Transients in Hydraulic Systems - Bentley Hammer

ADVANCED WATER DISTRIBUTION MODELING AND MANAGEMENT Authors Thomas M. Walski Donald V. Chase Dragan A. Savic Walter Grayman Stephen Beckwith Edmundo Koelle Contributing Authors Scott Cattran, Rick Hammond, Kevin Laptos, Steven G. Lowry, Robert F. Mankowski, Stan Plante, John Przybyla, Barbara Schmitz Peer Review Board Lee Cesario (Denver Water), Robert M. Clark (U.S. EPA), Jack Dangermond (ESRI), Allen L. Davis (CH2M Hill), Paul DeBarry (Borton-Lawson), Frank DeFazio (Franklin G. DeFazio Corp.), Kevin Finnan (Bristol Babcock), Wayne Hartell (Bentley Systems), Brian Hoefer (ESRI), Bassam Kassab (Santa Clara Valley Water District), James W. Male (University of Portland), William M. Richards (WMR Engineering), Zheng Wu (Bentley Systems ), and E. Benjamin Wylie (University of Michigan)

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C H A P T E R

13 Transients in Hydraulic Systems

A hydraulic transient is the flow and pressure condition that occurs in a hydraulic system between an initial steady-state condition and a final steady-state condition. When velocity changes rapidly because a flow control component changes status (for example, a valve closing or pump turning off), the change moves through the system as a pressure wave. If the magnitude of this pressure wave is great enough and adequate transient control measures are not in place, a transient can cause system hydraulic components to fail. This chapter presents the basic concepts associated with transient flow, discusses various methods to control hydraulic transients, and introduces aspects of system design that should be considered during transient analysis. Special attention is given to the specification of system equipment and devices that are directly related to causing and controlling hydraulic transients. The primary objectives of transient analysis are to determine the values of transient pressures that can result from flow control operations and to establish the design criteria for system equipment and devices (such as control devices and pipe wall thickness) so as to provide an acceptable level of protection against system failure due to pipe collapse or bursting. Because of the complexity of the equations needed to describe transients, numerical computer models are used to analyze transient flow hydraulics. An effective numerical model allows the hydraulic engineer to analyze potential transient events and to identify and evaluate alternative solutions for controlling hydraulic transients, thereby protecting the integrity of the hydraulic system.

13.1 INTRODUCTION TO TRANSIENT FLOW System flow control operations are performed as part of the routine operation of a water distribution system. Examples of system flow control operations include opening and closing valves, starting and stopping pumps, and discharging water in response to fire emergencies. These operations cause hydraulic transient phenomena, especially if they are performed too quickly. Proper design and operation of all

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aspects of a hydraulic system are necessary to minimize the risk of system damage or failure due to hydraulic transients. When a flow control operation is performed, the established steady-state flow condition is altered. The values of the initial flow conditions of the system, characterized by the measured velocity (V) and pressure (p) at positions along the pipeline (x), change with time (t) until the final flow conditions are established in a new steady-state condition. The physical phenomenon that occurs during the time interval TT between the initial and final steady-state conditions is known as the hydraulic transient. In general, transients resulting from relatively slow changes in flow rate are referred to as surges, and those resulting from more rapid changes in flow rate are referred to as water hammer events. Evaluating a hydraulic transient involves determining the values during the time interval TT of the functions V(x, t) and p(x, t) that result from a flow control operation performed in a time interval TM. Changes in other physical properties of the liquid being transported, such as temperature and density, are assumed to be negligible. The evolution of a transient is represented at incremental positions in the system through a graph like the one shown in Figure 13.1. In this graph, pressure (p) is represented as a function of time (t) resulting from the operation of a flow control valve. Note that the figure represents a view of the transient at a fixed point (x) just upstream of the valve that is being shut. In the figure, p1 is the initial pressure at the start of the transient event, p2 is the final pressure at the end of the event, pmin is the minimum transient pressure, and pmax is the maximum transient pressure.

Impacts of Transients A wave is a disturbance that transmits energy and momentum from one point to another through a medium without significant displacement of matter between the two points. For example, a wave caused by a boat moving across a lake will disturb a distant boat, but water is not directly transported from the moving boat to the other boat. As can be seen in Figure 13.1, a transient pressure wave subjects system piping and other facilities to oscillating high and low pressure extremes. These pressure extremes and the phenomena that accompany them can have a number of adverse effects on the hydraulic system. If transient pressures are excessively high, the pressure rating of the pipeline may be exceeded, causing failure through pipe or joint rupture, or bend or elbow movement. Excessive negative pressures can cause a pipeline to collapse or groundwater to be drawn into the system. Low-pressure transients experienced on the downstream side of a slow-closing check valve may result in a very fast, hard valve closure known as valve slam. This low-pressure differential across the valve can cause high-impact forces to be absorbed by the pipeline. For instance, a 10-psi (69-kPa) pressure differential across the face of a 16-in. (400-mm) valve results in a force in excess of 2,000 lb (8,900 N). This situation is common where hydropneumatic tanks are used on pump station header systems, but it can also result from elevated tanks that are in close proximity to pump stations.

Section 13.1

Introduction to Transient Flow

575

Figure 13.1 Hydraulic transient at position x in the system

Some flow control operations that initially cause a pressure increase can lead to significant pressure reductions when the wave is reflected. The magnitude of these pressure reductions is difficult to predict unless appropriate transient analysis is performed. If subatmospheric pressure conditions result, the risk of pipeline collapse increases for some pipeline materials, diameters, and wall thicknesses. Although the entire pipeline may not collapse, subatmospheric pressure can still damage the internal surface of some pipes by stripping the interior lining of the pipe wall. Even if a pipeline does not collapse, column separation (sudden vaporous cavitation) caused by differential flow into and out of a section could occur if the pressure in the pipeline is reduced to the vapor pressure of the liquid. Two distinct types of cavitation can result. Gaseous cavitation involves dissolved gases such as carbon dioxide and oxygen coming out of the water, and vaporous cavitation is the vaporization of the water itself. When the first type of cavitation occurs, small gas pockets form in the pipe. Because these gas pockets tend to dissolve back into the liquid slowly, they can have the effect of dampening transients if they are sufficiently large. With vaporous cavitation, a vapor pocket forms and then collapses when the pipeline pressure increases due to more flow entering the region than leaving it. Collapse of the vapor pocket can cause a dramatic high-pressure transient if the water column rejoins very rapidly, which can in turn cause the pipeline to rupture. Vaporous cavitation can also result in pipe flexure that damages pipe linings. Cavitation can and should be avoided by installing appropriate protection equipment or devices in the system, as described later in this chapter on page 607.

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When pressure fluctuations are very rapid, as is the case with water hammer, the sudden changes can cause pipelines and pipeline fittings (bends and elbows) to dislodge, resulting in a leak or rupture. In fact, the cavitation that commonly occurs with water hammer can—as the phenomenon’s name implies—release energy that sounds like someone pounding on the pipe with a hammer.

Overview of Transient Evaluation For typical water distribution main installation, transient analysis may be necessary even if velocities are low. System looping and service connections may amplify transient effects and need to be studied carefully. Transient analysis should be performed for large, high-value pipelines, especially those with pump stations. A complete transient analysis, in conjunction with other system design activities, should be performed during the initial design phases of a project. Normal flow control operations and predicable emergency operations should of course be evaluated during the design. However, uncommon flow control activities can occur once the system is in operation, making it important that all factors that could affect the integrity of the system be considered. Evaluating a system for potential transient impacts involves determining the values of head (Hmax and Hmin) at incremental positions in the system. These head values correspond to the minimum and maximum pressures of the transient pressure wave, depicted as pmax and pmin in Figure 13.1. Computation of these head values through the system allows the engineer to draw the grade lines for the minimum and maximum hydraulic grades expected to occur due to the transient. If the elevation (z) along the pipe is known, then the pipe profile can be plotted together with the hydraulic grades and used to examine the range of possible pressures throughout the system. Figure 13.2 shows a pumping system in which an accidental or emergency pump shutdown has occurred. The extreme values indicated by the hydraulic grade lines in Figure 13.2 were developed by reviewing the head versus time data at incremental points along the pipeline. The grade lines for Hmin and Hmax, which define the pressure envelope or head envelope, provide the basis for system design. If the Hmin grade line drops significantly below the elevation of the pipe, as shown in a portion of the system in Figure 13.2, then the engineer is alerted to a vacuum pressure condition that could result in column separation and possible pipeline collapse. Pipe failure can also result if the transient pressure in the pipe exceeds the pipe’s pressure rating. Maximum (or minimum) transient pressure can be determined for any point in the pipeline by subtracting the pipe elevation (z) from Hmax (or Hmin) and converting the resulting pressure head value to the appropriate pressure units. Specialized programs are necessary to perform transient analysis in water distribution systems. The extended-period simulation (EPS) discussed elsewhere in this book does not consider momentum in the system and is therefore incapable of detecting or analyzing hydraulic transients. Such simulations are sufficient to analyze hydraulic systems that undergo velocity and pressure changes slowly enough that significant inertial forces are not mobilized. If a system undergoes large changes in velocity and pressure in relatively short time periods, then transient analysis is required.

Section 13.2

Physics of Transient Flow

577

Figure 13.2 Grade lines for a pumping system during an emergency shutdown

13.2 PHYSICS OF TRANSIENT FLOW When a flow control device is operated rapidly in a hydraulic system, the flow momentum changes as a result of the acceleration of the liquid being transported and a transient is generated. This hydraulic transient is analyzed mathematically by solving the velocity [V(x, t)] and pressure [p(x, t)] equations for a well-defined elevation profile of the system, given certain initial and boundary conditions determined by the system flow control operations. In other words, the main goal is to solve a problem with two unknowns, velocity (V) and pressure (p), for the independent variables position (x) and time (t). Alternatively, the equations may be solved for flow (Q) and head (H). The continuity equation and the momentum equation are needed to determine V and p in a one-dimensional flow system. Solving these two equations produces a theoretical result that usually reflects actual system measurements if the data and assumptions used to build the numerical model are valid. Transient analysis results that are not comparable with actual system measurements are generally caused by inappropriate system data (especially boundary conditions) and inappropriate assumptions.

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Hydraulic transients can be analyzed using one of two model types: a rigid model or an elastic model. These models and their limitations are discussed in the next subsections.

The Rigid Model The rigid model assumes that the pipeline is not deformable and the liquid is incompressible; therefore, system flow control operations affect only the inertial and frictional aspects of transient flow. Given these considerations, it can be demonstrated using the continuity equation that any system flow control operations will result in instantaneous flow changes throughout the system, and that the liquid travels as a single mass inside the pipeline, causing a mass oscillation. In fact, if the liquid density and the pipeline cross-section are constant, the instantaneous velocity is the same in all sections of the system. These rigidity assumptions result in an easy-to-solve ordinary differential equation; however, its application is limited to the analysis of surge (see the next subsection on Limitations). The rigid model is established for each time instant (t) of the transient period using the fundamental rigid model equation: L dQ fL H 1 – H 2 = ----------------2- Q Q + ------- ------gA dt 2gDA

where

H1 H2 f L g D A Q dQ/dt

= = = = = = = = =

(13.1)

total head at position 1 in a pipeline (ft, m) total head at position 2 in a pipeline (ft, m) Darcy-Weisbach friction factor length of pipe between positions 1 and 2 (ft, m) gravitational acceleration constant (ft/s2, m/s2) diameter (ft, m) area (ft2, m2) flow (cfs, m3/s) derivative of Q with respect to time

If a steady-state flow condition is established — that is, if dQ/dt = 0 — then Equation 13.1 simplifies to the Darcy-Weisbach formula for computation of head loss over the length of the pipeline. However, if a steady-state flow condition is not established because of flow control operations, then three unknowns need to be determined: H1(t) (the upstream head), H2(t) (the downstream head), and Q(t) (the instantaneous flow in the conduit). To determine these unknowns, the engineer must know the boundary conditions at both ends of the pipeline. Using the fundamental rigid model equation, the hydraulic grade line can be established for each instant in time. The instanteneous slope of this line indicates the hydraulic gradient between the two ends of the pipeline, which is also the head necessary to overcome frictional losses and inertial forces in the pipeline. For the case of flow reduction caused by a valve closure (dQ/dt < 0), the slope is reduced. If a valve is

Section 13.2

Physics of Transient Flow

opened, the slope increases, potentially allowing vacuum conditions to occur in the pipeline. Limitations. The rigid model has limited applications in hydraulic transient analysis because the resulting equation does not accurately interpret the physical phenomenon of pressure wave propagation caused by flow control operations, and because it is not applicable to rapid changes in flow. With the rigid model, the slope change is directly proportional to the flow change. According to the model, if an instantaneous flow change (even a minor one) occurs as a result of a rapid flow control operation, the resulting head is immediately and excessively changed. Therefore, rigid model results are not realistic for analyzing rapid changes in the system. The slow-flow transient phenomenon to which the rigid model may be applied is called surge. With surge, head changes occur slowly and are relatively minor in magnitude, allowing changes of the liquid density and/or elastic deformation of the pipeline to be neglected.

The Elastic Model The elastic model assumes that changing the momentum of the liquid causes deformations in the pipeline and compression in the liquid. Because liquid is not completely incompressible, it can experience density changes. Based on these model assumptions, a wave propagation phenonemon will occur. The wave will have a finite velocity that depends on the elasticity of the pipeline and of the liquid. Elasticity of a Liquid. The elasticity of any medium is characterized by the deformation of the medium due to the application of a force. If the medium is a liquid, this force is a pressure force. The elasticity coefficient (also called the elasticity index, constant, or modulus) describes the relationship between force and deformation and is a physical property of the medium. Thus, if a given liquid mass in a given volume (V) is submitted to a static pressure rise (dp), a corresponding reduction (dV < 0) in the fluid volume occurs. The relationship between cause (pressure increase) and effect (volume reduction) is expressed as the bulk modulus of elasticity (Ev) of the fluid, as shown in Equation 13.2: dp dp E v = – -------------- = -----------------dV eV  dU e U

where

(13.2)

E v = volumetric modulus of elasticity (M/LT2) dp = static pressure rise (M/LT2) dV eV

= incremental change in liquid volume with respect to initial volume

dU eU

= incremental change in liquid density with respect to initial density

A relationship between a liquid’s modulus of elasticity and density yields its characteristic wave celerity, as shown in Equation 13.3.

579

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a =

where

E -----v = U

dp----dU

(13.3)

a = characteristic wave celerity of the liquid (L/T)

The characteristic wave celerity (a) is the speed with which a disturbance moves through a fluid. Its value is approximately equal to 4,716 ft/s (1,438 m/s) for water and approximately 1,115 ft/s (340 m/s) for air. For water with a one-percent volume of free air, this value is approximately 410 ft/s (125 m/s) due to the decreased elasticity of the air–liquid mixture. The next subsection explains the physical meaning of the characteristic wave celerity and the reason that the speed varies so widely depending on the medium. „ Example — Computing Modulus of Elasticity for a Fluid. Assume that a 0.26-gal (1-liter) volume of water at ambient temperature with a density of 1.94 slugs/ft3 (1,000 kg/m3) is subjected to a pressure of approximately 290 psi (20 bar). In this case, the volume would decrease by approximately 0.055 in3 (0.9 cm3), or by 0.09%. Compute the modulus of elasticity for water. Using Equation 13.2, the modulus of elasticity can be computed as Ev = -290 psi/-0.0009 = 3.2 u 105 psi or Ev = -20 bars/-0.0009 = 2.2 u 104 bars = 2.2 u 109 Pa = 2.2 GPa

Wave Propagation in a Liquid. Temporal changes in liquid density and deformations of system pipelines are not considered in steady-state flow analysis, even if considerable spatial changes in pressure exist due to frictional head losses or elevation differences in the system. Steady-state flow analysis assumes that to move a molecule of liquid in the pipeline system, a simultaneous displacement of all other liquid molecules in the system must occur. It also assumes that the liquid density is constant throughout the system. In reality, however, some distance exists between molecules, and a small disturbance to a fluid molecule is transmitted to an adjacent molecule only after traveling the distance that separates them. This movement produces a small local change in the density of the fluid, which in turn produces a wave that propagates through the system. The approach used to analyze transient waves depends on the perspective from which the equations are written. They can be written from the perspective of a stationary observer, an observer traveling with the velocity of the water, or an observer traveling with the velocity of the wave. For example, consider a liquid flowing with a velocity (V) in a nondeformable pipe that is subjected to a pressure force (dp) in the direction of flow caused by a system operation at the left end of the pipe (see Figure 13.3). The force applied to the liquid molecules on the left transmits as a molecular action to the adjacent molecules on the right, which characterizes a mechanical wave propagating in the direction of the flow. In Figure 13.3, the flow is to the right at velocity V, and the observer and the distur-

Section 13.2

Physics of Transient Flow

581

bance are moving to the right at velocity c. [The term c represents the speed of the wave relative to a fixed point, and is equal to the characteristic wave celerity (a) plus the velocity of the moving fluid (V).] The flow velocity in front of the moving observer relative to the observer is therefore (c - V) = a. Figure 13.3 Wave propagation in a liquid, assuming the observer is moving at velocity c

After a period of time, the wave will have traveled a distance (x), and a disturbed zone will exist behind the wave. In front of the wave, the initial flow condition is not yet affected and maintains its initial properties. The flow properties in the pipeline will appear variable to a stationary observer because the flow conditions change along the length of the pipeline. The observer moving with a control volume at velocity c will see the liquid flowing into the control volume at a velocity (c - V) and out of the control volume at velocity [c - (V + dV)], where dV is the disturbance of the absolute velocity of the flow caused by the pressure force. Water Hammer Theory. Water hammer refers to the transient conditions that prevail following rapid system flow control operations. It can be used beneficially, as in the case of a hydraulic ram, which is a pump that uses a large amount of flowing water to temporarily store elastic energy for pumping a small amount of water to a higher elevation. More commonly, the destructive potential of water hammer is what attracts the attention of water engineers. The concept of propagation of a wave in a liquid within a pipeline is needed to understand the water hammer phenomenon. The preceding subsection on “Elasticity of a Liquid” described pressure waves propagating in fluid only. This explanation can be used to describe wave speed in a completely rigid pipeline; however, most pipelines are made of deformable materials for which elasticity must be taken into account. To generate equations describing the water hammer phenomenon, the unsteady momentum and mass conservation equations are applied to flow in a frictionless, horizontal, elastic pipeline. First, the momentum equation is applied to a control volume at the wave front following a disturbance caused by downstream valve action. The following equation may be developed, which is applicable for a wave propagating in the upstream direction:

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a 'p = – Ua 'V or 'H = – --- 'V g

where

(13.4)

'p = change in pressure (psi, Pa) U = fluid density (slugs/ft3, kg/m3) a = characteristic wave celerity of the fluid (ft/s, m/s) 'V = change in fluid velocity (ft/s, m/s) 'H = change in head (ft, m)

The equation makes intuitive sense in that a valve action causing a positive velocity change will result in reduced pressure. Conversely, if the valve closes (producing a negative 'V ), the pressure change will be positive. By repeating this step for a disturbance at the upstream end of the pipeline, a similar set of equations may be developed for a pulse propagating in the downstream direction: a 'p = Ua 'V or 'H = --- 'V g

(13.5)

These equations are valid at a section in a pipeline in the absence of wave reflection. They relate a velocity pulse to a pressure pulse, both of which are propagating at the wave speed a. To be useful, a numerical value for the wave propagation velocity in the fluid in the pipeline is needed. Assume that an instantaneous valve closure occurs at time t = 0. During the period L/a (the time it takes for the wave to travel from the valve to the pipe entrance), steady flow continues to enter the pipeline at the upstream end. The mass of fluid that enters during this period is accommodated through the expansion of the pipeline due to its elasticity and through slight changes in fluid density due to its compressibility. The following equation for the numerical value of a is generated by applying the equation for conservation of mass to the entire pipeline for L/a seconds and combining it with Equation 13.4

a =

where

Ev ----U ----------------------E v 'A 1 + ------------A 'p

(13.6)

E v = volumetric modulus of elasticity of the fluid (lbf/ft2, Pa) 'A = change in cross-sectional area of pipe (ft2, m2)

For the completely rigid pipe, the pipe area change, 'A , is zero and Equation 13.6 reduces to Equation 13.3. For real, deformable pipelines, the wave speed is reduced, since a pipeline of area A will be deformed 'A by a pressure change 'p . The solid mechanics problem of finding this area change for a given pressure change is all that is needed to determine the wave propagation speed a of any pipeline. On page 586 of

Section 13.2

Physics of Transient Flow

this chapter, Korteweg’s equation (Equation 13.11) presents the form of this equation for a thin-walled elastic pipeline. By using Equation 13.6 to calculate a numerical value for the wave speed in the pipeline, Equations 13.4 and 13.5 may be used with confidence at any section in the pipeline in the absence of reflections. Given that a is roughly 100 times as large as g, a 1ft/s (0.3-m/s) change in velocity can result in a 100-ft (30-m) change in head. Because changes in velocity of several feet or meters per second can occur when a pump shuts off or a hydrant or valve is closed, it is easy to see how large transients can occur readily in water systems. Full Elastic Water Hammer Equations. Derivation of the complete equations for transient analysis is beyond the scope of this book but can be found in other references, such as Almeida and Koelle (1992) and Wylie and Streeter (1993). The water hammer equations are one-dimensional unsteady pressure flow equations given by 2

a wQ wH ------- + ------- ------- = 0 wt gA wx

(13.7)

wH wQ Q = 0 ------- + gA ------- + fQ ------------wx 2DA wt

(13.8)

Transient modeling essentially consists of solving these equations for a wide variety of boundary conditions and system topologies. The equations cannot be analytically solved, so various approximate methods have been developed over the years. Today, solutions for all but the simplest problems are performed using computers. The following subsection describes some of the approaches that have been used.

History of Transient Analysis Methods Various methods of analysis were developed for the problem of transient flow in pipes. They range from approximate analytical approaches whereby the nonlinear friction term in the momentum equation is either neglected or linearized, to numerical solutions of the nonlinear system. These methods can be classified as follows: Arithmetic method: This method neglects friction (Joukowski, 1904; Allievi, 1903 and 1925). Graphical method: This method neglects friction in its theoretical development but includes a means of accounting for it through a correction (Parmakian, 1963). Method of characteristics: This method is the most popular approach for handling hydraulic transients. Its thrust lies in its ability to convert the two partial differential equations (PDEs) of continuity and momentum into four ordinary differential equations that are solved numerically using finite difference techniques (Gray, 1953; Streeter and Lai, 1962; Chaudhry, 1987; Elansary, Silva, and Chaudhry, 1994).

583

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Algebraic method: The algebraic equations in this method are basically the two characteristic equations for waves in the positive and negative directions in a pipe reach, written such that time is an integer subscript (Wylie and Streeter, 1993). Wave-plan analysis method: This method uses a wave-plan analysis procedure that keeps track of reflections at the boundaries (Wood, Dorsch, and Lightner, 1966). Implicit method: This implicit method uses a finite difference scheme for the transient flow problem. The method is formulated such that the requirement to maintain a relationship between the length interval ' x and the time increment ' t is relaxed (Amein and Chu, 1975). Linear methods: By linearizing the friction term, an analytical solution to the two PDEs of continuity and momentum may be found for sine wave oscillations. The linear methods of analysis may be placed in two categories: the impedance method, which is basically steady-oscillatory fluctuations set up by some forcing function, and the method of free vibrations of a piping system, which is a method that determines the natural frequencies of the system and provides the rate of dampening of oscillations when forcing is discontinued (Wylie and Streeter, 1993). Perturbation method: With this method, the nonlinear friction term is expanded in a perturbation series to allow the explicit, analytical determination of transient velocity in the pipeline. The solutions are obtained in functional forms suitable for engineering uses such as the determination of the critical values of velocity and pressure, their locations along the pipeline, and their times of occurrence (Basha and Kassab, 1996).

Section 13.3

Magnitude and Speed of Transients

13.3 MAGNITUDE AND SPEED OF TRANSIENTS Using Equations 13.4 and 13.5, an engineer can calculate the magnitude of the change in pressure for a given change in velocity. These pressure changes can be very large. For example, for water in a pipeline with a = 3,200 ft/s (980 m/s), a change in velocity of 3.3 ft/s (1 m/s) (not uncommon at every pump switch) can result in a pressure surge of 330 ft (100 m) or 143 psi (980 kPa). With distribution system pressures on the order of 60 psi (410 kPa), a positive pressure wave of this magnitude can raise the pressure beyond the bursting strength of the pipe, while a negative pressure wave can drop the pressure below the vapor pressure of the liquid. A criterion used in determining which equation to use to evaluate a transient is the pipeline characteristic time. The significance of this attribute is explained in the first subsection. Next, this section introduces the Joukowsky equation, which is a formula used to predict the magnitude of a transient. Transient magnitude depends on the wave speed, which was introduced previously. This presentation is followed by a comparison of rigid and elastic water hammer magnitude calculations and a discussion of boundary and reflection methods.

Characteristic Time The pressure wave generated by a flow control operation propagates with speed a and reaches the other end of the pipeline in a time interval equal to L/a seconds. The same time interval is necessary for the reflected wave to travel back to its origin, for a total of 2L/a seconds. The quantity 2L/a is termed the characteristic time for the pipeline. It is used to classify the relative speed of a maneuver that causes a hydraulic transient. If a flow control operation produces a velocity change dV in a time interval (TM) less than or equal to a pipeline’s characteristic time, the operation is considered “rapid.” Flow control operations that occur over an interval longer than the characteristic time are designated “gradual” or “slow.” The classifications and associated nomenclature are summarized in Table 13.1. Table 13.1 Classification of flow control operations based on system characteristic time Operation Time

Operation Classification

TM = 0

Instantaneous

T M d 2L e a

Rapid

T M ! 2L ea

Gradual

T M » 2L ea

Slow

The characteristic time is significant in transient flow analysis because it dictates which method is applicable for evaluating a particular flow control operation in a given system. The rigid model provides accurate results only for surge transients generated by slow flow control operations that do not cause significant liquid compres-

585

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sion or pipe deformation. Instantaneous, rapid, and gradual changes must be analyzed with the elastic model.

Joukowsky’s Equation In 1897, Joukowsky demonstrated the applicability of Equations 13.4 and 13.5 by correctly predicting maximum line pressures and disturbance propagation times in his experiments to determine the maximum velocities that should be allowed in the Moscow water system with valves and other protection devices (surge tanks and relief valves). Noting that Ugh = p , Equations 13.4 and 13.5 can be rewritten to relate explicit head and flow changes as a a dH = r --- dV = r ------- dQ = r BdQ g gA

where

H a g V A Q B

= = = = = = =

(13.9)

head (ft, m) characteristic wave speed of the liquid (ft/s, m/s) gravitational acceleration constant (ft/s2, m/s2) fluid velocity (ft/s, m/s) area (ft2, m2) flow (cfs, m3/s) characteristic impedance, a/gA (s/ft2, s/m2)

The characteristic impedance factor, B, relates head changes to changes in flow. The value of B depends on liquid and pipe characteristics and is defined as equal to (a/gA). If the flow control change is executed rapidly (that is, the duration of the control change is less than 2L/a), the time interval can be subdivided into shorter intervals, and the individual head changes are then added to determine the total head change: 'H =

¦ dH

= r B ¦ dQ = r B 'Q

(13.10)

Celerity and Pipe Elasticity In 1848, Helmholtz demonstrated that wave celerity in a pipeline varies with the elasticity of the pipeline walls. Thirty years later, Korteweg developed an equation similar to Equation 13.11 that allowed for determination of wave celerity as a function of pipeline elasticity and liquid compressibility. When performing transient analyses today, an elastic model formulation with a correction to account for pipeline elasticity should be used.

a =

where

E -----v U ----------------------DE v 1 + ---------- \ eE

a = characteristic wave speed of the liquid (ft/s, m/s) E v = bulk modulus of elasticity for the liquid (lbf/ft2, Pa)

(13.11)

Section 13.3

Magnitude and Speed of Transients

U = liquid density (slugs/ft3, kg/m3) D = diameter (in., mm) e = wall thickness (in., mm) E = Young’s modulus for pipe material (lbf/ft2, Pa) \ = pipeline support factor

Equation 13.11 is valid for thin walled pipelines (D/e > 40). The factor \ depends on pipeline support characteristics and Poisson’s ratio. If a pipe is anchored throughout against axial movement, \ = 1 - P 2, where P is Poisson’s ratio. If the pipe has functioning expansion joints throughout, \ = 1. If the pipe is supported at only one end and allowed to undergo stress and strain both laterally and longitudinally, \ = 5/4 - P (ASCE, 1975). For thick-walled pipelines, there are theoretical equations proposed to compute celerity; however, field investigations are needed to verify these equations. The values shown in Table 13.2, and Table 13.3 for various pipeline materials and liquids are useful to calculate celerity during transient analysis. Figure 13.4 provides a graphical solution for celerity, given pipe wall elasticity and various diameter/thickness ratios. Table 13.2 Physical properties of some common pipe materials Young’s Modulus

Material

9

Poisson’s Ratio, P

(10 lbf/ft2)

(GPa)

Steel

4.32

207

0.30

Cast Iron

1.88

90

0.25

Ductile Iron

3.59

172

0.28

Concrete

0.42 to 0.63

20 to 30

0.15

Reinforced Concrete

0.63 to 1.25

30 to 60

0.25

Asbestos Cement

0.50

24

0.30

PVC (20 )

0.069

3.3

0.45

Polyethylene

0.017

0.8

0.46

Polystyrene

0.10

5.0

0.40

Fiberglass

1.04

50.0

0.35

Granite (rock)

1.0

50

0.28

o

Table 13.3 Physical properties of some common liquids Bulk Modulus of Elasticity

Density

Liquid

Temperature (oC)

(10 lbf/ft )

(GPa)

(slugs/ft )

(kg/m3)

Fresh Water

20

45.7

2.19

1.94

998

6

2

3

Salt Water

15

47.4

2.27

1.99

1,025

Mineral Oils

25

31.0 to 40.0

1.5 to 1.9

1.67 to 1.73

860 to 890

Kerosene

20

27.0

1.3

1.55

800

Methanol

20

21.0

1.0

1.53

790

587

588

Transients in Hydraulic Systems

Chapter 13

Figure 13.4 Celerity versus pipe wall elasticity for various D/e ratios

For pipes that exhibit significant viscoelastic effects (for example, plastics such as PVC and polyethylene), Covas et al. (2002) showed that these effects, including creep, can affect wave speed in pipes and must be accounted for if highly accurate results are desired. They proposed methods that account for such effects in both the continuity and momentum equations.

Comparing the Elastic and Rigid Models To compare the elastic and rigid models, it is only necessary to consider the frictionless flow condition and verify that the pressure changes using each model follow the relation: dV L ------'p rigid dt --------------------- v -------------'p elastic  a dV

(13.12)

where 'p rigid = change in pressure computed with rigid model 'p elastic = change in pressure computed with elastic model dV edt

= fluid acceleration

Although similar, the differences between the elastic and rigid model equations are significant and can be compared by examining the effect of a flow control operation performed at the end of a pipe over a time interval dt that causes a velocity change dV. In the elastic model, pressure changes depend on the flow control operation’s execution time compared to the pipeline’s characteristic time. In the rigid model, when a rapid flow control operation occurs (dt o 0), the computed pressure change will be excessive and will increase with pipe length (dx o L), even for small velocity changes.

Section 13.3

Magnitude and Speed of Transients

Both models produce similar results when dV o 0, which corresponds to a near steady-state flow condition, or to a transient flow condition characterized by slow pressure changes of the same magnitude as the head loss in the pipeline. Thus, as stated earlier, the rigid model is acceptable for slow operational changes for which TM is much longer than the characteristic time. If a flow control operation produces a velocity change dV in a time interval shorter than the characteristic time 2L/a (that is, the operation is “rapid”), the corresponding pressure change is practically the same as an “instant” flow control change in low friction systems. This pressure change can be determined by using Equation 13.4 or 13.5.

Wave Reflection and Transmission In addition to the equations describing transient flow, it is necessary to know about the boundaries—such as tanks, dead ends, and pipe branches—that control the behavior of the transient phenomena. Hydraulic systems commonly have interconnected pipelines with differing characteristics such as material and diameter. These pipeline segments and connection points (nodes) define a system’s topology. When a wave, defined by a head pulse ' Ho and traveling in a pipe, comes to a node, it transmits itself with a head value ' Hs to all other connected pipes and reflects in the initial pipe with a head value ' HR. The wave reflection occurring at a node changes the head and flow conditions in each of the pipes connected to the node. Figure 13.5 shows a node with four pipes connected to it. Part (a) shows the node as the transient wave approaches, and part (b) shows the node after wave reflection and transmission. If the distances between the pipe connections are small, the head at all connections can be assumed to be the same (that is, the head loss through the node is negligible), and the transmission (s) and reflection (r) factors can be defined as A0 2 -----'H s a0 s = ---------- = -------------n A 'H 0 -i ¦ ---ai i=0

where

s = transmission factor (dimensionless) 'H s = head of transmitted wave (ft, m) 'H 0 = head pulse (ft, m) A 0 = incoming pipe area (ft2, m2) a 0 = incoming wave speed (ft/s, m/s) A i = area of i-th pipe (ft2, m2) a i = wave speed of i-th pipe (ft/s, m/s) n = number of outgoing pipes i = pipe number index

(13.13)

589

590

Transients in Hydraulic Systems

Figure 13.5 Pipe connections

Chapter 13

Section 13.3

Magnitude and Speed of Transients

'H r = ----------R- = s – 1 'H 0

where

(13.14)

r = reflection factor (dimensionless) 'H R = head of reflected wave (ft, m)

These factors are used to determine how waves are reflected and transmitted at each branch and boundary. The expressions for r and s are obtained using Joukowsky’s equation (Equation 13.9) for several pipes connected to the same node and considering flow continuity before and after the arrival of the wave. Calculation approaches for evaluating transmitted and reflected waves in typical hydraulic system scenarios follow. • Pipe connected to a reservoir: In this case, n = 1 and A 1 o f. So, s = 0 and r = -1. In other words, a wave reaching a reservoir reflects with the opposite sign. Because 'HR = - 'H 0 and H f = H 0 + 'H 0 + 'H R , H f = H o in this case. ( H f represents final head—the head after wave transmission/reflection.) This scenario is depicted in Figure 13.6(a). • Pipe connected to a dead-end or closed valve: In this case, n = 1, and through the derivation of an equation for r similar to Equation 13.13, it can be shown that r = 1. In other words, a wave reflects at a closed extremity of a pipe with the same sign, and therefore, head amplification occurs at that extremity. If a flow control operation causes a negative pressure wave that reaches a closed valve, the wave’s reflection causes a further reduction in pressure. This transient flow condition can cause liquid column separation and in low head systems, potential pipeline collapse. Figure 13.6(b) shows that at a dead end, the wave is reflected with twice the pressure head of the incident wave. • Pipe diameter reduces (celerity increase): In this case, A1 < A0, and s > 1, so the head that is transmitted is amplified. For example, if A1 = A0/4 (or D1 = D0/2), then s = 8/5=1.6 and r = s – 1 = 0.6, and the head transmitted to the smaller pipeline is 60 percent greater than the incoming. The larger pipeline will also be subjected to this head change after the wave partially reflects at the node. The effect of a contraction is illustrated in Figure 13.6(c). • Pipe diameter increases (celerity decrease): In this case, an attenuation of the incident head occurs at a pipeline diameter increase. The smaller pressure wave is transmitted to the larger pipeline, and after the reflection, the smaller pipeline is subjected to the lower final head. Figure 13.6(d) shows that at an expansion, only some of the wave is reflected.

591

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Transients in Hydraulic Systems

Figure 13.6 Transmission and reflection factors

Chapter 13

Section 13.3

Magnitude and Speed of Transients

• Pipe with a lateral withdrawal: This case refers to a pipeline arrangement in which a withdrawal pipe or surge tank designated as pipe 1 is connected to a secondary network. In this case, n = 3, and with a0 = a1 = a2 and A0 = A2, it can be shown that 1 s = ------------------2 D1 1 + --------2 2D 0 1 r = s – 1 = – ------------------2 2D 0 ---------- + 1 2 D1

where

(13.15)

(13.16)

D1 = diameter of the lateral pipe (ft, m) D0 = incoming pipe diameter (ft, m)

The existence of a lateral withdrawal or feed connection will always reduce the head transmitted and decrease the system head (0 < s < 1). In this case, the transmitting factor can be referred to as a smoothing factor and can be used to determine the preliminary diameter of a surge tank needed to absorb incident waves without significant pressure wave transmission downstream. For example, if 0.05 < s < 0.10, then 6.2D0 > D1 > 4.2D0. With this preliminary estimate of the surge tank size, the engineer can apply the rigid model to analyze the transient flow condition downstream, because the elastic propagation effect is minimal. The example below illustrates the relationship between head and flow for the case of a reservoir and valve on either end of a frictionless pipeline (that is, for an ideal case). „ Example — Relationship Between Head and Flow. Figure 13.7 shows the evolution of a hydraulic transient that is initiated by the complete and instant closure of a valve and causes expansion and contraction of the pipeline and the liquid, which has a specific weight J 0 . A single wave is followed through a period 4L/a as it travels through a single frictionless pipe with the closed valve at one end and a reservoir at the other. The wave reflections at the reservoir and at the closed valve show the head and flow direction changes that occur with time. Descriptions of the individual steps in the progression of the transient wave follow. These steps correspond to those shown in Figure 13.7. a) At time 0 < t < L/a, the wave front is moving toward the reservoir. To the right of the front, the water has stopped and the pressure has increased. To the left of the front, the water does not yet “know” that the valve was shut, so it continues to move to the right at the initial head. b) At time t = L/a, the wave front has reached the reservoir and all the water in the pipe has stopped and is compressed. However, the head in the pipe is above the water level in the reservoir. This difference in head must be relieved, so the water begins to move to the reservoir. c) At time L/a < t