomas, 1975)
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HYDRAULIC DESIGN OF SPILLWAYS
17.36
Chapter Seventeen
and the siphon’s capacity. An alternative called an air-regulated siphon is designed to operate in a steady-state condition for any discharge between priming and capacity and flowing full of a water-air mixture without reaching the blackwater condition. Therefore, it produces a more stable flow condition with a smoother transition during priming. This type of siphon was developed more than 50 years ago and has been used extensively in the Far East to pass floods of up to 2200 m3/s (Ackers and Thomas, 1975). Its performance can be affected by waves, floating debris, and ice. Recent designs have provided corrections for these problems: for example, by installing of a vertical baffle wall at the upstream face of the siphon. Figure 17.26 shows a typical layout, and Fig. 17.27 shows a typical discharge rating curve. Except for the air intake, the design of an air-regulated siphon spillway is generally similar to that of a standard siphon spillway. A preliminary design can be developed by scaling from a design that has been model tested (e.g., Fig. 17.26). However, because approach and exit flow conditions vary from project to project, each design should always be confirmed by model tests. In particular, the mixed-flow conditions, along with potential cavitation at the siphon crest (as with the standard siphon), make this design more complicated than the standard siphon. However, this design is preferable when flow conditions are highly variable. A summary of experience with air-regulated siphons is available in Ackers and Thomas (1975).
17.9 TUNNEL SPILLWAY Tunnel spillways are used with embankment dams, where there is no suitable location for a chute spillway. A competent rock abutment is required. Tunnel spillways can be gated or ungated, depending on topographic and geologic constraints at the tunnel entrance. In some cases, gates may be required, as shown in Fig. 17.28. A tunnel spillway generally consists of the following elements: entrance structure, inclined tunnel section, flat tunnel section, and flip-bucket.
17.9.1 Entrance Structure The entrance structure serves to provide the required discharge capacity and to transition the flow to the inclined tunnel section. If the entrance structure is ungated, it generally will be a side-channel crest with a trough designed in accordance with Sec. 17.4. If the entrance is gated, it may be by ogee crest gates or by orifice gates. If ogee crest gates are used, the ogee is shaped in accordance with Sec. 17.2, Step 9. If the entrance is a gated orifice, use Sec. 17.5. The transition from an entrance structure to an inclined shaft must be gradual to maintain accelerating, supercritical, open-channel flow. The transition is usually from a rectangular section to a circular tunnel section. Angles of convergence should not exceed approximately 3° (1:20).
17.9.2 Inclined Tunnel Section The inclined tunnel provides for acceleration of the flow from the entrance transition to the flat-tunnel section downstream. The inclined shaft is connected to the flat tunnel by a vertical curve with a large radius (R/D / 5). The elevation of the bottom of the curve should provide velocity and depth to satisfy the following energy consistent with head-
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HYDRAULIC DESIGN OF SPILLWAYS
FIGURE 17.28 Typical arrangement of a tunnel spillway. (Harza Engineering Co., 1996).
Hydraulic Design of Spillways 17.37
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HYDRAULIC DESIGN OF SPILLWAYS
17.38
Chapter Seventeen
water and head losses up to and including the vertical bend, and uniform flow at a depth of 75 percent with long-term tunnel roughness (assume Manning’s n 0.016) in the flattunnel section downstream. If the head is sufficient to produce a velocity of 25 to 30 m/s at the end of the vertical curve, an aeration ramp must be provided upstream of the curve. The location and geometry of the aeration ramp generally are determined by a physical model because of the circular shape of the tunnel.
17.9.3 Flat-Tunnel Section The flat-tunnel section is generally situated as low as possible to minimize the tunnels size while maintaining the downstream end above the tailwater. If the downstream end is fixed—say, 3 meters above maximum tailwater—then the tunnel’s size and slope are proportioned to produce a depth and velocity at the vertical bend that is consistent with headwater and energy losses in the entrance structure and the inclined shaft. In some cases, the length of the tunnel may be such that the downstream tunnel extends directly head and to the entrance structure without an inclined tunnel section. In such a case, the aeration ramp would be placed in the down stream tunnel if the velocity reaches approximately 25 to 30 m/s. It is most convenient for hydraulics and construction if the tunnel section has a square bottom, which provides more flow area and simplifies the design of the aeration ramp. It also eliminates the transition from the downstream tunnel to the flip-bucket.
17.9.4 Flip-Bucket The energy dissipator for a tunnel spillway will almost always be a flip-bucket because it is generally the most economical solution. Any transition from the tunnel to the flip-bucket must be gradual. The simplest flip-bucket is the straight cylindrical type that has the same width as does the tunnel. However, if the impact of the jet with the tailrace is not acceptable, a special bucket may be required. Such a bucket might turn or spread the flow, to limit or localize the plunge-pool scour. A special bucket generally requires a physical model. Because the tunnel spillway, as a general rule, will have special features, a physical model is usually recommended.
17.10 SPILLWAY CHUTE 17.10 .1 Smooth Chute The spillway chute connects the crest structure with an energy dissipator. In plan, it may be straight or curved, have a uniform width, or be tapered. The most common design is a straight chute with a gradual taper. More complex designs require physical model tests. In section, it can have a uniform slope or have more than one slope connected by vertical curves. The most common chute profile is a flat upper chute and a steep lower chute connected by a vertical curve. Chute friction losses are generally calculated assuming that the minimum Manning’s n 0.010 (for energy dissipator design) and that the maximum Manning’s n 0.016 (for wall heights)
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HYDRAULIC DESIGN OF SPILLWAYS
Hydraulic Design of Spillways 17.39
FIGURE 17.29 Typical RCC spillway section with stepped chute. (Zipparro and Hansen, 1993)
Vo2/2g
h/l ˜1.4 Detail a
FIGURE 17.30 Notation sketches of a stepped chute: (Christodoulou, 1993)
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HYDRAULIC DESIGN OF SPILLWAYS
17.40
Chapter Seventeen
Figure 17.31 Relationship between head loss and critical depth for a stepped-chute spillway. (Christodoulou, 1993)
Side-wall freeboard can be taken as 20% of the calculated depth for a straight chute and 30% of calculated depth for a gradually tapered chute (3° maximum, each wall). For curved chutes, model studies are required. When chutes are gated, without interior divide walls, models are required to check ride-up of the flow on walls from nonuniform gate operations. Interior divide walls are sometimes used to separate normal-release bays from flood-release bays. The divide walls are normally sized for some part-gate condition to limit their height and cost.
17.10.2 Stepped Chutes With the introduction of roller-compacted concrete dams, it has become convenient to leave steps on the downstream face of the spillway sec. (RCC) of a gravity dam. The steps not only save money on the chute but also dissipate energy that would remain to be dissipated by the stilling basin at the base of the dam (Zipparro and Hasen, 1993). Fig. 17.29 is a typical spillway section in a roller-compacted concrete dam. The design procedure for hydraulic design of the spillway is as follows: 1. Set the ungated crest length so that the maximum crest head is no more than 10 ft. (3 m). 2. Shape the ogee in accordance with Step 9. in Sec. 17.2. 3. Set the step height so that the ratio of critical depth at the crest over the step height is yc/h 4 (Fig. 17.30). 4. Determine the head loss from headwater to stilling basin level based on Figs. 17.30 and 17.31.(N number of steps). 5. Size the stilling basin in accordance with Chap. 18.
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HYDRAULIC DESIGN OF SPILLWAYS
Hydraulic Design of Spillways 17.41
Figure 17.32 Aeration ramps and air supply systems. (Falvey, 1990).
17.11 SPILLWAY AERATION RAMPS In recent years, aeration has become the standard for cavitation protection for spillways (as well as for outlet works and other release facilities) for structures with a height greater than 50 m (Falvey, 1990; ICOLD, 1992; USACE, 1995; Zipparo and Hasen, 1993). Aeration ramps of various types have been used on spillway chutes (Fig. 17.32) as well as on tunnel spillways (Fig. 17.33). Air is supplied to the ramps in various ways (Fig. 17.32). The aeration ramp requirement is generally determined on the basis of an assessment of the cavitation potential along the entire length of the spillway. The cavitation potential can be expressed in terms of the cavitation number (or cavitation index) σ as po pv σ V2 ρwo 2g
(17.12)
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HYDRAULIC DESIGN OF SPILLWAYS
17.42
Chapter Seventeen
Figure 17.33
Aeration ramp and slot of the Yellowtail Dam tunnel spillway. (ICOLD, 1992)
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HYDRAULIC DESIGN OF SPILLWAYS
Hydraulic Design of Spillways 17.43
FIGURE 17.34 Cavitation Characteristics of Local Irregularities. (continued).
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HYDRAULIC DESIGN OF SPILLWAYS
17.44
Chapter Seventeen
where po pa + pg reference pressure, pa atmospheric pressure, pg gauge pressure, pv vapor pressure of water, ρw density of water, and Vo reference velocity. In general, cavitation damage is expected, at locations where the value of the cavitation number is less than 0.2. In addition to the flow velocity and depth, the occurrence of cavitation damage also depends on the expected or existing local irregularities in the spillway’s surface, the strength of the surface material, the elevation of the structure, and the length of operation. For a specific surface irregularity, the critical cavitation number can be determined from Fig. 17.34. An aeration ramp separates the flow from the boundary and forms a cavity so that air can be entrained underneath the surface of the free jet to provide protection against cavitation damage. Since the cavitation number depends on the local velocity and pressure, the maximum discharge is not necessarily the flow rate that produces the highest cavitation potential or the lowest value of the cavitation number of the flow. Aeration ramps for tunnels must be verified by a physical model unless an extremely close approximation can be made to an existing design that has been model tested. However, aeration ramps for chute spillways can largely be designed using computer models because the flow is primarily two-dimensional (Brater et al., 1996; DeFazio and Wei, 1983; Falvey, 1990; ICOLD, 1992; Wei and DeFazio, 1982; Zipparo and Hasen, 1993). Brief descriptions of several computer models for analysis of spillway aeration ramps can be found in Brater et al. (1996). However, if significant three-dimensional effects are produced by a flat chute slope or tapered side walls, where return flow or accumulation of flow in the cavity beneath the jet might reduce the effective jet trajectory, the design should be confirmed by physical model tests. The recommended procedure for the design of ramp and air vent on a simple chute is as follows: 1.
Perform an analysis of the spillway’s water surface profile for flows up to the maximum discharge in approximately 20-percent increments.
2.
For the preliminary assessment, find the upstream-most site where the mean velocity is approximately 30 m/s for maximum discharge and where protection by aeration is likely to be needed. Experience has shown that significant cavitation damage usually will not occur upstream of this point.
3.
Determine the cavitation numbers for the entire length of the spillway for each flow rate based on the flow velocities and depths obtained in Step 1. The computer program provided in Falvey (1990) can be used.
4.
Determine the upstream-most site where the computed value of the cavitation number is less than 0.20, or the critical cavitation number for the expected local irregularities, and find an appropriate site for the first ramp so that the impact point of the jet is at the desired location.
5.
Based on a consideration of the frequency and discharges of the spillway operation, each ramp should be sized to draw quantity of air equal to approximately 10% of the maximum water flow.
6.
The required cavity length L for maximum flow can then be determined using the following equation (see also Fig. 17.35):
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HYDRAULIC DESIGN OF SPILLWAYS
Hydraulic Design of Spillways 17.45
qa 0.022 VL
(17.13)
where qa air discharge per unit width, V mean water velocity approaching the ramp, and L cavity length. 7.
Use a computer model to determine a suitable ramp geometry to provide the required cavity length L, assuming a free-jet underpressure of –1.0 m (water). The computer models described in Falvey (1990), Harza Engineering Co. (1996), and Wei and DeFazio (1982) can be used.
8.
Size the air vent to provide the air flow from Step 6, with no more than –1.0 m under-pressure and an air vent velocity that does not exceed 80 m/s to avoid excessive noise and choking of the air flow.
9.
Use the computer model to analyze the performance of the ramp/air vent for the full range of flow. If the resulting air vent velocity or underpressure exceeds the allowable 80 m/s or –1.0 m, respectively, resize the ramp to provide acceptable conditions over the full range of flows.
10. Place the succeeding ramps no more than 50 m apart. The last ramp should be no closer than approximately 20 m to the entrance to the energy dissipator (stilling basin or flip-bucket). Check the concentration of air downstream of the aeration ramp in accordance with the following equation (Falvey, 1990):
Figure 17.35
Definition sketch of a free-jet from an aeration ramp. (Wei and DeFazio, 1982)
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HYDRAULIC DESIGN OF SPILLWAYS
17.46
Chapter Seventeen
(a)
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HYDRAULIC DESIGN OF SPILLWAYS
Hydraulic Design of Spillways 17.47
(c)
(d)
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HYDRAULIC DESIGN OF SPILLWAYS
17.48
Chapter Seventeen
(e)
(f) Exhibit 17.7 Guri hydroelectric project, Venezuela (Courtesy CVG–EDELCA, Caracas). (a) General view of the spillway showing aeration ramps and flip buckets. (b) Layout of the right spillway chute showing aerators and flip bucket. (c) General view of the spillway in operation. (d) Close–up view of the spillway in operation. (e) Close–up view of the flow condition at an intermediate aeration ramp. (f) Close–up view of flow separation at the end of a trestle pier for aeration. Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
HYDRAULIC DESIGN OF SPILLWAYS
Hydraulic Design of Spillways 17.49
Cx Coe K (LLx Li)
(17.14)
where CX mean air concentration at distance X, Co mean air concentration at beginning of aeration, LX slope distance downstream from aerator, Li slope distance downstream from aerator to beginning of aeration, and K 0.017 dimensional constant per meter (i.e., 0.017 m-1). On a straight section, the concentration of air decreases approximately 0.15 to 0.20% per meter (Falvey, 1990). 11. Install the ramp design from Step 7, in a physical model, if necessary. The model should be 1:20 or larger and should include all geometric details that could reduce the effective jet trajectory. The ramp design might need adjustment as a result of the model studies. In the model, air flow will be reduced because of scale effects. Therefore, use the ramp underpressure as input in the computer model to confirm the jet trajectory. Note that the above procedure is a rule-of-thumb approach based on experience over the past 30 years or so. Design of ramps over this period has varied significantly within the United States and around the world. Although considerable model information is available, prototype data are limited. The most critical piece of data relates air concentration at the chute’s surface to distance downstream of the ramp. This determines the required spacing of ramps. The other important criterion is how much air should be input at each ramp. Current thinking is that the concentration of air just downstream of the ramp should not exceed approximately 50% in the bottom flow layer. The guidelines above assume that the bottom 10% of depth should be equal parts of air and water. Exhibit 17.7 illustrates a spillway aeration ramp.
17.12 SAMPLE DESIGN Determine the geometry of the spillway crest and the discharge rating curve for an ungated overflow spillway. A bridge over the spillway will be supported on piers 1.8 m thick, with a maximum span width of 12 m between the centerline of piers. The reservoir and flood data are as follows: Maximum flood discharge
2800 m3/sec
Maximum flood pool elevation
110 m
Maximum normal pool elevation
100 m
Approach channel invert elevation
80 m
Downstream channel elevation
20 m
Maximum flood tailwater elevation
40 m
Assume that the overflow crest becomes tangent to a spillway chute that is slopes at 1h:1v.
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HYDRAULIC DESIGN OF SPILLWAYS
17.50
Chapter Seventeen 110 109
Reservoir Elevation (m)
108 107 106 105 104 103 102 101 100 0
250
500
750
1000
1250
1500
1750
2000
2250
2500
2750
3000
FIGURE 17.36 Spillway discharge rating curve.
17.12.1 Design Head In this example, the spillway crest elevation is the same as the normal maximum pool elevation, and the maximum head available to pass the maximum flood discharge is HMAX 110 100 10 m. The design head H0 will be selected as H0 0.8H HMAX 8 m. By selecting a design head that is less than the maximum head, there will be a region of negative pressure on the spillway crest during the maximum discharge, which results in an increased discharge coefficient. Negative pressure is acceptable during maximum discharge provided that it does not exceed one-half atmosphere.
17.12.2 Discharge Coefficient The basic discharge coefficient C0 is determined using Fig. 17.1. P 100 80 20 m P/H0 20/8 2.5 P/ C0 (0.552)(3.945) 2.178 Figure 17.2 is used to determine discharge coefficients for a range of heads to complete the discharge rating curve (Fig. 17.36). This spillway will have a vertical upstream face. Since the maximum tailwater elevation is well below the spillway crest, there will be no tailwater effect and no apron effect.
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HYDRAULIC DESIGN OF SPILLWAYS
Hydraulic Design of Spillways 17.51
No further corrections to the discharge coefficient will be required.
17.12.3 Crest Length Pier nose shape Type 3A from Fig. 17.6 is selected for the bridge piers. The pier contraction coefficient Kp can be assumed to be 0.0. The headwall will be 90° to the direction of flow with rounded abutments. Ka 0.1 HMAX/H /H0 1.25 CMAX/C C0 1.03 (Fig. 17.2) CMAX (1.03)(2.178) = 2.243 QMAX CMAXLHMAX3/2 (2.243)L(10)3/2 2800 m3/s L 39.48 m Three piers will be required to support the bridge. The net length of the crest L’ is determined from L = L’ – 2(0.1)(10) = 39.48 m L’ = 41.48 m (use four bays at 10.5 m each). The total spillway crest length, including three piers at 1.8 m thickness, is 47.4 m.
17.12.4 Checking Minimum Pressure on the Crest From Figs. 17.7 and 17.8 the minimum pressure at maximum discharge (H/Hd = 1.25) occurs along the pier and is about –2.8 m, which is less than one-half atmosphere or 5 m head. This is acceptable.
17.12.5
Discharge Rating Curve
The discharge rating curve is developed in the following table: Elevation
He
100.0
0.0
101.0
1.0
102.0
2.0
103.0
3.0
104.0
He/H H0
C/C C0
C
L
Q
0.0
0.78
1.70
42.0
0
0.125
0.83
1.81
41.8
76
0.250
0.87
1.89
41.6
223
0.375
0.89
1.94
41.4
417
4.0
0.5
0.92
2.00
41.2
660
105.0
5.0
0.625
0.95
2.07
41.0
948
106.0
6.0
0.25
0.97
2.11
40.8
1267
107.0
7.0
0.825
0.99
2.16
40.6
1621
108.0
8.0
1.0
1.00
2.18
40.4
1991
109.0
9.0
1.125
1.02
2.22
40.2
2411
110.0
10.0
1.25
1.03
2.24
40.0
2838
Source:Plot of Discharge Rating Curve
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HYDRAULIC DESIGN OF SPILLWAYS
17.52
Chapter Seventeen
102 101
Elevation (m)
100 99 98 97 96 95 -4
-3
-2
-1
0
1
2
3
4
5
6
Distance (m) FIGURE 17.37 Crest Shape
17.12.6 Crest Geometry Unit discharge approaching crest at design head H0 8m: q 1991/47.4 42 m3/s/m. Approach velocity: Va q/(P H0) 42/28 1.5 m/s. Approach velocity head: ha 0.115 m and ha/H /H0 0.014. Parameters for the crest geometry are determined from Fig. 17.9. K 0.503 N 1.865 Xc/H /H0 0.277 Yc/H /H0 0.12 R1/H /H0 0.525 R2/H /H0 0.225 The origin of the X-Y axis is at the crest of the spillway, and Xc is the distance from the upstream face to the crest. Upstream of the origin:
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HYDRAULIC DESIGN OF SPILLWAYS
Hydraulic Design of Spillways 17.53
Xc 2.216 m, Yc 0.960 m, R1 4.200 m, and R2 1.800 m. Downstream of the origin: Y 0.08325 X.865 Location of the tangent point is determined by Y’ 0.08325(1.865)X X0.865 1.0 Xt 8.614 Yt 4.619 The crest geometry is plotted on Fig. 17.37. Plot of crest geometry.
REFERENCES Ackers, P., and A. R. Thomas, “Design and Operation of Air-Regulated Siphons for Reservoir and Head-Water Control,” Proceedings of the Symposium on Design and Operation of Siphons and Siphon Spillways, London, UK, May 1975. Brater, E. F., H. W., King, J. E. Lindell, and C. Y. Wei, Handbook of Hydraulics, 7th ed., McGrawHill, New York, 1996. Chow, V. T., Open-Channel Hydraulics, McGraw-Hill, New York, 1966. Christodoulou, G. C., “Energy Dissipation on Stepped Spillways,” Journal of Hydraulic Engineering, ASCE, 119, (5): May 1993. Coleman, H. W., “Prediction of Scour Depth from Free Falling Jets,” Proceedings, of the ASCE Hydraulics Division Conference on Applying Research to Hydraulic Practice, Jackson, Ms, 1982, DeFazio F. G., and C. Y. Wei, “Design of Aeration Devices on Hydraulic Structures,” Frontiers in Hydraulic Engineering, American Society of Civil Engineers, New York, 1983. Falvey, H. T., Cavitation in Chutes and Spillways, USBR Engineering Monograph No. 42, USBR, 1990. Harza Engineering Co., Internal Report, July 1996. Hinchliff D. L., and K. L. Houston, Hydraulic Design and Application of Labyrinth Spillways, Research Engineering and Research Center, U.S. Bureau of Reclamation, 1984. ICOLD, Spillways, Shockwaves and Air Entrainment—Review and Recommendations, Bulletin No. 81, ICOLD, Paris, 1992. ICOLD, “Spillways for Dams,” Bulletin No. 58, ICOLD, Paris, 1987. Institute of Civil Engineers, “Mangla,” Proceedings of the Institute of Civil Engineers, Binnie & Partners, Westminster, 1968. Tacail, F. G., B. Evans, and A. Babb, “Case Study of a Labyrinth Weir Spillway,” Canadian Journal of Civil Engineering, 17, 1990. Tullis, J. P., N. Amanian, and D. Waldron, “Design of Labyrinth Spillways,” Journal of Hydraulic Engineering, ASCE 121 (3): March 1995. USACE, Hydraulic Design of Spillways, U.S. Army Corps of Engineers, No. EM 1110-2-1603, American Society of Civil Engineers, New York, 1995. USACE, Hydraulic Design Criteria, U.S. Army Corps of Engineers Waterways Experiment Station, Vicksburg, Ms, 1988.
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HYDRAULIC DESIGN OF SPILLWAYS
17.54
Chapter Seventeen
USBR, Design of Small Dams, U. S. Bureau of Reclamation, Denver, CO, 1987. Vischer, D. L., and W. H. Hager, Energy Dissipators—Hydraulic Design Considerations, IAHR Hydraulic Structures Design Mannual No. 9, A. A. Balkema, Rotterdam, Netherlands, 1995. Wei, C. Y., and F. G. DeFazio, “Simulation of Free Jet Trajectories for the Design of Aeration Devices on Hydraulic Structures,” Proceedings of the 4th International Conference on Finite Elements in Water Resources, Hannover, Germany, June 1982. Whittaker, J. G., and A. Schleiss, “Scour Related to Energy Dissipators for High Head Structures,” Nr. 73, Mitteilungen der Versuchsanstalt fur Wasserbau, Hydrologie und Glazioloie, Zurich, 1984 Zipparro, V. J., and H. Hasen, Davis' Handbook of Applied Hydraulics, 4th ed., McGraw-Hill, New York, 1993.
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Source: HYDRAULIC DESIGN HANDBOOK
CHAPTER 18
HYDRAULIC DESIGN OF STILLING BASINS AND ENERGY DISSIPATORS C. Y. Wei and James E. Lindell Harza Engineering Company Chicago, Illinois.
18.1 INTRODUCTION Stilling basins and energy dissipators are usually provided in conjunction with development of spillways, outlet works, and canal structures. It is often necessary to perform hydraulic model studies of individual structures to be certain that these energy dissipating devices will operate as anticipated. A relatively large volume of data is available from many laboratory and field studies performed in the past (Blaisdell, 1948; Bowers and Toso, 1988; Bowers and Tsai, 1969; Chadwick and Morfett, 1986; Chaudhry, 1993; Chow 1959; French, 1985; George, 1978; Hendreson, 1966; International Commission on Large Dams (ICOLD), 1987; Novak et al., 1990; Peterka, 1964; Robert son et al., 1988; Senturk, 1994; Toso and Boweis, 1988; U.S. Bureau of Reclamation (USBR), 1974, 1987; Vischer and Hager, 1995, 1998). Based on the results of many intensive studies, in 1958, A. J. Peterka (1964) of the U.S. Bureau of Reclamation (USBR) published a summary report of USBR’S studies entitled Hydraulic Design of Stilling Basins and Energy Dissipators, Engineering Monograph No. 25. Since then, this publication has been referenced widely within the hydraulic engineering community and still is one of the best references on this subject available today. Energy dissipators are used to dissipate excess kinetic energy possessed by flowing water. An effective energy dissipator must be able to retard the flow of fast moving water without damage to the structure or to the channel below the structure. Vischer (1995) classified various types of energy dissipators by their features as: (1) by sudden expansions, (2) by abrupt deflections, (3) by counterflows, (4) by rough walls, (5) by vortex devices, and (6) by spray inducing devices. The stilling basins and energy dissipators discussed in this chapter are related to energy dissipation by expansion and deflection. There are two basic types of energy dissipators. They are hydraulic jump-type dissipators and impact-type dissipators. The hydraulic jump type energy dissipators dissipate excess energy through formation of highly turbulent rollers within the jump. The impacttype dissipators direct the water into an obstruction that diverts the flow in all directions and generates high levels of turbulence and in this manner dissipates the energy in the flow. In other cases, the flow is directed to plunge into a pool of water where the energy
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HYDRAULIC DESIGN OF STILLING BASINS AND ENERGY DISSIPATORS
18.2
Chapter Eighteen
(a)
(b) Exhibit 18.1: Wanapum project, Washington (a) General view of the spillway in operation. (b) Layout of the spillway and stilling basin.
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HYDRAULIC DESIGN OF STILLING BASINS AND ENERGY DISSIPATORS
Hydraulic Design of Stilling Basins and Energy Dissipators 18.3
is diffused and dissipated. The impact-type energy dissipators include check drops and vertical drops, baffled outlets, baffled aprons, and vertical stilling wells. Generally, the use of an impact-type energy dissipator results in smaller and more economical structures.
18.2 STILLING BASINS Using six test flumes, USBR conducted model studies for five stilling basin designs. The results are summarized and presented in the Engineering Monograph No. 25 mentioned above (Peterka, 1964). In Basin I tests, all test flumes were used and the test data obtained provides basic hydraulic information concerning hydraulic jumps on a horizontal apron. The Type II basin was developed for high dam and earth dam spillways and large canal structures where the approach velocity is high and the corresponding Froude number exceeds 4.5. Type III stilling basin is suitable for general canal structures, small outlet works, and small spillways where the approach velocity is moderate or low and does not exceed 50–60 ft/s (15–18 m/s) and the unit discharge is less than 200 ft3/s/ft (18 m3/s). For smaller canal structures, outlet works, and diversion dams where the approach Froude number is relatively low (between 2.5 and 4.5) and the heads of the structures do not exceed 50 ft (15 m), Type IV stilling basin may be used. However, the jumps in the basin may be unstable and alternative design such as the modified Type IV basin may be considered. To achieve greater structure economy for high dam spillways, Type V stilling basin with sloping apron may be considered. Photos of several stilling basin in operation are given in Exhibits 17.2, 17.4, 18.1, and 18.2.
18.2.1 General Hydraulic Jump Basin (Basin I) The basic elements and characteristics of a hydraulic jump on horizontal aprons (Fig. 18.1) is provided to aid designers in selecting more practical basins such as Basins II, III, IV, V, and VI. Jump occurs on a flat floor with no chute blocks, baffled piers or end sill in the basin. Usually, it is not a practical basin because of its excessive length. For a high–velocity flow down a spillway chute with known terminal velocity (Fig. 18.1) and depth entering the basin, the required tail water depth, the length of jump, and loss of energy can be determined based on the curves provided in Fig. 18.2a–e.
18.2.2 Stilling Basins for High Dam and Earth Dam Spillways and Large Canal Structures (Basin II) This stilling basin was developed for use on high spillways, large canal structures, and so forth for approach Froude numbers above 4.5. With chute blocks and dentated end sill, the jump and basin length can be reduced by about 33 percent. The basic design features of Basin II stilling basin are given in Fig. 18.3a. For preliminary designs, the curves for estimating required tailwater depth, and length of jump are given in Figures 18.3b and c. The water surface and pressure profiles can be determined based on Fig. 18.3d and e. The water surface profile in this basin can be closely approximated by a straight line making an angle a (jump angle) with the horizontal. This line can also be considered as a pressure profile. The USBR guidelines for designing this type of stilling basins are given as follows:
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HYDRAULIC DESIGN OF STILLING BASINS AND ENERGY DISSIPATORS
Chapter Eighteen
(a)
18.4
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HYDRAULIC DESIGN OF STILLING BASINS AND ENERGY DISSIPATORS
Hydraulic Design of Stilling Basins and Energy Dissipators 18.5
(b)
(c) Exhibit 18.2 Mayfield hydroelectric project, Washinton (a) Layout of the spillway including flip bucke. (b) General view of the spillway and the stilling pool (c) General view of the spillway and stilling pool.
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HYDRAULIC DESIGN OF STILLING BASINS AND ENERGY DISSIPATORS
18.6
Chapter Eighteen
FIGURE 18.1 Curves for velocity entering stilling basins from 0.8:1 to 0.6:1 steep slopes.(From Peterka, 1964)
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HYDRAULIC DESIGN OF STILLING BASINS AND ENERGY DISSIPATORS
Hydraulic Design of Stilling Basins and Energy Dissipators 18.7
FIGURE 18.2 Basic hydraulic jump basins on horizontal aprons. (Basin I) (From Peterka, 1964)
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HYDRAULIC DESIGN OF STILLING BASINS AND ENERGY DISSIPATORS
18.8
Chapter Eighteen
FIGURE 18.3 Stilling basins for high dam and earth dam spillways and large canal structures. (Basin II)(From Peterka, 1964)
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HYDRAULIC DESIGN OF STILLING BASINS AND ENERGY DISSIPATORS
Hydraulic Design of Stilling Basins and Energy Dissipators 18.9
1.
Determine velocity V1 of flow entering the jump. Figure 18.1 may be used. This chart represents a composite of experience, computation, and a limited amount of experimental information obtained from prototype tests on Shasta and Grand Coulee Dams. The chart provides a fair degree of accuracy for chute having slopes of 0.8:1 or steeper, where computation is a difficult and arduous procedure. The asymptotic nature of the terminal velocity curves is also depicted in Fig. 18.1. For a constant head of 2.5 ft (0.8m) on the spillway crest, the terminal velocity does not increase significantly (from 51 ft/s or 15.5 m/s to 53 ft/s or 16.2 m/s) as the vertical distance (fall) from the reservoir level to stilling basin floor increases from 200 to 600 ft (61–183 m).
2.
Set apron elevation to utilize full conjugate tail water depth. Add a factor of safety if needed. A minimum margin of safety of 5 percent of tailwater depth (D2) is recommended.
3.
Exercise caution with effectiveness of the basin at lower values of the Froude number (V V1 / (gD1)1/2) of 4 or lower. D1 is the depth of the flow entering the basin.
4.
Determine the length of basin using the curve shown in Fig. 18.3c.
5.
Use the depth of flow entering the basin, D1 as the height of chute blocks. The width and spacing should be equal to approximately D1 but can be varied to avoid fractional blocks. A space equal to D1/2 is preferable along each side of wall to reduce spray and maintain desirable pressures.
6.
As shown in Fig. 18.3a, set the height of the dentated sill equal to 0.2D2 and the maximum spacing approximately 0.15D2. For narrow basins, the width and spacing may be reduced but they should remain equal.
7.
It is not necessary to stagger the chute blocks with respect to the sill dentates.
8.
It is recommended that the sharp intersection between chute and basin apron be replace with a curve of reasonable radius of at least 4D1 when the chute slope is 1:1 or greater. Chute blocks can be incorporated on the curve surface as readily as on the plane surfaces. The chute slope (0.6:1–2:1) does not have significant effect on the stilling basin action unless it is nearly horizontal. Following the above rules should result in a safe, conservative stilling basin for spillways up to 200 ft (60 ms) high and for flows up to about 500 (ft3/s/ft) [46.5 (m3/s/m)] basin width, provided that jet entering the basin is reasonably uniform both as to velocity and depth. For greater falls, larger unit discharges, or possible asymmetry, a model study of the specific design is recommended.
18.2.3 Short Stilling Basins for Canal Structures, Small Outlet Works, and Small Spillways [Basin III and the St. Anthony (SAF) Basin] For structures carrying relatively small discharges at moderate velocities, a shorter basin having a simpler end sill may be used if baffled piers are placed downstream from the chute blocks (Fig. 18.4). In this section, stilling basins for smaller structures in which velocity at the entrance to the basin are moderate or low ( up to 50–60 ft/s or 15–18 m/s) and discharges of up to 200 ft3/s/ft of width or 18 m3/s/m of width are discussed. The stilling basin action is very stable for this design. It has a large factor of safety against sweepout of the jump and operates equally well for all values of the Froude number above 4.0.
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HYDRAULIC DESIGN OF STILLING BASINS AND ENERGY DISSIPATORS
18.10
Chapter Eighteen
FIGURE 18.4 Short stilling basins for canal structures, small outlet works and small spillways.(Basin III) (From Peterka, 1964)
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HYDRAULIC DESIGN OF STILLING BASINS AND ENERGY DISSIPATORS
Hydraulic Design of Stilling Basins and Energy Dissipators 18.11
This basin should not be used for velocities above 50 ft/s or 15 m/s to avoid potential cavitation damages aginst baffle piers. Instead, Basin II type stilling basin should be considered or hydraulic model studies should be performed. The following USBR guidelines pertain to the design of the Basin III type stilling basin: 1. The stilling basin operates best at full conjugate tail water depth, D2. A reasonable factor of safety is inherent in the conjugate depth for all values of the Froude number and it is recommended that this margin of safety not be reduced. 2. Determine the length of basin using the design curve given in Fig. 18.4c. It is less than one-half the length of the natural jump. It should be noted that an excess of tail water depth does not substitute for basin length or vice versa. 3. Exercise caution with effectiveness of the basin at lower values of the approach Froude number [V V1 / (gD1)1/2] of 4.5 or lower. 4. Height, width, and spacing of chute blocks should equal the average depth of flow entering the basin, or D1. Width of blocks may be decreased, provide spacing is reduced a like amount. Should D1 proved to be less than 8 in or 20 cm, the blocks should be made 8 in or 20 cm high. 5. The height of the baffle piers (Fig. 18.4a) varies with the Froude number and is given in Fig. 18.4d. In narrow structures, block width and spacing may be reduced, provided both are reduced a like amount. A half space is recommended adjacent to the walls. 6. The upstream face of the baffle piers should be set at a distance of 0.8D2 from the downstream face of the chute blocks. This dimension is important. 7. The height of the solid end sill is given in Fig. 18.4d. The slope is 2:1 upward in the direction of flow. 8. It is undesirable to round or streamline the edges of the chute blocks, end sill, or baffle piers. It reduces the effectiveness of the energy dissipation. However, small chamfers on the block edges to prevent chipping of the edges and to reduce cavitation erosion may be used. 9. It is recommended that a radius of reasonable length greater than 4D1 be used at the intersection of the chute and basin apron for slopes of 45º or greater. 10. As a general rule, the slope of the chute has little effect on the stilling basin action unless long flat slopes are involved. 11. Experience indicates that the Type III basin works well for flow less than 200 ft3/s/ft or 18 m3/s/m based on basin width and approach velocity at the entrance of up to 50–60 ft/s or 15–18 m/s. The St. Anthony Falls (SAF) Hydraulic Laboratory of the University of Minnesota had also developed a similar basin for small spillways, outlet works, and small canal structures for approach Froude numbers ranging from 1.7 to 17 (Blaisdell, 1948; Chow, 1959). This basin was developed to achieve about 70 to 90 percent reduction of the jump lengths. This basin is commonly known as the SAF stilling basin (Fig. 18.5). Since the basin is relatively short so that a significant amount of residual energy can still exist downstream from the end sill, the channel reach downstream from the stilling basin should be allowed to erode until a stable scour depth is reached. Otherwise riprap protection should be provided to minimize scour (Sec. 18.7). The guidelines for designing this basin are summarized as follows (Blaisdell, 1948; Chow, 1959):
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HYDRAULIC DESIGN OF STILLING BASINS AND ENERGY DISSIPATORS
18.12
Chapter Eighteen
FIGURE 18.5
The SAF stilling basin. (From Blaisdell, 1948)
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HYDRAULIC DESIGN OF STILLING BASINS AND ENERGY DISSIPATORS
Hydraulic Design of Stilling Basins and Energy Dissipators 18.13
1.
The length L of the stilling basin is determined by the following equation.
4.5y2 L 0. F1 76 where y2 is the theoretical sequent depth of the jump corresponding to the approach flow depth y1 and F1 is the approach Froude number. 2.
The height of the chute blocks and floor blocks is y1, and the width and spacing are approximately 0.75y1.
3.
The distance from the upstream end of the stilling basin to the floor blocks is L/3. L
4.
No floor blocks should be placed closer to the side-wall than 3y1/8.
5.
The floor blocks should be placed downstream from openings between the chute blocks.
6.
The total width of the floor blocks should occupies about 40 to 55 percent of the stilling basin width.
7.
The widths and spacings of the floor blocks for diverging stilling basins should be increased in proportion to the increase in stilling basin width at the floor block location.
8.
The height of end sill is given by c 0.07y2.
9.
The depth of tailwater above the stilling basin floor is given by F12 y'2 1.10 y 1 20 2
for F1 1.7 to 5.5
y'2 0.85y2
for F1 5.5 to 11.0
F12 y'2 1.00 y 8 00 2
for F1 11 to 17
10. The top of the side-wall above the maximum tailwater level to be expected during the life of the structure is given by z y2/3. 11. Wing-walls should be equal in height to the stilling basin side-walls. The top of the wing-wall should have a slope of 1H:1V. V 12. The wing-wall should be placed at an angle of 45º to the outlet center line. 13. The stilling basin side-walls may be parallel for a rectangular stilling basin or they may diverge as an extension of the transition side-walls for a trapezoidal stilling basin as shown in Fig. 18.5. 14. A cutoff wall of nominal depth should be used at the end of the stilling basin. 15. The effect of entrained air should be neglected in the design of the stilling basin.
18.2.4 Low Froude Number Stilling Basins (Basin IV and Modified Basin IV) This stilling basin was developed for canal structures, outlet works, and diversion dams where the approach Froude number of the basin is relatively low (between 2.5 and 4.5)
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HYDRAULIC DESIGN OF STILLING BASINS AND ENERGY DISSIPATORS
18.14
Chapter Eighteen
FIGURE 18.6 Low Froude number (2.5–4.5) stilling basin design (Basin IV).(From Peterka, 1964)
and the heads of the structures are about 50 ft (or 15 m). In this case, the jump is not fully developed and unstable and the methods of design discussed previously do not apply. Alternative design and/or wave suppressors or Basin VI type stilling basin with a hanging baffle for energy dissipation may be considered. Guidelines for developing a low Froude number stilling basin (Basin IV) as depicted in Fig. 18.6 are given as follows (Peterka, 1964): 1. A model study of the stilling basin is imperative. 2. Reduction of excessive waves created in the unstable jump is the main problem concerning the design of the stilling basin. 3. A tailwater depth of 10 percent greater than the conjugate depth is strongly recommended. 4. Place as few appurtenances as possible in the path of the flow, as volume occupied by appurtenances helps to create a backwater problem, thus requiring higher training walls. 5. Use Fig. 18.6 to develop the design of the stilling basin. The number of deflector blocks shown in the figure is a minimum requirement. 6. The length of basin can be obtained from Fig. 18.2c. No baffle piers are needed in the basin. 7. The recommended maximum width of the deflector blocks is equal to D1 but 0.75D1 is preferable from a hydraulic standpoint. The ratio of block width to spacing should be maintained as 1:2.5. 8. The extreme tops of the deflector blocks are 2D1 above the floor of the stilling basin. 9. To accommodate the various slopes of chutes and ogee shapes encountered, the horizontal top length of the deflector blocks should be at least 2D1. The upper surface of each block is sloped at 5º in a downstream direction for better operation especially at lower discharges.
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HYDRAULIC DESIGN OF STILLING BASINS AND ENERGY DISSIPATORS
Hydraulic Design of Stilling Basins and Energy Dissipators 18.15
10. The addition of a small triangular sill placed at the end of the apron for scour control is desirable. An end sill of the type developed for short stilling basins (Basin III) can be used. The slope of the upstream face of the sill is 2:1 and the height of the sill can be determined based on Fig. 18.4d. 11. Basin IV stilling basin is applicable to rectangular cross sections only to minimize potential wave–related problems. Type IV stilling basin performs effectively in dissipating the energy at low Froude number flows for small canals and for structures with small unit discharges. It is also effective in minimizing wave problems. Based on additional model tests, the U.S. Bureau of Reclamation (USBR) has developed a modified stilling basin for low Froude number approach flows (George, 1978). This stilling basin is suitable for approach flows with Froude numbers ranging from 2.5 to 5.0. The basin is relatively short and is provided with chute blocks, baffle piers, and a dentated end sill as shown in Fig. 18.7a. The guidelines for designing Modified Type IV stilling basin are given as follows (George, 1978):
FIGURE 18.7(a) Low Froude number stilling basin (Modified Basin IV).(From George, 1978)
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HYDRAULIC DESIGN OF STILLING BASINS AND ENERGY DISSIPATORS
18.16
Chapter Eighteen
1.
A hydraulic model study is recommended to confirm the design. Erosion tests should be included. Such tests should be made over a full range of discharges to determine erosion potential downstream from the basin and to determine the potential for the abrasive bed materials to move upstream into the basin.
2.
Determine the theoretical D2 based on the known unit discharge and the approach flow depth D1.
FIGURE 18.7(b) Design curves for modified Basin IV stilling basin.(From Georges, 1978)
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HYDRAULIC DESIGN OF STILLING BASINS AND ENERGY DISSIPATORS
Hydraulic Design of Stilling Basins and Energy Dissipators 18.17
3.
Determine tailwater depth as TW 1.05 D2.
4.
Set the length of the basin L 3D2 (approximately).
5.
Use Fig. 18.7a to develop the basic dimensions of the basin.
6.
Determine the distance X from the chute blocks to the baffle piers. X varies from 1.3 to 0.7 times D2 as the approach Froude number varies from 2.5 to 5.6 as shown in Fig. 18.7b.
7.
Determine the distance L1 from the toe of the chute to the upstream face of the end sill from Fig. 18.7b.
8.
If (L1 the length of the end sill) is longer than L then the stilling basin should be extended to include the end sill.
9.
Set the widths of the baffle piers equal to 0.7D1 and heights equal to 1.0D1.
10. Determine the number of chute blocks and baffle piers by the following equations. The total number of chute blocks and spaces N = (width – 2kW)/ W W where k fractional width of block equal to side clearance, 0.375 k 0.50 width total width of stilling basin W 0.70D1 The N value obtained should be rounded to the nearest odd number and then adjust values of W and k should be adjusted. 11. Use 0.2D1 as the top length of the baffle piers. 12. Determine end sill dimensions. height 0.2D2 width, W 0.15D2 top length of end sills 0.2 height The number of blocks and spaces N (basin width)/W (N should be rounded to the nearest odd number and then the value of W should be adjusted)
18.2.5 Stilling Basin with Sloping Apron To achieve greater structural economy, a stilling basin with a sloping apron can be considered. This type of stilling basin is usually used on high dam spillways. It needs greater tail water depth than horizontal apron. The energy dissipation is as effective as occurs in the true hydraulic jump on a horizontal apron. The primary concern in sloping apron design is the tail water depth which is required to move the front of the jump up the slope to the location where the jump is expected to start. It may not be economically feasible to design the basin to confine the entire jump, especially when sloping aprons are used in
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HYDRAULIC DESIGN OF STILLING BASINS AND ENERGY DISSIPATORS
18.18
Chapter Eighteen
FIGURE 18.8 Stilling basins with sloping aprons. (From Peterka, 1964)
conjunction with medium or high overfall spillways where the rock foundation is in fairly good condition. When shorter aprons are used, the riverbed downstream must act as part of the stilling basin. On the other hand, when the quality of foundation material is questionable, it is desirable to make the apron sufficiently long to confine the entire jump. The total apron length may range from about 40 to 80 percent of the length of jump. The hydraulic jump may occur in several ways on a sloping apron, as depicted in Fig. 18.8. The jump may have its toe form on the slope and the jump itself ends over the horizontal apron (Case B), or ends at the junction of the slope and the horizontal apron (Case C), or the entire jumps forms on the slope (Case D). For practical purposes the action in Cases C and D is the same. Guidelines for the design of sloping aprons are given below: 1.
Determine an apron arrangement that will give the best economy for the maximum discharge condition.
2.
The first consideration should be to determine the apron slope that will require the minimum amount of excavation, the minimum amount of concrete, or both, for the maximum discharge and tailwater condition.
3.
Position the slope so that the front of the jump will form at the upstream end of the slope for the maximum discharge and tailwater condition (Fig. 18.9). It may be necessary to raise or lower the apron, or change the slope entirely. Data obtained from 13 existing spillways are also shown in Fig. 18.9. Each point in the figure has been connected with an arrow to the tan(θ) curve corresponding to the apron slope. The adequacy of the tailwater depth of these spillways can then be evaluated.
4.
Use Fig. 18.10 to determine the length of the jump for maximum or other flows. Shorter basins may be used where a solid bed exists. For most installations, an apron length of about 60 percent of the length of jump for the maximum discharge condition should be sufficient. Longer basins are needed only when the downstream riverbed is in very poor condition.
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HYDRAULIC DESIGN OF STILLING BASINS AND ENERGY DISSIPATORS
Hydraulic Design of Stilling Basins and Energy Dissipators 18.19
5.
Ascertain that the tailwater and length of basin available for energy dissipation are sufficient for, say 1/4, 1/2, and 3/4 capacity. If the tailwater depth is deficient, a different slope or a new position of the sloping portion of the apron should be considered.
6.
Horizontal and sloping aprons will perform equally well for high values of the Froude number if the proper tail water depth is provided.
7.
The slope of the chute upstream from a stilling basin has no significant effect on the hydraulic jump when the velocity distribution and depth of flow are reasonably uniform on entering the jump.
8.
A small solid triangular sill should be provided at the end of the apron to lift the flow as it leaves the apron for scour protection. The most effective height is between 0.05D2 and 0.10D2 and a slope of 3:1–2:1. Several existing stilling basins with sloping aprons are shown in Fig. 18.11. All stilling basins shown were designed with the aid of model studies.
9. The stilling basin should be designed to operate with as nearly symmetrical flow in the stilling basin as possible to avoid formation of large circulating eddies and transport of riverbed material into the apron area, and the potential undermining of the wing walls and riprap. 10. A model study is advisable where the discharge over high spillways exceeds 500 ft3/s/ft or 46.5 m3/s/m based on the apron width, where there is any form of asymmetry involved, and for the high values of the Froude number where stilling basins become more costly and the performance becomes less acceptable.
FIGURE 18.9 Comparison of existing sloping apron designs with experimental results. (From Peterka, 1964)
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HYDRAULIC DESIGN OF STILLING BASINS AND ENERGY DISSIPATORS
Chapter Eighteen
FIGURE 18.10 Jump length in terms of conjugate depth, D2 for stilling basins with sloping aprons.(From Peterka, 1964)
18.20
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HYDRAULIC DESIGN OF STILLING BASINS AND ENERGY DISSIPATORS
FIGURE 18.11(a) Existing stilling basins with sloping aprons.(From Peterka, 1964)
Hydraulic Design of Stilling Basins and Energy Dissipators 18.21
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HYDRAULIC DESIGN OF STILLING BASINS AND ENERGY DISSIPATORS
Chapter Eighteen
FIGURE 18.11(b) Existing stilling basins with sloping aprons.(From Peterka, 1964)
18.22
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HYDRAULIC DESIGN OF STILLING BASINS AND ENERGY DISSIPATORS
Hydraulic Design of Stilling Basins and Energy Dissipators 18.23
18.2.6 Other Types of Stilling Basins Other types of stilling basins that may be considered include: (1) positive step basin, (2) negative step basin, (3) baffle–sill basin, (4) baffle-block basin, (5) expanding stilling basin, and (6) bucket stilling basin. Detailed discussions of these basins have been provided by Vischer and Hager (1995). These basins are briefly discussed as follows: 1. Positive–step basin. An upward step of a given height is provided in a prismatic channel. No end sills are included. The required basin length is significantly longer than that of a classical jump basin. 2 Negative–step basin. A downward step is provided. No end sills are included. It requires a slightly longer basin length than the positive step basin. Basins with steps have not been popular because it is easier to use sills or blocks in a horizontal apron than to change the apron elevation at the step section. 3 Baffle–sill basin. A weir-type sill is provided to form a basin. The flow over the sill may be submerged or free. The sill is capable of stabilizing the jump in a shorter basin and with lower tailwater than is the classical jump basin. Sills can be economical and effective devices for energy dissipation even without additional appurtenance included. 4. Baffle–block basin. Baffle blocks are normally arranged in one or several rows that are oriented perpendicular to the direction of approach flow. Standard baffle blocks such as the USBR blocks should be used. Baffle blocks are prone to cavitation damage and should not be used for approach velocities above 20 m/s. For velocities between 20 and 30 m/s, a chamfer on the block edges should be provided to reduce the cavitation potential. 5. Expanding stilling basin. There are two types of expanding basins, namely gradually expanding basin and abruptly expanding basin. The gradually expanding basin requires less tailwater depth and can be used for highly variable tailwater. This type of basin is suitable for approach flow with Froude numbers less than 4. Very few basins of this type have been built. An abruptly expanding basin has been studied and reported by Vischer and Hager (Novak et al., 1990). No practical applications have been reported.
18.2.7 Fluctuating Pressures on Stilling Basin Floors When designing a stilling basin to achieve highest possible hydraulic efficiency in terms of energy dissipation, one should also consider the structural aspects of the stilling basin. The effect of transient pressures caused by turbulence in the jump can be significant and should be considered in the design of the structure. Extensive discussions of this subject have been provided by International Commission on Large Dams (ICOLD, 1987), Toso and Bowers (1988), and Visher and Hager (1995), and so on. The hydromechanic characteristics and the turbulence level of the jump in a stilling basin depends not only on the relative tailwater level but also on the geometry and the concrete finish conditions of the basin floor and training walls. The pressure fluctuations resulting from intense macro-scale turbulence in the jump must be carefully considered during the design of the structure. The pressure fluctuations vary widely in amplitude at all locations within the jump. The maximum halfamplitude of the fluctuation has been determined to be approximately 40 percent of the mean approach velocity head with a frequency of about 1 Hz. The dominant pulsating components have frequencies between 0 and 10 Hz. When the pressure becomes negative
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HYDRAULIC DESIGN OF STILLING BASINS AND ENERGY DISSIPATORS
18.24
Chapter Eighteen
at a point on the apron surface, a dangerous local instability may develop with respect to the uplift pressure at the bottom of the concrete slab. Some projects have experienced high uplift pressures under large areas of the basin floor and resulted in complete floor concrete slabs being torn up (Bowers and Toso, 1988; ICOLD, 1987; Toso and Bowers, 1988). In addition, cavitation, abrasion, and vibration due to intense turbulence and pressure fluctuations can also contribute significantly to the damage of a stilling basin. Based on the model studies of USBR Type II and Type III stilling basins, Toso and Bowers (1988) obtained the following useful conclusions: 1. The pressure fluctuations in the jump tend to approach a definite limit, on the order of 80 to 100 percent of the approach velocity head. This is on the order of 10–20 times the root-mean square (rms) of the pressure fluctuation. 2. Addition of chute blocks, intermediate blocks, and end sills did not result in significantly higher maximum negative and positive deviations than those for basins without blocks and sills. The energy dissipation was quicker. 3. Side-wall pressure fluctuations are very significant, and peak at one to two inflow depths above the floor. 4. The longitudinal extent of extreme pressure pulsation in the zone of maximum turbulence is on the order of eight times the inlet flow depth. The lateral extent of a characteristic pulse is approximately 1.6 times the longitudinal extent or 13 times the inlet flow-depth. ICOLD (1987) recommended, as a minimum precaution, that the following two conditions be considered when designing the stilling basin apron: 1. Full downstream uplift pressure applied over the entire area of the floor with basin empty. 2. Full uplift pressure equals 12 percent of the approach velocity head applied under the whole basin, with the basin full. If necessary, the basin floor can be strengthened by providing anchors or using thicker slabs which may be held in place by the side walls. ICOLD (1987) also recommended following structural arrangements to minimize potential uplift damages due to undesirable turbulent flow induced pressure fluctuations. 1. All contraction joints should be fitted with properly located and embedded seals. 2. There should be no drain openings in the training wall inside the basin. However, drain outlets in a dentated sill at the beginning of a stilling basin have performed satisfactorily. 3. Keep the areas of the floor slabs as large as possible. 4. Connect slabs by means of dowels, shear keys, and reinforcement across the joints. 5. Keep horizontal construction joints to a minimum, with dowels across them. 6. If drainage is necessary, keep it well away (1–1.5 m at least) from the wetted surfaces so that abrasion or cavitation erosion will not make it accessible to the turbulent flow.
18.3 DROP-TYPE ENERGY DISSIPATORS For small drops in canals with values of the Froude number between 2.5 and 4.5, a drop–type energy dissipator which is in the form of a grating is particularly applicable to reduce wave actions and dissipating energy. The device causes the over falling water jet to separate into a number of long, thin sheets of water which falls nearly vertical into the canal below. It has excellent capability in dissipating energy and eliminating wave problems. Guidelines for developing a drop-type energy dissipator are given as follows (Peterka, 1964): Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
HYDRAULIC DESIGN OF STILLING BASINS AND ENERGY DISSIPATORS
Hydraulic Design of Stilling Basins and Energy Dissipators 18.25
1. The device is highly recommended for approach flow with the Froude number below 3.0. 2. The Froude number is computed at the top of the drop. 3. The dissipator consists of a series of steel rails, channel irons, or timber beams in the form of grating installed at the drop (Fig. 18.12). 4. The spacing beams may vary from 2/3 to the full width of the beams. The narrower spacing is more effective. Use the following expression to compute the length of beams: Q L CSN 兹 兹2苶苶gy苶 where Qtotal discharge (ft3/s or m3/s, C experimental coefficient (dimensionless), S width of a space in feet or meters, N the number of spaces, g the acceleration of gravity (ft/s2 or m/sec2), and y the depth of flow in the canal upstream (ft or m). The value of C is about 0.245. 5. The length of the beams varies from about 2.9 to 3.6 times the depth of the approach flow. 6. The rails or beams may be tilted downward at an angle of 3º or more to provide some self-cleaning capability. It may also be made adjustable and tilted upward to act as a check to maintain a certain level in the canal upstream. However, more frequent cleaning of the device may be required.
18.4 WAVE SUPPRESSORS A wave suppressor is used to provide greater wave reduction to a proposed structure or an existing waterway. Two types of wave suppressors may be considered. They are raft-type and underpass-type wave suppressors. Both are applicable to most open-channel waterways having rectangular, trapezoidal, or other cross-sectional shapes. Both types may be used without regard to the Froude number. The underpass-type suppressor provides greater degrees of wave reduction but may be less economical than the raft-type.
FIGURE 18.12 Drop-type energy dissipator for small drop canals.(From Pterka, 1964)
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HYDRAULIC DESIGN OF STILLING BASINS AND ENERGY DISSIPATORS
18.26
Chapter Eighteen
FIGURE 18.13 Raft-type wave suppressor. (From Peterka, 1964)
18.4.1 Raft-Type Wave Suppressors A number of rafts of different designs were tested by USBR (Peterka, 1964). The most effective raft arrangement was found to consist of two rigid stationary rafts 20 ft (6.10 m) long by 8 ft (2.45 m) wide, made from 6- by 8-in timbers, placed in the canal downstream from the stilling basin as shown in Fig. 18.13. The arrangement is also applicable for suppressing waves having a regular period such as wind waves or waves produced by operation of pumps. Guidelines for designing a raft–type wave suppressor are provided as follows: 1. A space should be left between timbers and lighter crosspieces are placed on the rafts parallel to the flow. It creates many open spaces resembling rectangular holes. 2. The rafts should be perforated in a regular pattern and there should be some depth to these holes. 3. The ratio of hole area to total area of the raft may vary from 1:6 to 1:8. 4. The 8 ft (2.5 m) width, W W, as shown in Fig. 18.13, is a minimum dimension. 5. The raft must have sufficient thickness so that the troughs of the waves do not break free from the underside. 6. At least two rafts should be used, and the rafts should be rigid and held stationary. 7. The top surfaces of the rafts are set at the mean water surface in a fixed position so that they cannot move. 8. Spacing between rafts should be at least three times the raft dimension, measured parallel to the flow. Each raft can decrease the wave height about 50 percent.
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HYDRAULIC DESIGN OF STILLING BASINS AND ENERGY DISSIPATORS
Hydraulic Design of Stilling Basins and Energy Dissipators 18.27
9. For suppressing waves having regular periods, the second raft should be placed downstream at some fraction of the wave length to maintain its effectiveness. It may be necessary to make the second raft portable for narrow canals.
18.4.2 Underpass-Type Wave Suppressors Based on numerous studies conducted, USBR determined that the most effective wave dissipator to be located downstream from a stilling basin is the short-tube type underpass wave suppressor (Peterka, 1964). When it becomes necessary to make the raft-type wave suppressors adjustable or portable, or a moderate increase in depth in the stilling basin can be tolerated, consideration should be given to the underpass-type wave suppressors. It may be added to an existing structure or included in the original construction. It can be used to prevent wave overtopping of the canal lining or bank erosion due to waves. The structure consists of a horizontal roof placed in the flow channel with a headwall sufficiently high to cause all flow to pass beneath the roof as shown in Fig. 18.14a. Three main factors should be considered when designing an underpass-type suppressor. They are the submergence of the roof, the length of the underpass, and the increase in flow depth upstream of the underpass. The following guidelines may be used to design an underpasstype suppressor: 1. The height of the roof above the channel floor may be set to reduce wave heights effectively for a considerable range of flows or channel stages. 2. The maximum wave reduction occurs when the roof is set 33 percent of the flow depth below the water surface for maximum discharge. The submergence and the percent reduction in wave height becomes less, in general, for smaller-than-maximum discharges. 3. Fig. 18.14c can be used to estimate the wave reduction. The upper curve shown in the figure was obtained from the study conducted for the short tube underpass wave suppressor of the Carter Lake Dam No.1 Outlet Works. The lower curve shows the model test results of the Friant-Kern Canal (Fresno, California) underpass type suppressor for less than maximum discharges with smaller wave heights and shorter periods. The wave period greatly affects the performance of a given underpass. The suppressor provides a greater percentage reduction on short period waves. The wave action below a stilling basin usually has no measurable period and the water surface is choppy and consists of generated and reflected waves. The waves found downstream from hydraulic jumps or energy dissipators usually have a period of not more than 5 s. There is a tendency for the wave period to become less with decreasing discharge. 4. The underpass is most effective when the velocity beneath the underpass is less than about 10 ft/s or 3 m/s and the channel length downstream from the underpass is three to four times the length of the underpass. 5. The minimum length of underpass required depends on the amount of wave reduction considered necessary. For nominal wave reduction to prevent canal lining overtopping or bank erosion due to waves, a length 1.0–1.5D2 will provide about 60 to 75 percent wave height reduction. For greater wave reduction, a longer underpass is necessary.
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HYDRAULIC DESIGN OF STILLING BASINS AND ENERGY DISSIPATORS
18.28
Chapter Eighteen
FIGURE 18.14 Underpass-type wave suppressor (From Peterka, 1964)
For wave periods up to about 5 s, an underpass 2.0–2.5D2 long may provide up to 88 percent wave reduction. Up to about 93 percent of wave height reduction can be achieved by using an underpass 3.5–4.0D2 long. This length includes a 4:1 sloping roof extending from the underpass roof elevation to the tail water surface. The sloping portion should not exceed one-quarter of the total underpass length and slopes flatter than 4:1 provide better draft tube action and are more desirable. 6. The greatest wave reduction occurs in the first D2 of underpass length, the construction of two short underpasses rather than one may be considered. An additional wave reduc-
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HYDRAULIC DESIGN OF STILLING BASINS AND ENERGY DISSIPATORS
Hydraulic Design of Stilling Basins and Energy Dissipators 18.29
tion of 10 percent may be achieved but the extra cost of an additional headwall should be considered. 7. The backwater effect of the underpass can be determined based on Fig. 18.14b. 8. For design purposes, pressures along the underside of the roof may be considered to be atmospheric. The average pressures on the headwall and the downstream vertical wall may be considered as hydrostatic.
18.5 IMPACT-TYPE STILLING BASIN FOR PIPE OR OPEN CHANNEL OUTLETS This is an impact-type energy dissipator equipped with a hanging-type ᑦ-shaped baffle, contained in a relatively small boxlike structure, which requires no tail water for successful performance (Fig. 18.15). The energy dissipation is accomplished by flow striking the vertical hanging baffle and being turned upstream by the horizontal portion of the baffle and by the floor, in vertical eddies, and is greater than in a hydraulic jump of the same Froude number. It may be used to substitute Basin IV-type stilling basin for low Froude
FIGURE 18.15 Basic design of an impact-type stilling basin (From Peterka, 1964)
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HYDRAULIC DESIGN OF STILLING BASINS AND ENERGY DISSIPATORS
18.30
Chapter Eighteen
FIGURE 18.16 Selection of width for an impact-type stilling basin. (From Peterka, 1964)
FIGURE 18.17 Comparison of energy losses – impact basin and hydraulic jump.(From Peterka, 1964)
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HYDRAULIC DESIGN OF STILLING BASINS AND ENERGY DISSIPATORS
Hydraulic Design of Stilling Basins and Energy Dissipators 18.31
number applications as discussed in Sec. 18.2.4. The impact-type stilling basin generally provides greater efficiency than that of a jump on horizontal floor (Fig. 18.17). Based on hydraulic model test results, generalized design rules and procedures have been developed by USBR (Peterka, 1964) and are given below to allow determination of the proper basin size and all critical dimensions for a range of discharges up to 339 ft3/s (9.6 m3/s) and velocities up to about 30 ft/s (9.1 m/s). 1. The use of the impact-type stilling basin discussed in this section is limited to installation where the velocity at the entrance to the stilling basin does not greatly exceed 30 ft/s (9.1 m/s). 2. The basin operates as well whether a small pipe flows full or a larger pipe flows partially full is used. An open channel having a width less than the basin width will perform equally well. 3. Determine the stilling basin dimensions using Figs. 18.15 and 18.16 and Table 18.1, Columns 3–13 for the maximum expected discharge. For discharges exceeding 339 ft3/s (or 10 m3/s), it may be more economical to consider multiple units side by side. 4. Compute the necessary pipe area from the velocity and discharge. The values in Table 18.1, Columns 1 and 2, are suggested sizes based on a velocity of 12 ft/s (3.7 m/s) and the desire that the pipe run full at the discharge given in Column 3. The relationship between discharge and basin size given in the table should be maintained regardless of the pipe size chosen. An open–channel entrance may be used in place of a pipe. The approach channel should be narrower than the basin with invert elevation the same as the pipe. 5. A moderate depth of tail water will improve the performance although tail water is not a key factor for successful operation. For best operation, set the basin so that maximum tail water does not exceed d g/2. 6. Recommended thickness of various parts of the basin are given in Columns 14-18, Table 18.1. 7. Determine the minimum size of individual riprap protective stones which will resist movement when critical velocity occurs over the end sill. Most of the riprap should consist of the sizes given in Table 18.1, Column 19 or larger. The following empirical equation, which was developed based on studies performed by Marvis and Laushey, and Berry as reported by USBR (Peterka, 1964), may also be used to determine the stone size with reasonable accuracy. 兹d 苶 Vb 2.6 兹 where Vb bottom velocity (ft/s), and d diameter of rock (in). The rock is assumed to have a specific gravity of about 2.65. The accuracy of the equation for velocities above 16 ft/s (4.9 m/s) is not known. 8. The entrance pipe or channel may be tilted downward about 15º without affecting performance adversely. For greater slopes use a horizontal or sloping pipe (up to 15º) two or more diameters long just upstream from the stilling basin. Proper elevation of the invert at entrance is maintained as shown in Fig. 18.15.
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19.63
28.27
60
72
5–6
21✝
339
236
191
151
115
85
59
16–6
14–3
13–0
11–9
10–6
9–3
8–0
6–9
(4)
(3)
38
W
Max. dis– charge Q
12–3
10–9
9–9
9–0
8–0
7–3
6–3
5–3
4–3
(5)
H
22–0
19–0
17–4
15–8
14–0
12–4
10–8
9–0
7–4
(6)
K
9–3
8–0
7–4
6–9
6–0
5–3
4–7
3–11
3–3
(7)
a
Impact–Type Stillling Basin Dimensions.
12–9
11–0
10–0
8–11
8–0
7–1
6–1
5–1
4–1
(8)
b
6–11
5–11
5–5
4–11
4–5
3–10
3–4
2–10
2–4
(9)
c
d
2–9
2–5
2–2
2–0
1–9
1–7
1–4
1–2
0–11
(10)
Feet and Inches
1–3`
1–0
1–0
0–10
0–10
0–8
0–8
0–6
0–6
(11)
e
Source: From Peterka (1964). Abbreviation: a, ;b, ;c, ;d, ;e, ;;ff, ;g, ;k, ;ttb, ;ttf, ;tp, ;ttw,. (see Fig. 18.15) *Suggested pipe will run full when velocity is 12 ft/sec or half full when velocity is 24 ft/s. Size be modified for other velocities by Q AV, V but relation between Q and basin dimensions shown must be manteined. ✝For discharge less than 21 ft/s, obtain width from curve of Fig. 18.14. Other dimensions proportional to W; H 3W/4, W L 4W/3, W dW W/6, etc. Use curve of Fig. 18.21 to determine riprap size.
15.90
54
7.07
36
9.62
4.91
30
12.57
3.14
24
48
1.77
18
42
Area (ftt3) (2)
Dia (In) (1)
Suggested Pipe Size*
TABLE 18.1
3–0
3–0
3–0
3–0
3–0
3–0
2–6
2–0
1–6
(12)
f
6–2
5–4
4–11
4–5
3–11
3–6
3–0
2–6
2–1
(13)
g
12
11
10
9
8
7
6
6
6
(14)
tw
11
10
10
9
8
7
6
6
(16)
tb
121/2 121/2
111/2
101/2
91/2
81/2
71/2
61/2
61/2
61/2
(15)
tf
8
8
8
8
8
8
7
6
6
(17)
tp
Inches
6
6
4
4
4
3
3
3
3
14.0
13.0
12.0
10.5
9.5
9.0
8.5
7.0
4.0
Suggested Riprap Size (18) (19)
K
HYDRAULIC DESIGN OF STILLING BASINS AND ENERGY DISSIPATORS
18.32
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HYDRAULIC DESIGN OF STILLING BASINS AND ENERGY DISSIPATORS
Hydraulic Design of Stilling Basins and Energy Dissipators 18.33
9. The invert of the entrance pipe, or open channel, should be held at the elevation in line with the bottom of the baffle and the top of the end sill, regardless of the size of the pipe selected. 10. If a hydraulic jump is expected to form in the downstream end of the pipe and the pipe is sealed by the incoming flow, install a vent about one-sixth the pipe diameter at any convenient location upstream from the jump. 11. For the best possible operation of basin, use an alternative end sill and 45º wall design as shown in Fig. 18.15. Erosion tendencies will be reduced.
18.6 BAFFLED APRON FOR CANAL OR SPILLWAY DROPS (BASIN IX) Baffled aprons or chutes have been used in many irrigation projects for being practical and economical. The chute is constructed on an excavated slope, 2:1 or flatter, extending to below the channel bottom. The multiple rows of baffle piers on the chute prevent excessive acceleration of the flow and provide a reasonable terminal velocity. Initial tailwater is not a prerequisite for the structure to be effective. Backfill is placed over one or more rows of baffles to restore the original streambed elevation. It prevents excessive acceleration of the flow entering the channel when scour or downstream channel degradation occur. Through extensive model studies, the hydraulic design of the energy dissipators
FIGURE 18.18 Basic design of a baffled chute (From Peterka, 1964)
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HYDRAULIC DESIGN OF STILLING BASINS AND ENERGY DISSIPATORS
18.34
Chapter Eighteen
FIGURE 18.19 Recommended baffle pier heights and allowable velocities for baffled chutes (From Peterka, 1964)
with baffled aprons have been generalized (Peterka, 1964). Basic proportions of a baffled chute are given in Fig. 18.18 and a simplified design procedure has been developed and is outlined as follows: 1. The baffled apron should be designed for the maximum expected discharge, Q. 2. The unit discharge q Q/W may be as high as 60 ft3/s/ft [or 5.6 m3/s/m] based on chute width, W. W 3. Approach velocity, V1, should as low as practical. Use recommended approach velocity (Curve D) shown in Fig. 18.19. 4. The vertical offsets between the approach channel floor and the chute is used to create a stilling pool or desirable V1 and will vary in individual installations. See Fig. 18.20 for examples of approach pool arrangements. Use a short-radius curve to provide a crest on the sloping chute. Place the first row of baffle piers close to the top of the chute no more than 12 inches or 30 cm in elevation below the crest. 5.
Use the recommended height for baffled pier Curve B, Fig. 18.19.
6.
Baffle pier widths and spaces should be equal and about 1.5 H but not less than H. Partial blocks, width 1/3 H to 2/3 H, should be placed against the training walls in Rows 1, 3, 5, 7, and so forth, alternating with spaces of the same width in Rows 2, 4, 6, and so on.
7.
The slope distance (along a 2:1 slope) between rows of baffle piers should be 2H, twice the baffle height H. When the baffle height is less than 3 ft (or 91.5 cm), the row spacing may be greater than 2 H but should not exceed 6 ft or 183 cm.
8.
The baffle piers may be constructed with their upstream faces normal to the chute surface.
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HYDRAULIC DESIGN OF STILLING BASINS AND ENERGY DISSIPATORS
Hydraulic Design of Stilling Basins and Energy Dissipators 18.35
FIGURE 18.20a Examples of existing baffled chute designs (From Peterka, 1964)
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HYDRAULIC DESIGN OF STILLING BASINS AND ENERGY DISSIPATORS
18.36
Chapter Eighteen
FIGURE 18.20a Examples of existing baffled chute designs (From Peterka, 1964)
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HYDRAULIC DESIGN OF STILLING BASINS AND ENERGY DISSIPATORS
Hydraulic Design of Stilling Basins and Energy Dissipators 18.37
FIGURE 18.20b Examples of existing baffled chute designs (From Peterka, 1964)
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HYDRAULIC DESIGN OF STILLING BASINS AND ENERGY DISSIPATORS
18.38
9.
Chapter Eighteen
Four rows of baffle piers are required to establish full control of flow. The chute should be extended to below the normal downstream channel elevation and at least one row of baffles should be buried in the backfill.
10. The chute training wall should be three times as high as the baffle piers to contain the main flow and splash. 11. Riprap consisting of (6 to 12-in) (15 to 38-cm) stone should be placed at the downstream ends of the training walls to prevent eddies from working behind the chute. The riprap should not extend appreciably into the flow area.
18.7 RIPRAP FOR STILLING BASIN DOWNSTREAM PROTECTIONS Riprap stones are placed on the channel bottom and bank downstream of a stilling basin to prevent bank erosion caused by surges and residual energy from the stilling basin to reduce the possible undermining of the structure by the erosive currents. Factors affecting design of the riprap include size or weight of the individual stones, the shape of the large stones, the gradation of the entire mass of riprap, the thickness of the layer, the type of filter or bedding material placed beneath the riprap, the slope of the riprap layer, velocity and direction of currents, and eddy action and waves, etc. Based on published material, laboratory observations and field experience, a design curve (Fig. 18.21) was developed for the determination of the individual stone size to resist a range of velocities (Reference 3). Use the estimated bottom velocity or the average velocity at the end sill of the stilling basin to find the maximum stone size in Fig. 18.21. Specify riprap so that most of the graded mixture consists of this size. Place the riprap in a layer at least 1.5 times as thick as the maximum stone size. It is recommended that the riprap be placed over a filter, or bedding, composed of gravel or graded gravel having the larger particles on the surface.
18.8 SUBMERGED DEFLECTOR BUCKETS There are occasions that it is desirable to deliver the spillway discharge directly to the river without additional streambed protection works, the jet may be projected beyond the structure by a deflector bucket which acts as an energy dissipator at the base of a steep open chute spillway. USBR had developed both slotted and solid deflector buckets (Fig. 18.25) for high, medium, and low dam spillways. Both types require a greater depth of tailwater than a hydraulic jump stilling basin. However, the hydraulic action and the resulting performance of the two buckets are different. In general, the slotted bucket is an improvement over the solid type, particularly for lower ranges of tail water depths. USBR (Peterka, 1964) developed a simplified seven-step design procedure for the slotted bucket as follows: 1. Determine Q, q (per foot or meter of bucket width), V1, D1; compute Froude number from F V1/(g D1)1/2 for maximum flow and intermediate flows. 2. Enter Fig. 18.22 with F to find bucket radius parameter R/(D1 V12/2g) from which minimum allowable bucket radius, R, may be computed.
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HYDRAULIC DESIGN OF STILLING BASINS AND ENERGY DISSIPATORS
Hydraulic Design of Stilling Basins and Energy Dissipators 18.39
FIGURE 18.21 Curve to determine maximum stone size in riprap mixture. (From Peterka, 1964)
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HYDRAULIC DESIGN OF STILLING BASINS AND ENERGY DISSIPATORS
18.40
Chapter Eighteen
FIGURE 18.22 Minimum allowable bucket radius for slotted and solid buckets. .(From Peterka, 1964)
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HYDRAULIC DESIGN OF STILLING BASINS AND ENERGY DISSIPATORS
Hydraulic Design of Stilling Basins and Energy Dissipators 18.41
FIGURE 18.23 Minimum tail waterlimit for slotted and solid buckets..(From Peterka, 1964)
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HYDRAULIC DESIGN OF STILLING BASINS AND ENERGY DISSIPATORS
18.42
Chapter Eighteen
FIGURE 18.24 Maximum tail water limit for slotted and solid buckets. .(From Peterka, 1964)
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HYDRAULIC DESIGN OF STILLING BASINS AND ENERGY DISSIPATORS
Hydraulic Design of Stilling Basins and Energy Dissipators 18.43
FIGURE 18.25 Examples of submerged bucket designs. .(From Peterka, 1964)
FIGURE 18.26 Average water surface profiles for submerged buckets. (From Peterka, 1964)
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HYDRAULIC DESIGN OF STILLING BASINS AND ENERGY DISSIPATORS
18.44
Chapter Eighteen
FIGURE 18.27 Water surface profile characteristics for slotted buckets (From Peterka, 1964).
3. Enter Fig. 18.23 with R/(D1 V12/2g) and F to find Tmin/D / 1 from which minimum tailwater depth limit Tmin, may be computed. 4. Enter Fig. 18.24 as in Step 3 above to find maximum tailwater depth limit, Tmax. 5. Set bucket invert elevation so that tail water curve elevations are between tailwater depth limits determined by Tmin and Tmax. Keep apron lip and bucket invert above riverbed, if possible. For best performance, set bucket so that the tailwater depth is near Tmin. Check factor of safety against sweep out. 6. Complete the design of the bucket, using Fig. 18.25 to obtain tooth size, spacing, dimensions, and so on. 7. Use Figs. 18.26 and 18.27 to estimate the water surface profile in and downstream from the bucket.
18.9 FLIP BUCKETS Flip bucket or ski-jump energy dissipators are often used in association with high overflow dams to reduce the project cost when spray from the jet can be tolerated and the erosion by the plunging jet can be controlled. Most of the energy is dissipated when the jet plunges into the tailwater. Factors affecting the horizontal throw distance from the bucket lip to the point of jet impact are the exit velocity of the jet at the bucket lip, the bucket lip
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HYDRAULIC DESIGN OF STILLING BASINS AND ENERGY DISSIPATORS
Hydraulic Design of Stilling Basins and Energy Dissipators 18.45
angle, and the difference in elevation between the lip and the tailwater. With the origin of the coordinates taken at the lip of the bucket, the trajectory of the jet may be expressed by the following equation:
(a)
(b)
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HYDRAULIC DESIGN OF STILLING BASINS AND ENERGY DISSIPATORS
18.46
Chapter Eighteen
(c) Exhibit 18.3: Strontia springs project, Colorado (Courtesy Denver Water Department, Denver, Colorado) (a) General view of the spillway with low-level-outlet-work in operation. (b) General view of the spillway with low-level-outlet-work in operation. (c) Close-up view of the spillway in operation showing free trajectory and impact at the plunge pool.
(a)
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HYDRAULIC DESIGN OF STILLING BASINS AND ENERGY DISSIPATORS
Exhibit 18.4 Mossyrock Hydroelectric Project, Mossyrock, Washington (a) A view of the spillway in operation showing free tajectory and imact at the plunge pool. (b) Layout of the dam showing spillway, plunge pool, power intakes, power house and diversion tunnels.
(b)
Hydraulic Design of Stilling Basins and Energy Dissipators 18.47
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HYDRAULIC DESIGN OF STILLING BASINS AND ENERGY DISSIPATORS
18.48
Chapter Eighteen
FIGURE 18.28 Flip bucket and toe curve pressures. (From USACE, 1998)
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HYDRAULIC DESIGN OF STILLING BASINS AND ENERGY DISSIPATORS
Hydraulic Design of Stilling Basins and Energy Dissipators 18.49
x2 y x tan θ K[4 (d hv) cos2 θ] where θ angle of edge of lip with horizontal, K factor usually assumed as 0.9 to compensate for loss of energy, d depth of water on bucket, hv velocity head of jet at the lip of the bucket In general, the exit angle at the lip should not exceed 30º and the minimum radius of curvature should not be less than 5 times the depth of water on the bucket. The pressure distribution on spillway flip buckets associated with high–overflow dams can be estimated based on the Corps of Engineers test data (USACE, 1988) as shown in Fig. 18.28. For design purposes, allowance for spillway energy losses should be included in the computation of the energy head, HT at the invert of the bucket. A discussion of the plunge pool hydrulics including scour depth and jet diffusion is given in section 17.3. Photos of several flip bucket type energy dissipators are given in Exhibits 17.2, 17.3, 17.11, 18.3, and 18.4
18.9.3 Gas Supersaturation Gas supersaturation problems occur at dams with spillways designed with deep plunge pools and with deep stilling basins that operate submerged hydraulic jumps. When spilled water with entrained air plunges to depths where the pressures can significantly exceed one atmosphere, the flow becomes supersaturated with gasses. Fish exposed to these gas supersaturated conditions develop gas emboli in the tissues. This condition known as gas bubble disease, cause injury to the fish, and can result in death.. When considering the selection and design of an energy dissipator for use in a dam project, gas supersaturation must be considered. Deflectors that direct discharges along the surface and energy dissipating devices that disperse the flow to reduce the depth of the plunge, such as Howell-Bunger valves, are considerations. Stilling basins that are designed for high unit discharges, but primarily operate for lower discharges often have deep basins and excess tailwater depths for the lower discharges. In this condition the hydraulic jump is submerged, with the flow plunging to the bottom of a deep pool in the stilling basin. In large spillways these conditions can cause supersaturation. In situations where it is not practical to use alternative energy dissipators or design the spillway and stilling basin with lower unit discharges, it may be necessary to divide the spillway and stilling basin with walls. This permits operation of a portion of the structure at higher unit discharge for lower releases, thus effectively reducing the tailwater excess.
18.9.4 Abrasion in Stilling Basins Many stilling basins are subject to at least some wear due to abrasion from material that gets washed into the basin and circulates in contact with the concrete surfaces with the flow. To minimize problems due to abrasion, stilling basins should be operated with uniform discharge. Spillways with crest gates should be operated with all gates opened equally to avoid recirculation in the stilling basin. When only one or a few gates are opened on a spillway with a wide stilling basin, circulation patterns develop in the spillway which can transport streambed material from downstream into the stilling basin. It is necessary for the designer to consider all conditions under which the spillway and energy
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HYDRAULIC DESIGN OF STILLING BASINS AND ENERGY DISSIPATORS
18.50
Chapter Eighteen
dissipator will operate. If nonuniform operation of gates is expected to be required to pass low discharges, consideration should be given to dividing the spillway and stilling basin with guide walls to provide a portion that could be used to pass low discharges without creating recirculation patterns in the stilling basin.
18.10 STILLING BASIN DESIGN EXAMPLES 18.10.1 Design Example 1 The crest of an overfall spillway is 200 ft (61 m) above the horizontal floor of the stilling basin and the slope of the spillway chute is 0.7:1. The head (H) H on the spillway crest is 30 ft (9.14 m) and the maximum unit discharge (q) is 480 (ft3/s/ft) [44.6 (m3/s/m] based on the the stilling basin width. Design a Type II stilling basin for these conditions (Peterka, 1964). Step 1. Determine approach conditions including velocity (V V1) of flow entering the basin. a. Compute the total distance from the reservoir level to the basin floor (total fall) Z. Z head on the crest (H) H + vertical distance from crest to basin floor 30 200 230 ft (70.1 m) b. Entering Fig. 18.1 with Z ( 230 ft) and H ( 30 ft) and determine the ratio of VT) that is, actual velocity (V VA) versus theoretical velocity (V VA 0.92 VT c. Compute the theoretical velocity based on the equation given in Fig. 18.2. VT
冪2莦g莦莦2莦3莦0莦莦莦3莦20莦 117.6 ft/s (35.8 m/s)
d. Compute the actual velocity VA ( V1 of the jump) and the corresponding depth D1 and the approach Froude number F1. V1 VA 117.6 0.92 108.2 ft/s (33.0 m/s) q 480 4.44 ft (1.35 m) D1 108.2 V1 F1
V1 108.2 9.04 兹g苶苶 兹 兹苶 D1苶 兹 兹3苶苶2苶 .2苶 苶4苶 .44苶
Step 2. Set basin apron elevation: a. Determine tailwater depths. Entering Fig. 18.3b with the Froude number (F1) of 9.04, the heavy dashed line for TW/ W/D2 1.0 gives
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HYDRAULIC DESIGN OF STILLING BASINS AND ENERGY DISSIPATORS
Hydraulic Design of Stilling Basins and Energy Dissipators 18.51
TW/ W/D1 12.3 b. Compute D2. D2 TW 12.3 D1 12.3 4.44 54.6 ft (16.6 m) c. Check factor of safety (FS) with the minimum tailwater depth required as given in Fig. 18.3b. For F1 9.04, TW Wmin/D / 1 11.85, TW Wmin 11.85 4.44 52.6 ft (16.0 m) FS (TW TW Wmin)/D / 2 (54.6 52.6) / 54.6 4.0 percent 5 percent (recommended minimum margin of safety) To satisfy 5 percent minimum margin of safety: Use TW TW Wmin 0.05 D2 52.6 0.05 54.6 55.3 ft (16.9 m) Reposition the stilling basin apron accordingly. Step 3. Check the effectiveness of the stilling basin: F1 9.04 4.0 The jump should be fully developed for effective energy dissipation. Step 4. Determine the basin length: a. Entering Fig. 18.3c with F1 9.04 and determine the corresponding value of L/ L/D2. L 4.28 D2 b. LII L 4.28 54.6 234 ft (71.3 m) Step 5. Determine chute block height, width, and spacing. Referring to Fig. 18.3a, the recommended height, width, and spacing of the chute block is D1. Height width spacing D1 4.44 ft 4 ft 5.3 in (use 4 ft 6 in or 1 m 35 cm) Step 6. Determine the height, width, and of the dentated sill based on the recommended dimensions shown in Fig. 18.3a. a. Height 0.2D2 0.2 54.6 10.92 ft (use 11 ft or 3 m 33 cm) b. Width spacing 0.15D2 0.15D2 0.15 54.6 8.19 ft (2.50 m) (use 8 ft 3 in or 2 m 50 cm)
18.10.2 Design Example 2 In this example (Peterka, 1964), the dimensions of the Type III stilling basin of a small dam are to be determined. The width of the basin is 50 ft (15.24 m) and the flow is symmetrical. Based on the design of the spillway, values of V1 and D1 for the range of discharges to be considered have been determined and are given as follows:
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HYDRAULIC DESIGN OF STILLING BASINS AND ENERGY DISSIPATORS
18.52
Chapter Eighteen
FIGURE 18.29 Tailwater and jump elevation curve for design example 2. (From peterka, 1964)
Q ftt3/s (m3/s)
q ftt3/s/ft [(m3/s/m]
V1 ft/s (m/s)
D1 ft/s (m/s)
3,900 (110.5)
78.00 (7.25)
69.0 (21.0)
1.130 (0.344)
3,090 ( 87.5)
61.80 (5.74)
66.0 (20.1)
0.936 (0.285)
2,022 ( 57.3)
40.45 (3.76)
63.0 (19.2)
0.642 (0.196)
662 ( 18.7)
13.25 (0.87)
51.0 (15.5)
0.260 (0.079)
Resulting from a backwater analysis of the downstream channel, the tailwater rating curve is also available as shown in Fig. 18.29. The tailwater elevation for 3900 ft3/s (110.5 m3/s) is at elevation 617.50 ft (188.22 m). Step 1. Compute the “jump elevation curve.” a. Compute F1 based on given V1 and D1 values. (computed F1 values are shown in the table below) b. Determine D2 by entering Fig. 18.4b with the computed values of F1. c. Assume the most adverse operating condition occurs at the maximum discharge of 3900 ft3/s (110.5 m3/s) and set the apron elevation accordingly. D2 17.8 ft (5.43 m)
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HYDRAULIC DESIGN OF STILLING BASINS AND ENERGY DISSIPATORS
Hydraulic Design of Stilling Basins and Energy Dissipators 18.53
Apron elev. 617.5 17.8 El. 599.7 ft 188.22 5.43 El. 182.8 m d. Compute jump elevations for the remaining three discharges as shown in the table below for “jump elevation Curve a.” Q ftt3/s
F1
D1 ft
D2 / D1
D2 ft
Jump Elev. Curve a, ft
Jump Elev. Curve a', ft
3900 3090 2022 662 or
11.42 12.02 13.85 17.62
1.130 0.936 0.642 0.260
15.75 16.60 19.20 24.50
17.80 15.54 12.33 6.37
617.5 615.2 612.0 606.1
615.0 612.7 609.5 603.6
Q m3/s
F1
D1 m
D2 / D1
D2 m
Jump Elev. Curve a, m
Jump Elev. Curve a', m
110.5
11.42
0.344
15.75
5.43
188.22
187.45
87.5
12.02
0.285
16.60
4.74
187.52
186.75
57.3
13.85
0.196
19.20
3.76
186.54
185.78
18.7
17.62
0.079
24.50
1.94
184.74
183.98
Step 2. Compare the jump elevation curve with the tailwater rating curve as shown in Fig. 18.29. It indicates tailwater depth deficiency for smaller discharges especially at approximately 2850 ft3/s (80.7 m3/s) where the curvature of the tailwater rating curve is concave upward. Step 3. Shift the apron elevation curve downward such that the full conjugate depth is realized at the most adverse 2850 ft3/s (80.7 m3/s) tailwater condition. A downward shift of 2.5 ft (0.76 m) is required as indicated by “jump elevation Curve a'” in Fig. 18.29 and the accompanying table. Step 4. Reset the apron elevation: Apron elev. 599.7 2.5 El. 597.20ft 182.79 0.76 El. 182.03 m Step 5. Determine the remaining stilling basin details based on the maximum discharge of 3900 ft3/s (110.5 m3/s). Step 6. Determine basin length based on conjugate depth: Entering Fig. 18.4c with F1 11.42.
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HYDRAULIC DESIGN OF STILLING BASINS AND ENERGY DISSIPATORS
18.54
Chapter Eighteen
LIII 2.75 D2 The basin length LIII 2.75 17.80 48.95 ft (14.9 m) Step 7. Determine the height, width, and spacing of chute blocks in accordance with Fig. 18.4a. h1 W1 S1 1.0D1 1.13 ft (use 13 or 14 in) (35 cm) Step 8. Determine height of the baffle piers in accordance with Fig. 18.4d. h3 2.5D1 2.5 1.13 2.825 ft (use 34 in) (86 cm) Step 9. Compute the spacing of the baffle piers as 0.75h3. Baffle pier spacing 0.75 34 25.5 in (65 cm) Step 10.Compute the distance between the baffle piers and the chute blocks as 0.8D2. Distance 0.8 17.8 14.24 ft (4.34 m) Step 11.Compute the height of the solid end sill h4 based on Fig. 18.4d. h4 1.60D1 1.60 1.13 1.81 ft (use 22 in) (55 cm) The final dimensions of the Type III stilling basin are shown in Fig. 18.29.
REFERENCES Blaisdell, F. W., “Develop and Hydraulic Design—Saint Anthony Falls Stilling Basin,” Transactions, ASCE, 113, P.334 1948. Bowers, C. E., and J. W. Toso, “Karnafuli Project, Model Studies of Spillway Damage,” Journal of Hydraulic Engineering, ASCE, 114 (5), 1988. Bowers, C. E., and F. Y. Tsai, “Fluctuating Pressures in Spillway Stilling Basins,” Journal of Hydraulic Engineering, ASCE, 95 (HY6), 1969. Chadwick, A. J., and J. C. Morfett, Hydraulics in Civil Engineering, Allen & Unwin, London, 1986. Chaudhry, M. H., Open-Channel Flow, Prentice-Hall, Englewood Cliffs, NJ, 1993. Chow, V. T., Open-Channel Hydraulics, McGraw-Hill, New York, 1959. French, R. H., Open-Channel Hydraulics, McGraw-Hill, New York, 1985. George, R. L., Low Froude Number Stilling Basin Design, REC-ERC-78-8, U.S. Bureau of Reclamation, 1978. Henderson, F. M., Open Channel Flow, Macmillan, New York, 1966. International Commission on Large Dams (ICOLD), Spillways for Dams, Bulletin 58, ICOLD, Paris, 1987. Novak, P., A. I. B. Moffat, C. Nalluri, and R. Narayanan, Hydraulic Structures, Unwin Hyman, London, 1990. Peterka, A. J., Hydraulic Design of Stilling Basins and Energy Dissipators, Engineering
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HYDRAULIC DESIGN OF STILLING BASINS AND ENERGY DISSIPATORS
Hydraulic Design of Stilling Basins and Energy Dissipators 18.55 Monograph No. 25, U.S. Bureau of Reclamation, Denver, Co, 1964. Roberson, J. A., J. J. Cassidy, and M. H. Chaudhry, Hydraulic Engineering, Houghton Mifflin, Boston, 1988. Senturk, F., Hydraulics of Dams and Reservoirs, Water Resources Publications, Highlands Ranch, COl 1994. Toso, J. W., and C. E. Bowers, “Extreme Pressures in Hydraulic-Jump Stilling Basins,” Journal of Hydraulic Engineering, ASCE, 114 (8), 1988. U.S. Army Corps of Engineers (USACE), Hydraulic Design Criteria, U.S. Army Corps of Engineers Waterways Experiment Station, Vicksburg, MS, 1988. U.S. Bureau of Reclamation (USBR), Small Canal Structures, U.S. Bureau of Reclamation, Denver, CO, 1974. U.S. Bureau of Reclamation (USBR), Design of Small Dams, U.S. Bureau of Reclamation, Denver, CO, 1987. Vischer D. L., and W. H. Hager, Energy Dissipators—Hydraulic Design Considerations, IAHR Hydraulic Structures Design Manual No. 9, A. A. Balkema, Rotterdam, Netherlands, 1995. Vischer D. L., and W. H. Hager, Dam Hydraulics, John Wiley & Sons, New York, 1998.
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Source: HYDRAULIC DESIGN HANDBOOK
CHAPTER 19
FLOODPLAIN HYDRAULICS Roy D. Dodson Dodson & Associates, Inc. Houston, Texas.
This chapter deals with the practical considerations involved in identifying the floodplain for a stream channel, and analyzing the flow characteristics of the channel system, including its floodplain. In general, any land area that is susceptible to inundation by rising or flowing flood waters from any source could be considered to be a floodplain area. This would include areas affected by coastal or lacustrine (lake) flooding. However, the emphasis of this chapter will be on riverine floodplains, which are affected by flood waters from a stream or river. The determination of flood elevations and floodplain boundaries along stream channels has become increasingly important as more development has occurred in floodplain areas, or in areas that have been reclaimed from the floodplain. A floodplain analysis requires a large amount of data in order to be accurate and complete. Cross sections of the stream channel and detailed geometric descriptions of bridges and other structures are required. In addition, experience is usually required to accurately assess the roughness characteristics of a channel, to properly lay out cross sections, and to adequately address many other aspects of a good flood plain analysis. Sophisticated computer programs are available to analyze the data and produce detailed computations of water surface elevations, floodplain boundaries, and other results. However, these programs can produce misleading results without a properly planned and executed analysis. Because of the subjective judgments required for some of the input data and the uncertainty associated with certain values such as flow rates, some people have the attitude that "there is no right answer" to the question of determining floodplain elevations and boundaries. However, it is important not to allow this to degenerate into the attitude that "one answer is as good as another." This is clearly not true. A floodplain study performed well produces superior results. This chapter describes an approach toward floodplain studies that emphasizes quality control and thoroughness in all aspects of the study effort. The effort is focused on those aspects of the study that will produce the best overall results within given constraints of time and budget. This chapter provides guidance in the following areas: • Sources of information for the floodplain analysis; • Planning data collection operations to obtain required information; • Selecting the analytical approach for a particular floodplain situation; • Performing the analysis and assessing the results. 19.1 Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
FLOODPLAIN HYDRAULICS
19.2
Chapter Nineteen
19.1 LOCATING EXISTING DATA SOURCES FOR FLOODPLAIN STUDIES The first step in any floodplain analysis is to collect all available information that might be useful in the analysis. Available mapping and/or cross section data should be pursued before requesting new surveys. One or two weeks of contacting key government agencies, as well as local engineers, surveyors, aerial photographers, and mapping companies could save significant project funds. The principal sources of information which may be available include previous studies, topographic data, aerial photography, highway or street maps, construction drawings, stream gage data, and personal observations from local residents. Within the United States, federal agencies should be contacted to determine if they have data pertinent to the project being initiated. Table 19.1 lists the chief types of floodplain
✓
Bureau of Land Management (BLM)* Bureau of Reclamation† Federal Emergency
Floodplain
Storm Surge Data
Flood
Historic Floods
Streamflow Data
Meteorologic Data
Federal Government Data Sources Hydrologic Data
TABLE 19.1
✓ ✓
✓ ✓
✓
✓
✓
✓ ✓
✓ ✓
✓ ✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
Management Agency (FEMA)‡ National Weather Service Natural Resources
✓ ✓
✓
✓
✓
✓
Conservation Service (NRCS)|| Tennessee Valley Authority
✓
(TVA)
§
U.S. Army Corps of
✓
✓
Engineers (USACE) U.S. Geological Survey
✓
(USGS) *The BLM operates in the states of Alaska, Arizona, California, Colorado, Idaho, Michigan, Nevada, Oregon, Vermont, and Wyoming. †The Bureau of Reclamation operates only in the states of Arizona, California, Colorado, Idaho, Kansas, Montana, Nebraska, North Dakota, New Mexico, Nevada, Oregon, Oklahoma, South Dakota, Texas, Utah, Washington, and Wyoming. ‡FEMA Flood Insurance Studies are an especially important source of information, as described below. ||The NRCS was formerly called the Soil Conservation Service (SCS). §The TVA operates only in the states of Alabama, Georgia, Kentucky, Mississippi, North Carolina, Tennessee, and Virginia.
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FLOODPLAIN HYDRAULICS
Floodplain Hydraulics 19.3
study data typically available from each of the major federal government agencies. In addition, local and state agencies often have study data available. The types of information which may be derived from previous studies include • Surveyed cross section data • Data on channel structures such as bridges and culverts • Survey benchmarks used in previous field survey work • Computed or measured flow rates • Accounts of previous flooding, including high-water marks An assessment of the usability and technical accuracy should be made of all available information, including historical hydrologic data, high-water marks, flooding problems within the community, flood control measures, hydraulic structures that affect flooding, available community maps showing and naming all roads in the floodplains, topographic maps, digital data files, and elevation control data (including consideration of land subsidence* where applicable). Photographs of past major floods, if available, should be obtained. Within the United States, federal flood insurance studies are often valuable sources of information on floodplain hydraulics. It is always advisable to obtain the data from previous flood insurance studies whenever it is available. In fact, if a study is being prepared for submittal to the Federal Emergency Management Agency (FEMA) to replace or revise an existing flood insurance study, the data from the existing flood insurance study must be used as much as possible. Flood insurance study information may include hydrologic and hydraulic models, engineering and construction plans, floodplain maps, and flood profiles. In addition, any information should be obtained that may provide data for evaluating changes to the effective hydrologic or hydraulic models. Flood insurance study data may be requested from FEMA. If the original hydraulic models and work maps used in the Flood Insurance Study are not available, FEMA requires that the flood insurance study results be recreated from the information in the flood insurance study report book and the Flood Insurance Rate Maps. The data available from the published Flood Insurance Study report should be supplemented as needed and entered into the same hydrologic and/or hydraulic model used to create the original Flood Insurance Study. This model should then be calibrated to the Flood Insurance Study flood profiles to obtain 100-year water-surface elevations within an 0.1-ft tolerance, if possible. FEMA requires this to ensure a logical transition between revised and unrevised data.
19.1.1 Sources of Topographic Data Topographic maps for the area should be obtained whenever available. Although local and state governments sometimes compile detailed topographic maps, the most widely available topographic maps within the United States are the U.S. Geological Survey's (USGS) 7.5- minute or 15-minute quadrangle maps. USGS quadrangle maps may provide all the following information:
Land subsidence is the sinking or settling of land to a lower level in response to various factors, both natural and of human orign, such as earth movements, lowering of fluid pressure (or lowering of ground water level), removal of underlying supporting materials by mining or solution of solids, either artificially or from natural causes, compaction caused by wetting, oxidation of organic matter in soils, or added load on the land surface. *
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FLOODPLAIN HYDRAULICS
19.4
Chapter Nineteen
1. Survey control points. The locations of permanent vertical and horizontal control points (monuments or benchmarks) are indicated on the maps. 2. Topography. Topography is prepared to national standards and the accuracy of each map is keyed to the contour interval. The contour interval will vary depending on the amount of relief at the individual locations. 3. Ground cover. The color coding indicates the general type of ground cover, which may be useful in planning survey operations and in confirming estimates of roughness coefficients. 4. Development features. The locations of road crossings, developments, and other features are indicated on the maps. In the United States, the USGS is actively converting existing maps to digital form. Planimetric information is represented using digital raster graphics (DRG), digital orthophoto quads (DOQ), or digital line graphs (DLG). A digital raster graphic is a carefully scanned image of a USGS topographic map, including the borders. The map image is georeferenced to the surface of the Earth, so that most geographic information system (GIS) software can automatically position the DRG image correctly with respect to other types of geographic data. Figure 19.1 illustrates a portion of a USGS DRG file. A digital orthophoto quad is a digital image of an aerial photograph in which displacements caused by camera orientation and terrain have been removed. Orthophotos combine the image characteristics of a photograph with the geometric qualities of a map. The standard digital orthophoto produced by the USGS is a black-and-white or colorinfrared 1–m ground resolution quarter quadrangle (3.75-minute) image. Figure 19.2 illustrates a portion of a USGS DOQ File. Digital line graphs are vector files containing line data, such as roads and streams, digitized from USGS topographic maps. DLGs offer a full range of attribute codes, are highly accurate, and are topologically structured, which makes them ideal for use in GIS. A digital elevation model (DEM) is a digital file consisting of terrain elevations for ground positions at regularly spaced horizontal intervals. DEMs are developed from stereo models or digital contour line files derived from USGS topographic quadrangle maps. The
FIGURE 19.1 Example portion of USGS DRG file.
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FLOODPLAIN HYDRAULICS
Floodplain Hydraulics 19.5
FIGURE 19.2 Example of USGS DOQ file.
USGS produces five different digital elevation products. The most useful for floodplain studies consists of 7.5 minute ⫻ 7.5 minute blocks corresponding to the standard 1:24,000 scale USGS quadrangle maps. The data have a resolution (grid interval) of 30 m (approximately 100 ft). Figure 19.3 illustrates ground elevation contours computed using the data from a USGS DEM file.
derived from USGS DEM file.
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FLOODPLAIN HYDRAULICS
19.6
Chapter Nineteen
Canada and most Western European nations are undertaking similar programs for the development of digital elevation data from existing topographic data. Digital data are also being developed for other regions of the earth. Digital terrain data have two major applications to floodplain hydraulics. First, if the data are sufficiently detailed, they may be used as a basis for the channel and floodplain cross sections for hydraulic studies. This is emerging as a common practice when digital terrain models (DTMS) are available. However, the grid-based data available from a digital elevation model may not be suitable for use as channel cross section data, because the spacing between adjacent points (at least about 30 m for most DEM data) is not dense enough to provide a sufficiently detailed channel cross section. The use of DEM data for extending field-surveyed cross sections into floodplain areas is more promising, because less detail is generally needed in these areas. Digital topographic data may also be used for floodplain mapping. The results of a hydraulic model analysis only provides the floodplain boundaries at the locations where cross sections are available. Between these cross sections, the floodplain must be mapped using available topographic data. The availability of digital topographic data is dramatically improving the efficiency and accuracy of floodplain mapping, through links between GIS software and hydraulic analysis software. Figure 19.4 illustrates a stream channel for which two floodplains have been automatically calculated and displayed using software that links GIS software with the U.S. Army Corps of Engineers Hydrologic Engineering Center River Analysis System (HEC)-RAS water surface profiles computer program.
d and mapped using data from digita
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FLOODPLAIN HYDRAULICS
Floodplain Hydraulics 19.7
FIGURE 19.5 Example of USGS National Aerial Photography Project photograph.
19.1.2 Aerial Photography Aerial photography and topographic mapping may also be available from commercial aerial survey and mapping companies. Some local and state government agencies, such as transportation departments or taxing authorities, may obtain a complete set of aerial photographs on a regular basis. Aerial photography may also be available through the National Aerial Photography Project (NAPP), distributed through the USGS (Fig. 19.5). Up-to-date aerial photographs may indicate recent channel improvements, road crossings and other special structures, channel condition, land uses in the floodplain, and other information. It is important to note that areas and distances measured from aerial photographs may not be accurate unless the photograph has been orthorectified to correct for the effects of the camera angle, the curvature of the earth, and the apparent displacement of points resulting from differences in elevation at different locations in the photograph.
19.1.3 Highway or Street Maps Highway maps from state or local highway agencies or street maps from private map companies may contain information which may be missing from the USGS quadrangle maps, including recently constructed road crossings or other structures. Like topographic data and aerial photographs, highway and street data are also becoming available in digital form.
19.1.4 Construction Drawings Construction drawings may be available for channel modifications, bridges, or other projects affecting the stream channel or floodplain. Local or state transportation agencies and
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FLOODPLAIN HYDRAULICS
19.8
Chapter Ninetee
FIGURE 19.6 Example of digital street map (Census Bureau TIGER File).
railroad companies generally retain the construction drawings for all bridges and many other types of projects. As-built construction drawings are those that have been revised to reflect the actual dimensions of a completed construction project. As-built drawings for existing bridges are normally available from local or state transportation agencies. Local or federal agencies normally have as-built construction drawings for channel modification projects. The use of information from as-built construction drawings can greatly reduce the amount of survey information required for existing bridges or channel modifications. However, the survey benchmark used for each set of construction drawings should be determined, to be sure that all information is based on the same horizontal and vertical datum. 19.1.5 Stream Gage Data Stream gage data from stream gages operated by the USGS or other agencies can provide good estimates of the floodplain elevation at a particular point. Many floodplain analysis computer programs can accept a channel rating curve constructed from the stream gage data for use in computing the water surface elevation corresponding to a given flow rate. Figure 19.7 illustrates a channel rating curve. If the purpose of the analysis is to model a very extreme event, it is important to note that the rating curve available for the stream gage location may not have any historical basis for the stages at extremely high flow rates. These stages may simply be extrapolated from the historical record of lower flow rates.
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FLOODPLAIN HYDRAULICS
Floodplain Hydraulics 19.9
FIGURE 19.7 Channel rating curve.
19.1.6
Personal Observations
Personal observations by local residents, such as high-water marks, can be valuable in estimating the floodplain elevation for calibration of the computed water surface profile. Often the best observations are available from employees of city or county engineering departments, or other agencies with responsibilities for public facilities such as drainage channels, roads, or utility systems.
19.2 OBTAINING FIELD SURVEY DATA FOR FLOODPLAIN STUDIES Even though considerable information may be available from previous studies and other sources, it is often necessary to perform field surveys to provide additional information. Survey operations can be difficult to manage, but they often determine the success of a floodplain study. The following steps should be completed before field survey operations are initiated: 1. Channel stationing. The stream length should be accurately determined from the mouth (or other beginning point). Normally, this is done using a recent aerial photograph or other map which indicates the current channel morphology. Channel stationing should begin with 0 ⫹ 00 and increase in the upstream direction. Stationing should be measured along the thalweg.*
The thalweg is the line following the lowest part of a valley, whether under water or not. Usually this is the line following the deepest part of the bed or channel of a river. *
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FLOODPLAIN HYDRAULICS
19.10
Chapter Nineteen
2. Locate structures. The stream stations of all pertinent stream crossings, tributaries, and all other significant features should be summarized in table form, including those features found on only one map. The stream stations should be transferred to a topographic "work map" for the analysis. 3. Preliminary stream profile. If a stream profile showing the channel flow-line is not already available from a previous analysis, it may be useful to prepare a preliminary flowing profile from the contours on the topographic map. 4. Preliminary floodplain map. If a floodplain map is not already available from a previous analysis, the floodplain width at various points along the channel should be estimated. The estimate may be based on the readings of stream flow gages, preliminary computations using data from the topographic work map, or observations by local residents. For major studies, it may even be advisable to set up a preliminary hydraulic analysis using data from the topographic work map only. 5. Survey control points. The location and elevation of all known survey benchmarks in the area of the study should be indicated on the topographic work map. 6. Site reconnaissance. The engineer should then visit the study area, using the work map as a reference. Each road crossing and other structure should be inspected, photographed, and measured with a tape, if possible. Channel and floodplain conditions should be noted. Special notes and photograph locations should be recorded on the topographic work map. 7. Preliminary cross section locations. The proposed location and alignment of all surveyed cross sections should be indicated on the topographic work map. Finally, the engineer should meet with the survey coordinator to discuss the data requirements for the project, using the topographic work map, photographs, and the engineer's knowledge of the project area and computer program data requirements. Even after the survey work begins, the engineer should maintain frequent contact with those performing the survey work to be aware of their progress and any special problems.
19.2.1
Vertical and Horizontal Control for Field Surveys
The Global Positioning System (GPS), when used in the differential mode, is currently the best method for extending any survey control network unless satellite visibility is obscured (e.g., because dense forest) or severe radiofrequency disturbances are present. Differential GPS or third-order leveling* can be used to tie temporary benchmarks to an established datum; to determine the elevation of high-water marks; and, where needed, to establish horizontal and vertical control for aerial survey work. Whenever possible, available benchmarks should be used instead of field surveys, to reduce the survey cost. In particular, benchmarks shown on official floodplain maps should be used if possible. As a general rule, there should be approximately two temporary benchmarks per mile of stream length or four per square mile of floodplain, as appropriate (FEMA, 1995). Most benchmarks established in North America are referenced to the National Geodetic Vertical Datum of 1929 (NGVD29), which was formerly referred to as Mean Sea Level of 1929. This datum was originally established as a vertical reference representing the average sea level as measured at a series of tide gages throughout North America. As a result of additional work since 1929, a revised vertical reference has been established Closures within ± 0.05 ft ⫻ square root of distance in miles.
*
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FLOODPLAIN HYDRAULICS
Floodplain Hydraulics 19.11
which better represents the actual gravitation force of the earth as it exists in various locations throughout the North American continent. This reference is called the North American Vertical Datum of 1988 (NAVD88). FEMA and other federal government agencies are adopting NAVD88 as the standard vertical datum for new studies. The significance of the conversion from NGVD29 to NAVD88 depends on the particular location. The National Geodetic Survey has produced a computer program called VERTCON which will compute the conversion from NGVD29 to NAVD88 at any location in North America, given the coordinates of latitude and longitude. Field surveys should normally be accomplished by differential leveling or differential GPS methods, with vertical error tolerances of ⫾ 0.5 ft across the 100-year floodplain. Horizontal control may be unnecessary since cross sections can be located simply by visual reference to identifiable points on a map or aerial photograph. However, if crosssectional data will be combined with other data sets, it is necessary to establish and maintain adequate horizontal control.
19.2.2 Cross Section Locations The survey information required for a floodplain analysis generally consists of cross sections of the channel and floodplain along the stream. These cross sections may be field-surveyed or taken from digital terrain models. Some water surface profile computer programs treat each cross-section as representing a reach of the river and use only one section at the midpoint of the reach to calculate losses through the entire reach.* However, the most commonly used computer programs use cross sections to define break points in the geometry, and properties of adjacent sections are averaged to calculate losses through the reach. The objective of either computation scheme is to describe flow boundaries accurately enough to predict energy losses due to friction and changes in flow velocity. If only a few cross sections are available and they are located far apart, a greater amount of engineering judgment is required to satisfactorily analyze the problem. Any deviation from a smooth profile must be explained, and in some cases it can be traced back to inadequate cross-sectional data. A water surface profile in nature is really a curvilinear surface that follows the general slope of the channel. When the water surface profile is computed, the shape of this curvilinear surface is approximated by a series of straight-line segments. The endpoints of these line segments are the channel cross section locations. If many channel cross sections are included, the line segments will be short, and the true curvilinear shape of the natural water surface profile can be better represented. If only a few channel cross sections are used, the line segments will be long, and the true curvilinear shape of the natural water surface profile will not be represented well. Figure 19.8 illustrates a water surface profile that is well represented (using many channel cross sections) and the same profile that is poorly represented (using only a few channel cross sections). As indicated, if an adequate number of cross sections are not provided for the hydraulic analysis, the computed water surface elevation can vary considerably from the actual value, especially in areas in which there is a sudden change in the channel slope. One study indicated that reach lengths should be limited to a maximum of 800 m (about 0.5 mi) for wide floodplains and for slopes less than about 0.04 percent, 550 m (about 1800 ft) for slopes equal to or less than 0.06 percent, and 350 m (about 1200 ft) for *A reach is a length of a channel that is fairly uniform with respect to discharge, depth, area, and slope; more generally, any length of a river or drainage course.
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FLOODPLAIN HYDRAULICS
19.12
Chapter Nineteen
FIGURE 19.8 Example of profile plot.
slopes greater than 0.06 percent (Beasley, 1973). For flood insurance studies, maximum spacing is generally 150 m (about 500 ft) for unimproved channels and 600 m (about 2000 ft) for improved, regular channels. However, the actual spacing requirements vary according to energy considerations. A preliminary computer run using cross sections from topographic maps can be very valuable in determining the required spacing of surveyed cross sections. These cross section spacings are only guidelines, and should not be used as fixed spacing requirements. Instead, cross sections should be placed at all of the following locations along the stream channel: • Changes in slope. Cross-sections are needed at distinct changes in bed slope. • Changes in flow area. Cross-sections are needed at points of contraction or expansion of the channel. • Changes in flow rate. Cross-sections are needed in the main channel immediately above and below a confluence and in the main stream. A tributary cross section may also be obtained just above the confluence if the tributary is to be studied. • Changes in roughness. Additional cross-sections are needed at the upstream and downstream ends of each channel segment that has a significantly different Manning’s roughness coefficient, such as changes in plant cover, channel improvements, and so on. • Control sections. Cross sections are usually required immediately above and below control sections (such as bridges, drop structures, weirs, and so on) to adequately model the changing conditions in these flow transitions. A weir is a low-overflow dam or sill for measuring, diverting, or checking flow. • Encroachments. For encroachments into the floodplain (such as landfills, bridges, road embankments, dams, or levees), cross sections are needed at the upstream and downstream ends of the encroachment, and at regular intervals within the reach of the encroachment. Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
FLOODPLAIN HYDRAULICS
Floodplain Hydraulics 19.13
• Bends. Additional cross sections may be required in and around channel bends to properly represent the reach lengths in the chanwnel overbank areas. If an elevated road or dam is found in the field that was not indicated on the cross section layout map, the survey field crew should insert additional cross sections to represent these additional structures. However, this does not apply to roads that are not elevated and will not block large amounts of flow. After cross sections are placed at all of the locations identified above, then additional cross sections may be placed between these as needed to reduce the spacing between adjacent cross sections to the values recommended above. Laying out cross sections in this way, rather than at a fixed spacing, generally provides a superior analysis.
19.2.3 Cross Section Alignment and Orientation Cross sections should tie into high ground, so that the maximum elevation of each end of a cross section should be at least 1 ft higher than the anticipated maximum water surface elevation, if possible. In addition, the cross section locations should be accessible and practical. For example, pipeline and power line crossings make good locations for surveyed cross sections in wooded areas, because the right-of-way is usually cleared. However, cross sections must represent the average ground profile. Roadside ditches, washouts, gravel pits, raised areas, and other nonrepresentative conditions should be avoided. The actual orientation and extent of the surveyed cross section should be indicated on work maps. Most computer programs used for floodplain analysis require that cross sections should be surveyed from left to right, with these directions determined while looking
FIGURE 19.9 Example of work map showing cross section locations for field survey.
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FLOODPLAIN HYDRAULICS
19.14
Chapter Nineteen
downstream. The ground elevation as well as the distance from the channel flowline should be determined and recorded for each point on the cross section. Negative cross section stationing may be unacceptable to the computer programs used for floodplain analysis. Therefore, it is advisable to begin cross section stationing at 10 ⫹ 00 or 100 ⫹ 0 ⫹ so that the cross section can be extended later without having to restation the entire cross section. Cross sections should be placed perpendicular to the direction of flow. Cross sections can and should have angles or "doglegs" in them as needed so that the channel portion of the cross section can be perpendicular to the flow in the channel while the floodplain portions of the cross section can be perpendicular to the flow in the floodplain. Figure 19.10 provides an example. In the case of wide floodplains in very flat areas, where flows exceed the channel capacity and spill into the floodplains, the effective cross section is not easily defined. One way to solve this problem is to shape the cross section alignment so that the floodplain portions curve upstream. This curvature will contain the water in a reasonable width since the cross section elevations on the edge of the flow will be higher than in the case of the straight cross section alignment. A minimum of five points is usually required for the channel portion of a surveyed crosssection. This includes one point at the top of each channel bank, one point at the toe of each side slope, and one point at the channel flowline, as illustrated in Fig. 19.11. Additional points may be required when discontinuities in channel cross sections are encountered. The number of cross section points required for floodplain areas is dependent on the width of the cross section and on the character of the terrain in the floodplain. As a general rule, enough points should be shot to give a true representation of the floodplain terrain and to define any breaks in topography. Valley cross sections that are intended to show the typical floodplain may be relocated in the field by the survey party to avoid heavy brush or to take advantage of power line or pipeline easements, fence lines, or pasture lands. Any cross sections that are relocated must be clearly marked on the work maps. The channel station at which the cross-section intersects the channel center-line should be determined for each cross section. The stream stations determined using aerial pho-
FIGURE 19.10 Cross-sectional layout and reach lengths for meandering channel.
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FLOODPLAIN HYDRAULICS
Floodplain Hydraulics 19.15
FIGURE 19.11 Minimum survey points for channel only.
tographs or other maps for major structures such as bridges can be used as a reference in determining the stream stations of individual cross sections between the structures. Distance measurement to the nearest foot or meter is usually acceptable. The measured distances between cross sections are defined with reach lengths. Three reach lengths are required: left floodplain, right floodplain, and channel, which are measured from the current cross section to the previous cross section looking downstream. Channel reach lengths are measured along the channel invert. The floodplain reach lengths are measured along the anticipated flow path of the center of mass of flow in the floodplain area. This is generally assumed to be 1/3 the distance from the channel bank to the edge of the floodplain. The channel and floodplain reach lengths will often equal each other if the reach contains a straight channel with parallel flood plains. Reach lengths will not equal each other in channel bends. One floodplain reach length will be greater than the channel reach length while the other floodplain reach length will be less than the channel reach length (Fig. 19.12). Reach lengths for the channel and flood plains may be unequal where the flood plains are parallel and the channel meanders (Fig.19.12). The reach lengths may vary depending on the severity of the flood event. Therefore, a preliminary hydraulic analysis may be required. If multiple profiles (different flow rates or storm frequencies) are analyzed using the same reach lengths, the reach lengths for all the profiles should be determined by the most important profile, which is usually the 100-year profile. If the reach lengths of floodplains and channel differ, the most commonly used stream analysis computer programs calculate one effective reach length that is used in the Standard Step Method. This is a discharge-weighted reach length based on the discharges in the main channel and the two floodplains over the whole reach.
FIGURE 19.12 Cross-sectional layout and reach lengths for channel bends.
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FLOODPLAIN HYDRAULICS
19.16
Chapter Nineteen
(QLOB) (XLOB) ⫹ (QCH) (XLCH) ⫹ (QROB) (XLOBR) 苶⫽ L ᎏᎏᎏᎏᎏᎏ Q
(19.1)
where 苶 L ⫽ the weighted average reach length QLOB ⫽ the total flow in the left floodplain XLOB ⫽ the specified reach length for the left floodplain QCH ⫽ the total flow in the channel XLCH ⫽ the specified reach length for the channel QROB ⫽ the total flow in the right floodplain XLOBR ⫽ the specified reach length for the right floodplain Q ⫽ the total flow in the entire cross section.
19.2.4 Use of Aerial Topography and Contour Map Data Field survey costs for a flood plainstudy may be reduced in several ways. For example, the spacing of cross sections along the stream channel could be increased so as to reduce the number of cross sections required. As an alternative, the width of each cross section could be reduced. Studies have indicated that one of the best ways to improve the accuracy of a computed water surface profile is to provide more cross sections (HEC, 1986). This implies that one of the worst ways to save money on a floodplain study is to reduce the number of cross sections. In general, it is better to reduce the number of survey points within each cross section rather than to eliminate some cross sections entirely and thus extend the reach lengths between adjacent cross sections. Several methods may be used to maintain the required density of cross sections along the stream channel. For example, surveyed cross sections could be used more than once in the analysis. These repeated cross sections may be acceptable if field observations indicate that the channel and floodplain conditions are highly uniform through the stream reach represented by the repeated cross sections. Intermediate cross sections may also be synthesized (interpolated) from field-survey cross sections. Interpolated cross sections are described later in this chapter. During the data collection phase of the project, however, the most common method of reducing the cost of obtaining cross section data is to obtain all or part of the data points for the cross-section from aerial topography or contour maps. In some circumstances, aerial topography or contour maps can be used for the entire cross section, including the channel portion. Usually, however, the accuracy and level of detail available from aerial topography or contour maps are not sufficient for channel data. It is preferable to obtain at least the channel portion of the cross section by field survey methods, if at all possible. Sometimes, with little or no increase in field survey costs, some additional ground shots can also be made in the floodplain, using the same setup of the survey instrument already used for the channel survey. FEMA generally recommends the use of aerial topography or contour maps for obtaining cross sections (instead of section) for the portion of the cross section between the 100- and 500-year floodplain boundaries (FEMA, 1995). Aerial topography is very cost-effective if the size of the project justifies the fixed costs of mobilizing the aircraft and other related activities. Usually, the mapping data obtained by aerial photography are useful for many purposes other than the floodplain analysis, and therefore the full cost may be shared with other projects, departments, or even other organizational entities. Newer technologies such as LIDAR (Light Imaging raDAR) may also be very cost–effective, If a contour map meets commonly accepted accuracy standards, then the contour accuracy is slightly better than half of the contour interval while the accuracy of spot elevations is approximately 20 to 30 percent of the contour interval of the map being prepared. Therefore, cross sections developed using contours from topographic maps will not be as
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FLOODPLAIN HYDRAULICS
Floodplain Hydraulics 19.17
accurate as cross sections prepared from spot elevations. Whenever possible, spot elevations along the cross section should be determined directly using the source data set, rather than indirectly using contours.
19.2.5 Road Crossing Data Necessary dimensions and elevations of all hydraulic structures and underwater sections along the streams should be obtained from available sources or by field survey where necessary. Dimensions and elevations of hydraulic structures should not be established by aerial survey methods. Most computer programs used for floodplain analysis require four user-defined cross sections in the computations of energy losses due to the structure. These cross sections may be identified by number (1 through 4), beginning downstream of the bridge. A plan view of the basic cross section layout is shown in Fig. 19.13. Cross section 1 is located sufficiently downstream from the structure so that the flow is not affected by the structure (i.e., the flow has fully expanded from the narrow bridge opening). This distance should generally be determined by field investigation during high flows. If field investigation is not possible, there are two sets of criteria for locating the downstream section. The USGS has established a criterion for locating cross section 1 a distance downstream from the bridge equal to one times the bridge opening width (the distance between points B and C on Fig. 19.13). Traditionally, the Corps of Engineers criterion has been to locate the downstream cross section about four times the average length of the side constriction caused by the structure abutments
FIGURE 19.13 Cross section locations at a bridge or culvert.
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FLOODPLAIN HYDRAULICS
19.18
Chapter Nineteen
(the average of the distance from A to B and C to D on Fig. 19.13). The expansion distance will vary depending on the degree of constriction, the shape of the constriction, the magnitude of the flow, and the velocity of the flow. (A constriction is a local obstruction narrowing a waterway.) Both criteria should be used as rough guidance for placing cross section 1. Cross sections 1 and 2 should be close enough to one another so that friction losses can be adequately modeled. If the expansion reach requires a long distance, intermediate cross sections should be placed within the expansion reach to adequately model friction losses. Cross section 2 is located immediately downstream from the bridge (within a few feet). This cross section should represent the effective flow area of the natural channel just below the roadway crossing. It should not include the road fills, road ditches, mounds, or depressions that are not typical of the floodplain. Cross section 3 should be located just upstream from the bridge. The distance between cross section 3 and the bridge should be relatively short. This distance should only reflect the length required for the abrupt acceleration and contraction of the flow that occurs in the immediate area of the opening. Cross section 3 should represent the effective flow area of the natural channel just above the roadway crossing. Like cross section 2, it should not include the road fills, road ditches, mounds, or depressions that are not typical of the floodplain. Cross section 4 is an upstream cross section where the flow lines are approximately parallel and the cross section is fully effective. The USGS recommends that the distance between cross section 3 and 4 should be roughly the same as the average width of the bridge opening. However, this criteria for locating the upstream cross section may result in too short a reach length for situations where the width of the bridge opening is very small in comparison to the floodplain. The Corps of Engineers generally recommends that cross-section 4 be located a distance upstream equal to the average contraction width (the average of the distance from A to B and C to D on Fig. 19.13). According to the Corps of Engineers, the distance between cross sections 3 and 4 should be less than the distance between cross sections 1 and 2 for constricted flow conditions, because flow contractions can occur over a shorter distance than flow expansions. Both of these recommendations should be considered; the Corps of Engineers recommendation is probably better when the bridge opening is narrow compared with the width of the floodplain; the USGS recommendation is probably acceptable for all other cases.
19.2.6 Using Repeated Cross Sections for Roadway Crossings The engineer directing the floodplain analysis should determine, during a preliminary visit to the study area, if the channel is relatively uniform in the area of each road crossing structure. If the channel is highly uniform in the area of a particular road crossing, it may be possible to survey a single channel and floodplain cross section at the upstream or downstream face of the bridge, and repeat this cross section for the other three or more cross sections in the bridge model. If the channel is relatively uniform through the bridge itself, but changes upstream or downstream of the structure, a single field surveyed cross section may be used to represent the channel at both faces of the bridge, but different field-surveyed cross sections may be needed at other locations through the bridge model. If the channel is not uniform through the bridge, separate cross sections are required at the downstream face of the bridge and at the upstream face of the bridge. It may even be necessary to survey a cross section of the channel at the centerline of the bridge
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FLOODPLAIN HYDRAULICS
Floodplain Hydraulics 19.19
(underneath the bridge superstructure), where special conditions cause the channel underneath the bridge to differ significantly from the cross section at the upstream or downstream face.
19.2.7 Obtaining Bridge Survey Data Plan and profile sketches of the bridge on which all pertinent data are illustrated and identified should always be included in the survey notes. Figure 19.14 is a representative sketch of the type of data that will be required by the hydraulic engineer. Several pieces of information about the actual bridge itself are required. Two of the most important of these are top of road and low chord elevations. The top of road is the highest part of the roadway that forms a significant obstruction to flow over the roadway. The top of road can be the crest of the bridge roadway, the top of a sidewalk, or the crest of a solid bridge railing. When taking a cross-section on a multilane highway, the cross section should be obtained along the higher lane. If the median or shoulder is higher than the centerline, the high points should be located by side shots on the median or shoulder. Cross sections on railroads should be obtained on top of the rail. If the track is superelevated, the highest rail should be shot. The low chord (or low steel) of a bridge refers to the lowest part of the bridge that forms a significant obstruction to flow under the bridge. The low chord of most bridges is formed by the bottom of the bridge girders. Whenever any doubt exists as to the identification of the top of road and low chord of a bridge, elevations defining the shapes of all structures which form obstructions to flow should be determined and recorded. For example, if a water main or other pipe is suspended from the bridge, this condition should be recorded.
FIGURE 19.14 Example of sketch of bridge cross section.
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FLOODPLAIN HYDRAULICS
19.20
Chapter Nineteen
Natural ground elevations should be used instead of top of road elevations when the roadway is in a cut; that is, when natural ground is higher than the top of the roadway. In such cases, it is necessary to visualize the total obstruction to flood flow at the road. This will be a combination of the roadway and natural ground. Figure 19.15 illustrates this situation. Bridge railings or curbs should sometimes be considered when defining the top of roadway. If a railing or curb forms a substantial obstruction to flow over the bridge, the top of the rail or curb should be considered as the effective top-of-road. Figures 19.15 and 19.16 illustrate bridge decks with solid and open rails. Other important bridge data include the following: • Channel station: the channel station at which the bridge center-line (or one of the bridge faces) intersects the channel flowline.
FIGURE 19.15 Defining the bridge deck for roadways in open cuts.
FIGURE 19.16 Defining the bridge deck for bridges with solid rails.
FIGURE 19.17 Defining the bridge deck for bridges with open rails.
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FLOODPLAIN HYDRAULICS
Floodplain Hydraulics 19.21
• Bridge dimensions: the length and width of the bridge. • Skew angle: the angle between the bridge roadway centerline and the flowline of the channel. • Piers: the size, shape, number, and location of piers supporting the bridge.
19.2.8
Culvert Data
Data requirements for culvert crossings are very similar to those for bridges. Channel cross section locations for culvert crossings are the same as those for single bridges. The top of road of a culvert crossing is determined in the same way. For each culvert, the culvert size and shape, length, the downstream and upstream flowline elevations and centerline stations, and the type of headwall and wingwalls, and their angle with the main axis of the culvert should be recorded. If a culvert contains an accumulation of silt, the depth of silt at the upstream and downstream ends, and the consistency of the accumulated silt (compacted, loose, and so forth) should also be noted.
19.2.9 Channel Structures Channel structures such as weirs, drop structures, and sections of slope paving can have very significant effects on channel hydraulics. The locations, sizes, and configurations of these structures should therefore be carefully measured and recorded. If a channel section is fully or partially lined with concrete or some other type of slope paving, the survey information should record the type of paving (concrete lining, riprap, and so on), the beginning and ending channel station for the slope paving, and the vertical extent of the slope paving (that is, how far up the sides of the channel the paving extends).
19.3 SELECTING THE BEST APPROACH FOR A FLOODPLAIN STUDY There are three alternative approaches that are generally available for the computation of water surface elevations in a stream channel system: • One-dimensional steady, gradually varied flow conditions; • Two-dimensional, steady, gradually varied flow conditions; • One-dimensional unsteady flow conditions. It is also possible to model two-dimensional unsteady-flow conditions, although this approach is not common. The following sections describe these various approaches.
19.3.1 One-Dimensional and Two-Dimensional Flows In a constructed channel with a regular cross section, most of the lines of flow will be roughly parallel to the channel's longitudinal axis (the axis that follows the alignment of the channel from upstream to downstream). This is one-dimensional flow. However, in natural channels with irregular cross-sections, and especially when flow overtops the
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FLOODPLAIN HYDRAULICS
19.22
Chapter Nineteen
channel and enters the floodplain, there may be significant components of flow in other directions at certain locations. For example, flow may travel in a vertical direction for a short distance to overtop a roadway surface and spill into a lower area on the downstream side. Flow may also travel in a lateral direction from the floodplain into the channel to pass through a narrow culvert opening. Energy is required to set water in motion in any direction and overcome friction losses. The computer programs used for floodplain studies account for friction losses and other changes in energy potential. In fact, the computation of water surface elevation is really a by-product of the computation of the total energy level (often called the "energy grade line") at each location in the stream channel and floodplain. For those situations where energy losses are not the primary consideration, such as at a weir or a hydraulic jump, the programs use alternate solutions. (A hydraulic jump is a sudden transition from supercritical flow to the complementary subcritical flow, conserving momentum and dissipating energy.) One-dimensional computer programs used for floodplain studies account for energy losses in the downstream direction only (longitudinal flows). Two-dimensional computer programs consider lateral flows as well as longitudinal flows. Therefore, they analyze flow in both dimensions of the horizontal plane. A one-dimensional analysis is adequate for most purposes. Two–dimensional computer models are useful at complex roadway crossings, in shallow flooding areas, and whenever there is significant flow in more than one direction.
19.3.2 Changes in Flow Depth With Respect to Time and Distance Steady flow occurs whenever the depth of flow at a cross section, for a given discharge, is constant with respect to time. Unsteady flow occurs whenever the depth of flow at a crosssection varies with respect to time, for a given discharge. During a flood event, the flow depth in a stream channel increases from base flow conditions, reaches a peak value, and declines to the base flow again, generally within the span of a few hours or less. However, at any one moment, an observer standing on the edge of the floodplain would see what appears to be a steady flow condition. Steady flow can further be classified as uniform or nonuniform. Uniform flow occurs when the depth of flow and quantity of water are constant at every section of the channel under consideration. Under these conditions, the water surface and flowlines will be parallel to the stream bed and a hydrostatic pressure condition will exist (the pressure at a given section will vary linearly with depth). Uniform flow conditions are rarely attained in natural channels. Gradually varied flow occurs whenever the depth of water changes slowly over the length of the channel. Under this condition, the streamlines of flow are practically parallel. Therefore, uniform flow principles can be used to analyze the flow conditions, even though the flow is nonuniform. With rapidly varied flow conditions, there is a pronounced curvature of the flow streamlines. The assumption of hydrostatic pressure distribution is not valid. Therefore, uniform-flow principles are not adequate for the analysis of rapidly varied flow. Most computer programs used for floodplain analysis are based on the analysis of steady, gradually varied flow conditions. Some programs have special capabilities to analyze rapidly varied flow conditions, such as hydraulic jumps. A floodplain analysis that assumes steady flow is a "snapshot" of the channel system, usually at peak flow conditions. However, some channel flow situations are dominated by rapidly changing flow rates or other dynamic conditions. For example, a
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FLOODPLAIN HYDRAULICS
Floodplain Hydraulics 19.23
floodplain area may have substantial floodplain storage capacity that affects the flow rate in the stream channel; streams may experience a reversal of flow under certain conditions; the stream may be subject to sudden changes in flow rate due to the opening or closing of gates or other similar structures; streams may discharge into marine bodies that are affected by tidal movements; or the stream may be a part of a complex system of pipes, channels, ponds, and reservoirs. All of these situations may produce sudden changes in flow rate which are not adequately represented by the assumption of steady flow. The capability to analyze unsteady flow conditions is sometimes required.
19.3.3 Critical Flow and Critical Depth Critical depth is an important hydraulic parameter because it is always a hydraulic control. (Critical depth occurs when the specific energy is a minimum for a given discharge.) Hydraulic controls are points along the channel where the water level or depth of flow is limited to a predetermined level or can be computed directly from the quantity of flow. When the depth of flow is greater than critical depth, the velocity of flow is less than critical velocity for a given discharge and hence, the flow regime is subcritical. Conversely, when the depth of flow is less than critical depth, the flow regime is supercritical. Flow must pass through critical depth in going from a subcritical flow regime to a supercritical flow regime. Typical locations of critical depth are at (1). abrupt changes in channel slope when a flat (subcritical) slope is sharply increase to a steep (supercritical) slope, (2). a channel constriction such as a culvert entrance under some conditions, (3). the unsubmerged outlet of a culvert on subcritical slope, discharging into a widechannel or with a free fall at the outlet, and (4). the crest of an overflow dam or weir. Critical depth for a given channel is dependent on the channel geometry and discharge only, and is independent of channel slope and roughness.
19.3.4 Types of Stream Systems Stream systems may be divided into three main categories according to the complexity of the interconnections among the various flow paths within the stream system: • Simple channels. These are single-stream reaches with no tributary streams. Therefore, there are no confluences or junctions. The flow rates may vary from one cross section to the next along the length of the stream channel, because of local inflows or stream channel losses. The direction of flow in the channel is known. However, the flow regime within the channel is not known, and in fact, may vary along the length of the channel (changing from subcritical to supercritical or from supercritical to subcritical).(Fig. 19.18). • Dendritic channel systems. These include tributary streams that combine at confluences or junctions to form major streams. This process may be repeated through several steps. Streams within a dendritic channel system are sometimes identified by their “order.” First-order streams are the smallest tributaries, while higher-order streams are larger streams and rivers. In a dendritic system, each of the individual streams is analyzed as a simple channel (as described in the category above) (Fig. 19.19). ■
Network channel systems. These include all of the features of dendritic channel system. In addition, full network systems (sometimes called fully–looped systems) can
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FLOODPLAIN HYDRAULICS
19.24
Chapter Nineteen
FIGURE 19.18 Example of simple channel system.
have flow splits, where flow in a single channel (or a group of channels) can be divided among two or more flow paths which may either recombine downstream or discharge at completely different destinations. In some network channel systems, the direction of flow may be known in advance for each stream; for others, the direction of flow may be determined during the analysis and may even change during the analysis. A full network model can also include storage areas that can either provide water to, or divert water from, a channel. A full network model is required to properly deal with storage areas, because they represent flow splits (since water can be diverted into the storage area). See Fig. 19.20.
FIGURE 19.19. Example of dendritic channel system.
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FLOODPLAIN HYDRAULICS
Floodplain Hydraulics 19.25
19.3.5 Computer Programs Widely Used in Floodplain Analysis Flood elevations for riverine areas are normally determined by step-backwater* computer models, including HEC-RAS, HEC-2, WSPRO, and WSP-2. HEC-RAS River Analysis System [developed by the U.S. Army Corps of Engineers (USACE) Hydrologic Engineering Center] computes water surface profiles for onedimensional steady, gradually varied flow in rivers of any cross section (HEC, 1998a–c). HEC-RAS can simulate flow through a single channel, a dendritic system of channels, or a full network of open channels (sometimes called a fully looped system). However, HECRAS cannot currently determine how much flow follows each flow path at a flow split. In addition, the direction of flow in each stream must be provided to HEC-RAS as part of the input data. HEC-RAS can easily model sub– or supercritical flow, or a mixture of each within the same analysis. A graphical user interface provides input data entry, data modifications, and plots of stream cross sections, profiles, and other data. Program options include floodway computations, inserting trapezoidal excavations on cross sections, and analyzing the potential for bridge scour. The water surface profile through structures such as bridges, culverts, weirs, and gates can be computed. The program includes sophisticated routines
FIGURE 19.20 Example of network channel system.
Backwater is an unnaturally high stage in stream caused by obstruction or confirnement of flow, as by a dam, a bridge, or a levee. Its measure is the excess of unnatural over natural stage, not the difference in stage upstream and downstream from its cause. *
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FLOODPLAIN HYDRAULICS
19.26
Chapter Nineteen
for the analysis of roadways with multiple bridges and culverts. Variable channel roughness and variable reach length between adjacent cross sections can be accommodated. HEC-2 (developed by the USACE Hydrologic Engineering Center) was the predecessor to HEC-RAS, and has some of the same capabilities (HEC, 1990a). However, HEC-2 does not accommodate multiple bridge or culvert openings in most situations, and is restricted to either subcritical or supercritical flow computations in a single analysis. Because HEC-2 was widely used for almost 30 years, many existing floodplain studies were performed using HEC-2. HEC-RAS will read HEC-2 input data files and produce very similar results under most situations, but HEC-2 may still continue to be used for some time because of the need to match existing study results. HEC-2 is generally restricted to simple channel systems, but may be used for dendritic channel systems under limited circumstances. WSPRO Water Surface Profile [developed by the U.S. Geological Survey for the Federal Highway Administration (FHWA)] is a step backwater program for natural channels with an orientation to bridge constrictions (USGS, 1990). The water surface profile computation model has been designed to provide a water surface profile for six major types of open–channel flow situations: unconstricted flow, single opening bridge, bridge opening(s) with spur dikes, single opening embankment overflow, multiple alternatives for a single job, and multiple openings. WSPRO was developed for use primarily around roadway crossings. The bridge analysis routines originally used in WSPRO have now been incorporated into the HEC-RAS computer program. WSP2 Water Surface Profile 2 (developed by the USDA Natural Resources Conservation Service) is a step backwater program for natural channels [Natural Resources Conservation Service (NRCS, 1993)]. WSP2 estimates head loss at restrictive sections, including roadways with either bridge openings or culverts. 19.3.6
Two-Dimensional Water-Surface Computer Models
The most commonly used two-dimensional models include TABS-MD (developed by ACE Waterways Experiment Station) and FESWMS-2DH Two-Dimensional Flow in a Horizontal Plane computer program (developed by USGS).The TABS-MD (Multidimensional) Numerical Modeling System is a collection of generalized computer programs and utility codes, designed for studying multidimensional hydrodynamics in rivers, reservoirs, bays, and estuaries. The primary computational program in the TABSMD system is the RMA-2 model [Waterways Experiment Station (WES) Hydraulic Laboratory, 1997]. The TABS-MD system can be used to study project impacts on flows, sedimentation, constituent transport, and salinity. FESWMS-2DH Finite Element Surface Water Modeling System–Two–Dimensional Flow in a Horizontal Plane simulates steady and unsteady flow and is useful for simulating two-dimensional flow at width constrictions and highway crossings of rivers and floodplains (FHWA, 1989). FESWMS-2DH is based on the momentum balance approach, with consideration of shear stresses due to friction between the moving water and the fixed boundaries. 19.3.7 One-Dimensional Unsteady-Flow Models There are several computer programs available for simulating one–dimensional unsteadyflow conditions in stream channels. These include UNET, Storm Water Management Model (SWMM), DAMBRK, and NETWORK. UNET (supported and maintained by USACE HEC is a one-dimensional unsteady flow model that can simulate flow through a single channel, a dendritic system of channels, or a full network of open channels (HEC, 1990b). In UNET, storage areas can be the
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FLOODPLAIN HYDRAULICS
Floodplain Hydraulics 19.27
upstream or downstream boundaries for a river reach. In addition, the river can overflow laterally into the storage areas over a gated spillway, weir, levee, through a culvert, or a pumped diversion. In addition to solving the one-dimensional unsteady flow equations in a network system, UNET One–Dimensional Unsteady Flow Through a Full Network of Open Channels provides the user with the ability to apply several external and internal boundary conditions, including flow and stage hydrographs, gated and uncontrolled spillways, bridges, culverts, and levee systems. UNET can read channel cross sections that are input in a modified HEC-2 format. Future versions of the HEC-RAS computer program are expected to incorporate most of the functions of the UNET computer program, and will likely provide access to these functions through a user interface that is more convenient to use than the original UNET user interface, which relies heavily on the HEC-DSS (Data Storage System) for data entry and output. Storm Water Management Model (SWMM) (developed by the U.S. Environmental Protection Agency) is a comprehensive water quantity and quality simulation model developed primarily for urban areas. SWMM is a complete hydrology and hydraulics model that is most commonly applied to urban areas with closed storm sewer systems (Huber and Dickinson, 1988). The model uses different modules (or “blocks”) for different types of computations. SWMM is not widely used for floodplain studies on natural stream channels. However, the Extran and Transport blocks of the SWMM program can use natural channel cross section data in the same form as required by the HEC-2 computer program. The Transport block provides simple hydraulic computations, but the more sophisticated Extran block can perform a dynamic backwater analysis (Roesner et al., 1988). This can be useful when a sophisticated hydraulic analysis is required, particularly when the open channel is a part of an urban drainage network. Dam Break Flood Forecasting Model (DAMBRK) (developed by the National Weather Service) is an unsteady flow dynamic routing model which develops an outflow discharge hydrograph due to spillway and/or dam failure flows (Fread, 1982). This hydrograph is routed through the downstream river valley. This can be a useful method of floodplain analysis for such events. NETWORK: Enhanced Dynamic Wave Model (developed by the National Weather Service) is an unsteady flow dynamic routing model for a single channel or network (dendritic and/or bifurcated) of channels for free surface or pressurized flow (Fread & Lewis, 1986, Fread & Lewis, 1988 and Fread, 1993).
19.3.8 Selecting a Computer Program for a Floodplain Analysis A computer program that is suitable for one-dimensional steady, gradually varied flow conditions is adequate for most floodplain studies. The most widely used programs in this category are the Corps of Engineers HEC-RAS and the older HEC-2 computer programs. If conditions dictate, a two-dimensional analysis or a dynamic analysis (or, in some cases, a two-dimensional dynamic analysis) may be required. If a floodplain study is being done to revise a FEMA flood insurance study, FEMA normally requires that revised study be completed using the same computer program used for the original study. However, a different computer program may be accepted if documentation can be submitted justifying why a change would be an improvement. It is important to discuss such issues with FEMA in advance (FEMA, 1995).
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FLOODPLAIN HYDRAULICS
19.28
Chapter Nineteen
19.4 PERFORMING A FLOODPLAIN STUDY Water flowing in an open channel possesses two kinds of energy: (1) potential energy and (2) kinetic energy. Potential energy is due to the position of the water surface above some datum. Kinetic energy is caused by the movement of the water. The key to water surface profile calculation is the maintenance of an energy budget from cross section to cross section. All of the energy at one cross section, including potential energy and kinetic energy, must be accounted for at the next cross section. The difference in total energy level between one cross section and the next must be attributable to energy losses due to friction and to flow expansion or contraction. It is the determination of energy losses that imposes constraints on the permissible distance between cross sections and consequently on the number of cross sections required for a study.
19.4.1 Computing Water Surface Profiles For the gradually varied flow condition, the depth of flow may be established through a water surface profile analysis. The basic principles in water surface profile analysis are as follows: (1) The water surface approaches the uniform depth line asymptotically; (2) The water surface approaches the critical depth line at a finite angle; (3) Subcritical flow is controlled from the downstream end of the stream reach; and (4) Supercritical flow is controlled from the upstream end of the stream reach.
19.4.2
Starting Conditions for Water Surface Computations
The starting conditions at the upstream and/or downstream ends of each stream reach are called “boundary conditions.” These are sometimes difficult to determine. There are four common methods for determining the starting conditions for water surface profile computations: 1. Known or assumed elevation. If the purpose of the floodplain analysis is to reconstruct conditions that existed during a historical flood, for example, the analysis might begin with a water surface elevation that was observed or measured during that flood event. Streams that discharge into lakes or marine bodies of water may also start with a known water surface elevation in the body of water under certain conditions. 2. Rating curve. A water surface elevation may be computed from a rating curve representing the hydraulic capacity of a particular channel section (e.g., at the location of a stream flow gage). Alternatively, the rating curve may represent the hydraulic characteristics of a channel structure such as a weir. 3. Normal depth. Uniform flow may be assumed, based on the average slope of the energy grade line in the reach of channel near the beginning point. This is sometimes called the “slope-area” method. 4. Critical depth. A control section may be identified in which flow must pass through critical depth. This could provide a convenient starting condition for the water surface profile. The best method of starting a water surface profile is to use a known water surface elevation. A rating curve is also a good option, if one is available. However, it is common for neither of these sources to be available. Therefore, the normal depth and critical depth options are frequently used. Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
FLOODPLAIN HYDRAULICS
Floodplain Hydraulics 19.29
Natural channels do not usually operate under uniform flow conditions, because of variations in channel slope, roughness, and cross sectional shape. However, if the variations are not excessive, normal depth is often a reasonable estimate of the actual water surface elevation. It is important to obtain a good estimate for the slope of the energy grade line, because this has a large effect on the computed value of normal depth. It is common to assume that the slope of the energy grade line will equal the average slope of the channel bed. This average slope could be measured in several ways, and then the most representative slope value could be selected. For example, the channel bed slope between the starting cross section and the next cross section in the analysis could be used, if this represents typical conditions throughout the stream reach close to the starting location. If possible, the floodplain analysis should be arranged so that the results are relatively independent of assumed starting conditions. This may be done by starting the water surface profile several cross sections away from the area of interest. The computed water surface profiles resulting from a wide range of starting conditions will converge to the same profile after several cross sections. The channel length required for computed water surface elevations to converge at a particular location depends on the flow rate, the roughness coefficients, and the slope and shape of the cross section. Studies (HEC, 1986) have indicated that the channel length required for water surface profiles to converge from different starting elevations to a single profile may be estimated by the following equations: HD Ldc ⫽ 125ᎏᎏ S KH HD0.8 Ldn ⫽ ᎏᎏ S
(19.2) (19.3)
where Ldc ⫽ the channel length for a critical depth starting condition (m or ft), Ldn ⫽ the channel length for a normal depth starting condition (m or ft), HD ⫽ the average hydraulic depth for the 100-year flood event (ft or m) = the cross sectional flow area (m2 or ft2) divided by the top width of flow (m or ft), S ⫽ the average reach slope (percent), and K ⫽ constant ⫽ 120 for m; 150 for ft. Consider the stream profile illustrated in Fig. 19.21. If the purpose of the analysis is to determine the 100-year water surface elevation at the bridge, it is then important to verify that the starting water surface elevation downstream is adequate. Assume that the preliminary hydraulic analysis indicates that the top width of the 100-year floodplain at the five cross sections below the bridge ranges from about 500 m (about 1600 ft) to about 600 m (about 1900 ft), with an average top width of about 530 m (about 1750 ft). The cross sectional area of flow for these five cross sections ranges from about 465 m2 (about 5000 ft2) to about 880 m2 (9500 ft2), with an average value of about 700 m2 (7000 ft2). Therefore, the average hydraulic depth (HD) in this reach is about 700/530 ⫽ 1.3 m (about 4 ft). The first three cross-sections of the stream profile indicate that the channel bed is horizontal. Therefore, the fourth and fifth cross sections must be used to determine the channel bed slope. The average channel bed slope from the fourth cross section to the downstream end of the profile is (201.83 ⫺ 199.9)/(1265.3) ⫽ 0.153 percent. The average channel bed slope from the fifth cross section to the downstream end of the profile is (202.7 ⫺ 199.9)/(2043) ⫽ 0.137 percent. Therefore, a reasonable value for the starting slope of the energy grade line is about 0.15 percent. Using Eq. (19.2) and (19.3), the distance needed from the start of the profile to the bridge is about 125 ⫻ (1.3/0.15) ⫽ 125 ⫻ 8.67 ⫽ 1080 m (about 3300 ft), for starting conditions based on critical depth. For starting conditions based on normal depth, the distance is estimated to be about 120 ⫻ (1.30.8/0.15) ⫽ 120 ⫻ (1.23/0.15) ⫽ 990 m (about
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FLOODPLAIN HYDRAULICS
19.30
Chapter Nineteen
3000 ft). However, since the available channel length downstream of the bridge is only about 600 m (about 2000 ft), it appears that the channel distance downstream of the bridge is not long enough to ensure a stable water surface elevation at the bridge location, for either starting condition. As Fig. 19.21 indicates, the starting water surface affects the computed water surface elevation at the bridge. However, as Fig. 19.22 shows, starting the analysis further downstream isolates the bridge from the effects of the starting conditions.
19.4.3 Starting Conditions for Tributary Stream Analysis The computed normal depth at the mouth of a tributary channel may be well below the computed depth of flow in the receiving stream for the storm event of the same frequency. In such a case, the tributary profile computations could be started using the computed depth of flow in the receiving channel. However, this would only be appropriate if the peak flow conditions for the tributary channel coincide with those on the receiving stream. The assumption of coincident peaks may be appropriate if the following conditions are all met: (1) the ratio of the drainage areas lies between 0.6 and 1.4; (2) the times of peak flows are similar for the two combining watersheds; and (3) the likelihood of both watersheds being covered by the storm being modeled are high. Figure 19.23 provides some guidelines for coincidental occurrence of storms over the watershed of a tributary channel and the receiving stream for a 100-year design storm. This figure can be used to establish an appropriate design tailwater elevation for a tributary channel based on the expected coincident storm frequency of the receiving stream channel. For example, if the receiving stream has a drainage area of 200 ha and the tributary channel has a drainage area of 2 ha, the ratio of receiving area to storm drainage area is 200:2 which equals 100:1. From Fig. 19.23 and considering a 100-year design storm occurring over both areas, the flow rate in the main stream will be equal to that of a 25-year storm when the drainage system flow rate reaches its 100-year peak flow at the outfall. Conversely, when
FIGURE 19.21 Example of inadequate distance from starting conditions.
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FLOODPLAIN HYDRAULICS
Floodplain Hydraulics 19.31
FIGURE 19.22 Example of adequate distance from starting conditions.
FIGURE 19.23. Storm frequencies for coincidental occurrence for 100-year analysis of tributary channel. (From: FHWA, 1996)
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FLOODPLAIN HYDRAULICS
19.32
Chapter Nineteen
the flow rate in the main channel reaches its 100-year peak flow rate, the flow rate from the storm drainage system will have fallen to the 25-year peak flow rate discharge. This is because the drainage areas are different sizes, and the time to peak for each drainage area is different. When the peak flow rates for the main channel and the tributary stream do not coincide, it is more appropriate to compute the two streams separately. The tributary channel profile is computed using normal depth or other appropriate starting condition at the downstream end, rather than starting with the main channel water surface elevation. Figure 19.24 illustrates a profile plot for a typical tributary stream. As shown, the controlling water surface profile for the tributary channel is plotted using the higher of the following two elevations: (1). Main channel backwater. the water surface elevation from the main channel, extended horizontally back up the channel of the tributary stream; (2). Independent tributary profile. the water surface elevations independently computed for the tributary. When a stream discharges into a standing body of water, the normal pool elevation should be used as a starting condition. Mean high tide* is generally used as a starting condition for streams that discharge into a body of water that is tidally influenced. In both cases, the cross-section data for the stream channel should extend far enough down into the standing body of water so that the cross sectional area of at least the first two cross sections is very large. This will make the flow velocity, and therefore the energy loss, very low at these cross sections. The full area of each cross section should be included; cross sections should never be truncated below the pool elevation of the water body.
FIGURE 19.24 Profile plot for tributary stream. *The mean high tide is also called the mean high water (MHW). It is the average height of the maximum elevation reached by each rising tide over a specific 19-year period. MHW is the reference base for structure heights, bridge clearances, and so on.
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FLOODPLAIN HYDRAULICS
Floodplain Hydraulics 19.33
19.4.4 Standard Step Computations All of the computer programs commonly used for determining water surface elevations in stream channels use the “standard step method” of computation. This is an iterative method, in which the program works from one end of the stream channel to the other, one cross section at a time. At the first cross section, a starting condition is specified as part of the input data. At each succeeding cross section, the program uses the following steps: • A water surface elevation is estimated and used to compute the cross-sectional flow area. This allows the computation of the flow velocity and velocity head. The total energy head is the sum of the water surface elevation and the velocity head.* • The estimated water surface elevation is also used to compute the wetted perimeter, conveyance, and friction slope. • The friction slope values of the current and preceding cross section are averaged (by any one of several different methods, as described below). • The average friction slope is multiplied by the weighted average reach length to obtain the total energy loss in this stream reach. • The energy loss is added to the total energy at the preceding cross section to produce a revised estimate of the total energy at the current cross-section. • The computed total energy elevation is compared to the assumed total energy elevation from the first step. The steps above are repeated (with additional details to ensure computational stability and convergence, and to account for other factors) until the program determines that no further adjustments are necessary in the energy head at the current cross section. The program then computes various other output values before beginning the same procedure for the next cross section.
19.4.5 Roughness Coefficients The roughness coefficient, Manning’s n value, is an experimentally derived constant which represents the effect of channel roughness in Manning’s equation. Considerable care must be given to the selection of an appropriate n value for a given channel due to its significant effect on the results of Manning’s equation. Even experienced engineers may produce estimated roughness coefficients that vary by 25 percent or more for some channels and floodplains. The estimates should include the consideration that roughness may vary with flood stages, depending on such factors as the width-to-depth ratio of streams, vegetation in the channel and floodplains, and materials of the channel bed. The n values selected should be typical of the stream reach, and not just the particular location of the cross section itself. Often, land use in the immediate area of the cross section may be cropland; however, immediately downstream, the area may revert to woodland and brush. Here the appropriate n value would reflect the woodland and brush rather than the agricultural area. Aerial photographs and field inspections can be important in avoiding errors in n values. Recording the land use and n value across the floodplain on the plotted cross section and/or work maps is helpful in documentation.
Head represents an available force equivalent to a certain depth of water. This is the motivating force in effecting the movement of water. It is the height of water above any datum. *
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FLOODPLAIN HYDRAULICS
19.34
Chapter Nineteen
Tables 19.2 through 19.5 provide a listing of n values for various channel conditions. (Chow, 1959). Photographs and other information is available for stream channels with various values of n (Arcement and Schneider, 1984; Barnes, 1967). 19.4.6 Representative Friction Slope For a Channel Reach Most computer programs used for floodplain analysis compute p the friction loss between adjacent cross sections as the product of Sf and L, where Sf is the representative friction slope for a reach and L is the weighted average reach length defined by Eq. (19.1). The representative rate of friction loss is an average of the computed friction slope at each of the adjacent cross sections. Many programs provide more than one method of averaging the friction slope. The following averaging methods are typically available:
冢
冣
Q1 + Q2 2 Average conveyance equation: S苶f ⫽ ᎏ ᎏ K1 + K2 Sf1 ⫹ Sf2 Average friction slope equation: S苶f ⫽ ᎏᎏ 2
(19.4) (19.5)
Geometric mean friction slope equation: S苶f ⫽ 兹 兹S苶S苶 苶Sf2 f1•S苶
TABLE 19.2
(19.6)
Manning’s Roughness Coefficient for Floodplains
Type of Channel and Description Pasture, no brush Short grass High grass Cultivated areas No crop Mature row crops Mature field crops Brush Scattered brush, heavy weeds Light brush and trees, in winter Light brush and trees, in summer Medium to dense brush, in winter Medium to dense brush, in summer Trees Dense willows, summer, straight Cleared land with tree stumps, no sprouts Same as above, but with heavy growth of sprouts Heavy stand of timber, a few down trees, little undergrowth, flood stage below branches Same as above, but with flood stage reaching branches
Minimum
Normal
Maximum
0.025 0.030
0.030 0.035
0.035 0.050
0.020 0.025 0.030
0.030 0.035 0.040
0.040 0.045 0.050
0.035 0.035 0.040 0.045 0.070
0.050 0.050 0.060 0.070 0.100
0.070 0.060 0.080 0.110 0.160
0.110 0.030
0.150 0.040
0.200 0.050
0.050
0.060
0.080
0.080
0.100
0.120
0.100
0.120
0.160
Source: From Chow (1959).
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FLOODPLAIN HYDRAULICS
Floodplain Hydraulics 19.35 TABLE 19.3
Manning’s Roughness Coefficient for Excavated Channels
Type of Channel and Description
Min
Norm
Max
0.016
0.018
0.020
Earth, straight and uniform Clean, recently completed Clean, after weathering
0.019
0.022
0.025
Gravel, uniform section, clean
0.022
0.025
0.030
With short grass, few weeds
0.022
0.027
0.033
No vegetation
0.023
0.025
0.030
Grass, some weeds
0.025
0.030
0.033
Dense weeds or aquatic plants in deep channels
0.030
0.035
0.040
Earth, winding and sluggish
Earth bottom and rubble sides
0.028
0.030
0.035
Stony bottom and weedy banks
0.025
0.035
0.040
Cobble bottom and clean sides
0.030
0.040
0.050
No vegetation
0.025
0.028
0.033
Light brush or banks
0.035
0.050
0.060
Smooth and uniform
0.025
0.035
0.040
Jagged and irregular
0.035
0.040
0.050
0.050
0.080
0.012
Dragline-excavated or dredged
Rock cuts
Channels not maintained, weeds and brush uncut Dense weeds, high as flow depth Clean bottom, brush on sides
0.040
0.050
0.080
Same, highest stage of flow
0.045
0.070
0.110
0.080
0.100
0.140
Dense brush, high stage Source: From Chow (1959).
TABLE 19.4
Manning’s Roughness Coefficient for Minor Natural Streams*
Type of Channel and Description
Minimum
Normal
Maximum
0.025
0.030
0.033
Streams on plain 1. Clean, straight, full stage, no rifts or deep pools
A minor stream is one that has a top width of less than 100 ft at flood stage.
*
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FLOODPLAIN HYDRAULICS
19.36
Chapter Nineteen
TABLE 19.4
(Continued)
Type of Channel and Description
Min
Norm
Max
2. Same as (1), but some stones and weeds
0.030
0.035
0.040
3. Clean, winding, some pools and shoals
0.033
0.040
0.045
4. Same as (3), but some weeds and stones
0.035
0.045
0.050
5. Same as (4), lower stages, more
0.040
0.048
0.055
6. Same as (4), but more stones
0.045
0.050
0.060
7. Sluggish reaches, weedy, deep pools
0.050
0.070
0.080
8. Very weedy reaches, deep pools, or
0.075
0.100
0.150
0.030
0.040
0.050
0.040
0.050
0.070
ineffective slopes and sections
floodways with heavy stands of timber and underbrush Mountain streams, no vegetation in channel, banks usually steep, trees and brush along banks submerged at high stages Bottom: gravels, cobbles, and few boulders Bottom: cobbles with large boulders Source: From Chow (1959).
TABLE 19.5
Manning’s Roughness Coefficient for Major Natural Streams*
Type of Channel and Description
Minimum
Normal
Maximum
Regular section with no boulders or brush
0.025
—
0.060
Irregular and rough section
0.035
—
0.100
Source: From Chow (1959).
2Sf1•S Sf2 Harmonic mean friction slope equation: S苶f ⫽ ᎏᎏ (19.7) Sf1 + Sf2 where S苶f ⫽ the average friction slope; Q1 and Q2 ⫽ the flow rates at each of the two cross sections; respectively and Sf1 and Sf2 ⫽ the computed friction slope at each of the two cross sections, respectively. The average conveyance equation is commonly used, because it gives good results across a wide range of flow regimes and profile conditions. Actually, any of these friction loss equations will produce satisfactory estimates if the reach lengths between cross sections are not too long. However, some equations provide better results for a certain flow
A major stream is one with a top width of more than 100 ft at flood stage. The n value is less for major streams than for minor streams of similar description because banks offer less effective resistance. †
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FLOODPLAIN HYDRAULICS
Floodplain Hydraulics 19.37
regime (sub or supercritical) or for a certain type of profile. Research indicates that the average friction slope equation provides the most accurate determination of a known profile with the least number of cross sections for a "backwater" profile in a channel with a subcritical slope (Reed and Wolfkill, 1976). This equation is also preferred for some supercritical flow profiles (HEC, 1997c). The harmonic mean friction slope equation is the most suitable for drawdown* profiles in a subcritical channel (Reed and Wolfkill, 1976). The geometric mean friction slope equation is the most suitable for certain types of supercritical flow profiles (HEC, 1997c).
19.4.7 Cross Section Interpolation Occasionally it is necessary to supplement surveyed cross section data by interpolating cross sections between two surveyed sections. Interpolated cross sections are often required when the change in velocity head is too large. However, other criteria such as considering the percentage change in energy slope can also be useful. An adequate depiction of the change in energy gradient is necessary to accurately model friction losses as well as contraction and expansion losses. When cross sections are spaced too far apart, the program may end up defaulting to critical depth. Some of the computer programs available for floodplain studies can generate interpolated cross sections quickly and easily. However, interpolated cross sections should be used with caution. The interpolated cross section is based on the shape of the upstream and downstream cross sections. Because of this, interpolated cross sections often appear so realistic that it is easy to forget that they do not include any “real-world” data. Whenever an interpolated cross section is generated, it should be carefully reviewed and documented. Some computer programs allow the interpolated cross sections to be adjusted manually to better depict the information from the topographic map.
19.4.8 Supercritical Flow Regime Calculations Channel reaches that operate under a supercritical flow regime present special challenges in floodplain analysis. Because of the unstable nature of supercritical flow, especially in natural streams, critical depth should be computed and used for plotting the computed water surface profile. In a constructed channel, the supercritical flow condition may be more common (because of the more uniform slope, cross section, and roughness characteristics of most constructed channels). However, it is important for the designer to project possible future channel conditions, such as the condition that occurs when sediment accumulates on the bottom of a concrete-lined channel. Such a condition may change the composite roughness of the channel section enough to disturb the supercritical flow regime and cause a hydraulic jump under certain flow conditions.
19.4.9 Mixed-Flow Regime Calculations Some floodplain analysis computer programs have the ability to perform subcritical, supercritical, or mixed-flow regime calculations. A mixed-flow regime occurs when a reach contains sections with both supercritical and subcritical flows. The change from sub- to supercritical flow occurs in a drawdown curve, while the transition from super- to Drawdown occurs at changes from mild to steep channel slopes and weirs or vertical spillways.
*
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FLOODPLAIN HYDRAULICS
19.38
Chapter Nineteen
subcritical occurs in a hydraulic jump. The concept of "specific force" is used to determine which flow regime is controlling, as well as locating any hydraulic jumps. The total specific force of a particle of water is the sum of a dynamic component (the momentum of the flow passing through the channel cross section per unit time) and a static component (the force exerted by the hydrostatic pressure of the water) (Chow, 1959). Mixed-flow regime calculations are generally performed as follows: 1. Subcritical analysis. First, a subcritical water surface profile is computed starting from a known downstream boundary condition. During the subcritical calculations, all locations where the program defaults to critical depth are flagged for further analysis. 2. Supercritical boundary condition. Next the program begins a supercritical profile calculation beginning with the upstream boundary condition. If the boundary condition is supercritical, the program checks to see if it has a greater specific force than the previously computed subcritical water surface at this location. If so, then it is assumed to control, and the program will begin calculating a supercritical profile from this section. 3. Supercritical reaches. If the subcritical answer has a greater specific force, then the program begins searching downstream to find a location where the program defaulted to critical depth in the subcritical run. When a critical depth is located, the program uses it as a boundary condition to begin a supercritical profile calculation. 4. The program calculates a supercritical profile in the downstream direction until it reaches a cross section that has both a valid sub- and supercritical answer. When this occurs, the program calculates the specific force of both computed water surface elevations. Whichever answer has the greater specific force is considered to be the correct solution. If the supercritical answer has a greater specific force, the program continues making super-critical calculations in the program reaches a cross section whose subcritical answer has a greater specific force than the supercritical answer, the program assumes that a hydraulic jump occurred between that section and the previous cross-section. 5. The program then goes to the next downstream location that has a critical depth answer and continues the process. An example of a mixed-flow profile is shown in Fig. 19.25. The most downstream section of channel, which discharges into a reservoir, has a supercritical slope. However, subcritical flow is maintained in the lower part of this reach by the water level in the reservoir. The middle reach of the channel is subcritical, but there is a supercritical reach just above it. Where these two reaches meet, the flow passes through a hydraulic jump, which changes the flow regime from super- to subcritical. The most upstream reach of channel is subcritical. The transition between this reach and the super-critical reach just below it provides an example of a drawdown. Before a mixed-flow run can be made, the appropriate boundary conditions must be set at both the upstream and downstream ends of the stream. For example, the known water surface elevation in the reservoir provides the downstream boundary condition for the example described above. The upstream boundary condition could be a known water surface elevation, normal depth, or critical depth.
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FLOODPLAIN HYDRAULICS
Floodplain Hydraulics 19.39
FIGURE 19.25. Example of mixed–flow regimes.
19.5 ENSURING THE QUALITY OF A FLOODPLAIN ANALYSIS A high-quality floodplain analysis is never achieved by simply checking the results; it is achieved only by a consistent devotion to quality throughout the process of data collection and analysis. However, a final check of the results can eliminate some types of problems.
19.5.1 Reviewing Program Messages Most computer programs used for floodplain analysis can generate notes, warnings, and error messages during the analysis. Error messages describe problems that prevent the program from being able to complete the run. Warning messages identify conditions that require attention and possible correction. Notes simply provide more detailed information; they do not necessarily indicate problems in the analysis. In general, whenever a warning is set at a location, the hydraulic results should be reviewed at that location to ensure that the results are reasonable. If the hydraulic results are found to be reasonable, then the message can be ignored. However, in many instances, a warning level message may require some action to eliminate the message. Many of the warning messages are caused by inadequate or incorrect data. Some common problems that cause warning messages to occur are the following: • Cross-sections too far apart. This can be the root cause of several different warnings and error messages. The distance between cross-sections is too long if hydraulic properties of the flow change too radically from cross section to cross section. If, from one cross section to the next, the slope of the energy grade line decreases by
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FLOODPLAIN HYDRAULICS
19.40
Chapter Nineteen
more than 50 percent or increases by more than 100 percent, the reach length may be too long for accurate determination of energy losses caused by boundary friction. For example, Fig. 19.26 illustrates a plot of energy slope versus channel station. Each point on the chart represents the energy slope computed at a particular cross section along the stream channel. Note that there are three cross section locations (including the starting cross section) at which there are “spikes” in the computed energy grade line slope. Inserting additional cross sections reduces one of these “spikes” considerably, as the figure also indicates. S • Bad starting water surface elevation. If the user specifies a boundary condition that is not physically possible for the specified flow regime, the program will take action and set an appropriate warning message. However, even if the specified boundary condition is possible, it may still be inappropriate, as illustrated in Fig. 19.26. In this profile plot, the assumed value of the energy slope for the downstream boundary condition appears to be too high. • Cross-section starting and ending stations not high enough. If a computed water surface is higher than either endpoint of the cross section, a warning message will appear. A small vertical extension may not affect the results of the analysis significantly. However, excessive extensions can lead to serious errors in the analysis, because the program is not able to consider the additional flow area and capacity which may be available beyond the range of cross section data. No significant cross section extensions should be allowed if floodway computations are to be performed, because the conveyance capacity beyond the point of extension should be considered in the floodway computations (Fig. 19.27). • Bad cross section data. This can cause several problems, but most often the program will not be able to balance the energy equation and will default to critical depth.
Slope of Energy Grade Line
• Critical depth. During water surface profile calculations, the computer program may default to critical depth at a cross section to continue the calculations. Critical depth can occur for several reasons, including bad cross section data, excessively long reach lengths, or specification of the wrong flow regime (i.e., specifying subcritical flow when the flow is actually supercritical). On occasion, when the program is bal-
FIGURE 19.26 Example of excessive cross section spacing.
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FLOODPLAIN HYDRAULICS
Elevation
Floodplain Hydraulics 19.41
FIGURE 19.27 Example of extended water surface elevations.
ancing a water surface that is close to the top of a levee or similar structure, the program may go back and forth (above and below the levee) without being able to balance the energy equation. When this occurs, most computer programs will default to critical depth. With some computer programs, it is possible to suppress the critical depth warning messages by using a mixed-flow regime analysis. However, this should not be used unless the accuracy of the cross section data and reach lengths, and other common sources of isolated critical depth messages, have first been investigated. • Divided flow conditions. Divided–flow messages indicate that a portion of the cross section extends above the water surface, between areas of flow on either side. If several consecutive divided flow messages occur, the input data may have to be revised through this reach to provide a separate analysis for each flow area. Using two separate analyses, the profiles for this “island flow” situation may be determined (Fig. 19.28). By reviewing these messages, it should be possible to correct all of the obvious errors in a preliminary floodplain analysis. It may not be possible to eliminate all warning messages under all conditions. For example, some computer programs for floodplain analysis issue a warning message when the conveyance changes more than about 40 percent from one cross section to the next. Whenever this warning message is encountered, the cross section data and reach lengths should be checked. However, even with accurate cross section data and very short reach lengths, it is not possible to fully eliminate such a warning message for cross sections that represent sudden changes in the cross sectional area (at a roadway crossing, for example).
19.5.2 Reviewing the Stream Profile To evaluate the analysis further, examine a stream profile plot that shows the computed water surface elevation, energy grade line, and critical depth at each cross section along Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
FLOODPLAIN HYDRAULICS
19.42
Chapter Nineteen
FIGURE 19.28. Example of divided– flow conditions.
the length of the channel. If the profile indicates large changes in either the water surface elevation or the energy grade line, these changes should be investigated. Are such changes physically possible? Does the channel profile appear reasonable? Are there portions where the computed water surface coincides with the critical depth elevation? Does the flow regime (subcritical, supercritical, or mixed) appear to be correct? Examine the profile at all structures; are the structures correctly analyzed?
19.5.3 Reviewing Output Summary Tables Most computer programs used for floodplain analysis also provide summary tables listing the computed results at each cross section. Examine these summary tables and note locations where critical depth occurs, where there are significant changes in top width or flow distribution from one cross section to the next, where the energy slope (or water surface elevation) changes significantly from one cross section to the next, where there are crossing profiles (multiple profiles in which the computed water surface profile for a lower flow rate equals or exceeds the water surface profile for a higher flow rate, at one or more cross sections), and where there are other unusual conditions. If there is a large change in the water surface elevation or velocity head between consecutive cross sections, examine the distribution of discharge in the channel and floodplains. The distribution of flow in the cross section should also not vary too greatly
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FLOODPLAIN HYDRAULICS
Floodplain Hydraulics 19.43
from one cross section to the next. For example, it may be unreasonable for 80 percent of the flow to be in the channel at one cross section and only 20 percent of the flow to be in the channel at the next cross section. Check the minimum elevation in the cross section and the bank elevations; a large change in these elevations may explain a large change in water surface elevation. At locations where radical changes in top width of flow occur from one cross section to the next, determine with the aid of a topographic map the paths that the flow is likely to follow between the two cross sections. Flow cannot generally contract at a rate greater than 1:1, or expand at a rate greater than 4:1. In other words, the top width of flow should not decrease at a rate greater than 1 ft laterally for each foot flow travels in the downstream direction. Similarly, the top width of flow should not increase at a rate greater than 1 ft laterally for each 4 ft flow travels in the downstream direction. If the results of the floodplain analysis indicates that the top width of flow is changing at a greater rate, the cross sections or reach lengths may need to be modified accordingly. At locations where radical changes in energy slope occur (e.q., a change of 50 percent to 100 percent from one cross section to the next), the computation for friction loss may be inaccurate. It is probable that cross section geometry is not properly described and/or additional cross sections are required to limit the change in energy slope to acceptable levels.
19.5.4 Reviewing the Input Data A detailed review of the input data for the floodplain analysis should be performed on a section-by-section basis. The input data should be checked for accuracy and completeness, including the geometric data used for cross sections and structures, the reach lengths, the methods used to analyze bridges and other structures, the locations of all changes in flow rates, and the values of all loss coefficients. Some computer programs provide specialized output tables that list certain input data values (such as reach lengths or loss coefficients), to facilitate quick checking of these important data values. Loss coefficients should be consistent throughout the analysis. This does not mean that the loss coefficients should not vary; it means that the same physical conditions should be represented by the same loss coefficients throughout the entire analysis. Normal values of 0.1 to 0.3 should be used for the contraction and expansion loss coefficients at most cross sections. Values should generally be increased at bridges or at other locations where drastic increases or decreases in the floodplain cross section geometry occur. Extremely abrupt contractions or expansions, such as at culverts, should be represented using values of 0.6 to 0.8 or even higher. The graphical output provided by some floodplain analysis computer programs should be used as much as possible to get a quick view of the results. All cross sections should be plotted and reviewed, at least on the computer screen. The cross section plots will help find mistakes in the input data and in the approach used for the analysis.
19.5.5 Skewed Cross Sections In some cases, the surveyed cross section may not represent the actual cross sectional area of the channel perpendicular to the direction of flow, because the cross section was surveyed at a skew. Skew is corrected by multiplying each cross section station value by the
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FLOODPLAIN HYDRAULICS
19.44
Chapter Nineteen
cosine of the skew angle, which reduces the distance between adjacent cross section points. In effect, the cross section is projected onto a plane that is perpendicular to the direction of flow in the stream. An adjustment should be made where the angle of the cross section is 18 degrees or more different from perpendicular to the flow. There may be cross sections in which the direction of flow in the floodplains differs from the direction of flow in the channel. If so, only a portion of the channel cross section may have to be adjusted for skew.
19.5.6 Detailed Analysis of Roadway Crossings Roadway crossings are the source of many difficulties and errors in floodplain analysis. Although the capabilities of floodplain analysis computer programs have improved substantially, it is occasionally extremely difficult, if not impossible, to exactly model flow patterns through a roadway crossing. For all roadway crossings, the reasonableness of the computed energy losses should be reviewed. In addition, the various items of input data, including loss coefficients, roadway profiles, bank stations, effective area option, and reach lengths should be checked again after the analysis. If cross sections have been repeated through the stream reach around the roadway crossing, the uniformity of the channel through this reach should be checked. 19.5.7 Verification and Adjustment of Floodplain Analysis Where the data exist, a floodplain analysis should be verified against high water marks from actual past floods and/or the recorded rating curve at each stream gage in the study. Where no data exist, a sensitivity test should be performed to determine how critical the n value estimate and other key input variables might be. The following changes should be considered to allow the floodplain analysis to better match actual data: 1.
2. 3. 4. 5. 6. 7.
8.
Roughness coefficients. Change channel or floodplain roughness coefficients (within reasonable limits; if it is necessary to set roughness coefficients to unreasonable values to match historical results, there must be unresolved problems with other input data values, as listed below). Cross section errors. Revise survey data to correct for errors. Ineffective flow areas. Eliminate flow from ineffective areas. Starting conditions. Use a different starting water surface elevation or conditions to better match observed conditions. Cross sections. Insert additional cross sections to better reflect the true channel shape, or to eliminate excessive reach lengths. Bridge models. Correct the bridge models. Special conditions. Modify the input data to reflect special conditions during the observed flood (such as debris at bridge, levee or dam failures, diversions, and so on). Revised conditions. Modify the input data to reflect changed conditions since the recording of the data (such as a new bridge, channel clean-out, encroachment, land use changes, and the like).
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FLOODPLAIN HYDRAULICS
Floodplain Hydraulics 19.45
9.
10
Flow rates. Adjust the flow rates according to the results of a hydrologic analysis. Carefully consider how to apply available rainfall gage data to different portions of the watershed. Use weather radar data, if available, to provide more detailed information on rainfall distributions. (Weather radar data must generally be calibrated against actual rainfall gage readings to provide adequate accuracy for a hydrologic analysis.) Evaluate observed data. Consider the possibility that data supplied by local residents concerning actual flooding conditions may be faulty.
19.6 Floodway Determination A floodway is a corridor of effective flow area. It usually consists of the stream channel and any adjacent floodplain areas needed to convey a particular flood event within acceptable limits of water surface elevation and flow velocity. However, an overland flow path with no defined channel may also be called a floodway. An example of a floodway with no defined channel would be an overflow zone where flows spill over from one watershed to another during a flood event. For a floodway to be effective, it must be relatively clear of obstructions that would interfere with flows during a flood event. This may be accomplished in several ways. A public agency may obtain ownership of the floodway to preserve or enhance its conveyance capacity, or the public agency may obtain an easement that allows the periodic inundation of the floodway as needed. However, within the United States, the most common method of maintaining the conveyance capacity of floodways is by land use restrictions enacted through local floodplain management regulations. These regulations are required for local communities that desire to participate in the National Flood Insurance Program (NFIP). In the NFIP, a specific definition of the floodway is used for stream channel systems. This definition requires that the conveyance capacity of the floodway be sufficient for the 100-year storm event. However, the NFIP regulations recognize that it is neither practical nor desirable to preserve the entire 100-year floodplain as a floodway in all locations. Therefore, some obstructions in the 100-year floodplain are allowable under NFIP regulations. These obstructions, called encroachments, might include fill or structures associated with private development or public works projects such as roadways. Under NFIP regulations, three limits on encroachments are imposed: 1. The cumulative effect of all encroachments must not result in an increase of more than 1.00 ft in the 100-year water surface profile. It should be noted that local communities have the right to enact floodplain management regulations that are more stringent than the minimum NFIP requirements. 2. Encroachments cannot extend into the channel. 3. Encroachments must be represented as an equal loss of conveyance on both sides of the stream channel. Figure 19.29 illustrates this concept. The conveyance capacity is based on Manning’s equation described in Chap. 3. It may be represented as follows: 2/3 K = ᎏᎏAR (19.8) ᎏ n where K is the conveyance (more properly called “Manning’s conveyance”) in flow units, ⫽ 1 for SI units and 1.49 for American units, n is the Manning roughness coefficient, A
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FLOODPLAIN HYDRAULICS
19.46
Chapter Nineteen
FIGURE 19.29 Illustration of floodway.
is the cross-sectional area, and R is the hydraulic radius of the cross section (equal to the cross-sectional area of flow divided by the wetted perimeter of flow) in length units. The Manning’s conveyance value includes every term from Manning’s equation that pertains to the characteristics of the cross section. Only the slope of the energy grade line is omitted, because it depends on conditions downstream. Therefore, the Manning’s conveyance values serve as a measure of the flow-carrying capacity of a particular cross section or segment of a cross section, disregarding downstream or upstream conditions. Various computer programs can be used to assist in determining a floodway, but the HEC-RAS and HEC-2 programs are the most commonly used. The following steps are involved: 1. Natural conditions profile. Compute a water surface profile for conditions existing prior to encroachment. 2. Floodway profile. Compute another (higher) water surface profile representing the floodway, with encroachments. Using the multiple-profile capability of HEC-RAS or HEC-2, both these profiles can be computed in the same run, and the software will automatically provide a constant comparison of the two profiles. Both HEC-RAS and HEC-2 provide several methods of specifying encroachments for floodway studies. The two most commonly used are called Method 4 and Method 1. Method 4 provides the most direct means for delineating a floodway. The user prepares program input that specifies a target incremental increase in water surface elevation (normally 1 ft) at each cross section. This is often called the “target surcharge” The computer program then inserts an artificial encroachment into the left and right floodplain portions of each cross section, to remove enough total conveyance from each cross section to compensate for the increased channel conveyance provided by the increased water surface elevation. The program will not encroach into the channel as defined by the left and right bank stations. Because the program considers only the cross section conveyance in its determination of the encroachment stations, a water surface profile computed for the encroached floodway will not have elevations exactly equal to the desired value (natural elevation plus specified incremental amount). Flow rate and velocity are often redistributed whenever a cross section is modified. These changes, plus the backwater effects from downstream encroachments, can produce a result significantly different than the specified incremental change in water surface elevation. Therefore, it is generally necessary to make several trial
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FLOODPLAIN HYDRAULICS
Floodplain Hydraulics 19.47
runs using Method 4, making adjustments to the target surcharges with each run. At cross sections where the floodway profile exceeds the natural conditions profile by more than the desired amount, the target surcharge must be reduced. Conversely, where the program was not able to achieve the desired surcharge, the target should be increased for the next trial. Because the results at a cross section can be affected by downstream results, it is important to make adjustments to the target surcharge beginning from the downstream end of the stream system. After the floodway profile meets the desired conditions as closely as possible, the floodway computations can be completed with Method 1. In Method 1, the user simply specifies the stations for the encroachment limits on the left and right side of the stream channel. Initially, these would be set to the values computed in the final Method 4 analysis, but might be adjusted to make the floodway boundary more uniform. The stations of encroachment should be plotted in plan view and also on cross section plots to further evaluate the results. The encroachment stations must be between the channel bank station and the limit of the floodplain for each cross-section, without exception. Plotting or mapping of the floodway is crucial at this stage because the floodway must be a “smooth” corridor with fairly gradual transitions in its width and orientation, and no great undulations. The floodway is also subject to the 4:1 expansion and 1:1 contraction limitations. Additionally, since the floodway is defined as a zone of relatively high flow, this zone should be able to sustain high flow and not be blocked with obstacles such as dense development.(Fig. 19.30). The floodway surcharge should be close to but never greater than the maximum limit of 1.00 ft. Acceptable surcharges are usually not less than 0.8 ft as long as the floodway boundary is “smooth” and reasonable. Negative surcharge values may be caused by excessive encroachment, errors in the bridge modeling, or insufficient encroachment at the downstream section. Floodway encroachments must not be located outside the base 100-year floodplain boundaries, or within the channel bank stations. To ensure against these errors, the floodway stations should be checked against the floodplain stations and the channel bank stations.
FIGURE 19.30 Floodway Method 4 and Method 1
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FLOODPLAIN HYDRAULICS
19.48
Chapter Nineteen
Flow velocities should not greatly increase in the floodway run. In channels with high velocities it is possible to encroach the floodway boundaries and force very high velocities. This, in turn, can lower the floodway profile below the base 100-year profile by creating extremely high-velocity heads. This is not a valid floodway model and the encroachments should be widened to remove the anomaly. When flow is in the supercritical regime, or where velocity conditions are such that normal encroachment analyses are not possible or are inappropriate, the allowable rise should be applied to the energy grade line instead of the water-surface elevation. When performing a flood insurance restudy, the existing floodway configuration should be retained wherever possible. The floodway shown on the map, for most communities, has been adopted via ordinances and, thus, has gained regulatory status. This regulatory floodway is intended to remain static. However, floodway revisions are justifiable and necessary if restudy data indicate an increase in surcharge above the maximum limit, or if, as a result of improved data, the width or configuration of the floodway necessitates a change from that shown on the effective map. Note that while topographic mapping may guide the configuration of the floodway, it does not entirely dictate it and will not typically override the regulatory status of an adopted floodway. Floodway profiles must start with a known water surface elevation. Normally, this will be 1 ft above the base flood elevation at the starting cross section. However, if the base 100-year water surface profile is contained within the banks of the channel at its mouth, then the starting water surface elevation for the floodway will equal the base 100-year water surface elevation (because no encroachment—and therefore no surcharge—is possible). Even if the base 100-year starting water surface elevation is out of its banks at the mouth, the starting water surface elevation for the floodway will be the result of encroaching the base floodplain to either the maximum allowable surcharge (usually 1 ft) or the banks of the channel, whichever occurs first. The computation of a floodway on a tributary stream should be based on the 100-year flood discharge and elevation of that stream only and normally should not include consideration of any backwater flooding from the main stream. Therefore, the floodway elevations in the lower reach of a tributary subject to backwater flooding may be lower than those used to plot the flood profiles.
19.7 CONCLUSION This chapter has provided a fairly detailed overview of the practical considerations involved in identifying the floodplain for a stream channel. The following steps have been included: • Identifying and obtaining sources of information already available • Planning field data collection operations • Identifying the computer program best suited to the analysis • Performing the analysis and assessing the results Flooding worldwide is a serious problem. It causes more deaths and property damage worldwide than any other type of natural disaster. Flooding causes twice as many deaths worldwide as tropical cyclones, and as many as earthquakes, drought, and other disasters combined (McDonald and Kay, 1988).
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FLOODPLAIN HYDRAULICS
Floodplain Hydraulics 19.49
In the United States alone, approximately 7 percent of the land area is within the 100year floodplain. However, the percentage of urban area within the floodplain is much higher: about 16 percent. The total property value within the floodplain, already hundreds of billions of dollars, is growing at a rate of about 1.5 percent to 2.5 percent per year (McDonald and Kay, 1988). Flooding is not a temporary problem; it is a permanent feature of the environment that human society is learning to accept, understand, and accommodate. The techniques described in this chapter should allow for a better understanding of the extent and characteristics of flooding along stream channels. Through this improved understanding, better decisions concerning the appropriate use of floodplain property should be possible.
REFERENCES Arcement, G. J. Jr., and V. R. Schneider, Guide for Selecting Manning's Roughness Coefficients for Natural Channels and Flood Plains, Report No. FHWA-TS-84-204, U.S. Department of Transportation, Federal Highway Administration Washington, DC. Barnes, H. H., Jr., “Roughness Characteristics of Natural Channels,” Geological Survey WaterSupply Paper 1849, U.S. Department of the Interior, USGS, Washington, DC, 1967. Beasley, J. G., “An Investigation of the Data Requirements of Ohio for the HEC-2 Water Surface Profiles Model,” master's thesis, Ohio State University, 1973.Columbus OH. Brown, S. A., S. M. Stein, and J. C. Warner, Urban Drainage Design Manual, Hydraulic Engineering Circular No. 22, Federal Highway Administration, U.S. Department of Transportation, Washington, DC, 1996. Chow, V. T., Open Channel Hydraulics, McGraw-Hill, Inc. New York, 1959. Federal Highway Administration (FHWA), FESWMS-2DH Finite Element Surface-Water Modeling System: Two-Dimensional Flow in a Horizontal Plane Users Manual, Publication No. FHWARD-88-177, 1989.Washington, DC Federal Emergency Management Agency (FEMA), Guidelines And Specifications for Study Contractors, 1995.Washington, DC Federal Highway Administration, Urban Drainage Design Manual, Hydraulic Engineering Circular No. 22, U.S. Department of Transportation, Washington, DC, 1996. Fread, D. L., DAMBRK: The NWS Dam-Break Flood Forecasting Model, National Weather Service, Office of Hydrology, Silver Spring, MD, 1982. Fread, D. L., and J. M. Lewis, “FLDWAV: A Generalized Flood Routing Model,” Proceedings of National Conference on Hydraulic Engineering, Colorado Springs, CO, 1988. Fread, D. L., and J. M. Lewis, “Parameter Optimization for Dynamic Flood Routing Applications with Minimal Cross-Sectional Data,” Proceedings: ASCE Water Forum '86, World Water Issues in Evolution, Long Beach, CA, 1986, pp. 443–450. Fread, D. L., and J. M. Lewis, “FLDWAV: A Generalized Flood Routing Model,” Proceedings of National Conference on Hydraulic Engineering, Colorado Springs, CO, 1988. Fread, D. L., “NWS FLDWAV Model: The Replacement of DAMBRK for Dam-Break Flood Prediction,” Proceedings: 10th Annual Conference of the Association of State Dam Safety Officials, Inc., Kansas City, Mo, 1993, pp. 177–184. Huber, W. C., and R. E. Dickinson, Storm Water Management Model, Version 4: User's Manual, Environmental Research Laboratory, Environmental Protection Agency, Athens, GA, 1988. Hydrologic Engineering Center (HEC), Accuracy of Computed Water Surface Profiles, Research Document 26, U.S. Army Corps of Engineers Water Resources Support Center, Davis, CA, 1986. Hydrologic Engineering Center (HEC), HEC-2 Water Surface Profiles Users Manual, U.S. Army Corps of Engineers Water Resources Support Center, Davis, CA, 1990a.
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FLOODPLAIN HYDRAULICS
19.50
Chapter Nineteen
Hydrologic Engineering Center (HEC), UNET: One-Dimensional Unsteady Flow Through a Full Network of Open Channels, Users Manual, U.S. Army Corps of Engineers Water Resources Support Center, Davis, CA, 1990b. Hydrologic Engineering Center (HEC), HEC-RAS River System Analysis System,User’s Manual, Version 2.2, U.S. Army Corps of Engineers Water Resources Support Center, Davis, CA, 1998a, 1998b, and 1998c Hydrologic Engineering Center (HEC), HEC-RAS River Analysis System, Hydraulic Reference Manual, Version 2.0, U.S. Army Corps of Engineers Water Resources Support Center, Davis, CA, 1997b. Hydrologic Engineering Center (HEC), HEC-RAS River Analysis System, Applications Guide, Version 2.0, U.S. Army Corps of Engineers Water Resources Support Center, Davis, CA, 1997c. McDonald, A. T., and D. Kay, Water Resources Issues & Strategies, Longman Scientific & Technical, Essex, UK, 1988. Natural Resources Conservation Service (NRCS), “Computer Program for Water Surface Profiles.” Part 630, in National Engineering Handbook, 1993.Washington, DC. Reed, J. R. and A. J. Wolfkill 1976, “Evaluation of Friction Slope Models”, River ‘76 Symposium on Inland Waterways for Navigation Flood Control and Water Diversions, Colorado State University, Fort Collins, CO. Roesner, L. A., et al., Storm Water Management Model User's Manual Version 4: EXTRAN Addendum, EPA/600/3-88/001b, Environmental Research Laboratory, Protection Agency, Athens, GA, 1988. U.S. Geological Survey, Water Surface PROfiles, WSPRO, User's Manual, Reston, VA, 1990. Waterways Experiment Station (WES) Hydraulics Laboratory, Users Guide to RMA2 WES Version 4.3, U.S. Army Corps of Engineers, Vicksburg, MS, 1997.
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Source: HYDRAULIC DESIGN HANDBOOK
CHAPTER 20
FLOW TRANSITIONS AND ENERGY DISSIPATORS FOR CULVERTS AND CHANNELS Larry W. Mays Department of Civil and Environmental Engineering Arizona State University Tempe, Arizona
20.1 FLOW TRANSITIONS FOR CULVERTS Flow transitions are changes in the cross section of an open channel over short distances. They are designed to have a minimum amount of flow disturbance. Figure 20.1 illustrates the various types of transitions; the two most common ones are the abrupt (headwall) and the straight line (wingwall). Highway culverts typically are designed to operate with an upstream headwater pool that dissipates the of the channel approach velocity. This type of situation does not require an approach flow transition. Outlet transitions (expansions) should be considered in the design of all culverts, channel protection, and energy dissipators. Transition design can be categorized as • culverts with outlet control (subcritical flow) and • culverts with inlet control (supercritical flow).
20.1.1 Culverts with Outlet Control Culverts with outlet control can have abrupt expansions or gradual transitions. In an abrupt expansion, the water surface plunges or drops rapidly and the flow spreads out. The potential energy stored as depth is converted to kinetic energy with a higher velocity of flow. The transition (apron) end velocity can be determined using the experimental results of Watts (1968) (Figs. 20.2 and 20.3). Figure 20.2 relates the average depth brink / 0) for a rectangular outlet to the Froude number. Figure 20.3 is a similar depth ratio (yyA/y relationship for pipe culverts. These curves in Figs. 20.2 and 20.3 were developed for Fr’s ranging from 1.0 to 2.5, the applicable range for most abrupt outlet transitions. Usually, a low tailwater is encountered at culvert outlets and the flow is supercritical on the outlet apron. 20.1 Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
FLOW TRANSITIONS AND ENERGY DISSIPATORS FOR CULVERTS AND CHANNELS
20.2
Chapter Twenty
FIGURE 20.1 Transition types. (From Corry et al., 1975)
The design procedure for abrupt expansions is as follows (Corry et al., 1975): 1. Determine V0 and y0 at the culvert outlet using Fig. 20.4 for box culverts or Fig. 20.5 for circular culverts. vo 2. Compute Froude number, Fr gyo 3. Determine optimum flare angle (θ) using tan θ Fr /3 (Blaisdell and Donnelly, 1949). If the selected wingwall flare (θw) is greater than θ, consider reducing θw to θ. 4. Use Fig. 20.2 (for box culverts ) or Fig. 20.3 (for pipe culverts) to determine the downstream average depth (yA) knowing Fr and the desired distance L (expressed in multiples of D, diameter).
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FLOW TRANSITIONS AND ENERGY DISSIPATORS FOR CULVERTS AND CHANNELS
FIGURE 20.2 Average depth for abrupt exapnsion below rectangular culvert outlet. (From Corry et al., 1975)
Flow Transitions and Energy Dissipators for Culverts and Channels 20.3
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FLOW TRANSITIONS AND ENERGY DISSIPATORS FOR CULVERTS AND CHANNELS
Chapter Twenty
FIGURE 20.3 Average depth for abrupt expansion below circular culvert outlet. (From Corry et al., 1975).
20.4
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FLOW TRANSITIONS AND ENERGY DISSIPATORS FOR CULVERTS AND CHANNELS
Flow Transitions and Energy Dissipators for Culverts and Channels 20.5
FIGURE 20.4 Dimensionless rating curves for the outlets of rectangulars culverts on horizontal and mild slopes (From Simons et al; 1970)
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FLOW TRANSITIONS AND ENERGY DISSIPATORS FOR CULVERTS AND CHANNELS
20.6
Chapter Twenty
FIGURE 20.5 Dimensionless rating curve for the outlets of circular culverts on horizontal and mild slopes (From Simons, et al., 1970)
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FLOW TRANSITIONS AND ENERGY DISSIPATORS FOR CULVERTS AND CHANNELS
Flow Transitions and Energy Dissipators for Culverts and Channels 20.7
5. Determine the average velocity (V VA) for box culverts using VA 1.65 3Fr V0 and for pipe culverts, (for L 3D) using Q VA 1.65 0.45 5 . V g D
6. Compute downstream width W2 = Wo 2L tan θ, where tanθ Fr/3. r If θw , then use θw to compute W2. 7. If θ is used to compute W2, then compute the downstream depth y2 using W2 and VA. Because the flow prism is laterally confined, y2 will be larger than yA. If θW is used, y2 yA, and the average flow width is WA Q/V VAyA. If WA W2, use W2 to compute y2 Q/V VAW2. Example 1 (adapted from Corry et al., 1975). Determine the width of an abrupt expansion and the hydraulic condition (y and V V) at the end of an abrupt expansion for a 5 ft by 5 ft, 200 ft-long reinforced concrete culvert on a 0.2 percent slope (S0 0.002 ft/ft). The discharge is Q 270 cfs and a wingwall flare (θW 45°) with a 10 ft apron is considered. Given
TW 0 dc 5 ft and D
Solution. Step 1.
QB1g 2750321.2 4.5 ft
Q 270 4.83; dc 3 BD3/2 5(5)3/2
2
2
3
TW and 0. From Fig. 20.4, y0 / D 0.68; therefore, y0 and V0 can be deterD mined to be y0 0.68(5) 3 .40 ft and V0 270 [3.4 (5)] 15.88 ft/s.
/
Step 2. The outlet Fr V0 g .2(3 .4) 1.52 y0 15.88 32
/
Step 3. The optimum flare angle is determined using tan θ 1/3 Fr 1 (1.52) 3 0.507, so θ 26.9o. Step 4. With the apron length/width 10/5 2 and L 2.0B, use Fig. 20.2 to compute the average depth where ya y0 0.26 and ya 0.26 (3.4) 0.88 ft.
/
Step 5. Compute the average velocity, VA/V V0 1.65 0.3 Fr 1.65 .3 (1.52) 1.19, so VA 1.19 V0 1.19 (15.88) 18.90 ft/s Step 6. Determine the downstream width W2 W0 2L tan θW using θ θW = 45° because θW θ and W0 5 ft, then W2 5 2 (10) (tan 45°) 25 ft. Step 7. θW is used above, so y2 yA 0.88 ft and W2 Q/(V VA yA) 16.1 25 ft. When subcritical flow is maintained throughout a culvert, gradual transitions can be used. Referring to Fig. 20.6, upstream of section 1 where some backwater exists because of the culvert, flow is transitional from a channel into the culvert and out. According to the Federal Highway Administration (FHWA, 1978), a flare angle of 17.5o (4.5 to 1) or flatter provides a gradually varied transition that can be analyzed using the energy equation. Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
FLOW TRANSITIONS AND ENERGY DISSIPATORS FOR CULVERTS AND CHANNELS
Chapter Twenty
FIGURE 20.6 Definition sketch. (From Corry et al., 1975)
20.8
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FLOW TRANSITIONS AND ENERGY DISSIPATORS FOR CULVERTS AND CHANNELS
Flow Transitions and Energy Dissipators for Culverts and Channels 20.9
Referring again to Fig. 20.6, the headloss in the contraction HLC is V 2 V12 2 HLC CC 2g 2g
(20.1)
and the headloss in the expansion is V 2 V42 3 HLE Ce , 2g 2g
(20.2)
where transition loss coefficients are listed in Table (20.1) for the transition types (USACE, 1970). The design procedure for gradual transitions is as follows (Corry et al 1975): 1. Use Manning’s equation to compute y4 yn and V4, knowing Q, So, n and the outlet geometry. 2. Compute the critical depth using yc k1 (Q/W) W 2/3. For box culvert, k1 0.315 where Q is in cfs and W is the box culvert width in ft. In SI units k1 0.467 with Q in m3/s and W in m. Compare yn and yc to ensure subcritical flow. 3. For the chosen transition type, use Table 20.1 to obtain the transition loss coefficients Cc and Ce. 4. Choose y3 1.1 yc to have a culvert with a flow depth conservatively above. yc 5. Compute the culvert width using the following energy equation, ignoring the headloss caused by friction: V 2 V42 V42 V32 3 Z4 y4 Ce Z3 y3 2g 2g 2g 2g
V3 Q/W W3y3 where y3 1.1 yc 1.1k1 (Q/W W3)2 For box culverts and V4 Q/W W4y4. Neglecting Z3 Z4 for a short reach and a mild bottom slope,
Q 2 1 Q 1 y4 (1 Ce) y3 (1 Ce) W4y4 2g W3y3 2g 2
2 Q 2 1 Q 2/3 1 Q y4 (1 Ce) 1.1k1 Ce) 2/3 (1 C W3) ] 2g W3 W4y4 2g W3 [1.1k1 (Q/W TABLE 20.1
Transition Loss Coefficients
Transition Type
Contaraction Cr
Expansion Ce
Warped
1.10
0.20
Cylindrical quadrant
0.15
0.25
Wedge
0.30
0.50
Straight line
0.30
0.50
Square end
0.30
0.75
Source: Corry, et al (1975).
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FLOW TRANSITIONS AND ENERGY DISSIPATORS FOR CULVERTS AND CHANNELS
20.10
Chapter Twenty
Q 1 2/3 1 y4 (1 Ce) 1.1k1Q W3 W4y4 2g
2/3
2
2/3
2/3
1 Q 2 1.21k1 W3
1 (1 Ce) 2g
Q 1 1 Q 1 2/3 y4 (1 Ce) 1.1k1Q 2 (1 Ce) 1.21k1 2g W3 W4y4 2g 2
2/3
2/3
2/3
[ ] [B] 1 [A W3
and W3 ([B]/[A])3/2 where
(20.3)
Q [ ] y4 1 (1 Ce) [A W4y4 2g 2
and 2/3
Q 1 [B] 1.1k1Q2/3 (1 Ce) 1.21k12 2g 6. Compute y1 with y2 y3 using the energy equation V22 V22 V12 V12 Z2 y2 Cc Z1 y1 . 2g 2g 2g 2g
(20.4)
7. If the amount of backwater (y1- yn) exceeds a preferred or required freeboard, then select a larger culvert width and calculate y3 using Eq. (20.3) given in Step 5 (return to Step 5). 8. If the culvert width is acceptable, use the flow conditions to compute the transition length using a 4.5:1 flare (USACE, 1970): (W W4 W3) LT 4.5 2
(20.5)
or
(W W1 W2) LT 4.5 (20.6) 2 9. Compute the water surface profile through the structure using a standard step backwater analysis. This will include an evaluation of friction losses. Example 2. Determine the dimensions for a culvert and gradual transition needed for a 3 m-wide rectangular flood control channel at a slope of 0.001 m/m. The culvert length is 30.5 m, and is to convey 8.5 m3/s. Use n 0.02. Solution: Step 1. Assuming normal depth at Section 4 (refer to Fig. 20.6), then use Manning’s equation to compute y4 yn 1.99 m and Vn 1.42 m/s.
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FLOW TRANSITIONS AND ENERGY DISSIPATORS FOR CULVERTS AND CHANNELS
Flow Transitions and Energy Dissipators for Culverts and Channels 20.11
2/3
2/3
Q 8.5 Step 2. The critical depth for section 3 is yc 0.467 0.467 W 3 Step 3. Use a straight line transition with Ce 0.5 and Cc 0.3.
0.935 m.
Step 4. y3 1.1yc 1.1 (0.935) 1.03 m. Step 5. Compute W3 using Eq. (20.3). W3 ([B]/[A [ ])
3/2
2/3
2/3
[B] 1.1k1 Q
Q 1 (1 Ce) 1.21k12 2g 2/3
1.1(0.467)8.5
2/3
8.5 1 (1 0.5) 1.21(0.467)2 2(9.81)
2.542 2
2
Q 1 8.5 1 [ ] y4 (1 Ce) 1.99 [A (1 0.5) W4y4 2g 3(1.99) 2(9.81)
= 2.042 W3 (2.54, 42/2.042)3/2 1.39 m. Take W3 1.4 m. Step 6. Compute y1 with y2 = y3 using Eq. (20.4): 2
2
Q 1 Q 1 Z2 Z1 y2 (1 Cc) y1 (1 Cc) W2y2 2g W1y1 2g 2
1.03 (1 0.3)
8.5 1.4(1.03)
2
8.5 1 1 y1 (1 0.3) 2(9.81) 3y1 2(9.81)
0.53 . 3.33 y1 y12 Then y1 3.28 m, which is 1.29 m above yn. Step 7. Because y1 is 1.29 m above the normal depth and because only 0.6 m free board is available, we go back to Step 5 and compute y3. Assuming that the culvert width is 1.45 m: Repeat step 5, 2
2
Q Q y4 1(1 Ce) y3 1(1 Ce) W4y4 2g W3y3 2g 2
2
8.5 8.5 1 1 1.99 (1 0.5) y3 (1 0.5) 81) 81) 3(1.99) 2(9.81) 1.45y3 2(9.81)
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FLOW TRANSITIONS AND ENERGY DISSIPATORS FOR CULVERTS AND CHANNELS
20.12
Chapter Twenty
0.876 2.042 y3 y32 y3 1.76 m. Then, returning to Step 6, y1 using Eq. (20.4): 2
2
8.5 8.5 1 1 1.76 (1 0.3) y1 (1 0.3) 1.45(1.76) 2(9.81) 3y1 2(9.81)
0.53 2.50 y1 y12 y1 2.41 m. y1 is 0.42 m above the normal depth. Because a free board of 0.6 m is available; the design is OK. Step 8. Determine transition length using Eq. (20.5) for a 4.5:1 flare. (W W1 W2) (3 1.45) LT 4.5 4.5 3.49 m, or 3.5 m. 2 2 Step 9. Compute the water surface profile.
20.1.2 Culverts with Inlet Control Culverts with inlet control require the transitioning of supercritical flow. (see Fig. 20.7) Because supercritical flow is difficult to control without creating a hydraulic jump or other surface irregularities, the full flow area should be maintained. Because changing the flow smoothly requires a long structure, model studies should be performed to determine the transition geometry if a hydraulic jump is not desired. When a hydraulic jump is acceptable, Figs. 20.8 and 20.9 can be used (USACE, 1970). Such a design requires a rectangular channel and a long transition. The design procedure for supercritical flow contractions is as follows (Corry et al, 1975): 1. Flow conditions for the approach channel should be computed assuming normal flow (yn, Vn, Fr) using Manning's equation or other design aids. 2. Approach sections that are not rectangular should have a transition to the rectangular section with a bottom width of W1. This bottom width should be approximately equal to the average of the water surface top width (T) T and the trapezoidal section base width (Bw): i.e., W (T Bw) / 2. Compute the normal flow condition for this rectangular section using Manning's equation. 3. Assume a trial culvert width W2. Refer to Figure 20.7 for a definition of parameters. 4. Compute the contraction length needed to reduce W1 to W2. This is accomplished by varying the contraction using wall angle (θw) until L (W W1 W2) / 2 tan θw is equal to L1 L2 where L1 W1 / 2 tan β1 and L2 W2 / 2 tan (β2 θw). To minimize surface disturbances, L should be equal to L1 L2 (Corry et al, 1975). Choose W Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
FLOW TRANSITIONS AND ENERGY DISSIPATORS FOR CULVERTS AND CHANNELS
Flow Transitions and Energy Dissipators for Culverts and Channels 20.13
FIGURE 20.7 Supercritical inlet transition for rectangular channel. (From USACE, 1970)
a. Compute L (W W1 W2)/2 tan θw b. Find β1, y2 / y1, and Fr1 from tan β1 (1 ( 8 F r12 sin2β 1 3) tanθw 2 2 tan β1 1 8 F r12 sin2 β 1 1
(20.7)
y2 2 1 1 2 8 F s in β 1 r 1 1 y1 2
(20.8)
and
and
y y1 y2 y2 Frr22 1 Fr12 1 1 . y2 2y2 y1 y1 2
(20.9)
Alternatively, Figs. 20.8 or 20.9 can be used to approximately solve Eqs. (20.7), (20.8), and (20.9). c. Calculate L1 W1 / 2 tan β. / 2, and Frr2 using the same procedure as in Step 4b (that is, using β2 d. Compute β2, y3/y for β1, Fr2 for Fr1, y3/y / 2 for y2/y / 1 and the same θw). e. Calculate L2 W2 / 2 tan (β2 θw) Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
FLOW TRANSITIONS AND ENERGY DISSIPATORS FOR CULVERTS AND CHANNELS
20.14
Chapter Twenty
FIGURE 20.8 Supercritical inlet transition design curves for rectangular channels. (USACE, 1970)
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FLOW TRANSITIONS AND ENERGY DISSIPATORS FOR CULVERTS AND CHANNELS
Flow Transitions and Energy Dissipators for Culverts and Channels 20.15
FIGURE 20.9 Supercritical inlet transition design curves for rectangular channels (From Ippen, 1951)
f. Compare L with L1 + L2 if L L1 L2, then increase θW and repeat Steps 4a—f until L L1 L2. y y g. Compute y3 from y3 y1 2 3 . y1 y2
5. Compare depth y3 with the width W2. Culvert should be of a standard dimension. If not return to Step 3 using another W2 and repeat the design process until a better combination of y3 and W2 are found. Example 3 (adapted from Corry et al., 1975). Design the transition contraction and select the culvert size for a discharge of 300 cfs in a trapezoidal channel (6 ft bottom width, 2:1 side slope, S0 0.02 ft/ft and n 0.012) Solution: Step 1. The normal depth, velocity, and Fr are yn 1.67 ft, Vn 19.2 ft/s, and 19.2 Fr V 3.05m g g A/T 32 .2 (5 .6 )/12.7 Step 2. Because (T Bw )/2 (12.7 6) /2 9.4 ft, use W1 10 ft, a rectangular chan-
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FLOW TRANSITIONS AND ENERGY DISSIPATORS FOR CULVERTS AND CHANNELS
20.16
Chapter Twenty
nel. Use Manning’s equation to compute yn 1.54 ft and Vn 19.5 ft/s, then compute the Fr: 19 .5 Fr V 2.77. gy 32.2 ( 1 .54) Step 3. Assume a trial culvert width W2 5 ft. Step 4. Try θw 14° for Fr 2.8: L (10 5 ) / 2 tan 14° 10 ft β1 35°, y2/y / 1 1.8, Fr2 1.8 L1 10 / 2tan 35° 7.1 ft. β2 55°, y3/y / 2 1.6, Fr3 1.1 L2 5 / 2 tan (55° 14°) 2.9 ft. L1 L2 10 ft L, O.K. y3 1.54 (1.6) (1.8) 4.4 ft. Use w 14°, y3 4.4 ft., V3 13.6 ft/s, Fr3 1.1 Step 5. Because y3 4.4 ft. and W3 5 ft, a 5 5 box culvert will be satisfactory.
20.2 ENERGY DISSIPATION FOR CULVERTS AND CHANNELS 20.2.1 Hydraulic Jump Basins For supercritical flow expansions, the procedure outlined in Sec. 20.1 is applicable if the exit Fr is less than 3, if the location where the flow conditions are desired within three culvert diameters of the outlet and if So is less than 10 percent (Corry, et al 1975).For expansions outside these limits, a hydraulic jump basin (Fig. 20.10) should be used. This type of basin allows the flow to expand, drop or both, resulting in depth decreases, velocity increases, and an Fr increase in. Higher Fr's result in more efficient jumps and shorter basins. The design procedure for supercritical flow expansions with hydraulic jump basin is as follows (Corry, et al; 1975): 1. Compute the culvert brink depth yo using Figs. 20.4 or 20.5. 2. Compute the tailwater depth Tw in the downstream channel assuming normal flow (using Manning’s equation) or perform backwater analysis. 3. To determine the basin elevation, first select Z1 and then use the following steps. W1 in Fig. 20.10) and basin slopes Ss and ST (Fig. 20.10). a. Select basin width WB (W Slope of Ss or ST 0.5(2:1) or 0.33 (3:1) are satisfactory (Corry et al, 1975). b. Check WB using
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FLOW TRANSITIONS AND ENERGY DISSIPATORS FOR CULVERTS AND CHANNELS
Flow Transitions and Energy Dissipators for Culverts and Channels 20.17
FIGURE 20.10 Definition sketch basin transition. ( From Corry et al., 1975)
2L S 2 1 T T WB W0 3Fr0
(20.10)
where LT (Z Z0 Z1)/ST and the right-hand side is the limit that flares naturally in the slope distance L. c. Compute y1 using the following equation derived from the energy equation from the culvert outlet brink to the basin (Sec. 1 in Fig. 20.10). Use V1 Q / y1WB, to determine y1 and then V1: 1/2
Q y1WB 2g (Z Z0 Z1 y0 y1) V02
(20.11)
d. Compute the Fr V1 Fr1 gy1 e. Compute y2 using Eq. (20.12) for the hydraulic jump basin: y y2 1 1 8 F r 2 1 1 2
(20.12)
f. Compute Z3 from geometry using Z0
Z LT LB 2 S0 Ss So Z3 1 Ss
(20.13)
where (Z Z0 Z1) LT ST
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FLOW TRANSITIONS AND ENERGY DISSIPATORS FOR CULVERTS AND CHANNELS
20.18
Chapter Twenty
and from Fig. 20.11, determine L, which is defined as LB LB f( f y1, Fr1). g. Check value of Z1 by computing y2 Z2 and Z3 TW. If y2 Z2 Z3 TW, select another Z1 and repeat steps 4a to 4g until a balance is reached. 4. Compute LS and L: (Z Z3 Z2) LS Ss (Z Z0 Z3) L LT LB LS S0 Example 3. A supercritical flow expansion is to be designed for a reinforced concrete box culvert measuring 3 by 2 m. Determine the dimensions for the hydraulic jump basin using a design discharge of Q 11.8 m3/s. The slope is 6.5 percent, the invert outlet elevation is 30.5 m, the downstream channel has a bottom width of 3 m and side slopes of 2:1, and Manning’s n = 0.03. The brink depth is supercritical, y0 0.457 m and V0 8.47 m/s. Ss 0.5 and ST 0.5. (Refer to Figure 20.12)
FIGURE 20.11 Length of jump in terms of y1, rectangular channel. (From Corry et al., 1975)
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FLOW TRANSITIONS AND ENERGY DISSIPATORS FOR CULVERTS AND CHANNELS
FIGURE 20.12 Example problem 18.2.1 hydraulic jump basin. (Net to scale).
Flow Transitions and Energy Dissipators for Culverts and Channels 20.19
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FLOW TRANSITIONS AND ENERGY DISSIPATORS FOR CULVERTS AND CHANNELS
20.20
Chapter Twenty
Solution: Step 1. The brink depth and velocity are given: Fr0 4.0. Step 2. The tailwater depth is computed assuming normal depth. Manning’s equation is solved to obtain yn 0.57 m tailwater depth and Vn 4.85 m/s. (Refer to Fig. 20.12) Step 3. Assuming a basin elevation of Z1 25.9 m: a. Select WB 3 m and Ss ST 0.5 b. Check WB using Eq. 20.10, where W0 3 m,
/
2 WB 3 3 2LT S 1 3Frr0, OK. T
c. Compute y1 using Eq. (20.11) 1/2 2
11.8 y1(3)2(9.81)(30.5 25.9 0.457 y1) (8.47)
Solving y1 0.306 m and V1 Q/A / 11.8/[3(0.306)] 12.85 m/s. d. The Froude number is Fr 12.85/9 .81 (0 .306) 7.42
/
e. Compute y2 using Eq. (20.12). y2 0.306 1 8 (7 .42 )2 1 2 3.06 m. f. Compute Z3 using eq. (20.15). First compute LT (Z Z0 Z1)/2
(30.5 25.9)/0.5 9.2 m. From Fig. 20.11, LB/y / 1 63, so LB 63y1 63(0.306) 19.28 m, then Z3
[30.5 (9.2 19.28 25.9/0.5) 0.065] 28.34 m (0.065/0.5 1)
g. Check value of Z1: y2 Z2 3.06 25.9 28.96 and Z3 TW 28.34 0.58 28.92, y2 Z2 Z3 TW; therefore, Z1 25.9 is OK. 4. Compute Ls and then L. Ls (Z Z3 Z2) Ss (28.33 25.9)/0.5 4.86 m. Then
/
L LT LB Ls 9.2 19.28 4.86 33.34 m. Refer to Fig. 20.12 20.2.2 Forced Hydraulic Jump Basins 20.2.2.1 Saint Anthony Falls stilling basin. The Saint Anthony Falls, (SAF) stilling basin is a generalized design based upon model studies conducted by the U.S. Soil Conservation Service at the St. Anthony Falls Hydraulic Laboratory. University of Minnesota. Figure 20.13 illustrates the SAF stilling basin design which is recommended for small structures such as spillway outlet works, and for canals where the Fr ranges from 1.7 to 17 (at the dissipator entrance). Through the use of chute blocks, baffle or floor blocks, and an end sill, the basin length is about 80 percent of the free hydraulic jump length. The design procedure for SAF basins is as follows (Corry et al 1975): 1. Choose basin configuration and flare dimension, Z. (Refer to Fig. 20.13) Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
FLOW TRANSITIONS AND ENERGY DISSIPATORS FOR CULVERTS AND CHANNELS
Flow Transitions and Energy Dissipators for Culverts and Channels 20.21
2. Use the design procedure presented in Sec. 20.2.1 for supercritical expansions into hydraulic jump basins to determine basin width (W WB), elevation (Z1), length (L LB), total length (L), incoming depth (y1), incoming Fr (Fr1), and jump height (y2). Steps 5e and 5f in Sec. 20.2.1 are modified; for Step 5e, determine y2 using the sequent depth yj: y yj 1 1 8 F r 21 1 2
(20.14)
Fr12 y2 1.1 y 12 0 j
Fr1 1.7 to 5.5
(20.15)
y2 0.85yyj
Fr1 5.5 to 11
(20.16)
Fr1 11 to 17
(20.17)
2 1
Fr y2 1.0 yj 800
FIGURE 20.13 St. Anthony Falls stilling basin. (Form Blaisdell, 1959)
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FLOW TRANSITIONS AND ENERGY DISSIPATORS FOR CULVERTS AND CHANNELS
20.22
Chapter Twenty
For Step 55ff, compute LB: 4.5yj LB Fr10.76
(20.18)
3. Determine the dimensions of the chute block: Height:
h1 y1
Width:
W2 0.75y1 and W1 spacing
Wb Nc < (rounded) 2W1 WB Adjusted: W1 W2 (N N includes the 1/2 block at each wall) 2Nc c 4. Determine the dimensions of the baffle block: Number:
Height: h3 y1 Width: W3 spacing and W4 0.75y1 Basin width at baffle blocks: WB2 WB 2L LB/3Z Number of blocks: NB WB2/2W W3 rounded Adjusted W3 W4 WB2/2N NB Check total block width to insure that at least 40 to 55 percent of WB2 is occupied by blocks. Distance from chute blocks to baffle blocks = LB/3 5. End sill height: h4 0.07yyj. 6 Side wall height: y2 yj /3. Example 4. Determine the dimension of an SAF basin for the supercritical flow expansion described in Example 3. Solution: Step 1 Select a rectangular basin with no flare. Step 2 Steps 1 through 3(a — f) (in Example 3) for a supercritical flow expansion into a hydraulic jump basin. a. Given V0 8.47 m/s, y0 0.457 m, and Fr0 4.0. b. The tailwater depth TW yn 0.57 m and Vn 4.85 m/s. c. Assume that Z1 Z0 30.5 m. (Refer to Fig. 20.14) i. Compute yj using Eq. 20.14 with y1 y0 0.457 m and Fr1 Fr0 = 4.0. This assumes that Z1 Z0: i.e., the basin floor is the same elevation as the culvert outlet. y 1 8 F r 2 1 yj 1 1 2
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FLOW TRANSITIONS AND ENERGY DISSIPATORS FOR CULVERTS AND CHANNELS
FIGURE 20.14 Example 4 St. Anthony Falls stilling basin. (Not to scale)
Flow Transitions and Energy Dissipators for Culverts and Channels 20.23
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FLOW TRANSITIONS AND ENERGY DISSIPATORS FOR CULVERTS AND CHANNELS
20.24
Chapter Twenty 9.457 1 8 (4 )2 1 2
2.37 m. ii. Next, use Eq. (20.15) for Fr1 1.7 to 5.5 to compute y2: Fr12 y2 1.1 y 120 i (4)2 1.1 (2.37) 120
2.29 m Because y2 TW 0.57, we can lower the elevation of the basin. Use Z1 27.9 m with WB 3 m and ST Ss 0.5. WB is OK and no flare is used. iii. Compute y1 using the energy Eq. (20.11). The basin has been lowered so now y1 is not y0, the brink depth. 1/2
Q y1 WB 2g(Z Z0 Z1 y0 y1) V02 1/2
11.8 y1(3)2(9.81)(30.5 27.9 0.457 y1) (8.47)2 Solving y1 0.348 m. V1 11.8/(3 3 0.348) 5 11.3 m/s
/
/
iv. The Fr1 V V1 g y1 11.3 9 .81 (0 .348) 6.1 0.348 y v. Compute yj 1 1 8 Fr2 1 1 8 (6 .1 )2 1 2.83 m. 1 2 2 Fr12 (6.1)2 and y2 1.1 yi 1.1 2.83 2.24m. 120 120
vi. Compute LB using Eq. (20.18): LB 4.5yyj Fr 10.76 4.5 (2.83) 6.10.76 3.22m.
/
/
Compute LT using LT (Z Z0 Z1) ST (30.5 27.9) 0.5 5.2m.
/
/
Compute Z3 using Eq. (20.13) Z0 (LT LB Z2/Ss)S0 Z3 (S0/Ss 1) 30.5 (5.2 3.22 27.9/0.5)0.065 (0.065/0.5 1 29.7 Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
FLOW TRANSITIONS AND ENERGY DISSIPATORS FOR CULVERTS AND CHANNELS
Flow Transitions and Energy Dissipators for Culverts and Channels 20.25
vii. Check the assumption of Z1 27.9 m: y2 Z2 2.24 + 27.9 30.14 and TW Z3 0.57 29.7 30.2 m: y2 Z2 TW Z3. Because they are so close, then Z1 will be OK. Z3 Z2) Ss (29.7 27.9) 0.5 3.6 m, viii.LT 5.2 m, LB 3.22 m, Ls (Z and
/
/
L LT LB Ls 5.2 3.22 3.6 12 m. Step 3. Chute blocks: h1 y1 0.35, W1 0.75y1 0.26 m W4, Nc WB/2W1 3/ (2 ,26) 5.77 so use 6 blocks. Adjusted W1 WB (2N Nc) 3 (2 6) 0.25 m. This provides 5 blocks, 6 spaces, and a half block at each wall.
/
/
Step 4. Baffle blocks: h3 y1 0.35 m, W3 0.75y1 0.26 m W4. Basin width, WB2 WB 2L LB (3Z) Z 3 0 3 m (no flare), NB WB2 (2W W3) 3 (2
0.26) 5.77 so use 6 blocks. Adjusted W3 W4 3 (2 6) 0.25 m. Total block width 6(0.25) 1.5 m. Check percentage: 0.4 1.5 3.0 0.55, OK. This provides six blocks, five spaces, and a half-space at each wall. Distance from chute block LB/3 3.22/3 1.07 m.
/
/
/
/
/
Step 5. End sills: h4 0.07yyj 0.07(2.83) 0.2 m. Step 6. Side walls: height y2 yj 3 2.24 2.83 3 3.2 m. Refer to Fig. 20.14 with the dimension for the chute block shown.
/
/
20.2.2.2 Type II, III, and IV basins. The U.S. Bureau of Reclamation's Type II, III, and IV basins are illustrated in chapter 18, in Figs. 18.3, 18.4 and 18,7 respectively. Type II basin design. Use the design procedure presented in Sec. 20.2.1 for supercritical expansion into a hydraulic jump basin to determine WB, Z1, LB, L, y1, Fr1 and y2.
/
2 For Step 3e in that section, use C 1.1 to find y2 C1y1 1 F 8 F r 1 1 2. For Step 3f, use Fig. 20.15 to determine LB.
Determine the dimensions for the chute blocks and dentated sill height using the relations in Fig. 18.3. Type III basin design. Use the design procedure in Sec. 20.2.1 to determine basin
/
2 dimension. For Step 3e use C 1.0 to determine y2 C1y1 1 8 F r 1 2 For Step 1 3f, use Fig. 20.15 to determine LB. Use dimensions in Fig. 20.16 to determine chute block dimensions and spacing.
Use Fig. 18.4 to determine dimensions for baffle blocks and the end sill height. Type IV basin design. Use the same design procedure presented in Sec. 20.2.1 forsupercritical expansions into a hydraulic jump basin to determine the dimensions the basin.
/
2 1 8 F For Step 3e in the section, use C1 1.0 to determine y2 C1y1 r1 1 2. For Step 3f, use Fig. 20.15 to determine LB.
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FLOW TRANSITIONS AND ENERGY DISSIPATORS FOR CULVERTS AND CHANNELS
Chapter Twenty
FIGURE 20.15 U.S. Bureau of Reclamation Type II basin. (Corry et al, 1975)
20.26
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FLOW TRANSITIONS AND ENERGY DISSIPATORS FOR CULVERTS AND CHANNELS
FIGURE 20.16 Height of baffle piers and end sill (Type III basin). (From U.S. Bureau of Reclamation, 1987)
Flow Transitions and Energy Dissipators for Culverts and Channels 20.27
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FLOW TRANSITIONS AND ENERGY DISSIPATORS FOR CULVERTS AND CHANNELS
20.28
Chapter Twenty
Use dimensions in Fig. 18.7 to determine chute block dimensions and spacing. Use Fig. 18,7 to determine the end sill height. Example 5. Determine the dimensions of U.S. Bureau of Reclamation's Type II basin for the supercritical flow expansion described in Example 3. Solution: Step 1. Use the design procedure for a supercritical flow expansion into a hydraulic jump basin. (Refer to Examples 3 and 4). a. Given V0 8.47 m/s, y0 0.457 m, and Frr0 4.0. (Figure 20.17) b. The tailwater depth is TW yn 0.57 m, and Vn 4.85 m/s. c. Assume Z1 Z0 30.5 m, compute y2
/
y2 C1y11 8 Fr20 1 2
/
1.1(0.457) 1 8 (4 )2 1 2 2.6 m y2 TW (2.6 0.57). Therefore we need to lower elevation Z1 of the basin floor. i. Use a basin floor elevation of Z1 Z2 25.76 m, with WB 3 m, ST Ss 0.5. ii. WB is OK, no flare. iii. Compute y1 using energy Eq. (20.11). 1/2
Q y1WB 2g(Z Z0 Z1 y0 y1) V 02 11.8
y1(3)2(9.81)(30.5
1/2
25.76 0.457 y1) 8.472
Solving y1 0.3 m and V1 13.1 m/s.
/
iv. The Fr 13.1 9 .81 (0 .3) 7.6 1
/
v. Compute y2 1.1(0.3) 1 8 (7 .6 )2 1 2 3.39 m. vi. Using Fig. 20.15, LB y2 4.3, so LB 4.3(3.39) 14.6 m:
/
LT (Z Z0 Z1) ST (30.5 25.76) 0.5 9.5 m.
/
/
Using Eq. (20.13), compute Z3
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FLOW TRANSITIONS AND ENERGY DISSIPATORS FOR CULVERTS AND CHANNELS
Flow Transitions and Energy Dissipators for Culverts and Channels 20.29
Z3
[30.5 (9.5 14.6 25.76/0.5)0.065] 28.57 m. (0.065/0.5 1)
vii. Check Z1, y2 Z2 3.39 25.76 29.15 and Z3 TW 28.57 0.57 29.14, OK Z3 Z2) Ss (28.57 25.76) 0.5 5.6 m and viii. Compute Ls (Z
/
/
L LT LB Ls 9.5 14.6 5.6 29.6 m. Step 2. Chute blocks: h1 W1 W2 y1 0.3 m, NC 3 5, OK. Side-wall 2(0.3) spacing y1/2 0.15 m. Step 3. Dentated sill: h2 0.2y2 0.2(3.39) 0.7 m. W3 W4 0.15y2 0.15(3.39) 0.51 m. Ns WB W3 = 3/0.51 m = 5.88.
/
Use 5, which provides three blocks and two spaces, each of which is 0.6 m wide. Refer to Fig. 20.17 for the dimensions of the basin.
20.2.3 Impact-Type Energy Dissipation (USBR Type VI Basin) Figure 20.18 illustrates the U.S, Bureau of Reclamation's Type VI impact-type energy dissipator, which can be used with culverts. This basin is contained in boxlike structures requiring no tail water for operation. The structures can be used for open channels as well as culverts, and the basin can be used at sites where the entrance velocity to the basin does not exceed 50 ft/s, and the discharge is less than 400 cfs. This dissipator should not be used if the buildup of debris or ice can cause substantial clogging. The design procedure is as follows (Corry et al, 1975). 1. Compute the flow area at the end of the culvert using the maximum design discharge and velocity. Compute the equivalent depth of flow entering the dissipator from the culvert as 1/2 ye A 2
(20.19)
where A is the cross-sectional area of flow in the culvert. This converts the cross-sectional area of flow of a pipe into an equivalent rectangular cross section with a width twice the depth of flow. The culvert preceding the dissipator can be open, closed, or have any cross section. This approach ignores the size and shape of the culvert entirely except for the determination of flow entering the dissipator. 2. Compute Fr and the energy at the end of the culvert Ho: V2 H0 ye 0 . 2g Then use Figure 20.19 to determine the basin width. Enter Figure with Fr to determine Ho/W). W Ho/W then W Ho / (H 3. Use Table 20.2 to determine the dimensions of the dissipator structure.
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FLOW TRANSITIONS AND ENERGY DISSIPATORS FOR CULVERTS AND CHANNELS
Chapter Twenty
FIGURE 20.17 Example 5 U.S. Barean of reclamation type II basin
20.30
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FLOW TRANSITIONS AND ENERGY DISSIPATORS FOR CULVERTS AND CHANNELS
FIGURE 20.18 Baffle — wall energy dissipator of a U.S. Bureaun of Reclamation type IV basin. (From Corry et al., 1975)
Flow Transitions and Energy Dissipators for Culverts and Channels 20.31
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FLOW TRANSITIONS AND ENERGY DISSIPATORS FOR CULVERTS AND CHANNELS
20.32
Chapter Twenty
FIGURE 20.19 Design curve for a baffle-wall dissipator. (Corry et al., 1975)
20.2.4 Drop Structures 20.2.4.1 Straight-drop spillway. The straight-drop spillway shown in Figs 20.20 and . 20.21 is generally used for subcritical flow in the upstream as well as the downstream channel. To describe the flow geometry, the following drop number is used: q2 ND 3 , (20.20) gh0 where q is the discharge per unit width of the crest overfall, g is the acceleration caused by gravity, and ho is the height of the drop. The dimensions L1, y1, y2, and y3 in Fig. 20.22 are determined using the following: L1 ND0.27 (20.21) 4.30N h0 y1 ND0.22 1.0N h0
(20.22)
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FLOW TRANSITIONS AND ENERGY DISSIPATORS FOR CULVERTS AND CHANNELS
Flow Transitions and Energy Dissipators for Culverts and Channels 20.33 TABLE 20.2 Baffle Wall Dissipator: Dimensions of The Basin in Feet and Inches W
h1
L
h2
Dimensions of Basin in Feet and Inches h3 L1 L2 h4 W1 W2 t3
4–0
3–1
5–5
1–6
0–8
2–4
t2
t1
t4
t5
3–1 1–8
0–4 1–1
0–6 0–6
0–6
5–0 3–10
6–8 1–11 0–10 2–11 3–10 2–1
0–5 1–5
0–6 0–6
0–6
0–6 0–3 0–6 0–3
6–0
4–7
8–0
2–3
1–0
3–5
4–7 2–6
0–6 1–8
0–6 0–6
0–6
0–6 0–3
7–0
5–5
9–5
2–7
1–2
4–0
5–52-11
0–6 1–11
0–6 0–6
0–6
0–6 0–3
8–0
6–2 10–8
3–0
1–4
4–7
6–2 3–4
0–7 2–2
0–7 0–7
0–6
0–6 0–3
9–0 6–11 12–0
0–7 0–3
3–5
1–6
5–2 6–11 3–9
0–8 2–6
0–8
0-7
0–7
10-0
7–8 13–5
3-9
1–8
5–9
7–8 4–2
0–9 2–9
0–9 0–8
0–8
0–8 0–3
11–0
8–5 14–7
4–2 1–10
6–4
8–5 4–7
0–10
0–9 0–9
0–8
0–8 0–4
12-0
9–2 16–0
4–6
2–0 6–10
3-0
9–2 5–0
0–11 3–0
0–10 0–10
0–8
0–9 0–4
13–0 10–0 17–4 4–11
2–2
7–5 10–0 5–5
1–0 3–0
0–10 0–11
0–8
0–10 0–4
14–0 10–9 18–8
5–3
2–4
8–0 10–95–10
1–1 3–0
0–11 1–0
0–8
0–11 0–5
15–0 11–6 20–0
5–7
2–6
8–6 11–6 6–3
1–2 3–0
1–0 1–0
0–8
1–0 0–5
16–0 12–3 21–4
6–0
2–8
9–1 12–3 6–8
1–3 3–0
1–0 1–0
0–9
1–0 0–6
17–0 13–0 22–6
6–4 2–10
9–8 13–0 7–1
1–4 3–0
1–0 1–1
0–9
1–0 0–6
18–0 13–9 23–11
6–8
3–0 10–3 13–9 7–6
1–4 3–0
1–1 1–1
0–9
1–1 0–7
19–0 14–7 25–4
7–1
3–210–10 14–77–11
1–5 3–0
1–1 1–2 0–10
1–1 0–7
20–0 15–4 26–7
7–6
3–4 11–5 15–4 8–4
1–6 3–0
1–2 1–2 0–10
1–2 0–8
Source: Corry, et al; (1975)
y2 ND0.425 0.54N h0
(20.23)
y3 ND0.27 (20.24) 1.66N h0 L1 is the length of the jump, L1 can be determined using Fig. 20.22. The sequent depth and the tailwater depth TW must be compared to determine whether TW y3, or TW y3, or TW y3. If TW y3, the hydraulic jump moves downstream. In this case, it is necessary
FIGURE 20.20 Flow geometry of a straight-drop spillway. (From Corry et al., 1975)
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FLOW TRANSITIONS AND ENERGY DISSIPATORS FOR CULVERTS AND CHANNELS
20.34
Chapter Twenty
FIGURE 20.21 Straight-drop spillway stilling basin. (From Rand, 1955)
to construct the apron at the bed level and an end sill or baffles or to construct the apron below the downstream bed level and an end sill. If TW y3, the hydraulic jump may become submerged. If TW y3, the hydraulic jump begins at depth y2; there is no supercritical flow on the apron, and L1 is a minimum. 20.2.4.2 Grated energy dissipators. Energy dissipators with grates (Fig. 20.23) also can be used in conjunction with drop structures. The U.S.Bureau of Reclamation (1987) developed the following design recommendations for grates for incoming subcritical flow: 1. Select slot width with a full slot width at each wall. 2. Compute beam length LG using LG Q C(W)( W N) N 2 gy0,
(20.25)
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FLOW TRANSITIONS AND ENERGY DISSIPATORS FOR CULVERTS AND CHANNELS
Flow Transitions and Energy Dissipators for Culverts and Channels 20.35
FIGURE 20.22 Design chart for determination of L1. (From Corry et al., 1975)
where C is a coefficient equal to 0.245, W is the width of the slots in feet, and N is the number of slots or spaces between beams.Then compute the beam width 1.5W. W The quantity (W)( W N N) can be adjusted until an acceptable beam length, LG is determined. 3. For self-cleaning, the grate can be tilted appoximately 3o in the downstream direction. 20.2.4.3 Straight drop structures. The straight drop structure shown in Fig. 20.21 consists of a horizontal apron with blocks and sills to dissipate energy. This structure is for drops of less than 15 ft (4.57 m) and for sufficient tailwater. The design parameters include the length of the basin, the position and size of the floor blocks, the position and height of the end sill, the position of the wingwalls, and the geometry of the approach channel. This structure was developed by the Agricultural Research Service at the Saint Anthony Falls Hydraulic Laboratory. The design procedure for a straight-drop structure is as follows (Corry et al 1975): 1. Compute the minimum length of stilling basin LB LB L1 L2 L3 L1 2.55yc
(20.26)
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FLOW TRANSITIONS AND ENERGY DISSIPATORS FOR CULVERTS AND CHANNELS
20.36
Chapter Twenty
FIGURE 20.23. Energy dissipator with grate ( Corry et al, 1975)
where distances are illustrated in Fig. 20.21. The distance from the headwall to the point where the surface of the upper nappe strikes the still basin floor L1 is (L Lf Ls) L1 2 where
Lf 0.406
(20.27)
3.1954.368hyy 0 c
c
2 L h 0.691 0.228 t o yc yc yc Ls L 0.185 0.456 t yc
(20.28)
(20.29)
and
Lt 0.0406
3.1954.368hyy 2 c
c
(20.30)
where h2 h0 y3 (Fig. 20.21). Alternatively, L1 can be determined using Fig. 20.22. The distance from the point where the surface of the upper nappe strikes the
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FLOW TRANSITIONS AND ENERGY DISSIPATORS FOR CULVERTS AND CHANNELS
Flow Transitions and Energy Dissipators for Culverts and Channels 20.37
stilling basin floor to the upstream face of the floor blocks L2 is determined using L2 0.8 yc. The distance between the upstream face of the floor blocks and the end of the stilling basin, L3, is determined using L3 1.75yc. 2. Floor blocks are proportioned as follows: a) height 0.8yc; b) width and spacing should be 0.4yc with a variation of 0.15yc permitted, and c) blocks should be square in plan, and d) blocts should occupy between 50 to 60 percent of the stilling basin width. 3. Compute the end sill height as 0.4yc. 4. If longitudinal sills (for structural purposes only) are used, they should be constructed through the floor block, not between the floor blocks. 5. Compute sidewall height above the tailwater level as 0.85yc. 6. Wingwalls are constructed at an angle of 45° with the outlet center line with a top slope of 1:1. 7. Compute the minimum height of the tailwater surface above the floor of the stilling basin y3 using y3 2.15yc. 8. Modification, to the approach channel are as follows: The crest of the spillway should be at the same elevation as the approach channel, the bottom width should be equal to the spillway notch length Wo at the headwall, and protection with riprap or paving should be provided for a distance upstream of the headwall of 3yc. 9. Using the recommendations in Step 8, no special provision for aeration is needed. Example 6 Determine the dimensions of a straight-drop spillway stilling basin for a discharge of 7.08 m3/s. The downstream trapezoidal channel has a 3:1 side slope with a 3.05 m bottom width S0 = 0.002 m/m, n = 0.03, and a normal depth of 1.024 m. The drop h0 = 1.83 m. Solution: Step 1. Determine the minimum basin length LB. The critical depth yc is determined as yc 0.655 m. Then h0/y / c 1.83/0.655 2.79; h2 h0 2.15yc 1.83 2.15(0.655) 0.422 m; h2/y / c 0.422/0.655 0.644. Using Fig. 20.22, L1/y / c 8.2 for h0/y / c 2.79 and h2/y / c 0.644. Then L1 8.2(0.655) 5.37 m; L2 0.8yc 0.8(0.655) 0.52 m; and L3 1.75yc 1.75(0.655) 1.15 m; thus, LB L1 L2 L3 5.37 0.52 1.15 7.04 m. Step 2. Proportions the floor blocks are height 0.8yc 0.8(0.655) 0.524 m, width 0.4yc 0.262 m, and spacing 0.4yc 0.262 m. Step 3. Calculate the end sill height 0.4yc 0.262 m. Step 4. Use the longitudinal sill passing through the floor blocks. Step 5. Calculate the side-wall height above the tailwater 0.85yc 0.85(0.655) 0.557 m. Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
FLOW TRANSITIONS AND ENERGY DISSIPATORS FOR CULVERTS AND CHANNELS
20.38
Chapter Twenty
Step 6. Local wingwalls are at a 45° angle with the outlet center line. Step 7. Calculate the minimum height of tailwater above the floor of the stilling basin: y3 2.15yc 2.15(0.655) 1.41 m. The basin must be placed 1.43 1.024 0.406 m below the downstream bed level. 20.2.4.4 Box-inlet drop structure. The box-inlet drop structure shown in Figure 20.24 consists of two different sections that are effective in controlling flow: the crest of the box inlet and the opening in the headwall. This structure is based on experiments by the U.S. Soil Conservation Service at the Saint Anthony Falls Hydraulic Laboratory (Blaisdell and Donnelly, 1956). The design procedure of box-inlet drop structures is as follows: (FHWA, 1978): 1. Select ho 2. Select L1, W2, and Lc where Lc is the length of the box-inlet crest, Lc W2 2L1,
FIGURE 20.24 Box-inlet drop structure. (From Corry et al., 1975)
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FLOW TRANSITIONS AND ENERGY DISSIPATORS FOR CULVERTS AND CHANNELS
Flow Transitions and Energy Dissipators for Culverts and Channels 20.39
where W2 is the width of the box inlet, and L1 is the length of the box inlet. 3. Compute the head yo for the crest using the discharge equation for a rectangular weir: 2/3
Q yo 3.43Lc
(20.31)
ho 4. Compute and determine the coefficient of discharge C2 from Fig. 20.25. W2 L 5. Compute 1 and determine the relative head correction, CH, from Fig. 20.26. ho
6. Compute y0 for the headwall opening using 2/3
y0 Q CH 2g C2W2
(20.32)
which is based upon the rectangular weir equation . 7. Compare yo from Step 3 for the crest and yo from Step 6 for the headwall opening. The larger value of yo controls. If the crest controls, adjust yo from Step 3 using the following procedure:
/ /
1) Compute yo W2 and determine the correction for the head, C1, using Fig. 20.27. 2) Compute L1 W2 and determine the correction for the box inlet shape, CS, using Fig. 20.28
/
3) Compute W1 Lc and determine the correction for the approach channel width CA, using Fig. 20.29
/
4) Compute W4 W2 and determine the correction for dike effect (proximity of dike to box inlet crest) CE, using Table 20.3
FIGURE 20.25 Coefficient of discharge, with control at headwall opening. (From Corry et al., 1975)
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FLOW TRANSITIONS AND ENERGY DISSIPATORS FOR CULVERTS AND CHANNELS
20.40
Chapter Twenty
FIGURE 20.26 Relative head correction for h0/w2 ≥ 1\4 with control at headwall opening. (From Corry et al., 1975)
5) Determine the adjusted yo: 2/3 Q yo 3.43C C C C L 1 S A E C
(20.33)
8. Compute the critical depth in the straight section yc using 1/3
Q 2 yc 1 W2 g
(20.34)
9. Compute the critical depth at the exit of the stilling basin yc3 using 1/3
Q 2 yc3 1 W3 g
(20.35)
10. Compute the minimum length of the straight section L2 using
0.2 1 L2 yc L1 W2
(20.36)
for values of L1/W W2 0.25. 11.Compute the minimum length of the stilling basin using
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FLOW TRANSITIONS AND ENERGY DISSIPATORS FOR CULVERTS AND CHANNELS
Flow Transitions and Energy Dissipators for Culverts and Channels 20.41
FIGURE 20.27 Discharge coefficient and correction for head, with control at box-inlet crest. (From Corry et al., 1975)
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FLOW TRANSITIONS AND ENERGY DISSIPATORS FOR CULVERTS AND CHANNELS
20.42
Chapter Twenty
W FIGURE 20.28 Correction for box-inlet shape, with control at the box-inlet crest (1 ≥ 3). Lc From Corry et al., 1975)
FIGURE 20.29 Correction for approach-channel width, with control at box-inlet crest. (From Corry et al., 1975)
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FLOW TRANSITIONS AND ENERGY DISSIPATORS FOR CULVERTS AND CHANNELS
Flow Transitions and Energy Dissipators for Culverts and Channels 20.43 TABLE 20.3
Correction for Dike Efect CE: Control at the Box-Inlet Crest (Control at Box–Inlet Crest) W4 /W W2
L1/W2
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.5
0.90
0.96
1.00
1.02
1.0
.80
.88
.93
.96
1.04
1.05
1.05
.98
1.00
1.01
1.5
.76
.83
.88
2.0
.76
.83
.88
.92
.94
.96
.97
.92
.94
.96
.97
Source: Cerry et al (1975)
Lc L3 2L1 W2
(20.37)
Z W3 W2) Z( L3 2
(20.38)
and
and choose the larger value of L3. 12.Compute the minimum tailwater depth over the basin floor using y3 1.6yc3
if
W3 11.5yc3
(20.39)
or W3 y3 yc3 0.052W
if
W3 11.5yc3
(20.40)
13.Compute the height of the end sill h4 using h4 y3/6. 14.Determine the number of longitudinal sills: W2, use two sills; If W3 2.5W W2, use four sills. if W3 2.5W When two sills are used they should be located at a distance W5 on each side of the centerline. When four sills are used, the two additional sills should be located parallel to the outlet centerline and midway between the center sills and the sidewalls at the stilling basin exit. 15.Compute the minimum height of the sidewalls above the water surface at the exit of the stilling basin h3 using h3 y3/3. Sidewalls should extend above the tailwater surface under all conditions. 16.Wingwalls should be triangular in elevation and have a top slope of 45° with the horizontal. The top slopes can be as flat as 30°. Wingwalls should flare in plan at an angle of 60° with the outlet centerline. The flare-wall angle can be as small as 45°. Wingwalls parallel to the outlet centerline should not be used.
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FLOW TRANSITIONS AND ENERGY DISSIPATORS FOR CULVERTS AND CHANNELS
20.44
Chapter Twenty
20.2.5 RIPRAP BASINS The riprap basin recommended in Corry et al (1975) for culverts is shown in Fig. 20.30, which is a preshaped basin lined with riprap. The surface of the riprapped floor of the energy dissipating pool is at an elevation hs below the culvert invert, where hs is the approximate depth of scour that would occur in a thick pad of riprap if subjected to the design discharge (Corry et al 1975). The length of the pool is the larger of 10hs or Wo, and the overall length of the basin is the larger of 15hs or 4W Wo. The ratio hs /dd50 should be less than 4 [(hs/dd50) 4].Figure 20.31 provides design detaills for riprapped culvert energy basins.
FIGURE 20.30
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FLOW TRANSITIONS AND ENERGY DISSIPATORS FOR CULVERTS AND CHANNELS
Flow Transitions and Energy Dissipators for Culverts and Channels 20.45
FIGURE 20.31 Details of riprapped culvert energy basin. (Corry et al., 1975)
REFERENCES Blaisdell, F. W., Flow Through Diverging Open Channel Transitions at Supercritical Velocities, SCS Report No. SCS–TR–76, U.S. Department of Agriculture, April 1949. Blaisdell, F. W., and C. A. Donnelly, “The Box Inlet Drop Spillway and Its Outlet”, Transactions, of the American Society of Civil Engineers, 121:955–986, 1956. Blaisdell, F. W., The SAF Stilling Basin, U. S. Goverment Printing Office, 1959 Corry, M. L., P. L. Thompson, E. J. Watts, J. S., Jones, and D. L. Richards, Hydraulic Design of Energy Dissipators for Culverts and Channels, Enqineering Circular 145, Federal Highway Administration, U.S. Department of Transportation, Washington DC, 1975. Fedeard Highinay Administratin , Hydraulics of Bridge Waterways, Hydraulic Design Series No. 1, Federal Highway Administration, U.S. Department of Transportation, Washington, DC, 1978. Ippen, A. T., “Mechanics of Supercritical Flow”, Transactions of the American society of Civil Engineers,, 11b; 268–295, 1951 Rand, W., “Flow Geometry at Straight Drop Spillways,” Paper No. 791, Proceedings of the American Society of civil Engineers, Vol, 81, pp. 1-13, September 1955.
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FLOW TRANSITIONS AND ENERGY DISSIPATORS FOR CULVERTS AND CHANNELS
20.46
Chapter Twenty
Simons, D.B. M.A. stevens, F.J. watts, floos protection at Calrect autlets, Cer 69—70 DBS—MAS — FJW4, Colorado State University, Fort Collins, Colorado, 1970, Simons, D. B., M. A. Stevens, and F3. J. Watts, Flood Protection at Culvert Outlets, CER No. 69–70, DBS–MAS–FJWA, Colorado State University, Fort Collins, CO. 1970. U.S. Army Corp pf Enqineers, Hydraulic Design of Flood Control Channels, Engineering and Design Manual EM1110–2–1601, pp.20—26, July 1970. U.S. Bureau of Reclamation, (USBR), Design of Small Dams, U.S. Government Printing Office, Denver, CO, 1987. Watts, F. J., Hydraulics of Rigid Boundary Basins, doctoral dissertation, Colorado State University, Fort Collins, CO, 1968.
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Source: HYDRAULIC DESIGN HANDBOOK
CHAPTER 21
HYDRAULIC DESIGN OF FLOW MEASURING STRUCTURES John A. Replogle and Albert J. Clemmens USDA–ARS Water Conservation Laboratory Phoenix, Arizona
Clifford A. Pugh US Bureau of Reclamation, Denver, Colorado
21.1 INTRODUCTION Experienced water providers and users can use this chapter as a quick review of hydraulic principles related to water measurement and its relation to hydraulic design for environmental considerations. The hydraulic design of flow measuring structures usually confronts the engineer with two opportunities. One is the design of measurement structures in a retrofit situation and the other is in original project design. The retrofit mode is usually difficult and requires much innovation just to obtain passable function within the space and sizing limitations and other constraints usually imposed. Because of the increasing emphasis on quantifying flow rates and volumes in most aspects of water resource planning and management, the retrofit applications currently dominate the design problems. Most textbooks deal with recommending ideal installation situations and retrofit projects appear to be unable to comply without great economic impact. This too frequently can lead to arbitrary compromises that produce poor measurement performance. Even new installations may be limited by space requirements. This may force design decisions into the final construction that compromise accuracy. This chapter will strive to show the design concepts available, particularly those useful for designing both new and retrofit installations, and will point out measurement behaviors to be expected from various compromises. This chapter suggests those deviations that cause least impact and guides the designer to choices that may be hydraulically acceptable and still meet structural goals. Of the numerous flowmetering methods available to the hydraulic engineer, most are based on well-established hydraulic principles and are amenable to design manipulations of size, shape, and response. While this aspect of flow measurement is documented in several handbooks and texts, the design and retrofit of sites to accommodate and facilitate measurement is not as well described or is described in a scattered assortment of books and articles. Pipeline flows of water are usually less complicated to measure than open-channel flows, most obviously because the flow area does not change significantly with flow rate. 21.1 Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
HYDRAULIC DESIGN OF FLOW MEASURING STRUCTURES
21.2
Chapter Twenty-One
Consequently, many applications of pipeline flows are held to stricter accuracy standards than channel flows can reasonably achieve. Thus, channel flows and their measurement are usually limited to large delivery volumes and to accuracies acceptable to the related activities, such as sewer flows and irrigation deliveries. The purpose of this chapter is to consolidate design information for evaluating a flow measurement site, selecting a flow measuring system, and adapting the measuring site to optimize measuring and other functions that may be desired from the site. Emphasis will be on open-channel flow measurements because that is a likely need of the hydraulic engineer. Pipe flowmeters in water supply will also be discussed in lesser detail because the major application of the many types of pipe flowmeters is well covered in the chemical and petroleum industry literature. Experienced readers may wish to further investigate and seek more advanced references in hydraulics and fluid mechanics. Extensive information on fluid meter theory and detailed material for determining coefficients for tube-type meters is given in American Society, of Mechanical Engineers (ASME) (1959, 1971) and revisited with modern updates in books by Spitzer (1990) and Miller (1996). Brater and King (1982) have a thorough discussion of general critical depth relations and detailed relationships for most common hydraulic flow section shapes in open channels. Bos (1989) covers a broad segment of openchannel water measurement devices.
21.2 HYDRAULIC CONCEPTS RELATED TO WATER MEASUREMENT 21.2.1 Basic Concepts for Pipe and Channel Flows Flow can be classified into closed conduit flow and open-channel flow. Open-channel flow conditions occur whenever the flowing stream has a free or unconstrained surface that is open to the atmosphere. Flows in canals or in vented pipelines that are not flowing full are typical examples. In hydraulics, a pipe is any closed conduit that carries water under pressure. The filled conduit may be square, rectangular, or any other shape, but is usually round. If flow is occurring in a conduit but does not completely fill it, the flow is not considered pipe or closed conduit flow, but is classified as open-channel flow. Flow rate in a pipeline responds mainly to the pressure gradient or head difference that exists between two points along the pipeline, modified by the frictional resistance to flow caused by pipe length, pipe roughness, bends, restrictions, changes in conduit shape and size, the nature of the fluid flowing, and the cross-sectional area of the pipe. In open-channel flows, the pressure gradient, or energy grade line, is controlled mainly by the force due to gravity, which is influenced by the channel slope, resistance from the channel wall roughness, the channel shape, and the flow area. The fluid is usually water. Basic flow metering in both pipe flow and open channels depends on determining an average flow velocity by some means and combining it with the flow cross-sectional area. For open channels, a common means involves current meter measurements where metered point velocities are applied to their applicable subareas and summed over a flow cross section. Exceptions include tracer-dilution techniques that do not require flow area or velocity. The uses of tracer techniques are applicable to special pump calibrations and some difficult channel flows (mountain streams). They are avoided for most city water distribution systems, sewer flows and irrigation applications because of the general expense with han-
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HYDRAULIC DESIGN OF FLOW MEASURING STRUCTURES
Hydraulic Design of Flow Measuring Structures 21.3
dling the equipment and doing the analysis. The most used techniques applicable to openchannel systems, including sewer flows and irrigation canal flow measurements, depend on exploiting the special velocity properties of critical flow, as discussed in a section 21.2.3. Continuity equation. The first basic equation for water flowing in either pipes or channels is the continuity equation, which simply states that discharge rate (volumetric flow per unit time), Q, is equal to flow cross-sectional area, A, times flow mean velocity, V, V through the flow cross section, or Q AV
(21.1)
Bernoulli energy equation. Another basic equation involves energy relations and is also applicable to both pipe and channel flows. The most familiar form is for closed pipe flow, wherein the basic energy principles are described by the Bernoulli energy equation. For two locations along a pipe at stations 1 and 2,(Fig. 21.1), the Bernoulli equation can be expressed as V2 V2 z1 h1 1 z2 h2 2 constant 2g 2g
(21.2)
where the terms are expressed in length dimensions as z the height from an arbitrary reference plane (datum) h the pressure head V average velocity through the pipe cross-section at the designated location V2/2g the velocity head g the gravitational constant 1 ,2 subscripts denoting the respective locations along the pipeline. This equation is based on uniform velocity across the conduit area and no energy losses. However, in real fluid flows, nonuniform velocities exist and friction causes energy conversion to heat. Typically, these velocities are zero at the walls and reach a maximum profile velocity near the center of the flow. If the flow is viscous flow in a round pipe, the flow profile is parabolic, that is, “bullet-shaped.” If the velocity is fully turbulent, the bullet-shape is much flattened, with steep velocity gradients near the wall and nearly uniform profile across the remainder of the pipe. These idealized profiles can be skewed drastically by regulating valves, structures, conduit bends and other flow obstructions. Therefore, application of these equations depends on knowing, or controlling, the velocity profile so that the average velocity in the conduit cross section can be inferred.
FIGURE 21.1 Energy balance in pipe flow.
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HYDRAULIC DESIGN OF FLOW MEASURING STRUCTURES
21.4
Chapter Twenty-One
Equation (2.2) requires some adjustments to convert it to the energy equation, which is useful in analyzing flows in pipes or open channels with a small slope (Chow, 1959). First we introduce correction factors, α1 and α2, called the velocity distribution coefficients, to account for the computational expediency of using the average velocities, V1 and V2, to compute the kinetic energy term, V2/2g, at the respective locations 1 and 2 along a channel. These values for the usual range of turbulent flows in water usually range from about 1.01 to 1.05, although for thick petroleum products in pipe flows and low velocity flows, the value can approach a value of 2. Second, a term, hf, for the loss of energy between the two points is included. The result is V2 V2 z1 h1 1 1 z2 h2 2 2 hf 2g 2g
(21.3)
21.2.2 Pipe Hydraulics Reynolds number. The behavior of flow in pipes is governed primarily by the viscosity of the fluid. In pipeline flows, the ratio between the dynamic forces and the viscus forces is important for defining the limits between laminar and turbulent flows and other functions of pipe flow. This ratio is called the Reynolds number, Rn, and is defined as VL Rn c (21.4) v where V the velocity of the flow, Lc characteristic length, typically the pipe diameter, D and v the kinematic viscosity. Headloss characteristics in pipes. The Reynolds number, Rn, defined above, represents the effect of viscosity relative to inertia and is used to define appropriate flow ranges for headloss equations in pipe flow. For example, headloss is proportional to the square of the velocity, when the velocities and pipe size combinations defined by a pipe-diameterbased Reynolds number, Rn, greater than about 1000. Most of the flows of interest in general hydraulic engineering have Reynolds numbers greater than 1000. Some exceptions are found in drip or trickle irrigation systems common in agricultural and urban landscape settings. The headloss, hf, for Rn greater than the minimum value of about 1000 is traditionally expressed in terms of a friction factor, ff, the pipe diameter, D, pipe length, L, and the velocity head, V2/2g, where g is the gravitational constant, and V is the average velocity, as 2 hf f L V D 2g
(21.5)
The value for f is usually obtained from a Moody diagram which is a graphical representation of the f value in terms of the Reynolds number, the roughness height of the pipe wall material, , and the pipe diameter, D. The Moody diagram is a graphical solution of the Colebrook function ε/D 1 2.51 2log 3.7 兹f 兹苶 Rn兹f苶
(21.6)
The values range from 0.0000015 m for smooth plastic pipe to 0.00026 m for cast iron pipe. Concrete pipe ranges from about 0.0003 m to 0.003 m (Daugherty and Ingersoll, 1954). The equation can be readily solved by iteration techniques using a computer spreadsheet.
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HYDRAULIC DESIGN OF FLOW MEASURING STRUCTURES
Hydraulic Design of Flow Measuring Structures 21.5
21.2.3 Channel Hydraulics Hydraulic mean depth. The hydraulic mean depth, Dm [U.S. Bureau of Reclamation (USBR), 1997] is the flow cross-sectional area, A, divided by the flow surface width, T T, or Dm A T
(21.7)
For conduits such as pipes flowing nearly full, the surface flow width may be narrow, and Dm may be a larger value than the physical water depth. For the usual natural channels and most canals, Dm is interchangeable with average depth. Sometimes it is simply called the hydraulic depth (Chow, 1959). Froude number. Open-channel flow behavior is governed primarily by gravity forces. The ratio of the inertial forces to the gravity forces is called the Froude number, Fr, and is defined by Fr V (21.8) 兹苶苶 兹苶 兹g D苶 m where V the velocity of the flow, g the gravitational constant, Dm the hydraulic mean depth. The Froude number applies to most open channel flows and is used for defining model scale ratios and estimating stable flow characteristics in open channels. Specific energy. It is useful to define the energy equation in terms of the local channel bottom instead of an arbitrary datum. This is called the specific energy, E, and is given by: 2 E y V (21.9) 2g That is, the specific energy is equal to the sum of the depth of flow y and the velocity head (Fig. 21.2).
Critical flow and critical depth. In open channels a flow phenomenon occurs that does not happen in closed pipe flows. The process is called critical flow. Critical flow is defined for openchannel flows as the maximum discharge for the minimum specific energy, that is, critical flow represents the minimum combination of potential energy (depth of flow, y) and kinetic energy (velocity head, V2/2g) for the given discharge (Chow, 1959). The depth of flow then is the critical depth. By virtue of the continuity equation, for a constant discharge at critical flow, an increase in depth must necessarily be accompanied by a decrease in velocity, which is called subcritical velocity. Conversely, a decrease in depth for the same flow rate necessarily requires an increase in velocity, which is called supercritical velocity. When critical flow occurs in an open channel it can be shown (Chow, 1959) that V2 Dm αc (21.10) 2g 2 where Vc mean flow velocity, g gravitational constant, Dm hydraulic mean depth, and α velocity distribution coefficient (Chow, 1959). This can further be combined with the continuity equation, Eq. (21.4), to express the critical flow discharge rate, Qc, as
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HYDRAULIC DESIGN OF FLOW MEASURING STRUCTURES
21.6
Chapter Twenty-One
FIGURE 21.2 Specific energy balance.
Qc A
冪莦α莦 g Dm
(21.11)
In practice, the water surface slope in a contraction is relatively steep and the precise plane of the critical flow section is not easily or reliably located. Thus, the data for accurately evaluating the hydraulic depth, Dm, is not readily obtained. For critical flow flumes, the flow depth is therefore not measured at this critical section, but instead a depth is measured in the upstream channel, where the velocity head is computable or is minimal. The critical depth is then mathematically derived based on energy principles described by Bos et al. (1991). These flumes, sometimes referred to as the computable flumes that rely on critical flow theory, will be discussed in more detail in Section 21.7. For maximum discharge for minimum energy, the condition described above for critical flow in open channels, it can be shown that Dm Vc2 α A 2g 2T 2
(21.12)
where: A = the channel flow area, T = the top width of the channel flow, Dm = the hydraulic mean depth, and Vc = the critical velocity. Thus, the velocity head at critical flow is equal to half the hydraulic mean depth, sometimes called hydraulic depth, Dm A/T (Chow, 1959). From the above, Vc 1 Fr (21.13) 兹苶苶 兹苶 兹g D/苶 α where Fr is the Froude number defined above. Thus, at critical flow the Froude number is unity. Also note that the Froude number can be defined by Fr V/ V Vc for velocities other than critical. Normal depth. Yet another depth is associated with open channel flows, the normal depth. When the flow in an open channel does not change from station to station, the flow is said to be uniform and the bottom slope, the hydraulic grade line, and the energy grade line are all parallel to each other. Figure 21.2 shows the condition when the flow is not uniform. Modular limit. If the downstream depth in a channel is too deep, the backwater will prevent critical depth from occurring. The flow is considered to be submerged whenever the downstream water surface exceeds the crest elevation of a channel control, such as a
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HYDRAULIC DESIGN OF FLOW MEASURING STRUCTURES
Hydraulic Design of Flow Measuring Structures 21.7
weir or flume. For flumes, particularly, this submergence has little effect on critical depth, and free flow exists until a certain limiting submergence for that particular flow module called the modular limit is reached. At some point of submergence, the upstream flow depth is affected. and the modular limit is exceeded, and free flow does not occur. The modular limit is defined as that limiting submergence ratio, and is based on the ratio of the downstream depth to upstream depth. The modular limit occurs when the downstream backwater causes more than1 percent change in the calculated discharge in a particular flow module, or device (Bos, 1989). When the modular limit is exceeded, the flow is called nonmodular. 21.2.4 Energy Balance Relationships in Channels Hydraulic problems concerning fluid flow are commonly described in terms of conserving kinetic and potential energy, and are conveniently expressed using the classical Bernoulli equation in combination with the Continuity equation. The applications of these equations are generally well documented, particularly for pipe flows, in texts and handbooks and are not repeated here (Brater and King, 1982; Miller, 1996). The case for open channels is less complete, but is given considerable treatment in Brater and King (1982), Chow (1959), and Herschy (1985). The computational uncertainties evolve from the effects of friction and viscosity that distort the classic assumptions of a uniform velocity profile across the fluid stream. When accountings for friction and flow profile are successfully applied, the results for discharge computations are usually good to excellent for both pipes and open channels (Bos et al., 1991). Headloss characteristics in channels. In terms of frictional headlosses, the wetted perimeter, Pw, of the flow is important. Hydraulic radius, Rh, is defined as the area of the flow section, A, divided by the wetted perimeter, Pw, or Rh A (21.14) Pw Conversely, the wetted perimeter times the hydraulic radius is equal to the area of an irregular flow section. The hydraulic radius of a channel can be compared to the radius of a pipe, r, with a cross-sectional area A πrr2 and a circumference or wetted perimeter Pw 2 πr. Under these conditions, the hydraulic radius compares to the pipe radius, and to the pipe diameter, D, as r D Rh (21.15) 2 4 The Manning’s formula. Canal and stream discharge rates are usually estimated with use of the Manning’s formula. Many open-channel flow equations have been proposed, but the most used is the Manning’s formula. This expression is partly rational and uses an empirical coefficient, n, that is used in both the SI and American unit systems. In general form it becomes Cm 2/3 1/2 V R S (21.16) n h e where V average velocity, n the Manning’s roughness coefficient, Rh the hydraulic radius, Se the energy-line slope, and Cm conversion of units: 1.0 for metric units and 1.486 for American units. The factor Se is the slope of the energy line. Note that the bed slope of the channel, So, and the slope of the water surface, Sw, are not to be used. These parameters are, however, equal to Se when uniform flow, with the resulting normal depth occurs. As defined above, Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
HYDRAULIC DESIGN OF FLOW MEASURING STRUCTURES
21.8
Chapter Twenty-One
normal depth occurs when a channel flow approaches uniformity from station to station along the channel (Chow, 1959). For design purposes, the n value for concrete lined canals is usually about 0.014. A good finish can lower it to 0.012, while concrete in poor condition and channels constructed with shot crete or gunite, usually have n values from 0.016 to 0.018. In some instances, concrete lined canals, with significant algae growth, have experienced n values as high as 0.032. This latter value approaches the values usually experienced with unlined channels, 0.03–0.04. Thus, for reliable application, the use of Manning’s formula requires field experience and on-site inspection of the channel being computed.
21.2.5 Modeling Characteristics for Open Channels For flowing water in open channels, fluid friction is a factor as well as gravity and inertia. This would seem to present a problem for hydraulic scale modeling, because both dynamic and kinematic similarity are difficult to achieve simultaneously. Fortunately for most open-channel flows, there is usually fully developed turbulence. Thus, the fluid friction losses are nearly proportional to V 2, and are nearly independent of Reynolds number, Rn, with rare exceptions. This means that in open–channel flows, inertia and gravity forces dominate over viscous forces (associated with pipe flows) and are a function of the Froude number, Fn, alone. Geometric similarity between a model and a prototype then provides kinematic similarity. For kinematic similarity the ratios of the respective velocities are everywhere the same. The velocity ratio, Vr, is the velocity in the prototype, Vp, divided by the velocity in the model, Vm, or V Vr p (21.17) Vm For Froude modeling, and from the definition for Fn, we note that V is proportional to the square-root of a length, L (for open channels we used the hydraulic depth, Dm) with the gravitational constant, g, assumed to be constant. Thus, the above equation can be written as V 兹L 兹 苶r苶 p (21.18) Vm where Lr = the length ratio between prototype and model dimensions, Lp: Lm Because the velocity varies as 兹 兹L 苶苶r and the cross-sectional area as Lr2 it follows that Qp : Qm L5/2 r : 1
(21.19)
Lp 5/2 Qp Qm Lm
(21.20)
or
This equation is valid when all the physical structure dimensions and the heads are of the same ratio. For example, it can be used to convert a flume rating for one size to that
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HYDRAULIC DESIGN OF FLOW MEASURING STRUCTURES
Hydraulic Design of Flow Measuring Structures 21.9
of a similar flume of another size. Scale modeling works best for determining calibrations in a range of Lp:Lm less than about 10:1, although ranges exceeding 50:1 have sometimes been used for studying special situations.
21.3 BASIC PRINCIPLES OF WATER MEASUREMENT Flow is usually measured by determining an average flow velocity and using the flow area to compute the volume discharge. Flow meters then have the function of detecting this velocity and combining it with the physical information of the conduit to produce a useable readout. This is easily demonstrated for closed conduits. Propeller meters, ultrasonic meters, laser-Doppler velocimeters, electromagnetic meters, Venturi meters, and orifice meters all are based on inferring a basic velocity measurement applied to a flow area for a discharge rate. For open channels, many flumes depend on determining the velocity based on energy principles of critical flow. Weirs are usually described in terms of orifice flow integrated over the weir width and the crest depth. Again these are basically velocity expressions for flow through a defined area. Dilution techniques applicable to both closed pipe and open channel flows depend on detecting the amount of fluid added to a known starting amount of tracer material. The dilution ratio determines the discharge ratio, in the case of constant injection of a tracer. The tracer may be a chemical or even injected heat or heated fluid. Electromagnetic meters depend on generating voltages by flowing a conductive fluid, usually water, through a magnetic field to produce a velocity indication.
21.3.1 Water Meter Classification Flow measuring devices are commonly classified into those that are rate meters and measure discharge rate as the primary reported indication and those that are quantity meters and measure volume as the primary indication. The latter include weighing tanks and batch volume tanks and are used mostly in laboratory settings as flow rate standards. Devices in either of these broad classes can again be divided according to the physical principle that is used to detect that primary indication (ASME, 1959). The meter part that interacts with the flow to produce the primary indication is referred to as the primary device. This interaction exploits one or more of a few physical principles, such as pressure force, energy conversion, weight, electrical properties, mixing properties, sonic properties, and so on, to generate a signal. Primary devices are thus limited in number and variety. Secondary devices convert the primary interaction into useable readout. These secondary devices are numerous and relatively unlimited in configuration and variety. The function of one class can be converted into the response of the other with suitable secondary devices. Some water measuring devices particularly suitable to municipal water supply, wastewater treatment, agricultural irrigation, and drainage applications are the historical rate meters that are treated in most hydraulic text. These include (1) weirs, (2) flumes, (3) orifice meters, and (4) Venturi meters. Head, h, or upstream depth, commonly is used for the open channel devices such as flumes and weirs. Either pressure, p, head, h, or differential head, ∆h, or differential pressure, ∆p ∆ , is used with tube-type devices, such as Venturi meters and orifice meters.
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HYDRAULIC DESIGN OF FLOW MEASURING STRUCTURES
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Chapter Twenty-One
Venturi meters in pipelines and long throated flumes in open-channel flows are examples where the energy principles and the flow accountings mentioned above give good to excellent computational results with minor dependency on empirical coefficients (Bos, 1989; Bos et al., 1991).
21.3.2 Installation Requirements Special difficulties arise in applying velocity profile and friction accountings when insufficient pipe or channel exists upstream from a flow measuring device. This is needed to ensure that predictable and acceptable velocity profiles are presented to the meter. Frequently pipe or channel lengths can be significantly shortened by special structural flow conditioners. These structural measures then become a design option. Some of these are discussed below and in section 21.3.3 Designs for pipe discharges are well described in textbooks and in standard handbooks. The design difficulties center around selecting appropriate metering candidates for accomplishing the measuring function and in providing an appropriate environment for economical, accurate, and serviceable operation. In the case of pipe flows, recommended straight pipe lengths, in terms of pipe diameter, are to be provided upstream of the meter to assure reasonable operating accuracy. These lengths depend on the flow pattern presented to the meter primarily caused by valves and pipe elbows upstream from the meter. The number and orientation of elbows greatly influence the circulation patterns and flow profile distortions presented to the meter. Open-channel flow water measurement generally requires that the Froude number of the approach flow be less than 0.5 to prevent wave action that would hinder or possibly prevent an accurate head determination. Energy concepts are used to describe Venturi meters in pipe flows based on the Bernoulli equation in which part of the pipe forms a contracted throat that necessarily changes the flow velocity and hence converts some of the static pressure to velocity head. The decrease in static pressure is the basis for flow detection. A similar concept can be applied to open channels. A historical version is the so–called Venturi flume (Brater and King, 1982) that detects the change in water surface elevation between an upstream station and in a contracted section. However, this small change is difficult to accurately detect, so the direct concept is not used. Rather, contractions are designed to be severe enough to force critical flow velocities in the contracted section. Thus, only an upstream head is needed to define the flow energy and flow area which can be converted to discharge rate. These are generally called critical-flow flumes. The flow condition where only one head measurement is needed is called free flow. The critical-flow flumes themselves consist of those called long-throated flumes that force parallel flow in the contracted, or control, section, called the throat, and those that have curvilinear flow in the throat and are called short-throated flumes. The limiting throat control section is the sharp-crested weir consisting of a thin plate. Thus, for flumes and weirs one unique head value exists for each discharge, simplifying the calibration procedure. However, if the downstream flow level submerges critical depth enough to affect the upstream reading, the modular limit is exceeded, and free flow does not occur. When exceeded, separate calibrations at many levels of submergence are then required, and two head measurements are needed to measure flow. This condition generally is to be avoided in meter site design because it reduces the accuracy of the measurement and increases the difficulty of flow determination. The modular limit for sharp-crested weirs, in practice, is less than zero, requiring full clearance of the overfall nappe of at least 3 cm, while short-throated flumes can usually tolerate 65 percent to 70 percent submergence. LongDownloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
HYDRAULIC DESIGN OF FLOW MEASURING STRUCTURES
Hydraulic Design of Flow Measuring Structures 21.11
throated flumes can tolerate from 70 percent to 90 percent depending on flow conditions and flume size. Designing flumes for submerged flow beyond the modular limit decreases the accuracy of the flow measurement. Sometimes flumes and weirs can be overly submerged unintentionally by poor design, construction errors, structural settling, attempts to supply increased delivery needs with increasing downstream heads, accumulated sediment deposits, or weed growths. Sometimes use of the submerged range beyond the modular limit is an economic compromise. Approach flow conditions for pipes. Water measurement devices are generally calibrated with certain approach flow conditions. The same approach conditions must be attained in field applications of measuring devices. Poor flow conditions in the area just upstream of the measuring device can cause large discharge indication errors. For open channels, the approaching flow should generally be subcritical. The flow should be fully developed, mild in slope, and free of curves, projections, and waves. Pipeline meters commonly require 10 or more diameters of straight pipe approach. Fittings and combinations of fittings, such as valves and bends, located upstream from a flow meter can increase the number of required approach diameters. Several references (ASME, 1971; ISO, 1991) give requirements for many pipeline configurations and meters. These are discussed in detail by Miller (1996). Flow conditioning options. Many installations, especially in retrofit situations, do not provide for sufficient lengths of straight pipe to remove velocity profile distortions and swirl to an acceptable level. Therefore, the designer may need to use flow conditioners in combination with straight pipe lengths. Swirl sensitivity varies widely. Some meters are particularly sensitive to swirl, such as the propeller and turbine meters. Magnetic flow meters are somewhat less sensitive to radial velocities than single-path ultrasonic flow meters. Venturi meters are less sensitive than orifice meters. For a swirl angle of 20° , the discharge coefficient changes by about 1 percent for a Venturi meter with β 0.32 (β is the ratio of meter throat diameter to the pipe diameter) and about 10 percent for a similar orifice. Thus a swirl can increase the discharge through an orifice for the same differential head reading (Miller, 1996). In pipeline flows, contractions can produce a central jet and also increase an incoming swirl, while expansions tend to slow swirls and produce enough secondary flow to restore flow profiles to some semblance of acceptability. These characteristics can modify the straight pipe lengths needed or the type of flow conditioner to recommend (Miller, 1996). Rough pipes also tend to reduce a swirl. For flows, such as that encountered in sewage discharges and irrigation pipeline deliveries that originate from open channels, many of the tube-bundle types of flow conditioners can gather trash and cause maintenance problems. Many meter providers in these situations use fins or vanes that protrude from the wall and have sloped upstream edges that shed trash. The vanes protrude about one-fourth of the pipe diameter into the flow, leaving the center core of the flow open. While these vanes can vary in number and length, the logic being that the fewer the vanes the longer they should be in the direction of flow, common configurations are four vanes that are about two or three pipe diameters long. Vanes in themselves do not condition wall jets well. Field experience, has shown that troublesome flow profiles can be conditioned significantly by inserting an orifice into the pipe. The orifice diameter is about 90 percent of the pipe diameter and is used to control wall jets and force them to mix with the general flow. The orifice in itself tends to cross-mix
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HYDRAULIC DESIGN OF FLOW MEASURING STRUCTURES
21.12
Chapter Twenty-One
the jets and would appear to reduce spin. However, if the jets are symmetrical and an initial swirl exists, orifices tend to increase the swirl. Inserting an orifice appears to be supported by recent recommendations of Miller (1996) where it is stated: “To achieve a fully developed profile, it is important that the flow be blocked or restricted close to the wall, with the central core having the larger flow area.” The addition of vanes when space permits is recommended. Because orifices, in general, tend to force the flow to the pipe center while increasing spin, it appears best to place the vanes upstream from the orifice. If they are placed downstream, the spin not only may be increased, but the spinning central flow may not be touched by the vanes.
21.3.3 Examples of Flow Conditioning in Field Situations Flow conditioning in an irrigation delivery pipeline. As mentioned previously, measuring devices frequently must be installed in flow situations that are less than optimal. A field example occurred in Arizona where a large pipe was used as an outlet to a secondary canal and a single-path ultrasonic meter placed in it was subjected to flow profile distortions. The pipe was about 0.75 m in diameter and delivered approximately 400 L/s. The flow rate readout was unstable, with fluctuations varying by about 15 percent. The problem appeared to be caused by slowly spiraling flow induced by the bottom jet from a partly open pipe o inlet gate and a 45 elbow. This is similar to two closely spaced pipe elbows that are not in the same plane, which can cause a spiral flow pattern (ASME, 1971). A successful attempt to modify the jet and cause it to cross mix so that the jet effects and the strength of the spiral flow were reduced, was accomplished by inserting a large βratio orifice in the pipe (Fig. 21.3). This consisted of an annular metal ring with the outside radius approximately that of the pipe and an inside diameter about 10 percent less, or an orifice with β 90 percent. The orifice was installed about three diameters downstream from the elbow. The slight increase in headloss was compensated by increasing the upstream gate opening. The orifice can be constructed by cutting notches from an appropriately sized piece of angle iron or aluminum and bending it to a polygon that approximates the circle diameter of the pipe interior. Some leakage around the ring is acceptable. For propeller meters, additional vanes projecting from the walls may be needed to further
FIGURE 21.3 An orifice plate with a large opening is used to condition a flow profile.
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HYDRAULIC DESIGN OF FLOW MEASURING STRUCTURES
Hydraulic Design of Flow Measuring Structures 21.13
reduce spiral flow. These vanes would be placed upstream from the orifice. In this installation, the fluctuation was reduced to within about 3 percent. Flow conditioning in channels. By analogy and using a minimum of 10 pipe diameters of a straight approach channel, open channel flow would require 40 hydraulic radii of straight, unobstructed, unaltered approach, based on the calculation of hydraulic radius for circular pipes being equal to one-fourth the pipe diameter, (Eq. 21.15). This would translate for very wide channels into approximately 40 times the flow depth. For narrow channels that are as deep as they are wide, this would compute to be about 13 channel depths or top widths. Other recommendations on approach channel criteria are presented by Bos (1989) and USBR (1997). Major features of that criteria follow: • If the control width is greater than 50 percent of the approach channel width, 10 average approach flow widths of straight, unobstructed approach are required. • If the control width is less than 50 percent of the approach width, 20 control widths of straight, unobstructed approach are required. • If upstream flow is below critical depth, a jump should be forced to occur. In this case, 30 measuring heads of straight, unobstructed approach after the jump should be provided. • If baffles are used to correct and smooth out approach flow, then 10 measuring heads (10 h1) should be placed between the baffles and the measuring station. Approach flow conditions should be continually checked for deviation from these conditions as described in Bos (1989) and USBR (1997). The baffles described above can become unacceptable maintenance problems in open channels. Some field expediencies are therefore described that have been found to work in specific instances, but have not been studied for assured design generalizations. Nevertheless, these constructions are but small extensions to currently accepted practices in pipe flows. Applications for openchannel flow conditioners include abrupt channel turns, sluice gate outflows, and channels downstream from a hydraulic jump. The abrupt turns may benefit from floor and wall mounted vanes or fins. Based on pipe flow experience, and assuming the channel is half of a closed conduit, these fins or vanes would probably be about 10 percent to 15 percent of the channel depth. As in pipe flow, wall jets that can develop downstream from sluice gates appear to need treatment. This can be in the form of a structural angle bolted on the channel floor and up the walls. Suggested size, based on the pipe flow analogy, is for the angle to be about 5 percent into the channel flow depth. Whether the sidewalls need larger angles when the channels are wide has not been tested.
21.3.4 Wave Suppression Of special concern in open channels is wave suppression downstream from a sluice gate, hydraulic jump, or an abrupt turn. Thus, the flow conditioners in channels have the additional task not present in pipe flows of surface wave suppression. Excessive waves in irrigation canals make reading sidewall gages difficult. These waves are usually caused by a jet entry from a sluice gate or by a waterfall situation. The unstable surface can be 10–20 cm high and extend for tens of meters downstream. Wave suppression in canals. A surface wave suppressor was tested by Schuster as reported in USBR (1997). It basically was a constructed roof over the canal for a distance Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
HYDRAULIC DESIGN OF FLOW MEASURING STRUCTURES
21.14
Chapter Twenty-One
FIGURE 21.4 Wave suppresor design (From USBR, 1997).
equal to about four times the flow depth. The roof structure is inserted into the flow about one-third the flow depth. All flow is forced to pass under the structure. Wave suppression is between 60 percent and 93 percent (Fig. 21.4). For canals that usually flow at one level, this wave-suppression method is appropriate. The wave suppressor shown in Figure 21.4 has been successfully used in both large and small channels (USBR, 1997). An important aspect is that the structure is fixed and not allowed to float. Floating suppressors are not effective. Successful field applications of wave suppressors include some installations in trapezoidal irrigation channels, with 1:1 side slopes and 60-cm bottom width. They were flowing about 400 L/S at about 45 cm deep. While the velocity was not high, about 0.8 m/s, the agitation from a flow entry gate was producing waves about 15 cm high. The suppressor “roof” was only about 60 cm in the direction of flow, and penetrated the flow by about 15 percent. Another version that has worked in small channels is illustrated in Figure 21.5. This can work with a single cross–member if the flow is usually at a fixed discharge rate and becomes similar to the suppressor described above. In severe jet cases an additional floor sill, about 10 percent of the flow depth in height, has been used successfully. The length of the roof in the flow direction has not been well studied, but field observations seem to support a length greater than two lengths of the surface wave, if that can be estimated, otherwise, use two to four times the maximum flow depth as described above. To suppress waves in canals that do not always flow at the same depth, a staggered set of baffles may help (Replogle, 1997). Because these will be submerged part of the time, they must have a thickness that overlaps slightly to accommodate the vertical depth of interest. To avoid obstructing the channel severely, these baffles probably should not obstruct more than about 20 percent of the channel at any particular location. Staggering them as shown in Fig. 21.5 would accomplish this without excessive obstruction. Rounding the upstream edges will help shed trash, but may be less effective in suppress-
FIGURE 21.5 Wave suppressor for variable-depth flows in a canal. (From Replogle, 1997)
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HYDRAULIC DESIGN OF FLOW MEASURING STRUCTURES
Hydraulic Design of Flow Measuring Structures 21.15
ing waves. Observe in the sequence of drawings in Fig. 21.5 that the staggering is upward in the downstream direction. Note that the next baffle slightly overlaps the horizontal flow lines so that flow passing over the top of one baffle is not allowed to free-fall and start another wave. Fig. 21.5 a–c illustrate, the general behavior as the flow becomes less deep.
21.4 MEASUREMENT ACCURACY Accurate application of water measuring devices generally depends upon standard designs or careful selection of devices, careful fabrication and installation, good calibration data and adequate analysis. Also needed is proper user operation with appropriate inspection and maintenance procedures. During operation, accuracy requires continual verification that all measuring systems, including the operators, are functioning properly. Thus, good training and supervision are required to attain measurements within prescribed accuracy bounds. Accuracy is the degree of conformance of a measurement to a standard or true value. The standards are selected by users, providers, governments, or compacts between these entities. All parts of a measuring system, including the user, need to be considered in accessing the system's total accuracy. As mentioned above, a measurement system usually consists of a primary element, which is that part of the system that creates what is sensed, and is measured by a secondary element. For example, weirs and flumes are primary elements. A staff gage is a secondary element. Designers, purchasers, and users of water measurement devices generally rely on standard designs and manufacturers to provide calibrations and assurances of accuracy. A few water users and providers have the facilities to check the condition and accuracy of flow measuring devices. These facilities have comparison flow meters and/or volumetric tanks for checking their flow meters. These test systems are used to check devices for compliance with specification and to determine maintenance needs. However, maintaining facilities such as these is not generally practical. Various disciplines and organizations do not fully agree on some of the definitions related to measuring device specifications, calibration, and error analysis. Therefore, it is important to verify that a clear and mutual understanding of the specifications, calibration terminology, and the error analysis processes is established when discussing these topics with others.
21.4.1 Definitions of Terms Related to Accuracy Error. Error is the deviation of a measurement, observation, or calculation from the truth. The deviation can be small and inherent in the structure and functioning of the system and be within the bounds or limits specified. Lack of care and mistakes during fabrication, installation, and use can often cause large errors well outside expected performance bounds. Because the true value is seldom known, some investigators prefer to use the term uncertainty. Uncertainty describes the possible error or range of error which may exist. Investigators often classify errors and uncertainties into spurious, systematic, and random types. Precision. Precision is the ability to produce the same measurement value within given accuracy bounds when successive readings of a specific quantity are measured. Precision represents the maximum departure of all readings from the mean value of the readings. Thus, a single observation of a measurement cannot be more accurate than
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HYDRAULIC DESIGN OF FLOW MEASURING STRUCTURES
21.16
Chapter Twenty-One
the inherent precision of the combined primary and secondary precision. It is possible to have good precision of an inaccurate reading. Thus, precision and accuracy differ. Spurious errors. Spurious errors are commonly caused by accident, resulting in false data. Misreading and intermittent mechanical malfunctions can cause discharge readings well outside of expected random statistical distribution about the mean. Spurious errors can be minimized by good supervision, maintenance, inspection, and training. Experienced, well-trained operators are more likely to recognize readings that are significantly out of the expected range of deviation. Unexpected spiral flow and blockages of flow in the approach or in the device itself can cause spurious errors. Repeating measurements does not provide information on spurious error unless repetitions occur before and after the introduction of the error. On a statistical basis, spurious errors confound evaluation of accuracy performance. Systematic errors. Systematic errors are errors that persist and cannot be considered random. Systematic errors are caused by deviations from standard device dimensions, anomalies to the particular installation, and possible bias in the calibration. Systematic errors cannot be removed or detected by repeated measurements. They usually cause persistent error on one side of the true value. The value of a particular systematic error for a particular device may sometimes be considered as a random error. For example, an installation error in the zero setting for a flume might be + 1 mm for one flume and 2 mm for another. For each flume the error is systematic, but for a number of flumes it would be a random error. Random errors. Random errors are caused by such things as the estimating required between the smallest division on a head measurement device and water surface waves at a head measuring device. Loose linkages between parts of flowmeters provide room for random movement of parts relative to each other, causing subsequent random output errors. Repeated readings decrease the average expected error resulting from random errors by a factor of the square root of the number of readings. Total error. Total error of a measurement is the result of systematic and random errors caused by component parts and factors related to the entire system. Sometimes, error limits of all component factors are well known. In this case, total limits of simpler systems can be determined by computation (Bos et al., 1991). In more complicated cases, it may be difficult to confidently combine the limits. In this case, a thorough calibration of the entire system as a unit can resolve the difference. In any case, it is better to do error analysis with data where entire system parts are operating simultaneously and compare discharge measurement against an adequate discharge comparison standard. Expression of errors. Instrument errors are usually expressed by manufacturers as either a percent of reading or a percent of full scale. The secondary devices based on electronic outputs are more frequently expressed in terms of percent full scale. The designer must be aware that a probable error value of say 1 percent full-scale can exceed 10 percent for small value readings on the output device. When used with weirs, for example, the head reading of h1.5 in the weir equation can increase this 10 percent head measurement error to a 15 percent flow measurement error.
21.4.2 Terms Related to Measurement Capability Linearity. Linearity usually means the maximum deviation in tracking a linearly varying quantity, such as measuring head, and is generally expressed as percent of full scale.
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HYDRAULIC DESIGN OF FLOW MEASURING STRUCTURES
Hydraulic Design of Flow Measuring Structures 21.17
Discrimination. Discrimination is the number of decimals to which the measuring system can be read. Precision is no better than the discrimination. Repeatability. Repeatability is the ability to reproduce the same reading for the same quantities. Thus, it is related to precision. Sensitivity. Sensitivity is the ratio of the change of a secondary measurement, such as head, to the corresponding change of discharge. Range and Rangeability. Range is fully defined by the lowest and highest value that the device can measure without damage and comply within a specified accuracy. The upper and lower range bounds may be the result of mechanical limitations, such as friction at the lower end of the range and possible overdriving damage at the higher end of the range. Range can be designated in other ways: (1) as a simple difference between maximum discharge (Qmax) and minimum discharge (Qmin), (2) as the ratio (Qmaxx/Qmin), called rangeability, and (3) as a ratio expressed as 1:(Qmin/Qmax). Neither the difference nor the ratios fully define range without knowledge of either the minimum or maximum discharge. Additional terms (hysteresis, response, lag, rise time). Additional terms related more to dynamic variability might be important when continuous records are needed or if the measurements are being sensed for automatic control of canals and irrigation. Hysteresis is the maximum difference between measurement readings of a quantity established by the same mechanical set point when set from a value above and reset from a value below. Hysteresis can continually get worse as wear of parts increases friction or as linkage freedom increases. Response has several definitions in the instrumentation and measurement field. For water measurement, one definition for response is the smallest change that can be sensed and displayed as a significant measurement. Lag is the time difference of an output reading when tracking a continuously changing quantity. Rise time is often expressed in the form of the time constant, defined as the time for an output of the secondary element to achieve 63 percent of a step change of the input quantity from the primary element.
21.4.3 Comparison Standards Water providers may want, or may be required, to have well-developed measurement programs that are highly managed and standardized. If so, water delivery managers may wish to consult American Society for Testing Materials Standards (ASTM, 1988), Bos (1989), International Organization for Standardization (ISO, 1983: ISO, 1991), and the National Handbook of Recommended Methods for Water Data Acquisition (USGS, 1980). Research laboratories, organizations, and manufacturers that certify measurement devices may need to trace accuracy of measurement through a hierarchy of increasingly rigid standards. The lowest standards in the entire hierarchy of physical comparison standards are called working standards, which are shop or field standards used to control quality of production and measurement. These standards might be gage blocks or rules used to ensure proper dimensions of flumes during manufacturing or devices carried by water providers and users to check the condition of water measurement devices and the quality of their output. Other possible working standards are weights, volume containers, and stopwatches. More complicated devices are used, such as surveyors’ levels, to check weir staff gage zeros. Dead weight testers and electronic standards are needed to check and maintain more sophisticated and complicated measuring devices, such as acoustic flow meters and devices that use pressure cells to measure head. Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
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21.18
Chapter Twenty-One
For further measurement assurance and periodic checking, water users and organizations may keep secondary standards. Secondary standards are used to maintain integrity and performance of working standards. These secondary standards can be sent to government laboratories, one of which is the National Bureau of Standards in Washington, D.C., to be periodically certified after calibration or comparison with accurate replicas of primary standards. Primary standards are defined by international agreement and maintained at the International Bureau of Weights and Measurements in Paris, France. Depending on accuracy needs, each organization should trace their measurement performance up to and through the appropriate level of standards. For example, turbine acceptance testing, such as in the petroleum industry, might justify tracing to the primary standards level.
21.5 SELECTION OF PRIMARY ELEMENTS OF WATER MEASURING DEVICES 21.5.1 General Requirements Design considerations involve the selection of the proper water measurement device for a particular site or situation. Site-specific factors and variables must be considered in extended detail. Each system has unique operational requirements and installation concerns. Knowledge of the immediate measurement needs and reliable estimates on future demands of the proposed system is advantageous. Possible selection constraints may be imposed by laws and compact agreements and should be consulted before selecting a measurement device. Contractual agreements for the purchase of pumps, turbines, and water measuring devices for water supply, sewage and drainage districts often dictate the measurement system required for compliance prior to payment. These constraints may be in terms of accuracy, specific comparison devices, and procedures. Bos (1989) provides an extensive and practical discussion on the selection of open channel water measurement devices. Miller (1996) provides a recent compilation of selection criteria for pipe flow-meters suited to liquids and steam and other gas flows. Bos (1989) provides a selection flow chart and a table of water measurement device properties to guide the selection process for the open channel devices. Miller (1996) describes each meter in detail for the pipe systems, but is more general in leaving the selection to the designer. Because the design engineers for civil engineering projects are most likely to be dealing with irrigation water supply, waste water, or drainage and flood flows, the emphasis is placed on the measuring systems deemed most appropriate to these processes. Large closed-pipe systems for water supplies are frequently encountered, so installation situations appropriate to these will also be included. Gas flows, including steam, are more likely to be encountered by mechanical and chemical engineers and those readers are referred to Miller (1996) and ASME (1959, 1971).
21.5.2 Types of Measuring Devices System operators for water supply, drainage, and waste water commonly use many types of standard water measuring devices, usually in open channels with limited applications in closed conduits. Particularly prominent uses of open channel devices are found in irrigation delivery systems and farm distribution systems, although these measuring devices
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HYDRAULIC DESIGN OF FLOW MEASURING STRUCTURES
Hydraulic Design of Flow Measuring Structures 21.19
are frequently used for sewer flows and even flood flows. However, the latter two areas of application are frequently more difficult because of the likelihood of heavy bed loads and floating debris. In pipe flowmeters, the most commonly installed devices in industry are the orifice meters, accounting for up to 80 percent of all industrial meters (Miller, 1996). Venturi meters and flow tubes provide much of the remainder. In absolute numbers, the household meters, based on various technologies from nutating disks to paddle-wheel turbines, dominate. For open–channel flows, weirs, flumes, submerged and free orifices, and current meters dominate the flow measuring methods. Pipe flow meters, propeller and turbine, acoustic, magnetic, and vortex-shedding meters are used on large water supply wells such as those used in irrigation and municipal water supply. Differential head meters, such as orifice meters, Venturi meters, and flow tubes, are also used in these applications. The meters considered herein are 1. Open-channel flow devices a. b. c. d. e. f.
Current metering (cup, propeller, and electromagnetic probes) Weirs Flumes Acoustic (transonic and Doppler) Tracers Miscellaneous
2. Pipe flow devices a. b. c. d. e. f.
Differential head meters Acoustic (transonic and Doppler) Tracers Turbine/propeller/other insert mechanical Vortex-shedding Miscellaneous
The main factors which influence the selection of a measuring device include (USBR, 1997): a. Accuracy requirements b. Cost c. Legal constraints d. Range of flow rates e. Head loss f. Adaptability to site conditions g. Adaptability to variable operating conditions h. Type of measurements and records needed I. Operating requirements j. Ability to pass sediment and debris k. Longevity of device for given environment l. Maintenance requirements m. Construction and installation requirements
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HYDRAULIC DESIGN OF FLOW MEASURING STRUCTURES
21.20
Chapter Twenty-One
n. o. p. q. r.
Device standardization and calibration Field verification, troubleshooting, and repair User acceptance of new methods Vandalism potential Impact on environment
Accuracy requirements. The desired accuracy of the measurement system is an important consideration in the selection of a measurement method. Most water measurement installations, including the primary and secondary devices, can produce accuracies of 5 percent. Some systems are capable of 1 percent under laboratory settings. However, in the field, maintaining such accuracies usually requires considerable expense or special effort in terms of construction, secondary equipment, calibration in-place, and stringent maintenance. Selecting a device that is not appropriate for the site conditions can result in a nonstandard installation of reduced accuracy, sometimes exceeding 10 percent. Accuracies are frequently reported that relate only to the primary measurement method or device. However, many methods require secondary measurement equipment that produces the actual readout. This readout equipment typically increases the overall error of the measurement. Cost. The cost of the measurement method includes the cost of the device itself, the installation, secondary devices, operation, and maintenance. Measurement methods vary widely in their cost and in their serviceable life span. Measurement methods are often selected based on the initial cost of the primary device with insufficient regard for the additional costs associated with providing the desired records of flows over an extended period of time. Legal constraints. Governmental or administrative water board requirements may dictate the water measurement devices or methods. Water measurement devices that become a standard in one geographic area may not necessarily be accepted as a standard elsewhere. In this sense, the term “standard” does not necessarily signify accuracy or broad legal acceptance. Many water agencies require certain water measurement devices used within their jurisdiction to conform to their standard for the purpose of simplifying operation, employee training, and maintenance. Flow range. Many measurement methods have a limited range of flow conditions for which they are applicable. This range is usually related to the need for certain prescribed flow conditions which are assumed in the development of calibrations. Large errors in measurement can occur when the flow is not within this range. For example, using a bucket and stopwatch for large flows that engulf the bucket is not very accurate. Similarly, sharp-edged devices, such as sharpcrested weirs, typically do not yield good results with large channel flows. These are measured better with large flumes or broad-crested weirs, which in turn are not appropriate for trickle flows. Certain applications have typical flow ranges. Irrigation supply monitoring seldom demands a low-flow-to-high-flow range above about 30, while this range on natural stream flows may exceed 1000. In some cases, secondary devices can limit the practical range of flow rates. For example, with devices requiring a head measurement, the accuracy of the head measurement from a visually read wall gage may limit the measurement of low flow rates. For some devices, accuracy is based on percent of the full-scale value. While the resulting error may be well within acceptable limits for full flow, at low flows, the resulting error may become excessive, limiting the usefulness of such measurements. Generally, the device should be
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HYDRAULIC DESIGN OF FLOW MEASURING STRUCTURES
Hydraulic Design of Flow Measuring Structures 21.21
selected to cover the desired range. Choosing a device that can handle an unnecessary large flow rate may result in compromising measurement capability at low flow rates, and vice versa. This choice depends on the objective of the measurement. For example, in irrigation practice, usually choose a device that can measure the most common flow range at the expense of poorly measuring extremes, such as flood flows. For urban drainage, the flood peak may be important. For practical reasons, different accuracy requirements for high and low flows may be chosen. This is reasonable when an annual total is the primary goal and the low flows contribute a small percentage to that total. Also, if the inaccurate low flow readings are truly a random error then this error approaches zero with large accumulations of readings. Thus the designer needs to know if management decisions are made from individual readings or from long term averages. Headloss. Most water measurement devices require a drop in head. On retrofit installations, for example, to an existing irrigation project, such additional head may not be available, especially in areas that have relatively flat topography. On new projects, incorporating additional headloss into the design can usually be accomplished at reasonable cost. However, a tradeoff usually exists between the cost of the device and the amount of headloss. For example, acoustic flow meters are expensive but require little headloss. Sharp-crested weirs are inexpensive but require a relatively large headloss. The head loss required for a particular measuring device usually varies over the range of discharges. In some cases, head needed by a flow measuring device can reduce the capacity of the channel at that point. Adaptability to site conditions. The selection of a flow measuring device must address the site of the proposed measurement. Several potential sites may be available for obtaining a flow measurement. The particular site chosen may influence the selection of a measuring device. For example, discharge in a canal system can be measured within a reach of the channel or at a structure such as a culvert or check structure. A different device would typically be selected for each site. The device selected ideally should not alter site hydraulics so as to interfere with normal operation and maintenance. Also, the shape of the cross-sectional flow area may favor particular devices. Adaptability to variable operating conditions. Flow demands for most water delivery systems usually vary over a range of flows and flow conditions. The selected device must accommodate the flow range and changes in operating conditions, such as variations in upstream and downstream head. Weirs or flumes should be avoided if downstream water levels can, under some conditions, cause excessive submergence. Also, the information provided by the measuring device should be conveniently useful for the operators performing their duties. Devices that are difficult and time consuming to operate are less likely to be used and are more likely to be used incorrectly. In some cases, water measurement and water level or flow control are desired at the same site. A few devices are available for accomplishing both (e.g., constant-head orifice, vertically movable weirs, and Neyrpic flow module; Bos, 1989). However, separate measurement and control devices are typically linked for this purpose and usually can exceed the performance of combined devices in terms of accuracy and level control, if care is exercised to assure that the separate devices are compatible and achieve both functions when used as a system. Type of measurements and records needed. An accurate measure of instantaneous flow rate is useful for system operators in setting and verifying flow rate. However, because flow rates change over time, a single (instantaneous) reading may not accurately reflect the total volume of water delivered. Where accounting for water volume is desired,
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HYDRAULIC DESIGN OF FLOW MEASURING STRUCTURES
21.22
Chapter Twenty-One
a method of accumulated individual flow measurements is needed. Where flows are steady, daily measurements may be sufficient to infer total volume. Most deliveries, however, require more frequent measurements. Meters that accumulate total delivered volume are desirable where water users take water on demand. Totalizing and automatic recording devices are available for many measuring devices. For large structures, the cost for water-level sensing and recording hardware is small relative to the structure cost. For small structures, these hardware costs remain about the same and thus become a major part of the measurement cost, and may often exceed the cost of the primary structure itself. Many water measuring methods are suitable for making temporary measurements (flow surveys) or performing occasional verification checks of other devices. The method chosen for such a measurement might be quite different from that chosen for continuous monitoring. Although many of these flow survey methods are suited for temporary operation, the focus here is on methods for permanent installations. Operating Requirements. Some measurement methods require manual labor to obtain a measurement. Current metering requires a trained staff with specialized equipment. Pen-and-ink style water-stage recorders need operators to change paper, add ink, and verify proper functioning. Manual recording of flows may require printed forms to be manually completed and data to be accumulated for accounting purposes. Devices with manometers require special care and attention to assure correct differential-head readings. Automated devices, such as ultrasonic flowmeters and other systems that use transducers and electronics, require operator training to set up, adjust, and troubleshoot. Setting gatecontrolled flow rates by simple canal level references or by current metering commonly requires several hours of waiting between gate changes for the downstream canal to fill and stabilize. However, if a flume or weir is installed near the control gate, that portion of the canal can be brought to the stable, desired flow level and measured flow rate in a few minutes, and the canal downstream of the flume or weir can then fill to the correct level over a longer time without further gate adjustments. Thus, the requirements of the operating personnel in using the devices and techniques for their desired purposes must be considered in meter selection. Some measuring devices may inherently serve an additional function applicable to the operation of a water supply system. For example, weirs and flumes serve to hydraulically isolate upstream parts of a canal system from the influence of downstream parts. This occurs for free overfall weirs and flumes flowing below their modular limits. Acoustic, propeller, magnetic, and vortex-shedding flowmeters do not provide this function without additional structural measures such as a downstream overfall. If these meters are used, and the isolation function is desired, then the designer should be made aware of the requirement and provide a free overfall. Isolating the influence of upstream changes from affecting downstream channels, is less easily accomplished. However, it can be partly implemented with orifices that have a differential head that is large compared to the upstream fluctuations. The designer should be aware that a sharp-crested weir overfall requires a relatively high head drop and may need to be excessively wide to provide the isolation function with low absolute head drop. While a long board can be used downstream from a propeller meter to provide the necessary width of flow that will pass a required quantity of water at small head, that small head, and the crude board would not be well suited for measuring flow rate. The designer may wish to take advantage of broad-crested weir behavior and provide a thick crest that can withstand in excess of about 80 percent submergence, which usually translates into low absolute head loss. When used with a propeller meter, for example, the broad-crested weir need not be well defined and can be economically installed (Replogle, 1997). Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
HYDRAULIC DESIGN OF FLOW MEASURING STRUCTURES
Hydraulic Design of Flow Measuring Structures 21.23
Ability to pass sediment and debris. Canal systems often carry a significant amount of sediment in the water. Removal of all suspended solids from the water is usually prohibitively expensive. Thus, some sediment will likely be deposited anywhere the velocities are reduced, which typically occurs near flow measuring structures. Whether this sediment causes a problem depends on the specific structure and the volume of sediment in the water. In some cases, this problem simply requires routine maintenance to remove accumulated sediment; in others, the accumulation can make the flow measurement inaccurate or the device inoperative. Sediment deposits can affect approach conditions and increase approach velocity in front of weirs, flumes, and orifices. Floating and suspended debris such as aquatic plants, washedout bank plants, and fallen tree leaves and twigs can plug some flow measurement devices and cause significant flow measurement problems. Many of the measurement devices which are successfully used in closed conduits (e.g., orifices, propeller meters, and so on) are not usable in culverts or inverted siphons because of debris in the water. Attempting to remove this debris at the entrance to culverts is an additional maintenance problem. Flumes, especially long-throated flumes, can be designed to resist sedimentation. The design options available are to select a structure shape that will maintain velocities that assure erosion of sediments, or at least continued movement of incoming sediments through the flume, at important flow rates. In large broad-crested weirs (a class of longthroated flumes) for capacities greater than 1 m3/s per m of flume width, velocities greater than 1 m/s can be achieved for the upper 75 percent of the flow range, and is usually erosive enough to maintain flume function even for high-sediment bed loads. At the lower flow ranges and for heavy sediment bed loads, deposition is likely and frequent maintenance may be required. Trapezoidal sections tend to retain low velocities into the upper ranges of flow and are less sediment worthy. Long-throated flumes with flat bottoms throughout and side contractions maintain a high velocity for 0.5 m3/s per m width, and higher, but must have throat lengths that are 2 to 3 times the throat width in order to be accurately computable. The sediment worthiness of a flume design depends more on these absolute velocities than on whether the flume floor is flat throughout or raised as in a broadcrested weir. This prompts the designer to select shapes that can provide these velocities. One suggestion for broad crested weirs in a fixed sized channel is to construct a false floor in the head gage area to increase the velocity there and prevent changes in area of flow there. Also, sediments can accumulate in the upstream channel to a depth of the false floor without affecting the function of the flume. This can extend the time between mandatory channel cleaning. Device environment. Any measurement device with moving parts or sensors is subject to failure if it is not compatible with the site environment. Achieving proper operation and longevity of devices is an important selection factor. Very cold weather can shrink moving and fixed parts differentially and solidify oil and grease in bearings. Water can freeze around parts and plug pressure ports and passageways. Acidity and alkalinity in water can corrode metal parts. Water contaminants such as waste solvents can damage lubricants, protective coatings, and plastic parts. Mineral encrustation and biological growths can impair moving parts and plug pressure transmitting ports. Sediment can abrade parts or consolidate tightly in bearing and runner spaces in devices such as propeller meters. Measurement of wastewater and high sediment transport flow may preclude the use of devices that require pressure taps, intrusive sensors, or depend upon clear transmission of
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HYDRAULIC DESIGN OF FLOW MEASURING STRUCTURES
21.24
Chapter Twenty-One
sound through the flow. Water measurement devices that depend on electronic devices and transducers must have appropriate protective housings for harsh environments. Improper protection against the site environment can cause equipment failure or loss of accuracy. Maintenance requirements. The type and amount of maintenance varies widely with different measurement methods. For example, current metering requires periodic maintenance of the current meter itself and maintenance of the meter site to assure that is has a known cross section and velocity distribution. When the flow carries sediment or debris, most weirs, flumes, and orifices require periodic cleaning of the approach channel. As mentioned above, design and meter selection can mitigate the maintenance problems with sediments, but are not likely to eliminate them. Electronic sensors need occasional maintenance to ensure that they are performing properly. Regular maintenance programs are recommended to ensure prolonged measurement quality for all types of devices. Construction and installation requirements. In addition to installation costs, the difficulty of installation and the need to retrofit parts of the existing conveyance system can complicate the selection of water measurement devices. Clearly, devices that can be easily retrofitted into the existing canal system are much preferred because they generally require less down time, and usually present fewer unforeseen problems. Device standardization and calibration. A standard water measurement device infers a documented history of performance based on theory, controlled calibration, and use. A truly standard device has been fully described, accurately calibrated, correctly constructed, properly installed, and sufficiently maintained to fulfill the original installation requirements and flow condition limitations. Discharge equations and tables for standard devices should provide accurate calibration. Maintaining a standard device usually only involves a visual check and measurement of a few specified items or dimensions to ensure that the measuring device has not departed from the standard. Many standard devices have a long history of use and calibration, and thus are potentially more reliable. Commercial availability of a device does not necessarily guarantee that it satisfies the requirements of a standard device. When measuring devices are fabricated onsite or are poorly installed, small deviations from the specified dimensions can occur. These deviations may or may not affect the calibration. The difficulty is that unless an as-built calibration is performed, the degree to which these errors affect the accuracy of the measurements is largely unknown. All too frequently, design deviations are made under the misconception that current metering can be used to provide an accurate field calibration. In practice, calibration by current metering to within 2 percent is difficult to attain. An adequate calibration for free-flow conditions requires many current meter measurements at several discharges. Changing and maintaining a constant discharge for calibration purposes is often difficult under field conditions. Field verification, troubleshooting, and repair. After construction or installation of a device, some verification of the calibration is generally recommended. Usually, the methods used to verify a permanent device (e.g., current metering) are less accurate than the device itself. However, this verification simply serves as a check against gross errors in construction or calibration. For some devices, errors occur as components wear and the calibration slowly drifts away from the original. Other devices have components that simply fail, that is, you get the correct reading or no reading at all. The latter is clearly preferred. However, for many devices, occasional checking is required to ensure that they are still performing as intended. Selection of devices may depend on how they fail and how easy it is to verify that they are performing properly.
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HYDRAULIC DESIGN OF FLOW MEASURING STRUCTURES
Hydraulic Design of Flow Measuring Structures 21.25
User acceptance of new methods. Selection of a water measurement method must also consider the past history of the practice at the site. When improved water measurement methods are needed, proposing changes that build on established practice are generally easier to institute than radical changes. It can be beneficial to select a new method that allows conversion to take place in stages to provide educational examples and demonstrations of the new devices and procedures. Vandalism potential. Instrumentation located near public access is a prime target for vandalism. Where vandalism is a problem, measurement devices with less instrumentation, or instrumentation that can be easily protected, are preferred. When needed, instrumentation can be placed in a buried vault to minimize visibility. Impact on environment. During the selection of a water measurement device, consideration must be given to potential environmental impacts. Water measurement devices vary greatly in the amount of disruption to existing conditions that is needed for installation, operation, and maintenance. For example, installing a weir or flume constricts the channel, slows upstream flow, and accelerates flow within the structure. These changes in the flow conditions can alter local channel erosion, local flooding, public safety, local aquatic habitat, and movement of fish up and down the channel. These factors may alter the cost and selection of a measurement device.
21.5.3 Selection Guidelines Selection of a water measurement method can be a difficult, time-consuming design process if one were to formally evaluate all the factors discussed above for each measuring device. This difficulty is one reason that standardization of measurement devices within water agency jurisdictions is often encouraged by internal administrators. However, useful devices are sometimes overlooked when devices similar to previous purchases are automatically selected. Therefore, some preliminary guidance on selection is offered to the designer so that the number of choices can be narrowed down before a more thorough design analysis of the tradeoffs between alternatives is performed. Short list of devices based on application. The list of practical choices for a water measuring device is quickly narrowed by site conditions because most devices are applicable to a limited range of channel or conduit conditions. Economics also limits applicable devices. For example, few irrigation deliveries to farms can justify expensive acoustic meters. Likewise, using current meters for manual flow measurement in a channel is appropriate for intermittent information but is usually too labor intensive for use on a continuous basis. Table 21.1 provides a list of commonly used measurement methods that are considered appropriate for each of several applications. Table 21.2 provides an abbreviated table of selection criteria and general compliance for categories of water measurement devices. The symbols (), (0), and () are used to indicate relative compliance for each selection criterion. The () symbol indicates positive features that might make the device attractive from the standpoint of the associated selection criteria. A () symbol indicates negative aspects that might limit the usefulness of this method based on that criterion. A (0) indicates no strong positive or negative aspects in general. A (V) V means that the suitability varies widely for this class of devices. The letters NA mean that the device is not applicable for the stated conditions. A single negative value for a device does not mean that the device is not useful or appropriate, but other devices would be preferred for those selection criteria. Vortex-shedding flow meters are not specifically rated in this grouping. They are expected to compete with orifice meter applications. They generally offer less head loss Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
HYDRAULIC DESIGN OF FLOW MEASURING STRUCTURES
21.26
Chapter Twenty-One
TABLE 21.1
Application–Based Selection of Water Measurement Devices
1. Openchannel conveyance system a. Natural channels (see Herschy, 1985) (1) Rivers Periodic current metering of a control section to establish stagedischarge relation Broad–crested weirs Long–throated flumes Shortt–crested weirs Acoustic velocity maters (AVM—transient time) Acoustic Doppler velocity profiles Floatt–velocity/area method Slope–area methods (2) Intermediate–sized and small streams Current metering/control section Broad–crested weirs Long–throated flumes Shortt–crested weirs Shortt–throated flumes Acoustic velocity meters (AVM—transient time) Floatt–velocity/area method b. Regulated channels (see USBR, 1997) (1.) Spil ways (a) Gated Sluice gates Radial gates (b) Ungated Broad–crested weirs (including special crest shapes, Ogee crest, etc.) Shortt–crested weirs (2) Large canals (a) Control structures Check gates Sluice gates Radial gates Overshot gates (b) Other Long–throated flumes Broad–crested weirs Shortt–throated flumes Acoustic velocity meters
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HYDRAULIC DESIGN OF FLOW MEASURING STRUCTURES
Hydraulic Design of Flow Measuring Structures 21.27 TABLE 21.1
(Continue)
(3.) Small canals (including openchannel fluid conduit flow) Long–throated flumes Broad–crested weirs Shortt–throated flumes Sharp–crested weirs Rated flow control structures (check gates, radial gates, sluice gates, overshot gates) Acoustic velocity meters (c) Other Floatt–velocity/area methods (4) Irrigation delivery to farm turnout (a) Pipe turnouts (short inverted siphons, submerged culverts, etc.) Metergates Current meter Weirs Short–throated flumes Long–throated flumes (b) Other Constant head orifice Rated sluice gates Movable weirs 2. Closed conduit conveyance systems (see Brater and King, 1982; Miller, 1996) (a.) Large pipes Venturi meters, venturi tubes, nozzles Rated control gates (orifice) Acoustic velocity meters (transit time) (b.) Small and intermediate–sized pipelines Venturi meters, Venturi tubes, nozzles Orifices (in–line, end–cap, shunt meters, etc.) Propeller and turbine meters Magnetic meters Acoustic meters (transit–time and Doppler) Pitot tubes Elbow meters Vortex – shedding Trajectory methods (e. g., full-pipe trajectory; California pipe method for part full pipe) Other commercially available meters (household types) SOURCE: From USBR (1997).
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0
0
2
Debris pass Longivity Moving parts
Electr. requir.
0
0
Construction
Field verify
Standarization
Short
0
NA
NA
0
0
0
0
Flumes
Throated
0
0
NA
NA
0
0
0
0
(Channel)
Orifice
Submerg
Current
Acoustic
0
0
0
0
0
0
NA
0
0
NA
NA
NA
0
0
0
0
(Channel)
Vel Meter
(Channel)
Metering
Radial
†
*
Venturi, orifice, pilot tube, etc. Propeller meters, turbine meters, paddle wheel meters, etc.
Differ.
0
0
0
0
0
0
0
0
0
V
0
0
NA
NA
V
(Pipes)
0
NA
Mechan.
0
0
0
0
V
V
0
NA
NA
0
0
0
(Pipes)
Head meters* Head meters†
0
0
0
(Pipe Exit)
Meters
Propeller
0
0
Gates
and Sluice
Source: Adapted from USBR (1997). Symbols 1, 0, 2 are used as relative indicators comparing application of the listed water measuring device to the listed criteria Symbol V denotes that situability varies widely. Symbol NA denotes “not applicable” to criteria
0
0
0
NA
Maintenance
Volume Sediment Sediment pass
NA
NA
0
NA
Closed conduit Measurement Type Rate
0
Short, full pipe
NA
0
NA
Unliner canal
0
Flows 0.25 m3/s
0
Flows 5 m3/s
0
0
0
0
0
Flumes
0
Accuracy
Cost
Weirs
Long–
Throated
Flow span
Weirs
Criteria
Broad
Crested
Headloss Site Condition Lined canal
Crested
Selection
Sharp
Selection Guide for Water Measuring Devices.
Devise
TABLE 21.2
0
0
0
0
0
0
NA
NA
0
0
0
0
(Pipes)
meters
Magnetic
0
0
0
0
0
NA
NA
0
(Pipes)
Doppler
Acoustic
Acoustic
Acoustic
Pipe
Transon.
0
2
0
0
0
0
0
NA
NA
0
0
0
0
0
0
0
NA
NA
0
(Pipe, 1 Path) (Multipath)
Transonic
HYDRAULIC DESIGN OF FLOW MEASURING STRUCTURES
21.28
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HYDRAULIC DESIGN OF FLOW MEASURING STRUCTURES
Hydraulic Design of Flow Measuring Structures 21.29
that orifice meters and can cover a wider discharge range for a particular installation. Although they have been around for many years, they have only recently been offered in a configuration that makes them competitive with orifice meters, which they are generally expected to replace because they can produce less pipe head loss. Open-channel applications for vortex-shedding meters are not considered practical.
21.6 SELECTION OF SECONDARY DEVICES FOR DISCHARGE READOUT AND CONTROL While the emphasis of this chapter is on hydraulic design, it is important that the output of the design be translated into useful information for the user.
21.6.1 Intended Uses The secondary device that is used with a flowmeter depends strongly on the use of the information. Immediate management decisions, such as adjusting a valve or canal gate, require nearly instant feedback to the operator at an accuracy and precision that fully utilizes the available accuracy of the primary device. As discussed above, random errors over many measurements tend to cancel, and long-term totals can often absorb large random detection errors that would not be tolerable for decisions depending on a single reading. The designer should be cognizant of the user needs and be prepared to provide an appropriate output.
21.6.2 Quality Assurance Secondary devices provide a variety of functions, primary of which are data recording and data quality assurance. The secondary devices necessary for these may not be the same. It is good practice to provide manual, instantaneous flow rate output at the meter site so that the servicing personnel can quickly know that the main secondary instrument is functioning. For weirs and flumes wall gages that show flow rates directly are recommended as a quick visual check of secondary instrumentation. However, a wall gage that shows a head reading that is converted to a discharge rate by the technician using a table or equation will usually suffice. Sometimes it is practical to provide field check capability to a secondary device by special treatment of the installation. For example, if a pressure transducer is used to detect head on a flume, the transducer can be mounted in a stilling well attached to the flume. If it is further mounted on a movable rack, it can allow servicing personnel to raise and lower the transducer a prescribed amount to verify that the output signal correctly reports the change (Replogle, 1997). In pipe flows, differential head meters, such as Venturi and orifice meters should be fitted with manometer ports that can be easily accessed by servicing personnel to verify the detection and transmission of data by electronic devices. Propeller, turbine, acoustic, and magnetic meters are among meters that are usually closely integrated with their secondary devices so that separate verification of the primary device function is more challenging. These checks of the secondary devices do not, however, necessarily detect a malfunction of the primary device, such as scale growth on Venturi surfaces, worn orifice plates,
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HYDRAULIC DESIGN OF FLOW MEASURING STRUCTURES
21.30
Chapter Twenty-One
or sediment filled weirs and flumes. It is sometimes advisable to provide measurement in a main line and in both lines of a bifurcation, thus providing redundancy that can help detect meter malfunctions. Regular inspection is a necessary operational requirement and an important design and selection feature is often the ease of inspection.
21.7 APPLICATIONS OF LONG-THROATED FLUMES Long-throated flumes are replacing the older open-channel devices, such as Parshall flumes, because of their design flexibility in terms of size, shape, accuracy and economics. Versions for large canals and small irrigation furrows are in use. Shapes vary from rectangular to trapezoidal, with circular versions useful in partly full pipes and complex versions for exceptionally wide flow ranges (Bos et al., 1991; Clemmens et al., 1993; USBR, 1997; Wahl and Clemmens, 1998). Long-throated flumes and broad-crested weirs operate by using a channel contraction to cause critical flow. If there is not enough contraction, critical flow does not occur. Flow is then nonmodular and gauge readings become meaningless. If there is too much contraction, the water surface upstream may be raised excessively and cause canal over-
TABLE 21.3 Choices of Broad – Crested Weir Sizes and Rating Tables for Lined Canals, Metric Units.*,† Canal Shape Max Side Bottom Slope Width z1 b1 (m) (1) (2)
Canal Depth† d (m) (4)
1.0
0.25
0.70
1.0
0.30
0.75
1.0
0.50
0.80
1.0
0.60
0.90
Range of Canal Capacities Lower‡ Upper (m3/s) (m3/s) (4) (5) 0.08 0.09 0.10 0.11 0.12 0.13 0.09 0.10 0.11 0.12 0.13 0.16 0.11 0.12 0.12 0.16 0.18 0.20 0.12 0.13
||
0.14 0.24|| 0.38 0.43 0.37 0.32 0.21 0.34 0.52 0.52 0.44 0.31 0.33|| 0.52|| 0.68|| 0.64 0.46 0.29 0.39|| 0.62
Selection Table 21.4 (6) Am Bm Cm Dm1 Em1 Fm1 Bm Cm Dm1 Em1 Fm1 Gm1 Dm2 Em1Em2 Fm1Fm2 Gm1 Hm Im Em2 Fm2
Weir Weir Shape Crest Sill Width Ht. bc p1 (m) (m) (7) (8) 0.5 0.6 0.7 0.8 0.9 0.1 0.6 0.7 0.8 0.9 1.0 1.2 0.8 0.9 1.0 1.2 1.4 1.6 0.9 1.0
0.125 0.175 0.225 0.275 0.325 0.375 0.15 0.20 0.25 0.30 0.35 0.45 0.15 0.20 0.25 0.35 0.45 0.55 0.15 0.20
Min Head Loss
H (m) (9) 0.015 0.018 0.022 0.026 0.030 0.033 0.017 0.021 0.025 0.029 0.033 0.039 0.019 0.024 0.029 0.037 0.043 0.048 0.021 0.025
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HYDRAULIC DESIGN OF FLOW MEASURING STRUCTURES
Hydraulic Design of Flow Measuring Structures 21.31 TABLE 21.3
(Continued)
Canal Shape Max Side Bottom Slope Width z1 b1 (m) (1) (2)
Canal Depth† d (m) (4) 0.16
1.0
0.75
1.0
1.5
0.60
1.2
1.5
0.75
1.4
1.5
1.00
1.6
1.5
1.25
1.7
1.5
1.50
1.8
Range of Canal Capacities Lower‡ Upper (m3/s) (m3/s) (4) (5) 1.09 0.18 0.20 0.22 0.16 0.18 0.20 0.22 0.20 0.24 0.27 0.29 0.32 0.35 0.24 0.27 0.29 0.32 0.35 0.38 0.29 0.32 0.35 0.38 0.43 0.32 0.35 0.38 0.43 0.49 0.55 0.35 0.38 0.43 0.49 0.55
Gm1 0.86 0.64 0.43 0.91 1.51 1.22 0.94 1.3|| 2.1|| 2.5 2.2 1.8 1.4 1.8 2.8 3.9|| 3.5 3.1 2.6 3.4|| 4.7 5.7 5.1 3.9 4.1|| 5.6|| 7.2 5.9 4.5 3.3 4.8|| 6.5|| 8.1|| 6.6 5.1
Selection Table 21.4 (6) 1.2 Hm Im Jm Gm2 Hm Im Jm Km Lm Mm Nm Pm Qm Lm Mm Nm Pm Qm Rm Nm Pm Qm Rm Sm Pm Qm Rm Sm Tm Um Qm Rm Sm Tm Um
Weir Weir Shape Crest Sill Width Ht. bc p1 (m) (m) (7) (8) 0.30 1.4 1.6 1.8 1.2 1.4 1.6 1.8 1.50 1.75 2.00 2.25 2.50 2.75 1.75 2.00 2.25 2.50 2.75 3.00 2.25 2.50 2.75 3.00 3.50 2.50 2.75 3.00 3.50 4.00 4.50 2.75 3.00 3.50 4.00 4.50
0.035 0.40 0.50 0.60 0.225 0.325 0.425 0.525 0.300 0.383 0.467 0.550 0.633 0.717 0.333 0.417 0.500 0.583 0.667 0.750 0.417 0.500 0.583 0.667 0.833 0.417 0.500 0.583 0.750 0.917 1.083 0.417 0.500 0.667 0.833 1.000
Min Head Loss
H (m) (9) 0.043 0.050 0.049 0.030 0.038 0.047 0.053 0.031 0.38 0.044 0.050 0.056 0.059 0.036 0.042 0.049 0.055 0.062 0.066 0.046 0.052 0.059 0.065 0.081 0.048 0.055 0.061 0.074 0.084 0.089 0.051 0.058 0.071 0.083 0.092
Source: Adapted From Replogte et al., (1990). a La Hmax:Lb 2 to 3p 3 1:x La Lb 2 to 3H Hmax L 1.5H1max, but within range given in Table 25.4 d 1.2 h1max p1; ∆H ∆ .0.1H1 † Limited by sensitivity ‡ Maximum recomended canal depth || Limited by Froude number, otherwise limited by canal depth
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HYDRAULIC DESIGN OF FLOW MEASURING STRUCTURES
21.32
Chapter Twenty-One
TABLE 21.4 Rating Equations and Ranges of Application for Trapezoidal Broad–crested Weirs* (Adapted from Replogle, Clemmens and Bos, 1990) Discharge equation: Q C1(h1 C2)U Units: (h1, m; Q, m3/s) Coef for Eq.
Weir Am
Weir Bm
Weir Cm
Weir Dm1
Weir Dm2
Weir Em1
Weir Em2
Weir Fm1
0.23 L 0.34 0.30 L 0.42 0.35 L 0.58 040 L 0.58 0.30 L 0.45 0.38 L 0.56 0.38 L 0.56 0.42 L 0.61
C1 C2 U h1min h1max Q1min Q1max Coef. for Eq.
2.145 0.0067 1.8667 0.030 00.220 0.005 0.140
2.365 0.0079 1.8599 0.030 0.280 0.005 0.240
2.276 0.0045 1.7597 0.040 0.340 0.010 0.380
2.837 0.0125 1.8765 0.040 0.390 0.010 0.520
2.913 0.0101 1.8555 0.040 0.300 0.010 0.250
2.921 0.0090 1.8152 0.030 0.370 0.010 0.520
3.037 0.0089 1.8252 0.030 0.370 0.010 0.520
3.122 0.010 1.813 0.030 0.370 0.010 0.680
Weir Fm2
Weir Gm1
Weir Gm2
Weir Hm
Weir Im
Weir Jm
Weir Km
Weir Lm
0.42 L 0.61 0.50 L 0.75 0.45 L 0.68 0.56, L 0.84 0.48 L 0.71
C1 C2 U h1min h1max Q1min Q1max Coef. for Eq.
0.40 L 60
0.48 L 0.72 0.58 L 0.87
3.319 0.0107 1.8376 0.030 0.410 0.010 0.680
3.601 0.0130 1.8271 0.040 0.500 0.020 1.100
3.695 0.0108 1.8141 0.040 0.450 0.020 0.920
3.854 0.0067 1.7522 0.040 0.560 0.020 1.500
4.135 0.0096 1.7438 0.040 0.480 0.020 1.200
4.281 0.0048 1.6708 0.030 0.400 0.020 0.940
4.369 0.0170 1.8324 0.060 0.470 0.040 1.300
Weir Mm
Weir Nm
Weir Pm
Weir Qm
Weir Rm
Weir Sm
Weir Tm
5.389 0.0151 1.8581 0.060 0.580 0.050 2.100 Weir Um
0.65 L 0.97 0.75 L 1.10 0.80 L 1.20 0.85 L 1.28 0.95 L 1.40 0.95 L 1.40 0.85 L 1.20 0.68 L 1.00
C1 C2 U h1min h1max Q1min Q1max
5.831 0.0169 1.8528 0.060 0.650 0.050 2.800
6.284 0.0268 1.8986 0.090 0.750 0.100 3.900
6.713 0.0190 1.8397 0.080 0.790 0.100 4.700
7.115 0.023 1.8331 0.070 0.850 0.100 5.700
7.563 0.0243 1.8477 0.070 0.930 0.100 7.200
8.397 0.0209 1.8043 0.060 0.950 0.100 8.200
9.144 0.0129 1.7301 0.060 0.810 0.100 6.600
9.789 0.0080 1.6711 0.060 0.660 0.100 5.100
Source: Adapted from Replogle et al. (1993). Calibrations developed with computer model (Clemmens et al., 1993).
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HYDRAULIC DESIGN OF FLOW MEASURING STRUCTURES
Hydraulic Design of Flow Measuring Structures 21.33
topping or other problems. The problem facing the designer is to select the shape of the control section, or throat, so that critical flow occurs throughout the full range of discharges to be measured. Also, the designer must provide acceptable sensitivity and accuracy while not causing too much disruption in upstream flow conditions (e.g., sediment deposition, canal overtopping). This appears to be a difficult task, but existing design aids and rating tables make this task more manageable. A selection of these aids and tables are presented herein. A multitude of possible designs have been sorted, based on practical experience and theory, to a relatively few selected structures from which the designer may choose. Stage-discharge equations or direct rating tables for three types of channel conditions are presented in this chapter. These are: (l) lined trapezoidal channels, (2) earthen canals or lined rectangular channels, and (3) circular pipes and conduits not flowing full. Some offerings for natural streams are presented in Bos et al., (1991) and Replogle et al., (1990). Trapezoidal broad-crested weirs (i.e., having only a bottom contraction) were used for the development of selected standard sizes for lined trapezoidal channels. Calibration equations developed from tables computed with the computer model (Clemmens et al., 1993) are presented in the following series of tables (Tables 21.3 and 21.4). Trapezoidal flumes with side contractions were not selected for general use in standard irrigation situations because they are usually more difficult to construct and require more head loss. A design aid was also developed to help ensure sufficient sensitivity, limit the Froude number, and otherwise assist in selection. The designer needs to only select a weir width with its corresponding sill height. Rectangular broad-crested weirs were again chosen for unlined canals. However, for these weirs the designer must select a channel (and throat) width as well as a sill height, and must be more aware of the other design considerations. For lined rectangular channels, only the sill height must be selected. These tables and equations can also be used to determine the rating for side-contracted, rectangular flumes by appropriate adjustments to handle changing velocity-of-approach problems (Bos et al., 1991). Circular pipes can also be accommodated with broad-crested weirs extending across the pipe. Convenient sizes have been built using sill heights ranging from 0.2 to 0.5 of the pipe diameter. The designer selects a sill height and pipe diameter to handle the desired flow rate and backwater conditions. For natural streams, V shaped flumes were chosen because of the wide range of discharges that must be measured. The only design choice here is the throat side slope. The designer has the option of designing a flume shape or size, not presentad here, by using the theoretically based computer program (Clemmens et al., 1993). A new version of this program for the Windows environment is currently in the beta-test phase (Wahl and Clemmens, 1998).
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HYDRAULIC DESIGN OF FLOW MEASURING STRUCTURES
21.34
Chapter Twenty-One
FIGURE 21.6. Broad-crested weir in trapezoidal, concrete-lined canal.
Sediment carrying capabilities are sometimes important design characteristic. As mentioned in a previous section, flumes that are V shaped do not pass bedoad sediment well. The rating tables are developed with the assumption of a particular known approach channel cross section (or flow area). However, any particular control section size and shape can be used with any approach section size and shape. The discharge can be adjusted with an approach velocity coefficient, as discussed in detail in Bos et al., (1991), or custom calibrated with the flume program (Clemmens et al., 1993; Wahl and Clemmens, 1998). The rating tables given here automatically limit the Froude number. If smaller approach areas are used, the designer must determine that the Froude number remains less than about 0.45. Frequently, the site conditions may call for flumes that would have dimensions beyond the ranges provided by the ratings in this chapter. To extend beyond these limits and for further information refer to Ackers et al., (1978), Bos (1989); Bos et al., (1991); Ciemmens et al., (1984, 1993); and Wahl and Clemmens, (1998).
21.7.1 Structures for Lined Trapezoidal Canals Standard broad-crested weir sizes are recommended for use in slip-formed canals of convenient metric dimensions. In lined canals, the canal itself furnishes the entrance channel section La, and the tailwater section (Fig. 21.6). Basic requirements become a converging transition to the throat section and the throat section itself. Calibrations for these standard sizes are included herein so that the subsequent designer may select one of these weirs to be built into an existing lined channel as shown in Fig. 21.6. In selecting these standardsized canals and the related flow rates, consideration was given to proposals by the International Commission on Irrigation and Drainage, to the construction practices of the U.S. Bureau of Reclamation, U.S. Department of the Interior (USDI), and to design criteria for small canals used by the Natural Resources Conservation Service, U.S. Department of Agriculture (USDA) (Bos et al., 1991). Present practice leans or tends toward side slopes of 1:1 horizontal to vertical for small, monolithic, concrete-lined canals with bottom widths less than about 0.8 m and Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
HYDRAULIC DESIGN OF FLOW MEASURING STRUCTURES
Hydraulic Design of Flow Measuring Structures 21.35
depths less than about 1 m. Deeper and wider canals tend toward side slopes of 1.5:1. When the widths and depths are greater than about 3 m, the trend is more toward 2:1 side slopes. This is particularly observed if canal operating procedures may allow rapid dewatering of the canal. In some soil conditions this can cause hydrostatic pressures on the underside of the canal walls that lead to wall failure. Most of the lined canals used in a tertiary irrigation unit or on large farms are of the smaller size. They have 0.3- to 0.6-m bottom widths, 1:1 side slopes, and capacities below 1 m3/s (35 ft3/s). In Table 21.3 precomputed broad-crested weir selections are given for canals with bottom widths at quarter-meter increments, with special insertions for 0.3 m (approximately 1 ft) and 0.6 m (approximately 2 ft). The offering of many precomputed sizes will aid in retrofitting older canal systems and yet not prevent the adoption of standard sized canals as proposed by the International Commission on Irrigation and Drainage (ICID). Canal sizes with bottom widths in excess of 1.5 m or 5 ft, respectively, are avoided in the precomputed tables on the assumption that these sizes deserve special design consideration. Table 21.3 shows a number of precomputed weirs in SI units that may be used for the various combinations of bottom widths and sidewall slopes as given in the first two columns. American units are available in USBR (1997). The third column lists recommended values of maximum canal depth, d, for each side-slope, bottom-width combination. For each canal size, several standard weirs can be used (Column 6). Columns 4 and 5 give the limits on canal capacity for each canal-weir combination. These limits on canal capacity originate from three sources: 1. The Froude number in the approach channel is limited to a maximum of 0.45 to ensure water surface stability. 2. The canal freeboard Fb upstream from the weir should be greater than 20 percent of the upstream sill-referenced head, h1. In terms of canal depth this limit becomes d 1.2 h1 P1. 3. The sensitivity of the weir at maximum flow should be such that a 0.01 m change in the value of the sill-referenced head h1 causes less than 10 percent change in discharge. Also indicated in the last column of Table 21.3 is a minimum headloss ∆H ∆ that the weir must provide. Excessive downstream water levels may prevent this minimum headloss, which means that the weir exceeds its modular limit and no longer functions as an accurate measuring device. The required head losses for the various broad-crested weirs were evaluated by the method described earlier and, for design purposes, listed for each weir size with the restriction that the computed modular limit will not exceed 0.90. Thus, the design headloss is either 0.1 H1 or the listed value for ∆H ∆ , whichever is greater. For these calculations, it was assumed that the weir was placed in a continuous channel with a constant cross section (e.g., p1 p2, b1 b2, and z1 z2) and that the diverging transition was omitted (abrupt expansion). Technically, the modular limit is based on the drop in total energy head through the weir (i.e., including velocity head). In the above continuous channels, the velocity head component is usually similar in magnitude upstream and downstream from the structure when p1 is approximately equal to p2. This means that h may be substituted for H in most cases. Table 21.3 is primarily intended for the selection of these standard weirs. It is also useful for the selection of canal sizes. The Froude number in the canal is automatically limited to 0.45. Selecting the smallest canal for a given capacity will give a reasonably effi-
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HYDRAULIC DESIGN OF FLOW MEASURING STRUCTURES
21.36
Chapter Twenty-One
cient section. For instance, if the design capacity of the canal is to be 1.0 m3/s, the smallest canal that can be incorporated with a measuring structure has b1 0.60 m, z1 1.0, and d 0.90 m. Larger canals can also be used. The hydraulic grade line of the channel should also be checked to ensure an adequate design. Each standard weir can be used for a range of bottom widths. This is possible because the change in flow area upstream from the weir causes only a small change in velocity of approach and a corresponding small change in energy head. The width ranges have been selected so that the error in discharge caused by the change in flow area is less than 1 percent. This is a systematic error for any particular approach channel size, and the extent of this error varies with discharge. A weir suitable for several of the listed canal bottom widths is also suitable for any intermediate width. For example, in Table 21.3, Weir Gm1 can be used in canals with bottom widths of 0.30, 0.50, and 0.60 m, or any intermediate width, for example, b1 0.45 m. The user will need to calculate the sill height, headloss, and maximum design discharge for these intermediate sizes. The rating equations for the weirs are given in Table 21.4 and will reproduce the values presented in the original tables produced by computer modeling (Bos et al., 1991) to within about 1 percent. The original tables were computed using the following criteria (Fig. 21.6): 1. Each weir has a constant bottom width bc and a sill height p1 that varies with the canal dimensions. 2. The ramp length can be chosen such that it is between 2 and 3 times the sill height. The 3:1 ramp slope is preferable. 3. The gage is located a distance at least H1max upstream from the start of the ramp. In addition, it should be located a distance of roughly two to three times H1max from the entrance to the throat. 4. The throat length should be greater than 1.5 times the maximum expected sill-referenced depth h1max, but should be within the limits indicated in Table 21.4. 5. The canal depth must be greater than the sum of ( p1 h1max Fb), where Fb is the freeboard requirement, or roughly 0.2 times h1max. Occasionally, a weir cannot be found from these tables that will work satisfactorily. The user must then judge between several options, for example: 1. Find a new site for the flume with more vertical head available. 2. Add to the canal wall height upstream from the site so that more backwater effect can be created. 3. Try one of the other weir shapes. 4. Use the tables to interpolate and get a rating for an intermediate width, probably with some sacrifice in accuracy. 5. Produce a special design using the computer model. Example. Given: An existing canal has a bottom width of b1 0.30 m, side slopes z1 1, and a total depth d 0.55 m. The discharge depth relationship was estimated for one flow rate by using surface flows, and using Manning’s formula (Eq. 16) to estimate the depth at the maximum expected flow. For this example the maximum flow was estimated to be Qmax 0.15 m3/s at a canal flow depth y2 0.43 m prior to installation of the measuring device. Required: Select a broad-crested weir structure from Table 21.4.
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HYDRAULIC DESIGN OF FLOW MEASURING STRUCTURES
Hydraulic Design of Flow Measuring Structures 21.37
Procedure: The basic strategy for selecting a flume for an existing canal is to cause enough rise in the existing flowing water surface to create the necessary water surface drop needed to allow the flume to function properly. Because the rise in water surface is nearly equal to the required energy loss, ∆H ∆ , this value is used to approximate the water surface drop, or, ∆h ⬇ ∆H ∆ . This minimum water surface rise usually must be provided for the maximum expected flow rate, and is then safe for lesser flow rates for this type of installation in a trapezoidal channel. (Note: Placing a rectangular structure in a trapezoidal channel may require checking the low flow also, because as flow rate decreases, the depth drops faster in a rectangular channel than in a trapezoidal channel, and discharge may be nonmodular at low flow rates.) Use Table 21.3 and find the rows for z1 1 and b1 0.30 m. For a flow rate of 0.15 m3/s, weirs Bm through Fm1 can be tried because Qmax falls within the discharge ranges for each of these weirs. Weir Bm is tried first because it has the lowest sill height, p1 0.15 m, and should be the most economical to build. The minimum required head loss listed in column 9, ∆H ∆ , for this weir is 0.017 m. The actual headloss through the weir should be the greater of the listed ∆H ∆ and 0.1h1. To find the head on the flume, h1, use the discharge equation for long-throated flumes given at the top of Table 21.4: Q C1(h1 C2)U From Table 21.4 the appropriate coefficients for the discharge equation for flume Bm are C1 2.365 C2 0.0079 U 1.8599 Solving for h1 and with Q 0.150 m3/s: 1
Q 0.15 1.8599 h1 C2 0.0079 C1 2.365 1 u
h1 0.219 m In this case, 0.1 h1 0.0219 m exceeds the listed value of 0.017 m, and the larger value of 0.0219 m is used as the needed drop in water surface. For the flume to function properly, the difference in the upstream water depth, h1 p1, and the downstream channel depth, y2, should be greater than the required water surface drop, or (h1 p1 ) y2 (0.219 0.150) 0.430 0.061 m 0.0219 m The computed value is actually negative, which means this flume would be completely submerged Now examine the nextsized weir. However, because the next size can be estimated to raise the water surface by something slightly less than the sill increase of 0.05 m, it is unlikely that raising the sill by only 0.05 m will work because the previous sill produced no drop, by virtue of the negative value calculated. Thus, move on to weir Dm1 with a p1 0.21 m. In Table 21.4, we find ,with the corresponding values from Table 21.4 for C1,C C2 and U, that h1 0.196 and ∆H ∆ = 0.025 m in Table 21.3. However, 0.1 h1 0.020 m, so use the listed value for ∆H ∆ 0.025 m as the needed drop. Now again check the difference in the upstream depth, h1 p1, and downstream channel depth, y2 to see if (h1 p1 ) y2 ∆H 0.025 m (the required drop in water surface). Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
HYDRAULIC DESIGN OF FLOW MEASURING STRUCTURES
21.38
Chapter Twenty-One
This time, (h1 p1) y2 (0.196 0.250) 0.430 0.016 m, which still does not provide the needed drop of 0.025 m. Next try weir Em1 with p1 0.30 m. This time, h1 is calculated to be 0.186 m and, from Table 21.3, ∆H ∆ 0.029 m. Then (h1 p1) y2 (0.186 0.30) 0.430 0.056 m, which exceeds the needed 0.029 m value. Thus, this weir meets the primary criterion and can be checked further. Weir
C1
C2
U
h1
0.1 h1
Table
H
Req. Drop
h1+p1 Adequate –y2
Bm
2.365
0.0079 1.8599
0.219
0.0219 0.217 0.0219 20.061
No
Dm1
2.913
0.0101 1.8555
0.196
0.0219 0.025
0.025
0.016
No
Em1
2.921
0.0090 1.8152
0.186
0.0186 0.029
0.029
0.056
Yes
The minimum required canal depth, d, including freeboard, can be calculated from dmin 1.2 h1 P1 1.2 (0.186) 0.30 0.523 m, which is less than the 0.550 m canal depth given in this example. This weir is acceptable, and the remainder of the hydraulic dimensions can be obtained from Tables 21.3 and 21.4. They are The length from the gauge to the ramp La = H1max = 0.37 m The ramp length Lb 3 p1 3 (0.3 m) 0.9 m The throat length L 3 p1 1.5 (0.186) 0.28 m However, from Table 21.4, L 0.38 m, so use L 0.38 m, to use the given calibration values. A downstream expansion is not necessary. The flume width bc 0.90 m. This example is modified from Bos et al. (1991). Other examples and detailed discussion of this example are presented therein.
FIGURE 21.7. Flow measuring structure for earthen channel with rectangular control section. (From Bos et al., 1991)
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HYDRAULIC DESIGN OF FLOW MEASURING STRUCTURES
Hydraulic Design of Flow Measuring Structures 21.39 TABLE 21.5
Rating equations and Ranges of Application for Rectangular Weirs*
Lb 2 to 3 times p1; La H1max; La Lb 2 to 3 times H1max ∆ 0.1H ∆H H1, or value listed Discharge, q, is given im m3/s per meter width Coef. for Eq.
0.10,bc,0.20m L 5 0.2m p1 5 p1 5 0.05m 8
C1 C2 U h1min h1max qmin qmax ∆ ∆H
2.449 0.0003 1.608 0.014 0.130 0.003 0.092 0.012
Coef. for Eq. C1 C2 U h1min h1max qmin qmax ∆ ∆H
Coef. for Eq. C1 C2 U h1min h1max qmin qmax ∆ ∆H
p1 0.1 m
0.20,bc,0.30m L 5 0.35m p1 5 p1 5 0.1 8
1.817 0.000 1.530 0.026 0.130 0.003 0.079 —
2.271 0.0014 1.612 0.025 0.235 0.006 0.221 0.025 m
0.5 bc .0m L 0.75m p1 p1 2m 0.3 m
0.30,bc,0.50m
1.744 0.000 1.517 0.025 0.330 0.006 0.192 —
p1 ∞
p1 0.2 m 2.095 0.004 1.627 0.070 0.670 0.030 1.110 0.046 m
2.316 0.003 1.641 0.050 0.360 0.019 0.438 0.028 m
2.081 0.003 1.611 0.050 0.500 0.018 0.689 0.048 m
1.973 0.003 1.594 0.050 0.500 0.018 0.660 0.063 m
1.709 0.000 1.516 0.050 0.500 0.018 0.595 —
p1 0.1m
bc 2.0m L 1.0m p1 0.2m
p1 0.3m
p1 ∞
2.098 0.0056 1.637 0.100 0.700 0.070 1.191 0.047 m
1.925 0.006 1.609 0.100 1.000 0.051 1.960 0.87 m
1.848 0.004 1.581 0.100 1.000 0.051 1.877 0.124 m
1.691 0.000 1.515 0.100 1.000 0.051 1.689 —
p1 5 0.1
p1 5 0.2
L 5 0.5m p1 5 8
2.276 0.0013 1.615 0.035 0.330 0.011 0.381 0.027 m
2.017 0.0007 1.574 0.035 0.330 0.011 0.353 0.044 m
1.731 0.000 1.517 0.035 0.330 0.011 0.353 —
1.0 bc 2.0m L 1.0m p1 p1 0.3 m 0.4 m 1.976 0.0027 1.598 0.070 0.670 0.030 1.059 0,066 m
1.976 0.0027 1.599 0.070 0.670 0.030 1.028 0.086 m
p1 ∞ 1.702 0.000 1.519 0.070 0.670 0.030 0.925 —
Source: Adapted from Replogle et al. (1990). Calibrations developed with computer model (Clemmens et al., 1993). *
21.7.2 Rectangular Structures for Unlined Canals Weirs and flumes for earthen (unlined) channels require a structure that contains the following basic parts: entrance to approach channel, approach channel, converging transition, throat, diverging transition, stilling basin, and riprap protection. As illustrated in Fig. 21.7, the discharge measurement structure for an earthen channel is longer, and thus more expensive, than a structure in a concrete-lined canal (Fig. 21.6).
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HYDRAULIC DESIGN OF FLOW MEASURING STRUCTURES
21.40
Chapter Twenty-One
In the latter, the approach channel and sides of the control section already exist and the riprap is not needed. The approach canal of Fig. 21.7 provides a known flow area and velocity of approach. The rating equations for the rectangular weirs given in Table 21.5 assume that the approach section is rectangular and has the same width as the throat. If the upstream sillreferenced head is not measured in a rectangular approach canal of this same width, but instead is measured in the upstream earthen section, then these tables require correction to the discharge, Q, for the change in the approach velocity. Procedures for this are given by Bos (1989). The full-length structure of Fig. 21.7 can be further shortened by deleting the diverging transition or the rectangular tail water channel. The diverging transition may be deleted if the available head loss over the structure exceeds 0.4 ∆H1, so that no velocity head needs to be recovered. The rectangular tail water channel may be deleted if, at maximum flow, the Froude number at the beginning of the exit channel is less than 1.7 (Bos et al.,1991). A rectangular broad-crested weir discharges nearly equal quantities of water over equal widths. The major differences are associated with the friction along the walls. Thus the flow is nearly two-dimensional over the weir, so that rating tables can provide the flow rate, q, in m3/s per meter width of sill for each value of h1. This allows a wide variety of sizes for rectangular broad-crested weirs. For each width, bc, of the weir, an accurate rating table for the total discharge, Q, can be developed by multiplying the table discharges by bc. Thus, Q bcq
(21.21)
Table 21.5 presents rating equations for a series of rectangular broad-crested weirs that were developed from the computer-modeled tables given in Bos et al. (1991). These equations will reconstruct those computer-derived tables to within about 1.5 percent. This adds an additional uncertainty to the 3 percent error claimed for the original tables. That 3 percent includes the added error caused by averaging small groupings of weir widths. The weirs were selected to keep the error of side wall effects to within 1 percent. Ratings are given for several sill heights, pl, to aid in design. Interpolation between sill heights gives reasonable results. If the approach area, A1, is larger than that used to develop these rating tables, either because of a higher sill or a wider approach channel, the ratings must be adjusted for Cv. To simplify this process, the discharge over the weir for a Cv value of 1.0 is given in the far right column of each grouping. This column is labeled Pl ∞, because that would cause the velocity of approach to be zero and Cv 1.0. This approximates a weir at the outlet of a reservoir or lake. The complete correction procedure is given in Bos et al., (1991) and Bos (1989). The design procedure for lined rectangular canals, is relatively straightforward. It consists of selecting a sill height, P1, that causes modular flow throughout the discharge range, and provides sufficient freeboard at the maximum discharge. For unlined canals, an appropriate width must be chosen. There are usually several widths that will work. Extremely wide, shallow flows are subject to measurement errors due to low head detection sensitivity. Extremely narrow, deep flows require long structures and large head losses. Because of the wide variety of shapes that can be encountered in earthen channels and in the range of discharges to be measured, it is rather complicated to determine the inter-
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HYDRAULIC DESIGN OF FLOW MEASURING STRUCTURES
Hydraulic Design of Flow Measuring Structures 21.41
related values of h1max, p1, and bc of the structure. While this makes the design process somewhat more complicated, it allows the designer greater flexibility and expands the applicability of the weirs. 1. The discharges to be measured (per meter width) must be within the range of discharges shown in the rating table for the selected weir if these flume dimensions and tables are to be used. 2. The allowable measurement error should not be exceeded. This allowable error may be different at different flow rates (see Bos et al., 1991). 3. Flow should be modular at all flow rates to be measured. 4. Placing a weir in the canal should not cause overtopping upstream. 5. The measuring structure should be placed in a straight section with a relatively uniform cross section for a distance of about 10 times the width of the channel. 6. The Froude number should not exceed 0.45 for a distance of at least 30 times h1 upstream from the weir. The following criteria should be considered by the designer. If these criteria are followed, the designer should obtain a satisfactory structure that will operate as intended. For a rectangular structure in an earthen canal, the rectangular section need not extend 10 times its width upstream from the flume if a gradual taper is used to guide the flow into the rectangular section. For the flumes given here, it is recommended that the rectangular section extend upstream from the head measurement location (gauging station), as shown in Fig. 21.7. It is also recommended that riprap be placed downstream from the structure for a distance of 4 times the maximum downstream water depth. A step should be provided at the transition between the rectangular section and the riprap section to avoid local erosion from floor jets. Sizing of riprap and filters is discussed by Bos (1989) and Bos et al. (1991). An analysis of head measurement errors is presented by Bos (1989) and Bos et al., (1991) and will not be repeated here. For lined channels, a freeboard criterion of 0.2 h1max has been used satisfactorily. For unlined channels it may be more appropriate to specify a
FIGURE 21.8 Long-throated flume in a partly filled pipe.
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HYDRAULIC DESIGN OF FLOW MEASURING STRUCTURES
21.42
Chapter Twenty-One
maximum water depth, y1max. Submergence or modular flow should be checked at both minimum and maximum expected discharges. If the channel is rectangular, or the length of the rectangular throated structure downstream from the weir sill is as in Fig. 21.7, then we can use the lower value of ∆H ∆ 0.1 H1 or the ∆H ∆ value given at the bottom of Table 21.5. If a shorter length in an earthen channel is used and the tailwater channel is significantly larger than the stilling basin would be, then considerably more headloss will probably be required. The designer could use the headloss value for the discharge into a lake or pool, ∆H ∆ 0.4 H1. This may represent a drastic difference in the value of headloss. The designer may decide to use the shortened structure and calculate the actual modular limit by use of the computer model (Bos et al., 1993; Wahl and Clemmens, 1998). Another alternative is to build a prototype in the field and set the crest to the appropriate level by trial and error. For a more comprehensive background and design discussion for rectangular flumes, refer to Bos et al. (1991).
21.7.3 Structures for Circular Channels The broad-crested weir has been adapted for flow measurements in pipes with a free water surface because of its relatively simple construction, Fig. 21.8. Other shapes, such as side contractions and center line piers can be and have been used. The reader is referred to the computer program and design procedure for most calibrations (Clemmens et al., 1993; Wahl and Clemmens, 1998). There are two situations of particular interest for designing a weir in a pipe. The weir can be placed somewhere in the middle of a straight pipe, or at the end of a pipe, such as the entrance to a deep manhole, which is probably more common because the site is more accessible. When the weir is placed in the interior of a straight pipe, the presence of the weir must cause a rise in the upstream water surface to develop the required head loss. This increase in flow area upstream causes a proportional decrease in velocity and may cause subsequent sediment deposition. Where this is a problem, a downstream ramp should be considered to reduce the required head loss and thus accommodate a smaller sill height. Sediment problems are further aggravated at low flows because the free-flowing (normaldepth) water surface drops proportionally faster than the water surface upstream from the weir. Thus, the velocity difference becomes greater and greater with decreasing discharge. For situations where wide fluctuations in flow rates exist and sedimentation is a problem, an alternative weir shape or location should be considered. An alternative for the measuring location is the end of a pipe, particularly where there is a large drop in water surface. A weir can often be designed so that the water level upstream from the weir either matches, or is below, the normal depth of water in the pipe. This may avoid aggravating sedimentation problems. Portable flumes using plastic pipe. Portable flumes for flow rates up to about 50 L/S with trapezoidal or rectangular shape are described in Bos et al. (1991). More recently, calibrations have been presented for sills placed in circular pipe sections. Small versions are convenient to construct and use as portable flumes. Dimensionless ratings for average roughnesses and profile lengths were previously presented for partly full circular culverts and pipes fitted with similar sills (Clemmens et al., 1984). However, direct computation by the computer model (Clemmens et al., 1993) for the specific roughness of the construction materials provides a slightly more accurate rating. These portable versions can be used for measuring drain tile outlet flows or to measure irrigation furrow flows. An equation for these circular flumes was developed for a group with the following relative proportions:
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HYDRAULIC DESIGN OF FLOW MEASURING STRUCTURES
Hydraulic Design of Flow Measuring Structures 21.43
La L 3 D/4 Lb 3 p1 where D the diameter of the pipe. The equation is limited to the following ranges of application: 0.2 P1 0.5 0.1 m D 10 m 0.1 L h1max 0.9 (D p1) The developed expression is h U Q(m3/s) D2.5C11 C2 D
(21.22)
where p C1 2.6308 1.7823 1 D p p 2 C2 0.018561 0.070370 1 0.065893 1 D D p U 2.0492 1.8092 1 1.5162 ((p1 /D / )2 The forgoing expressionsDwere derived by computing a series of flume sills and ramps
冢 冣
(broad-crested weirs) in a pipe 1 m in diameter with sill heights between 0.2 D and 0.5 D. The resulting direct computed tables for each sill height were curve-fitted to generate values for C1, C2, and U for each sill height in the 1-m-diameter pipe. These several values for C1, C2, and u as functions of p1 were further curve-fitted to obtain the above expressions. The D2.5 factor represents the Froude-modeling length ratio applied to generalize the equation to a range that we recommend should not exceed about 10:1 (0.1–10 times the direct-computed size of D 1 m). For the smallest size, deviations at the low flow ranges can be as high as 3 percent compared to direct computer-modeled tables. For the larger sizes this deviation is less than 1 percent from the direct computed tables. Example. Because of the relative complexity of applying the above equation to various sizes, an example for a common size in Enghish units is presented. The diameter is 1 ft and the sill height is D/3: Convert American measurements to meters: D 1 ft 0.3048 m p1 0.3333 ft 0.1016 m Then 0.1016 C1 2.6308 1.7823 2.0367 0.3048 0.1016 0.1016 2 C2 0.018561 0.07037 0.065893 0.002426 0.3048 0.3048 0.1016 0.1016 2 U 2.0492 1.8092 1.5162 1.6146 0.3048 0.3048
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HYDRAULIC DESIGN OF FLOW MEASURING STRUCTURES
21.44
Chapter Twenty-One 5/2
D
0.30485/2 0.05129
Thus, Eq. (21.22), for this pipe size and sill height becomes 1.6146 h1 Q (0.05129)(2.0367) .002436 0.3048
or 1.6146 h1 Q (0.1045) .002426 0.3048
where h1 is expressed in meters and Q is in m3/s. ( Multiply Q by 35.31 to convert to ft3/s.) As with the previously discussed broad-crested weirs, the width, bc, is one of the two most important dimensions in the flume. The other is the head measurement (Bos et al., 1991). For the head measurement, a translocated stilling well is recommended, Fig. 21.8. This method transfers the upstream depth reading to an easily measured location above the sill at the flume outlet. The translocated stilling well conveniently references the upstream head to the sill floor without the necessity of accurately leveling the flume. This makes it truly portable (Bos et al., 1991). The upstream gage reading location should be used only if the flume can be conveniently, or permanently, leveled. If the weir is located in a section of pipe, the normal depth, yn, equals the downstream depth y2. The maximum flow depth for stable, nonpriming flow is about 0.9 of the pipe diameter. Thus, y2 would need to be less than this by an amount equal to the needed head drop in the flume. Thus, y2 ∆H ∆ p1 h1 0.9D
(21.23)
Equation 21.23 gives the limits on design to provide for modular flow y1 y2 h and to keep the pipe from flowing full y1 0.9D at maximum flow. Here again we use ∆h ⬇ ∆ . If flow in the pipe is caused by downstream backwater effects (in excess of normal ∆H depth), these criteria should be checked at low flows. With a weir at the end of a pipe and sufficient overfall, the weir can be lower to keep approach velocities high for better sediment transport. Preferably, the normal depth, yn, should be larger than or equal to y1. Hence at maximum flow: y1 p1 h1 0.9D
(21.24)
All weir and pipe combinations can be considered to be Froude models of each other. This means that rating tables for sizes other than those shown in the table can be readily developed without the computer model, as long as all dimensions remain proportional.
21.8 FIELD SIMPLIFIED, EXPEDIENT MEASUREMENT TECHNIQUES Obtaining useful design information on an existing flow system in a retrofit situation is frequently a challenge. In open channel flows, reliance on historical flow levels and the uniform flow equations can sometimes provide the needed information (Chow, 1959).
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Hydraulic Design of Flow Measuring Structures 21.45
FIGURE 21.9 Suggested method to obtain accurate flow profile information for computing the energy slope of a flowing canal for accurately applying Manning’s formula.
Too frequently, true uniform flow does not occur even in canals that appear to have long straight reaches. Careful field surveys of the canal slope and the water surface slope over an extended distance may be required in order to calculate the needed energy slope for use in the required equations. Even then the friction value is usually nebulous. Good current metering records are valuable for both rivers and large canals. Accurate current metering in small canals is often not suitable. However, small canals can frequently be temporarily measured with portable flumes and weirs and the information used for permanent designs.
21.8.1 Channel Roughness Measurement and Water Surface Profile Measurements. An accurate and quick field technique to obtain a channel flow rate using Manning’s formula, or determine a channel flow area, or obtain data to calculate energy-line slope for computing the roughness value for Manning’s n is illustrated in Fig. 21.9. The technique uses a standard surveying level and two static pressure tubes inserted into the canal water at approximately equal distances upstream and downstream from the surveying instrument site. The static pressure tubes are described in Replogle (1997), and Bos et al. (1991). The distances are typically 50–100 m. Using equal distances in both directions is good surveying technique in case the surveying instrument is not in good repair. Stilling wells in the form of small cups placed slightly into the water surface provide a damped water surface for the survey data. The cups need to be tall enough to survive splash from velocity impact and surface waves. Manning’s formula (Eq. 21.16) can be used with an estimate of Manning’s n value and the channel energy line slope, Se (Chow, 1959). As mentioned earlier, Manning’s n value typically is 0.012–0.014 for new concrete canals in excellent condition, increasing to 0.018 for damaged concrete surfaces, and in excess of 0.030 when heavy grass-like algae growths are present. The method has been successfully applied to concrete-lined canals to evaluate channel roughness. For this process, an independent measurement of the flow rate was obtained with portable broad-crested weirs as described in another section, and Manning’s equation was solved for n. The channel flow areas and the flow velocities at the two points illustrated in Fig. 21.9, could be accurately calculated using the survey data for the water surface. These velocities then provided the velocity head that was added to the respective flow depths to establish the slope of the energy grade line, Se.
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HYDRAULIC DESIGN OF FLOW MEASURING STRUCTURES
21.46
Chapter Twenty-One
21.8.2 Portable Flow Measuring Flumes Portable measuring flumes can provide reliable site-specific data that can usually be translated into precise design information for permanent structures. Portable flow measuring flumes are described in Bos et al., (1991) and in Replogle and Wahlin (1998). The latter reference describes flumes with adjustable throats that are especially convenient to install in small unlined canals. They allow accurate measurements with canal ponding of less than 2–3 cm. The permanent structure design will often be one of the long-throated flumes described in earlier sections. This design process has been expedited by several design-oriented computer programs, which aid in the selection and installation of these devices. These have also been described in earlier sections.
21.8.3
Surface Floats
The velocity of surface floats is sometimes used to estimate flows in lined channels. Unless survey data exist to define a natural channel, the surface velocity times an unknown area of flow produces an unreliable discharge rate. Lined prismatic channels, after probing to determine sedimentation conditions, are a fair candidate for this method of flow estimation. Rather than using a single float, multiple floats consisting usually of buoyant ditchbank trash, popcorn, or even numerous ice cubes, provide a surface float method that is reliable to 10 percent. Ice cubes are particularly useful in windy conditions. The recommended procedure is to apply the floating material across the channel about two or more top widths upstream from a designated point A. Measure the time for the initial front of the floating material to pass from point A to a point B established 30–50 m downstream. Because of the many particles, the time for the front edge of the floating material to reach point B should provide the velocity of the fastest surface flow element. This flow element is a function of the channel shape and the channel roughness. Also, the roughness and the length between stations A and B contribute to the longitudinal dispersion of the floating material. While this length dispersion pattern has not been established well enough to firmly define Manning’s n, it does appear to be related. For field flow estimates, when the channel is rough or large, the average velocity is about 90 percent of the maximum surface velocity. When the channel is small or smooth, the value approaches 70 percent. While this trend is evident, precise values have not been established. The current information does support an average value of 80 percent. These values differ from the rod floats as used and reported in USBR (1997) where the average is represented by range from 66 percent to 80 percent with an average of 73 percent. However, those floats sampled at least the top quarter of the flow depth as opposed to only the surface velocity suggested here.
21.8.4 Checking a Flow Profile Sometimes there is a need to inexpensively check how the velocity profile is behaving near a measuring device, and to check if measures taken to condition it have been effective. One way to obtain quick and easy results is the rising bubble method. Trickle irrigation tubing is weighted so that it will stay in a straight line across the bottom of the channel of interest. Pressurized air or other gas is released at a rather fast rate from
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HYDRAULIC DESIGN OF FLOW MEASURING STRUCTURES
Hydraulic Design of Flow Measuring Structures 21.47
FIGURE 21.10 Air bubbles used to check velocity patterns.
the many small holes, Fig. 21.10. The predominant larger bubbles rise uniformly enough to define an undisturbed water surface area between the line of injection and the predominant emergence. The smaller bubbles rise more slowly and emerge in the downstream bubble trail. One immediate observation is the symmetry of the emergence line. A ragged, nonsymmetrical line in a prismatic channel indicates velocity profile distortions. Using rising bubbles as a flow measurement method. This same system can be used to measure discharge rate. The discharge is calculated quite simply by the product of the surface area defined by the emerging bubbles and the release line, multiplied by the average bubble rise velocity of 0.218 m/s (Herschy, 1985). A limitation of this rising-bubble method is the difficulty of measuring the surface area accurately, but this method will give good discharge estimates in poorly defined earth channels and automatically adjusts for both velocity profile and channel shape.
21.8.5 Low-Pressure Pipe Venturi The Venturi tube, although old compared to the modern sonic meters, is still a viable method of measuring fluid flows. It has good anticlogging characteristics and is sometimes used on pressure flows of sewage. In low-pressure pipelines associated with agricultural irrigation (less than 30 m.of pressure head) Venturi tubes were fashioned from standard Polyvinyl chloride (PVC) pipe fittings for 30.5 cm (12-in) main lines and a throat diameter of 20.3 cm (8 in) (Replogle and Wahlin, 1994). Figure 21.11 shows the basic construction. Standard pipe fittings were used to reduce to 10-in pipe then to the 8in pipe. Fig. 21.12 shows a method of making suitable manometer taps.
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HYDRAULIC DESIGN OF FLOW MEASURING STRUCTURES
21.48
Chapter Twenty-One
FIGURE 21.11 Configuration installed in the field.
FIGURE 21.12 Pressure tap showing glued-on boss and pipe fitting used in field units.
The basic expression for discharge, Q, is derived from the classical Bernoulli Equation and can be written in a form that is applicable to round pipes or other conduit shapes as
冪莦莦莦莦莦莦莦莦
2g Cd Ap At Q (hp ht) 兹苶 A2苶 苶 A2苶t p 苶 where Cd discharge coefficient, usually 0.90 Cd 0.99, Ap area of approach piping,,At area of contracted throat section, G gravitational constant (9.81 m/s2 = 32.16
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HYDRAULIC DESIGN OF FLOW MEASURING STRUCTURES
Hydraulic Design of Flow Measuring Structures 21.49
ft/s2), velocity distribution coefficient (estimated as 1.02), and hp – ht Differential head between upstream pipe tapping and throat tapping. These Venturi meters fashioned from plastic pipe fittings of the kind usually used by the irrigation industry have an expected accuracy of 2 percent, not including the errors of the readout method. It is recommended that a throat length of three times the throat diameter be used for these plastic Venturi meter constructions. Shorter throat lengths appear to cause difficulties in pressure detection due to flow separation. Longer throat lengths produce excessive headloss. The rate of contraction of the reducer fittings caused no significant change in Cd, and thus the meter calibration. However, commercial fittings vary in their contraction angle and the fittings with a 25 contraction angle (measured from pipe centerline) exhibited a greater total head loss through the meter than those with a less severe 15 contraction rate (18 percent and 25 percent of head reading respectively). These constructions conform to expected Venturi meter behavior with a discharge coefficient about 2 percent to 5 percent less than “standard” Herschel–type Venturi tubes. The discharge coefficient, Cd, can be estimated by the following empirical equation: Cd m1 m2 e(Rn/m3) where m1 0.964; m2 0.0466; m3 254,000; and Rn Reynolds number (based on pipe diameter). The most important construction factor is the fabrication of the pressure taps and the immediate connections. They should be drilled with appropriate backing blocks to reduce burrs and with a guide to assure that they are constructed perpendicular to the pipe wall. A slight rounding of the piezometer tap holes will help reduce burrs. It is recommended that the pressure taps be installed on the sides of the meter to prevent air bubbles from entering the pressure lines. It is not necessary that the pressure taps be on the same horizontal line and the meter can be mounted at any angle. Slow-setting PVC cement is recommended to allow workers sufficient time to uniformly assemble and adjust large pipe parts. For an accepted error of 2 percent, one properly constructed pressure tap at each of the two required piping locations is adequate.
25.9 ACKNOWLEDGMENTS We hereby acknowledge M. G. Bos who encouraged and contributed to a significant cooperative effort with Replogle and Clemmens to produce the first comprehensive treatment of long-throated flumes. That publications by Bos et al., (1991) formed the basis for much of the discussion presented on the newer “computable” flumes, including the computer program developed to compute them (Clemmens et al., 1993). Much of the remainder of the text for this chapter was condensed from the recent U.S. Bureau of Reclamation publication (USBR, 1997) to which the current authors contributed significantly, but to which we acknowledge the editorial efforts of Mr. Russ Dodge, and other Reclamation employees, who undertook much responsibility to bring that publication to fruition.
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21.50
Chapter Twenty-One
REFERENCES Ackers, P., W. R. White, J. A. Perkins, and A. J. M. Harrison, Weirs and Flumes for Flow Measurement, John Wiley & Sons, New York, 1978. ASME, Fluid Meters, Their Theory and Application, 5th ed., Report of ASME Research Committee on Fluid Meters, American Society of Mechanical Engineers, New York. 1959. ASME., Fluid Meters, Their Theory and Application, 6th ed., H. S. Bean, ed., Report of ASME Research Committee on Fluid Meters, American Society of Mechanical Engineers, New York. 1971. ASTM, Annual Book of ASTM Standards, Water Environment Technology, Section 11, Vol. 11.01 Water (1), American Society for Testing and Materials, 1988. Bos, M. G. ed.,. Discharge Measurement Structures. 3rd rev. ed., Publication 20,. International Institute for Land Reclamation and Improvement/ILRI, Wageningen, The Netherlands. 1989. Bos, M. G., J. A. Replogle, and A. J. Clemmens, Flow Measuring Flumes for Open Channel Systems, American Society of Agricultural Engineers, St. Joseph, MI, (republication of 1984 edition by John Wiley & Sons), 1991. Brater, E. F., and H. W. King, Handbook of Hydraulics, 6th ed., McGraw-Hill, New York, 1982. Chow, V. Te, Open Channel Hydraulics, McGraw-Hill, New York, 1959. Clemmens, A. J., M. G. Bos, and Replogle, J. A. “RBC Broad-Crested Weirs for Circular Sewers and Pipes,” Journal of Hydrology (Netherlands), 68:349-368, 1984; Stout, G. E., and G. H. Davis, eds., The Ven Te Chow Memorial Volume. 1984 Clemmens, A. J., M. G., Bos, and Replogle, J. A., FLUME: Design and Calibration of LongThroated Measuring Flumes (with computer disk, Version 3.0), Publication No. 54, International Institute for Land Reclamation and Improvement/ILRI, Wageningen, The Netherlands, 1993. Daugherty, R. L., and A. C. Ingersoll, Fluid Mechanics, 5th ed., McGraw-Hill, New York, 1954. Herschy, R. W., Streamflow Measurement, Elsevier Applied Science Publishers, London and New York, 1985. ISO Standard 5167, Measurement of Fluid Flow by Means of Plates, Nozzles and Venturi Tubes Inserted in Circular Cross-Section Conduits Running Full, ISO 5167-1980(E), Geneva, Switzerland, 1991. ISO, Measurement of Liquid Flows in Open Channels, Handbook No. 15, International Organization for Standardization. Geneva, Switzerland, 1983. Miller, R. W., Flow Measurement Engineering Handbook, 3rd ed., McGraw-Hill, New York, 1996. Reginald, W., Streamflow Measurement, Elsevier Applied Science Publishers, London and New York, 1985. Replogle, J. A., A. J., Clemmens, and Bos, M. G. “Measuring Irrigation Water,.” in T. A. Howell, K. H. Solomon, and G. Hoffman eds. Management of Farm Irrigation Systems, American Society of Agricultural Engineers, St. Joseph, MI, 1990. Replogle, J. A., and B. T. Wahlin, “Venturi Meter Construction for Plastic Irrigation Pipelines,” Applied Engineering in Agriculture,10(1):21–26, 1994. Replogle, J. and B. Wahlin, “Portable and Permanent Flumes with Adjustable Throats,” Irrigation and Drainage Systems, Vol. 12, No. 1, Kluwer Academic Publishers, Netherlands, págs. 23-34., 1998 Replogle, J. A., “Practical Technologies for Irrigation Flow Control and Measurement,” Journal of Irrigation and Drainage Systems, 11(3):241–259, 1997. Spitzer, D. W., Industrial Flow Measurement. Resources for Measurement and Control Series, Instrument Society of America, 1990.
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HYDRAULIC DESIGN OF FLOW MEASURING STRUCTURES
Hydraulic Design of Flow Measuring Structures 21.51 USBR, Water Measurement Manual, U.S. Department of Interior, Bureau of Reclamation, 3rd ed., Superintendent of Documents, U.S. Government Printing Office, Washington, DC, 1997. USGS, National Handbook of Recommended Methods of Water-Data Acquisition, U.S. Geological Survey, Office of Water Data Coordination, Government Printing Office, Washington, DC, 1980. Wahl, T. L., and A. J. Clemmens, Improved Software for Design of Long-Throated Flumes. in Proceedings of the 14th Technical Conference on Irrigation, Drainage and Flood Control, U.S. Committee on Irrigation and Drainage. Phoenix, AZ, 1998.
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Source: HYDRAULIC DESIGN HANDBOOK
CHAPTER 22
WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS Federico E. Maisch Sharon L. Cole David V. Hobbs Frank J. Tantone William L. Judy Greeley and Hansen Richmond, VA
22.1 INTRODUCTION Designers of water treatment plants and wastewater treatment plants are faced with the need to design treatment processes which must meet the following general hydraulic requirements: • Water treatment plants. Provide the head required to allow the water to flow through the treatment processes and to be delivered to the transmission/distribution system in the flow rates and at the pressures required for delivery to the users. • W Wastewater treatment r plants. Provide the head required to raise the flow of wastewater from the sewer system to a level which allows the flow to proceed through the treatment processes and be delivered to the receiving body of water. The above requires knowledge of open-channel, closed-conduit, and hydraulic machine flow principles. It also requires an understanding of the interaction between these elements and their impact on the overall plant (site) hydraulics. Head is either available through the difference in elevation (gravity) or it has to be converted from mechanical energy using hydraulic machinery. Distribution of flows using open channels or closed conduit is critical for proper hydraulic loading and process performance. This chapter brings together information on commonly used hydraulic elements and specific applications to water treatment plants and wastewater treatment plants. The development of hydraulic profiles through the entire treatment process with examples for water treatment plants and wastewater treatment is also presented. Many processes and flow control devices are similar in both water treatment plants and wastewater treatment plants. Both types of plants require flow distribution devices, gates and valves, and flowmeters. These devices are discussed in Section 22.2. The development of water treatment plant hydraulics, including examples from in-place facilities, are presented in Section 22.3. Wastewater treatment plant hydraulics are discussed in Section 22.4, and Section. 22.5 is devoted to non-Newtonian flow principles. 22.1 Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
22.2
Chapter Twenty-Two
22.2 GENERAL 22.2.1 Introduction This section addresses some elements which are common to both water treatment plants and wastewater treatment plants including: • Flow distribution–manifolds • Gates and valves • Flowmeters • Local losses
22.2.2 Flow distribution–manifolds In the design of water and wastewater treatment plants, proper flow distribution can be as critical as process design considerations, which typically receive much more attention. Plant failures resulting from unequal and unmanageable flow distribution are possibly as common and as serious as those resulting from errors in process design. Flow distribution devices, such as distribution channels, pipe manifolds or distribution boxes, are commonly used to distribute or equalize flow to parallel treatment units, such as flocculation tanks, sedimentation basins, aeration tanks, or filters. 22.2.2.1 Distribution boxes. The simplest of these devices, the distribution box, typically consists of a structure arranged to provide a common water surface as the supply to two or more outlets. The outlets are typically over weirs and the key to equal flow distribution is to provide independent hydraulic characteristics between the downstream system and the water level in the distribution box. In other words, provide a free discharge weir (nonsubmerged under all conditions) for each outlet to eliminate the impact of downstream physical system differences on the flow distribution. Velocity gradients across the distribution box must be nearly zero to equalize flow conditions over each outfall weir. Weirs clearly should be of uniform design in terms of physical arrangement length and materials of construction. They should also be adjustable to account for any minor flow differences noted in actual operation. The same principles apply if the designer wishes to distribute flows in specific proportions which are not necessarily equal. In this case the designer could control the proportions of flow distribution by varying the relative geometry of the weirs (i.e., change the width or invert of each weir to achieve a desired flow distibution). The specifics of weir hydraulics are covered in various texts in the literature. Attention should always be paid to the selection of the proper coefficients to model the specific weir geometry and the geometry of the approach flow. 22.2.2.2 Distribution channels and pipe manifolds. Distribution channels and manifolds are also common in plant design but a bit more complex in their function and design. The distribution of flow in these devices is impacted by the flow distribution itself. Since a portion of the flow leaves the channel or manifold along the length of the device, the velocity of flow and, therefore, the relationship of energy grade line, velocity head and hydraulic grade line varies along the length of the device. This is more clearly visible in a distribution channel of uniform cross section, using side weirs along its length for flow distribution. At each weir, flow leaves the channel, resulting in less velocity head in the channel
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
Water and Wastewater Treatment Plant Hydraulics 22.3
and possibly a higher water surface at each ensuing weir. Chao and Trussell (1980), Camp and Graber (1968), and Yao (1972) have presented comprehensive approaches for the design of distribution channels and manifolds and should be reviewed for details of design. As in distribution boxes, the most important consideration to achieving equalized flow distribution is to minimize the effects of unequal hydraulic conditions relative to each point of distribution. In channels this can be accomplished by tapering the channel cross section, varying weir elevations, making the channel large enough to cause velocity head changes to be insignificant or a combination of these. Similar considerations may be applied to manifolds with submerged orifice outlets. A reliable approach here is to provide a large enough manifold, resulting in a total headloss along the length of the distribution of less than one tenth the loss through any individual orifice. This approach essentially results in the orifices becoming the only hydraulic control and the accuracy of the flow distribution is then dependent on the uniformity of the orifices themselves.
22.2.3 Gates and Valves Gates and valves generally serve to either control the rate of flow or to start/stop flow. Gates and valves in treatment plants are typically subjected to much lower pressures than those in water distribution systems or sewage force mains and can be of lighter construction. 22.2.3.1 Gates. Gates are typically used in channels or in structures to start and stop flow or to provide a hydraulic control point which is seldom adjusted. Because of the time and effort required to operate gates, they are not suited for controlling flow when rapid response, frequent variation, or delicate adjustments are needed. Primary design considerations when using gates are the type of gate fabrication and the installation conditions during construction. There are many fabrication details including materials used, bottom arrangement, and stem arrangement. For instance, for solids bearing flows, a flush bottom, rising stem gate can be used to avoid creating a point of solids deposition and to minimize solids contact with the threaded stem. Gate manufacturers are a good source of information for gate fabrication details and can assist with advice regarding specific applications. Most commonly used gates are designed to stop flow in a single direction. They may use upstream water pressure to assist in achieving a seal (seating head), but typically also must be designed to resist static water pressure from downstream (unseating head). Both seating and unseating heads must be evaluated in design of a gate application. For most manufacturers, the seating or unseating head is expressed as the pressure relative to the center line of the gate. 22.2.3.2 Valves. Table 22.1 provides a summary of several types of valves and their applications. Valves are used to either throttle (control) flow or start/stop flow. Start/stop valves are intended to be fully open or fully closed and nonthrottling. They should present minimum resistance to flow when fully open and should be intended for infrequent operation. Gate valves, plug valves, cone valves, ball valves, and butterfly valves are the most common start/stop valve selections. Butterfly valves have a center stem, are most common in clean water applications and should not be used in applications including materials that could hang-up on the stem. Therefore, they are seldom used at wastewater plants prior to achieving a filter effluent water quality.
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
22.4
Chapter Twenty-Two
TABLE 22.1
Typical Valves and Their Application*
Type
Open/Close
Throttling
Water
Wastewater
Sluice gate
X
X
X
Slide gate
X
X
X
Gate valve
X
X
X
Plug valve
X
X
X
X
Cone valve
X
X
X
X
Ball valve
X
X
X
X
Butterfly valve
X
X
X
Swing check
X
X
Lift check
X
X
X
Ball check
X
X
X
Spring check
X
X
X
Globe valve
X
X
Needle valve
X
X
Angle valve
X
X
X
X
Pinch/diaphragm
X
X
*
Typical applications–exceptions are possible, but consultation with valve manufacturers is recommended.
Check valves are a special case of a start/stop valve application. Check valves offer quick, automatic reaction to flow changes and are intended to stop flow direction reversal. Typical configurations include swing check, lift check, ball check and spring loaded. These valves are typically used on pump discharge piping and are opened by the pressure of the flowing liquid and close automatically if pressure drops and flow attempts to reverse direction. The rapid closure of these valves can result in unacceptable “waterhammer” pressures with the potential to damage the system. A detailed surge analysis may be required for many check valve applications (see Chapter. 12). At times, mechanically operating check valves should be avoided in favor of electrically or pneumatically operated valves (typically plug, ball, or cone valves) to provide a mechanism to control time of closing and reduce surge pressure peaks. Throttling valves are used to control rate of flow and are designed for frequent or nearly continuous operation depending on whether they are manually operated or electronically controlled. Typical throttling valve types include globe valves, needle valves, and angle valves in smaller sizes, and ball, plug, cone, butterfly, and pinch/diaphragm valves in larger sizes. Throttling valves are typically most effective in the mid-range of loose line open/close travel and for best flow control should not be routinely operated nearly fully closed or nearly fully open.
22.2.4 Flow meters The most common types of flow meters used in water and wastewater treatment plants are summarized in Table 22.2 and fall into the following categories:
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
Water and Wastewater Treatment Plant Hydraulics 22.5 TABLE 22.2 Type
Common Types of Flow Meters Typical Accuracy
Size Range
Headloss
Cost
W
WW
Venturi
0.75% of rate
1–120 in
Low
Medium
X
X
Orifice plate
2% of scale
Any size
Medium
Low
X
X
Pitot tube
0.5–5% of scale
1/2–96 in
Low
Low
X
Parshall flume
5% of rate
Wide range
Low
Medium
X
X
Magnetic
0.5% of rate
1/10–120 in
None
High
X
X
Doppler
1–2.5% of rate
1/8–120 in
None
High
X
X
Propeller
2% of rate
Up to 24 in
High
High
X
Turbine
0.5–2% of rate
Up to 24 in
High
High
X
• Pressure differential/pressure measuring meters (e.g., Venturi, orifice plate, pitot tube, and Parshall flume meters) • Magnetic meters • Doppler (ultrasonic) meters • Mechanical meters (e.g., propeller and turbine meters) Accurate flow measurements require uniform flow patterns. Most meters are significantly impacted by adjacent piping configurations. Typically a specific number of straight pipe diameters is required both upstream and downstream of a meter to obtain reliable measurements. In some cases, 15 straight pipe diameters upstream and 5 straight pipe diameters downstream are recommended. However, different types of meters have varying levels of susceptibility to the uniformity of the flow pattern. Meter manufacturers should be consulted. 22.2.4.1 Pressure differential/pressure measuring meters. Pressure differential/pressure measuring flow meters include Venturi meters, orifice plates, averaging pitot meters, and Parshall flumes. These meters measure the change in pressure through a known flow cross section–or in the case of the pitot meter, measure the difference in pressure at a point in the flow versus static pressure just downstream in a uniform section of pipe. Venturi meters and orifice plates are commonly used in water and wastewater. Solids in wastewater could plug the openings of a pitot tube meter-limiting their use to relatively clean liquids. The Venturi meter and orifice plate meter use pressure taps at the wall of the device and can be arranged to minimize potential for debris from clogging the taps. The Parshall flume can be arranged with a side stilling well and level measuring float system or an ultrasonic level sensing device to measure water level. 22.2.4.2 Magnetic meters. In a magnetic flowmeter, a magnetic field is generated around a section of pipe. Water passing through the field induces a small electric current proportional to the velocity of flow. Because a magnetic meter imposes no obstruction to the flow, it is well suited to measuring solids bearing liquids as well as clean liquids and produces no headloss in addition to the normal pipe loss. Magnetic meters are among the least susceptible to the uniformity of the stream lines in the approaching flow.
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
22.6
Chapter Twenty-Two
22.2.4.3 Ultrasonic meters. In an ultrasonic flow meter, a pair of transceivers (transmitter/receiver) are positioned diagonally across from each other on the pipe wall. The transmitter sends out a signal which is affected by the speed of the flow. The receiver measures the difference between the speed of the signal when directed counter to the flow (slowed by the flow) and when directed with the flow (speeded up by the flow). The time difference is a function of fluid velocity, which is used to compute the flow. As with magnetic meters, no flow obstruction is imposed resulting in no headloss in addition to the normal pipe loss. Ultrasonic meters are also available for open-channel applications. 22.2.4.4 Mechanical meters. Mechanical meters include propeller and turbine-type equipment. The two meters are similar in function in that in each a device is inserted into the flowpath. The device is rotated by the flow and the speed of rotation is used to compute rate of flow. These devices impose an obstruction to flow, are recommended for clean water only, and generally result in significant headloss.
22.2.5 Local Losses In any piping system as flow travels along the pipe, pressure drops as a result of headloss due to friction along the pipe and local losses at bends, fittings, and valves. The local losses at bends, fittings, and valves are least significant in long, straight piping systems and most significant at treatment plants where the length of straight pipe is relatively short and therefore, the frictional pipe losses comprise a smaller fraction of the total losses when compared to the summation of all local losses. A term often used to refer to local losses is “minor losses,” however, because of the later consideration the term “minor losses” can be misleading. Traditionally, local losses have been computed in terms of “equivalent length” of straight pipe or in terms of multiples of velocity head. The “equivalent length” or loss factor K methods attempt to estimate the local losses based on the characteristic of the specific bend, fitting or valve. The K loss factor method is discussed here. Essentially, a local loss is computed as follows: 2 hL KV 2g
(22.1)
where hL local loss, K loss factor, V velocity, g gravitational acceleration. The values for K reported by various sources vary considerably for some local losses and are relatively consistent for others. See references. There are many literature sources for K values. The Bureau of Reclamation (1992) is one such source of information regarding energy loss equations. Table 22.3 shows a range of K factors from additional sources as well as a typically used value for each. Judgment must be applied in computing local losses, taking into account any unique system conditions. Throughout this chapter K values were obtained from equipment manufacturers when available. Values from Table 22.3 were used only as an approximation when more specific data were unavailable. The reader is cautioned that there are application-specific characteristics which have significant influence on the K factors. One of these characteristics, for example, is size. A K value of 0.6 is often encountered in literature to characterize the losses associated with flow through the run of a tee. However, for flow past tees in large pipes this factor can be very small and nearly zero.
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
Water and Wastewater Treatment Plant Hydraulics 22.7
Gate valve 100% open 0.39 75% open 1.1 50% open 4.8 25% open 27 Globe valve–open 10 Angle valve–open 4.3 Check valve–ball 4.5 Swing check Butterfly valve–open 1.2 Foot valve–hinged 2.2 Foot valve–poppet 12.5 Elbows 45° regular 45° long radius 90° regular 90° long radius 180° regular 180° long radius (flanged) Tees Std. teee–flowthrough run 0.6 Std. teee–flow-through branch1.8 Return bend 1.5 Mitre bend 90° 1.8 60° 0.75 30° 0.25 Expansion d/D = 0.75 0.18 d/D = 0.5 0.55 d/D = 0.25 0.88 Contraction d/D = 0.75 0.18 d/D = 0.5 0.33 d/D = 0.25 0.43 Entrancee–projecting 0.78 Entrancee–sharp 0.5 Entrancee–well rounded 0.04 Exit 1.0
0.19 1.15 5.6 24 10 5
2.1–3.1 65–70 0.6–2.3
0.30–0.42 0.18–0.20 0.21–0.3 0.14–0.23 0.38 0.25
0.19
0.1–0.3
10 5
4.0–6.0 1.8–2.9 06–2.2 0.16–0.35 1.0–1.4 5.0–14.0
0.6–2.5
0.18 0.25 0.18
0.5 0.7 0.6
0.6 1.8 1.8 2.2
0.3 0.75 0.4
1.8
1.129–1.265 0.471–0.684 0.130–0.165
0.8 0.35 0.1
Typically Used Value
Committee on Pipeline Planning (1975) 0.2 1.2 5.6 24 10 2.5
0.42
0.6 1.8 2.2
0.78 0.5 0.04 1.0
Sanks (1989)
Simon (1986)
Cameron Hydraulic Data
Daugherty (1977)
Crane Co. (1987)
Walski (1992)
Valve and Fitting Types
Bulletin No. 2552, University of Wisconsin
Typical K Factors for Computing Local Losses
Ten-State Standards (1978)
TABLE 22.3
0.2 1.2 5.6 25 10 5 5 2.5 0.5 2.2 14 0.42 0.2 0.25 0.19 0.38 0.25 0.6 1.8 2.2 1.3 0.6 0.16
0.19 0.56 0.92
0.2 0.6 0.9
0.2 0.6 0.9
0.19 0.33 0.42 0.83 0.5
0.2 0.3 0.4 0.78 0.5 0.04 1.0
0.2 0.33 0.43 0.8 0.5 0.04 1.0
0.8 0.5 0.04 0.04
0.8 0.5 0.25 1.0
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
22.8
Chapter Twenty-Two
22.3 HYDRAULICS OF WATER TREATMENT PLANTS 22.3.1 Introduction Water treatment comprises the withdrawal of water from a source of supply and the treatment of raw water through a series of unit processes for the beneficial use of the system customers. Raw water quality can vary widely. The ultimate uses of water by the system customer (e.g., drinking, fire protection, irrigation, aquifer recharge, etc.) can also vary and be subject to different treatment level requirements and regulations. Therefore, the selected treatment processes vary widely over a multitude of treatment technologies in use. Water treatment consists of a series of chemical, biological, and physical processes connected by channels and pipelines. Figures 22.1 and 22.2 illustrate process flow diagrams (flowsheets) for typical surface water and groundwater treatment plants, respectively. The designer of the water treatment process must carefully evaluate source water characteristics and desired water quality characteristics of the treated water to design treatment processes capable of purifying the source water to water suitable for the system customers. The objective of this chapter is to review the hydraulic considerations required to convey water through the treatment process. Design of a plant’s treatment process is closely linked with the hydraulic design of the treatment plant. This chapter presumes that the designer has evaluated and selected treatment processes for the water treatment plant. Although design flows are discussed below, we have also assumed that the designer has chosen a design flow requirement for the treatment process. For municipal treatment plants, design flows are based on the service area
FIGURE 22.1 Typical surface water treatment plant process flow diagram.
FIGURE 22.2 Typical ground water treatment plant process flow diagram with dual trains (#1 and #2).
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
Water and Wastewater Treatment Plant Hydraulics 22.9
population and the per capita use of water by the population served. The per capita use of water can be obtained from literature sources as an initial approximation. However, these initial estimations must be corroborated with actual site specific population counts and water usage. For nonmunicipal treatment facilities, treated water needs of the service area must be individually evaluated. 22.3.1.1 Sources of supply. Natural sources of supply include groundwater and surface water supplies. Groundwater supplies typically are smaller in daily delivery but serve more systems than surface water supplies. Groundwater supplies normally come from wells, springs, or infiltration galleries. Wells constitute the largest source of groundwater. Except in rare circumstances of artesian wells (wells under the influence of a confined aquifer) and springs, groundwater collection involves pumping facilities. Hydraulics of groundwater treatment plants are frequently based on hydraulics of conduits under pressure, such as pipelines, pressure filters, and pressure tanks. Raw water characteristics of groundwaters are uniform in quality compared with surface supplies. Surface water supplies are normally larger in daily delivery. Surface supplies are used to service larger population centers and industrial centers. In areas where groundwater supplies are limited in yield or where groundwater supplies contain undesirable chemical characteristics, smaller surface water treatment plants may be utilized. Surface water sources of supply include rivers, lakes, impoundments, streams, and ponds. The treatment processes chosen in plants treating surface water favor nonpressurized systems such as gravity sedimentation. The larger flow volumes characteristic of surface water supplies also favor open channel hydraulic structures for conveying water through the treatment process. Raw water characteristics of surface supplies can vary rapidly over short periods of time and also experience seasonal variation. 22.3.1.2 Treatment requirements. Treatment requirements for municipal water treatment plants are normally defined by regulatory agencies having authority over the plant’s service area. In the United States, regulatory agencies include national government regulations promulgated through the Environmental Protection Agency and state government regulations. Water treatment plants are designed to meet these regulations. Treatment regulations change through improved knowledge of health effects of water constituents and through identification of possible new water-borne threats. The designer therefore should attempt to select treatment processes which will also meet treatment requirements which are expected to be promulgated over the next few years. To the extent possible, treatment plant process design should provide flexibility for future plant expansions or for possible additional treatment processes. Because hydraulic design of plants must go hand-in-hand with the process selection, plant hydraulic design should provide for the flexibility to add future plant facilities. Treatment requirements for industrial water treatment plants are dictated by process needs of the industry and less by regulatory agency requirements. Industrial water treatment plants that result in contact between or ingestion of the treated water by humans must conform to the local regulatory requirements. 22.3.1.3 General design philosophy. Effective design of water treatment plant hydraulics requires that the hydraulic designer have a thorough knowledge of all aspects of the water system. The overall treatment system hydraulic design must be integrated and coordinated including the treatment plant, the raw water intake and pumping facilities, the treated water storage, and treated water pressure/head requirements. The design within the water treatment plant must also be integrated between the various treatment processes.
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
22.10
Chapter Twenty-Two
Additionally, design considerations must address the availability of operating personnel and hours of operation such that the process and hydraulic requirements conform to available resources.
22.3.2 Hydraulic Design Considerations in Process Selection Water treatment plant process selections are controlled principally by characteristics of the raw water and by the desired water quality characteristics of the finished water. Flow through each unit process and each conduit connecting processes results in loss of hydraulic head. Most treatment plants have limited head available. The selection of a particular unit process will include evaluation of numerous criteria including costs, operability, performance, energy use and similar items. One criteria which must also be evaluated for each process is the hydraulic head requirements of the process. 22.3.2.1 Head available. For the design flow to pass through a water treatment plant, the total available head must exceed the head requirements of the unit processes and connecting conduits. The head available is the difference in energy grade line (EGL) in the hydraulic profile between the head works of the plant and the end of the plant. Additional head may be provided by pumping or by lowering the elevation of treatment units at the end of the plant. See Figure 22.3 for a typical water treatment plant hydraulic profile. For most surface water plants, the hydraulic profile at the head of the plant is controlled by raw water pumps pumping from the intake facilities. The hydraulic profile at the head of a plant in a groundwater system is typically determined by the well pumps serving the plant. 22.3.2.2 Typical unit process head requirements. Following below is a table of typical head requirements for water treatment plant processes. This table may be used for initial evaluation of unit processes. More detailed hydraulic evaluations must be performed after plant operating modes and design flows are determined. Detailed hydraulic evaluations must also include headlosses in connecting conduits.
Unit Process Intakes, including bar screens Rapid mixing Flocculation Sedimentation Filtration – Gravity – Pressure Disinfection Aeration – Spray – Cascade – Compressed air Softening Ion exchange softening Iron and manganese removal
Head Requirement at Rated Capacity, m (ft) 0.3–0.9 (1–3) 0.15–0.30 (0.5–1) 0.06–0.15 (0.2–0.5) 0.6–2.4 (2–8) 3–4.6 (10–15) 3–7.6 (10–25) 0.15–0.6 (0.5–2) 3–4.6 (10–15) 3–4.6 (10–15) 0.15–0.6 (0.5–2) 0.15–0.6 (0.5–2) 0.6–1.5 (2–5) 0.6–1.5 (2–5)
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
FIGURE 22.3 Hydraulic profile.
Water and Wastewater Treatment Plant Hydraulics 22.11
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
22.12
Chapter Twenty-Two
22.3.3 Hydraulic Design Considerations in Plant Siting Plant sites are normally selected before the hydraulic designer initiates design of the treatment system. If a plant site has not been selected, the designer should be aware of hydraulic considerations which may influence site selection. Site elevation has the most significant impact on plant hydraulics. A plant site located above the service area will eliminate or reduce pumping requirements from the plant to the service areas. Typical municipal distribution system pressures are 40–70 psi, therefore the elevation of the treatment plant should be at least 100 ft above the service area to eliminate finished water pumping. Similarly, plant sites which permit gravity intake of the source water may reduce or eliminate raw water pumping. Few plants are able to meet these optimal conditions. The typical surface water plant must pump both raw and finished water. Raw water (low-lift) pumps are used to pump water from the water source into the treatment facilities and finished water (high-lift) pumps are used to pump from the treatment plant into the service area distribution system.
22.3.4 Hydraulic Design Consideration in Plant Layout After the plant site has been identified, the plant design may be arranged for optimal hydraulic benefit. In particular, arrangement of treatment processes to allow flow to move down gradient minimizes excavation needs for structures. Arrangements which are designed for future expansion should consider the hydraulic needs of the expanded plant as well as the process needs. Grouping of processes together facilitates movement of water through the treatment process train. The designer should also consider secondary hydraulic systems for optimal design. Chemical feed systems and dewatering systems are examples of secondary hydraulic systems which must be coordinated with the treatment flow system. Normally it is desirable to minimize the length of chemical piping systems. Dewatering systems are usually based on gravity drainage of basins and conduits.
22.3.5 Bases for Design After evaluation and selection of a source of supply and development of the treatment plant process train, the designer is prepared to develop the plant Bases for Design. The Bases for Design is a summary of design flow and capacity, and proposed treatment processes, including the chemical storage and feed facilities. 22.3.5.1 Design flows. Design flows for water treatment plants serving municipalities are typically based on the projected population within the water service area for the design life of the treatment facilities. Population data is normally determined from census records, land use zoning information, and studies of existing and projected population densities. Service area per capita demands are affected by the mix of domestic, commercial, and industrial water users which are unique to each service area.Typically water consumption records are available for water service areas. For new facilities, the use of generalized water consumption data may be needed. In the United States, water consumption varies widely but generally ranges between 100–200 gallons per capita per day.
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Water and Wastewater Treatment Plant Hydraulics 22.13
From studies of projected population and per capita demand, planned design flows for the water treatment facilities may be developed. These demands include the following: • Annual average demand. The average daily water consumption for the water service areas, generally computed by multiplying the average daily consumption (gallons per capita) by the projected population of the service area. • Maximum demand. Maximum demand experienced by the water plant throughout its service life. The maximum hour demand is generally 200 to 300 percent of the average demand but numerous factors affect the peak demand experienced by water treatment plants. These factors include seasonal demands (particularly for plants where service areas are located in extremes of hot and cold temperatures), normal daily flow variations, the community size, industrial usage, and system storage. Normally system storage is provided to service peak hour demands, allowing the treatment facilities to be designed on peak day demands. Peak day demands generally range between 125 and 200 percent of the average demand. • Minimum flow. As the name suggests, the minimum flow expected to be processed through the treatment facilities. Minimum flow depends upon system operations. In general, minimum flows for municipal plants may be estimated as 50 percent of the average demand, but range between 25 and 75 percent of the average demand. 22.3.5.2 Rated treatment capacity. The rated treatment capacity of a plant is that capacity for which each of the unit processes are designed. For municipal treatment plants with adequate system storage, the rated treatment capacity is the system’s maximum day demand. Where storage is limited, the rated treatment capacity may be greater, for example, the system maximum hour demand or greater. Smaller systems may be designed to produce the rated treatment capacity in one or two 8-h shifts rather than over the entire 24-h day. 22.3.5.3 Hydraulic treatment capacity. Treatment plants are normally designed for a hydraulic capacity greater than the rated treatment capacity. Hydraulic treatment capacities are normally equal to 125 to 150 percent of the rated treatment capacity. The hydraulic treatment capacity provides flexibility for future process changes or alternative flow routings through the plant. Hydraulic capacities in excess of the rated treatment capacity provide some margin of safety for operations which may not be optimal (e.g., control gates inadvertently left partially open). 22.3.5.4 Treatment process bases for design. The development of the water treatment plant’s “Bases for Design” is a key step in establishing the criteria to which the plant will be designed. This document must be reviewed carefully with the water treatment plant owner representatives and understood and agreed to by all before the final design proceeds. The Bases for Design presents a summary of each treatment process including design flows (minimum, average, rated capacity), specification of dimension of major elements (e.g., tanks, pumps), both hydraulic and process loading characteristics, required performance, and design data for the chemical storage and feed system. Table 22.4 presents an example of the bases for design for sedimentation basins (one of the many unit processes in a water treatment plant).
22.3.6 Plant Hydraulic Design As noted above, a water treatment plant consists of a series of treatment processes connected by free surface flow channels and pipelines. During development of the plant’s
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
22.14
Chapter Twenty-Two
TABLE 22.4
Treatment Process Bases for Design—Sedimentation Basins Item
Stage I
Stage II
StageIII
Annual Average
Maxi– mum Day
Annual Average
Maxi– mum Day
Annual Average
Maximum Day
4
4
8
8
12
12
3.17
1.99
3.17
1.99
3.17
1.99
Surface loading [(galⴢm)/ft ]
0.47
0.75
0.47
0.75
0.47
0.75
Flowthrough velocity (ft/min)
1.21
1.93
1.21
1.93
1.21
1.93
8
8
8
8
8
8
1
1
1
1
1
1
4
4
4
4
4
4
Number of basins Basin characteristics Plan–75 ft 230–6 in Nominal side water depth–12 ft (SWD) Surface area/basin–17,288 ft2 Volume/basin–207,456 f3 Channels/basin–2 L:W ratio–6.1:1 Displacement time (h) 2
Sludge collectors Longitudinal collectors Type: chain flight Number per basin Cross collectors Type: chain flight Number per basin Settled sludge pumps Type: progressive cavity Number: 100 gal/min capacity 400 gal/min capacity
4
4
4
4
4
4
200 gal/min capacity
—
—
8
8
16
16
Capacity (gal/min) Installed
2000
2000
3600
3600
5200
5200
Firm
1600
1600
3200
3200
4800
4800
Bases for Design, the designer determines the rated treatment capacity, average flow, minimum flow and hydraulic capacity of the plant. Following development of the Bases for Design, the designer must evaluate plant operating modes to develop a detailed plant flow diagram and hydraulic profile through the plant.
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Water and Wastewater Treatment Plant Hydraulics 22.15
22.3.6.1 Plant operating modes. Operating modes describe the sequence of treatment processes the water goes through to achieve the required level of purification. Operational modes are normally presented in the form of simplified block diagrams which illustrate the flow path through the plant from one process to the next. These operational mode block diagrams are useful in visualizing stages during construction, future planned plant expansions or simply alternative operating modes. Figures 22.4 through 22.9 show an example of a sequence of plant operating modes for a surface water treatment plant which illustrate three stages of a plant expansion program with alternatives for the flocculation and sedimentation basins to work in series or in parallel. Plant processes proposed include raw water control chambers, rapid mix chambers, flocculation/sedimentation basins, ozone contact chambers, and filters. In this example, the raw water control chambers are used to split flow between plant process groups and also as a rapid mix chamber for chemical addition.
FIGURE 22.4 Stage I—operational mode diagram.
FIGURE 22.5 Stage II—parallel operational mode diagram.
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22.16
Chapter Twenty-Two
The Stage I facilities including raw water control chamber, flocculation/sedimentation basins and filters are depicted in Fig. 22.4. Operational modes for a proposed plant expansion to double the plant capacity (Stage II) are shown in Figs. 22.5 through 22.7 and operating modes for a second plant expansion to triple the plant capacity (Stage III) are shown in Figs. 22.8 and 22.9. Settled water ozone contact chambers were added to the expanded plant, which illustrates treatment upgrades. Operational modes for the Stage II treatment plant include parallel and series flocculation/sedimentation. When the plant is operated in the parallel mode, influent raw water for each set of sedimentation basins flows by gravity from the raw water control chamber serving the basin set. Raw water flow is divided between each sedimentation basin in service at the raw water control chamber. Settled water from each set of basins is routed to an ozone contact chamber. Ozonated settled water is then combined prior to flowing to the filters.
FIGURE 22.6 Stage II—series flocculation/sedimentation basin operational mode diagram.
FIGURE 22.7 Stage II—split parallel operational mode diagram.
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Water and Wastewater Treatment Plant Hydraulics 22.17
FIGURE 22.8 Stage III—parallel operational mode diagram.
FIGURE 22.9 Stage III—split parallel operational mode diagram.
The Stage III split parallel operational mode is similar to the parallel operational mode except that the ozonated settled water from each set of basins is not combined prior to flowing to the filters. Side-by-side plant scale treatment studies are possible with the future split parallel mode since part of the flocculation/ sedimentation/filtration processes can be operated as a “control” while the remainder of the plant can be operated in a controlled experimental mode. The series flocculation/sedimentation operational mode is designed to permit operation of the sedimentation basins in two stages in lieu of the single–stage parallel mode. Under certain raw water conditions, operation of the basins in series may enhance performance of the basins. Chemical feed for the first and second sedimentation stages may be adjusted to respond to raw water conditions and settled water quality after the first–stage sedimentation. Series flocculation/sedimentation increases hydraulic losses through the
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
22.18
Chapter Twenty-Two
plant. Under this mode, twice as much flow is routed to each basin and the flow pattern is longer, since the settled water from the first sedimentation stage must be returned to the influent of the second sedimentation stage. Operational mode block diagrams are also a convenient means to illustrate the effect of side stream flows which may impact the overall plant flow. For example, removal of sludge from the sedimentation basins is accompanied by a decrease in flow leaving the basins compared with flow entering the basins. In a similar manner, filter backwash water removes a certain amount of flow. A plant designed to produce a certain rated capacity may have to treat more than the rated capacity through certain processes. The impact of these side stream flows must be evaluated on an individual basis. In many treatment plants, backwash water treatment facilities are installed to recycle backwash water to the head of the plant. 22.3.6.2 Plant flow diagrams. After establishing plant operating modes, more detailed flow diagrams are developed by the designer. The diagrams normally start with possible valving and gating arrangements and are then expanded with tentative valve, sluice gate, pipeline, and conduit sizes. Valving arrangements are designed to enable any of the major operational units (e.g., sedimentation basin, ozone contact chamber) to be removed from service. The arrangement may include design of temporary flow stop devices, such as stop logs (sectional barriers which were originally constructed of logs but are now commonly metal plates). The arrangement should be designed to permit maintenance work on major valves and sluice gates while minimizing the impact on plant process. Major channel sections should be designed so they can be removed from service and dewatered while minimizing impacts on the rest of the plant. The designer should distinguish between units taken out of service frequently (such as filters), periodically (such as sedimentation basins), or rarely (such as conduits). Filter backwashing occurs so frequently that the rated treatment capacity can be met with one filter out for backwashing. Sedimentation basins may be removed from service once or twice per year for equipment maintenance. Since the basins outages occur at widely scattered intervals, it is reasonable to design the units to be removed from service during lower flow periods. Conduits and pipelines are rarely removed from service, but the hydraulic impacts can be significant. Depending on the conduit location, removal of a conduit can remove a portion of the plant from service. Effective design will provide redundant conduits so that a portion of the plant can remain in service during conduit dewatering. The focus of this section has been on the main plant hydraulics, but the hydraulic designer must also design for hydraulic subsystems. An important group of these subsystems include dewatering of all basins and conduits. Where plant elevations will allow, gravity dewatering is recommended. In most cases, dewatering pumps are necessary. These pumps may be located in the unit being dewatered or may be located in a separate structure connected to the process unit by dewatering pipelines. 22.3.6.3 Hydraulic Profile. One of the most important tools in the hydraulic design of a water treatment plant is the development of a hydraulic profile. The hydraulic profile is a diagram showing the energy grade line (EGL) at each unit process. For open tanks with flows at minimal velocities, which is the case in most water treatment plants, the velocity head is negligible and the hydraulic grade line (HGL) or water surface elevation (WSEL) provide an adequate representation of the EGL. Profiles normally include critical structural elevations of processes and conduits. The profile may also include ground surface profiles and other site information.
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
Water and Wastewater Treatment Plant Hydraulics 22.19
Hydraulic profiles are developed for each of the design flows. In the case of water treatment plants, the design flows may include rated treatment capacity, hydraulic capacity, average flow, and minimum flow. Hydraulic profiles should also take into consideration unit processes or conduits which may be taken out of service. Hydraulic profiles are valuable design and operational tools to assist in scheduling routine maintenance activities and for evaluating the impact to the treatment plant capacity during outages of process units or conduits. Computations of hydraulic profiles begin at control points where there is a definite relationship between the plant flow and water surface depth. For gravity flow plants, the most common forms of control points are weirs and tank water surface elevations (e.g., clear well water surface elevations), but other types of control points may be used. From each control point, head losses associated with local losses, plant piping, and open channel flow are added to the control water surface. Since flows in water treatment plant’s are mostly in the subcritical regime (Froude number 1), most hydraulic designers will work upstream from the control point. For pressure plants, control points are typically pressure regulating or pressure control points, frequently in the service area distribution system. From these control points and knowledge of the flow velocity, both the EGL and HGL may be computed back to the treatment facilities. Hydraulic profiles are valuable design tools to identify overall losses through the plant. Profiles are also valuable to identify units with excessive losses. Since total head available is normally limited, units with excessive losses should be considered for redesign to reduce local loss coefficients or to reduce velocities. Figure 22.3 is an example hydraulic profile for a gravity surface water treatment plant with conventional treatment processes. The method of computing headlosses is presented in Section 22.3.7.
22.3.7 Water Treatment Plant Process Hydraulics In this section calculations required to establish the WSEL through a medium-sized water treatment plant will be presented. A schematic of the water treatment plant is shown in Fig. 22.10. Notice that future growth has been considered in the initial design. Three examples are included which illustrate typical hydraulic calculations. The first example calculates the WSEL from the sedimentation basin effluent chamber back through the flocculation/sedimentation basins to the Raw Water Control Chamber. The second follows the flow from the clear well back through the filters. Filter hydraulics are illustrated in the third example. All examples are presented in a spreadsheet format which is designed to facilitate calculating the EGL, HGL, and WSEL at various points through the treatment process and for multiple flow rates (i.e., minimum, daily average, peak hour, future conditions). 22.3.7.1 Coagulation. Process criteria and key hydraulic design parameters. The coagulation process, used to reduce particulates and turbidity, is carried out in three steps: mixing (often referred to as rapid or flash mixing), flocculation, and sedimentation. Each of these steps is briefly discussed below. Rapid mixing. The mixing process imparts energy to increase contact between existing solids and added coagulants. Possible mixer types include turbine, propeller, pneumatic, and hydraulic. Headloss that occurs in mixing chambers depends on the chosen mixing device. Most mechanical mixers do not create significant head losses. The headloss coefficient (K) K associated with a specific mixer can be obtained from the manufacturer. Pneumatic mixing, which is not common, has associated losses similar to those
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
22.20
Chapter Twenty-Two
FIGURE 22.10 Schematiz of a water treatment plant.
for aeration (see table in Section 22.3.2.2, above). Hydraulic mixing takes place using weirs, swirl chambers, throttled valves, Parshall flumes, or other devices to induce turbulence. Head loss coefficients for these devices can be obtained from the manufacturer. Important considerations during the initial design of a mixing chamber include: • Velocity gradient. This is mixer—specific information and can be obtained from the manufacturer. The system should be designed to provide a velocity gradient that is optimal for the coagulation process taking place. • Dead spots and short circuiting. An ideal mixing system will have minimal dead spots and short circuiting. These can be avoided with proper sizing and placement of mixers. Flocculation. Coagulated particles form larger particles (flocs) during the gentle mixing of flocculation, where the flow travels slowly through a series of flocculator paddles, baffles, or conduits. Inlets and weirs are designed to provide low turbulence for protection of the flocs. The energy provided to the system by the flocculators (manufacturer-specific) or baffling is decreased as the flow approaches the sedimentation basins. Sedimentation. Gravity sedimentation removes coagulated solids prior to filtration. There are four zones in a clarifier as shown in Fig. 22.11 and listed below: • Inlet zone—where upstream flow conditions transition smoothly to uniform flow settling conditions • Sedimentation zone—where sedimentation takes place • Sludge zone—where solids collect and are removed • Outlet zone—where settling conditions smoothly transition to downstream flow conditions
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Water and Wastewater Treatment Plant Hydraulics 22.21
FIGURE 22.11 Hypothetical zones in a rectangular sedimentation basin.
Each of the zones is designed to minimize turbulence and avoid short circuiting. The velocity in the sedimentation zone is limited to 0.3 m/s (1 ft/s) for average flow. Sludge removal equipment moves slowly so that settling patterns are not disturbed. Because the process is designed for smooth flow and minimal turbulence, very little head loss occurs in sedimentation basins. Ports at the inlet and outlet produce the greatest head losses in this process. Hydraulic design example. Table 22.5 illustrates the calculation of the WSEL, using metric units, through the coagulation process at the medium-sized water treatment plant shown in Fig. 22.10. Figs. 22.12 through 22.14 show plan views and details of the
TABLE 22.5
Hydraulic Calculations of a Typical Coagulation Process, SI Units Initial Operation Parameter
1. Plant Flow (m3/s) Note: For Points 1 through 8, see Fig. 22.12 2. WSEL at Point 1 (Calculation done in Table 22.6) (m) 3. Point 1 to Point 2 Average flow 21Q/32 (m3/s) Flow depth WSEL @ 1 – invert (106.60 m) (m) Flow area 5.13 m width depth (m2) Velocity flow/area (m/s) Hydraulic Radius r A/P / (P w 2d) (m) Conduit loss [(V n)/(rr2/3)]2 L (m) where Manning’s n 0.014 and L 28.96 m WSEL at Point 2 (m)
Min. Day. Avg. Day
Design Operation Avg Day Max. Hour
2.19
3.06
3.28
4.38
109.73
109.73
109.74
109.74
1.44 3.13 16.05 0.09 1.41
2.01 3.13 16.06 0.13 1.41
2.15 3.13 16.07 0.13 1.41
2.87 3.14 16.10 0.18 1.41
0.00 109.73
0.00 109.73
0.00 109.74
0.00 109.74
4. Point 2 to Point 3 Average Flow 5Q/16 (m3/s) Flow depth WSEL @ 2 invert (106.60 m) (m) Flow area 5.13 m width depth (m3) Velocity flow/area (m/s) r A/P / (P w 2d) (m) Conduit loss [(V n)/(rr2/3)]2 L (m) where Manning’s n 0.014 and L 14.63 m WSEL at Point 3 (m)
0.68 3.13 16.05 0.04 1.41
0.96 3.13 16.06 0.06 1.41
1.03 3.13 16.07 0.06 1.41
1.37 3.14 16.10 0.08 1.41
0.00 109.73
0.00 109.73
0.00 109.74
0.00 109.74
5. Point 3 to Point 4 Average flow Q/8 (m3/s) Flow depth WSEL @ 3—invert (106.60 m) (m) Flow area 5.13 m width depth (m3)
0.27 3.13 16.05
0.38 3.13 16.06
0.41 3.13 16.07
0.55 3.14 16.10
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
22.22
Chapter Twenty-Two
TABLE 22.5
(Continued) Initial Operation Parameter
Velocity flow/area (m/s) r = A/P / (P w 2d) (m) Conduit loss [(V n)/(rr2/3)] 2 L (m) where n 0.014 and L 21.95 m WSEL at Point 4 (m) 6. Point 4 to Point 5 Flow Q/32 (m3/s) Port area 0.30 m deep 0.76 m wide (m2) Velocity flow/area (m/s) Submerged entrance loss 0.8 V2/2g (m) WSEL at Point 5 (in Sedimentation Tank) (m)
Design Operation
Min. Day. Avg. Day Avg. Day Max. Hour 0.02 1.41
0.02 1.41
0.03 1.41
0.03 1.41
0.00 109.73
0.00 109.73
0.00 109.74
0.00 109.74
0.07 0.23 0.29 0.00 109.73
0.10 0.23 0.41 0.01 109.74
0.10 0.23 0.44 0.01 109.74
0.14 0.23 0.59 0.01 109.76
23.16 0.77 105.97 3.77 87.36
23.16 0.82 105.97 3.77 87.41
23.16 1.09 105.97 3.79 87.68
9.71 0.01
9.71 0.01
9.74 0.01
7. Point 5 to Point 6 Width of sedimentation basin (W) W (m) 23.16 Flow (Q/4) (m3/s) 0.55 Invert elevation of sedimentation baffles (m) 105.97 Flow depth (H) H (WSEL at Point 5—baffle invert) (m) 3.76 Area downstreams of baffle (W H H) (m2) 87.21 Horizontal openings in baffle, 2.54 cm wide spaced every 22.86 cm. Area of openings A W .0254 H/.2286 (m2) 9.69 Velocity of downstream baffle (V downstream) 0.01 (Q/A) (m/s) Velocity of 2.54 cm opening section (V1) (Q/A / ) (m/s) 0.06 Local losses sudden expansion (1.0 (V downstream)2/2g) 2 sudden contraction (0.36 (VI) V / 2g) (m) 0.00 WSEL at Point 6 (Upstream of sedimentation baffles) (m) 109.73
0.08
0.08
0.11
0.00 109.74
0.00 109.74
0.00 109.76
8. Point 6 to Point 7 Loss per stage (provided by flocculator manufacturer) (m) 0.01 Total loss (three stages) (m) 0.04 WSEL at Point 7 (m) 109.77
0.01 0.04 109.78
0.03 0.09 109.83
0.05 0.15 109.91
9. Point 7 to Point 8 Flow Q/24 (m3/s) Port area 0.30 m deep 0.46 m wide (m2) Velocity flow / area (m/s) Entrance loss 1.25 V2/2g (m) WSEL at Point 8 (inlet port) (m)
0.09 0.14 0.65 0.03 109.80
0.13 0.14 0.92 0.05 109.83
0.14 0.14 0.98 0.06 109.89
0.18 0.14 1.31 0.11 110.02
0.09 0.68 0.62 0.15 0.27
0.13 0.72 0.65 0.19 0.28
0.14 0.77 0.71 0.19 0.29
0.18 0.90 0.82 0.22 0.30
0.00 109.80
0.00 109.83
0.00 109.89
0.00 110.02
0.18
0.26
0.27
0.36
Note: For Points 8 through 14, see Fig. 22.13 10. Point 8 to Point 9 Average flow Q/24 (m3/s) Flow depth WSEL @ 8 – invert (109.12 m) (m) Flow area 0.91 m width depth (m2) Velocity flow/area (m/s) r = A/P (P w 2d) (m) Conduit loss [(V n)/(rr2/3)] 2 L (m) where n 0.014 and L 3.86 m WSEL at Point 9 (m) 11. Point 9 to Point 10 Average flow Q/12 (m3/s)
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
Water and Wastewater Treatment Plant Hydraulics 22.23 TABLE 22.5
(Continued) Initial Operation Parameter
Design Operation
Min. Day. Avg. Day Avg. Day Max. Hour
Flow depth WSEL @ 9 – invert (109.12 m) (m) Flow area 0.91 m width depth (m2) Velocity flow/area (m/s) r = A/P / (P w 2d) (m) Conduit loss [(V n)/(rr2/3)]2 L (m) where n 0.014 and L 3.86 m WSEL at Point 10 (m)
0.68 0.62 0.29 0.27
0.72 0.65 0.39 0.28
0.77 0.71 0.39 0.29
0.90 0.82 0.44 0.30
0.00 109.80
0.00 109.84
0.00 109.89
0.00 110.02
12. Point 10 to Point 11 Flow Q/8, m3/s Flow depth WSEL @ 10 invert (109.12 m) (m) Flow area 0.91 width depth (m2) Velocity flow/area (m/s) 2 Loss at two 45° bends 2 0.2 V /2g (m) WSEL at Point 11 (m)
0.27 97.34 89.01 0.00 0.00 109.80
0.38 97.38 89.04 0.00 0.00 109.84
0.41 97.44 89.09 0.00 0.00 109.89
0.55 97.56 89.21 0.01 0.00 110.02
13. Point 11 to Point 12 Flow Q/4 (m3/s) Flow depth WSEL @ 11 invert (109.12 m) (m) Flow area 1.52 m width depth (m2) Velocity flow/area (m/s) Loss at two 45° bends 2 0.2 V 2/2g (m) r = A/P / (P w 2d) (m) Conduit loss [(V n)/(rr2/3)]2 L (m) where n 0.014 and L 9.75 m WSEL at Point 12 (m)
0.55 0.68 1.04 0.52 0.00 0.36
0.77 0.72 1.09 0.70 0.00 0.37
0.82 0.78 1.18 0.69 0.00 0.38
1.09 0.90 1.37 0.80 0.00 0.41
0.00 109.81
0.00 109.84
0.00 109.90
0.00 110.03
14. Point 12 to Point 13 Flow Q/4, (m3/s) Flow depth WSEL @ 12 invert (109.12 m) (m) Inlet area 1.52 m width depth (m2) Velocity flow/area (m/s) Inlet loss 1 V 2/2g (m) WSEL at Point 13 (Mixing Chamber No. 2 outlet) (m)
0.55 0.69 1.05 0.52 0.01 109.82
0.77 0.72 1.10 0.69 0.02 109.87
0.82 0.78 1.19 0.69 0.02 109.92
1.09 0.91 1.38 0.79 0.03 110.06
15. Point 13 to Point 14 Note: Mixers provide negligible head loss 0.55 Flow Q/4 (m3/s) Chamber area 1.83 m 1.83 m (m2) 3.34 Velocity flow/area (m/s) 0.16 Losses Mixer (1 V 2/2g) Sharp bend (1.8 V 2/2g) (m) 0.00 WSEL at Point 14 (Mixing Chamber No. 2 inlet) (m) 109.82
0.77 3.34 0.23 0.01 109.87
0.82 3.34 0.25 0.01 109.93
1.09 3.34 0.33 0.02 110.07
1.53 2.79 0.55 0.40
1.64 2.79 0.59 0.40
2.19 2.79 0.78 0.40
0.01
0.01
0.02
0.01 109.89
0.01 109.95
0.02 110.11
Note: For Points 14 through 21, see Fig. 22.14 16. Point 14 to Point 15 1.09 Flow Q/2 (m3/s) Conduit area 2.29 m wide 1.22 m deep (m2) 2.79 Velocity flow/area ( m/s) 0.39 R = A/P / (P 2w 2d) (m) 0.40 Conduit losses L [V/(0.849 V C R0.63)] 1/0.54 (m) where L 47.24 m and Hazen-Williams C 120 0.00 Local losses flow split (0.6 V 2/2g) contraction 0.01 (0.07 V 2/2g) 0.67 V 2/2g (m) WSEL at Point 15 (at Mixing Chamber No. 1) (m) 109.83
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
22.24
Chapter Twenty-Two
TABLE 22.5
(Continued) Initial Operation Parameter
17. The above calculations (for Points 1 through 15) have been for flow routed through Tank No. 4. When the flow is routed through Tank No. 1. the WSEL (m) is: In reality, the headloss through each basin is equal. The flow through the basin naturally adusts to equalize headlosses, i. e. flow through Tank No. 1 is greater than Q/4 and flow through Tank No. 4 is less than Q/4. The actual headloss through each basin can be estimated as the average of: Losses through Tank No’s. 1 and 4 and the WSEL (m) at Point 15 is: 18. Point 15 to Point 16 Flow Q (m3/s) Conduit area 2.29 m wide 1.22 m deep (m2) Velocity flow/area (m/s) R A/P / (P 2w 2d) (m) Conduit losses L [V/(0.849 V C R0.63)] 1/0.54 (m) where L 125.58 m and Hazen-Williams C 120 WSEL at Point 16 (m) 19. Point 16 to Point 17 Flow Q (m3/s) Conduit area @ 16 2.29 m wide 1.22 m deep (m2) Conduit area @ 17 1.68 m wide 1.68 m deep (m2) Average area (m2) Velocity flow / Area (m/s) R @ 16 A16/ (2 (2.29 m 1.22 m)) (m) R @ 17 A17/ (2 (1.68 m 1.68 m)) (m) Average R, (m) Conduit losses L [V/(0.849 V C R0.63)]1/0.54 (m) where L 9.14 m and Hazen-Williams C 120 WSEL at Point 17 (m) 20. Point 17 to Point 18 Flow Q (m3/s) Conduit area @ 17 1.68 m wide 1.68 m deep (m2) Velocity 17 flow/area 17 (m/s) 2 Pipe area @ 18 (D) (m) where D 1.68 m 4 Velocity 18 flow/area 18 (m) Exit losses V182/2g – V172/2g (m/s) WSEL at Point 18 (m) 21. Point 18 to Point 19 R = A/P / (P d ) (m) Local losses 3 elbows (3 0.25V 2/2g) entrance (0.5 V 2/2g) 1.25 V 2/2g (m) Conduit losses L [V/(0.849 V C R0.63)]1/0.54 (m) where L 138.68 m and Hazen-Williams C 120 WSEL at Point 19 (exit of Control Chamber) (m)
Design Operation
Min. Day. Avg. Day Avg. Day Max. Hour
109.82
109.88
109.94
110.08
109.83
109.89
109.95
110.10
2.19 2.79 0.78 0.40
3.06 2.79 1.10 0.40
3.28 2.79 1.18 0.40
4.38 2.79 1.57 0.40
0.04 109.87
0.08 109.97
0.10 110.04
0.16 110.26
2.19 2.79 2.81 2.80 0.78 0.40 0.42 0.41
3.06 2.79 2.81 2.80 1.09 0.40 0.42 0.41
3.28 2.79 2.81 2.80 1.17 0.40 0.42 0.41
4.38 2.79 2.81 2.80 1.56 0.40 0.42 0.41
0.00 109.88
0.01 109.98
0.01 110.05
0.01 110.27
2.19
3.06
3.28
4.38
2.81 0.78 2.21 0.99 0.02 109.90
2.81 1.09 2.21 1.39 0.04 110.01
2.81 1.17 2.21 1.49 0.04 110.09
2.81 1.56 2.21 1.98 0.8 110.35
0.42
0.42
0.42
0.42
0.06
0.12
0.14
0.25
0.07 110.03
0.13 110.27
0.15 110.39
0.26 110.86
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
Water and Wastewater Treatment Plant Hydraulics 22.25 TABLE 22.5
(Continued) Initial Operation Parameter
22. Point 19 to Point 20 Weir elevation (m) Depth of flow over weir (WSEL @ 19 – weir elevation), (m) Length of weir, L, (m) 3/2 0.385 Flow over weir q 1.71 h3/2 [ 1 (d / n) ] L Note: Rather than solve for h, find an h by trial and error that gives a q equal to the flow for the given flow scenarios (given in Item 1) assume h (m) then q (m3/s) assume h (m) then q (m3/s) Note: These q’s equal the flows for the given scerios (Item 1) h (m) WSEL at Point 20 (h WSEL @ Point 19) (m) 23. Point 20 to Point 21 Flow Q (m3/s) Sluice gate area 1.37 m 1.37 m (m2) Velocity Flow/Area (m/s) Gate Losses 1.5 V 2/2g (m) WSEL at Point 21 (Raw Water Control Chamber) (m) The overflow weir in the Raw Water Control Chamber is 3.05 m long and is sharp crested Q = 1.82 L h3/2 so h (Q/1.82L)2/3 (m) The water surface must not rise above elevation 112.78 m The overflow weir elevation may be safely set at 111.86 m
Design Operation
Min. Day. Avg. Day Avg. Day Max. Hour
109.73
109.73
109.73
109.73
0.30 2.74
0.54 2.74
0.66 2.74
1.13 2.74
0.60 1.84 0.66 2.18
0.90 3.14 0.89 3.07
0.95 3.12 0.97 3.27
1.35 4.21 1.37 4.42
0.66 110.39
0.89 110.62
0.97 110.70
1.37 111.10
2.19 1.88 1.16 0.10
3.06 1.88 1.63 0.20
3.28 1.88 1.74 0.23
4.38 1.88 2.33 0.41
110.49
110.82
110.93
111.51
0.54
0.67
0.70
0.85
hydraulic reaches analyzed in the example. The circled numbers indicate points at which the WSEL is calculated. Hydraulic calculations start downstream of the sedimentation basins (Fig. 22.12) and proceed upstream through the mixing chamber (Fig. 22.13) and the Raw Water Control Chamber (Fig. 22.14). Mechanical mixers and mechanical flocculators are used. Conduit losses between the rapid mix chambers and the Raw Water Control Chamber are also calculated in the example. Three different flow rates (i.e., minimum day, average day, and, maximum hour) are used in the calculations. This is a range of design flow conditions that a design engineer would typically take into consideration. The longest path through the flocculation and sedimentation processes, through Basin No. 4, is followed (Points 1 through 15). Although not shown, losses along the shortest path have also been calculated. As would be expected, the calculated head loss is smaller for the shorter path. The actual losses are equal for each path. The flows through each path naturally adjust to equalize losses. The flow through the longest path is slightly smaller than the flow through the shortest path. In the example, the WSEL at Point 15 is adjusted to reflect the average losses through the basins. The WSEL calculations upstream of Point
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
22.26
Chapter Twenty-Two
FIGURE 22.12 Flocculation/ sedimentation basin
15 are based on the adjusted WSEL. Alternatively the weirs or ports feeding flow into each basin may be adjusted to create an equal distribution of flows in all basins as discussed in Sec. 22.2.1. 22.3.7.2.2 Filtration. Process criteria. Suspended solids are removed from the water as it passes through a porous medium during filtration. Filters operate under either gravity or pressure. Filters also differ in the type and distribution of the media used (fine, course, uniformly graded, graded coarse to fine, etc.) and the direction of flow through the media (upflow, downflow, and biflow). Pressure filter hydraulics information is very product specific and should be obtained from the manufacturer. The design engineer using pressure filters should then apply this information to the project using project–specific hydraulic considerations. This section presents information on gravity filters. Key hydraulic design parameters. The headloss through a filter increases with use as the voids become filled with solid particles. When the headloss reaches a certain point (terminal headloss), the filter is backwashed to remove the solids. The rate of headloss buildup is dependent on several factors, including how the filter is graded (the arrangement of media particle sizes). The rate of headloss buildup is reduced (and filtration is more effective) when the flow first goes through the coarse media and then the fine media.
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
Water and Wastewater Treatment Plant Hydraulics 22.27
FIGURE 22.13 Mixing chamber
FIGURE 22.14 Raw water control chamber
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
22.28
Chapter Twenty-Two
However, during backwash, the high rate of flow expands the filter bed and, over time, the media are regraded so that the more coarsely graded grains are located at the bottom and the fines are located at the top. To benefit from the coarse-to-fine grading, an upward flow pattern can be used, but is very uncommon. More often the filter media are selected such that the fine media have a higher specific gravity than the coarse media to maintain the course-to-fine gradation during backwash. The most commonly used filter media are natural silica sand and crushed anthracite coal; however garnet and ilmenite are used in mixed media beds. Granular carbon is often used if taste and odor control is desired. The terminal headloss is determined by a combination of factors including filter breakthrough (when the filter bed loses its adsorptive capacity), available static head, and outlet pressure required. The filter should be designed so that the headloss in any level of the filter bed does not exceed the static pressure. A negative head can result in air binding in the filter which will, in turn, further increase headloss. Filter influent piping is sized to limit velocities to about (0.6 m/s). Wash-water and effluent piping flow velocities are kept below (1.8 m/s) so that hydraulic transients(waterhammer) and excessive headlosses are minimized and controlled to within tolerable limits. Hydraulic design example. Table 22.6 illustrates the calculation of the WSEL from the clear well back upstream to the Sedimentation Basin effluent at the medium-sized water treatment plant shown in Fig. 22.10. Figures 22.15 and 22.16 show details of the hydraulic reaches analyzed in the example. Table 22.7 illustrates the filter hydraulic calculation, the details of which are shown in Figs. 22.17 and 22.18. The hydraulic profile of the plant (based on hydraulic calculations done in Tables 22.5, 22.6 and 22.7) is shown in Figure 22.3.
TABLE 22.6 Hydraulic Calculations in a Medium–Sized Water Treatment Plant from the Filter Effluent to the Effluent Clearwell Initial Operation Parameter 1. Flow (m3s) Note: for Points 22 through 28, see Figure 22.15 2. Point 22 to Point 23 Maximum water level in Clearwell (Point 22) (m) Invert in Clearwell (m) Flow Q/2 (m3/s) Stop logs @ A Flow area (2 openings, 1.52 m wide, 3.66 m deep) (m2) Velocity flow/area (m/s) Loss 0.5 V 2/2g (m) Baffles Flow area (3.05 m wide, 3.66 m deep) (m2) Velocity flow/area (m/s) Loss 1.0 V 2/2g (m) Stop logs @ B and C Same as the losses @ A, times 2 (m)
Min Day
Avg Day
Design Operation Avg Day Max Hour
2.19
3.06
3.28
4.38
105.16 101.50 1.09
105.16 101.50 1.53
105.16 101.50 1.64
105.16 101.50 2.19
11.15 0.20 0.00
11.15 0.27 0.00
11.15 0.29 0.00
11.15 0.39 0.00
11.15 0.20 0.00
11.15 0.27 0.00
11.15 0.29 0.00
11.15 0.39 0.01
0.00
0.00
0.00
0.01
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
Water and Wastewater Treatment Plant Hydraulics 22.29 TABLE 22.6
(Continued) Initial Operation Parameter
WSEL at Point 23 (m)
Min Day
Avg Day
Design Operation Avg Day Max Hour
105.16
105.17
105.17
105.18
1.09
1.53
1.64
2.19
2.21 0.50 0.01 0.01 0.01
2.21 0.69 0.02 0.01 0.01
2.21 0.74 0.03 0.01 0.01
2.21 0.99 0.05 0.03 0.03
0.00 105.19
0.00 105.22
0.00 105.23
0.00 105.28
0.55 2.32 0.24 0.00
0.77 2.32 0.33 0.01
0.82 2.32 0.35 0.01
1.09 2.32 0.47 0.01
0.00
0.00
0.00
0.00
5. Point 25 to Point 26 Sluice Gate No. 1 flow area 1.22 m 0.91 m (m2) Velocity Q/A / (m/s) Loss 0.5 V 2/2g (m) WSEL at Point 26 (m)
1.11 0.49 0.01 105.20
1.11 0.69 0.01 105.24
1.11 0.74 0.01 105.24
1.11 0.98 0.02 105.32
6. Point 26 to Point 27 Sluice Gate No. 2 Loss 0.8 V 2/2g (m) WSEL at Point 27 (m)
0.01 105.21
0.02 105.25
0.02 105.27
0.04 105.36
3. Point 23 to Point 24 Flow Q/2 (m3/s) 1.68 (m) diameter pipe Flow area d 2/4 (m2) Velocity flow/area (m/s) Exit loss @ clearwell V 2/2g (m) o Loss @ 2 - 90 bends (0.25 V 2/2g) 2 (m) Entrance loss @ Filter Building 0.5 V 2/2g (m) Pipe loss (3.022 V 1.85 L)/ (C 1.85 D 1.165 ) where C 120 and L 57.91 m (m) WSEL at Point 24 (m) 4. Point 24 to Point 25 Flow Q/4 (m3/s) Flow area 1.52 m 1.52 m2 Velocity Q/A (m/s) Loss as flows merge 1.0 V 2/2g (m) Conduit loss [(V n)/(R 2/3 )]2 L (m) where n 0.013, L 16.76 m and R A/P / (P 6.10 m) WSEL at Point 25 (m)
7. Point 27 to Point 28 Port to Filter Clearwell: Calculate losses through port as if were a weir when depth of flow is below top of port. Port dimmensions 2.74 m wide by 0.813 m deep. Flow Q/4 (m3s) Weir (bottom of port) elevation (m) Depth of flow over weir (WSEL @ 27 – weir elevation) (m) Flow over submergedweir q 1.71 h3/2 [1 - (d/ d h)3/2]0.385 L Note: Rather than solve for h, find an h, by trial and error, that gives a q equal to the flow for the given flow scenario assume h (m) then q (m3/s) assume h (m) then q (m3/s) Note: These q’s equal the flows for the given scenarios h (m)
0.55 104.85
0.77 104.85
0.82 104.85
1.09 104.85
0.36
0.40
0.42
0.51
0.40 0.59 0.39 0.52
0.45 0.69 0.46 0.76
0.50 0.95 0.48 0.82
0.60 1.23 0.58 1.09
0.39
0.46
0.48
0.58
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
22.30
Chapter Twenty-Two
TABLE 22.6
(Continued) Initial Operation Parameter
WSEL at Point 28 (m) Filters—See Filter Hydraulics in Table 22.7 Note: for Points 29 through 33, see Fig. 22.16 8. Point 29 WSEL above filters (m) 9. Point 29 to Point 30 Entrance to Filter #4 Flow, Q/8 (m3/s) Channel velocity = flow/area (area 1.22 m 1.22 m) (m/s) Submerged entrance loss 0.8 V 2/2g (m) 1.22 m pipe velocity flow/area (area d 2/4 ) (m/s) Butterfly valve loss 0.25 V 2/2g (m) Sudden enlargement loss 0.25 V 2/2g (m) WSEL in influent channel (Point 30) (m) 10. Point 30 to Point 31 Flow depth WSEL @ 30 invert (107.29 m) (m) Flow area 1.83 m width depth (m2) Velocity flow/area (m/s) R = A/P / (P w 2d) (m) Conduit Loss [(V n)/(rr2/3)]2 L where n 0.014 and L 10.77 m (m) WSEL at Point 31 (m) 11. Point 31 to Point 32 Flow Q/4 (m3/s) Flow depth WSEL @ 31 - invert (107.29 m) (m) Flow area 1.83 m width depth (m2) Velocity flow/area (m/s) R = A/P / (P w 2d) (m) Conduit loss [(V n)/(r 2/3 )]2 L (m) where n 0.014 and L 10.77 m WSEL at Point 32 (m) 12. Point 32 to Point 33 Flow 3Q/8 (m3/s) Flow depth WSEL @ 32 – invert (107.29 m) (m) Flow area 1.83 m width depth (m2) Velocity flow/area (m/s) R A/P / (P w 2d) (m) Conduit loss [(V n)/(rr2/3)]2 L (m) where n 0.014 and L 10.77 m WSEL at Point 33 (m) 13. Point 33 to Point 1 Flow Q/2 (m3/s) Flow depth WSEL @ 33 – invert (107.29 m) (m) Flow area 1.83 m width depth (m2) Velocity flow/area (m/s) R = A/P / (P w 2d) (m) Conduit loss [(V n)/(r 2/3 )]2 L (m) where n 0.014 and L 11.07 m WSEL at Point 1 (m)
Design Operation
Min Day
Avg Day
Avg Day Max Hour
105.24
105.31
105.33
105.43
109.73
109.73
109.73
109.73
0.27
0.38
0.41
0.55
0.18 0.00
0.26 0.00
0.28 0.00
0.37 0.01
0.23 0.00 0.00 109.73
0.33 0.00 0.00 109.73
0.35 0.00 0.00 109.73
0.47 0.00 0.00 109.74
2.44 4.46 0.06 0.67
2.44 4.47 0.09 0.67
2.44 4.47 0.09 0.67
2.45 4.48 0.12 0.67
0.00 109.73
0.00 109.73
0.00 109.73
0.00 109.74
0.55 2.44 4.46 0.12 0.67
0.77 2.44 4.47 0.17 0.67
0.82 2.44 4.47 0.18 0.67
1.09 2.45 4.48 0.24 0.67
0.00 109.73
0.00 109.73
0.00 109.73
0.00 109.74
0.82 2.44 4.46 0.18 0.67
1.15 2.44 4.47 0.26 0.67
1.23 2.44 4.47 0.28 0.67
1.64 2.45 4.48 0.37 0.67
0.00 109.73
0.00 109.73
0.00 109.73
0.00 109.74
1.09 2.44 4.46 0.24 0.67
1.53 2.44 4.47 0.34 0.67
1.64 2.45 4.47 0.37 0.67
2.19 2.45 4.48 0.49 0.67
0.00 109.73
0.00 109.73
0.00 109.74
0.00 109.74
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
Water and Wastewater Treatment Plant Hydraulics 22.31
FIGURE 22.15 Clearwell to filter effluent
FIGURE 22.16 Filter effluent
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
22.32
Chapter Twenty-Two
TABLE 22.7
Example Hydraulic Calculation of a Typical Filter
Parameter Plant flow (m3/s) Filter loading, [(m ⴢ m)/m ] Filter area per filter—seven (7) out of eight (8) filters in operation (m2) Flow loading area (m3/s) Losses through filter effluent piping (Fig. 22.17) 0.51 m piping (Q): Pipe velocity Q/A / (m/s) Local losses Exit (0.5) butterfly o valves (2 0.25) 90 elbows (2 0.4) 2 tee (1.8) 3.6 V /2g (m) R A/P / (d 2/4 p)/(d p) dd/4 (m) Conduit losses L [V/(0.849 V C R0.63)] 1/0.54 where L 6.10 m and HazenWilliams C 120 (m) 0.51 m piping (Q/2): Pipe velocity Q/A (m/s) Local Losses Butterfly Valve (0.25) (m) R A/P / (d 2/4 p)/(d p) dd/4 (m) Conduit losses L [V/(0.849 V C R0.63)] 1/0.54 where L 3.05 m and HazenWilliams C 120 (m) 0.61 m piping (Q/2): Pipe velocity Q/A / (m/s) Local losses entrance (1.0) tee (1.8) 2.8 V 2/2g (m) Filter (clean) and underdrain losses (obtain from manufacturer) (m) Total losses (effluent pipe and clean filters) (m) 3
2
Initial Operation Min. Day Avg. Day.
Design Operation Avg. Day. Max. Hour.
2.19
3.06
3.28
4.38
0.083 115
0.167 115
0.250 115
0.334 115
0.16
0.32
0.48
0.64
0.79
1.58
2.37
3.16
0.11 0.13
0.46 0.13
1.03 0.13
1.83 0.13
0.01
0.03
0.06
0.11
0.40 0.00 0.13
0.79 0.01 0.13
1.19 0.02 0.13
1.58 0.03 0.13
0.00
0.00
0.01
0.02
0.27
0.55
0.82
1.10
0.01
0.04
0.10
0.17
0.09 0.23
0.15 0.70
0.23 1.45
0.34 2.50
Assume that headloss will be allowed to increase 2.44 m before the filters are backwashed. A rate controller will be used to maintain a constant flow through the filters. Determine the ranges of available head over which the rate controller will operate. Static Head (Fig. 22.18) WSEL above filters (m) WSEL in filter effluent conduit, Point 29 (see Example 22.2) break Maximum (m) Minimum (m) Static head WSEL above filters—WSEL at Point 29 (Filter effluent conduit-2) Maximum (m) Minimum (m) Available head static head 2.44 m Maximum (m) Minimum (m)
109.73
109.73
109.73
109.73
105.61 105.16
105.61 105.16
105.61 105.16
105.61 105.16
4.57 4.11
4.57 4.11
4.57 4.11
4.57 4.11
2.13 1.68
2.13 1.68
2.13 1.68
2.13 1.68
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
Water and Wastewater Treatment Plant Hydraulics 22.33
Head, meters
FIGURE 22.17 Filter effluent piping
FIGURE 22.18 Available head over which filter effluent rate controller operates—metric units.
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
22.34
22.3.8
Chapter Twenty-Two
Membrane Technology
Membranes are synthetic filtering media manufactured from a variety of materials including polypropylene, polyamide, polysulfone, and cellulose acetate. The membrane material can be arranged in various configurations, including the following: • Spiral wound • Hollow fiber • Tubular • Plate frame Examples of these configurations are presented in Fig. 22.19. In water and wastewater treatment applications, the most common configurations are spiral wound and hollow fiber. In general, there are four classes of membranes: microfilters (MF), ultrafilters (UF), nanofilters (NF), and hyperfilters. Treatment through hyperfilters is referred to as hyperfiltration, or reverse osmosis (RO). The hydraulics associated with membranes are membrane-specific and can be obtained from the manufacturer. This section presents general considerations pertinent to flow through membranes. As with natural particle media filters, clean membranes have a specific headloss and, over time, as the membranes become covered with a cake buildup, the effectiveness of the membrane decreases and headloss increases. Fouling (excessive buildup) may damage the membrane. The need for pretreatment ahead of membranes is determined by the raw water quality and the membrane type. In general, microfilters and ultrafilters do not require pretreatment for treating surface or groundwater. Nanofilters and reverse osmosis membranes may require pretreatment depending on the type of fouling. Membrane fouling can result from particulate blocking, chemical scaling, and biological growth within the membranes. An estimate of particulate blocking can be made using indices such as the Silt Density Index (SDI) and the Modified Fouling Index (MFI). These fouling indices are determined from simple bench membrane tests using 0.45 micron Millipore filters and monitoring flow through the filter at a given pressure, usually 30 psig. Approximate values of suitable SDIs for nanofiltration are 0–3 units, and for reverse osmosis, 0–2 units. Corresponding values of MFI are, for nanofiltration 0 to 10 s/L2, and for RO, 0–2 s/L2. Scaling control is essential in RO and nanofilter membrane filtration, especially when the filtration provides water softening. Controlling precipitation or scaling within the membrane element requires identification of limiting salt, acid addition for prevention of calcium carbonate precipitation within the membrane, and/or the addition of an antiscalant. The amount of antiscalant or acid addition is determined by the limiting salt. A diffusion controlled membrane process will naturally concentrate salts on the feed side of the membrane. As water is passed through the membrane, this concentration process will continue until a salt precipitates and scaling occurs. Scaling will reduce membrane productivity and, consequently, recovery is limited by the allowable recovery just before the limiting salt precipitates. The limiting salt can be determined from the solubility products of potential limiting salts and the actual feed stream water quality. Ionic strength must also be considered in these calculations as the natural concentration of the feed stream during the membrane process increases the ionic strength, allowable solubility and recovery. Calcium carbonate scaling is commonly controlled by sulfuric acid addition, although sulfate salts, such as barium sulfate and strontium sulfate, are often the limiting salt. Commercially available antiscalants can be used to control scaling by complexing the
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Water and Wastewater Treatment Plant Hydraulics 22.35
FIGURE 22.19 Membrane configurations. (a) Spiral wound, (b) hollow fiber, (c) tubular, (d) plate and frame.
metal ions in the feed stream and preventing precipitation. Equilibrium constants for these antiscalants are not available which prohibits direct calculation. However, some manufacturers provide computer programs for estimating the required antiscalant dose for a given recovery, water quality, and membrane. Biological fouling is controlled with some membranes such, as cellulose acetate, by maintaining a free chlorine residual of not more than 1 mg/L. Other membranes, such as the thin-film composites, are not chlorine tolerant and must rely on upstream disinfection by, for example, ultraviolet disinfection or chlorination-dechlorination. The extent of fouling for a specific application and its influence in the design of nanofiltration and RO membrane systems is best determined by pilot studies. It has been suggested that some buildup on the membrane may be beneficial to treatment by providing an additional filtering layer. At facilities operated by the Metropolitan Water District of Southern California (MWD), removal rates of 1.7–2.9 logs were
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22.36
Chapter Twenty-Two
observed for seeded virus MS2 bacteriophage through microfilters that had a pore size an order of magnitude larger than the nominal size of MS2 (1). The microfiltration system used by MWD utilizes an air backwash procedure whereby compressed air at 90–100 psig is introduced into the filtrate side of the hollow fiber membranes. Accumulated particulates dislodged by the compressed air are swept away by raw water introduced to the feed side of the membranes. The backwash sequence is carried out automatically at preset time intervals. MWD found the best interval to be every 18 minutes. The total volume of backwash represents approximately 5–7 percent of influent flow. The difference between influent and effluent pressure across the membrane is termed the transmembrane pressure (TMP). Despite the frequent air and water backwashes, the TMP gradually increases over time. Generally, when the TMP reaches approximately 15 psig, chemical cleaning of the membranes is carried out. If the TMP is allowed to increase beyond 15 psig, particulates can become deeply lodged within the lattice structure of the membranes and will not be removed, even by chemical cleaning. Chemical cleaning typically lasts 2–3 hours and involves circulating a solution of sodium hydroxide and a surfactant through the membranes, and soaking them in the solution. The membranes at the MWD microfilter plants have a surface loading rate of 40–67 ft2. The lower flux rate of 40 ft2 has the advantage that the rate of increase of TMP is reduced and the interval between chemical cleanings is increased. A possible explanation for this is that particulates are not forced as deeply into the lattice structure of the membranes, thereby allowing the air-water backwash to clean the membranes more effectively. By reducing the flux rate from 67–40 ft2, the interval between chemical cleanings was increased from 2 to 3 weeks to almost 20 weeks. However, MWD has instituted a maximum run time of 3 months between chemical cleanings to ensure the long-term integrity of the membranes. Nanofiltration is widely used for softening groundwaters in Florida. A typical nanofiltration plant would include antiscalant for scale control added to the raw water. Cartridge filters, usually rated at 5 microns, remove particles that may foul the membrane system. Feed water pumps boost the pretreated water pressure to about 90–130 pounds per square inch (psi) before entering the membrane system. The membranes typically are spiral wound nanofiltration membranes generally with molecular weight cutoff values in the 200–500 dalton range.
22.4 WASTEWATER TREATMENT Many factors and considerations influence the hydraulic design of a wastewater treatment plant. This section describes typical phases of wastewater treatment planning required for design of new plants or additions to existing plants, and then presents typical unit process hydraulic computations.
22.4.1 Wastewater Treatment Planning Hydraulic decision making for a new wastewater treatment plant or expansion of an existing plant involves several planning phases. Typical planning phases are presented below in their common order of consideration. 22.4.1.1 Service area and flows. More than 15,000 municipal wastewater treatment plants are in operation in the United States today. The plants are designed to treat a total of about
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Water and Wastewater Treatment Plant Hydraulics 22.37
140 million m3 of flow each day. Flow quantities requiring treatment change over time based on a number of factors related to service area. These factors include the following: Changes in service area size. Most often the service area will increase in size during the wastewater treatment plant service life. However, service area size may decrease, such as when wastewater in larger metropolitan areas is diverted to an alternate wastewater treatment plant. Information about anticipated changes in the size of a wastewater treatment plant service area can sometimes be found in “regional planning” documents. Changes in service area land use. Changes in the type of land use in the service area, such as from residential to industrial, will impact the flow rates to be served by the treatment plant. Also, the development of impervious areas within the wastewater treatment plant service area will reduce infiltration and increase runoff volume and rate. If this runoff then enters the sewer system it will impact the flow rate to the plant. A combined sewer system will be more susceptible to this type of change than a separate sewer system. Changes in service area density. Wastewater treatment plant flows are a function of the number of inhabitants and industries which generate the wastewater. An understanding of the regional planning issues which may affect the wastewater treatment plant service area assists in estimating future increases in flow and making appropriate provisions for future plant expansions. Such flow increases will likely be partially offset by increased water conservation in water-limited areas. Changes in service area infiltration/inflow. Most often the rates of infiltration/inflow will increase as the collection system becomes older. Such flow increases can generally be offset by periodic sewer rehabilitation, manhole rehabilitation, and enforcement of inflow control ordinances. The quantity of wastewater to be handled by a wastewater treatment plant is affected primarily by the type of wastewater produced in the service area and type of wastewater collection system used. The four types of wastewater which may be produced in a given sewer system service area include sanitary wastewater, industrial wastewater, stormwater, and infiltration/inflow. The three types of sewer systems used to collect some or all of these flow types include sanitary, storm, and combined-sewer systems. The types of wastewater are defined as follows: Sanitary flow. Wastewater discharged from residences and from institutional, commercial and similar facilities. Quantities of sanitary flow can be estimated on a per capita basis for each type and size of residence or facility producing the flow. Industrial flow. Wastewater discharged from industrial facilities. In a heavily industrialized area, industrial flow can make up a majority of a wastewater plant’s influent flow. Industrial wastewater quantities produced by a given facility can be estimated based on facility type, size, and rate of production. Stormwater. Stormwater is precipitation runoff. Stormwater enters storm or combined collection systems as surface or subsurface inflow. The rate of stormwater entering a storm or combined sewer system as inflow mirrors the intensity and quantity of the precipitation event, although if the precipitation is frozen the runoff will be delayed until melting occurs. Infiltration Water (including stormwater) that seeps into a wastewater collection system through the ground, usually through cracks or leaks in the collection system. Accordingly, infiltration rates typically vary both annually and seasonally. The age of the
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22.38
Chapter Twenty-Two
collection system should be considered when estimating infiltration rates because older collection systems are prone to higher infiltration rates. If the amount of infiltration is significant enough to affect plant influent water quality, the treatment processes must be selected accordingly. Inflow. Surface and subsurface stormwater discharging directly into a wastewater collection system. Precipitation events significantly impact inflow rates and can also impact infiltration rates by surcharging the groundwater table. The elevation of the groundwater table relative to the sewer elevation directly affects the infiltration flow rate. The types of sewer systems include sanitary-sewer systems which collect sanitary wastewater, industrial wastewater (if present in service area), and infiltration/inflow. Storm-sewer systems collect stormwater and infiltration/inflow, and combined-sewer systems collect sanitary wastewater, industrial wastewater (if present in service area), stormwater, and infiltration/inflow. Flows to wastewater treatment plants are conveyed by separate-sewer systems and, in some older systems, combined-sewer systems. Hydraulic design guidelines for sanitary-sewer systems have been compiled by the American Society of Civil Engineers and the Water Environment Federation (1982). 22.4.1.2 Effluent requirements. Treated wastewater can be discharged to rivers, lakes, oceans, and groundwater. There is also increasing re-use of wastewater for nonpotable applications, such as irrigation and industrial processing. Effluent quality requirements for wastewater treatment plants are generally established by regulatory agencies in the plant’s NPDES permit. Those minimum acceptable effluent characteristics and the anticipated influent characteristics determine what level of treatment is required and, thereby, determines to some degree what treatment processes are needed. Because each process type requires a different amount of head, the influent characteristics and effluent requirements also indirectly affect the plant head requirements. 22.4.1.3 Process selection. Each unit process in a wastewater treatment plant flow train treats the wastewater physically, chemically or biologically, or in some combination thereof. Because various combinations of unit processes are generally available to produce the desired effluent quality, the designer must choose among the options to select the optimum combination. In anticipation of future requirements, potential changes in effluent requirements and corresponding treatment train modifications should also be considered. Typical unit treatment processes for new wastewater treatment plants include screening, grit removal, primary sedimentation, aeration, secondary sedimentation, granular media filtration, disinfection, dechlorination, and postaeration. Figure 22.20 is a flow diagram showing how the typical processes are interconnected. Unless only one treatment process combination is capable of adequately treating the wastewater, pertinent factors must be used to select the process train. Typical factors include capital and operating costs, environmental impacts, aesthetics, and public acceptance. Process head requirements can directly affect capital costs, as those processes with higher head requirements are more likely to necessitate costly pumping facilities and deep structure excavation. 22.4.1.4 Hydraulic bases for design. Flow rates for the wastewater treatment plant must be established for the hydraulic design. Design year flow projections are often based on estimated conditions 15–20 years in the future. Providing sufficient treatment capacity to accommodate new development can be an important municipal commodity for expanding the municipal tax base. The design should also provide allowances for the initial plant operation when flow may be significantly less than the design flow, as well as expansion or rehabilitation to handle flows reasonably anticipated beyond the design year.
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Water and Wastewater Treatment Plant Hydraulics 22.39
Peak flow is used for hydraulic design, whereas average flow is used for treatment process design. Peak flow is defined as the maximum hour flow experienced by the wastewater treatment plant throughout its service life. The maximum hour flow is generally two to five times the average daily flow. Plants serving combined collection systems can experience even greater flow variations. Treatment plant unit processes must convey the maximum flow unless this flow would cause a hydraulic washout of the treatment plant. In this situation, the designer should consider the use of equalization basins to minimize negative impact on the treatment process. In addition, the plant must also be able to fully process minimum flow without undesirable settling of solids throughout the treatment train. Plants normally encounter diurnal fluctuation of pollutant loadings, as well as flow loadings. Fluctuation in pollutant loadings may impact treatment process selection and consequently impact process hydraulics. 22.4.1.5 Flow diagram. A flow diagram should be prepared to depict the results of process selection and hydraulic bases of design. Details in a flow diagram should include the type of unit processes, number of basins for process redundancy, flow distribution and junction chambers, piping, and conduits for interconnecting the unit processes and major recycle streams such as return-activated sludge (RAS). Figure 22.20, which was mentioned above, shows a typical flow diagram. 22.4.1.6 Plant siting. Several factors affect the plant site selection process, including site elevation, topography, geology, and hydrology; site access; utility availability; seismic activity; surrounding land use and future availability; noise, odor and air quality requirements at and near the site; existing collection system and receiving water proximity; and other environmental considerations. A site’s hydraulic suitability for a wastewater treatment plant is determined primarily by site elevation and topography. The typical site elevation is low-lying, which facilitates the flow of wastewater from the service area by gravity and minimizes costly pumping in the collection system. Such a site, however, may require flood protection. The difference in head between the plant influent sewer and the receiving water body is the head available for the treatment plant. If available head does not exceed the plant’s head requirements, additional head can be provided by pumping the wastewater. Selecting processes with lower head requirements can also reduce the need for pumping. Pumping of wastewater, especially untreated wastewater, should be avoided when possible due to potential operational difficulties of handling the associated rags, grit, stringy material and other large solids. A mild, continuous slope usually provides optimal gravity flow conditions. Relatively flat sites often necessitate higher pumping heads. Sites on a severe, uneven slope or slopes can require costly hydraulic and structural features, and should be avoided when possible. 22.4.1.7 Plant layout. The selected treatment processes establish the major space and hydraulic requirements needed to develop initial plant layouts. Also, provisions for future unit process additions and plant capacity expansions should be included both spatially and hydraulically. Support facilities, such as maintenance, laboratory and administrative facilities, must also be considered. Arranging process elevations to generally follow plant site topography minimizes the amount of structural excavation. Site geology constraints may limit the practical depth and elevation of the processes. In such cases, additional pumping facilities may be necessary to provide sufficient head for the required water surface elevation. When arranging treatment processes, a preliminary hydraulic profile should be developed as discussed below. The plant hydraulic profile and site topography and geology information together determine the location having the optimal elevation for each process.
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Chapter Twenty-Two
FIGURE 22.20 Schematic flow diagram of typical wastewater treatment plant.
22.40
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Water and Wastewater Treatment Plant Hydraulics 22.41
Other objectives when developing a plant layout at a selected site include: close proximity of processes to associated facilities; structure grouping according to process; transportation equipment and staff traffic pattern efficiency; minimization of process piping; and safe, isolated hazardous chemical and material locations. When preparing layouts for addition of a new process to an existing plant, the existing plant hydraulic profile should be consulted to determine the amount of head available for the new process. If adequate hydraulic head is not available for the new process, new pumping facilities will be necessary. 22.4.1.8 Hydraulic profile and calculations. A hydraulic profile should be prepared for the flow train to graphically depict the results of hydraulic calculations and site layouts. Details in a profile should include free water surface elevations throughout the flow train, including unit treatment processes, interconnecting piping and channels, junction chambers, flowmeters and flow control devices, as well as structural profiles. Figure 22.21 shows a typical hydraulic profile. Both high and low water levels are shown to illustrate the range of liquid levels anticipated at each structure. Sufficient freeboard must be provided to prevent liquid or floating material from splashing over the sides under conditions of high water level. Low water levels are important when designing devices requiring a mimimum amount of submergence, such as surface skimmers or baffles. In addition to normal high and low water levels, hydraulic calculations should address other potential conditions. For example, for each process having redundant structures, the largest capacity unit should be assumed to be out of service during maximum flow for consideration of a “worst case”. The process structure should always be hydraulically capable of accommodating the change in elevation due to the “worst case.” head requirements without liquid overtopping the walls. The process head requirement is the amount of head lost by the wastewater as it passes through a process at maximum flow. The head requirement for a specific process can vary with flow rate, influent water quality, process equipment size, process equipment layout, process equipment components included, and process equipment manufacturer.
22.4.2 Typical Unit Process Hydraulics 22.4.2.1 Bar screens. Process criteria. The first unit operation typically encountered in a wastewater treatment plant is screening. A schematic diagram of a typical bar screen system is shown in Fig. 22.22. A screen is comprised of a screening element with circular or rectangular openings designed to retain coarse sewage solids. The screens are designated as hand cleaned or mechanically cleaned based on the method of cleaning. Based on the size of the openings, screens are designated as coarse or fine. The general dividing line between coarse and fine screens is an opening size of 6 mm (1/4 in). A bar screen is a coarse screen designed to remove large solids or trash that could otherwise damage or interfere with the downstream operations of treatment equipment, such as pumps, valves, mechanical aerators, and biological filters. The bar screens are oriented vertically or at a slope varying from 30°– 80° with the horizontal. Key hydraulic design parameters. The key hydraulic design parameters for bar screens include the approach channel, effective bar opening, and operating head loss. Approach channel. Velocity distribution in the approach channel is an important factor in successful bar screen operation. A straight channel ahead of the channel provides good velocity distribution across the screen and promotes effectiveness of the device. Use of a configuration other than a straight approach channel has often resulted in uneven flow
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Chapter Twenty-Two
FIGURE 22.21 Typical hydraulic profile for wastewater treatment plant.
22.42
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Water and Wastewater Treatment Plant Hydraulics 22.43
distribution within the channel and accumulation of debris on one side of the screen. The velocity in the approach channel should be maintained at a self-cleaning value to dislodge deposits of grit or screenings. Ideally, the velocity in the screen chamber should exceed 0.4 m/s (1.3 ft/s) at minimum flows to avoid grit deposition if grit chambers follow bar screens. However, this is not always practical with the typical diurnal and seasonal fluctuation in wastewater flows. In general, common design practice provides velocities of 0.6–1.2 m/s (2–4 ft/s) for mechanically cleaned bar screens and 0.3–0.6 m/s (1–2 ft/s) with a velocity of 0.9 m/s (3 ft/s) at peak instantaneous velocity for manually cleaned bar screens. Effective bar opening. Various types of bar screens, including trash racks, manual screens and mechanically cleaned bar screens, employ a wide range of openings from 6 to 150 mm (14–6 in). The smaller screen openings collect larger quantities of screenings and generally produce higher head losses. The effective area of the screen openings equals the sum of the vertical projections of the screen openings. Operating head loss. As the screenings are collected, the openings in the screen become partially clogged and head losses increase. The maximum design allowance for headloss through the clogged screens is generally limited to 0.8 m (2.5 ft). Curves and tables for head loss through the screening device are usually available from the equipment manufacturer. To prevent flooding of the screening area caused by severe blinding of the screen during a power failure or similar disruption to cleaning, the design should provide for an overflow weir or gate and a parallel channel allowing overflows to flow around the screen. Hydraulic design example. The wastewater influent transported through the inlet sewer passes the bar screens prior to discharge into the pump well. Three bar screens are provided to handle hydraulic loadings varying from 1.0 m3/s (23 mgd) for minimum day flow during initial operation to 3.2 m3/s (73 mgd) for maximum hour flow during design operation. Sluice gates and stop logs are provided as part of the bar screen design so that any bar screen can be isolated for maintenance as required. Design hydraulic calculations for the bar screens are shown in Table 22.8. The WSEL at the pump well provides a downstream control point for the bar screens and channels. The WSEL at the pump well normally fluctuates between the pump control high water level and low water level. A high water level (HWL) of 100.60 m at the pump well is assumed. The channel bottom elevation of 99.50 m is determined to provide channel flow velocities in a range of 0.2–1.3 m/s for the flow range between the minimum and maximum day flow rates. The head requirements for the sample bar screen system is in the range of 0.17–0.36 m (0.56–1.2 ft) when the pump wet well level is at the maximum elevation of 100.60. 22.4.2.2 Grit tanks. Process criteria. Grit, consisting of sand, gravel, cinders, and other heavy solid materials, is present in wastewater conveyed by either separate or combined sewer systems, with far more in the latter. Grit removal prevents unnecessary abrasion and wear of mechanical equipment, grit deposition in pipelines and channels, and accumulation of grit in primary sedimentation basins or aeration basins and anaerobic digesters. Traditionally removal of 95 percent of grit particles larger than 0.21 mm (0.008 in or 65 mesh) has been the target of grit equipment design. Modern designs are now capable of removing up to 75 percent of 0.15 mm (0.006 in or 100 mesh) to avoid adverse effects on downstream processes. A variety of grit removal devices have been applied over the years. The basic types of grit removal processes include aerated grit chambers, vortex-type, detritus tank, horizontal flow type and hydroclone. Vortex systems are increasingly being selected. Detritus
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
Chapter Twenty-Two
FIGURE 22.22 Schematic diagram of bar screen system.
22.44
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Water and Wastewater Treatment Plant Hydraulics 22.45
tanks and aerated grit chambers are still popular. Depending on the type of grit removal process used, the removed grit is often further concentrated in a cyclone, classified, and then washed to remove light organic material captured with the grit. Key hydraulic design parameters. The key hydraulic design parameters for grit tanks include the inlet channel or inlet baffle, and effluent weir. Inlet channel/inlet baffle. For aerated grit chambers, the tank inlet and outlet should be positioned so that the flow through the tank is perpendicular to the roll pattern created by the diffused air. Inlet and outlet baffles serve to dissipate energy and minimize short circuiting. For vortex tanks, the flow into the vortex tank should be straight, smooth and streamlined. As a good practice, the straight inlet channel length should be seven times the width of the inlet channel or 15 ft, whichever is greater. The ideal velocity in the influent channel ranges from 0.6 to 0.9 m/s (2–3 ft/s) and should be used for flows between 40 and 80 percent of the peak flow. The minimum acceptable velocity for low flow is 0.15 m/s (0.5 ft/s). A baffle, located at the entrance, helps control the flow system in the tank and also forces the grit downward as it enters the tank. For detritus tanks, the performance relies on well-distributed flow into the settling basin. Allowances for inlet and outlet turbulence, as well as short circuiting, are necessary to determine the total tank area required. For horizontal flow grit chambers, velocity control throughout the chamber at approximately 0.3 m/s (1 ft/s) is important. An allowance for inlet and outlet turbulence is necessary to determine the actual length of the channel.
TABLE 22.8
Example Hydraulic Calculation of a Typical Bar Screen System Initial Operation Parameter
Min Day
1. Wastewater flow rate, Q (m3/s) 1.0 (mgd) 23 Bar screens Total number of units 3 Number of units in operation 2 Number of units on standby 1 Flow rate per screen in operation, q (m3/s) 0.5 Width of each bar screen, w (m) 2.5 2. At point 8 Pump wetwell HGL at high water level, HGL7 (m) 100.60 (pump starts at EL 100.60 and stops at EL 100.00) Pump well bottom EL (m) 99.00 Critical depth in a rectangular channel, Yc=(q2/g/w2)1/3 0.16 Bar screen channel depth= 1.10 pump WW HGL - channel bottom EL (m) (Water level at pump well controls upstream hydraulics if bar screen channel depth is higher than Yc) Is bar screen channel depth higher than Yc? yes 3. Point 8 to point 7 Channel bottom EL (m) 99.50
Design Operation
Avg.Day
Avg.Day
Max Hour Max Hour
1.6 36
2.0 46
3.2 73
3.2 73
3 2 1 0.8 2.5
3 2 1 1.0 2.5
3 2 1 1.1 2.5
3 2 1 1.6 2.5
100.60
100.60
100.60
100.60
99.00
99.00
99.00
99.00
0.22 1.10
0.25 1.10
0.26 1.10
0.35 1.10
yes
yes
yes
yes
99.50
99.50
99.50
99.50
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22.46
Chapter Twenty-Two
TABLE 22.8
(Continued) Initial Operation Parameter
Min Day
Depth in channel, y7 (m) 1.10 Velocity, V7 (m/s) 0.18 Exit loss from channel to pump well Exit loss coeficient, Kexit 1.0 Headloss, Hle7=K Kexit. V772/2g (m) 0.00 HGL at point 7, HGL7 HGL8+Hle7 (m) 100.60 4. Point 7 to Point 6 Friction headloss through channel Length of approach channel, L6 (m) 7 Manning’s number for concrete channel, n 0.013 Channel width, w6 (m) 2.50 Water depth, h6 (m) 1.10 Velocity, V6 (m/s) 0.18 Hydraulic radius, R6 (h6 w6)/(2 h6 w6) 0.59 Headloss, Hlf6 (V6 n/r662/3)2 L6 (m) 0.00 HGL at Point 6, HGL6 HGL7 + Hlf6 (m) 100.60 5. Point 6 to Point 5 Bar width (m) 0.010 Bar shape factor, bsf 2.42 Cross-sectional width of bars, w (m) 0.89 Clear spacing of bars, b (m) 1.61 Upstream velocity head, h (m) 0.0041 Angle of bar screen with horizontal, p (degrees) 60 (Kirschmer’s eq),. Hls bsf w/b 1.33 h sin p (m) 0.01 Allow 0.15 m head for blinding by screenings, Ha (m) 0.15 HGL upstream of bar screen, HGL5 HGL6 Hls Ha (m)
100.76
6. Point 5 to Point 4 Friction headloss through channel Length of approach channel, L4 (m) 7.00 Manning’s number for concrete channel n 0.013 Channel width, w4 (m) 2.50 Channel bottom elevation (m) 99.65 Water depth, h4 (m) 1.11 Channel velocity, V V4 (m/s) 0.18 Hydraulic radius R4 h4 w4/(2 h4 w4) 0.59 Headloss , Hlf4 f (V4* V n/R / 4 (2/3) 2 L4 (m) 0.00 HGL at Point 4, HGL4 HGL5 + Hlf4 f (m) 7. Point 4 to Point 3 Headloss at sluice gate contraction Kgate Sluice gate width (m) Sluice gate height (m) Velocity through sluice gate, Vs (m/s)
Avg.Day
Design Operation Avg.Day
Max Hour Max Hour
1.10 0.29
1.10 0.36
1.10 0.39
1.10 0.58
1.0 0.00 100.60
1.0 0.01 100.61
1.0 0.01 100.61
1.0 0.02 100.62
7 0.013 2.50 1.10 0.29 0.59 0.00 100.60
7 0.013 2.50 1.11 0.36 0.59 0.00 100.61
7 0.013 2.50 1.11 0.39 0.59 0.00 100.61
7 0.013 2.50 1.12 0.57 0.59 0.00 100.62
0.010 2.42 0.89 1.61 0.0104 60
0.010 2.42 0.89 1.61 0.0163 60
0.010 2.42 0.89 1.61 0.0186 60
0.010 2.42 0.89 1.61 0.0418 60
0.02
0.03
0.03
0.06
0.15
0.15
0.15
0.15
100.77
100.78
100.79
100.83
7.00 0.013 2.50 99.65 1.12 0.29
7.00 0.013 2.50 99.65 1.13 0.35
7.00 0.013 2.50 99.65 1.14 0.38
7.00 0.013 2.50 99.65 1.18 0.54
0.59
0.59
0.60
0.61
0.00
0.00
0.00
0.00
100.76
100.77
100.78
100.79
100.83
1.0 1.2 0.9 0.38
1.0 1.2 0.9 0.59
1.0 1.2 0.9 0.74
1.0 1.2 0.9 0.78
1.0 1.2 0.9 1.13
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
Water and Wastewater Treatment Plant Hydraulics 22.47 TABLE 22.8
(Continued) Initial Operation Parameter
Sluice gate headloss, Hls Kgate Vs2 /2g (m) HGL at Point 3, HGL3 (m) 8. Point 3 to Point 2 Water depth at point 2, h2 (m) Channel width, w2 (m) Channel velocity, V V2 (m/s) Fitting headloss through 45° bend Kbend 0.2 Headloss, Hlb2 = Kbend V 2 2/2g (m) Friction headloss through channel Length of approach channel, L2 (m) Manning’s, number for concrete channel n Hydraulic radius R2 h2 w2/(2 f2 f w2) (m) Headloss Hlf2 f (V2 V n/R / 2(2/3) 2 L2, (m) Entrance loss Kent 0.5 n Headloss, Hle2 = Kent V 2 2/2g(m) HGL at Point 2, HGL2 HGL3 Hlb2 Hlf2 f Hle2 (m) 9. Point 2 to Point 1 HGL at point 1, HGL 1 HGL2 (m) Invert EL of inlet sewer, INV1 (m) Crown EL of inlet sewer, CWN1 (m) Surcharge to inlet sewer?
Min Day
Avg.Day
Design Operation Avg.Day
Max Hour Max Hour
0.01
0.02
0.03
0.03
0.06
100.77
100.79
100.81
100.82
100.90
1.12 2.00 0.22 0.20 0.0005
1.14 2.00 0.35 0.20 0.0013
1.16 2.00 0.43 0.20 0.0019
1.17 2.00 0.46 0.20 0.0021
1.25 2.00 0.64 0.20 0.0042
4.00 0.013
4.00 0.013
4.00 0.013
4.00 0.013
4.00 0.013
0.53
0.53
0.54
0.54
0.56
0.0001
0.0002
0.0003
0.0003
0.0006
0.50 0.0013
0.50 0.0031
0.50 0.0047
0.50 0.0053
0.50 0.0105
100.77
100.79
100.82
100.83
100.91
100.77 99.50 101.65 No
100.79 99.50 101.65 No
100.82 99.50 101.65 No
100.83 99.50 101.65 No
100.91 99.50 101.65 No
Effluent weir. The effluent weir of the grit chamber provides the hydraulic control point of this process. With a free fall at the weir, critical depth occurs upstream near the weir and it affects the water surface profile upstream if the flow is subcritical. The effluent weir should be designed to keep the velocity below 0.3 m/s (1 ft/s) and to minimize turbulence in the outlet. Hydraulic design example. A schematic diagram of a typical vortex grit tank system is shown in Fig. 22.23. The effluent from the bar screen is pumped to the grit tank influent channel. The influent is distributed to three grit tanks. The hydraulic loading conditions are the same as those for the bar screens. Design hydraulic calculations for the vortex grit tank system is shown in Table 22.9. The head requirements for the sample grit tank system are in the range of 0.30–0.69 m (1.0–2.3 ft). 22.4.2.3 Sedimentation tanks. Process criteria. A typical municipal wastewater treatment system consists of primary sedimentation and secondary (or final) sedimentation tanks. The purpose of both type of sedimentation tanks is to separate the settleable solids from the liquid stream by gravity settling.
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
22.48
Chapter Twenty-Two
The primary sedimentation tank receives the wastewater passed through bar screens and/or grit tanks. The objectives of primary sedimentation are to produce a liquid effluent suitable for downstream biological treatment and to achieve solids separation. The solids result in a sludge that can be conveniently and economically treated before ultimate disposal. On an average basis, the primary sedimentation tank removes approximately 60 and 30 percent of influent total suspended solids (TSS) and 5-day biological oxygen demand (BOD5), respectively. The secondary sedimentation tank receives mixed liquor from the aeration tank. Mixed liquor is a suspended biological growth stream containing microorganisms and treated wastewater. The microorganisms settle with other settleable solids and the clear water is discharged from the sedimentation tank as an effluent. The sedimentation process also thickens the settled solids, a major part of which is returned to the aeration tank and the remainder is wasted as secondary sludge. Sedimentation tank performance is critical for meeting effluent limits for TSS and BOD5. The secondary sedimentation effluents are usually designed to produce 30 mg/L or lower for TSS or BOD5, depending on the effluent requirement. Both primary and secondary sedimentation tanks are commonly arranged in either rectangular or circular shape. Key design parameters include surface overflow rate (SOR), tank water depth, hydraulic detention time, and weir loading rate. Solids loading rate is another important parameter for the secondary sedimentation tank. A properly designed sedimentation tank will provide similar performance for both rectangular and circular shapes. Choice of the shape depends on the site constraints, construction cost, and designer preference. Key hydraulic design parameters. The key hydraulic design parameters for sedimentation tanks include the inlet conditions, inlet channel, inlet flow distribution, inlet baffle, outlet conditions, overflow weir, and effluent launder. Inlet conditions. Inlets should be designed to dissipate the inlet port velocity, distribute flow and solids equally across the cross-sectional area of the tank, and prevent short circuiting in the sedimentation tank. The minimum distance between the inlet and outlet should be 3 m (10 ft) unless the tank includes special provisions to prevent short circuiting. Inlet channel. Inlet channels should be designed to maintain velocities high enough to prevent solids deposition. The minimum channel velocity is typically 0.3 m/s (1 ft/s). Alternatively, inlet channel aeration or water jet nozzles can be designed to prevent solids deposition. Inlet flow distribution. Inlet flow can be distributed by inlet weirs, submerged ports, or orifices with velocities between 0.05 and 0.15 m/s (0.15–0.5 ft/s), and sluice gates or gate valves. Uniform flow to the sedimentation tanks can be achieved by locating inlet ports away from sides, adding partitions or baffles in the inlet zone to redirect the influent, and creating a higher head loss in the inlet ports relative to that in the inlet channel. Alternatively, splitter boxes are used for equally splitting the flow as well as solids contained in the liquid into multiple sedimentation tanks. Inlet baffle. Inlet baffles are designed to dissipate the energy of the inlet velocities. Baffles are usually installed 0.6–0.9 m (2–3 ft) downstream of the inlet port and submerged 0.45–0.6 (1.5–2 ft), depending on tank depth. The top of the baffle should be far enough below the water surface to allow scum to pass over the top. Circular tanks typically have a feed well with a diameter 15 to 20 percent of the tank diameter. The submergence varies depending on the manufacturer. Outlet conditions. Effluent should be uniformly withdrawn to prevent localized high velocity zones and short circuiting. Typically, effluent is withdrawn from a sedimentation
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
Water and Wastewater Treatment Plant Hydraulics 22.49
FIGURE 22.23 Schematic diagram of vortex grit tank system.
TABLE 22.9
Example Hydraulic Calculation of a Typical Vortex Grit Tank System Initial Operation Parameter
1. Wastewater flow rate, Q (m3/s) (mgd) 2. Vortex grit tanks Total number of units Number of units in operation Number of units on standby Flow rate per vortex grit tank in operation (m3/s)
Design Operation
Min Day
Avg.Day
Avg.Day
Max Hour
Peak
1.0 23
1.6 36
2.0 46
3.2 73
3.2 73
3 2 1
3 2 1
3 2 1
3 3 0
3 2 1
0.5
0.8
1.0
1.1
1.6
106.00 105.00 0.8
106.00 105.00 1.0
106.00 105.00 1.1
106.00 105.00 1.6
Control point is located at Point 8 (effluent channel weir) Hydraulic calculations upstream of control point 3. At Point 8 Headloss over sharp-crested weir Sharp-crested weir EL, weir EL (m) Effluent channel bottom EL (m) Flow rate over weir, q (m3/s)
106.00 105.00 0.5
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
22.50
Chapter Twenty-Two
TABLE 22.9
(Continued) Initial Operation Parameter
Min Day
Design Operation
Avg.Day
Avg.Day
Max Hour
Peak
3.00
3.00
3.00
3.00
3.00
0.20 0.20 0.00
0.28 0.28 0.00
0.32 0.32 0.00
0.34 0.34 0.00
0.45 0.45 0.00
106.20
106.28
106.32
106.34
106.45
3.00 105.00 1.20
3.00 105.00 1.28
3.00 105.00 1.32
3.00 105.00 1.34
3.00 105.00 1.45
1.0 0.0010
1.0 0.0022
1.0 0.0032
1.0 0.0036
1.0 0.0069
HGL at Point 7, HGL7 HGL8 Hle7 (m) 106.20
106.28
106.33
106.34
106.45
2.50 105.00 1.28 0.25
2.50 105.00 1.33 0.30
2.50 105.00 1.34 0.32
2.50 105.00 1.45 0.44
10.00 0.013
10.00 0.013
10.00 0.013
10.00 0.013
0.63 0.0002
0.64 0.0003
0.65 0.0003
0.67 0.0006
1.0 0.0032
1.0 0.0046
1.0 0.0051
1.0 0.0099
106.28
106.33
106.35
106.46
1.0 1.5 1.0 1.20
1..0 1.5 1.0 1.28
1.0 1.5 1.0 1.33
1.0 1.5 1.0 1.34
1.0 1.5 1.0 1.45
1.0
1.0
1.0
1.0
1.0
0.33
0.53
0.67
0.71
1.07
Length of weir, L (m) Head over end contracted weir, He (assumed) Headloss, He8 (q/1.84 (L – 0.2He)(2/3) (m) Hle8 – He (must be zero) HGL at Point 8, HGL8 weir EL Hle8 (m) 4. Point 8 to Point 7 Channel width, w7 (m) Channel bottom EL (m) Water depth, h7 (m) Velocity, V 7 (m/s) Exit headloss from channel to effluent weir Exit headloss coefficient Kexit 1.0 Headloss, Hle7 Kexit V72/2g (m)
5. Point 7 to Point 6 Channel width, w6 (m) 2.50 Channel bottom EL (m) 105.00 Water depth, h6 (m) 1.20 Velocity, V V6 (m/s) 0.17 Friction headloss through channel Length of approach channel, L6 (m) 10.00 Manning’s number for concrete channel n 0.013 Hydraulic radius, R6 (h6 w6)/ (2 x h6 w6) (m) 0.61 2 Headloss Hlf6 f [(V6 V n/R / 6 (2/3)] L6(m)0.0001 Fitting headloss through 90º bend Fitting headloss coefficient Kbend 1.0 1.0 Headloss, Hlb6 Kbend V6 V 2/2g(m) 0.0014 HGL at Point 6, HGL6 HGL7 Hlf6 f Hlb6 (m) 106.21 6. Point 6 to Point 5 Headloss through sluice gate Sluice gate headloss coefficient Kgate 1.0 Sluice gate width (m) Sluice gate height (m) Water depth, h5 (m) Sluice gate height or h5, whichever is smaller (m) Velocity through sluice gate, V5 (m/s) V
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
Water and Wastewater Treatment Plant Hydraulics 22.51 TABLE 22.9
(Continued) Initial Operation Parameter
Headloss, Hls5 Kgate V5 V 2/2g (m) HGL at Point 5, HGL5 HGL6 Hls5 (m)
Avg.Day
Avg.Day
Max Hour
0.0057
0.0145
0.0227
0.0258
0.0580
106.21
106.30
106.36
106.37
106.52
2.50 105.20 1.01 0.29
2.50 105.20 1.16 0.35
2.50 105.20 1.17 0.36
2.50 105.20 1.32 0.48
1.0 0.0043
1.0 0.0061
1.0 0.0067
1.0 0.0120
10.00 0.013
10.00 0.013
10.00 0.013
10.00 0.013
0.58 0.0003
0.60 0.0004
0.61 0.0004
0.64 0.0007
106.30
106.36
106.38
106.54
0.06
0.06
0.06
0.06
0.06
106.27
106.36
106.42
106.44
106.60
2.00 105.60 0.67 0.37
2.00 105.60 0.76 0.52
2.00 105.60 0.82 0.61
2.00 105.60 0.84 0.63
2.00 105.60 1.00 0.80
14.00 0.013
14.00 0.013
14.00 0.013
14.00 0.013
14.00 0.013
0.40
0.43
0.45
0.46
0.50
0.0011
0.0020
0.0025
0.0027
0.0039
1.0 1.5 1.0
1..0 1.5 1.0
1.0 1.5 1.0
1.0 1.5 1.0
1.0 1.5 1.0
7. Point 5 to Point 4 Channel width, w4 (m) 2.50 Bottom of channel EL (m) 105.20 Water depth, h4 (m) 1.01 Channel velocity, V V4 (m/s) 0.20 Fitting headloss through a 90º bend Fitting headloss coefficient Kbend 1.0 1.0 Headloss, Hlb4 Kbend V4 V 2/2g (m) 0.0020 Friction headloss through channel Length of channel, L4 (m) 10.00 Manning’s n for concrete channel 0.013 Hydraulic radius, R4 h4 w4/ (2 h4 w4) (m) 0.56 Headloss, Hlf4 f [(V4 V n/R / 4(2/3)]2 L4 (m)0.0001 HGL at Point 4, HGL4 HGL5 Hlb4 Hlf4 f (m) 106.21 8. Point 4 to Point 3 Headloss across vortex grit tank, Hltank (m) (per manufacturer recommendations) HGL at Point 3, HGL3 HGL4 Hltank (m) 9. Point 3 to Point 2 Channel width, w2 (m) Bottom of channel EL (m) Water depth, h2 (m) Channel velocity, V V2 (m/s) Friction headloss through channel Length of approach channel, L2 (m) Manning’s n for concrete channel Hydraulic radius, R2 h2 w2/ (2 h2 w2) (m) 2 Headloss, Hlf2 f [(V2 V n/R / 2(2/3)] L2 (m) Headloss through sluice gate Sluice gate headloss coefficient Kgate 1.0 Sluice gate width (m) Sluice gate height (m)
Design Operation
Min Day
Peak
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
22.52
Chapter Twenty-Two
TABLE 22.9
(Continued) Initial Operation
Design Operation
Parameter
Min Day
Avg.Day
Avg.Day
Max Hour
Peak
Water depth, h2 (m) Sluice gate height or h2, whichever is smaller Velocity through sluice gate,V V2 (m/s) Headloss, Hls2 Kgate V2 V 2/2g (m) HGL at Point 2, HGL2 HGL3 Hlf2 f Hls2 (m)
0.67
0.76
0.82
0.84
1.00
0.67 0.49 0.0125
0.76 0.70 0.0249
0.82 0.81 0.0335
0.84 0.85 0.0364
1.00 1.07 0.0586
106.29
106.39
106.46
106.48
106.66
2.00 105.65 0.64 0.39
2.00 105.65 0.74 0.54
2.00 105.65 0.81 0.62
2.00 105.65 0.83 0.64
2.00 105.65 1.01 0.79
1.0
1.0
1.0
1.0
1.0
0.0078
0.0149
0.0195
0.0210
0.0322
5.00 0.013
5.00 0.013
5.00 0.013
5.00 0.013
5.00 0.013
10. Point 2 to Point 1 Channel width, w1 (m) Bottom of channel EL (m) Water depth, h1 (m) Channel velocity, V1 (m/s) Fitting headloss through a 90º bend Fitting headloss coefficient Kbend 1.0 Headloss, Hlb1 Kbend 2 V1 /2g (m) Friction headloss through channel Length of approach channel, L1 (m) Manning’s n for concrete channel Hydraulic radius, R1 h1 w1/ (2 h1 w1) (m) (2/3)2 Headloss, Hlf1 f (V1 n/R / 1 L1 (m)
0.39
0.43
0.45
0.45
0.50
0.0005
0.0008
0.0009
0.0010
0.0013
(Influent channel may be aerated using diffused air to prevent solids settling or odor problem) HGL at Point 1, HGL1 HGL2 Hlc1 Hlf1 f (m)
106.30
106.41
106.48
106.50
106.69
tank over an effluent weir into a trough and/or effluent channel. Clarifier performance can often be improved by installation of interior baffles. For circular tanks, particularly for secondary sedimentation tanks, a baffle mounted on the wall beneath the effluent weir can deflect solids rising along the wall. Alternatively, mid-radius baffles supported by the sludge removal mechanism are also available. Overflow weir. The overflow weir must be level to promote uniform effluent withdrawal. Weirs may be either straight edged or “V”-notched. “V”-notched weirs have higher headloss, but provide better lateral distribution than straight-edged weirs that are imperfectly leveled. Effluent launder (or trough). Effluent launders may be designed with submerged orifices or free discharge into the collection chamber or channel from which the effluent flows to the effluent pipe. Disadvantage of the submerged launder is that it is not effective
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
Water and Wastewater Treatment Plant Hydraulics 22.53
in varying flow rates. Disadvantage of the free fall launder is potential release of odorous gases. Two principal approaches to weir and launder design are the long-launder and short-launder options. Long launders control the head loss over the weir within a narrow range. In cold regions, fluctuating water levels with short launders would minimize ice attachment to launders and basin walls. Hydraulic design example for primary sedimentation. A schematic diagram of typical circular primary sedimentation tank system is shown in Fig. 22.24. The primary sedimentation tanks receive the grit tank effluent and hydraulic loading conditions are the same as those of the grit tanks. A single primary sedimentation tank is shown for simplicity. Design hydraulic calculations for the primary sedimentation tank system is shown in Table 22.10. Note that the design locates Points 5 and 6 at elevations such that downstream flow conditions will not impact flow conditions in the effluent channel or overflow weir. The head requirements for the sample primary sedimentation tanks are in the range of 1.1–1.5 m (3.6–4.9 ft). Hydraulic design example for secondary sedimentation. A schematic diagram of typical rectangular secondary sedimentation tank system is shown in Figure 22.25. The secondary sedimentation tanks receive flows from the aeration tanks and hydraulic loading conditions are same as those of the aeration tanks. A single secondary sedimentation tank is shown for simplicity. Design hydraulic calculations for the secondary sedimentation tank system is shown in Table 22.11. The head requirements for the sample secondary sedimentation tanks are in the range of 1.6–1.7 m (5.2–6.2 ft). 22.4.2.4 Aeration tanks Process criteria. The most common aerobic suspended growth treatment system for municipal wastewater is the activated sludge system. Wastewater and biological solids (mixed–liquor suspended solids or MLSS) are combined, mixed, and aerated in the aeration tank. The biological MLSS solids take up the organics and nutrients contained in the wastewater and convert them into more biosolids and gaseous by-products. After sufficient time for biological reactions, the mixed liquor is transferred to the following secondary sedimentation tanks where biosolids are separated from the wastewater. The separated wastewater is discharged as an effluent. The separated biosolids are returned to the aeration tank (return activated sludge or RAS) while a predetermined amount of the separated biosolids is wasted as waste activated sludge (WAS). Factors that must be considered in the design of the activated sludge process include loading criteria, selection of reactor type, sludge production, oxygen requirements and transfer, nutrient requirements, environmental requirements, solid-liquid separation, and effluent characteristics. Sizing of aeration basins is based on two key factors: providing sufficient time for oxidation of organics or ammonia nitrogen; and maintaining of a flocculent, well-settling MLSS that can be effectively removed by gravity settling. Solids residence time (SRT) or mean cell residence time (MCRT) is often used to relate substrate removal time requirements to biological growth and biosolids production. Once an SRT is selected, calculation of aeration tank volume requires an estimation of biosolids production and selection of proper MLSS concentration. The selected MLSS concentration along with the solids settling characteristics is important to the final sedimentation tank performance. Therefore, sizing of the aeration tank is always optimized with the final sedimentation tank design. The aeration tank should be provided with sufficient oxygen required for the biological reaction and sufficient power required for thorough mixing of the biomass with the incoming wastewater stream. Although a variety of diffused aeration and mechanical aeration systems are available, diffused aeration systems are more popular in the municipal wastewater treatment. Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
Chapter Twenty-Two
FIGURE 22.24 Schematic diagram of primary sedimentation tank (PST) system.
22.54
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
Water and Wastewater Treatment Plant Hydraulics 22.55
Aeration basin configurations.. Common aeration basins include various process configurations, physical configurations and designs for process selectors. A schematic diagram of a typical rectangular aeration tank system is shown in Fig. 22.26. Process configuration Various aeration process configurations can be used depending on the range of loading conditions, design effluent quality, aeration system design requirements and flexibility of operation. Configurations often encountered include complete mix, plug flow, oxidation ditch, and a combination of these. For smaller plants, oxidation
TABLE 22.10 Tank System
Example Hydraulic Calculations of a Typical Primary Sedimentation Initial Operation Parameter
1. Wastewater flow rate, Q (m3/s) (mgd) 2. Primary sedimentation tanks (PSTs) Total number of units Number of units in operation Number of units on standby Flow rate per PTS in operation, q (m3/s)
Min Day Avg.Day
Design Operation Avg.Day Max Hour
Peak
1.0 23
1.6 36
2.0 46
3.20 73
3.20 73
3 2 1
3 2 1
3 3 0
3 3 0
3 2 1
0.5
0.8
0.7
1.1
1.6
104.46
104.46
104.46
104.46
104.46
0.10
0.10
0.10
0.10
0.10
104.56
104.56
104.56
104.56
104.56
45.0 2 0.50 0.25
45.0 2 0.80 0.40
45.0 2 0.67 0.33
45.0 2 1.07 0.53
45.0 2 1.60 0.80
0.20 1.00
0.20 1.00
0.20 1.00
0.20 1.00
0.20 1.00
69.87
69.87
69.87
69.87
69.87
0.14
0.14
0.14
0.14
0.14
Control points are located at Points 5 and 6 so that back up from down stream does not flood effluent channel or overflow weir. Hydraulic Calculations beginning at Point 7 1. At Point 7 HGL7 must be equal to HGL1 of aeration tank (m) 2. At Point 6 Allowance of 0.10 m from HGL at pipe entrance to bottom of PST effluent trough at discharge end (m) Elevation of PTS trough bottom at discharge end, ELdcb (m) Calculation of water depth in PST effluent trough Tank diameter, Dt (m) Number of channels per tank, nc Total flow through tank, q (m3/s) Flow per channel, qc q/nc (m3/s) Channel slope, Sc, (selected to prevent solids settling) Channel width, w6 (m) Channel length, Lc 3.14 (Dt (w6/2))/nc (m) Change in channel EL, EL dif Sc Lc (m)
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
22.56
Chapter Twenty-Two
TABLE 22.10
(Continued) Initial Operation Parameter
Min Day Avg.Day
Critical depth, yc (qc2/(g w62)0.33 (m) 0.19 Water depth at upstream end of channel, yu 2 (yc)2 (yc (S*L/3) L 2]0.5 (2 Sc L L/3) (m) 0.21 Channel bottom El at upstream end of trough, 104.70 ELucb ELdcb ELdif (m) HGL at trough downstream, HGL6d ELdcb yc (m) HGL at trough upstream, HGL6u ELucb yu (m) 3. Point 6 to Point 5 Allowance to Weir from high trough HGL (m) Weir elevation, Elwe, max. HGL6u allowance (m) Headloss over V V–notch weirs Number of weirs per tank, Nw Tank diameter, Dt, (m) Weir length, Lw (Dt) 3.14 (m) Hydraulic load, So q/Lw / , [(m3·/s)/m] Weir angle, A, (degrees) V-notch height, Vh (m) V-notch width, Vw 2 (TAN(A ( /2) Vh (m) Space between notches, Esv (m) Number of notches per weir, nv Lw/(Ew Esv) Flow per notch, Qcw q/nv (m3/s) Weir coefficient for 90º notch, Cw Water depth over the weir, hle5 (Qcw/Cw)(1/2.48) hle5 < Vh? (If not, need to readjust calculations) HGL at Point 5, HGL5 ELwe hle5 (m) 4. Point 5 to Point 4 Headloss through primary sedimentation tanks Number of tanks, Nt Flow per tank, q (m3/s) Tank diameter, Dt (m) Side water depth, Dsw (m) Tank bottom elevation, ELt HGL5 Dsw (m) Tank floor slope, St (%)
Design Operation Avg.Day Max Hour
Peak
0.26
0.23
0.31
0.41
0.33
0.28
0.42
0.58
104.70
104.70
104.70
104.70
104.75
104.82
104.79
104.87
104.97
104.91
105.03
104.98
105.12
105.28
0.10
0.10
0.10
.010
0.10
105.38
105.38
105.38
105.38
105.38
1 45.00 141.30 0.0035 90.00 0.10
1 45.00 141.30 0.0057 90.00 0.10
1 45.00 141.30 0.0047 90.00 0.10
1 45.00 141.30 0.0075 90.00 0.10
1 45.00 141.30 0.0113 90.00 0.10
0.20 0.03
0.20 0.03
0.20 0.03
0.20 0.03
0.20 0.03
614 0.0008 1.34
614 0.0013 1.34
614 0.0011 1.34
614 0.0017 1.34
614 0.0026 1.34
0.05
0.06
0.06
0.07
0.08
Yes
Yes
Yes
Yes
Yes
105.44
105.45
105.44
105.45
105.47
2 0.50 45.00 4.30
2 0.80 45.00 4.30
3 0.67 45.00 4.30
3 1.07 45.00 4.30
2 1.60 45.00 4.30
101.14 8.33
101.14 8.33
101.14 8.33
101.14 8.33
101.14 8.33
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
Water and Wastewater Treatment Plant Hydraulics 22.57 TABLE 22.10
(Continued) Initial Operation Parameter
Minimum floor tank elevation, ELtf 0.0833 (Dt/2) t EL (m) Headloss through tank, hlt4 t (m) (available from equipment manufacturer) HGL at Point 4, HGL4 HGL5 hlt4 t (m)
Min Day Avg.Day
Design Operation Avg.Day Max Hour
Peak
99.27
99.27
99.27
99.27
99.27
0.05
0.05
0.05
0.05
0.05
105.49
105.50
105.49
105.50
105.52
1.07 6.50
1.07 6.50
1.07 6.50
1.07 6.50
0.89 120 0.27
0.74 120 0.27
1.19 120 0.27
1.78 120 0.27
5. Point 4 to Point 3 Headloss through PST influent pier Pier diameter, Dp 1.07 m 1.07 Pier length, Lp (m) 6.50 Velocity, V3 V Q/(3.14 (Dp/2)2) (m/s) 0.56 Hazen-Williams coefficient, Cp 120 Hydraulic radius, Rp Dp/4 (m) 0.27 Slope, Sp [V3/(0.85 V Cp Rp(0.63)](1/0.54) (%) 0.03 Headloss, Hlf3 f Lp Sp (m) 0.0020 Exit headloss from pier Exit headloss coefficient Kexit 1.0 1 Headloss, hle3 K V3 V 2/2g (m) 0.0158
0.07 0.0047
0.05 0.0033
0.12 0.0079
0.26 0.0168
1 0.0404
1 0.0281
1 0.0719
1 0.1617
HGL at Point 3, HGL3 HGL4 Hlf3 f hle3 (m)
105.54
105.52
105.58
105.69
3
3
3
3
1 1.20 0.80 0.71
1 1.20 0.67 0.59
1 1.20 1.07 0.94
1 1.20 1.60 1.42
120 0.30 70.0
120 0.30 70.0
120 0.30 70.0
120 0.30 70.0
0.04 0.0287
0.03 0.0205
0.07 0.0490
0.15 0.1037
0.05 0.0128
0.05 0.0089
0.05 0.0227
0.05 0.0511
105.58
105.55
105.65
105.85
105.50
6. Point 3 to Point 2 Total number of pipes 3 Number of pipes per primary sedimentation tank 1 Pipe diameter, Dp (m) 1.20 Flow per pipe, q (m3/s) 0.50 Velocity, V V2 0.44 Friction headloss through primary sedimentation tank influent pipe Hazen-Williams coefficient, Cp 120 Hydraulic radius, Rp Dp/4 (m) 0.30 Length of pipe, Lp (m) 70.0 Slope, Sp [V2/(0.85 V Cp Rp(0.63)](1/0.54) (%) 0.02 Headloss, hlf2 f Lp Sp (m) 0.0120 Fitting headloss through two 45º bends Fitting headloss coefficient Kbend 0.5 0.05 Headloss, hlb2 K V2 V 2/2g (m) 0.0050 HGL at Point 2, HGL2 HGL3 hlb2 hlf2 f (m)
105.52
7. At Point 1
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
22.58
Chapter Twenty-Two
TABLE 22.10
(Continued) Initial Operation Parameter
Entrance headloss from primary sedimentation tank influent distribution box to influent pipe Pipe diameter, Dp (m) Flow per pipe, q (m3/s) Velocity, V1 (m/s) Entrance headloss coefficient Kentrance 0.5 Headloss, Hle1 Kentrance V12/2g (m) HGL at Point 1, HGL1 HGL2 Hle1 (m) Allowance to grit tank effluent weir from maximum HGL1, Hall (m) Grit tank effluent elevation, ELgr HGL1 Hall (m)
Min Day Avg.Day
Design Operation Avg.Day Max Hour
Peak
1.20 0.50 0.44
1.20 0.80 0.71
1.20 0.67 0.59
1.20 1.07 0.94
1.20 1.60 1.42
0.50
0.50
0.50
0.50
0.50
0.0050
0.0128
0.0089
0.0227
0.0511
105.52
105.60
105.56
105.68
105.90
0.10
0.10
0.10
0.10
0.10
106.00
106.00
106.00
106.00
106.00
ditches are more popular and for larger plants, plug flow is favored. Various modifications of plug flow systems include conventional, tapered aeration, step aeration, modified aeration, and contact stabilization. Physical configuration. Various physical configurations are used in the aeration tank design, including rectangular, circular, oval, and octagonal shapes. Selector design. Selectors are small compartments for aerobic, anoxic or anaerobic processing usually located in the front end of the aeration tank. The purpose of the selectors is to promote the growth of floc-forming microorganisms by providing a favorable food to microorganisms (F:M) ratio while suppressing filamentous growth. Typically selectors are designed with low HRTs and high F:M ratio. Key hydraulic design parameters. The key hydraulic design parameters for aeration tanks include the distribution box, inlet channel, inlet flow distribution, inlet baffles, aeration equipment, RAS, effluent weir, and effluent channel. Distribution box. Sluice gates, weirs, gate valves or orifices installed in a distribution box are often used to distribute the upstream flow to multiple aeration tanks and to a secondary treatment bypass line. Design should provide the desired rate of flow distribution at all flow conditions with minimum headloss. Provisions to minimize solids deposition in the distribution box and appurtenances should be considered. Inlet channel. Inlet channels should be designed to maintain velocities high enough to prevent solids deposition but low enough to minimize headloss. A velocity of 0.3 m/s (1 ft/s) is typically used to keep organic solids in suspension. Alternatively, inlet channel aeration with diffused air, fed at a rate of 0.5–0.8 m3/min (20–30 scfm), is often used. Inlet flow distribution. Inlet flow can be distributed by inlet weirs, submerged ports or orifices, and sluice gates or gate valves. Return activated sludge may be introduced prior Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
FIGURE 22.25 Schematic diagram of final sedimentation tank.
Water and Wastewater Treatment Plant Hydraulics 22.59
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
22.60
Chapter Twenty-Two
TABLE 22.11
Example Hydraulic Calculation of a Typical Final Sedimentation Tank System Initial Operation PARAMETER
1. Wastewater flow rate, Q (m3/s) (mgd) RAS flow, Qras (% of average day flow) RAS flow, Qras /100, (m3/s) Final sedimentation tank influent flow, Qin, (m3/s) Final sedimentation tank effluent flow, Qeff, f (m3/s) Final sedimentation tanks Total number of units Number of units in operation Number of units on standby Tank width (m) Influent per operating tank, qin, (m3/s) Effluent per operating tank, qeff, f (m3/s) 2. Select control point at Point 3 (where effluent wiers are located) Hydraulic calculations downstream of control point At Point 3 V-notch weir Number per tank, Nw Individual weir length, Lw (m) Total weir length, Lwt Lw Nw (m) Weir angle, A degrees V-notch height, Vh (m) V-notch width, Vw 2 (TAN( N(A/2) Vh (m) Space between notches, Esv (m) Total number of notches per tank, nv Lwt/( t Vw Esv) Flow per notch, Qcw qeff/ f nv Weir coefficient for 90º notch, Cw Water depth over the weir, hle3 (Qcw/Cw)(1/2.48) (m) hle3 Vh? (If not, need to readjust calculations) Weir EL (m) (Select weir elevation so that HGL1 equals aeration tank’s HGL6) EGL at Point 3, EGL3 Weir EL hle3 (m) Velocity head, HV3 0 (assume V3 V 0) (m)
Min Day Avg.Day
Design Operation Avg.Day Max Hour
Peak
1.0 23
1.6 36
2.0 46
3.2 73
3.2 73
20
50
50
100
100
0.32
0.80
1.00
2.00
2.00
1.32
2.40
3.00
5.20
5.20
1.00
1.60
2.00
3.20
3.20
4 3 1 16
4 3 1 16
4 3 1 16
4 4 0 16
4 3 1 16
0.44
0.80
1.00
1.30
1.73
0.33
0.53
0.67
0.80
1.07
20 7.0 140.0 90.0 0.10
20 7.0 140.0 90.0 0.10
20 7.0 140.0 90.0 0.10
20 7.0 140.0 90.0 0.10
20 7.0 140.0 90.0 0.10
0.20 0.03
0.20 0.03
0.20 0.03
0.20 0.03
0.20 0.03
608 0.0005 1.34
608 0.0009 1.34
608 0.0011 1.34
608 0.0013 1.34
608 0.0018 1.34
0.04
0.05
0.06
0.06
0.07
yes
yes
yes
yes
yes
103.37
103.37
103.37
103.37
103.37
103.41
103.42
103.43
103.43
103.44
0.00
0.00
0.00
0.00
0.00
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
Water and Wastewater Treatment Plant Hydraulics 22.61 TABLE 22.11
(Continued) Initial Operation PARAMETER
HGL at point 3, HGL3 Weir EL hle3 (m) 3. Point 3 to Point 4 Effluent troughs Number of troughs, nt Flow per trough, qt qeff/ f nt (m3/s) Trough slope, St (%) (select to prevent solids settling) Trough width, w6 (m) Approximate trough length, Lt (m) Change in trough EL due to slope difEL4 St* Lt (m) Critical depth at downstream end, yc (qt2/(gw62)0.33 (m) Water depth at upstream end of trough for free fall from trough into final effluent channel yu4 [2 (yc)2 (yc (S*L/3) L 2].5 (2 S L L/3) (m) Max water EL downstream of weir (occurring at max. hourly flow with one tank out of service) Elmax4 weir EL 0.1 (m) (see Point 3 for weirEL) Trough bottom EL at upstream end of trough, TbuEL4 (m) Tbu EL4 EL max4 yu for max hour flow with one tank out of service HGL at upstream end, HGL4u Tbu EL4 yu4 (m) Velocity head, HV4 V u0 (assume V 0) (m) EGL at upstream end, EGL4u HGL4u HV4 V u (m) Trough bottom EL at downstream end of trough Tbd EL4 Tbu EL4 dif EL4 (m) HGL at point 4, HGL4 TbdEL4 yc (m) Velocity head, HV4 V d Vc2/2g (m)
Min Day Avg.Day
Initial Operation
Avg.Day Max Hour
Peak
103.41
103.42
103.43
103.43
103.44
10 0.03
10 0.05
10 0.07
10 0.08
10 0.11
0.20 0.5 7.0
0.20 0.5 7.0
0.20 0.5 7.0
0.20 0.5 7.0
0.20 0.5 7.0
0.01
0.01
0.01
0.01
0.01
0.08
0.11
0.12
0.14
0.17
0.12
0.17
0.20
0.23
0.28
103.27
102.99
102.99
102.99
102.99
102.99
103.11
103.16
103.19
103.22
103.27
0.00
0.00
0.00
0.00
0.00
103.11
103.16
103.19
103.22
103.27
102.97
102.97
102.97
102.97
102.97
103.05 0.04
103.08 0.05
103.10 0.06
103.11 0.07
103.14 0.08
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
22.62
Chapter Twenty-Two
TABLE 22.11
(Continued) Initial Operation PARAMETER
EGL at downstream end, EGL4d HGL4d HV4 V d (m)
Min Day Avg.Day
Design Operation
Avg.Day Max Hour
Peak
103.09
103.13
103.16
103.18
103.22
102.87
102.87
102.87
102.87
102.87
HGL maximum at Point 5, HGL5ELmax5(m)102.87 Velocity head, HV5 V 0 (assume V 0) (m) 0.00 EGL maximum at Point 5, EGL5m HGL5m HV5 V (m) 102.87
102.87
102.87
102.87
102.87
0.00
0.00
0.00
0.00
102.87
102.87
102.87
102.87
1.00
1.60
2.00
3.20
3.20
0.20 3.0 64.0
0.20 3.0 64.0
0.20 3.0 64.0
0.20 3.0 64.0
0.20 3.0 64.0
0.13
0.13
0.13
0.13
0.13
0.23 0.29
0.31 0.43
0.36 0.52
0.49 0.74
0.49 0.74
102.13
102.13
102.13
102.13
102.13
102.42
102.56
102.65
102.87
102.87
0.00
0.00
0.00
0.00
0.00
4. Point 4 to Point 5 Effluent channel upstream Max. water surface level at upstream end of effluent channel, ELmax5 TbdEL4 0.1 (m)
5. Point 5 to Point 6 Effluent channel downstream Flow through channel, Qeff (m3/s) Channel slope, Sc (%) (select to prevent solids settling) Channel width, w6 (m) Approximate channel length, Lch (m) Change in channel EL, difEL6 Sc Lch (m) Critical depth, yc (q2/(gw62)0.33 (m) Water depth at upstream end of channel, yu6 [2 (yc)2 (yc (S L/3) L 2].5 (2 S L L/3) (m) Channel bottom EL at upstream end of channel, cbuEL6 HGL5- maximum yu6 (m) HGL at upstream end of channel, HGL5 cbuEL6 yu6 (m) Velocity head, HV5 V 0 (assume V 0) (m) EGL at upstream end of channel, EGL5 HGL5 HV5 V (m)
102.42
102.56
102.65
102.87
102.87
Channel bottom EL at downstream end of channel, cbdEL6 cbuEL6 difEL6 (m)
102.00
102.00
102.00
102.00
102.00
HGL at Point 6, HGL6 cbdEL6 yc (m) Velocity head, HV6 V Vc2/2g (m)
102.23 0.11
102.31 0.15
102.36 0.17
102.50 0.24
102.50 0.24
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
Water and Wastewater Treatment Plant Hydraulics 22.63 TABLE 22.11
(Continued) Initial Operation PARAMETER
EGL at Point 6, EGL6 HGL6 HV6 V (m) 6. At Point 7 Max water EL downstream of channel end free-fall HGL at Point 7, HGL7 cbdEL6 0.1 (m) (This must be the same as maximum elevation at Point 1 of multimedia filter)
Min Day Avg.Day
Design Operation
Avg.Day Max Hour
Peak
102.34
102.47
102.54
102.74
102.74
101.90
101.90
101.90
101.90
101.90
3
3
3
4
3
0.44 16.0 120.0
0.80 16.0 120.0
1.00 16.0 120.0
1.30 16.0 120.0
1.73 16.0 120.0
99.2 4.24
99.2 4.25
99.2 4.26
99.2 4.26
99.2 4.27
0.0
0.0
0.0
0.0
0.0
103.41
103.42
103.43
103.43
103.44
0.00
0.00
0.00
0.00
0.00
103.41
103.42
103.43
103.43
103.44
1.0 1.0 1.0 4
1.0 1.0 1.0 4
1.0 1.0 1.0 4
1.0 1.0 1.0 4
1.0 1.0 1.0 4
0.11
0.20
0.25
0.33
0.43
0.21 1.0
0.31 1.0
0.36 0.9
0.44 0.9
0.53 0.9
0.16
0.24
0.27
0.33
0.40
0.17
0.31
0.38
0.49
0.66
0.11
0.20
0.25
0.33
0.44
Hydraulic Calculations Upstream of Control Point 7. At Point 2 Final sedimentation tanks (Gould type) Number of tanks in operation, nt Flow per tank upstream of sludge collection, qin (m3/s) Tank width, Wt (m) Tank length, Lt (m) Tank bottom elevation at influent end (m) Side water depth (m) Assume friction losses, Hlf2, f through tank are negligible EGL at Point 2, EGL2 EGL3 Hlf2 f (m) Velocity head, HV2 V 0 (assume V 0) (m) HGL at Point 2, HGL2 EGL3 HV2 V (m) 8. Point 2 to Point 1 Tank influent sluice gates Height (m) Width, Ws (m) Area (m2) Number of sluice gates per tank, Nsg Flow per sluice gate, qsg qin/Nsg / (m3/s) Upstream head over weir, Du (select so Qsub qsg D) (m) Effective sluice gate width, Ws' Ws (0.1)(2 contractions)(Dd) (m) Downstream head over weir, Dd (qsg/1.84/Ws')(2/3) (m) Free–fall flow, Qfree 1.84 Ws' Du(3/2), (m3/s) Submerged flow, Qsub Qfree (1 (Dd/ d/Du)3/2)0.385 (m3/s)
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
22.64
Chapter Twenty-Two
TABLE 22.11
(Continued) Initial Operation PARAMETER
Difference, (Qsub qsg) (m3/s) (should be zero) Head difference between tank and channel, Hl 1 Du Dd (m) Top of sluice gate set elevation, Els HGL2 Dd (m) HGL at Point 1 (upstream of sluice gate), HGL1 HGL2 Hl1 (m) Velocity head, HV1 0 (assume V 0) (m) EGL at Point 1, EGL1 HGL1 HV1 (m) Maximum HGL1 (m)
Min Day Avg.Day
Design Operation
Avg.Day Max Hour
Peak
0.00
0.00
0.00
0.00
0.00
0.051
0.077
0.090
0.106
0.130
103.26
103.19
103.15
103.10
103.04
103.46
103.50
103.52
103.54
103.57
0.00 103.46
0.00 103.50
0.00 103.52
0.00 103.54
0.00 103.57 103.57
Max HGL1 should equal HGL6 for aeration tank
to or after the inlet flow distribution. Good mixing should be provided to promote uniform distribution of the influent flow and RAS flow. Wastewater flow split inlet design with a relatively high headloss is often used to provide reasonably equal distribution of flow to multiple aeration tanks or to multiple inlets in each aeration tank operating in a step feed mode. Sometimes influent distribution piping which is extended to and having an inlet port at each step feed point is used. Inlet baffles. Depending on the aeration tank configuration, inlet baffles are used to dissipate the energy from the inlet velocities. Inlet baffles are designed to direct uniform distribution of MLSS along the width of the aeration tank. Aeration equipment. Diffused aeration systems are predominantly used in the municipal treatment plants. Although the air bubbles dispersed in the wastewater occupy approximately 1 percent of the volume, no allowance is made in aeration tank sizing. The volume occupied by submerged piping and diffusers is usually negligible. If spiral-flow mixing with coarse bubble diffusers is used, the width-to-depth ratios vary from 1:1 to 2.2:1. The tank depth, most commonly 4–5 m (13–16 ft), is usually determined by desired oxygen transfer efficiency of various aeration equipment. Freeboard from 0.3 to 0.6 m (1 to 2 ft) above the water surface is normally provided. If surface mechanical aerators are used, a freeboard of more than 0.6 m (2 ft) may be required depending on the power input for the aeration and mixing. Freezing during the winter due to the mist should also be considered in the design. Return activated sludge (RAS). The rate of RAS is normally 30 to 50 percent of the wastewater flow. Peak rate of RAS may go up to 100 percent of the wastewater flow for large plants and up to 150 percent of the wastewater flow for small plants. Design should provide adequate mixing, hydraulic capacity, and uniform distribution where RAS is introduced to the incoming wastewater. Effluent weir. The effluent weir provides a fixed control elevation of hydraulics in the aeration tank. Sometimes effluent ports instead of effluent weir are used to minimize headloss.
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
Water and Wastewater Treatment Plant Hydraulics 22.65
FIGURE 22.26 Schematic diagram of aeration tank system. (AT = aeration tank; PST = primary sedimentation tank).
Effluent channel. The design considerations described in the inlet channel also apply to the design of the effluent channel. Often the effluent channel from the aeration tanks is the same as the influent to the final sedimentation tanks. Hydraulic design example. The aeration tanks receive the primary sedimentation tank effluent and hydraulic loading conditions are the same as those of the primary sedimentation tanks. Design hydraulic calculations for the aeration tank system is shown in Table 22.12. The head requirements for the sample aeration tanks are in the range of 0.4–1.0 m (1.3–3.3 ft). 22.4.2.5 Granular media filter. Process criteria. Granular media filtration is usually used where the plant suspended solids effluent limit is equal to or less than 10 mg/L. It may also be applied following secondary biological treatment to remove particulate carbonaceous BOD5 and residual insolubilized phosphorus. The degree of suspended solids
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
22.66
Chapter Twenty-Two
removal when filtering secondary effluents without the use of chemical coagulation depends on the degree of bioflocculation achieved during secondary treatment. The presence of significant amounts of algae impedes filtration of lagoon effluents. Pretreatment with a coagulant is considered to be a good practice for such cases. There are many types of proprietary granular filters available. However, granular media filters are generally classified according to direction of flow, type, and number of media comprising the bed, the driving force, and method of flow control. Most wastewater filters are downflow units while some proprietary filters use various combinations of upflow and downflow. The driving force for filtration may be either gravity or pressure. Gravity filters are commonly used in large municipal treatment plants while pressure filters are often used in smaller plants. Gravity filters are generally sized for a filtration rate of 1.4–4 L/(m2ⴢs)/ (2–6 gal/(ft2ⴢmin) and terminal headlosses of 2.4–3.0 m (8–10 ft). Multiple units are used to allow continuous filtration during backwash or maintenance. Typical length to width ratio of gravity filters vary from 1:1 to 4:1. Key hydraulic design parameters. The key hydraulic design parameters for granular media filters include headlosses, filter operation, collection and distribution systems, and backwash requirements. Head losses. The head losses includes the losses associated with piping, valves, meters, bends, constrictions, filter media, underdrains, and collection systems. All losses vary with the square of the velocity. Clean water headloss for the filter media is influenced by media type, size, uniformity, and depth. As filtration rate increases within the terminal head loss range, less headloss capacity is available for solids storage. The head required for the filter is the sum of all headlosses including the terminal head loss of the filter media. If sufficient head is not available, pumping of filter influent is required. Filter operation. Three basic methods of filter operation are constant pressure, constant rate and variable declining rate. The constant pressure system requires a large upstream storage and is seldom used with gravity filters. The constant rate system requires a relatively costly rate control system and true constant-rate filtration is seldom used. In declining-rate filtration, the filtration rate may be kept constant using influent or effluent control weirs during the initial period of operation and, thereafter, declining rate of filtration. Generally, declining-rate filters are the best mode of gravity filter operation unless the design terminal headloss exceeds 3 m. Collection and distribution systems. (underdrain). In conventional downflow filters, the underdrain system serves to both collect the filtrate and distribute the backwash water. Traditional systems using gravel layers with perforated pipe are no longer commonly used. More popular underdrain materials include precast channels, poured-in-place concrete, or steel pipe with built-in nozzles and orifices. Porous plates made of aluminum oxide or stainless steel are also available but they are susceptible to clogging. Backwash requirements. Backwash is the cleaning of the filter by reversing the flow through the filter media at a controlled flow rate. Backwashing causes an expansion of the bed, normally no more than 10 percent of the depth, by allowing abrasive action among particles. The quantity of backwash water will generally be about 3000–4000 L/m2 (75–100 gal/ft2). Bachwashed water is collected in the wash-trough which is located about 0.9 m (3 ft) above the filter media. Biological solids in secondary effluent are strongly attached to the media and air scour before or during backwash is often required to promote successful cleaning. Air requirements for the air scour are on the order of 0.015–0.025 (m3/m2)/s [3–5 (ft3Ⲑft Ⲑ 2)/min].
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
Water and Wastewater Treatment Plant Hydraulics 22.67 TABLE 22.12
Example Hydraulic Calculation of a Typical Final Aeration Tank System Initial Operation PARAMETER
1. Wastewater flow rate, Q (m3/s) (mgd) RAS flow. (% of average flow) (added downstream of aeration tank influent sluice gates) RAS flow, Qras (m3/s) 2. Aeration tanks Total of nunber of units Number of units in operation Number of units on standby Flow rate per aeration tank in operation, q (m3/s) Flow rate per aeration tank in operation including RAS flow (downstream of influent sluice gate), qras (m3/s)
Design Operation
Min Day Avg. Day Avg. Day Max Hour
Peak
1.00 23
1.60 36
2.00 46
3.20 73
3.20 73
20
50
50
100
100
0.32
0.80
1.00
2.00
2.00
3 2 1
3 2 1
3 3 0
3 3 0
3 2 1
0.50
0.80
0.67
1.07
1.60
0.66
1.20
1.00
1.73
2.60
103.57
103.57
103.57
103.57
103.57
103.67 100.67 0.66 6.00 0.15
103.67 100.67 1.20 6.00 0.23
103.67 100.67 1.00 6.00 0.20
103.67 100.67 1.73 6.00 0.29
103.67 100.67 2.60 6.00 0.38
103.82
103.90
103.87
103.96
104.05
0.03
0.04
0.03
0.05
0.07
103.85
103.94
103.91
104.01
104.12
0.66 6.0 60.0
1.20 6.0 60.0
1.00 6.0 60.0
1.73 6.0 60.0
2.60 6.0 60.0
97.87
97.87
97.87
97.87
97.87
5.95 5
6.03 5
6.00 5
6.09 5
6.18 5
Control point is located at Point 5 (aeration tank effluent weir). 3. At Point 6 Set maximum HGL6 effluent weir elevation 0.10 (m) Hydraulic Calculations Upstream of Control Point 4. Point 6 to Point 5 Headloss over sharp-crested weir Sharp-crested weir EL (m) Effluent channel bottom EL (m) Flow rate over weir, qras (m3/s) Length of weir L (m) Headloss, Hle5 (q/1.84L)(2/3) (m) HGL at Point 5, HGL5 weir EL Hle5 (m) Velocity head, HV5 V (qras/ Wp/Hle / 5)2/2g (m) EGL at Point 5, EGL5 HGL5 HV5 V (m) 5. Point 5 to Point 4 Flow rate per aeration tank in operation, qras (m3/s) Pass width, Wp (m) Tank length, Lt (m) Tank bottom elevation, ELtb Avg. day WSEL - 6 (m) Water depth in tank at design average flow, Dt (m) Number of passes per tank, Np
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
22.68
Chapter Twenty-Two
TABLE 22.12
(Continued) Initial Operation PARAMETER
Effective length of tank, L (Lt)(Np) (m) 300.0 Velocity, V V4 (m/s) 0.02 Critical depth, yc ((q2/g / / Wp2)(0.333) (m) 0.11 Friction headloss through aeration tank channel Manning’s number for concrete channel n 0.013 Hydraulic radius, R (Dt Wp)/ (2 Dt Wp) (m) 1.99 Headloss, Hlf4 f (V4 V n/R / (2/3))2 L (m) 0.0000 Fitting headloss through 90º Fitting headloss coefficient Kbend 1.0 Number of bends, Nb 8 Headloss, Hlb4 Kbend* V4 V 2/2gNb (m) 0.0001 Velocity head, Hvsd (see below at Point 3) 0.07 EGL at Point 4, EGL4 EGL5 Hlf4 f Hvsd (m) Velocity head, Hvsd (see below at Point 3) HGL at Point 4, HGL4 EGL4 HV4 V (m) 6. Point 4 to Point 3 Headloss over aeration tank influent sluice gates Sluice gate width, Ws (m) Sluice gate height (m) Flow per sluice gate, q (m3/s) Upstream head over weir, Du (select so Zsub q 0) (m) Effective sluice gate width, Ws' Ws (0.1)(2 contractions) (Du) (m) Downstream head over weir, Dd (q/1.84/Ws') (2/3) (m) Free–fall flow, Qfree 1.84 Ws'Du(3/2), (m3s) Submerged flow, Qsub Qfree (1 (Dd/ d/Du)3/2)0.385, (m3/s)
Design Operation
Min Day Avg. Day Avg. Day Max Hour
Peak
300.0 0.03
300.0 0.03
300.0 0.05
300.0 0.07
0.16
0.14
0.20
0.27
0.013
0.013
0.013
0.013
2.00
2.00
2.01
2.02
0.0000
0.0000
0.0000
0.0001
1.0 8
1.0 8
1.0 8
8
0.0004
0.0003
0.0009
0.0020
0.10
0.08
0.12
0.16
103.92
104.03
103.99
104.13
104.28
0.07
0.10
0.08
0.12
0.16
103.85
103.94
103.91
104.01
104.12
1.2 1.0 0.50
1.2 1.0 0.80
1.2 1.0 0.67
1.2 1.0 1.07
1.2 1.0 1.60
0.52
0.73
0.64
0.91
1.24
1.10
1.05
1.07
1.02
0.95
0.39
0.55
0.49
0.69
0.94
0.76
1.21
1.01
1.62
2.43
0.50
0.80
0.67
1.07
1.60
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
Water and Wastewater Treatment Plant Hydraulics 22.69 TABLE 22.12
(Continued) Initial Operation PARAMETER
Design Operation
Min Day Avg. Day Avg. Day Max Hour
Difference, (Qsub q) (m3/s) (should be zero) 0.00 Head difference between tank and channel, Hl4 Du Dd (m) 0.13 Velocity head downstream of sluice gate, HVsd (q/Ws'/Dd / )2/2g, (m) 0.07 Velocity head upstream of sluice gate, HVsu (q/Ws'/'/Du)2/2g (m) 0.04 Top of sluice gate elevation, Els HGL4 Dd (m) 103.45 HGL upstream of sluice gate, HGLsu HGL4 Hl4 (m) 103.98 EGL upstream of sluice gate, EGLsu HGLsu HVsu (m) 104.02 Friction headloss through influent channel to tank #3 Average length of influent channel per tank, L3 31.5 Np Wp 3 tanks1/2 (m) Influent channel width, W W3 (m) 4.0 Manning’s number n for concrete channel n 0.013 Influent channel bottom elevation, Elb avg. EGLsu 3 (m) 101.1 Water depth in influent channel, h3 HGLs Elb (m) 2.87 Hydraulic radius, R (h3 w3)/ (2 h3 w3) (m) 1.18 Velocity, V3 V q/w3/h3 (m/s) 0.04 Headloss, Hlf3 f (V3 V n/R / (2/3))2 L3 (m) 0.0000 Friction headloss through influent channel to tank #2 Flow rate, q2 2 q (m3/s) 1.00 Velocity, V2 V q/w2/h2 (m/s) 0.09 Headloss, Hlf2 f (V2 V n/R / (2/3))2 L3 (m) 0.0000 Friction headloss through influent channel to tank #1 Flow rate, q1 3 q (m3/s) 1.00 Velocity, V1 q/w1/h1 (m/s) 0.09 Headloss, Hlf1 f (V1 n/R / (2/3))2 L3 (m) 0.0000
Peak
0.00
0.00
0.00
0.00
0.18
0.16
0.22
0.30
0.10
0.08
0.12
0.16
0.05
0.05
0.07
0.09
103.38
103.42
103.33
103.18
104.12
104.06
104.23
104.42
104.17
104.11
104.30
104.51
31.5
31.5
31.5
31.5
4.0 0.013
4.0 0.013
4.0 0.013
4.0 0.013
101.1
101.1
101.1
101.1
3.00
2.95
3.12
3.31
1.20 0.07
1.19 0.06
1.22 0.09
1.25 0.12
0.0000
0.0000
0.0000
0.0001
1.60 0.13
1.33 0.11
2.13 0.17
3.20 0.24
0.0001
0.0001
0.0001
0.0002
1.60 0.13
2.00 0.17
3.20 0.26
3.20 0.24
0.0001
0.0001
0.0003
0.0002
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
22.70
Chapter Twenty-Two
TABLE 22.12
(Continued) Initial Operation PARAMETER
HGL at Point 3, HGL3 HGLs Hlf3 f Hlf2 f Hlf1 f (m) 7. Point 3 to Point 2 Headloss through sluice gate Sluice gate headloss coefficient Kgate Sluice gate width, W W2 (m) Sluice gate height, Hg (m) Channel water depth, Dc (m) Gate opening depth, Hg or Dc, whichever is smaller (m) Velocity through sluice gate, V Q/W2 V5 W (m/s) Headloss, Hls2 Kgate V 2/2g (m) V5 HGL at Point 2, HGL2 HGL3 Hls2 (m)
Design Operation
Min Day Avg. Day Avg. Day Max Hour
Peak
103.98
104.12
104.06
104.23
104.42
1.0 1.80 1.80 2.87
1.0 1.80 1.80 3.00
1.0 1.80 1.80 2.95
1.0 1.80 1.80 3.12
1.0 1.80 1.80 3.31
1.80
1.80
1.80
1.80
1.80
0.31
0.49
0.62
0.99
0.99
0.0049
0.0124
0.0194
0.0498
0.0498
103.98
104.13
104.08
104.28
104.47
2.00 1.60 0.51 1.0
2.00 2.00 0.64 1.0
2.00 3.20 1.02 1.0
2.00 3.20 1.02 1.0
0.0132
0.0207
0.0529
0.0529
1.60 2.00 0.51 120.00 0.50 50.00
2.00 2.00 0.64 120.00 0.50 50.00
3.20 2.00 1.02 120.00 0.50 50.00
3.20 2.00 1.02 120.00 0.50 50.00
0.0001 0.0061
0.0002 0.0093
0.0004 0.0221
0.0004 0.0221
0.80 1.50 0.45 120.00 0.38 50.00
0.67 1.50 0.38 120.00 0.38 50.00
1.07 1.50 0.61 120.00 0.38 50.00
1.60 1.50 0.91 120.00 0.38 50.00
0.0001
0.0001
0.0002
0.0005
8. Point 2 to Point 1 Exit headloss from primary sed. tank effluent pipe to aeration tank influent channel Primary effluent pipe diameter, Dp (m) 2.00 All PST effluent flow, Q (m3/s) 1.00 Velocity, V1 (m/s) 0.32 Exit headloss coefficient Kexit 1.0 Exit headloss, hle1 (V12)/ 2g Kexit (m) 0.0052 Friction headloss through PST effluent pipe section 2 Flow per pipe, q (m3/s) 1.00 Pipe diameter, (Dp2) (m) 2.00 Velocity, V12 (m/s) 0.32 Hazen-Williams coefficient, Cp 120.00 Hydraulic radius, Rp2 (Dp2)/4 (m) 0.50 Length of pipe, Lp2 (m) 50.00 Slope Sp2[V12/(0.85CpRp2(0.63)](1/0.54) (%) 0.0001 Headloss, hlf2 f Lp2 Sp2 (m) 0.0026 Friction headloss through PST effluent pipe section 1 Flow per pipe, q (m3/s) 0.50 Pipe diameter, Dp1 (m) 1.50 Velocity, V11 (m/s) 0.28 Hazen-Williams coefficient, Cp 120.00 Hydraulic radius, Rp1 (Dp1)/4 (m) 0.38 Length of pipe, Lp1 (m) 50.00 Slope,Sp1[V11/(0.85CpRp1(0.63)](1/0.54) (%) 0.0001
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
Water and Wastewater Treatment Plant Hydraulics 22.71 TABLE 22.12
(Continued) Initial Operation PARAMETER
Pipe entrance headloss Ke Headloss, hen1 Ke V112/2g (m) HGL at upstream of PST effluent pipe, HGL1 HGL2 hle1 hlf2 f hlf1 f hen1 (m) HGL7 of PST must be maximum of HGL1 (m)
Min Day Avg Day
Design Operation Avg Day Max Hour
Peak
0.50 0.0020
0.50 0.0052
0.50 0.0037
0.50 0.0094
0.50 0.0209
103.99
104.16
104.12
104.38
104.59
104.59
104.59
104.59
104.59
104.59
Hydraulic design example. A schematic diagram of a typical granular media filter system is shown in Fig. 22.27. The granular media filters receive the secondary effluent either before or after chlorination and hydraulic loading conditions are the same as those of the secondary effluent. A single granular media filter is shown for simplicity. Design hydraulic calculations for the granular media filter system is shown in Table 22.13. The head requirements for the granular media filters are in the range of 2.8–3.2 m (9.3–10.6 ft). 22.4.2.6 Mixing and contact chambers. Process criteria. Physical and chemical wastewater treatment processes involve mixing, coagulation, flocculation, and sedimentation. Chemical coagulation is often used for enhanced treatment in primary sedimentation and for tertiary treatment after secondary treatment, and before or after filtration. Advantages of coagulation include greater removal efficiencies of total suspended solids, organic materials, phosphorus, and other pollutants. Disadvantages include an increased production of chemical sludge and an increased operating cost. Chemical coagulants are mixed with wastewater during rapid mix which is the first step of the coagulation process. The coagulants destabilize the colloidal particles which allows their agglomeration. Velocity gradients (G) or a mixing intensity of 300 (mⲐ mⲐm)/s are generally sufficient for rapid mix. The rapid mix can be accomplished with mechanical mixers, in-line blenders, pumps, or air mixers. Following the rapid mixing, flocculation takes place through gentle prolonged mixing which promotes the destabilized particles to grow and agglomerate. Typical detention times for flocculation range between 20 and 30 minutes. During this period, velocity gradients of 50–80 (mⲐ mⲐm)/s should be maintained. Following flocculation, the settleable solids are settled in the following sedimentation tank. Key hydraulic design parameters. The key hydraulic design parameters for mixing and contact chambers include the inlet channel, inlet baffles, mixing equipment, and outlet channel Inlet channel. Inlet channels should be designed to maintain velocities high enough to prevent solids deposition and to promote equal distribution of flow if multiple tanks are used. Inlet baffles. Inlet baffles should be designed to dissipate the energy from the velocities and to prevent short circuiting.
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
22.72
Chapter Twenty-Two
Mixing equipment. A sufficient freeboard should be provided to prevent liquid spillage over the walls due to intense mixing. Provisions for easily removing the mixing equipment for repair and maintenance should be considered. Tank geometry should be configured to minimize areas with inadequate mixing. Outlet channel. Velocity in the outlet channel which leads to the sedimentation tank should be high enough to prevent solids from settling but not too high to cause breakdown of flocculated solids. 22.4.2.7 Cascade aerators Process criteria. Cascade aeration is a physical unit process typically used for effluent aeration. The system employs a series of steps or weirs over which the effluent is discharged. The system is configured to maximize turbulence in order to increase oxygen transfer. The head requirements vary depending on the initial dissolved oxygen (DO) and the desired final DO. If the necessary head is not available, effluent pumping or mechanical aeration is required. Although cascade aeration is not a new concept, its application to wastewater treatment is relatively new. Design criteria for an efficient cascade aeration system design include a fall height at each step equal to or less than 1.2 m (4 ft); a flow rate equal to or less than 235 (m3Ⲑh)/m Ⲑ [315(gal/min)/ft] of width; and a pool depth after each fall equal to or less than 0.28 m (0.9 ft). Hydraulic design example. A schematic diagram of a typical cascade aeration system is shown in Figure 22.28. Cascade aerators normally receive the secondary treatment effluent and hydraulic loading conditions are the same as those of the secondary treatment effluent. Design hydraulic calculations for the cascade aeration system is shown in Table 22.14. The head requirements for this example of the cascade aerators is 4.6 m (15.1 ft). 22.4.2.8 Effluent outfall. Process Criteria. The treatment plant accomplishes as much pollutant removal as required to produce effluent meeting the criteria established by the regulatory agencies. Ultimate disposal of wastewater effluents are by dilution in receiving waters, by discharge on land, seepage into the ground, or reclamation and reuse. Of these, disposal into the receiving waters is the most common practice. The receiving waters include rivers, lakes, estuaries, and oceans. The outfall size is determined by the velocity, headloss, structural considerations, and the economics of the situation. Velocities of 0.6–0.9 m/s (2–3 ft/s) at average flow are normally recommended in pipeline design to avoid excessive head loss. If the effluent received preliminary treatment, lower velocities can be used. However, velocities higher than 2.4–3.0 m/s (8–10 ft/s) should be avoided due to excessive headloss. Key hydraulic design parameters. The key hydraulic design parameters for effluent outfalls include available head, mixing and dispersion, submerged discharge, and diffusers. Available head. Sufficient head for gravity flow from the point of plant effluent discharge to the receiving stream is not always possible. If sufficient head is not available, effluent pumping is required to prevent flooding of the plant area. Some plants require effluent pumping during storm events or where tidal waves cause salt water intrusion. Mixing and dispersion. The outfall should be designed to operate at an adequate velocity to promote rapid dispersion and mixing of the effluent with the receiving stream. This will minimize localized deposits of settleable solids and stratification of the residual organics and nutrients in the localized area, which may cause a DO deficit and algae growth.
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
Water and Wastewater Treatment Plant Hydraulics 22.73
FIGURE 22.27 Schematic diagram of multimedia filter system. (HWL = .)
Submerged discharge. An effluent discharge pipe terminated at the bank of a stream usually leads to development of foam under low-flow conditions. The problem of foam can be overcome simply by submerging the pipe discharge below the low-water level when physical conditions in the stream allows such an arrangement. Diffusers. Certain outfalls, such as an ocean disposal, are typically accomplished by submarine outfall that consists of a long section of pipe to transport effluent and a diffuser section to dilute the effluent with the receiving stream. When the effluent water is discharged from a single- or multiport diffuser, the exit velocity will provide turbulent mixing with the surrounding water. Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
22.74
Chapter Twenty-Two
TABLE 22.13
Example Hydraulic Calculation of a Typical Multimedia Filter System Initial Operation PARAMETER
1. Wastewater flow rate, Q (m3/s) (mgd) 2. Multimedia filters Total number of units Number of units in operation Number of units on standby Flow rate per operating multimedia filter, q (m3/s) Hydraulic Calculations at Filter Effluent 3. At Point 7 Max. HGL in filtered water storage tank, HGL7 (m) Velocity in storage tank, V V7 (m/s) Max. EGL in storage tank, EGL7 HGL7 V7 V 2/2g (m)
Min Day Avg Day
Design Operation Avg Day Max Hour
Peak
1.0 23
1.6 36
2.0 46
3.2 73
3.2 73
6 4 2
6 5 1
6 5 1
6 6 0
6 5 1
0.25
0.32
0.40
0.53
0.64
98.67 0.00
98.67 0.00
98.67 0.00
98.67 0.00
98.67 0.00
98.67
98.67
98.67
98.67
97.67
98.77 1.00 7.00 0.18 98.95
98.77 1.60 7.00 0.25 99.02
98.77 2.00 7.00 0.29 99.06
98.77 3.20 7.00 0.40 99.17
97.77 3.20 7.00 0.40 99.17
0.00
0.00
0.00
0.00
0.00
98.95
99.02
99.06
99.17
99.17
1.00 3.00 2.00 10.00 0.17
1.60 3.00 2.00 10.00 0.27
2.00 3.00 2.00 10.00 0.33
3.20 3.00 2.00 10.00 0.53
3.20 3.00 2.00 10.00 0.53
0.60 0.013
0.60 0.013
0.60 0.013
0.60 0.013
0.60 0.013
0.0001
00002
0.004
0.0009
0.0009
1.0 0.25 0.32
1.0 0.32 0.41
1.0 0.40 0.51
1.0 0.53 0.68
1.0 0.64 0.82
0.0052
0.0085
0.0132
0.0236
0.0339
98.96
99.03
99.07
99.19
99.20
4. At Point 6 Filtered water effluent channel weir Sharp-crested weir EL, Wel6 HGL7 0.1 (m) Flow rate over weir Q (m3/s) Length of weir (m) Headloss, Hlw6 (q/1.84L)(2/3) (m) HGL at Point 6, HGL6 Wel6 Hlw6 (m) Velocity in weir box, V6 (assume V 0) (m) V EGL at Point 6, EGL6 HGL6 V6 V 2/2g (m) 5. Point 6 to Point 5 Loss through effluent concrete conduit Flow rate, Q (m3/s) Width of conduit, Wc (m) Depth of conduit, Dc (m) Length of conduit, Lc (m) Velocity, Vc (m/s) Hydraulic radius, R (Wc Dc/2) /(Wc Dc) (m) Manning’s n Headloss, Hlc5 (Vc n/R / (2/3))2 Lc (m) Exit loss from pipe to concrete conduit Effluent pipe diameter, Dp (m) Pipe flow (for each filter) (m3/s) Velocity, Vp (m/s) Hle5 Vp2/2g for sharp concrete outlet (m) EGL at Point 5, EGL5 EGL6 Hlc5 Hle6 (m)
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
Water and Wastewater Treatment Plant Hydraulics 22.75 TABLE 22.13
(Continued) Initial Operation PARAMETER
Velocity head at Point 5, HV5 V Vp2/2g (m) HGL at Point 5, HGL5 EGL5 HV5 V (m)
Design Operation
Min Day Avg. Day Avg. Day Max Hour
Peak
0.01
0.01
0.01
0.02
0.03
98.95
99.02
99.06
99.17
99.17
0.90
0.90
0.90
0.90
0.90
0.25
0.32
0.40
0.53
0.64
0.39 120 0.23 15.00
0.50 120 0.23 15.00
0.63 120 0.23 15.00
0.84 120 0.23 15.00
1.01 120 0.23 15.00
0.0193 0.0029
0.0305 0.0046
0.0461 0.0069
0.0785 0.0118
0.1100 0.0165
0.30 0.90
0.30 0.90
0.30 0.90
0.30 0.90
0.30 0.90
0.0024
0.0039
0.0061
0.0108
0.0155
1.20 0.32 0.0062
1.20 0.41 0.0102
1.20 0.51 0.0159
1.20 0.68 0.0283
1.20 0.82 0.0407
0.50 0.0026
0.50 0.0042
0.50 0.0066
0.50 0.0118
0.50 0.0170
98.97
99.05
99.11
99.25
99.29
0.00
0.00
0.00
0.00
0.00
98.97
99.05
99.11
99.25
99.29
2.5
2.5
2.5
2.5
2.5
101.47
101.55
101.61
101.75
101.79
0.00
0.00
0.00
0.00
0.00
6. Point 5 to Point 4 Filter effluent pipe loss Pipe diameter, Dp (m) Max. flow through filter effluent pipe q (m3/s) Velocity of flow through pipe, Vp (m/s) Hazen-Williams coefficient, Cp Hydraulic radius, Rp Dp/4 (m) Length of pipe, Lp (m) Slope, Sp[Vp/(0.85 Cp Rp0.63)](1/0.54) (%) Head loss, Hlf4 f Lp Sp (m) Headloss through butterfly valve Kvalve (fully open) Valve diameter (m) Headloss, Hval4 Kvalve (Vp2/2g) (m) Flow rate controller Venturi throat-to-inlet ratio for long tube, Krate Inlet velocity, Vi Vp (m/s) Headloss, hrate = Krate (Vi2/2g) (m) (minimum headloss when control valve is fully open) Pipe entrance loss Kent Headloss, Hlent Kent (Vp2/2g) (m) EGL at Point 4, EGL4 EGL5 Hlf4 f Hval4 Hrate Hlent (m) Velocity head, HV4 V V4 V 2/2g (assume V 0) (m) HGL at Point 4, HGL4 EGL4 HV4 V (m) 7. Point 4 to Point 3 Dirty filter head requirement, Hldf (m) (assumed) (consult with filter manufacturer) Dirty filter EGL, EGLdf HGL4 Hldf (m) Velocity head, HV3 V 0 (m) (assume V3 V 0) (m)
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
22.76
Chapter Twenty-Two
TABLE 22.13
(Continued) Initial Operation PARAMETER
Dirty filter HGL, HGLdf EGLdf HV3 V (m) 101.47 Clean filter headloss Filter bed area (m2) 160 Flow per filter, q (m3/s) 0.25 Filter rate, qfilt m3(min ⴢ m2) 0.094 Media depth, Dm (m) 1.00 Effective media size, Md (mm) 0.50 Headloss through filter, Hlf 2.32 m loss per m3(min ⴢ m2) (consultant with manufacturer) 0.2175 Entrance headloss through underdrain flume, Hlu 0.45 m m3(min ⴢ m2) 0.0422 (consult with filter manufacturer) Clean filter EGL, EGLcf EGL4 Hlf Hlu (m) 99.23 Velocity head, HV3 V 0 (assume V 0) (m) V3 Clean filter HGL, HGLcf EGLcf HV3 V (m) EGL required at Point 3, EGL3 EGLdf (m) 101.47 HGL required at Point 3, HGL3 HGLdf (m)101.47 (Head required for dirty filter controls) 8. Point 3 to Point 2 Filter inlet discharge loss Keff 1.0 Flow rate, q (m3/s) 0.25 Pipe diameter, Dp 2 (m) 0.9 Velocity, Vp2 (m/s) 0.39 Headloss, Hld2 Keff (Vp22/2g) (m) 0.0079 EGL at Point 2, EGL2 EGL3 Hld2 (m) 101.48 Velocity head, HV2 V Vp22/g / (m) 0.01 HGL at Point 2, HGL2 EGL2 HV2 V (m) 101.47 9. Point 2 to Point 1 Headloss through butterfly valve Kval (fully open) Headloss, Hlv1 Kval (Vp22/2g) Headloss through inlet pipe Length of pipe, Lp1 (m) Hazen-Williams coefficient, Cp Hydraulic radius, Rp Dp2/4 (m) Headloss, Hlf1 f (Vp2/(0.85 Cp Rp1.63)(1/0.54) Lp (m) Headloss through entrance to pipe Kent Headloss, Hlent Kent Vp2/2g (m)
Design Operation
Min Day Avg. Day Avg. Day Max Hour
Peak
101.55
101.61
101.75
101.79
160 0.32 0.120 1.00 0.50
160 0.40 0.150 1.00 0.50
160 0.53 0.200 1.00 0.50
160 0.64 0.240 1.00 0.50
0.2784
0.3480
0.4640
0.5568
0.0540
0.0675
0.0900
0.1080
99.38
99.52
99.81
99.95
101.55 101.55
101.61 101.61
101.75 101.75
101.79 101.79
1.0 0.32 0.9 0.50 0.0129 101.56 0.01 101.55
1.0 0.40 0.9 0.63 0.0202 101.63 0.02 101.61
1.0 0.53 0.9 0.84 0.0359 101.79 0.04 101.75
1.0 0.64 0.9 1.01 0.0517 101.084 0.05 101.79
0.3 0.0024
0.3 0.0039
0.3 0.0061
0.3 0.0108
0.3 1.0155
20.0 120 0.23
20.0 120 0.23
20.0 120 0.23
20.0 120 0.23
20.0 120 0.23
0.0039
0.0061
0.0092
0.0157
0.0220
0.50 0.0039
0.50 0.0065
0.50 0.0101
0.50 0.0179
0.50 0.0258
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
Water and Wastewater Treatment Plant Hydraulics 22.77 TABLE 22.13
(Continued) Initial Operation PARAMETER
EGL at Point 1, EGL1 EGL2 Hlv1 Hlf Hlent (m) Velocity head, HV1 0 (assume V1 0) (m) HGL at Point 1, HGL1 EGL1 HV1 (m) Minimum required control HGL at Point 1 (m) (Max. HGL1 must equal HGL7 of final sedimentation tank)
Design Operation
Min Day Avg. Day Avg. Day Max Hour
Peak
101.49
101.58
101.65
101.83
101.90
0.00
0.00
0.00
0.00
0.00
101.49
101.58
101.65
101.83
101.90
101.90
101.90
101.90
101.90
101.90
22.4.2.9 Slurry and chemical pumping. Sludge solids. Typical needs for sludge pumping involve transporting sludge from primary and secondary clarifiers to and between thickening, conditioning, digestion or dewatering facilities, and from biological processes for recycle or further treatment. Several different types of sludge pumps are used since various types of sludge require a wide range of service conditions. The flow characteristics (rheology) of wastewater sludges vary widely from process to process and from plant to plant. Because rheological properties directly influence pipeline friction losses of pumped sludges, head loss characteristics of wastewater sludges also vary extensively. Minimizing pumping distance and applying a conservative multiplier to headlosses calculated for equivalent flows of water is the traditional approach to the design of sludge pumping and piping systems. However, this approach is often inadequate. As a result of past research of non Newtonian fluid characteristics of sludges, sludge pumping system design data based on specific measured rheological characteris-
FIGURE 22.28 Schematic diagram of cascade aeration system.
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22.78
Chapter Twenty-Two
tics of sludge and the characteristics on piping systems are now available. These data are presented in Section 22.5. Scum. Scum is collected from the surface of primary sedimentation tank or secondary sedimentation tank. Scum from the secondary treatment is more dilute and is usually returned to the head of the treatment plant or thickened prior to combining the thickened scum with that from primary treatment. The scum is collected to a scum wet well and pumped to another location for processing. Progressive cavity pumps, pneumatic ejectors, and recessed impeller centrifugal pumps are used to pump scum. Key design elements for the scum collection and handling system include sloping the bottom of the scum
TABLE 22.14
Example Hydraulic Calculation of a Typical Cascade Aeration System Initial Operation PARAMETER
1. Wastewater flow rate, Q (m3/s) (mgd) 2. Cascade aerator Total number of units Flow rate through aerator, Q (m3/s) Optimal flow rate per m width over step, q (m3/s) DO concentration of postaeration influent, CO (mg/L) Desired DO concentration of postaeration effluent, Cu (mg/L) Calculation of aerator dimensions with predetermined weir length 3. Weir length, W (m) Flow over weir, q Q/W, W (m3/s) Critical depth at upstream step edge, hc (q2/g / )1/3 (m) Optimal fall height of nappe, h, Length of downstream bubble cushion, Lo 0.0629(h0.134)(q0.666) (m) Length of downstream receiving channel, L 0.8Lo (m) Optimal tailwater depth, H' 0.236h (m) for h 1.2 m Deficit ratio log at 20º C, In(r20) 5.39(h1.31)(q0.363)(H H0.31) Deficit ratio, r20
Design Operation
Min Day Avg. Day Avg. Day Max Hour
Peak
1.0 23
1.6 36
2.0 46
3.2 73
3.2 73
1 1.00
1 1.60
1 2.00
1 3.20
1 3.20
0.0653
0.0653
0.0653
0.0653
0.0653
0.00 5.00
0.00 5.00
0.00 5.00
0.00 5.00
0.00 5.00
5.0
5.0
5.0
5.0
5.0
0.20
0.32
0.40
0.64
0.64
0.160 1.2
1.219 1.2
0.254 1.2
0.347 1.2
0.347 1.2
5.16
7.05
8.18
11.19
11.19
4.12
5.64
6.54
8.95
8.95
0.28
0.28
0.28
0.28
0.28
0.42 1.53
0.36 1.43
0.33 1.39
0.28 1.32
0.28 1.32
Calculate concentration of dissolved oxygen downstream of step. If concentration is less than desired downstream concentration, add another step and again calculate DO downstream concentration. Continue adding steps until the desired DO concentration is achieved.
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Water and Wastewater Treatment Plant Hydraulics 22.79 TABLE 22.14
(Continued) Initial Operation PARAMETER
Design Operation
Min Day Avg. Day Avg. Day Max Hour
Peak
Select cascade aerator dimensions corresponding to those calculated for average flow. 4. Calculation of number of steps to obtain desired DO Desired DO concentration at average flow, Cu (mg/L) Step 1 effluent DO, C1 9.07 (1 (1/r20)) CO/r20) (mg/L) Step 2 effluent DO, C2 9.07 (1 (1/r20)) C1/r20) (mg/L) Step 3 effluent DO, C3 9.07 (1 (1/r20)) C2/r20) (mg/L)
5.00 3.14
2.73
2.55
2.21
2.21
4.81
4.51
4.39
4.14
4.14
6.01
5.80
5.70
5.52
5.52
1.00 1.20
1.00 1.20
1.00 1.20
1.00 1.20
1.00 1.20
HGL at Point 1, HGL1 (m)
97.53
97.53
97.53
97.53
97.53
HGL at Point 2, HGL2 HGL1 h (m) HGL at Point 3, HGL3 HGL2 h (m) HGL at Point 4, HGL4 HGL3 h (m)
96.33 95.13 93.93
96.33 95.13 93.93
96.33 95.13 93.93
96.33 95.13 93.93
96.33 95.13 93.93
In this example, the desired downstream DO concentration for average flow is achieved after three steps. 5. Calculation of HGL at each step Head loss from filtered water storage tank to point 1 (m) Cascade fall height, h (m)
tank, use of smooth pipe such as glass-lined pipe, providing flushing connections, pigging stations and cleanouts. Grit slurry. Removal and conveyance of grit from the grit chamber can be accomplished with varying degrees of success by a number of different methods, including inclined screw or tubular conveyers, chain and bucket elevators, clamshell buckets, and pumping. Of these methods, pumping of grit from hoppers in the form of slurry offers distinct advantages over other methods but also has some disadvantages. The advantages include small space requirement and flexibility of service by any grit pump from any grit tank to any grit handling system with simple valve operation. A disadvantage is frequent maintenance required for piping and valves due to the abrasive grit. Considerations to be given in piping design include minimization of bends, providing cleanouts at critical bends, providing redundant piping at the location of likely clogging, and maintaining a velocity of 1–2 m/s (3–6 ft/s). Vortex or recessed impeller pumps and air lift pumps normally handle grit slurries. Frequent pumping and applying waterjets or compressed air to loosen the compacted grit in the hopper prior to pumping is a good practice for grit pumping. Chemical solutions. Chemicals used in municipal treatment plants are received in either liquid or solid form. The chemicals in solid form generally are converted to soluDownloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
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22.80
Chapter Twenty-Two
tion or slurry prior to feeding although dry feeding is also practiced. Design of solution feed systems mainly depends on liquid volume and viscosity. Liquid feed units include piston, positive-displacement, and diaphragm pumps, as well as liquid gravity feeders. The unit best suitable for a particular application depends on the required head, chemical corrosiveness, application rate, other liquid properties, and the type of control.
22.5 NON-NEWTONIAN FLOW CONSIDERATIONS This section addresses pipe transport of mixtures of solids in a liquid media. This is relevant to us for the analysis of wastewater sludge transport. When a fluid motion begins within a pipe, the velocities of flow at all points along the cross section of the pipe are equal. Over time, velocity gradients are established, beginning at the wall of the pipe due to the resistance forces developed at the fluid-solid interface. Eventually the velocity gradients extend throughout the cross section of the flow. The velocity gradients result from the relative movement between fluid layers and the resultant shear. Fluids resist shear and, therefore, shear stresses are caused within a fluid in motion in a pipe. For water and other newtonian fluids, the shear stress is directly proportional to the velocity gradient. Many suspensions behave in non-newtonian fashion, as plastic fluids. In thin suspensions, the suspended particles are not in contact and the suspension will exhibit the newtonian properties of water. When the concentration becomes sufficiently great to force the particles into contact with each other, a measurable stress is needed to produce motion. Experiments by Bingham (1922) and Babbitt and Caldwell (1939) demonstrated that sewage sludges exhibit both types of flow characteristics depending on the type of solids and the moisture content. At low solid concentrations, the solid particles are generally not in contact with one another. In this case the presence of the solids has negligible impact on the density and the viscosity of the liquid. As the solids concentration increases, the suspended particles come into contact with each other and the resultant shearing stress must be overcome before any movement can start. Under such conditions, the flow assumes plastic characteristics and the headloss varies almost directly with the reduction of moisture M. The headlosses associated with the two types of flow are different. The dividing point between these two is called the limiting moisture content ML, which is defined as the moisture content in percent where a measurable yield stress, Sy, first occurs. As described by Chou (1958), below ML, the flow is plastic, and, above it, the flow is in suspension only. Furthermore, it is generally recognized that in sludge flow, as in other fluid flow, there is a critical velocity and, consequently, the Reynolds number, which divides the flow into laminar and turbulent stages. With flow in suspension there is no yield stress value and the Reynolds number takes the form of ρV VD Re (22.2) µ where Re Reynolds number ρ specific weight, V velocity, D pipe diameter, µ coefficient of viscosity similar to that for water. In plastic flow the apparent viscosity decreases with the increase in velocity, as discussed by Hatfield (1938) and, in a given range, it may be expressed as 16Sy D µ η 3V
(22.3)
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Water and Wastewater Treatment Plant Hydraulics 22.81
where Sy yield stress η coefficient of rigidity and the corresponding Reynolds number becomes ρV VD 3ρDv2 R 3ηv 16SyD µ
(22.4)
Using the Babbitt and Caldwell (1939) recommendation, and taking 2000 as the lower and 3000 as the upper limits of R, the critical velocities are: For flow in suspension, where M ML (flow in suspension); 2000µ VLC ρD
(22.5)
3000µ VUC ρD
(22.6)
For plastic flow, where M ML (plastic flow): 1000η 103 兹 兹9 苶苶4苶 η2苶苶 苶 D2苶S苶 yρ VLC ρD
(22.7)
η2苶苶 苶 D2苶S苶 1500η 127 兹 兹1苶苶4苶0苶 yρ VUC ρD
(22.8)
where VLC lower critical velocity VUC upper critical velocity The yield stress value Sy, the coefficient of rigidity η, and the specific weight are the basic variables required in computing critical velocities and headlosses. These properties vary from sludge to sludge depending on characteristics such as moisture M, nature of the suspended particles, temperature, and extent of turbulence. These factors also influence each other making it difficult to develop a useful equation for engineering practice. To resolve this issue, Chou presented an approach using moisture content M as the principal index of sludge, while placing all other parameters into the general term “origin or kind of sludge,” such as “”primary,” “digested,” and “digested from Imhoff tank,” and so on. The following development was presented by Chou (1958). The graphical values were taken from Babbitt and Caldwell (1938) and Keefer (1940). G related to M. Specific gravity, G, is primarily used in computing specific weight as in ρ 62.4 G. Specific gravity for activated sludge was shown to be, G 1.007, at a typical moisture content of about 98 to 99 percent. Primary sludges are more variable, but the curve in Fig. 22.29 indicates a reasonable mean. In Figure 22.29, digested sludges have a cluster of points near G 1.025, but the curve shows the general tendency. The three points from Imhoff tanks are on a smooth curve. Sy related to M. Yield stress, Sy, has an important role to play in calculating headloss and critical velocities. In Fig. 22.30 the two curves marked with “Imhoff Tank” and “Good Digestion” were considered to be representative of true conditions. The “Primary” values based on two points are clearly an approximation. The rest of the points varied consider-
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22.82
Chapter Twenty-Two
ably and were designated as the curve of “Poor Digestion” in an attempt to represent the upper limit of the range. The limiting moisture ML can be determined from Fig. 22.30 as the point where Sy 0 cuts the curve. η related to M. The experimental determination of the coefficient of rigidity indicated its variation with moisture M to be less pronounced than Sy. Accordingly, the plots are more scattered. The two lines (shown in Fig. 22.31) of “High” and “Mean” are suggested for design purposes. Case 1—Suspension/Laminar Stage. For flow in suspension, the solid particles are free to move past one another and there is consequently no yield value to overcome. Reduction of moisture content only slightly increases the specific weight ρ (ρ 62.4 G) and the viscosity µ. Both remain close to the values for water. The yield stress, Sy, is zero for flow in suspension. The equation for headloss for laminar stage flow in suspension becomes ηV H 2 L 62.4G D
(22.9)
where G specific gravity in which both G and η for the corresponding M can be determined from the Figs. 22.29 and 22.31. Case 2—Suspension/Turbulent Stage. Streck (1950) and Winkel (1943) reported the headloss of turbulent flow in suspension may be computed as follows: HS G2HW
(22.10)
where HS the headloss of flow in suspension with moisture M HW the corresponding headloss of pure water G the specific gravity of the suspension (from Fig. 22.29) The headloss of flow in suspension for both laminar and turbulent conditions is not significantly greater than the corresponding headloss for water. Case 3—Plastic Flow/Laminar Stage. Plastic flow in the laminar stage is the most common case in sludge flow. As discussed above, the headloss is partly due to yield value and partly due to coefficient of rigidity, both of which are affected by the moisture M. Babbitt and Caldwell (1939) reported headloss for this case as follows: ηV H 16Sy 2 L 3ρD ρD
(22.11)
in which the values of ρ, Sy, and η may be determined from Figs. 22.29, 22.30 and 22.31, respectively. For any moisture below the limiting value, plastic flow conditions mean Sy
0 and a headloss occurs due to yield value, Sy, alone. As motion begins, headloss increases with the first power of velocity in the laminar stage. Hence, as soon as the applied head is greater than Sy, relatively little additional head is required to accelerate the flow to critical velocity. Therefore, it may be concluded that the most economical velocity of sludge flow is the critical velocity, above which the headloss increases rapidly with the velocity. Case 4—Plastic Flow/Turbulent Stage. Published data for turbulent plastic flow headloss are variable and inconsistent. Due to variation of sludge characteristics, the velocities, the results are extremely unpredictable.
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Water and Wastewater Treatment Plant Hydraulics 22.83
For fully turbulent flow, it seems reasonable that the headloss results primarily from kinetics and is proportional to v2/2g and the specific weight ρ and, therefore, will differ from that of water only slightly by the effect of ρ. This ideal condition of full turbulence rarely occurs for plastic flows. As the moisture drops below ML, the critical velocities increase and the thickness of the boundary layers is increased in proportion to moisture reduction. The velocity distribution in a cross section and the impacts of the boundary layers are not the same as the regular patterns of homogeneous liquids. Due to the complicated and variable phenomena occurring during turbulent plastic flow, it is difficult, if not impossible, to accurately anticipate headloss for flow in this condition. Designing for this
SPECIFIC GRAVITY G
FIGURE 22.29 Specific gravity G of sludge (From Chou, 1958)
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
22.84
Chapter Twenty-Two
condition is uncertain and not recommended. However, some experimental data are available for guidance when turbulent plastic flow is unavoidable. Brisbin (1957) compiled headloss data for raw, thickened sludge. Thus, from such complicated phenomena, uniform results can hardly be expected. The corresponding C in the Hazen-Williams formula V 1.318Cr0.63 s0.54
(22.12)
where r hydraulic radius and s H/ H/L hydraulic slope was computed from the observed headlosses. These C ' values are tabulated in Table 22.15 along with the ratio to water headloss.
Yield Stress, Sy FIGURE 22.30 Yield value of Sy of sewage sludges (From Chou, 1958)
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Water and Wastewater Treatment Plant Hydraulics 22.85
22.5.1 Headloss Computation With the source and M of the sludge known or assumed, the first step is to determine if the flow is a suspension or plastic. Empirically this can be done by the curves in Fig. 22.30. Values for G, Sy and η are then chosen from curves in Figs. 22.29, 22.30, and 22.31. Example. Given primary sludge, M 95. The flow is plastic since M ML (M ML 99.8 percent at point in Fig. 22.30 where Sy 0). From Figs. 22.29, 22.30 and 22.31, G 1.022, 1.022 62.4 63.77 lb/ft2 Sy 0.065 lb/ft/s η 0.0127 (lbⴢft)/s Critical velocities (22.13)
PERCENTAGE OF MOISTURE BY WEIGHT
103兹0 兹苶苶 .0苶1苶5苶1苶3苶 苶4苶 .1苶4苶5苶 D2苶 VLC 12.7 63.77D
FIGURE 22.31 Coefficient of rigidity n of sludge (From Chou, 1958)
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22.86
Chapter Twenty-Two
127兹0 兹苶苶 .0苶2苶2苶6苶 苶4苶 .1苶4苶5苶 D2苶 VUC 19.1 63.77D
(22.14)
The values are given in Table 22.16. Laminar stage H 0.00555 V 0.000204 2 L D D
(22.15)
The values are tabulated against the pipe diameter D for a range of laminar flow velocities in Table 22.17. Turbulent stage: Assume C 100 for M 100, and from a plot of Table 1 C' values, the corresponding C' 54.7 for M 95. V 72.09r 0.63s0.54
(22.16)
V1.85 V1.85 s H L 72.091.85r1.165 Constant The headlosses are computed in Table 22.18. It is useful to plot results as shown in Figs. 22.32 and 22.33 with critical velocities indicated. For laminar flow, values are taken from the left of VLC, and for turbulent flow, they are taken from right of VLC. It is also useful to tabulate results as shown in Table 22.19, including the minimum headloss to account for Sy as well as the operating headloss. Head losscomputations for solids bearing flows are not an exact science. Where the physical properties of the sludge cannot be measured, use of the data reproduced here in Figs. 22.29, through 22.31 and the methodology developed by Chou et al. (1958) and summarized here should provide reasonable results.
TABLE 22.15
C’ Values for Raw, Thickened Sludge
M
C’
Moisture Content (%)
Percentage of C at M 100%
Ratio to Water Headloss 100 1.85 C’
100
100
100
98
80.5
1.49
97
—
—
96
62.8
2.37
95
—
—
94
50.5
3.54
91.5
37.6
6.11
90
33.6
7.54
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Water and Wastewater Treatment Plant Hydraulics 22.87
FIGURE 22.32 Results of Headloss computation examples–laminar flow
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
22.88
Chapter Twenty-Two
FIGURE 22.33 Head loss for turbulent flow (m 95%)
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
Water and Wastewater Treatment Plant Hydraulics 22.89 TABLE 22.16
Example of Critical Velocities
D
8 in
10 in
14 in
20 in
VLC (ft/s) VUC (ft/s)
3.58
3.55
3.45
3.40
4.52
4.42
4.31
4.23
10 in
14 in
20 in
0.000293v
0.000149v
0.000073v
0.00666
0.00476
0.00333
0.00754
0.00521
0.00355
0.00770
0.00527
0.00358
TABLE 22.17
Example Hydraulic Slope for Laminar Stage
D
8 in
H ft 0.000458v L ft H V 0, 0.00833 L V 3, H 0.00970 L VLC , H 0.00997 L Varies (see Table 22.16)
TABLE 22.18
Example Hydraulic Slope for Turbulent Stage D:8 in
D:10 in
D:14 in
D:20 in
V
V1.85 338.9
V1.85 439.6
V1.85 650.5
V1.85 986.5
Varies Table 22.16 19.64 27.51 36.60 46.85 58.25 70.80
0.0481 0.0580 0.0813 0.108 0.138 0.172 0.209
0.0356 0.0447 0.0625 0.0832 0.106 0.133 0.161
0.0229 0.0302 0.0423 0.0562 0.072 0.0896 0.109
0.0146 0.0199 0.0279 0.0371 0.0474 0.0591 0.0718
gal/m
ft3/s
ft/s
2000 2000 4000 4000 600 500 1600 3300
4.46 4046 8.91 8.91 1.34 1.11 3.57 7.35
8.20 12.80 8.36 4.08 2.44 3.21 3.34 3.37
1.85
V fps VUC 5 6 7 8 9 10
TABLE 22.19 Summary of Results Pipes L ---------- D 16 ft ------- 0 in 5 ft ------- 8 in 11 ft ------- 14 in 20 ft ------- 20 in 20 ft ------- 10 in 50 ft ------- 8 in 30 ft ------- 14 in 40 ft ------- 20 in
Q
Headloss, feet Minimum(1) Operating 0.11 0.04 0.05 0.08 0.13 0.42 0.14 0.13
1.78 1.65 0.86 0.28 0.15 0.49 0.16 0.14
Minimum is the headloss required to overcome Sy and initiate flow.
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22.90
Chapter Twenty-Two
REFERENCES American Society of Civil Engineers, and Water Environment Federation, Gravity Sanitary Sewer Design and Construction, American Society Civil Engineers Manuals and Reports on Engineering Practice No. 60 and Water Environment Federation Manual of Practice No. FD-5, 1982. Babbitt, H. E., and Caldwell, David H., Laminar Flow of Sludges in Pipes with Special Reference to Sewage Sludge, University of Illinois, Bulletin 319, 1939. Bingham, E. C., Fluidity and Plasticity, McGraw-Hill, New York, 1922. Brisbin, S. G., “Flow of Concentrated Raw Sewage Sludges in Pipes,” Proceedings Paper 1274, American Society Civil Engineers 1957. Bulletin No. 2552 University of Wisconsin. Bureau of Reclamation, Design Standards No.3, Water Conveyance Systems, Chapter 11 General Hydraulic Considerations (Draft), (7-2071) (6-84), Sept. 30, 1992. Camp, T. R., and Graber, S. D., Dispersion Conduits, Journal of the Sanitary Engineering Division, American Society of Civil Engineer, 94(SA1), February 1968. Chao, J.–L., and Trussell, R. R., “Hydraulic Design of Flow in Distribution Channels,” Journal of Environmental Engineering Division, ASCE, 6(EE2), April 1980. Chou, T.–L., “Resistance of Sewage Sludge to Flow in Pipes,” Journal of Sanitary Engineering Div., American Society of Civil Engineer, Paper 1780, September 1958. Committee on Pipeline Planning, Pipeline Division, Pipeline Design for Water and Wastewater, American Society of Civil Engineers, New York, 1975. Crane Co., “Flow of Fluids Through Valves, Fittings, and Pipe”, Technical Paper No. 410-C, 23rd ed., Banford, Ontario, 1987. Daugherty, R. L., and J. B. Franzini, Fluid Mechanics with Engineering Applications, 7th ed., McGraw-Hill, New York, 1977. Hatfield, W. D., “Viscosity or Psendo-Plastic Properties of Sewage Sludges,” Sewage Works Journal, 10, 1938. Ito, H., and Imani, K., “Energy Losses at 90o Pipe Junctions.” Journal of the Hydraulics Division, American Society of Civil Engineer, HY9, 1973. Keefer, C. E., Sewage Treatment Works, McGraw-Hill, New York, 1940. Sanks, R. L., Pumping Station Design, Butterworths, Stoneham, MA, 1989. Shaw, G. V., and A. W. Loomis, eds., Cameron Hydraulic Data, Ingersoll-Rand Co., Cameron Pump Division, 14th Ed., 1970. Simon, A. L., Hydraulics, 3rd ed., John Wiley & Sons, New York, 1986. Streck, O., Grund und Wasserbrau in Praktischen Biespielen, Springer-Verlag, Berlin, 1950. Ten-State Standards, Recommended Standards for Sewage Works, Great Lakes–Upper Mississippi Board of Sanitary Engineers, Health Education Service, Inc., Albany, NY, 1978. Walski, T. M., Analysis of Water Distribution Systems, Krieger, Malabar, FL, 1992. Williamson, J. V., and Rhone, T. J., ßDividing Flow in Branches and Wyes,” Journal of the Hydraulics Division, American Society of Civil Engineer, No. HY5, 1973. Winkel, R., Angwandte Hydromechanik im Wasserbau, Ernst & Sohn, Berlin, 1943. Yao, K. M., Hydraulic Control for Flow Distribution, Journal of the Sanitary Engineering Division, American Society of Civil Engineer, 98 (SA2), April 1972
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
Water and Wastewater Treatment Plant Hydraulics 22.91
APPENDIX
WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS ENGLISH UNITS EXAMPLES
TABLE 22.5
Hydraulic Calculations of a Typical Coagulation Process Initial Operation Parameter
1. Plant Flow (mgd) (ft3/s)
Design Operation
Min Day
Avg Day
Avg Day Max Hour
50 77.36
70 108.31
75 116.04
100 154.72
360.1
360.01
360.02
360.04
50.77 10.26 172.72 0.29 4.62
71.08 10.27 172.89 0.41 4.63
76.15 10.27 172.94 0.44 4.63
101.54 10.29 173.25 0.59 4.63
0.00 360.01
0.00 360.02
0.00 360.02
0.00 360.04
24.18 10.26 172.72 0.14 4.62
33.85 10.27 172.89 0.20 4.63
36.26 10.27 172.94 0.21 4.63
48.35 10.29 173.25 0.28 4.63
0.00 360.01
0.00 360.02
0.00 360.02
0.00 360.04
9.67 10.26 172.72 0.06 4.62
13.54 10.27 172.89 0.08 4.63
14.51 10.27 172.94 0.08 4.63
19.34 10.29 173.25 0.011 4.63
Note: For Points 1 through 8, see Fig. 22.12. 2. WSEL at Point 1 (calculation done in Table 22.6) (ft) 3. Point 1 to Point 2 Average Flow 21Q/32 (ft3/s) Flow depth WSEL @ 1 invert (349 ft 9 in) (ft) Flow area 16ft 10in width depth (ft2) Velocity flow/area (ft/s) R = A/P / (P w + 2d) (ft) 2 Condiut loss [(V n )/(1.486 R2/3)] L (ft) where n 0.014 and L 95 ft WSEL at Point 2 (ft) 4. Point 2 to Point 3 Average flow 5Q/16 (ft3/s) Flow depth = WSEL @ 2 – invert (349 ft 9 in) (ft) Flow area 16 ft – 10 in width depth (ft2) Velocity flow/area (ft/s) R = A/P / (P w 2d) (ft) 2 Condiut loss [(V n)/(1.486b R2/3) ] L (ft) x L where n 0.014 and L 48 ft WSEL at Point 3 (ft) 5. Point 3 to Point 4 Average flow Q/8 (ft3/s) Flow depth WSEL @ 3 – invert (349 ft – 9 in) (ft) Flow area 16 – ft 10 in width depth (ft2) Velocity flow/area (ft/s) R= A/P / (P w 2d) (ft)
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
22.92
Chapter Twenty-Two
TABLE 22.5
(Continued) Initial Operation Parameter
Condiut Loss [(V n)/(1.486 R2/3)]2 L (ft) where n 0.014 and L 72 ft WSEL at Point 4 (ft) 6. Point 4 to Point 5 Flow Q/32 (ft3/s) Port area 1 ft deep 2.5 ft wide (ft2) Velocity flow/area (ft/s) Submerged entrance loss 0.8 V 2/2g (ft) WSEL at Point 5 (in sedimentation tank) (ft) 7. Point 5 to Point 6 Width of sedimentation basin (W) W (ft) Flow (Q/4) (ft3/s) Invert elevation of sedimentation baffles (ft) Fow depth (H) H (WSEL at Point 5 – baffle invert) (ft) Area downstreams of baffle (W H H) (ft2) Horizontal openings in baffles, 1 in wide, every 9 inches Area of openings, A W H H/ 9 (ft2) Velocity of downstream baffle (V downstream) (Q/A) (ft/s) Velocity of 1 in opening section (V1) (Q/A / ) (ft/s) Local losses Sudden expansion (1.0 V downstream2/2g) sudden contraction (0.36 V12/2g) (ft) WSEL at Point 6 (Upstream of sedimentation baffles) (ft)
Design Operation
Min Day
Avg Day
Avg Day Max Hour
0.00 360.01
0.00 360.02
0.00 360.02
0.00 360.04
2.42 2.50 0.97 0.01 360.02
3.38 2.50 1.35 0.02 360.04
3.63 2.50 1.45 0.03 360.05
4.84 2.50 1.93 0.05 360.09
76.00 19.34 347.67 12.35 938.77
76.00 27.08 347.67 12.37 940.39
76.00 29.01 347.67 12.38 940.88
76.00 38.68 347.67 12.42 943.83
104.31
104.49
104.54
104.87
0.02
0.03
0.03
0.04
0.19
0.26
0.28
0.37
0.00
0.00
0.00
0.00
360.02
360.04
360.05
360.09
8. Point 6 to Point 7 Loss per stage (provided by flocculator manufacturer) (ft) Total loss (three stages) (ft) WSEL at Point 7 (ft)
0.04 0.13 360.15
0.04 0.13 360.17
0.10 0.29 360.34
0.17 0.50 360.59
9. Point 7 to Point 8 Flow Q/24 (ft3/s) Port area 1 in deep 1 ft – 6 in wide (ft2) Velocity flow/area (ft/s) Entrance loss 1.25 V 2/2g (ft) WSEL at Point 8 (inlet port) (ft)
3.22 1.50 2.15 0.09 360.24
4.51 1.50 3.01 0.18 360.35
4.84 1.50 3.22 0.20 360.54
6.45 1.50 4.30 0.36 360.95
3.22 2.24 6.73 0.48 0.90
4.51 2.35 7.05 0.64 0.92
4.84 2.54 7.63 0.63 0.94
6.45 2.95 8.84 0.73 0.99
Note: For Points 8 through 14, see Fig. 22.13 10. Point 8 to Point 9 Average Flow Q/24 (ft3/s) Flow depth WSEL @ 8 - invert (358 ft) (ft) Flow area 3 ft width depth (ft2) Velocity flow/area (ft/s) R = A/P / (P w 2d) (ft)
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
Water and Wastewater Treatment Plant Hydraulics 22.93 TABLE 22.5
(Continued) Initial Operation Parameter
Min Day
Avg Day
Design Operation Avg Day Max Hour
2
Condiut loss [(V n)/(1.486 R 2/3)] L where n = 0.014 and L 12 ft – 8in (ft) WSEL at Point 9 (ft)
0.00
0.00
0.00
0.00
360.24
360.35
360.54
360.95
11. Point 9 to Point 10 Average flow Q/12 (ft3s) Flow depth WSEL @ 9 – invert (358 ft) (ft) Flow area 3 ft width depth (ft2) Velocity flow/area (ft/s) R = A/P / (P w 2d) (ft) Condiut loss [(V n)/(1.486 R 2/3)]2 L (ft) where n 0.014 and L 12 ft – 8 in (ft) WSEL at Point 10 (ft)
6.45 2.24 6.73 0.96 0.90
9.03 2.35 7.05 1.28 0.92
9.67 2.54 7.63 1.27 0.94
12.89 2.95 8.85 1.46 0.99
0.00 360.24
0.00 360.35
0.00 360.54
0.00 360.95
12. Point 10 to Point 11 Flow Q/8 (ft3/s) Flow depth WSEL @ 10 – invert (358 ft) (ft) Flow area 3 ft width depth (ft2) Velocity flow/area (ft/s) Loss at two 45o bends 2 0.2 V 2/2g (ft) WSEL at Point 11 (ft)
9.67 2.24 6.73 1.44 0.01 360.26
13.54 2.35 7.06 1.92 0.02 360.38
14.51 2.54 7.63 1.90 0.02 360.57
19.34 2.95 8.85 2.18 0.03 360.98
13. Point 11 to Point 12 Flow Q/4 (ft3/s) Flow depth WSEL @ 11 – invert (358 ft) (ft) Flow area 5 ft width depth (ft2) Velocity flow/area (ft/s) Loss at two 45o bends 2 0.2 V2/2 V g (ft) R A/P / (P w 2d) (ft) Condiut Loss [(V n)/(1.486 R 2/3)]2 L (ft) where n 0.014 and L 32 ft WSEL at Point 12 (ft)
19.34 2.26 11.28 1.71 0.02 1.19
27.08 2.38 11.88 2.28 0.03 1.22
29.01 2.57 12.83 2.26 0.03 1.27
38.68 2.98 14.90 2.60 0.04 1.36
0.01 360.28
0.01 360.42
0.01 360.61
0.01 361.04
14. Point 12 to Point 13 Flow Q/4 (ft3/s) Flow depth WSEL @ 12 – invert (358 ft) (ft) Inlet area 5 ft width depth (ft2) Velocity flow/area (ft/s) Inlet loss 1 V 2/2g (ft) WSEL at Point 13 (Mixing chamber No. 2 outlet) (ft)
19.34 2.28 11.41 1.70 0.04 360.33
27.08 2.42 12.09 2.24 0.08 360.50
29.01 2.61 13.05 2.22 0.08 360.69
38.68 3.04 15.18 2.55 0.10 361.14
19.34 36.00 0.54
27.08 36.00 0.75
29.01 36.00 0.81
38.68 36.00 1.07
0.01
0.02
0.03
0.05
360.34
360.52
360.71
361.19
15. Point 13 to Point 14 Note: Mixers provide negligible head loss Flow Q/4 (ft3/s) Chamber area 6 ft 6 ft (ft2) Velocity flow/area (ft/s) Losses Mixer (1 V 2/2g) Sharp bend (1.8 V 2/2g) (ft) WSEL at Point 14 (Mixing Chamber No. 2 inlet) (ft)
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
22.94
Chapter Twenty-Two
TABLE 22.5
(Continued) Initial Operation Parameter
Design Operation
Min Day
Avg Day
Avg Day Max Hour
38.68 30.00 1.29 1.30
54.15 30.00 1.81 1.30
58.02 30.00 1.93 1.30
77.36 30.00 2.58 1.30
0.02
0.03
0.03
0.06
0.02 360.37
0.03 360.58
0.04 360.79
0.07 361.31
360.31
360.51
360.70
361.17
360.34
360.54
360.74
361.24
77.36 30.00 2.58 1.30
108.31 30.00 3.61 1.30
116.04 30.00 3.87 1.30
152.72 30.00 5.16 1.30
0.15 360.49
0.28 360.82
0.31 361.06
0.53 361.77
77.36 30.00 30.25 30.13 2.57 1.30 1.38 1.34
108.31 30.00 30.25 30.13 3.60 1.30 1.38 1.34
116.04 30.00 30.25 30.13 3.85 1.30 1.38 1.34
154.72 30.00 30.25 30.13 5.14 1.30 1.38 1.34
0.01 360.50
0.02 360.84
0.02 361.08
0.04 361.81
Note: For Points 14 through 21, see Fig. 22.14 16. Point 14 to Point 15 Flow Q/4 (ft3/s) Condiut area 7.5 ft wide 4 ft deep (ft3) Velocity flow/area (ft/s) R A/P / (P 2w 2d) (ft) Condiut losses L [V/(1.318 V C R0.63)]1/0.54 (ft) where L 155 ft and Hazen-Williams C 120 Local losses Flow split (0.6 V 2/2g) contraction (0.07 V 2/2g) 0.67 V 2/2g (ft) WSEL at Point 15 (at Mixing Chamber No. 1) (ft) 17. The above calculations (for Points 1 through 15) have been routed through Tank No. 4 When the flow isrouted through Tank No. 1, the WSEL (ft) is: In reality, the headloss through each basin is equal. The flow through the basin naturally adjusts to equalize headlosses, that is flow through Tank No. 1 is greater than Q/4 and flow through Tank No. 4 is less than Q/4. The actual headloss through each basin is the average of Tank #’s 1 and 4 and the WSEL (ft) at Point 15 is: 18. Point 15 to Point 16 Flow Q (ft3/s) Condiut area 7.5 ft wide 4 ft deep (ft2) Velocity flow/area (ft/s) R A/P / (P 2w 2d) (ft) Condiut losses L [V/(1.318 V C R0.63)]1/0.54 (ft) where L 412 ft and Hazen-Williams C 120 WSEL at Point 16 (ft) 19. Point 16 to Point 17 Flow Q (ft3/s) Condiut area @ 16 7.5 ft wide 4 ft deep (ft2) Condiut area @ 17 5.5 ft wide 5.5 ft deep (ft2) Average area (ft2) Velocity flow/area (ft/s) R @ 16 A16/ (2 (7.5 ft 4 ft) (ft) R @ 17 A17/ (2 (5.5 ft 5.5 ft) (ft) Average R (ft) Condiut Losses L [V/(1.318 V C R0.63)]1/0.54 (ft) where L 30 ft and Hazen-Williams C 120 WSEL at Point 17 (ft)
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
Water and Wastewater Treatment Plant Hydraulics 22.95 TABLE 22.5
(Continued) Initial Operation Parameter
20. Point 17 to Point 18 Flow Q (ft3/s) Condiut area @ 17 5.5 ft wide 5.5 ft deep (ft2) Velocity 17 flow/area 17 (ft/s) Pipe area @ 18 5.5 ft2/4 (ft2) Velocity 18 flow/area 18 (ft/s) Exit Losses V182/2g V172/2g (ft) WSEL at Point 18 (ft) 21. Point 18 to Point 19 R A/P / (P d ) (ft) Local losses 3 elbows (3 0.25V 2/2g) entrance (0.5 V 2/2g ) 1.25 V 2/2g (ft) Condiut losses L [V/(1.318 V C R0.63)]1/0.54 where L 455 ft and Hazen-Williams C 120 (ft) WSEL at Point 19 (exit of Control Chamber) (ft)
Design Operation
Min Day
Avg Day
Avg Day Max Hour
77.36 30.25 2.56 23.76 3.26 0.06 360.56
108.31 30.25 3.58 23.76 4.56 0.12 360.96
116.04 30.25 3.84 23.76 4.88 0.14 361.22
154.72 30.25 5.11 23.76 6.51 0.25 362.06
1.38
1.38
1.38
1.38
0.21
0.40
0.46
0.82
0.24 361.00
0.44 361.81
0.50 362.18
0.85 363.74
360.00
360.00
360.00
360.00
1.00 9.00
1.81 9.00
2.18 9.00
3.74 9.00
22. Point 19 to Point 20 Weir elevation (ft) Depth of flow over weir (WSEL @ 19 - weir elevation) (ft) Length of weir, L (ft) Flow over weir q 3.1 h 3/2 x [1–(d/h)3/2]0.385 L Note: Rather than solve for h, find an h by trial and error that gives a q equal to the flow for the given flow scenario (given in Item 1) First Iteration assume h (ft) then q (ft3/s) Second Iteration assume h (ft) then q (ft3/s) Note: These q’s equal the flows for the given scerios (Item 1) h (ft) WSEL at Point 20 (h + WSEL @ Point 19) (ft)
2 66.63 2.17 77.11
3 113.72 2.93 108.41
3 99.77 3.2 116.04
4 90.40 4.5 154.28
2.17 362.17
2.93 362.93
3.2 363.20
4.5 364.50
23. Point 20 to Point 21 Flow Q (ft3/s) Sluice gate area 54 in 54 in (ft2) Velocity = flow/area (ft/s) Gate losses 1.5 V 2/2g (ft) WSEL at Point 21 (Raw Water Control Chamber) (ft)
77.36 20.25 3.82 0.34 362.51
108.31 20.25 5.35 0.67 363.60
116.04 20.25 5.73 0.76 363.96
154.72 20.25 7.64 1.36 365.86
1.76
2.21
2.31
2.80
The overflow weir in the Raw Water Control Chamber is 10 ft long and is sharp crested. Q 3.3 L h 3/2 so h (Q/3.3/L / )2/3 (ft) The water surface must not rise above elevation 370 ft – 0 in. The overflow weir elevation may be safely set at 367 ft – 0 in.
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
22.96
Chapter Twenty-Two
TABLE 22.6 Hydraulic Calculations in a Medium Sized Water Treatment Plant from the Filter Effluent to the Effluent Clearwell Initial Operation Parameter 1. Plant flow (mgd) (ft3/s)
Design Operation
Min Day
Avg Day
Avg Day Max Hour
50 77.36
70 108.31
75 116.04
100 154.72
345.00 333.00 38.68
345.00 333.00 54.15
345.00 333.00 58.02
345.00 333.00 77.36
120.00 0.64 0.00
120.00 0.90 0.01
120.00 0.97 0.01
120.00 1.29 0.01
120.00 0.64 0.01
120.00 0.90 0.01
120.00 0.97 0.01
120.00 1.29 0.03
0.01 345.02
0.01 345.03
0.01 345.04
0.03 345.06
38.68
54.15
58.02
77.36
23.76 1.63
23.76 2.28
23.76 2.44
0.02 0.02
0.04 0.04
0.05 0.05
23.76 3.26 0.16 0.08 0.08
0.03 345.08
0.05 345.16
0.06 345.19
0.10 345.49
Note: for Points 22 through 28, see Fig. 22.15 2. Point 22 to Point 23 Maximum water level in clearwell (Point 22) (ft) Invert in clearwell, (ft) Flow Q/2 (ft3/s) Stop logs @ A Flow area (2 openings, 5 ft wide, 12 ft deep) (ft2) Velocity flow/area (ft/s) Loss 0.5 V 2/2g (ft) Baffles Flow area (10 ft wide, 12 ft deep) (ft2) Velocity flow/area (ft/s) Loss 1.0 V2/2g (ft) Stop logs @ B and C Same as the losses @ A, times 2 (ft) WSEL at Point 23 (ft) 3. Point 23 to Point 24 Flow Q/2 (ft3/s) 66 inch diameter pipe Flow area d 2/4 p (ft2) Velocity flow/area (ft/s) Exit loss @ clearwell V 2/2g (ft) Loss @ 2 90o bends (0.25 V 2/2g) 2 (ft) Entrance loss @ filter building 0.5 V 2/2g (ft) Pipe loss (3.022 V 1.85 L)/ (C 1.85 D1.165) (ft) where C 120 and L 190 WSEL at Point 24 (ft) 4. Point 24 to Point 25 Flow Q/4 (ft3/s) Flow area 5 ft 5ft (ft2) Velocity Q/A / (ft/s) Loss as flows merge 1.0 V 2/2g (ft) Condiut loss [(V n)/(1.486 R 2/3)] 2 L (ft) where n 0.013, L 55 ft and R A/P / (P 20) WSEL at Point 25 (ft)
19.34 25.00 0.77 0.01
27.08 25.00 1.08 0.02
29.01 25.00 1.16 0.02
38.68 25.00 1.55 0.04
0.00 345.10
0.00 345.19
0.00 345.21
0.01 345.54
5. Point 25 to Point 26 Sluice Gate No. 1 flow area 48 in 36 in (ft2) Velocity Q/A / (ft/s) Loss 0.5 V 2/2g (ft) WSEL at Point 26 (ft)
12 1.61 0.02 345.12
12 2.26 0.04 345.23
12 2.42 0.05 345.26
12 3.22 0.08 345.62
6. Point 26 to Point 27 Sluice Gate No. 2 Loss 0.8 V 2/2g (ft) WSEL at Point 27 (ft)
0.03 345.15
0.06 345.29
0.07 345.33
0.13 345.75
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
Water and Wastewater Treatment Plant Hydraulics 22.97 TABLE 22.6
(Continued) Initial Operation Parameter
7. Point 27 to Point 28 Port to filter clearwell. Calculate losses through port as if it were a weir when depth of flow is below top of port. Port Dimmensions 9 ft wide by 2 ft – 8 in feet deep Flow Q/4 (ft3/s) Weir (bottom of port) elevation (ft) Depth of flow over weir (WSEL @ 27 weir elevation) (ft) Flow over submerged weir = q = 3.1 h3/2 [1 (d/ d h)3/2]0.385 L Note: Rather than solve for h, find an h by trial and error that gives a q equal to the flow for the given flow scenario (given in item 1). assume h (ft) = then q (ft3s) = assume h (ft) = then q (ft3/s) = Note: These q’s equal the flows for the given scerios (Item 1) h (feet) WSEL at Point 28, (ft)
Design Operation
Min Day
Avg Day
Avg Day Max Hour
19.34 344.00
27.08 344.00
29.01 344.00
38.68 344.00
1.15
1.29
1.33
1.75
1.3 20.8841 1.28 19.4646
1.4 20.2379 1.49 27.0883
1.5 25.5883 1.55 29.232
2 41.0387 1.97 38.485
1.28 345.28
1.49 345.49
1.55 345.55
1.97 345.97
360.00
360.00
360.00
360.00
9.67
13.54
14.51
19.34
0.60
0.85
0.91
1.21
0.00
0.01
0.01
0.02
0.77 0.00 0.00 360.01
1.08 0.00 0.00 360.02
1.15 0.01 0.01 360.02
1.54 0.01 0.01 360.04
8.01 48.05 0.20 2.18
8.02 48.11 0.28 2.18
8.02 48.12 0.30 2.18
8.04 48.22 0.40 2.18
0.00 360.01
0.00 360.02
0.00 360.02
0.00 360.04
Filters–See Filter Hydraulics in Table 22.7 Note: For Points 29 thruogh 33, see Fig. 22.16 8. Point 29 WSEL above filters (ft) 9. Point 29 to Point 30 Entrance to Filter #4 Flow Q/8 (ft3/s) Channel Velocity V Flow/ F /Area (area 4 ft 4 ft) (ft/s) Submerged entrance loss = 0.8 V 2/2g (ft) 48 in Pipe velocity flow/area (area d 2/4 ) (ft/s) Butterfly valve loss 0.25 V 2/2g (ft) Sudden elargement loss 0.25 V 2/2g (ft) WSEL in influent channel (Point 30) (ft) 10. Point 30 to Point 31 Flow depth WSEL @ 30 invert (352 ft) (ft) Flow area 6 ft width depth (ft2) Velocity flow/area (ft/s) R A/P / (P w 2d) (ft) Condiut loss [(V n)/(1.486 R 2/3)] 2 L where n 0.014 and L 35 ft – 4 in (ft) WSEL at Point 31 (ft) 11. Point 31 to Point 32
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
22.98
Chapter Twenty-Two
TABLE 22.6
(Continued) Initial Operation Parameter
Flow Q/4 (ft3/s) Flow depth WSEL @ 31 invert (352 ft) (ft) Flow area 6 ft width depth (ft2) Velocity flow/area (ft/s) R A/P / (P w 2d) (ft) Condiut loss [(V n)/(1.486 R 2/3)] 2 L (ft) where n 0.014 and L 35ft 4 in WSEL at Point 32 (ft) 12. Point 32 to Point 33 Flow 3Q/8 (ft3/s) Flow depth WSEL @ 32 invert (352 ft) (ft) Flow area 6 – ft width depth (ft2) Velocity flow/area (ft/s) R A/P / (P w 2d) (ft) Condiut loss [(V n)/(1.486 R 2/3)] 2 L (ft) where n 0.014 and L 35 ft – 4 in WSEL at Point 33 (ft) 13. Point 33 to Point 1 Flow Q/2 (ft3/s) Flow depth WSEL @ 33 invert (352 ft) (ft) Flow area 6 ft width depth (ft3) V Velocity flow/area (ft/s) R A/P / (P w 2d) (ft) Condiut loss [(V n)/1.486 R 2/3)] 2 L (ft) where n 0.014 and L 36 ft – 4 in WSEL at Point 1 (ft)
TABLE 22.7
Design Operation
Min Day
Avg Day
Avg Day Max Hour
19.34 8.01 48.06 0.40 2.18
27.08 8.02 48.11 0.56 2.18
29.01 8.02 48.12 0.60 2.18
38.68 8.04 48.22 0.80 2.18
0.00 360.01
0.00 360.02
0.00 360.02
0.00 360.04
29.01 8.01 48.06 0.60 2.18
40.61 8.02 48.11 0.84 2.18
43.52 8.02 48.13 0.90 2.18
58.02 8.04 48.22 1.20 2.18
0.00 360.01
0.00 360.02
0.00 360.02
0.00 360.04
38.68 8.01 48.06 0.80 2.18
54.15 8.02 48.11 1.13 2.18
58.02 8.02 48.13 1.21 2.18
77.36 8.04 48.23 1.60 2.18
0.00 360.01
0.00 360.02
0.00 360.02
0.00 360.04
Example Hydraulic Calculation of a Typical Filter Initial Operation Parameter
Plant Flow (mgd) Filter loading, gpm/ft2 Filter area per filter – 7 out of 8 Filters in Operation (ft2) Flow = loading area (gal/min) (mgd) (ft3/s) Losses through filter effluent piping (Fig. 22.17) 20 in piping (Q): Pipe velocity Q/A / (ft/s) Local losses Exit (0.5) butterfly valves (2 0.25) + 90o Elbows (2 0.4) tee (1.8) 3.6 V 2/2g (ft) R A/P / (d 2/4 )/(d ) dd/4 (ft) Condiut losses L [V/(1.318 V C R 0.63)]1/0.54 where L 20 ft and Hazen-Williams C 120 (ft) 20 in piping (Q/2):
Design Operation
Min Day
Avg Day
Avg Day Max Hour
50 2 1240
70 4 1240
75 6 1240
100 8 1240
2480 3.57 5.53
4960 7.14 11.05
7440 10.71 16.58
9920 14.29 22.10
2.53
5.07
7.60
10.13
0.36 0.42
1.43 0.42
3.23 0.42
5.74 0.42
0.03
0.09
0.20
0.34
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
Water and Wastewater Treatment Plant Hydraulics 22.99 TABLE 22.7
(Continued) Initial Operation Parameter
Min Day
Pipe velocity Q/A (ft/s) Local losses butterfly valve (0.25) (ft) R A/P / (d 2/4 )/(d ) dd/4 (ft) Condiut losses L [V/(1.318 V C R 0.63)]1/0.54 where L 10 ft and Hazen-Williams C 120 (ft) 24 in piping (Q/2): Pipe velocity Q/A, (ft/s) Local losses entrance (1.0) Tee (1.8) 2.8 V 2/2g (ft) Filter (clean) and underdrain losses (obtain from manufacturer) (ft) Total losses (effluent pipe and clean filters) (ft)
Avg Day
Design Operation Avg Day Max Hour
1.27 0.01 0.42
2.53 0.02 0.42
3.80 0.06 0.42
5.07 0.10 0.42
0.00
0.01
0.03
0.05
0.88
1.76
2.64
3.52
0.03
0.13
0.30
0.54
0.30 0.73
0.50 2.20
0.75 4.57
1.10 7.87
Assume that headloss will be allowed to incrase eight ft before the filters are backwashed. A rate controller will be used to maintain a constant flow through the filter. Determine the ranges of available head over which the rate controller will operate. Static head see figure 2.18 WSEL above filters (ft) WSEL in filter effluent conduit, Point 29 (see Example 22–2) Maximum (ft) Minimum (ft) Static head WSEL above filters – WSEL in filter effluent condiut Maximum (ft) Minimum (ft) Available head static head 8 ft Maximum (ft) Minimum (ft)
TABLE 22.8
360.00
360.00
360.00
360.00
346.50 345.00
346.50 345.00
346.50 345.00
346.50 345.00
15.00 13.50
15.00 13.50
15.00 13.50
15.00 13.50
7.00 5.50
7.00 5.50
7.00 5.50
7.00 5.50
Example Hydraulic Calculation of a Typical Bar Screen System Initial Operation Parameter
1. Wastewater flow rate (ft3/s) (mgd) Bar screens Total of number of units Number of units in operation Number of units in standby Flow rate per screen in operation, q (ft3/s) Width of each bar screen, w (ft) 2. At Point 8 Pump wetwell HGL at high water level, HGL7 (ft)
Min Day Avg Day
Design Operation Avg Day Max Hour Max Hour
35.3 23
56.5 37
70.6 46
113.0 73
113.0 73
3 2 1 17.1 8.2
3 2 1 28.3 8.2
3 2 1 35.3 8.2
3 3 0 37.7 8.2
3 3 1 56.5 8.2
330.05
330.05
330.05
330.05
330.05
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
22.100
Chapter Twenty-Two
TABLE 22.8
(Continued) Initial Operation Parameter
Min Day Avg Day
(pump starts at EL 330.05 ft and stops at EL 328.08 (ft)) Pump well bottom EL (ft) Check for critical depth in bar screen channel: Critical depth in a rectangular channel Yc (q2/g / /w2)(1/3) (ft) Bar screen channel depth pump WW HGL channel bottom ELl (ft) (water level at pump well controls upstream hydraulics if normal depth is higher than Yc) Io bar screen channel depth higher than yc 3. Point 8 to Point 7 Channel bottom EL (ft) Depth in channel, y7 (ft) Velocity, V V7 (ft/s) Exit loss from channel to pump well Exit loss coefficient Kexit 1.0 Headloss Kexit V7 V 2/2g, Hle7 (ft) HGL at Point 7, HGL7= HGL8 le(ft) 4. Point 7 to Point 6 Friction headloss through channel Length of approach channel, L6 (ft) Manning's number n for concrete channel Channel width, w6 (ft) Water depth, h6 (ft) Velocity, V V6 (fps) Hydraulic radius, R6 (h6 w6)/(2 h6w6) (ft) Headloss (V6 V xn/1.486 R6 (2/3))2 L6, Hlf6 f (ft) HGL at Point 6, HGL6 HGL7 Hlf6 f (ft) 5. Point 6 to Point 5 Calculate headloss through bar screen Space between bars (ft) Bar width (ft) Bar shape factor, bsf Cross sectional width of bars, w (ft) Clear spacing of bars, b (ft) Upstream velocity head, h (ft) Angle of bar screen with horizontal, p (degrees) Kirschmer’s eq. Hls bsf w/b 1.33 h sin p (ft) Allow 6 in head for blinding by screenings, Ha (ft)
Design Operation Avg Day Max Hour Max Hour
324.80
324.80
324.80
324.80
324.80
0.52 3.61
0.72 3.61
0.83 3.61
0.87 3.61
1.14 3.61
yes
yes
yes
yes
yes
326.44 3.61 0.60
326.44 3.61 0.95
326.44 3.61 1.19
326.44 3.61 1.27
326.44 3.61 1.91
1.0 0.01
1.0 0.01
1.0 0.02
1.0 0.03
1.0 0.06
330.06
330.07
330.07
330.08
330.11
23 0.013 8.20 3.61 0.60
23 0.013 8.20 3.62 0.95
23 0.013 8.20 3.63 1.19
23 0.013 8.20 3.63 1.26
23 0.013 8.20 3.67 1.88
1.92
1.92
1.93
1.93
1.94
0.00
0.00
0.00
0.00
0.00
330.06
330.07
330.08
330.08
330.11
0.06 0.033 2.42 2.93 5.27 0.0134
0.06 0.033 2.42 2.93 5.27 0.0342
0.06 0.033 2.42 2.93 5.27 0.0535
0.06 0.033 2.42 2.93 5.27 0.0608
0.06 0.033 2.42 2.93 5.27 0.1369
60
60
60
60
60
0.02
0.05
0.08
0.09
0.21
0.5
0.5
0.5
0.5
0.5
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
Water and Wastewater Treatment Plant Hydraulics 22.101 TABLE 22.8
(Continued) Initial Operation Parameter
HGL upstream of bar screen, HGL5 HGL6 HlsHa (ft)
Min Day Avg Day
Design Operation Avg Day Max Hour Max Hour
330.58
330.62
330.66
330.67
330.82
6. Point 5 to Point 4 Friction headloss through channel Length of approach channel, L4 (ft) 22.97 Manning’s n for concrete channel 0.013 Channel width, w4 (ft) 8.20 Channel bottom elevation (ft) 326.94 Water depth, h4 (ft) 3.64 Channel velocity, V V4 (ft/s) 0.59 Hydraulic radius R4h4w4/(2h4w4) 1.93 Headloss (V4 V n/1.486 R4(2/3) )2 L4, Hlf4 f (ft) 0.00
22.97 0.013 8.20 326.94 3.68 0.93 1.94
22.97 0.013 8.20 326.94 3.72 1.16 1.95
22.97 0.013 8.20 326.94 3.74 1.23 1.96
22.97 0.013 8.20 326.94 3.89 1.77 2.00
0.00
0.00
0.00
0.00
330.58
330.62
330.66
330.67
330.83
1.0 3.94 2.95 1.23
1.0 3.94 2.95 1.95
1.0 3.94 2.95 2.41
1.0 3.94 2.95 2.56
1.0 3.94 2.95 3.69
HGL at Point 4, HGL4 HGL5 Hlf4, f ft 7. Point 4 to Point 3 Headloss at sluice gate contraction Kgate Sluice gate width (ft) Sluice gate heigth (ft) Velocity through sluice gate, Vs (ft/s) Sluice gate headloss, Hls Kgate Vs 2/2g (ft) HGL at Point 3, HGL3 (ft) 8. Point 3 to Point 2 Water depth at Point 2, h2 (ft) Channel width, w2 (ft) Channel velocity, V V2 (ft/s) Fitting headloss through a 45° bend, Kbend 0.20 V 2/2g, Hlb2 (ft) Headloss Kbend V2 Friction headloss through channel Length of approach channel, L2 (ft) Manning’s n for concrete channel Hydraulic radius R2 h2 w2/ (2 f 2 w2) (ft) Headloss (V n/1.486 R2(2/3))2 L2, Hlf2 f (ft) Entrance loss Kent 0.5 2 Headloss Kent V 2 /2g, Hle2 (ft) HGL at point 2, HGL2 HGL3 Hlb2 Hlf2 f Hle2 (ft) 9. Point 2 to Point 1 HGL at Point 1, HGL 1 HGL2 (ft) Invert EL of inlet sewer, INV1 (ft) Crown EL of inlet sewer, CWN1 (ft) Surcharge to inlet sewer?
0.02
0.06
0.09
0.10
0.21
330.60
330.68
330.75
330.75
331.04
3.67 6.56 0.73
3.74 6.56 1.15
3.81 6.56 1.41
3.84 6.56 1.49
4.10 6.56 2.10
0.20 0.0017
0.20 0.0041
0.20 0.0062
0.20 0.0069
0.0137
13.12 0.013
13.12 0.013
13.12 0.013
13.12 0.013
13.12 0.013
1.73
1.75
1.76
1.77
1.82
0.00
0.00
0.00
0.00
0.00
0.50 0.0042
0.50 0.0103
0.50 0.0155
0.50 0.0174
0.50 0.0342
330.61
330.69
330.77
330.80
331.09
330.61 326.44 333.50 No
330.69 326.44 333.50 No
330.77 326.44 333.50 No
330.80 326.44 333.50 No
331.09 326.44 333.50 No
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
22.102
Chapter Twenty-Two
TABLE 22.9
Example Hydraulic Calculation of a Typical Vortex Grit Tank System Initial Operation Parameter
1. Wastewater flow rate, Q (cfs) (mgd) 2. Vortex grit tanks total number of units Number of units in operation Number of units in standby Flow rate per vortex grit tank in operation, q (cfs)
Min Day Avg Day
Design Operation Avg Day Max Hour
Peak
35.3 23
56.5 36
70.6 46
113.0 73
113.0 73
3 2 1
3 2 1
3 2 1
3 3 0
3 2 1
17.7
28.3
35.3
37.7
56.5
347.79 344.51 17.7 9.84
347.79 344.51 28.3 9.84
347.79 344.51 35.3 9.84
347.79 344.51 37.7 9.84
347.79 344.50 56.5 9.84
0.67 0.67 0.00
0.92 0.92 0.00
1.06 1.06 0.00
1.11 1.11 0.00
1.46 1.46 0.00
348.45
348.70
348.85
348.90
349.25
9.84 344.49 3.96 0.45
9.84 344.49 4.21 0.68
9.84 344.49 4.36 0.82
9.84 344.49 4.41 0.87
9.84 344.49 4.46 1.21
1.0 0.0032
1.0 0.0072
1.0 0.0105
1.0 0.0117
1.0 0.0226
348.46
348.71
348.86
348.91
349.27
8.20 344.49 3.97 0.54
8.20 344.49 4.22 0.82
8.20 344.49 4.37 0.98
8.20 344.49 4.42 1.04
8.20 344.49 4.78 1.44
32.81 0.013
32.81 0.013
32.81 0.013
32.81 0.013
32.81 0.013
2.02
2.08
2.12
2.13
2.21
0.0003
0.0006
0.0009
0.0010
0.0018
1.0
1.0
1.0
1.0
1.0
Control point is located at channel weir Hydraulic Calculations Upstream of Control point 3. At Point 8 Headloss over sharp-crested weir Sharp-crested weir EL, weir EL (ft) Effluent channel bottom EL (ft) Flow rate over weir, q (ft3/s) Length of weir, L (ft) Head over end contracted weir, He (assumed) [q/ 33.3(L–0.2 L He)](2/3) (ft) Hle8 – He (must be zero) HGL at Point 8, HGL8 weir EL Hle8 (ft) 4. Point 8 to Point 7 Channel width, w7 (ft) Channel bottom EL (ft) Water depth, h7 (ft) Velocity, V V7 (ft/s) Exit headloss from channel to effluent weir Exit headloss coefficient Kexit 1.0 2 Headloss, Hle7 Kexit V 7 /2g (ft) HGL at Point 7, HGL7 HGL8 Hle7 (ft) 5. Point 7 to Point 6 Channel width, w6 (ft) Channel bottom EL (ft) Water depth, h6 (ft) Velocity, V V6 (ft/s) Friction headloss through channel Length of approach channel, L6 (ft) Manning’s n for concrete channel Hydraulic radius, R6 (h6 w6)/ (2 h6 w6) (ft) 2 Headloss (V6 V n/R / 6(2/3)) L6, Hlf6 f (ft) Fitting headloss through 90° bend Fitting headloss coefficient Kbend 1.0
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
Water and Wastewater Treatment Plant Hydraulics 22.103 TABLE 22.9
(Continued) Initial Operation Parameter
Min Day Avg Day
Headloss Kbend V6 V /2g, Hlb6 (ft) 2
HGL at Point 6, HGL6 HGL7 Hlf6 f Hlb6 (ft) 6. Point 6 to Point 5 Headloss through sluice gate Sluice gate headloss coefficient Kgate 1.0 Sluice gate width (ft) Sluice gate height (ft) Water depth, h5 (ft) Sluice gate height or h5, whichever smaller Velocity through sluice gate, V V5 (ft/s) 2 Headloss, Hls5 Kgate V5 /2g (ft) HGL at Point 5, HGL5 HGL6 Hls5 (ft) 7. Point 5 to Point 4 Channel width, w4 (ft) Bottom of channel EL (ft) Water depth, h4 (ft) Channel velocity, V V4 (ft/s) Fitting headloss through a 90° bend Fitting headloss coefficient Kbend 1.0 Headloss, Hlb4 Kbend V4 V 2/2g (ft) Friction headloss through channel Length of approach channel, L4, (ft) Manning’s n for concrete channel Hydraulic radius R4 h4 w4/ (2 h4 w4) (ft) Headloss, Hlf4 f (V4 V n/1.486 R4(2/3))2 L4 (ft) HGL at Point 4, HGL4 HGL5 Hlb4 Hlf4 f (ft) 8. Point 4 to Point 3 Headloss across vortex grit tank, H1tank (ft) (per manufacturer recommendations) HGL at Point 3, HGL3 HGL4 H1tank (ft) 9. Point 3 to Point 2 Channel width, w2, (ft) Bottom of channel EL (ft) Water depth, h2 (ft) Channel velocity, V V2 (ft/s) Friction headloss through channel Length of approach channel, L2 (ft) Manning’s n for concrete channel Hydraulic radius R2 h2R4(2/3))2 w2/(2*h2 w2) (ft)
Design Operation Avg Day Max Hour
Peak
0.0046
0.0103
0.0151
0.0168
0.0322
348.46
348.72
348.88
348.93
349.31
1.0 4.92 3.28 3.97
1.0 4.92 3.28 4.22
1.0 4.92 3.28 4.37
1.0 4.92 3.28 4.42
1.0 4.92 3.28 4.78
3.28 1.09 0.0186
3.28 1.75 0.0475
3.28 2.19 0.0743
3.28 2.33 0.0845
3.28 3.50 0.1902
348.48
348.77
348.95
349.01
349.50
8.20 345.14 3.34 0.65
8.20 345.14 3.62 0.95
8.20 345.14 3.81 1.13
8.20 345.14 3.87 1.19
8.20 345.14 4.35 1.58
1.0 0.0065
1.0 0.0140
1.0 0.0199
1.0 0.0219
1.0 0.0389
32.81 0.013
32.81 0.013
32.81 0.013
32.81 0.013
32.81 0.013
1.84
1.92
1.97
1.99
2.11
0.0005
0.0009
0.0013
0.0014
0.0023
348.49
348.78
348.97
349.04
349.54
0.20
0.20
0.20
0.20
0.20
348.68
348.98
349.17
349.23
349.73
6.56 346.46 2.23 1.21
6.56 346.46 2.52 1.71
6.56 346.46 2.71 1.98
6.56 346.46 2.78 2.07
6.56 346.46 3.28 2.63
45.93 0.013
45.93 0.013
45.93 0.013
45.93 0.013
45.93 0.013
1.33
1.43
1.48
1.50
1.64
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
22.104
Chapter Twenty-Two
TABLE 22.9
(Continued) Initial Operation Parameter
Min Day Avg Day
Headloss, Hlf2 f (V2* V n/1.486 R ) L2 (ft) 0.0035 Headloss through sluice gate Sluice gate headloss coefficient Kgate 1.0 1.0 Sluice gate width (ft) 4.9 Sluice gate height (ft) 3.3 Water depth, h2 (ft) 2.23 Sluice gate height or h2, whichever smaller (ft) 2.23 Velocity through sluice gate (ft/s) 1.61 2 Headloss, Hls2 Kgate V2 V /2g (ft) 0.0403
Design Operation Avg Day Max Hour
Peak
(2/3) 2
HGL at point 2, HGL2 HGL3 Hlf2 f Hls2 (ft) 10. Point 2 to Point 1 Channel width, w1 (ft) Bottom of channel EL (ft) Water depth, h1 (ft) Channel velocity, V1 (ft/s) Fitting headloss through a 90° deg bend Fitting headloss coefficient Kbend 1.0 Headloss, Hlb1 Kbend V12/2g (ft) Friction headloss through channel Length of approach channel, L1 (ft) Manning’s n for concrete channel Hydraulic radius R1 h1 w1/(2 h1 w1) (ft) Headloss, Hlf1 f (V1 n/1.486 R1(2/3) )2 L1 (ft)
0.0064
0.0082
0.0087
0.0125
1..0 4.9 3.3 2.52 2.52 2.28 0.0804
1.0 4.9 3.3 2.71 2.71 2.65 0.1087
1.0 4.9 3.3 2.78 2.78 2.76 0.1181
1.0 4.9 3.3 3.28 3.28 3.50 0.1905
348.73
349.07
349.29
349.36
349.94
6.56 346.62 2.11 1.28
6.56 346.62 2.44 1.76
6.56 346.62 2.67 2.02
6.56 346.62 2.74 2.10
6.56 346.62 3.32 2.60
1.0 0.0253
1.0 0.0482
1.0 0.0633
1.0 0.0683
1.0 0.1046
16.40 0.013
16.40 0.013
16.40 0.013
16.40 0.013
16.40 0.013
1.28
1.40
1.47
1.49
1.65
0.0015
0.0025
0.0031
0.0032
0.0043
348.75
349.12
349.35
349.43
350.05
(Influent channel may be aerated using diffused air to prevent solids settling or odor problem) HGL at Point 1, HGL1 HGL2 Hlb1 Hlf1 f (ft)
TABLE 22.10 Example Hydraulic Calculation of a Typical Primary Sedimentation Tank System Initial Operation Design Operation Parameter 1. Wastewater flow rate, Q (ft3/s) (mgd) 2. Primary sedimentation tanks (PSTs) Total number of units Number of units in operation Number of units on standby Flow rate per PTS in operation, q (ft3/s)
Min Day Avg Day
Avg Day Max Hour
Peak
35.3 23
56.5 37
70.6 46
113.0 73
113.0 73
3 2 1 17.7
3 2 1 28.3
3 3 0 23.5
3 3 0 37.7
3 2 1 56.5
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
Water and Wastewater Treatment Plant Hydraulics 22.105 TABLE 22.10
(Continued) Initial Operation Parameter
Min Day Avg Day
Design Operation Avg Day Max Hour
Peak
Control points are located at Points 5 and 6 so that back up from down stream does not flood effluent channel or overflow weir. Hydraulic Calculations beginning at Point 7 3. At Point 7 HGL7 must be equal to HGL1 of aeration tank (ft) 4. At Point 6 Allowance of 0.33 ft from HGL at pipe entrance to bottom of PST effluent trough at discharge end (ft) Elevation of PTS trough bottom at discharge end, EL dcb (ft)
342.73
342.73
342.73
342.73
342.73
0.33
0.33
0.33
0.33
0.33
343.06
343.06
343.06
343.06
343.06
147.6
Calculation of water depth in PST effluent trough Tank diameter, Dt (ft)
147.6
147.6
147.6
147.6
Number of channels per tank nc
2
2
2
2
2
Total flow through tank, q (ft3/s)
17.66
28.25
23.54
37.67
56.50
8.83
14.13
11.77
18.83
28.25
Flow per channel, qc q/nc (ft3/s) Channel slope, Sc (selected to prevent solids setting) Channel width, w6 (ft)
0.20
0.20
0.20
0.20
0.20
3.28
3.28
3.28
3.28
3.28 229.23
Channel length, Lc 3.14 229.23
229.23
229.23
229.23
Change in channel EL, ELdif Sc Lc (ft)
(Dt-(w6/2))/nc (ft)
0.46
0.46
0.46
0.46
0.46
Critical depth, yc (qc2/(g w62))0.33 (ft)
0.62
0.84
0.75
1.02
1.33
Water depth at upstream end of channel, yu
0.69
1.07
0.91
1.38
1.92
343.52
343.52
343.52
343.52
343.52
343.68
343.91
343.81
344.08
344.39
344.21
344.59
344.43
344.90
345.44
0.33
0.33
0.33
.033
0.33
[2 (yc)2 (yc (S L/3) L 2 ]0.5 (2 S L L/3) (ft) Channel bottom El at upstream end of trough, ELucb ELdcb ELdif (ft) HGL at trough downstream, HGL6d ELdcb yc (ft) HGL at trough upstream, HGL6u ELucb yu (ft) 5. Point 6 to Point 5 Allowance to Weir from high trough HGL (ft)
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
22.106
Chapter Twenty-Two
TABLE 22.10
(Continued) Initial Operation Parameter
Weir elevation, Elwe, max. HGL6u allowance (ft) Headloss over V–notch weirs Number of weirs per tank, Nw Tank diameter, Dt (ft) Weir length, Lw (Dt) 3.14 (ft) Hydraulic load, So q/Lw / ,(ft3/s/s/ft) Weir angle, A (°) V–notch height, Vh (ft) V–notch width, Vw 2 (TAN(A ( /2)) Vh (ft) Space between notches, Esv (ft) Number of notches per weir, nv Lw/(Ew Esv) Flow per notch, Qcw q/nv Weir coefficient for 90° notch, Cw Water depth over the weir, hle5 (Qcw/Cw)(1/2.48) (ft) hle5 Vh? (if not, need to readjust calculations) HGL at point 5, HGL5 ELwe hle5, (ft) 6. Point 5 to Point 4 Headloss through primary sedimentation tanks Number of tanks, Nt Flow per tank, q (ft3/s) Tank diameter, Dt (ft) Side water depth, Dsw (ft) Tank bottom elevation, ELt HGL5 Dsw (ft) Tank floor slope, St (%) Minimum floor tank elevation, Eltf 0.0833 (Dt/2) t ELt (ft) Headloss through tank, hlt4 t (ft) (Available from equipment manufacturer) HGL at point 4, HGL4 HGL5 hlt4 t (ft) 7. Point 4 to Point3 Headloss through PST influent pier Pier diameter, Dp 42 in Pier length, Lp (ft) Velocity, V3 V Qt/(3.14 t (Dp/2)2 ) (ft/s) Hazen-Williams coefficient, Cp Hydraulic radius, Rp Dp/4 (ft)
Min Day Avg Day
Design Operation Avg Day Max Hour
Peak
345.77
345.77
345.77
345.77
345.77
1 147.64 463.58 0.0381 90.00 0.33
1 147.64 463.58 0.0609 90.00 0.33
1 147.64 463.58 0.0508 90.00 0.33
1 147.64 463.58 0.0813 90.00 0.33
1 147.64 463.58 0.1219 90.00 0.33
0.66 0.10
0.66 0.10
0.66 0.10
0.66 0.10
0.66 0.10
614 0.0288 2.43 0.17
614 0.0460 2.43 0.20
614 0.0383 2.43 0.19
614 0.0614 2.43 0.23
614 0.0920 2.43 0.27
Yes
Yes
Yes
Yes
Yes
345.93
345.97
345.95
345.99
346.03
2 17.66 147.64 14.11
2 28.25 147.64 14.11
3 23.54 147.64 14.11
3 37.67 147.64 14.11
2 56.50 147.64 14.11
331.85 8.33 325.70
331.85 8.33 325.70
331.85 8.33 325.70
331.85 8.33 325.70
331.85 8.33 325.70
0.16
0.16
0.16
0.16
0.16
346.10
346.13
346.12
346.16
346.20
3.51 21.33 1.83 120 0.88
3.51 21.33 2.92 120 0.88
3.51 21.33 2.43 120 0.88
3.51 21.33 3.89 120 0.88
3.51 21.33 5.84 120 0.88
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
Water and Wastewater Treatment Plant Hydraulics 22.107 TABLE 22.10
(Continued) Initial Operation Parameter
Min Day Avg Day
Design Operation Avg Day Max Hour
Peak
Slope, Sp [V3/(1.318 V Cp Rp(0.63))](1/0.54) (%) Headloss, Hlf3 f Lp Sp (ft) Exit headloss from pier Exit headloss coefficient Kexit 1.0 2 Headloss, hle3 K V3 V /2g (ft)
0.03 0.0064
0.07 0.0153
0.05 0.0109
0.12 0.0261
0.26 0.0552
1 0.0517
1 0.1324
1 0.0920
1 0.2354
1 0.5297
HGL at Point 3, HGL3 HGL4 Hlf3 f hle3 (ft)
346.16
346.28
346.22
346.42
346.78
3
3
3
3
3
1 3.94 17.66 1.45
1 3.94 28.25 2.32
1 3.94 23.54 1.93
1 3.94 37.67 3.10
1 3.94 56.50 4.64
120 0.98 229.7
120 0.98 229.7
120 0.98 229.7
120 0.98 229.7
120 0.98 229.7
0.02 0.0395
0.04 0.0942
0.03 0.0672
0.07 0.1605
0.15 0.3402
0.05 0.0164
0.05 0.0419
0.05 0.0291
0.05 0.0744
0.05 0.1674
346.21
346.42
346.32
346.65
347.29
3.94 17.66 1.45
3.94 28.25 2.32
3.94 23.54 1.93
3.94 37.67 3.10
3.94 56.50 4.64
0.50 0.0164
0.50 0.0419
0.50 0.0291
0.50 0.0744
0.50 0.1674
346.23
346.46
346.35
346.73
347.46
0.33
0.33
0.33
0.33
0.33
347.79
347.79
347.79
347.79
347.79
8. Point 3 to Point 2 Total number of pipes Number of pipes per primary sedimentation tank Pipe diameter, Dp (ft) Flow per pipe, q (cfs) Velocity, V V2 Friction headloss through primary sedimentation tank influent pipe Hazen-Williams coefficient, Cp Hydraulic radius, Rp Dp/4 (ft) Length of pipe, Lp (ft) Slope, Sp [V2/(1.318 V Cp (1/0.54) (%) Rp (0.63))] Headloss, hlf2 f Lp Sp (ft) Fitting headloss through two 45° bends Fitting headloss coefficient Kbend 0.5 Headloss, hlb2 K V2 V 2/2g (ft) HGL at Point 2, HGL2 HGL3 hlb2 hlf2 f (ft) 9. At Point 1 Entrance headloss from primary sedimentation tank influent distribution box to influent pipe Pipe diameter, Dp (ft) Flow per pipe, q (ft3/s) Velocity, V1 (ft3/s) Entrance headloss coefficient Kentrance 0.5 Headloss, Hle1 Kentrance V12/2g (ft) HGL at point 1, HGL1 HGL2 Hle1 (ft) Allowance to grit tank effluent weir from maximum HGL1, Hall (ft) Grit tank effluent elevation, ELgr HGL1 Hall (ft)
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
22.108
Chapter Twenty-Two
TABLE 22.11
Example Hydraulic Calculation of a Typical Final Sedimentation Tank Initial Operation Parameter
1. Wastewater flow rate, Q (ft3/s) (mdg) RAS flow, % of average day flow RAS flow, Qras Q RAS flow/100 (ft3/s) Final sedimentation tank influent flow, Qin (ft3/s)
Min Day Avg Day
Design Operation Avg Day Max Hour
Peak
35.31 23 20 11.30
56.50 37 50 28.25
70.63 46 50 35.31
113.01 73 100 70.63
113.01 73 100 70.63
46.62
84.76
105.94
183.64
183.64
Final sedimentation tank effluent flow, Qeff (ft3/s)
35.31
56.50
70.63
113.01
113.01
Final sedimentation tanks Total number of units Number of units in operation Number of units on standby Tank width (ft) Influent per operating tank, qin (ft3/s) Effluent per operating tank, qeff (ft3/s)
4 3 1 52 15.54 11.77
4 3 1 52 28.25 18.83
4 3 1 52 35.31 23.54
4 4 0 52 45.91 28.25
4 3 1 52 61.21 37.67
20 23.0 459.3 90.0 0.33
20 23.0 459.3 90.0 0.33
20 23.0 459.3 90.0 0.33
20 23.0 459.3 90.0 0.33
20 23.0 459.3 90.0 0.33
0.66 0.10
0.66 0.10
0.66 0.10
0.66 0.10
0.66 0.10
608 0.0194 2.43
608 0.0310 2.43
608 0.0387 2.43
608 0.0465 2.43
608 0.0620 2.43
0.14
0.17
0.19
0.20
0.23
Yes
Yes
Yes
Yes
Yes
339.16
339.16
339.16
339.16
339.16
339.30
339.33
339.35
339.36
339.38
0.00
0.00
0.00
0.00
0.00
339.30
339.33
339.35
339.36
339.38
2. Select control Point at Point 3 (where effluent wiers are located) Hydraulic calculations downstream of control point At Point 3 V-notch weir Number per tank, Nw Individual weir length, Lw (ft) Total weir length, Lwt Lw Nw (ft) Weir angle, A° V-notch height, Vh (ft) V-notch width, Vw 2 (TAN(A ( /2)) Vh (ft) Space between notches, Esv (ft) Total number of notches per tank, nv Lwt/( t Vw Esv) Flow per notch, Qcw qeff /nv Weir coefficient for 90° notch, Cw Water depth over the weir, hle3 (Qcw/Cw)(1/2.48) (ft) hle3 Vh? (if not, need to readjust calculations) Weir EL (ft) (select weir elevation so that HGL1 equals aeration tank’s HGL6) HGL at Point 3, EGL3 Weir EL hle3 (ft) Velocity head, HV 0 (assume V3 V 0) (ft) HGL at point 3, HGL3 weir EL hle3 (ft)
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
Water and Wastewater Treatment Plant Hydraulics 22.109 TABLE 22.11
(Continued) Initial Operation Parameter
Min Day Avg Day
3. Point 3 to Point 4 Effluent troughs Number of troughs, nt 10 Flow per trough, qt qeff /nt 1.18 Trough slope, St (%) (select to prevent solids settling) 0.20 Trough width, w6 (ft) 1.6 Approximate trough length, Lt (ft) 23.0 Change in trough EL, difEL4 St Lt (ft) 0.05 Critical depth, yc (qt 2/(gw62)0.33 (ft) 0.26 Water depth at upstream end of trough for free fall 0.41 from trough into final effluent channel yu4 [2 (yc)2 (yc-(S L/3)) L 2]0.5 (2 S L L/3) (ft) Max water EL downstream of weir (occuring at max, hour flow with one tank out of service), Elmax4 weir EL-0.33 ft (see Point 3 for weirEL) Trough bottom EL at upstream end of trough, TbuEL4 ft 337.90 TbuEL4 ELmax4 yu for max hour flow with one tank out of service HGL at upstream end, HGL4u TbuEL4 yu4 (ft) Velocity head, HV4 V u0 (assume V 0) (ft) EGL at upstream end, EGL4u HGL4u HV4 V u (ft) Trough bottom EL at downstream end of trough Tbd EL4 TbuEL4 dif EL4 (ft) HGL at Point 4, HGL4 TbdEL4 yc (ft) Velocity head, HV4 V d = Vc2/ 2g (ft) EGL at upstream end, EGL4u HGL4u HV4 V u, ft 4. Point 4 to Point 5 Effluent channel upstream Max. water surface level at upstream end of effluent channel, ELmax5 = TbdEL4-0.33 (ft) HGL at Point 5, HGL5 ELmax5 (ft) Velocity head, HV5 V 0 (assume V 0) (ft) EGL maximum at point 5, EGL5m HGL5m HV5 V (ft) 5. Point 5 to Point 6 Effluent channel downstream Flow through channel, Qeff (ft3/s)
Design Operation Avg Day Max Hour
Peak
10 1.88
10 2.35
10 2.83
10 3.77
0.20 1.6 23.0 0.05 0.35
0.20 1.6 23.0 0.05 0.41
0.20 1.6 23.0 0.05 0.46
0.20 1.6 23.0 0.05 0.56
0.57
0.67
0.76
0.93
338.83
337.90
337.90
337.90
337.90
338.31
338.47
338.57
338.66
338.83
0.00
0.00
0.00
0.00
0.00
338.31
338.47
338.57
338.66
338.83
337.86
337.86
337.86
337.86
337.86
338.12 0.39
338.21 0.54
338.27 0.63
338.32 0.71
338.41 0.87
338.51
338.75
338.90
339.03
339.28
337.53
337.53
337.53
337.53
337.53
337.53 0.00
337.53 0.00
337.53 0.00
337.53 0.00
337.53 0.00
337.53
337.53
337.53
337.53
337.53
35.31
56.50
70.63
113.01
113.01
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
22.110
Chapter Twenty-Two
TABLE 22.11
(Continued) Initial Operation Parameter
Min Day Avg Day
Channel slope, Sc (%) (select to prevent solids settling) 0.20 Channel width, w6 (ft) 9.8 Approximate channel length, Lch (ft) 210.0 Change in channel EL, difEL6 Sc Lch (ft) 0.42 Critical depth, yc (q2/(gw62)0.33 (ft) 0.75 Water depth at upstream end of channel, 0.94 yu6 [2 (yc)2 (yc (S L/3))2]0.5 (2 S L/3), ft Channel bottom EL at upstream end of channel, 335.10 cbuEL6 HGL5- maximum yu6 (ft) HGL at upstream end of channel, HGL5 cbuEL6 yu6 (ft) Velocity head, HV5 V 0 (assume V 0) (ft) EGL at upstream end of channel, EGL5 HGL5 HV5 V (ft) Channel bottom EL at downstream end of channel, cbdEL6 cbuEL6 difEL6 (ft) HGL at Point 6, HGL6 cbdEL6 yc (ft) Velocity head, HV6 V Vc2/2g (ft) EGL at Point 6, EGL6 HGL6 HV6 V (ft) 6. At Point 7 Max. water EL downstream of channel end free-fall HGL at Point 7, HGL7 cbdEL6 0.33 (ft) (This must be the same as maximum elevation at Point 1 of multi-media filter.)
Design Operation Avg Day Max Hour
Peak
0.20 9.8 210.0 0.42 1.02 1.41
0.20 9.8 210.0 0.42 1.18 1.69
0.20 9.8 210.0 0.42 1.61 2.43
0.20 9.8 210.0 0.42 1.61 2.43
335.10
335.10
335.10
335.10
336.04
336.51
336.79
337.53
337.53
0.00
0.00
0.00
0.00
0.00
336.04
336.51
336.79
337.53
337.53
334.68
334.68
334.68
334.68
334.68
335.42 1.17 336.60
335.70 1.62 337.31
335.86 1.88 337.74
336.29 2.59 338.88
336.29 2.59 338.88
334.35
334.35
334.35
334.35
334.35
3
3
3
4
3
15.54 52.5 393.7 325.4 13.92
28.25 52.5 393.7 325.4 13.95
35.31 52.5 393.7 325.4 13.97
45.91 52.5 393.7 325.4 13.98
61.21 52.5 393.7 325.4 14.01
Hydraulic calculations upstream of control point 7. At Point 2 Final sedimentation tanks (Gould type) Number of tanks in operation, nt Flow per tank upstream of sludge collection, qin (ft3/s) Tank width, Wt (ft) Tank length, Lt (ft) Tank bottom elevation at influent end (ft) Side water depth (ft) Assume friction losses, Hlf2, f through tank are negligible EGL at Point 2, EGL2 EGL3 Hlf2 f (ft) Velocity head, HV2 V 0 (assume V 0) (ft) HGL at Point 2, HGL2 EGL3 HV2 V (ft)
0.0
0.0
0.0
0.0
0.0
339.30 0.00 339.30
339.33 0.00 339.33
339.35 0.00 339.35
333.36 0.00 333.36
339.38 0.00 339.38
8. Point 2 to Point 1
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
Water and Wastewater Treatment Plant Hydraulics 22.111 TABLE 22.11
(Continued) Initial Operation Parameter
Tank influent sluice gates Height (ft) Width, Ws (ft) Area (ft2) Number of sluice gates per tank, Nsg Flow per sluice gate, qsg qin /Nsg / , (ft3/s) Upstream head over weir, Du (select so Qsub qsg 0) (ft) Downstream head over weir, Dd (qsg/3.33/Ws')(2/3) (ft) Effective sluice gate width, Ws' Ws (0.1)(2 contractions)(Dd) (ft) Free fall flow, Qfree 3.34 Ws' Du (3/2) (ft3/s) Submerged flow, Qsub Qfree (1 (Dd/ d/Du)3/2 )0.385 (ft3/s) Difference, (Qsub qsg), ft3/s Head difference between tank and channel, Hl 1 Du Dd (ft) Top of sluice gate set elevation, Els HGL2 Dd (ft) HGL at Point 1 (upstream of sluice gate), HGL1 HGL2 Hll (ft) Velocity head, HV1=0 (assume V 0) (m) EGL at point 1, EGL1 HV1 (m)
Min Day Avg Day
Design Operation Avg Day Max Hour
3.3 3.3 10.8 4
3.3 3.3 10.8 4
3.3 3.3 10.8 4
3.3 3.3 10.8 4
3.3 3.3 10.8 4
3.88
7.06
8.83
11.48
15.30
0.67
1.02
1.19
1.43
1.76
0.51 3.2
0.77 3.1
0.90 3.1
1.09 3.0
1.33 3.0
5.89
10.70
13.38
17.39
23.18
3.89 0.00
7.07 0.00
8.83 0.00
11.48 0.00
15.30 0.00
0.163
0.246
0.287
0.347
0.426
338.79 339.46
338.56
338.45
338.27
338.05
339.46 0.00 339.46
339.58 0.00 339.58
339.63 0.00 339.63
339.71 0.00 339.71
339.81 0.00 339.81
Maximum HGL1 (ft) Max HGL1 should equal HGL 6 for aeration tank.
TABLE 22.12
Peak
339.81
Example Hydraulic Calculation of a Typical Aereation Tank System Initial Operation Parameter
1. Wastewater flow rate, Q (ft3/s) (mgd) RAS flow, % of average flow (added downstream of aeration tank influent sluice gates) RAS flow, Qras Q RAS flow/100 (ft3/s) 2. Aeration tanks Total of nunber of units Number of units in operation Number of units on standby
Min Day Avg Day
Design Operation Avg Day Max Hour
Peak
35.31 23
56.50 37
70.63 46
113.01 73
113.01 73
20
50
50
100
100
11.30
28.25
35.31
70.63
70.63
3 2 1
3 2 1
3 3 0
3 3 0
3 2 1
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
22.112
Chapter Twenty-Two
TABLE 22.12
(Continued) Initial Operation
Parameter Flow rate per aeration tank in operation, q (ft3/s) Flow rate per aeration tank in operation including RAS flow (downstream of influent sluice gate), qras, (ft3/s)
Design Operation
Min Day Avg Day Avg Day Max Hour
Peak
17.66
28.25
23.54
37.67
56.50
23.31
42.38
35.31
61.21
91.82
339.79
339.79
339.79
337.79
339.79
340.12 330.28 23.31 19.69
340.12 330.28 42.38 19.69
340.12 330.28 35.31 19.69
340.12 330.28 61.21 19.69
340.12 330.28 91.82 19.69
0.50
0.75
0.66
0.95
1.25
340.63
340.87
340.79
341.08
341.37
0.09
0.13
0.11
0.17
0.22
340.71
341.00
340.90
341.24
341.59
23.31 19.7 196.9
42.38 19.7 196.9
35.31 19.7 196.9
61.21 19.7 196.9
91.82 19.7 196.9
320.94
320.94
320.94
320.94
320.94
19.69 5 984.3 0.06
19.94 5 984.3 0.11
19.85 5 984.3 0.09
20.14 5 984.3 0.15
20.44 5 984.3 0.23
0.35
0.52
0.46
0.67
0.88
0.013
0.013
0.013
0.013
0.013
6.56
6.59
6.58
6.61
6.64
Control point is located at Point 5 (aeration tank effluent weir). 3. At Point 6 Set maximum HGL6 effluent weir elevation0.33 (ft) Hydraulic calculations upstream of control point 4. Point 6 to Point 5 Headloss over sharp-crested weir Sharp-crested weir EL (ft) Effluent channel bottom EL (ft) Flow rate over weir, qras (ft3/s) Length of weir (ft) headloss, Hle5 (q/3.33L)(2/3) (ft) HGL at Point 5, HGL5 weir EL Hle5 (ft) Velocity head, HV5 V (gras/Wp/Hle / 5)2/2g (ft) EGL at Point 5, EGL5 HGL5 HV5 V (ft) 5. Point 5 to Point 4 Flow rate per aeration tank in operation, qras (ft3/s) Pass width, Wp (ft) Tank length, Lt (ft) Tank bottom elevation, ELtb avg. day WSEL 19.69 (ft) Water depth in tank at design average flow, Dt (ft) Number of passes per tank, Np Effective length of tank, L Lt Np (ft) Velocity, V V4 (ft/s) Critical depth, yc ((q2/g / /Wp2 ))(0.333) (ft) Friction headloss thruogh aeration tank channel Manning's n for concrete channel Hydraulic radius, R (Dt Wp)/(2 Dt Wp) (ft) Headloss, Hlf4 f (V4 V n/
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
Water and Wastewater Treatment Plant Hydraulics 22.113 TABLE 22.12
(Continued) Initial Operation
Parameter 1.486 R (2/3))2 L (ft) Fitting headloss through a 90° bend Fitting headloss coefficient Kbend 1.0 Number of bends, Nb Headloss, Hlb4 Kbend V 2/2g (ft) V4 Velocity head, Hvsd (see below at Point 3) RAS flow, % of average flow EGL at Point 4, EGL4 EGL5 Hlf4 f Hvsd (ft) Velocity head, Hvsd (see below at point 3) HGL at point 4, HGL4 EGL4 HV4 (ft)
6. Point 4 to Point 3 Headloss over aeration tank influent sluice gates Sluice gate width, Ws (ft) Sluice gate heigth (ft) Flow per sluice gate, q (ft3/s) Upstream head over weir, Du (select so Zsub q 0) (ft) Downstream head over weir, Dd (q/3.33/Ws')(2/3) (ft) Effective sluice gate width, Ws' Ws (0.33)(2 contractions)(Du) (ft) Free fall flow, Qfree 3.34 Ws'Du (3/2) (ft3/s) Submerged flow, Qsub Qfree (1-(Dd/ d/Du)0.385 (ft3/s) Difference, (Qsub q), ft3/s (should de zero) Head difference between tank and channel, Hl4 Du Dd (ft) Velocity head downstream of sluice gate, HVsd (q/ Ws'/Dd / )2/2g, Velocity head upstream of sluice gate, HVsu (q/ Ws'/Du / )2/2g (ft) Top of sluice gate elevation, Els HGL4 Dd (ft) HGL upstream of sluice gate, HGLsu HGL4 Hl4 (ft) EGL upstream of sluice gate, EGLsu HGLsu HVsu (ft) Friction headloss through influent channel to tank #3 Average length of influent channel per tank, L3 Np Wp 3 tanks1/2 (ft) Influent channel width, W W3 (ft)
Design Operation
Min Day Avg Day Avg Day Max Hour
Peak
0.0000
0.0001
0.0000
0.0001
0.0003
1.0 8
1.0 8
1.0 8
1.0 8
1.0 8
0.0004
0.0014
0.0010
0.0030
0.0065
0.74
1.03
0.90
1.28
1.76
341.45 0.74 340.71
342.03 1.03 341.00
341.81 0.90 340.90
342.53 1.28 341.25
343.35 1.76 341.60
3.9 3.3 17.66
3.9 3.3 28.25
3.9 3.3 23.54
3.9 3.3 37.67
3.9 3.3 56.50
1.71
2.39
2.10
2.97
4.07
1.29
1.82
1.59
2.25
3.08
3.60
3.46
3.52
3.34
3.12
26.75
42.80
35.67
57.07
85.61
17.66 0.00
28.25 0.00
23.54 0.00
37.67 0.00
56.51 0.00
0.41
0.58
0.51
0.72
0.98
0.22
0.31
0.28
0.39
0.53
0.13
0.18
0.16
0.22
0.31
339.42
339.19
339.31
339.00
338.51
341.13
341.58
341.41
341.96
342.58
341.25
341.76
341.57
342.19
342.89
103.3
103.3
103.3
103.3
103.3
13.1
13.1
13.1
13.1
13.1
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
22.114
Chapter Twenty-Two
TABLE 22.12
(Continued) Initial Operation
Parameter Manning's n for concrete channel Influent channel bottom elevation, Elb ave.EGLsu 9.84 (ft) Water depth in influent channel, h3 HGLs Elb (ft) Hydraulic radius, R (h3 w3)/(2 h3 w3) (ft) Velocity, V3 V q/w3/h3 (ft/s) Headloss, Hlf3 f (V3 V n/1.486 R (2/3) )2 L3 (ft) Friction headloss through influent channel to tank #2 Flow rate, q2 2 q (ft3/s) Velocity, V2 V q/w2/h2 (ft/s) Headloss, Hlf2 (V2 n/1.486 R(2/3) )2 L3, m Friction headloss through influent channel to tank #1 Flow rate, q1 3 q (ft3/s) Velocity, V1 q/w1/h1 (ft/s) Headloss, Hlf1 f (V1 n/1.486 R(2/3))2 L3 (ft) HGL at Point 3, HGL3 HGLs Hlf3 f Hlf2 f Hlf1 f (ft) 7. Point 3 to Point 2 Headloss through sluice gate Sluice gate headloss coefficient Kgate 1.0 RAS flow, % of average flow (added downstream of aeration tank influent sluice gates) RAS flow, Qras Q RAS flow/100, cfs Sluice gate width, W W2 (ft) Sluice gate heigth, Hg (ft) Channel water depth, Dc (ft) Gate opening depth, Hg or Dc whichever is smaller (ft) Velocity through sluice gate, V5 V Q/W2 W (ft/s) Headloss, Hls2 Kgate V5 V 2/2g (ft) HGL at point 2, HGL2 HGL3 Hls2 (ft) 8. Point 2 to Point1 Allowance Exit headloss from primary sed, tank effluent pipe to aeration tank influent channel Primary effluent pipe diameter, Dp (ft) All PST effluent flow, Q 9 (ft3/s)
Design Operation
Min Day Avg Day Avg Day Max Hour
Peak
0.013
0.013
0.013
0.013
0.013
331.7
331.7
331.7
331.7
331.7
9.40
9.86
9.68
10.24
10.86
3.86 0.14
3.94 0.22
3.91 0.19
4.00 0.28
4.09 0.40
0.0000
0.0001
0.0000
0.0001
0.0002
35.31 0.29
56.50 0.44
47.09 0.37
75.34 0.56
113.01 0.79
0.0001
0.0002
0.00020.
0004
0.0008
35.31 0.29
56.50 0.44
70.63 0.56
113.01 0.84
113.01 0.79
0.0001
0.0002
0.0004
0.0009
0.0008
341.13
341.58
341.41
341.97
342.58
1.0
1.0
1.0
1.0
1.0
20
50
50
100
100
11.30
28.25
35.31
70.63
70.63
5.91 5.91 9.40
5.91 5.91 9.86
5.91 5.91 9.69
5.91 5.91 10.24
5.91 5.91 10.86
5.91
5.91
5.91
5.91
5.91
1.01
1.62
2.03
3.24
3.24
0.0159
0.0408
0.0637
0.1630
0.1630
341.14
341.62
341.47
342.13
342.75
6.56 35.31
6.56 56.50
6.56 70.63
6.56 113.01
6.56 113.01
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
Water and Wastewater Treatment Plant Hydraulics 22.115 TABLE 22.12
(Continued) Initial Operation
Parameter Velocity, V1 (ft/s) Exit headloss coefficient Kexit 1.0 Exit headloss, hle1 (V12)/2/g / Kexit (ft) Friction headloss through PST effluent pipe section 2 Flow per pipe, Q (ft3/s) Pipe diameter, Dp2 (ft) Velocity, V12 (ft/s) Hazen–Williams coefficient, Cp Hydraulic radius, Rp2 Dp2/4 (ft) Length of pipe, Lp2 (ft) Slope, Sp2 [v12/(1.318 Cp Rp2(0.63))](1/0.54) (%) Headloss, hlf2 f Lp2 Sp2 (ft) Friction headloss through PST effluent pipe section 1 Flow per pipe, q (ft3/s) Pipr diameter, Dp1 (ft) Velocity, V11 (ft/s) Hazen-Williams coefficient, Cp Hydraulic radius, Rp1 Dp1/4 (ft) Length of pipe, Lp1 (ft) Slope, Sp1 [v11/(1.318 Cp Rp1(0.63))](1/0.54) (%) Headloss, hlf1 f Lp1 Sp1 (ft) Pipe entrance head loss Ke Head loss, hen1 Ke V112/2g (ft) HGL at upstream of PST effluent pipe, HGL1 HGL2 hle1 hlf2 f hlf1 f hen1 (ft) HGL7 of PST must be maximum of HGL1 (ft)
TABLE 22.13
Design Operation
Min Day Avg Day Avg Day Max Hour
Peak
1.04 1.0
1.67 1.0
2.09 1.0
3.34 1.0
3.34 1.0
0.0169
0.0434
0.0677
0.1734
0.1734
35.31 6.56 1.04 120.00 1.64 164.04
56.50 6.56 1.67 120.00 1.64 164.04
70.63 6.56 2.09 120.00 1.64 164.04
113.01 6.56 3.34 120.00 1.64 164.04
113.01 6.56 3.34 120.00 1.64 164.04
0.0001 0.0084
0.0001 0.0202
0.0002 0.0305
0.0004 0.0728
0.0004 0.0728
17.66 4.92 0.93 120.00 1.23 164.04
28.25 4.92 1.49 120.00 1.23 164.04
23.66 4.92 1.24 120.00 1.23 164.04
37.79 4.92 1.99 120.00 1.23 164.04
56.50 4.92 2.97 120.00 1.23 164.04
0.0001 0.0095
0.0001 0.0227
0.0001 0.0163
0.0002 0.0389
0.0005 0.0819
0.50 0.0067
0.50 0.0171
0.50 0.0120
0.50 0.0306
0.50 0.0685
341.18
341.73
341.60
342.44
343.14
343.14
343.14
343.14
343.14
343.14
Example Hydraulic Calculations of a Typical Multimedia Filter System Initial Operation Parameter
1. Wastewater flow rate, Q (ft3/s) (mgd) 2. Multimedia filters Total number of units Number of units in operation Number of units on standby Flow rate per operating multimedia filter, q (ft3/s)
Min Day Avg Day
Design Operation Avg Day Max Hour
Peak
35.3 23
56.5 37
70.6 46
113.0 73
113.0 73
6 4 2
6 5 1
6 5 1
6 6 0
6 5 1
8.83
11.30
14.13
18.83
22.60
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
22.116
Chapter Twenty-Two
TABLE 22.13
(Continued) Initial Operation
Parameter
Min Day Avg Day
Design Operation
Avg Day Max Hour
Peak
Hydraulic calculations at filter effluent 3. At Point 7 Max HGL in filtered water storage tank, HGL7 (ft) Velocity in storage tank, V V7 (ft/s) Max EGL in storage tank, EGL7 HGL7 V7 V 2/2g (ft)
323.72 0.00
323.72 0.00
323.72 0.00
323.72 0.00
323.72 0.00
323.72
323.72
323.72
323.72
323.72
4. At Point 6 Filtered water effluent channel weir Sharp-crested weir EL, Wel6 HGL7 0.33 (ft) Flow rate over weir Q (ft3/s) Length of weir (ft) Headloss, Hlw6 (q/3.33L)(2/3) (ft)
324.05 35.31 22.97 0.60
324.05 56.50 22.97 0.82
324.05 70.63 22.97 0.95
324.05 113.01 22.97 1.30
324.05 113.01 22.97 1.30
324.65
324.87
325.00
325.35
325.35
0.00
0.00
0.00
0.00
0.00
324.65
324.87
325.00
325.35
325.35
35.31 9.84 6.56 32.81 0.55
56.50 9.84 6.56 32.81 0.87
70.63 9.84 6.56 32.81 1.09
113.01 9.84 6.56 32.81 1.75
113.01 9.84 6.56 32.81 1.75
1.97 0.013
1.97 0.013
1.97 0.013
1.97 0.013
1.97 0.013
0.0003
0.0008
0.0012
0.0031
0.0031
3.3 8.83 1.04
3.3 11.30 1.34
3.3 14.13 1.67
3.3 18.83 2.23
3.3 22.60 2.67
0.0170
0.0278
0.0434
0.0772
0.1111
324.66
324.89
325.04
325.43
325.46
0.02
0.03
0.04
0.08
0.11
324.65
324.87
325.00
325.35
325.35
2.95
2.95
2.95
2.95
2.95
HGL at Point 6, HGL6 Wel6 Hlw6 (ft) Velocity in weir box, V V6, m (assume V 0) (ft) EGL at Point 6, EGL6 HGL6 V6 2/2g (ft) 5. Point 6 to Point 5 Loss through effluent concrete condiut Flow rate, Q (ft3/s) Width of condiut, Wc (ft) Depth of condiut, Dc (ft) Length of condiut, Lc (ft) Velocity, Vc (ft/s) Hydraulic radius, R Wc Dc/2/(Wc Dc) (ft) Manning's n Headloss, Hlc5 (Vc n/1.486 R(2/3) )2 Lc (ft) Exit loss from pipe to concrete conduit Effluent pipe diameter, Dp (ft) Pipe flow (for each filter) (ft3/s) Velocity, Vp (ft/s) Hle5 Vp 2/2g for sharp concrete outlet (ft) EGL at Point 5, EGL5 EGL6 Hlc5 Hle6 (ft) Velocity head at Point 5, HV5 V Vp2/2g (ft) HGL at Point 5, HGL5 EGL5 HV5 V (ft) 6. Point 5 to Point 4 Filter effluent pipe loss Pipe diameter, Dp (ft)
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
Water and Wastewater Treatment Plant Hydraulics 22.117 TABLE 22.13
(Continued) Initial Operation
Parameter Max flow through filter effluent pipe q (ft3/s) Velocity of flow through pipe, Vp (ft/s) Hazen-Williams coefficient, Cp Hydraulic radius, Rp Dp/4 (ft) Length of pipe, Lp (ft) Slope, Sp [Vp/(1.318 Cp Rp0.63)](1/0.54) (%) Head loss, Hlf4 f Lp Sp (ft) Headloss through butterfly valve Kvalve (fully open) Valve diameter (ft) Headloss, hval4 Kvalve (Vp2/2g) (ft) Flow rate controller Venturi throat-to-inlet ration for long tube, Krate Inlet velocity, Vi V Vp (ft/s) Headloss, hrate Krate (Vi 2/2g) (ft) (minimum headloss when control valve is fully open) Pipe entrance loss Kent Headloss, Hlent Kent (Vp2/2g) (ft) EGL at Point 4, EGL4 EGL5 Hlf4 f Hval4 Hrate Hlent (ft) Velocity head, HV4 V V 2/2g, (assume V 0) (ft) V4 HGL at Point 4, HGL4 EGL4 HV4 V (ft) 7. Point 4 to Point3 Dirty filter head requirement, Hldf, f (ft) (assumed) (consult with filter manufacturer) Dirty filter HGL, HGLdf HGL4 Hldf (ft) Velocity head, HV3 V 0 (assume V3 V 0) (ft) Dirty filter HGL, HGLdf EGLdf HV3 V (ft) Clean filter headloss Filter bed area (ft2) Flow per filter, q (ft3/s) Filter rate, qfilt, (ft3/ min/ft2) Media depth, Dm (ft) Effective media size, Md (in) Headloss through filter, Hlf 2.32 ft loss per (ft3/ min/ft2)(consultant with manufacturer)
Min Day Avg Day
Design Operation
Avg Day Max Hour
Peak
8.83
11.30
14.13
18.83
22.60
1.29 120 0.74 49.21
1.65 120 0.74 49.21
2.06 120 0.74 49.21
2.75 120 0.74 49.21
3.30 120 0.74 49.21
0.0193 0.0095
0.0305 0.0150
0.0461 0.0227
0.0786 0.0387
0.1102 0.0542
0.30 2.95 0.0078
0.30 2.95 0.0127
0.30 2.95 0.0198
0.30 2.95 0.0353
0.30 2.95 0.0508
1.20 1.04 0.0203
1.20 1.34 0.0333
1.20 1.67 0.0521
1.20 2.23 0.0926
1.20 2.67 0.1333
0.50 0.0085
0.50 0.0139
0.50 0.0217
0.50 0.0386
0.50 0.0555
324.71
324.97
325.16
325.63
325.75
0.00
0.00
0.00
0.00
0.00
324.71
324.97
325.16
325.63
325.75
8.2
8.2
8.2
8.2
8.2
332.91
33.17
333.36
333.83
333.96
0.00
0.00
0.00
0.00
0.00
332.91
33.17
333.36
333.83
333.96
1722 8.83 0.308 3.28 0.2
1722 11.30 0.394 3.28 0.2
1722 14.13 0.492 3.28 0.2
1722 18.83 0.656 3.28 0.2
1722 22.60 0.787 3.28 0.2
0.7136
0.9134
1.1417
1.5223
1.8268
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
22.118
Chapter Twenty-Two
TABLE 22.13
(Continued) Initial Operation
Parameter
Min Day Avg Day
Entrance headloss through underdrain flume, Hlu 0.45 ft lossper (ft3/ min/ft2) (ft)(consult with filter manufacturer) Clean filter EGL, EGLcf EGL4 Hlf+ f Hlu (ft) Velocity head, HV3 V 0 (m) (assume V3 V 0) (ft) Clean filter HGL, HGLcf EGLcf HV3 V (ft) EGL requierd at Point 3, EGL3 EGLdf (ft) HGL requierd at Point 3, HGL3 HGLdf (ft) (Head requierd for dirty filter controls) 8. Point 3 to Point 2 Filter inlet discharge loss Keff Flow rate, q (ft3/s) Pipe diameter, Dp2 (ft) Velocity, Vp2 (ft/s) 2 headloss, Hld2 Keff (Vp2 /2g) (ft) EGLat Point 2, EGL2 EGL3 Hld2 (ft) Velocity head, HV2 V Vp22/g / (ft) HGL at Point 2 HGL2 EGL2 HV2 V (ft) 9. Point 2 to Point 1 Head loss through butterfly valve Kval (fully open) Headloss, Hlv1 Kval (Vp2 2/2g) Headloss through inlet pipe Length of pipe, Lp1 (ft) Hazen-Williams coefficient (Cp) Hydraulic radius, Rp Dp2/4 (ft) Headloss, Hlf1 f (Vp2/(1.318 Cp Rp1.63))(1/0.54) Lp (ft) Headloss through entrance to pipe Kent Headloss, Hlent Kent Vp 2/2g (ft) EGL at Point 1, EGL1 EGL2 Hlv1 Hlf Hlent (ft) Velocity head, HV1 0 (assume V1 0) (ft) HGL at point 1, HGL EGL1 HV1 (ft) Minimum required control HGL at Point 1 (ft) (max. HGL1 must equal HGL7 of final sedimentation tank)
Design Operation
Avg Day Max Hour
Peak
0.1384
0.1772
0.2215
0.2953
0.3543
325.56
326.06
326.52
327.45
327.94
0.00
0.00
0.00
0.00
0.00
325.56
326.06
326.52
327.45
327.94
332.91
33.17
333.36
333.83
333.96
332.91
33.17
333.36
333.83
333.96
1.0 8.83 3.0 1.29 0.0258
1.0 11.30 3.0 1.65 0.0423
1.0 14.13 3.0 2.06 0.0661
1.0 18.83 3.0 2.75 0.1176
1.0 22.60 3.0 3.30 0.1693
332.94 0.03
333.21 0.04
333.43 0.07
333.95 0.12
333.13 0.17
332.91
333.17
333.36
333.83
333.96
0.3 0.0078
0.3 0.0127
0.3 0.0198
0.3 0.0353
0.3 1.0508
65.6 120 0.74
65.6 120 0.74
65.6 120 0.74
65.6 120 0.74
65.6 120 0.74
0.0127
0.0200
0.0303
0.0516
0.0723
0.50 0.0129
0.50 0.0212
0.50 0.0331
0.50 0.0588
0.50 0.0847
332.97
333.27
333.51
334.10
334.33
0.00 332.97
0.00 333.27
0.00 333.51
0.00 334.10
0.00 334.33
334.33
334.33
334.33
334.33
334.33
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
Water and Wastewater Treatment Plant Hydraulics 22.119 TABLE 22.14
Example Hydraulic Calculation of a Typical Cascade Aeration System Initial Operation Parameter
1. Wastewater flow rate, Q (ft3/s) (mgd) 2. Cascade aerator Total number of units Flow rate through aerator, Q (ft3/s) Optimal flow rate per ft width over step, q (ft2/s) DO concentration of postaeration influent, Co (mg/L) Desired DO concentration of postaeration effluent, Cu (mg/l)
Min Day Avg Day
Design Operation Avg Day Max Hour
Peak
35.3 23
56.5 37
70.6 46
113.0 73
113.0 73
1 35.31
1 56.50
1 70.63
1 113.01
1 113.01
0.7029
0.7029
0.7029
0.7029
0.7029
0.00 5.00
0.00 5.00
0.00 5.00
0.00 5.00
0.00 5.00
16.4 2.15
16.4 3.44
16.4 4.31
16.4 6.89
16.4 6.89
0.524 3.9
0.717 3.9
0.832 3.9
1.138 3.9
1.138 3.9
16.93
23.16
26.87
36.74
36.74
13.55
18.53
21.49
29.39
29.39
0.93
0.93
0.93
0.93
0.93
0.42 1.53
0.36 1.43
0.33 1.39
0.28 1.32
0.28 1.32
Calculation of aerator dimensions with with predetermined weir length 3. Weir length, W (ft) Flow over weir, q Q/W (ft3/s/ft) Critical depth at upstream step edge, hc (q2/g)1/3 (ft) Optimal fall height of nappe, h (ft) Length of downstream bubble cushion, Lo 0.0629(h0.134)(q0.666) (ft) Length of downstream receiving channel, L 0.8Lo (ft) Optimal tailwater depth, H' 0.236 h, ft for h 3.9 ft Deficit ratio log at 68 F, Inr68 1.86(h1.31)(q0.363)(H H0.31) Deficit ratio, r20 Calculate concentration of dissolved oxygen downstream of step. If concentration is less than desired concentration, add another step and again calculate DO downstream concentration. Continue adding steps until the desired DO oncentration is achieved. Select cascade aerator dimension corresponding to those calculated for average flow. 4. Calculation of number steps to obtain desired DO Desired DO concentration at average flow, Cu (mg/L) Step 1 effluent DO, C1 9.07(1 (1/r20)) Co/r20) (mg/L) Step 2 effluent DO, C2 9.07 (1 (1/r20)) Co/r20) (mg/L)
5.00 3.13
2.73
2.55
2.20
2.20
4.80
4.51
4.38
4.13
4.13
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WATER AND WASTEWATER TREATMENT PLANT HYDRAULICS
22.120
Chapter Twenty-Two
TABLE 22.14
(Continued) Initial Operation Parameter
Step 3 effluent DO, C3 9.07 (1 (1/r20)) Co/r20) (mg/L)
Min Day Avg Day
Design Operation Avg Day Max Hour
Peak
6.00
5.79
5.70
5.52
5.52
3.28 3.94
3.28 3.94
3.28 3.94
3.28 3.94
3.28 3.94
HGL at Point 1, HGL1 (ft)
319.98
319.98
319.98
319.98
319.98
HGL at Point 2, HGL2 HGL1 h (ft)
316.04
316.04
316.04
316.04
316.04
HGL at Point 3, HGL3 HGL2 h (ft)
312.11
312.11
312.11
312.11
312.11
HGL at Point 4, HGL4 HGL3 h (ft)
308.17
308.17
308.17
308.17
308.17
In this example, the desired downstream DO concentration for average flow is achieved after three steps. 5. Calculation of HGL at each step Head loss from filtered water storage tank to point 1 (ft) Cascade fall height, h (ft)
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Source: HYDRAULIC DESIGN HANDBOOK
CHAPTER 23
HYDRAULIC DESIGN FOR GROUNDWATER CONTAMINATION Paul C. Johnson Department of Civil and Environmental Engineering Arizona State University Tempe, Arizona
23.1 INTRODUCTION This chapter focuses on in situ treatment systems designed to eliminate or minimize the impacts of hazardous chemicals on groundwater quality. This chapter builds on the principles introduced in Chapter 4. First, typical soil and groundwater contamination scenarios of interest are introduced. Then a discussion of general remediation design principles and strategies follows. Finally, the design, monitoring, and refinement of selected in situ remediation technologies is presented. These include both conventional and developing technologies, and include containment, reaction barrier, active remediation, and passive remediation systems.
23.1.1 Unique Features of In Situ Treatment Technology Design The design of in situ treatment systems for groundwater contamination is uniquely different from the design of above-ground treatment and hydraulic conveyance systems. The subsurface is an environment that is difficult to describe precisely; it is naturally heterogeneous and, due to practical constraints, characterization data are generally limited. Thus, treatment systems must be designed under conditions of great uncertainty–usually the amount of chemical spilled is unknown, the timing of the release is unknown, and the system into which the spill occurred is poorly characterized. For this reason, the design of in situ treatment systems relies heavily on • • • •
conceptual models, screening-level calculations, empiricism, heuristics, and experience, and monitoring and refinement of the design.
A conceptual model is a realization of how the subsurface might look and how contaminants might move, based on the available data. Often, there are several plausible conceptual models, and the treatment system designer must allow for a range of possibilities in the design. It is also recognized that the conceptual model is continuously refined as new data become available. 23.1 Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
HYDRAULIC DESIGN FOR GROUNDWATER CONTAMINATION
23.2
Chapter Twenty-Three
Screening level calculations are used to estimate treatment performance limits. They are generally biased toward estimating best-case performance, and are useful for making rough estimates of treatment times and treatment limits. They are primarily used for feasibility screening and creating the initial treatment system design. More sophisticated models are available and can be used; however, given the typical level of site characterization data and gross approximations that must be made, their use is rarely warranted, except at those sites where treatment costs are projected to be very high. Empiricism and experience play a key role in the design as well. In many ways, each site is unique; yet, the collective experience from many sites is invaluable. This design field is still relatively young and most experiences are not well-documented. As is often the case, successful applications are more likely to be reported than failures, although the knowledge from both would be equally valuable. In practice, there are many “rules of thumb” that designers use and some are well founded in theory, while others are indefensible (yet they still persist in practice). In any case, it is useful to be aware of these rulesof-thumb even if their bases are not well documented in the literature. Finally, given the inherent large initial uncertainty, it is important to recognize that the design phase continues well past the construction, installation, and operation of the initial system. Following sound design practices does not guarantee success; it does, however, provide a higher probability of success at most sites. Appropriate monitoring is essential to verify assumptions built into the conceptual model and initial design, and system design refinement should be anticipated in the initial design. For this reason, it is important to build robust systems that are both flexible and expandable.
23.1.2
Overview
It has been estimated that costs associated with the remediation of known hazardous waste sites in the United States will exceed $700 billion (all costs are quoted in U.S. dollars) and that the current average rate of expenditure is about $20 billion (Bredehoeft, 1994). Despite the fact that society has been tackling this difficult problem since the 1970s, the environmental profession still struggles to find cost-effective remediation processes. Remediation costs for service station sites now range from $100,000 to $500,000 per site, while costs for the larger landfill and manufacturing sites often exceed $10,000,000 per site. Costs associated with sites where groundwater is contaminated generally exceed those where only soil is impacted by at least an order of magnitude. Clearly, these are not trivial expenses; thus, appropriate technology selection, design, and optimization are important activities.
23.1.3 Groundwater Contamination Scenarios–Point Versus Area Sources Groundwater contamination is prevalent in most industrial societies. Leaking fuel tanks, petroleum storage and refining, solvent degreasing activities, chemical manufacturing and storage, landfills, and mining activities have all contributed to groundwater contamination problems. Groundwater contamination is also prevalent in most modern agricultural areas, where the combination of irrigation and fertilizer, pesticide, and herbicide use have contributed to many large-scale groundwater contamination problems. Fig. 23.1 [US Enviromental Protection Agency (USEPA), 1990] summarizes the more common sources of groundwater contamination and the relative frequency at which they are expected to cause groundwater impacts. Septic tanks and leaking underground fuel storage tanks are considered to be the largest single contributor to the number of groundwa-
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HYDRAULIC DESIGN FOR GROUNDWATER CONTAMINATION
FIGURE 23.1 Common sources of groundwater contamination and the number of states and territories considering each to be major threat to groundwater quality. (USEPA, 1990, after Fetter, 1993).
Hydraulic Design for Groundwater Contamination 23.3
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23.4
Chapter Twenty-Three
ter contamination sources. Agricultural activities are likely the largest contributor to groundwater contamination in terms of the actual extent of impact. For the purposes of this chapter, we first segregate the universe of groundwater contamination scenarios into two main categories: • Point sources, • Area, or distributed sources Underground storage tanks, above-ground storage tanks, landfills, pipeline releases, chemical manufacturing and petroleum refining locations, wood treating facilities, and so forth are all considered to be point sources. Agricultural activities are often considered to be area, or distributed sources. By virtue of their large areal extent, the environmental engineering profession has yet to develop effective solutions for the remediation of area source contamination problems. These often involve widespread region- or basin-scale nitrate or pesticide contamination. Currently, the only practicable solution for these problems appears to be above-ground water blending and direct well-head treatment. In other words, contaminated groundwater is either diluted with water of higher quality as it is pumped from the aquifer, or else it is treated as necessary above-ground as it is removed from the aquifer. As stated, area source problems are rarely addressed with in situ treatment systems, and therefore, these will not be considered further in this chapter.
23.1.4 Groundwater Contamination Scenarios–Segregation by Contaminant Type Point source groundwater contamination problems are divided here into three main categories according to the contaminant properties. The three categories are • Light nonaqueous phase liquids (LNAPLs) • Dense nonaqueous phase liquids (DNAPLs) • Inorganics and other dissolved constituents LNAPLs, in their pure liquid form, are less dense than water (ρ < 1 g/cm3). Most LNAPL sites involve the release of petroleum products or crude oil (e.g., service stations, refineries, pipeline spills). DNAPLs, in their pure liquid form, are more dense than water (ρ > 1 g/cm3). DNAPL contamination is generally found near sites where dry cleaning and aviation, automobile, and electronic circuit board degreasing operations took place. Historically these used chlorinated solvents, such as trichloroethylene (TCE) and perchloroethylene (PCE). Metals and salts fall into the category of inorganics and other dissolved constituents. Mining operations, electroplating operations, leaking wastewater treatment facilities, and landfills are examples of sources that historically have added contaminants to groundwater in dissolved form.
23.1.5 Groundwater Contamination Scenarios–Subsurface Contaminant Distributions Figures 23.2a, 23.2b, and 23.2c schematically illustrates release scenarios involving LNAPLs, DNAPLs, and inorganics. While very simplistic, these figures highlight key characteristics that impact subsurface characterization activities, remediation design, and monitoring. Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
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Hydraulic Design for Groundwater Contamination 23.5
FIGURE 23.2 Example contaminant release scenarios: (a) LNAPL release.(b) DNAPL release.
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23.6
Chapter Twenty-Three
FIGURE 23.2 (Continued) (c) Example contaminant release scenarios: (c) dissolved contaminant release.
As their density is less than that of water, one can expect to find LNAPLs in the general vicinity of the water table. Provided that a given spill or leak is of sufficient size and that there are not any impermeable vertical flow barriers, LNAPLs travel downward and pool on top of the water table. Then over time, water table fluctuations cause a vertical smearing so that immiscible-phase LNAPL can be found trapped by capillary forces in the soil pores above and below the current groundwater table level. LNAPL solubilities are generally so low that dissolution (and other natural loss processes) occurs very slowly and pools of LNAPL will persist for decades. Dissolution of LNAPLs into groundwater leads to the formation of dissolved contaminant plumes. Aquifer characteristics and chemical properties control the growth and extent of the dissolved contaminant plume. Fortunately, most LNAPL sites involve the release of petroleum hydrocarbons and some of these are aerobically degradable by indigenous microorganisms. The result is that natural chemical degradation often limits the growth of the dissolved LNAPL plume. For example, two survey studies have shown that most benzene, toluene, ethylbenzene, and xylenes (BTEX) dissolved plumes at leaking underground storage tank (LUST) sites do not extend more than 300 ft (100 m) from the downgradient edge of the source zone (Mace et al., 1997; Rice et al., 1995). It is important to note, however, that not all fuel-related contaminants degrade at an appreciable rate. For example, some oxygenates (e.g., MTBE), are highly soluble and resistant to degradation. Consequently, these compounds generally migrate much further distances than the more degradable monoaromatic chemicals (e.g., benzene, toluene, xylenes). The behavior of these compounds is currently the focus of intense research. DNAPL sites are generally more complex and challenging. Provided that a given spill or leak is of sufficient size and that there are not any impermeable vertical flow barriers, DNAPLs travel downward to the water table. Initially, they too will pool on top of the water table; but, provided that the spill or leak is large enough, the weight
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Hydraulic Design for Groundwater Contamination 23.7
of the DNAPL pool will exceed the entry pressure of the medium and at that point the DNAPL will penetrate through the groundwater table. It will continue to migrate downward until a vertical flow barrier is encountered, or until all of the DNAPL liquid is bound by capillary forces. DNAPL vertical migration in aquifers appears to be very sensitive to small changes in soil structure, and as a DNAPL migrates downward, it may also spread horizontally along thin, more permeable, soil layers. DNAPL solubilities are also generally low so that dissolution (and other natural loss processes) occurs very slowly and DNAPL found below the water table will persist for decades. The assessment and remediation of DNAPL sites is generally much more challenging than that of LNAPL sites. Most DNAPLs of interest are chlorinated solvents (e.g., TCE, PCE, and so on) and these degrade slowly, if at all, under natural conditions. This results in dissolved plumes that are much longer than the dissolved LNAPL plumes that might be found in similar settings, and plumes > 1500 ft (500 m) in length are not uncommon; in fact, plumes several miles (kilometers) in length can be found in regions of concentrated historical aviation or electronics manufacturing. In the case of inorganics, such as those originating from the landfill in Fig. 23.2, the contaminants are already dissolved in solution when they reach the water table. Thus, these chemicals follow groundwater flow once they reach the aquifer. Dissolved inorganic chemical migration is governed by complex geochemical interactions; however, in some cases of interest (e.g., nitrate contamination), the contaminants typically are not transformed or retarded, and may also form long dissolved plumes that persist for extended periods of time. These solutions generally behave as neutrally buoyant liquids (ρ ≈ 1 g/cm3); however, if the total concentration of inorganics exceeds about 10,000 mg/L H2O, then density gradients will cause these solutions to migrate vertically downward within aquifers.
23.2 REMEDIATION GOALS When subsurface soil and/or groundwater contamination have been identified, decisions must be made as to whether or not treatment is necessary, and if so, to what levels. Often regulatory guidelines prescribe the actions that must be taken. In most cases, contaminant concentrations in soil and groundwater are compared with levels that have been deemed acceptable. These levels are sometimes referred to as target levels. If exceedences are noted, then treatment or further study is required; otherwise, no action other than monitoring is typically required. While the development and enforcement of target levels is not the focus of this chapter, it is important to recognize that target levels play a significant role in the selection and design of in situ treatment systems. There is no universal set of target levels, and they vary from state to state and country to country. Thus, it is worthwhile to briefly discuss the range of situations that might be encountered in practice. Target levels may be prescribed in terms of acceptable groundwater concentrations, or in terms of soil concentrations that are expected to be sufficiently protective of groundwater quality. In the United States, target levels are often linked to • maximum contaminant levels (MCLs) or maximum contaminant level goals (MCLGs) • risk-based considerations, or • resource protection goals
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23.8
Chapter Twenty-Three
23.2.1 Maximun Contaminant Levels (MCL) In the United States, maximum contaminant levels (MCLs) are concentrations that must not be exceeded at any drinking water supply point (e.g., the home faucet). Water suppliers are responsible for meeting these criteria. MCLs are promulgated by USEPA, and are based on considerations of health effects, aesthetic impacts, available treatment technologies, analytical limitations, and cost. Table 23.1 lists MCLs and MCLGs for some chemicals of interest (USEPA 1996a). Historically, it was common for regulatory agencies to uniformly adopt MCLs as target levels for groundwater quality.
23.2.2 Risk-Based Target Levels In recent years some regulatory agencies have moved away from using MCLs as fixed cleanup levels, and have begun to consider risk-based target levels [e.g., American Society of Testing and Materials (ASTM), 1995]. Like MCLs, risk-based target levels are concentrations also deemed to be protective; however, they differ from MCLs in that they take into account site-specific considerations, the beneficial uses of the aquifer (or soil), the distance from the contaminant source zone and actual water supply wells (or other potential receptors), and any dilution that might occur during groundwater pumping. Risk-based target levels may be larger, or smaller, than MCLs, depending on the site-specific conditions.
23.2.3 Resource Protection Goals In some areas aquifers are considered to be valued resources, and therefore must be protected from any impacts. In such areas, no level of impact is acceptable. These anti degradation policies require that all contamination must be remediated and the resource must be restored to pristine, or background, conditions. Of the three approaches, this is typically the most stringent as the cleanup goal is either zero, or the background concentration. It is also typically the most difficult goal to achieve.
23.2.4 Application of the Target Levels—Remediation, Points of Compliance, and Containment Once target levels are selected, one must choose how and where to apply them. Two obvious choices are • apply target levels everywhere in the soil and groundwater, or • apply the target levels to the boundary of a compliance zone. While total cleanup of soils and groundwater is always preferred, it is not always practicable. For this reason, a compliance zone is sometimes negotiated. Contaminant concentrations must not exceed the target levels outside of this compliance zone, but may exceed them within the compliance zone. This approach is similar to that taken when permitting mixing, or dilution zones for surface water discharges. The allowance of a compliance zone greatly impacts the range of technologies that can be selected at a site. In this case complete remediation of the source zone and groundwater is not always necessary and this opens the door for consideration of contaminant con-
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Hydraulic Design for Groundwater Contamination 23.9 TABLE 23.1 Regulatory and Human Health Benchmarks Used for Soil and Groundwater Target Cleanup Levels
Chemical
Acenapthene Acetone Aldrin Anthracene Antimony
Maximum Contaminant Level Goal (MCLG) ( (µg/L )
Maximum Contaminant Level (MCL) ( (µg/L )
Water HealthBased Limit* Mq/L
2,000 4,000 0.005 10,000 6
6
Arsenic Barium 2000 Benz(a)anthracene Benzene Benzo(b)fluoranthene Benzo(k)fluoranthene Benzoic acid Benzo(a)pyrene Beryllium 4 Bis (2-chloroethyl) ether Bis (2-ethylhexyl) phthalate Bromodichloromethane Bromoform (tribromomethane) Butanol Butyl benzyl phthalate Cadmium 5 Carbazole Carbon disulfide Carbon tetrachloride Chlordane p-Chloroaniline Chlorobenzene 100 Chlorodibromomethane 60 Chloroform 2-Chlorophenol Chromium 100 Chromium (III) Chromium (VI) Chrysene Cyanide (amenable) 200 DDD
50 2,000 0.1 0.005 0.1 1.0 100,000 0.2 4 0.08 6 100* 100* 4,000 7,000 5 4 4,000 5 2 100 100 100* 100* 200 100 40,000 100 10 200 0.4
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23.10
Chapter Twenty-Three
TABLE 23.1.
(Continued)
Chemical
DDE DDT Dibenze(a,h)anthracene Di-n-butyl phthalate 1,2-Dichlorobenzene 1,4-Dichlorobenzene 3,3-Dichlorbenzidene 1,1-Dichloroethane 1,2-Dichloroethane 1,1-Dichloroethylene cis–1,2-Dichloroethylene trans–1,2-Dichloroethylene 2,4-Dichlorophenol 1,2-Dichloropropane 1,3-Dichloropropene Dieldrin Diethylphthalate 2,4-Dimethylphenol 2,4-Dinitrophenol 2,4-Dinitrotoluene 2,6-Dinitrotoluene Di-n-octyl phthalate Endosulfan Endrin Ethylbenzene Fluoranthene Fluorene Heptachlor Heptachlor epoxide Hexachlorobenzene Hexachloro-1,3-butadiene -HCH (-BCH) -HCH (-BCH) HCH (Lindane) Hexachlorocyclopentadiene Hexachloroethane Indeno(1,2,3-cd)pyrene
Maximum Contaminant Level Goal (MCLG) ( (µg/L )
Maximum Contaminant Level (MCL) ((µg/L)
Water HealthBased Limit* Mq/L
0.3 0.3 0.01 4,000 600 75
600 75 0.2 4,000
7 70 100
5 7 70 100 100 5 0.5 0.005 30,000 700 40 0.1 0.1 700 200
2 700
2 700 1,000 1,000 0.4 0.2 1
1
0.2 50
1 0.01 0.05 0.2 50 6 0.1
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HYDRAULIC DESIGN FOR GROUNDWATER CONTAMINATION
Hydraulic Design for Groundwater Contamination 23.11 TABLE 23.1
(Continued)
Chemical
Maximum Contaminant Level Goal (MCLG) ( (µg/L )
Isophorone Mercury Methoxychlor Methylbromide Mthylene Chloride 2-Methylphenol (o-cresol) Napthalene Nickel Nitrobenzene N-Nitrosodiphenylamine N-Nitrosodi-n-propylamine Pentachlorophenol Phenol Pyrene Selenium Silver Styrene 1,1,2,2-Tetrachloroethane Tetrachloroethylene Thallium Toluene Toxaphene 1,2,4-Trichlorobenzene 1,1,1-Trichloroethane 1,1,2-Trichloroethane Trichloroethylene
Maximum Contaminant Level (MCL) ( (µg/L )
Water HealthBased Limit*
90 2 40
2 40 50 5 2,000 1,000 100 20 20 0.01 1 20 1,000
50
50
100
100
200 0.4
70 200 3
5 2 1,000 3 70 200 5
0
5
0.5 1,000
2,4,5-Trichlorophenol
4,000
2,4,6-Trichlorophenol
8
Vandium
300
Vinyl acetate
40,000
Vinyl chloride (chloroethene) m-, o-, and p-Xylene Zinc
2 10,000
10,000 10,000
* - proposed MCL=80 µg/L, Drinking Water Regulations and Health Advisories, USEPA (1995a)
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23.12
Chapter Twenty-Three
tainment systems. In summary, prior to selecting and designing in situ treatment technologies, the user must • define the target treatment levels (both in soil and groundwater), • define the points of compliance where these goals are to be applied, and • define the time frame within which compliance must be achieved.
23.3 INSITU TREATMENT TECHNOLOGIES–GENERAL CLASSIFICATIONS In situ treatment technologies are generally designed to perform one, or more of the following functions: • Contaminant source zone removal • Aquifer restoration • Prevent, or minimize continued contaminant migration Rarely does a single technology accomplish all three of these goals, and so combinations and sequences of complementary technologies are often used. 23.3.1 Source Zone Treatment Technologies Source zone treatment technologies target removal and destruction of the residual contaminants in soil that serve as the source for groundwater contamination. The goal here is to treat the cause of the groundwater contamination, rather than the symptom. Example source zone treatment technologies include • Free-product recovery • Excavation and disposal or above-ground treatment • • • • • • •
In situ soil venting Bioventing In situ air sparging Enhanced in situ soil venting with soil heating and/or soil fracturing In situ vitrification Phytoremediation Groundwater pump and treat systems
Fig 23.3 depicts simple process schematics of the most common of these source zone treatment technologies.
23.3.2 Aquifer Restoration Technologies Aquifer restoration technologies target treatment of dissolved contaminant plumes. These technologies may be employed before, during, or after source zone treatment. Examples of aquifer restoration technologies include
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FIGURE 23.3. Example source zone treatment technology processes schematic: (a) free free-product product recovery, (b) in situ air sparging, (c) soil vapor extraction, and (d) bioventing.
Hydraulic Design for Groundwater Contamination 23.13
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23.14
Chapter Twenty-Three
• Groundwater pump and treat systems • Natural attenuation • In situ air sparging • Enhanced bioremediation Fig. 23.4 displays simple process schematics of the first three of these aquifer restoration technologies. Restoration of groundwater quality and use of these technologies is only feasible if a complementary source zone treatment technology is also utilized.
23.3.3 Contaminant Migration Prevention Contaminant migration prevention technologies are designed to minimize future impacts of contaminants on groundwater. These technologies are often employed at sites where: a) the source zone location is not known and/or b) there are no practicable source zone and aquifer restoration options. Examples of these technologies, or strategies, include • Natural attenuation • Groundwater pump and treat systems • In situ reaction walls • In situ air sparging curtains • Infiltration barriers • In situ containment options (grout walls, sheet piling walls, and so on) Figure 23.5 displays simple process schematics of common contaminant migration prevention technologies.
23.4 GENERIC TECHNOLOGY SELECTION AND DESIGN PROCESS Fig. 23.6 presents a generalized sequence of actions to be followed when selecting and designing any in situ treatment technology. Each key step is discussed briefly below in this section, and then discussions specific to common source zone treatment, aquifer restoration, and contaminant migration prevention technologies follow in Secs. 23.5, 23.6, and 23.7.
23.4.1 Site Assessment and Conceptual Model Development As stated above in Sec. 23.1.1, the design of in situ treatment systems for groundwater contamination is challenging and unique because the subsurface is an environment that is difficult to describe precisely; it is naturally heterogeneous and, due to practical constraints, characterization data is generally limited. Thus, the first step of any design process involves characterizing the site and, from the limited data available, proposing a conceptual model (or models) of how the subsurface might look and how contaminants might move. At a minimum, the following data are essential to the design process:
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FIGURE 23.4. Example aquifer restoration process schematics: (a) groundwater pump and treat, (b) in situ air sparging, (c) natural attenuation.
Hydraulic Design for Groundwater Contamination 23.15
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Chapter Twenty-Three
FIGURE 23.5 Example contaminant migration technology processes schematics: (a) physical barriers, (b) in situ air sparging curtain, (c) reaction barrier.
23.16
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HYDRAULIC DESIGN FOR GROUNDWATER CONTAMINATION
Hydraulic Design for Groundwater Contamination 23.17
FIGURE 23.6 Generalized technology selection and design flowchart.
• A historic assessment • A geologic/hydrogeologic assessment • A contaminant distribution assessment • A receptor assessment • A political assessment The historic assessment reviews the history of activities at the site, including chemical inventory records, plot plans (locations of structures, pipelines, rail roads, storage areas, maintenance areas, and so on); records of known spills, leaks, and other releases, and any existing site assessment data and reports for this site and any other nearby sites. The geologic/hydrogeologic assessment often involves collecting soil cores, installing groundwater wells, performing aquifer characterization tests, monitoring groundwater elevations, and using other geotechnical tools (e.g., cone penetrometer) to characterize the subsurface. The contaminant distribution assessment involves conducting chemical analyses of soil, water, groundwater, and soil gas samples, and measuring free-product levels in monitoring wells. The receptor assessment involves visiting the site and nearby area and reviewing water well inventory records for the immediate vicinity. The goal is to identify persons or resources that have been, or could be impacted by the contamination.
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23.18
Chapter Twenty-Three
Finally, the political assessment involves identifying the appropriate regulatory authorities, the potentially responsible parties, and any other persons or groups that may be involved in the decision-making process for the site. The collective goal of these various assessment activities is to identify the key components of the conceptual model(s): • • • • • • • • • •
Primary sources (storage tanks, pipelines, degreasing operations, etc.) Time of release and amount of contaminant released Depth to groundwater Direction of groundwater flow and groundwater velocity Subsurface geology (soil type, distinct layers) Chemicals present Contaminant distribution in the subsurface Existence and location of any mobile free-product liquid Potential receptors and any existing adverse impacts Persons that will be involved in the decision making—process
Often the time of the release and the quantity released are unknown, unless a single catastrophic well-documented event took place (many slow leaks from underground storage tanks go undetected for years). However, if they are known, then this information can be used to better choose the number and locations of soil and groundwater samples. For example, for a spill size of volume Vspill (m3 or ft3), the maximum depth dspill (m or ft) penetrated by the spill can be estimated by
V ill dspill sp Aspill SR T dspill Vspill Aspill SR
(23.1)
maximum depth penetrated by the spill (m or ft) volume of liquid contaminant spilled (m3 or ft3) cross-sectional area (plan view) impacted by spill (m2 or ft2) residual saturation of contaminant liquid in soil (m3 fluid/m3 pores or ft3 fluid/ft3 pores). stet
T
total soil porosity ( ≈ 0.40 m3 pores/m3 soil or ft3 pores/ft3 soil) Where the residual saturation SR is specific to the contaminant fluid and porous medium. SR increases with increasing surface/interfacial tension and decreasing pore size (finer soils). Table 23.2 [American Petroleum Institute (API), 1989] provides reasonable values for SRT for petroleum fuels as a function of soil type. Also given are equivalent soil concentrations Csoil (mgcontaminantt/kgsoil). While these values are specific to the petroleum mixtures and soils given in the table, they also provide reasonable estimates for many other liquid contaminants of concern. As an example, Fig. 23.7 presents depths of spill penetration as a function of spill size and infiltration area, for a range of soil types. In addition to soil concentration measurements, groundwater and soil gas samples can also be used to help define contaminant source zone locations. A general rule of thumb is to look for areas where groundwater or soil gas concentrations exceed 1 percent of the maximum chemical-specific concentration expected for the contaminants released.
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Hydraulic Design for Groundwater Contamination 23.19 TABLE 23.2
Ranges of Residual Liquid Hydrocabon Concentrations in the Unsaturated Zone.
Residual Liquid Hydrocarbon Concentrations in Unsaturated Zone Soils ( SRT) Gasolines Soil
(gal/ftt3) (L/m3) (mg/kg)
Middle Distillates (gal/ftt3) (L/m3) (mg/kg)
Fuel Oils (gal/ftt3) (L/m3) (mg/kg)
Coarse gravel
0.02
2.5
950
0.04
5
2200
0.07
10
4,800
Coarse sand
0.06
7.5
2800
0.1
15
6500
0.22
30
15,000
0.15
20.0
7500
0.3
40
17000
0.6
80
39,000
Fine sand/ silts
Abreviations: (gal/ft3) gallons of liquid per ft3 of soil; L/m3] b L of liquid/m3 of soil, (mg/kg) mg of liquid/kg of soil; (mg/kg) concentrations based on ρsoil 1.85 g soil/cm3 soil, ρliquid 0.7, 0.8, and 0.9 g,-liquid/cm3 liquid for gasoline, middle distillates, and fuel oils, respectively. Source: API, 1989.
In the case of single-component spills, the maximum dissolved concentration CWmax [mg/L /LH2O] is the pure component solubility S (mg/L H2O), while the maximum vapor concentration CVmax (mg/L / vapor) is derived from tabulated values of vapor pressure PV (torr), the molecular weight MW (g/mol), and the Ideal Gas Law. In summary, for single-component spills: W [mg/L cmax /LH2O] S
(23.2)
(PV/760 torr / atm) (MW 103 mg/g / ] V [mg/L C max / vapor] 0.0821 L atm /mol / K 293 K
(23.3)
Srt 0.20
FIGURE 23.7 Depth of spill penetration vs. spill size and infiltration area.
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23.20
Chapter Twenty-Three
w and CV Table 23.3 contains Cmax max for a range of chemicals of interest. For chemicalspecific properties of other chemicals, the reader is referred to the USEPA Superfund Chemical Data Matrix databases (USEPA, 1996b). For the case of mixtures, the appropriate equations for estimating the maximum dissolved and vapor concentrations of each compound are
W cmax, /LH ] XiS [mg / LH ) i (mg/L 2O
2O
(23.4)
Xi (PV/760torr / atm) (MW 103 mg / g) W cmax, / vapor) (23.5) i (mg/L 0.0821L atm/mol /K / 293 K
TABLE 23.3 Chemical Properties of Selected Soil and Groundwater Contaminants of Interest (derived from USEPA SCDM 1996b)
Chemical Acetone Aldrin Anthracene Atrazine Benz(a)anthracene Benzene Benzo(a)pyrene Benzo(b)fluoranthene 1,3-Butadiene Butanol Carbon tetrachloride Chlordane Chlorobenzene Chloroform Chloromethane Chrysene m-Cresol Cyclohexane DDD DDE DDT 1,3 Dichlorobenzene 1,2-Dichloroethane trans 1,2-Dichloroethylene 1,2-Dichloropropane Dieldrin
Molecular Weight (g/mole) 58 365 178 216 238 78 252 252 54 74 154 410 113 119 51 228 108 84 320 318 354 147 99 97 113 381
Pure Component Solubility (mg/L LH2O)
Vapor Pressure
106 0.017 0.043 70 0.009 1800 0.002 0.002 740 106 790 0.056 470 7900 5300 0.006 23,000 55 0.09 0.12 0.025 130 8500 6300 2800 0.2
23 6 10-6 3 10-6 3 10-7 1 10-7 9 6 10-9 5 10-7 2100 74,000 120 5 10-5 12 200 4300 9.5 10-5 0.14 97 7 10-7 6 10-6 2 10-7 2.2 79 330 52 5 10-6
(torr)
Maximun Vapor Conc.
(mg/Lvapor) 730 0.0001 0.00003 0.000004 0.000001 405 0.00000008 0.000007 6200 300,000 1010 0.001 14 1300 12,000 0.001 0.83 450 0.00001 0.0001 0.000004 18 430 1800 320 0.0001
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HYDRAULIC DESIGN FOR GROUNDWATER CONTAMINATION
Hydraulic Design for Groundwater Contamination 23.21 TABLE 23.3
(Continued)
Chemical
Molecular Weight (g/mole)
Endrin Ethylbenzene Ethylene glycol Formaldehyde Heptachlor Hexane Isobutanol Lindane Methanol Methylethylketone Napthalene PCBs Phenol Styrene Tetrachloroethylene Toluene Toxaphene Trichlorbenzene 1,1,2-Trichlorethane Trichloroethylene Vinyl chloride m-xylene Gasoline (fresh)* Gasoline (weathered)*
381 106 62 30 373 86 74 291 32 72 128 356 94 104 166 92 413 181 133 131 63 106 95 111
Pure Component Solubility (mg/L LH2O) 0.2 170 106 550,000 0.18 12 85,000 7.3 106 220,000 31 0.07 83,000 310 200 530 0.74 49 4400 1500 8800 160
Vapor Pressure
Maximun Vapor Conc.
(Torr)
(mg/Lvapor) 3 10-6 9.6 0.092 5200 0.0004 152 10 0.0004 130 95 0.085 7.7 10-5 0.28 6.1 19 28 9.8 10-7 0.43 23 73 3000 8.5
0.00006 0.056 0.31 8500 0.008 720 40 0.006 230 370 0.60 0.0015 1.4 35 170 141 0.00002 4.3 170 523 10,000 49 1300 220
Source: Johnson, et al., (1990a, b).
where Xi(molesi/total moles in mixture) is the mole fraction of the chemical of interest in the mixture. In many cases, when the complete composition of the mixture is unknown, the mole fraction can be approximated by the mass fraction Wi (massi/mass of mixture):
Xi (moles i/total moles) ⬇ Wi (mass i/total mass)
(23.6)
For example, if a spill of benzene liquid occurred, we would expect to see dissolved and soil gas vapor benzene concentrations approaching 1800 mg/L / H20 and 400 mg/L / vaporr, respectively, near the source zone. If, on the other hand, a gasoline spill occurred, then we would expect to see dissolved and soil gas vapor benzene concentrations approaching 18 mg/L / H2O and 4 mg/L / vaporr, respectively, for a gasoline containing ⬇ 1 percent by weight benzene. For the site assessment, how much data collection is enough? Clearly, it is desirable to be able to completely characterize a site, but this option is cost-prohibitive. There are no universal guidelines that define minimum numbers of soil borings, samples, and so on; however, some useful questions to consider are
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HYDRAULIC DESIGN FOR GROUNDWATER CONTAMINATION
23.22
Chapter Twenty-Three
• Have you adequately defined primary sources, depth to groundwater, direction of groundwater flow, groundwater velocity, subsurface geology (soil type, distinct layers), chemicals present, contaminant distribution in the subsurface, existence and location of any mobile free-product liquid, potential receptors and any existing adverse impacts? • Is it likely that the additional data will change the proposed site conceptual model? • Are additional data critical to the prescreening of treatment technologies? • What are the risks of designing a system based on the current data? One other important factor to consider is that some treatment technologies are more robust than others. Here “robustness” is a measure of how easily an initial system design can be refined once operation begins. It is also a measure of how sensitive treatment effectiveness is to small variations in subsurface conditions and contaminant distributions. In other words, a very robust technology is one that can be designed with little site-specific information, and that can be made to perform as desired through minor operating condition changes rather than major system equipment changes. Table 23.4 provides a qualitative indication of the robustness of many treatment technologies in use today.
23.4.2 Select Target Treatment Levels At this step, appropriate target treatment levels are identified for the chemicals that have been identified at the site. The discussion in Sec. 23.2 outlines the range of target treatment goals that might be encountered. Briefly, at this stage the user must • define the target treatment levels (both in soil and groundwater), • define the points of compliance where these goals are to be applied, and • define the time frame within which compliance must be achieved. After this step is completed, site concentrations are compared with the target treatment values. Any exceedences will trigger further assessment, and treatment may be required.
23.4.3 Identify Potential Technologies If treatment is necessary, a gross pre-screening of technologies takes place. This prescreening considers the site conditions, documented performance of the available technologies at similar sites, the state of understanding of the technology, economic factors, and the treatment goals (target levels and target time frame). Table 23.4 contains information that can be used to provide a rough screen of technologies. At the end of this step the user is generally considering one to three treatment options.
23.4.4 Screening Level Calculations At this stage a more refined screening of technologies occurs based on basic fundamental screening calculations. These calculations are generally biased toward predicting optimal system performance. Typical calculations involve estimating maximum treatment rates. Examples of these screening level calculations are presented in Secs. 23.5 and 23.7. Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
HYDRAULIC DESIGN FOR GROUNDWATER CONTAMINATION
Hydraulic Design for Groundwater Contamination 23.23 TABLE 23.4
Example Technology Prescreening Matrix.
Technology
Application
Relative Time Frame
Robustness
Typical Settings Shallow contamination (
