[Detlev_G_Kroger]_Air-Cooled_Heat_Exchangers_and_...
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AIR-COOLED HEAT EXCHANGERS AND COOLING TOWERS THERMAL-FLOW PERFORMANCE EVALUATION AND DESIGN
VOLUME 1 By DetJev G. KrIfger
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Copyright© 2004 by PennWell Corporation 1421 South Sheridan Road/P. O. Box 1260 Tulsa, Oklahoma 74101 800.752.9764 +1.918.831.9421
[email protected] www.pennwell-store.com www.pennwell.com Managing Editor: Kirk Bjornsgaard Production Editor: Sue Rhodes Dodd Book design by Wes Rowell Cover design by Matt Berkenbile Library of Congress Cataloging-in-Publication Data Kröger, Detlev G. Air-cooled heat exchangers and cooling towers : thermal-flow performance evaluation and design / by Detlev G. Kröger,. p. cm. Includes bibliographical references. ISBN 0-87814-896-5 1. Factories--Cooling. 2. Petroleum refineries--Cooling. 3. Electric power plants--Cooling. 4. Heat--Transmission. 5. Refrigeration and refrigerating machinery. I. Title. TH7688.F2K76 2004 621.402'2--dc22 2003022429 All rights reserved. No part of this book may be reproduced, stored in a retrieval system, or transcribed in any form or by any means, electronic or mechanical, including photocopying and recording, without the prior written permission of the publisher. Printed in the United States of America 1 2 3 4 5 08 07 06 05 04
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To my family and my students.
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List of Symbols A ATD a B b C c cp cv D DALR d de E Ey e F f G g H h hD hd I i ifg J K k L Lhy M m N
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Area, m2 Air travel distance, m Coefficient; constant; length, m; surface area per unit volume, m-1 Breadth, m Exponent; constant; length, m; defined by Equation 3.3.4 Coefficient; heat capacity rate mcp, W/K; Cmin/Cmax; cost Concentration, kg/m3 Specific heat at constant pressure, J/kgK Specific heat at constant volume, J/kgK Diffusion coefficient, m2/s Dry adiabatic lapse rate, K/m Diameter, m Equivalent or hydraulic diameter, m Elastic modulus, N/m2; energy, J Characteristic pressure drop parameter, m-2 Effectiveness Force, N; fan; correction factor Friction factor Mass velocity, kg/sm2 Gravitational acceleration, m/s2; gap, m Height, m Heat transfer coefficient, W/m2K Mass transfer coefficient defined by Equation 4.1.3, m/s Mass transfer coefficient defined by Equation 4.1.13, kg/m2s Insolation; Bessel function Enthalpy, J/kg Latent heat, J/kg Bessel function Loss coefficient; incremental pressure drop number Thermal conductivity, W/mK Length, m Hydraulic entry length, (x/deRe) Molecular weight, kg/mole; torque, Nm; mass, kg Mass flow rate, kg/s Revolutions per minute, minute-1; NTU
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LIST OF SYMBOLS
NTU Ny n P Pe p pcr Q q R Ry r s st T Tu t U u V v W w X x Y y z
Number of transfer units, UA/Cmin Characteristic heat transfer parameter, m-1 Number; exponent Pitch, m; power, W Perimeter, m Pressure, N/m2 Critical pressure, N/m2 Heat transfer rate, W Heat flux, W/m2 Gas constant, J/kgK; thermal resistance, m2K/W Characteristic flow parameter, m-1 Radius, m; recirculation factor defined by Equation 8.4.1 Blade tip clearance, m Yield or ultimate stress, N/m2 Temperature, °C or K Turbulence intensity Thickness, m Overall heat transfer coefficient, W/m2K Internal energy, J/kg Volume flow rate, m3/s; molecular volume; volume, m3 Velocity, m/s Work, J; width, m Humidity ratio, kg water vapor/kg dry air Mole fraction Co-ordinate; elevation, m; distance, m; quality Defined by Equation 5.2.4 Co-ordinate Co-ordinate; elevation, m; exponent
Greek Symbols α αe αm αQ β
Thermal diffusivity, k/rcp ; thermal expansion coefficient; void fraction Kinetic energy coefficient defined by Equation 1.4.5 Momentum velocity distribution correction factor defined by Equation 1.4.25 Defined by Equation 9.2.9 Volume coefficient of expansion, K-1; porosity
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Γ γ ∆ δ ε η θ j λ µ m n ρ σ τ ϕ
w
Flow rate per unit length, kg/sm cp /cv ; as defined by Equation 3.4.39 Differential Boundary layer thickness, m; condensate film thickness, m Surface roughness, m; expansibility factor Efficiency; degree of separation Angle, °; temperature differential, K; potential temperature, °C Von Karman constant Eigenvalue; defined by Equation 2.7.4; defined by Equation 4.4.19 Dynamic viscosity, kg/ms Kinematic viscosity, m2/s; Poisson’s ratio Temperature lapse rate, K/m Density, kg/m3 Area ratio; surface tension, N/m Shear stress, N/m2; time, s Potential function; angle, °; defined by Equation 3.2.21 or Equation 3.3.13; relative humidity defined by Equation 4.1.21; expansion factor defined by Equation 5.2.3; dimensionless temperature difference Defined by Equation 2.7.5
Dimensionless Groups Eu Fr FrD Gr Gz Ku j Le Lef Me Nu Oh
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Euler number, ∆p/(ρv2) Froude number, v2/(dg) Densimetric Froude number, ρv2/(∆ρdg) Grashof number, gρ2L3β∆T/µ2 for a plate or gρ2d3β∆T/µ2 for a tube Graetz number, RePrd/L for a tube Kutateladze number, ifg/(cpDT) Colburn j-factor, StPr0.67 Lewis number, k/(ρcpD) or Sc/Pr Lewis factor, h/(cphd) Merkel number, hd afi Lfi /Gw Nusselt number, hL/k for a plate or hd/k for a tube Ohnesorge number, µ/(ρdσ)0.5
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LIST OF SYMBOLS
Pe Pr Re Sc Sh St
Péclet number, RePr Prandtl number, µcp/k Reynolds number, ρvL/µ for a plate or ρvd/µ for a tube Schmidt number, µ/(ρD) Sherwood number, hDL/D for a plate or hDd/D for a tube Stanton number, h/(ρvcp) or Nu/(RePr)
Subscripts a abs ac acc al amm av app b c cd cf cp cr ct ctc cte cu cv D d do db de ds
Air or based on air-side area Absolute Adiabatic cooling Acceleration Aluminum Ammonia Mixture of dry air and water vapor Apparent; approach Base; bundle; bend; boundary layer Concentration; convection heat transfer; combining header; casing; contraction; cold; critical; condensate Conservative design Counterflow Constant properties Critical Cooling tower Cooling tower contraction Cooling tower expansion Copper Control volume Darcy; drag; drop; diffusion Diameter; diagonal; drop; dynamic; dividing header; dry section; diffusion; mass transfer Downstream Drybulb Drift or drop eliminator Steam duct
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dif e F F/dif Fhe f fi fr fs g gen H h he i isen id i iso j m m max min mo n na nu o ob P p p q r re
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Diffuser Energy; expansion; effective; equivalent; evaporative Fan Fan/diffuser Fan to heat exchanger distance Fin; friction; fluid; factor Fill Frontal; face Fill support Gas; ground Generator Height Hot; header; hub Heat exchanger Inlet; inside Isentropic Ideal Inlet louver Isothermal jet; junction Laminar; longitudinal; liquid; lateral; large Logarithmic mean Mean; momentum; model; mass transfer; mixture Maximum Minimum Monin-Obukhov Nozzle; normal Noise attenuator Non-uniform Outlet; outside; initial; oil; original Obstacle Poppe Constant pressure; production; plate; process fluid; passes; plume Plenum chamber Constant heat flux Root; row; radial co-ordinate; refrigerant; reference; recirculation; ratio Effective root
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LIST OF SYMBOLS
rec red rz s sc si sp ss T ∆T t tp tr ts tus ud up v vc w wb wd x y z θ π ∞
Recovery Reducer Rain zone Screen; steam; static; saturation; shell; support; superficial; steel; soil; scaling; spray Settling chamber; surface condenser Inlet shroud Spray Supersaturated Constant temperature; temperature; T-junction; test Constant temperature difference Total; tube; tape; transversal; turbulent; transition; terminal; blade tip; fin tip Two-phase Tube row Tube cross section; tower support Wind tunnel upstream cross section Upstream and downstream Upstream Vapor Vena contracta Water; wall; wind; walkway; wet section Wetbulb Water distribution system Co-ordinate; quality Co-ordinate Co-ordinate; zinc Inclined; yawed At 180° Infinite; free stream
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Preface The objective of these volumes is to provide modern analytical and empirical tools for evaluation of the thermal-flow performance or design of air-cooled heat exchangers and cooling towers. People who can make use of this information include students, design engineers, manufacturers, contractors, planners, plant managers, and end users. They may be in the fields of air-conditioning, refrigeration, mining, processing, chemicals, petroleum, power generation, and many other industries. They will be able to prepare improved specifications and evaluate bids more critically with respect to thermal performance of new cooling systems. Possible improvements through retrofits of existing cooling units can be determined, and impacts of plant operations and environmental influences can be predicted. Reasons for poor performance will be better understood, and where necessary, a plant can be optimized to achieve the lowest life cycle cost. The format and presentation of the subject matter has evolved from courses offered at universities and from industry-based research, development, and consultation over many years. Volume I consists of chapters 1 through 5; Volume II consists of chapters 6 through 10. The Table of Contents for the companion volumes is listed in Appendix D in each volume. An attempt is made to maintain a meaningful compromise between empirical, analytical, and numerical methods of analysis to achieve a satisfactory solution without introducing an unnecessary degree of complexity or cost. In some cases, sophisticated numerical methods have to be used in order to obtain sufficient insight into aspects of a particular problem. Such programs and the related infrastructure and manpower may be expensive, so it is desirable to stress analytical or empirical methods where meaningful. The reader is introduced systematically to the literature, theory, and practice relevant to the performance evaluation and design of industrial cooling systems. Problems of increasing complexity are presented. Many of the procedures and examples presented are not only of academic interest, but are applicable to actual systems, and have been tested in practice. The design engineer is supplied with an extensive and up-to-date source of information. In order to be an informed planner or client, it is important to understand the factors that influence the type and thermal-flow design of any cooling system in order to prepare clear and detailed specifications. A lack of insight
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and poor specifications often lead to serious misunderstandings between suppliers and clients and can result in significant increases in the ultimate cost of the plant. The merits of a particular cooling system should be evaluated critically. Often this can only be done in a holistic and interdisciplinary approach. This approach takes into consideration the entire cycle or plant and its environment when an optimization exercise is performed. For those interested in further reading, an extensive list of references is included at the end of every chapter. In view of the iterative nature of solving most heat exchanger performance evaluation or thermal-flow design problems, computers are essential tools. An exception is simple first approximations. In the numerical examples, values are often given to a large number of decimal places. These numbers are usually from the computer output and do not necessarily imply a corresponding degree of accuracy. However, increasingly improved designs are essential to reduce system costs in the design of large systems or where mass production is involved. In view of the increasing competition, access to computers and more reliable design information will lead to more refined and sophisticated designs. With a better understanding of the performance characteristics of a cooling system, control can be improved in different operating conditions. The worked problems not only show how to apply various equations but each problem forms a part of the learning process and introduces important additional information. The problems gradually lead up to more extensive and complex evaluations. I am grateful to many friends and colleagues in both the academic and industrial worlds who directly or indirectly contributed to this work. However, this text would not have been written without the support and patience of my family and the valuable input of my graduate students.
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CONTENTS Volume I Preface ........................................................................................X List of Symbols ..........................................................................XII 1
Air-Cooled Heat Exchangers and Cooling Towers ......................1 Introduction..........................................................................1 Cooling Towers ....................................................................2 Air-cooled Heat Exchangers ..............................................12 Dry/Wet and Wet/Dry Cooling Systems ............................27 Conservation Equations ....................................................42 References ..........................................................................49
2
Fluid Mechanics ....................................................................55 Introduction........................................................................55 Viscous Flow ......................................................................56 Flow in Ducts ......................................................................61 Losses in Duct Systems ......................................................75 Manifolds ............................................................................94 Drag ....................................................................................98 Flow through Screens or Gauzes ....................................101 Two-phase Flow................................................................106 References ........................................................................122
3
Heat Transfer ......................................................................131 Introduction......................................................................131 Modes of Heat Transfer ..................................................131 Heat Transfer in Ducts ....................................................142
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Extended Surfaces ............................................................162 Condensation ..................................................................169 Heat Exchangers ..............................................................191 References ........................................................................216 4
Mass Transfer and Evaporative Cooling ................................223 Introduction......................................................................222 Mass Transfer....................................................................233 Heat and Mass Transfer in Wet-cooling Towers ............236 Fills or Packs......................................................................249 Effectiveness-NTU Method Applied to Evaporative System ..............................................................274 Closed-Circuit Evaporative Cooler ..................................282 Rain Zone..........................................................................299 Drift Eliminators ..............................................................309 Spray and Adiabatic Cooling ..........................................313 Visible Plume Abatement ................................................314 References ........................................................................319
5
Heat Transfer Surfaces ........................................................329 Introduction......................................................................329 Finned Surfaces ................................................................330 Test Facilities and Procedures..........................................337 Interpretation of Experimental Data..............................341 Presentation of Data........................................................346 Heat Transfer and Pressure Drop Correlations ..............376 Oblique Flow through Heat Exchangers ........................387 Corrosion, Erosion and Fouling ......................................401 Thermal Contact and Gap Resistance ............................407 Free Stream Turbulence ..................................................420 Non-uniform Flow and Temperature Distribution ........428 References ........................................................................430
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APPENDICES..............................................................................441 A Properties of Fluids ....................................................441 B
Temperature Correction Factor ................................459
C
Conversion Factors ....................................................465
D Contents, Volume II....................................................473
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1 Air-Cooled Heat Exchangers and Cooling Towers 1.0 Introduction In any power generating or refrigeration cycle, heat has to be discharged. This is also true in many chemical and process plant cycles, internal combustion engines, computers, and electronic systems. The efficiency of a modern automobile engine is such that most of the energy contained in the fuel is rejected through the exhaust and the radiator. In a fossil-fired power plant with an efficiency of about 40%, more than 40% of the heat input has to be rejected through the cooling system. Even more heat has to be rejected in less efficient nuclear power plants. Considerably less heat is rejected in a modern combined cycle power plant. In the past, the hydrosphere has been the commonly used heat sink at industrial plants. The simplest and cheapest cooling method was to direct water from a river, dam, or ocean to a plant heat exchanger and to return it, heated, to its source. In industrialized countries, the permissible rise in temperature of such cooling water is often limited and restricts the use of natural water for once-through cooling.
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The task of choosing the source of cooling for large industrial plants is becoming increasingly complex. Many things contribute to the problem:
•
dwindling supplies of cooling water and adequate plant sites
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rapidly rising water costs at well beyond inflation rates in most industrialized countries
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noise restrictions (Hill) and other environmental considerations
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proliferating legislation (Steele, and Vereinigung der Grosskraftwerksbetreiber)
Because of restrictions on thermal discharges to natural bodies of water, most new generating capacity or large industries requiring cooling will have to make use of closed cycle cooling systems. Cooling towers, evaporative- or wetcooling systems, generally are the most economical choice for closed cycle cooling where an adequate supply of suitable water is available at a reasonable cost to meet the makeup water requirements of these systems. Unfortunately, according to Burger, many cooling towers have in the past failed to meet design specifications in part due to outdated design methods. Air-cooled heat exchangers are found in the electronics industry, vehicles, air conditioning, and refrigeration plants as well as chemical and process plants where fluids at temperatures of approximately 60 °C or higher are to be cooled. The use of air-cooled or dry-cooling systems in industry or in power plants is often justified where cooling water is not available or is very expensive. In certain applications, dry/wet or wet/dry cooling systems offer the best option according to Bartz and Mitchell. An appropriate and well-designed cooling system can have a very significant positive impact on plant performance and profitability.
1.1 Cooling Towers The development, practice, and performance of evaporative cooling systems or cooling towers have been described in numerous publications such as McKelvey, Berliner, Cheremisinoff, Hill and Burger. A cooling tower is a device that uses a combination of heat and mass transfer to cool water. The water to be cooled is distributed in the tower by spray nozzles, splash bars, or
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film fill in a manner that exposes a very large water surface to atmospheric air. The movement of the air is accomplished by fans (mechanical draft), natural draft, or the induction effect from water sprays. A portion of the water is evaporated because the moisture content of the air is less than saturated at the temperature of the water. Since this process of evaporation requires energy to change the water from liquid to vapor, the water is cooled. Figure 1.1.1 shows a typical cooling circuit at a power plant. Turbine exhaust steam condenses in a surface condenser where heat is given up to the cooling water circulating through the condenser tubes. The hot water leaving the condenser is piped to the cooling tower distribution basin and flows downward through the fill or packing. This serves to break up the water into small droplets or spreads it in a thin film in order to maximize the surface contact between the water and the cooling air. Finally, it is drawn through the fill by the axial flow fan. The water, after being cooled by a combination of evaporation and convective heat transfer, is pumped through the condenser in a continuous circuit. Approximately 1–3% of the circulating water is lost due to evaporation.
Fig. 1.1.1 Mechanical Draft Cooling Tower Installed in a Power Plant
Over the years, the combination of theoretical and experimental studies as well as extensive practical experience has led to the improved design and operation of such cooling systems. Unfortunately, these developments are not always fully exploited during the life of a particular system due to poor maintenance and a lack of operating experience according to Willa.
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Mechanical draft Different types of evaporative cooling systems or cooling towers are distinguished by the method used to move air through the system, mechanical draft or natural draft, and the arrangement of the fill section, crossflow or counterflow of air and water streams. Mechanical draft towers may be a forced draft or induced draft design. A forced draft tower has a fan or blower located where the ambient airstream enters the tower and forces the air through the fill shown in Figure 1.1.2(a).
a
b
Fig. 1.1.2 Forced Draft Cooling Tower (a) Cooling Tower (b) Temperature Distribution
Hot water is introduced through spray nozzles located above the fill and flows downward in counterflow with the airstream. Small droplets that are entrained by the upward flowing airstream are collected in a drift eliminator where they accumulate to form larger drops that are returned to the fill. The region below the fill contains falling droplets and is called the rain zone. The recooled water is collected in a basin and returned to the plant. Figure 1.1.2(b) shows the temperature relationship between water and air as they pass through the cooling tower. The curves indicate a drop in water temperature and a rise in the air wetbulb temperature during their passage through the tower. The temperature difference between the water entering and leaving the tower is defined as the range. The difference between the
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temperature of the water leaving the tower and the wetbulb temperature of the entering air is known as the approach. Forced draft towers are characterized by relatively high air inlet velocities and low exit velocities and are susceptible to recirculation of the hot, moist plume air. Examples of induced draft towers are shown in Figure 1.1.3. They may be of the crossflow or counterflow type. In a crossflow tower, the fill is installed at some angle to the vertical to make provision for the inward motion of droplets due to drag forces caused by the entering cooling air. Other arrangements of the fill are possible according to Stupic. Plume recirculation is less of a problem in induced draft towers than it is in forced draft towers. More fan power is required to move the same mass of air because the air has a lower density, i.e., it is warmer and contains more water vapor than inlet air.
a
b
Fig. 1.1.3 Induced Draft Cooling Towers (a) Mechanical Draft Crossflow (b) Mechanical Draft Counterflow
A section through a large single-cell circular induced draft cooling tower is shown in Figure 1.1.4. The fan-drive equipment is located in a chamber isolated from the water system. Fans having diameters of up to 28 m are employed in these towers.
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Fig. 1.1.4 Single-Cell Circular Induced Draft Cooling Tower
Mechanical draft towers have traditionally been used in an in-line arrangement of individual cells to form a rectangular bank (Fig. 1.1.5a). A more recent development is the round mechanical draft tower with multiple fans (Fig. 1.1.5b).
a
b
Fig. 1.1.5 Alternative Mechanical Draft Cooling Tower Arrangements (a) In-Line (b) Round
Natural draft In natural draft cooling towers, the required airflow through the fill is created by the difference in density between the heated humid air inside the tower and the denser ambient air outside the tower. Crossflow and counterflow fill arrangements (Fig. 1.1.6) are encountered.
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a
b
Fig. 1.1.6 Natural Draft Cooling Towers (a) Crossflow (b) Counterflow
Crossflow towers have a fill configuration in which the air flows perpendicular to the downward falling water. The hot water is delivered through risers to distribution basins above the fill and is distributed by gravity through low-pressure nozzles in the floor of the basin. A modern concrete cooling tower has a hyperbolic shaped shell, which may be up to 180 m in height (Goldwirt). It is possible to reduce the size of the tower by installing axial flow fans at its base (Moore and Gardner). Although the cost of the structure is reduced, this is offset by the capital cost of the fan installation and the running costs. Fan-assisted natural draft cooling towers may be considered where excessive plume recirculation in alternative multibank mechanical draft units make these unacceptable. Figure 1.1.7 from Gardner illustrates the design proposal for the Ince B power plant fan-assisted draft tower (Gardner and Jones). The tower is 117 m tall and has a shell base diameter of 86 m. The roofed structure is 172 m in outside diameter. It surrounds the base and houses the crossflow fill and 35 fans. The fans consume 6 MWe at the design point, 0.6% of the station output. The fill consists of a prefabricated design of timber-lath splash bars. Two arrays of corrugated louvers are necessary to cope with the high air and water mass fluxes of an assisted draft tower. Compared with the usual design, the trailing edges of the louvers are slightly extended to ensure a close approach to axial flow at the fan entry.
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Fig. 1.1.7 Fan-Assisted Crossflow Cooling Tower
The relative merits of different cooling towers and fill arrangements are outlined by Lefevre. In view of environmental considerations, cooling towers incorporating ducts that introduce desulphurized flue gas into the plume for better dispersion are operational at a number of power plants. An example from Petzel of such a system is shown in Figure 1.1.8.
Fig. 1.1.8 Crossflow Cooling Tower with Flue Gas Desulphurization Unit
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In a modern fossil-fuelled power plant equipped with a wet-cooling system, an average of 1.6–2.5 liters of cooling water will be required for cooling per kWh(e) of net generation. A 600 MWe coal-fired plant operating at 70% annual capacity factor would require between 5 x 106 m3 and 10 x 106 m3 of makeup water annually to replace cooling tower evaporation losses alone. In addition, a portion of the circulating cooling water must be systematically discharged as blowdown in order to limit the buildup of dissolved solids in the circulating water. A small amount of cooling water will be lost as drift, i.e., the carryover of entrained water droplets by the air passing through and out of the tower. For conventional wet-cooling towers operating in non-zero discharge plants, blowdown and drift losses combined will range from about 20–50% of evaporative losses, corresponding to 6–3 cycles of concentration of dissolved solids in the circulating water. A cycle of concentration is the ratio of dissolved solids in the circulating water to that of the makeup water. Blowdown will account for all of these losses since drift losses can be as low as 0.01% of the circulating water flow rate. On this basis, total wet-cooling system makeup water requirements for the 600 MWe coal-fired plant used in this example could exceed 11 x 106 m3 per year with a waste stream averaging nearly 10,000 m3 per day requiring disposal if low quality makeup water is used. Figure 1.1.9 shows a number of natural draft cooling towers at a power plant.
Fig. 1.1.9 Natural Draft Cooling Towers
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Nuclear generating units reject 45–50% more heat to the condenser cooling water per kWh(e) of net generation than fossil-fuelled units. The heat rejection per kWh(e) of net generation from geothermal power plants will be four or more times as great as from fossil-fuelled plants. Wet-cooling system makeup water requirements and blowdown will be greater for nuclear and geothermal power plants. Makeup water requirements and blowdown for combined cycle power plants, in which only about one-third of the total electrical output is generated in the steam cycle, generally will be less than one-half of those for conventional fossil-fuelled plants of comparable size. If the water supply used to provide makeup is variable, a storage reservoir may be required in order to ensure an adequate supply is available at all times. Evaporation and seepage losses from such a reservoir can add as much as 20% to overall make-up water requirements. Environmental requirements for limiting temperature rise of surface water and the maximum temperature limit of returning cooling water have resulted in greater use being made of once-through helper-cooling towers. River or other surface water may be passed through a surface condenser to achieve the required cooling before being cooled in a helper-cooling tower and returned to its source. Depending on the seasonal availability of cooling water and environmental considerations, plants incorporating cooling towers may be operated in •
an open circuit requiring no towers.
•
a closed circuit relying on the cooling towers
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an open circuit in which the cooling tower functions as a helpertower according to Chapelain
In the past, water costs have been a very small component of total busbar energy production costs. With municipalities and developers seeking to acquire water rights to meet anticipated growth and future needs, water costs have increased dramatically in some areas. When other increased costs associated with the use of water for power plant cooling are added in, water-related costs become more significant. These costs include:
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•
pumping
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water treatment
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blowdown disposal
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environmental study
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permit acquisition
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Restrictive legislation that could establish water use priorities unfavorable to utilities or chemical plants is potentially of even greater consequence to industry than rising water costs. The options available currently for reducing or eliminating plant cooling system makeup requirements and waste water include the use of •
wet-cooling systems designed to operate with high cycles of concentrating dissolved solids in the circulating water
•
various types of dry-cooling systems making no consumptive use of water
•
various types of cooling tower systems that combine dry- and wetcooling technology
General studies to determine the comparative economics of alternative heat rejection systems should not fail to consider all of the potential advantages offered by the use of water conserving systems. For example, dry-cooled or dry/wet-cooled plants need not be located at the same site as the base case wet-cooled plant with which they are being compared. These plants should take into account the siting flexibility afforded by the use of the water conserving systems. Fuel cost savings resulting from locating a coal-fired plant at the mine mouth where there may not be enough water available to permit the use of wet-cooling could be substantially greater than the accompanying increase in transmission costs. The use of a water-conserving heat-rejection system could permit expansion of existing generating facilities at a site lacking sufficient water for wet-cooling and take advantage of existing support and service facilities and rights-of-way. Even with an adequate water supply at a given site, the use of a waterconserving system could, in some cases, reduce indirect project costs and lead times by reducing environmental study, public hearing, and permit requirements. Other factors cannot be ignored in practice. These include: •
changes in micro climate
•
corrosion of equipment, piping, and structural steel
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emission of chemicals
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poor visibility
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freezing of ground or road surfaces located near cooling tower plumes
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potential health hazards, such as legionnaires’ disease, in poorly maintained systems according to both Crunden and Cuchens
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The impact of all these factors on the comparative economics of alternative heat-rejection systems will depend upon the unique circumstances of each particular application. For the foreseeable future, wet-cooling towers are expected to remain the economical choice in most cases where an adequate supply of suitable makeup water is available at a reasonable cost. However, decreasing water availability, increasing water costs, and more stringent environmental and water use or accessibility regulations will make a water-conserving heat-rejection system a practical and economical choice for more power plant and other applications. This is especially true if the effectiveness of such systems can be improved according to several sources including Surface, Kosten, Mathews, and McHale.
1.2 Air-Cooled Heat Exchangers In an air-cooled heat exchanger, or air cooler, heat is transferred from the process fluid to the cooling airstream via extended surfaces or finned tubes. While the performance of wet-cooling systems is dependent on the ambient wetbulb temperature, the performance of air-cooled heat exchangers is determined by the drybulb temperature of the air. The drybulb temperature is higher than the wetbulb temperature and experiences more dramatic daily and seasonal changes. Small air-cooled heat exchangers (compact heat exchangers as described by Kays) find application in many areas including computers and other electronic equipment, vehicles (radiators, oil coolers, intercoolers), air-conditioning and refrigeration plants (condensers), etc. These are illustrated in works by U.S. Army Material Command, Plank, and also McQuiston. Larger air-cooled heat exchangers are found in refrigeration and chemical plants, various process industries, and power plants. Movement of the cooling air is achieved by mechanical means, fans, or buoyancy effects, e.g., natural draft dry-cooling towers. Although the capital cost of an industrial air-cooled heat exchanger is higher than a water-cooled alternative, this is not always the case. The cost of providing suitable cooling water and other running expenses may be such that the former is more cost effective over the projected life of the system. Other considerations are also of importance depending on the process or application according to Maze. In arid areas where insufficient or no cooling water is available, air cooling is the only effective method of heat rejection.
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Mechanical draft Various air-cooled heat exchanger configurations are found in practice. In some situations, however, the choice of design is critical to the proper operation of the plant. Air-cooled heat exchangers may be forced draft or induced draft types. In forced draft types, the fans are installed in the cooler inlet airstream below the finned tube heat exchanger bundle (Fig. 1.2.1). The result is that power consumption for a given air mass flow rate is less than for the induced draft configuration. The fan drives located in the cooler airflow below the unit are also easier to maintain, and the fans are not exposed to high temperatures, which makes the choice of construction material less critical.
Fig. 1.2.1 Forced Draft Air-Cooled Heat Exchanger Bay
Since the escape velocity of the air from the top of the bundle is a low 2.5–3.5 m/s, the unit is susceptible to hot plume air recirculation. This problem may be accentuated by the proximity of similar heat exchangers or other structures. Anti-recirculation fences or windwalls are often fitted in such cases. Generally, the airflow distribution through the heat exchanger is not as uniform as for the induced draft installation. Since the heat exchanger is open to the atmosphere, the performance can change measurably due to wind, rain, hail, or solar radiation. Hail screens may be required to protect the finned surfaces. The induced draft system shown in Figure 1.2.2 is less sensitive to certain changes in weather conditions. The airflow distribution through the heat exchangers is more uniform than in a forced draft unit. Because of the relatively high escape velocity of the air from the fan, this type of system
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is less susceptible to crosswinds and plume recirculation. The higher fan power consumption for a given air mass flow rate and the fact that the fan and its drive system are exposed to the warm airstream are disadvantages of this configuration.
Fig. 1.2.2 Induced Draft Air-Cooled Heat Exchanger
In large air-cooled condensers, the finned tube bundles may be sloped at some angle up to 60° with the horizontal or A-frame (Fig. 1.2.3) in order to reduce land area. However, this arrangement has a higher air-side pressure drop.
Fig. 1.2.3 A-Frame Air-Cooled Condenser
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For practical reasons, other configurations (Fig. 1.2.4) may be preferred for particular applications. The rectangular arrangement (Fig. 1.2.4a) is very compact and finds application in closed circuit cooling plants, while the vertical arrangement (Fig. 1.2.4b) is suitable for smaller plants. The V-configuration (Fig. 1.2.4c) is often used with counterflow condensers.
a
b
c
Fig. 1.2.4 Air-Cooled Heat Exchanger Configurations (a) Rectangular (b) Vertical (c) V-configuration
An example of an air-cooled refrigerant condenser is shown in Figure 1.2.5. Air-cooled heat exchangers, usually referred to as radiators, find application in vehicles ranging from passenger cars to military vehicles, power generating sets, etc. (U.S. Army Command Engineering Design Handbook).
Fig. 1.2.5 Air-Cooled Refrigerant Condenser
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There are two basic types of air-cooled or dry-cooling systems that find application in power plants. In the direct system, also referred to as the GEA System, the turbine exhaust steam is piped directly to the air-cooled finned tube condenser (Fig. 1.2.6). The finned tubes are arranged in the form of an A-frame or delta to reduce the required land area. The steam exhaust pipe has a large diameter and is required to be as short as possible to minimize pressure losses. A forced or induced flow of cooling air through the finned tube bundles is created by axial flow fans. The application of the direct cooling system in small power generating units became a reality in the 1930s according to both Happel and Heeren.
Fig. 1.2.6 Direct Air-Cooled Condensing System
In 1970, a 160 MWe direct dry-cooled power plant was commissioned at Utrillas/Teruel in Spain. This relatively arid region is located 1200 m above sea level. Exhaust steam leaves the turbine through two 3.5 m diameter pipes and is fed to heat exchanger bundles located above the turbine house (Fig. 1.2.7) according to March. The bundles consist of galvanized elliptical finned tubes arranged in a staggered pattern (Fig. 1.2.8). Forty 5.6 m diameter axial flow fans with their drive units are suspended from vibration-proof bridges on the condenser platform below the A-frames.
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Fig. 1.2.7 Utrillas Power Plant, Spain
Fig. 1.2.8 Galvanized Elliptical Finned Tubes
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The 365 MWe Wyodak power plant near Gillette, Wyoming in the United States, was for many years the largest direct air-cooled plant in operation. The plant is located in an arid, coal-rich area 1240 m above sea level where extreme climactic conditions are experienced (-40 °C to 43 °C). The plant shown in Figure 1.2.9 became operational in 1978. Details of its dimensions, operation, and performance are reported by Schulenberg and Kosten et al.
Fig. 1.2.9 Air-Cooled Condenser at Wyodak Power Plant, United States
In 1987, the 4 x 150 MWe Touss direct air-cooled plant was commissioned in Iran. Since then, many other larger plants have been constructed. Presently, the world’s largest direct air-cooled power plant, Matimba, became operational in 1991 at Lephalale (Ellisras) in the Republic of South Africa according to Von Cleve and Knirsch. Large reserves of coal justified the erection of a dry-cooled plant in this arid part of the country. A photo of the 6 x 665 MWe plant is shown in Figure 1.2.10.
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Fig. 1.2.10 Matimba Power Plant, South Africa
The air-cooled condenser consists of 384 heat exchanger bundles per unit, each almost 3 m wide and 9.6 m long, made up of two rows of galvanized plate finned elliptical tubes shown in Figure 5.1.1(4), and arranged in the A-frame configuration with an apex angle of 56°. Air is forced through the bundles by 48 axial flow fans per unit, each 9.1 m in diameter, located underneath the bundles and about 45 m above ground level. Each fan is driven by a 270 kW electric motor through a bevel spur gearbox. Each condenser unit covers a plot area of 72 m x 85 m. A total of 905 MW heat is rejected per condenser unit at a turbine outlet pressure of 17.9 x 103 N/m2, an ambient air temperature of 18 °C, and a pressure of 91.33 x 103 N/m2. A system similar to the Matimba air-cooled condenser is also employed in three of the six 665 MWe units at the Majuba power plant located east of Johannesburg, South Africa. Cooling for the remaining three turbines is provided by natural draft wet-cooling towers. The development of a large flattened steel tube having aluminum fins (Fig. 5.1.10) has made it possible to construct air-cooled steam condensers having only one row of finned tubes. According to Rathje, numerous power plants employing these tubes have been constructed.
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Natural draft A schematic drawing of an indirect dry-cooling tower incorporating a direct contact spray condenser is shown in Figure 1.2.11. Recooled water from the cooling tower is introduced into the condenser via nozzles, and the turbine exhaust steam condenses on the droplets or water jet as described by Buxmann and Mikovics.
Fig. 1.2.11 Indirect Dry-Cooling System
A part of the condensate is returned to the boiler, but most is pumped at a positive gauge pressure to finned tube heat exchangers located at the base of a natural draft cooling tower. The recooled water returns to the condenser via an energy recovery turbine, through which its pressure drops to below ambient conditions. This particular layout is also referred to as the Heller system after the Hungarian engineer who first proposed the concept. This is described by Heller and Szabo. The Heller system has found application at numerous power plants throughout the world. In 1962, a 120 MWe turbine rejecting 169 MW heat through a hyperbolic concrete natural draft dry-cooling tower was commissioned at the Rugeley power plant in Great Britain. An aerial view of the tower is shown in Figure 1.2.12. Note the size of the tower and the relatively high inlet compared to the wet counterflow cooling towers according to Christopher.
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Fig. 1.2.12 Rugeley Power Plant, Great Britain
A similar design was commissioned in 1967 at the Ibbenbüren power plant in the Federal Republic of Germany where a 150 MWe turbine had been installed according to Scherf. During the period 1969–1972, a total of 2 x 100 MWe and 2 x 220 MWe generating capacity was installed at the Gagarin power plant, also known as Visonta or Màtra, at Gyöngyös, Hungary. In the natural draft towers at this plant, the heat exchanger deltas were arranged vertically around the base of the tower in all cases to maximize the air-side surface area. Since the bundles are self-supporting, water distribution is simple and installation is straightforward (Fig. 1.2.13), resulting in reduced cost when compared with other layouts. Each delta is approximately 15 m in length Fig. 1.2.13 Installing a Heat and consists of slotted aluminum Exchanger Delta plate finned tubes (Fig. 1.2.14a).
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a
b
Fig. 1.2.14 Finned Tubes (a) Aluminum Slotted Plate Fin (b) Aluminum Fin Wrapped on Steel Tube
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Measurements at the Rugeley and at other similar towers indicate this bundle arrangement tends to be sensitive to winds and results in a reduction in cooling capacity. Alternative arrangements shown in Figure 1.2.15 were later considered with a view to reducing the sensitivity to wind.
Fig. 1.2.15 Heat Exchanger Bundle Arrangements
The cooling tower erected at the Grootvlei power plant in the Republic of South Africa in 1971 has its heat exchanger deltas installed horizontally at two different levels in the inlet cross section of the tower (Fig. 1.2.15b). The heat transfer surface in the Grootvlei tower (Grootvlei 5) consists of galvanized steel tubes onto which an aluminum fin is tension wound (Fig. 1.2.14b). Regularly spaced zinc collars prevent fins from unwinding. Details of the system are given by van der Walt et al.
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During the period 1971–1974, four dry-cooling towers (Fig. 1.2.16) were constructed at the Razdan power plant in Armenia to cool 2 x 210 MWe and 2 x 200 MWe turbines. The towers have welded steel frames covered with corrugated aluminum sheets. This type of construction was utilized due to earthquake hazards in the region. The towers are 120 m high and have an outlet diameter of 60 m. Heller-type finned-tube bundles 15 m high are located around the periphery at the base of the towers. Motor operated louvers mounted before the deltas protect the finned surfaces and allow the airflow to be controlled.
Fig. 1.2.16 Dry-Cooling Towers at the Razdan Power Plant, Armenia.
Other more recent Heller systems include the 8 x 250 MWe Shahid Rajai power plant in Iran described by Ludvig, the 2 x 200 MWe Datong power plant in China, and the 2 x 200 MWe Teshrin plant that commenced operation in 1993 according to Spilko. In some indirect cooling systems, a conventional surface condenser is employed instead of the spray condenser. Due to the additional barrier offered by the surface condenser, thereby reducing radiation hazards, this is the only system likely to be considered where dry-cooling is required for a nuclear plant.
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An indirect dry-cooling system incorporating a surface condenser to cool a 200 MWe turbine was commissioned at the Grootvlei power plant in South Africa in 1978 (Grootvlei 6). The heat exchanger bundles, consisting of staggered steel tubes with wrapped-on aluminum fins similar to those at Grootvlei 5, are arranged horizontally in a radial pattern in the inlet cross section of the 120 m high concrete tower (Fig. 1.2.15c). Another variation in heat exchanger layout is found at the 134 m high Candiota cooling tower in Brazil, which has bundles made up of plate-finned elliptical tubes arranged with an ineffective cylindrical section in its center (Fig. 1.2.15e). The Kendal power plant in the Republic of South Africa is the largest indirect dry-cooled plant in the world and has a total of 6 x 686 MWe turbines according to Trage. The six hyperbolic concrete natural-draft cooling towers are each 165 m high with a base diameter of 163 m, and each tower is equipped with 500 heat exchanger bundles arranged in concentric circles at the base of the tower (Fig. 1.2.17). The helically wound, galvanized, elliptical finned tubes shown in Figure 5.1.1(3) have a total length of approximately 2000 km per tower. The circuit includes a conventional surface condenser. Compared to a wet-cooling system, approximately 50 x 106 m3 water is saved annually.
Fig. 1.2.17 Kendal Cooling Tower, South Africa
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A novel cable tower was erected to serve the 300 MWe turbine at the Schmehausen nuclear plant in Germany. Figure 1.2.18 shows the cable cooling tower shell supported by a reinforced concrete pylon 180 m high. The cable net, which is covered by aluminum sheets, is held in position by two rings supported by the pylon. The heat exchanger bundles are arranged in a radial pattern on three concentric rings in the inlet cross-sectional area of the tower. Each bundle is 15 m long and consists of galvanized elliptical finned tubes according to Hirschfelder and Von Cleve.
Fig. 1.2.18 Schmehausen Cooling Tower, Federal Republic of Germany
Technical and economic considerations suggest direct, natural-draft, air-cooled condensers may offer an alternative option for rejecting heat in large plants according to Nagel and Schrey. As in the case of wet-cooling towers, it is also possible to exploit the good dispersion characteristics of the plume by locating the flue gas stack inside a dry-cooling tower according to Techet.
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1.3 Dry/Wet and Wet/Dry Cooling Systems Dry/wet cooling systems have been developed to save water in arid regions while avoiding the high cost of full dry-cooling systems and ensuring low process fluid temperatures where necessary. An excessive rise in cooling water temperature during periods of peak ambient temperature and demand will result in a loss of efficiency of a turbogenerator set. In such a case, the dry section of the system may be sized to reject the total heat load at a low ambient temperature while maintaining the turbine back pressure within specified limits. The heat-rejection capacity of the dry section at the peak ambient temperature is then determined. The difference between the heat dissipation capacity and the dry section capacity at peak ambient temperature is the required capacity of the wet section of the cooling system. The heat dissipation capacity is needed to avoid exceeding the specified turbine back pressure according to Englesson and Larinoff. Another way of sizing wet sections of a dry/wet cooling system may be to limit the quantity of makeup water based on the local water availability. Wet/dry cooling systems designed for plume abatement are wet systems with just enough dry-cooling added to reduce the relative humidity of the combined effluent from the wet and dry section below the point where a visible plume will form under cool, high relative humidity, ambient conditions. This definition is according to Alt, Lindahl, and Ergebnisse der Projektgruppe. When a single cooling tower incorporates a wet and a dry section, it is sometimes referred to as a hybrid system. According to Fischer, hybrid systems may consist of other combinations, e.g., an evaporative condenser built into a wet-cooling tower. By combining features of an air-cooled heat exchanger and a wet-cooling tower, it is possible to create a hybrid evaporative cooler, which may offer reduced operating costs for particular duties. The evaporative cooler in conventional form may be regarded as a cooling tower in which the fill or packing is replaced by a bank of corrosion resistant smooth or finned tubes carrying the process fluid. This is illustrated in Figure 1.3.1. Air is drawn upward over the tubes while water falls downward over the tubes. Some water is lost by evaporation while the remainder falls into a sump from which it is recirculated. The loss of water by evaporation is about
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the same as from a wet-cooling tower and has the same duty. Of course, the secondary circulation loop through a heat exchanger with its additional resistance to heat transfer is eliminated. If a bank of finned tubes is used in an evaporative cooler, it may be possible to reduce the annual water consumption by operating the unit in the dry mode during winter months when the ambient temperature is low.
Fig. 1.3.1 Closed Circuit Evaporative Cooler
There are other ways of combining dry- and wet-cooling in a single heat rejection system according to Von Cleve, Nowosad, and Mitchell. These include deluge enhancement, combinations of dry- and wet-cooling units, and adiabatic cooling, precooling the entering air by humidification. In the case of the former, the performance of a dry-cooled system is enhanced during periods of high ambient temperature and/or high cooling demand by deluging the air side of the heat transfer surface with water. The air flowing over the water causes evaporation and lowers the air/water interface temperature. The resultant increase in temperature difference between the internal hot fluid and the external deluge film increases the rate of heat transfer. The rate of heat transfer can be increased by a factor of up to five by deluging the air-side surface of the heat exchangers compared to a dry-cooled system at equivalent temperature and air-side pressure-drop conditions. A concept that incorporates this form of cooling
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is the Heller/EGI Advanced Dry/deluged (HEADd) combined cooling system, which is the indirect dry-cooling system shown in Figure 1.2.11. In addition, it includes auxiliary and preheater/ peak dry/wet cooling units according to Bodás. An example of this type of system is found at the 1200 MWe combined cycle Trakya power station in Turkey, which consists of two 600 MWe plants (Fig. 1.3.2) according to Szabó and Jarosi. Each plant has two identical units consisting of two 2 x 100 MWe gas turbine sets connected to separate heat recovery steam generating boilers, each in turn connected to a 100 MWe steam turbine. Cooling water from the cooling tower is injected into direct contact spray condensers to condense steam from the double exhaust turbines. The mixed cooling water and condensate is then extracted from the bottom (hot-well) of each condenser by two 50% duty circulating water pumps.
Fig. 1.3.2 Trakya Cooling Towers with Power Plant on Right
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About 3% of this flow, which corresponds to the amount of steam condensed, is fed to the boiler feed water system by condensate booster pumps, while the remaining water is returned to the natural draft cooling tower. The concrete cooling tower is 135 m high with a diameter at the base of 121.6 m. Selfsupporting 20 m high aluminum heat exchanger bundles are arranged in a delta configuration with an apex angle of 50° around the tower base circumference. The airflow through the louvers can be controlled by electrically adjustable louvers. In addition to the vertical cooling deltas consisting of a modified version of Forgo-type perforated plate-fin MBV treated aluminum surfaces (Fig. 5.1.9), the cooling system incorporates mechanical draft dry/wet preheater/peak coolers located inside the tower and auxiliary dry/wet coolers outside the tower. The auxiliary dry/wet cooling cells shown in Figure 1.3.3 are required to ensure effective cooling is maintained even during the hottest peak load periods. These coolers consist of heat exchanger bundles arranged in a V-configuration below an axial flow fan. The heat exchanger bundles are also made up of Forgo-type perforated plate-fin MBV treated aluminum surfaces similar to those employed in the dry-cooling deltas, except the tubes run horizontally. With a V-inclined angle of approximately 50°, the plate fins are covered with a thin uniform film of water when deluged from above along the upper edge of the bundles. The water is collected in trays at the bottom of the “V” where it is recirculated by deluging pumps and makeup water is added as required.
Fig. 1.3.3 Mechanical Draft Dry/Wet Auxiliary Cooler
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Deluge water may be obtained from the main cooling water circuit. Because of the high or boilerfeed quality of this water, it is expensive. Other pretreated water may be considered if regular flushing with circuit water is ensured to avoid fouling or scaling. Preheater/peak coolers similar to the auxiliary coolers, are installed inside the cooling tower shown in Figure 1.3.4. These water-to-air heat exchangers are also made up of V-bundles consisting of Forgo-type perforated plate-fin MBV treated aluminum surfaces. The preheater/peak coolers, which have an effective area of 5% of the total heat transfer surface area, are connected in parallel with the main cooling deltas and operate in the natural draft mode. During the hottest peak periods, they enhance the cooling capacity by being deluged with condensate quality water and by operating in an induced mechanical draft mode (Fig. 1.3.4a). During start-up of the cooling plant during cold winter periods, these coolers are used to preheat the cooling deltas before filling. During this operation, the rotation of the fan is reversed, and the cooling delta louvers are closed (Fig. 1.3.4b).
a
b
Fig. 1.3.4 Preheater/Peak Coolers inside Cooling Tower (a) Louvers Open (b) Louvers Closed
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The water flow rate in the preheater/peak coolers is controlled depending on its mode of operation. During the preheating and deluged peak cooling periods, high flow rates are maintained, but these are reduced during normal operation. Measurements show a flow of approximately 60 m3/h deluging water will increase the output of the 100 MWe unit by more than 2 MWe compared to a dry operation for an ambient temperature of 38 °C. The circuiting in the cooling tower is such that when one of the steam turbines is not in operation, the entire tower is available for cooling the second turbine. The external surfaces of the vertical cooling deltas can be cleaned periodically by pressurized water jets installed on a 20 m high water distributor which moves on rails along the tower perimeter. Deluge cooling is also incorporated in the cooling towers in the Isfahan power plant in Iran. The four aluminum clad welded steel cooling towers shown in Figure 1.3.5 are similar to those at the Razdan plant described by Kravets.
Fig. 1.3.5 Isfahan Cooling Towers during Construction, Iran
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The dry/wet cooling system at the 500 MWe San Juan power plant in New Mexico consists of two induced draft cooling towers. Each tower consists of five cells, and each cell contains sixteen air-cooled heat exchanger modules and two evaporative sections. Water in the towers flows in series through the dry heat exchanger to the wet sections, while the airflow is in parallel through the dry and wet sections (Figure 1.3.6). At design conditions of a drybulb temperature of 35 °C and a wetbulb temperature of 18.9 °C, approximately 27% of the heat load is discharged as sensible heat. However, at lower ambient temperatures, the wet sections can be bypassed for fully dry operation.
Fig. 1.3.6 San Juan Dry/Wet Cooling Tower
Dry/wet cooling using an ammonia phase-change system designated the Advanced Concepts Test (ACT) was tested at Pacific Gas and Electric Company’s Kern Station at Bakersfield, California. The facility is capable of condensing approximately 7.5 kg/s of steam from a small house turbine. Details of the facility are given in a number of publications published by EPRI. In the ammonia heat transport system, the exhaust steam from the last stage of the turbine is condensed in a double heat-transfer-enhanced steam condenser/ammonia reboiler located directly below the turbine shown in Figure 1.3.7.
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Fig. 1.3.7 Schematic of ACT Facility
Liquid ammonia is boiled as it is pumped through the tubes under a pressure set by the operating temperature in the air-cooled condenser. To avoid substantial reduction in the heat transfer coefficient by complete evaporation of ammonia, the flow rate is set to yield a vapor quality of less than 0.8. This two-phase mixture is passed through a vapor-liquid separator from which the vapor is sent to the air-cooled condenser. The liquid is combined with the ammonia condensate from the dry tower and recycled back through the condenser/reboiler. The reboiler and vapor-liquid separator could be designed to provide thermosyphon recirculation of the ammonia. At the ACT facility, a recirculation pump is part of the loop to provide independent control of the liquid flow for experimental purposes. From the vapor-liquid separator, vapor is transported to the cooling tower by the temperature difference and the associated vapor pressure difference of the saturated ammonia. The ammonia vapor is condensed in the air-cooled (dry) tower. The liquid flows to an ammonia hot well (liquid receiver) and is then returned to the condenser/reboiler. During periods of high ambient air temperature, heat rejection is augmented in one of the following three ways demonstrated at the ACT facility.
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1.
In the deluge system, water is sprayed over the extended surfaces of the air-cooled heat exchanger (ammonia condenser). The ensuing increase in heat transfer in the presence of an evaporating film on the extended surfaces increases the capability of the heat exchanger to accept the heat load imposed by the condenser/reboiler. The percentage of the heat exchanger surface that is wetted is automatically established by the pressure of the ammonia in the heat transport system.
2.
In the augmentation condenser system, a portion of the ammonia vapor from the vapor-liquid separator is routed to a separate evaporative condenser in parallel with the air-cooled ammonia condenser. The fraction of ammonia is controlled by the ammonia pressure in the system.
3.
In the capacitive-cooling system, a fraction of the steam from the last stage of the turbine is routed to a parallel water-cooled condenser. The cooling water is modulated to maintain the desired ammonia pressure in the system. The water is in a closed loop with a large storage tank and piped in a fashion that develops a thermocline in the tank. The thermocline moves down the tank as water is circulated through the condenser. During periods of lower ambient air temperature and lower loads on the heat rejection system, the hot water in the storage tank is cooled by an ammonia heat pump that rejects heat to the air-cooled tower. During this phase, the thermocline moves upward in the water storage tank.
If operated over the period of a year, each of the dry systems would use only 25% of the water required to reject this heat load in an evaporative cooling tower. The third would consume no water, evaporative cooling being replaced by the delayed cooling of the closed system water supply. The cooling tower shown in Figure 1.3.8 is provided with two sets of heat exchangers of different types:
•
a deluged all-aluminum plate fin-tube heat exchanger
•
an aluminum skived-fin-tube heat exchanger
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Fig. 1.3.8 ACT Test Facility
Located on the long vertical sides of the tower are two all-dry skived-fin heat exchangers sloped 1° toward the outlet to promote drainage. Fourteen dry/wet heat exchangers are arranged in seven A-frame or delta assemblies on the elevated floor of the tower. To allow deluging of the latter, the fin plates are oriented vertically with tubes running horizontally. Deluge water is introduced at a controlled rate at the apex of the delta configuration from a series of nozzles located in a bonnet running the length of the heat exchanger, which seals off airflow out the ends. The water sprays from the nozzles in a flat pattern, impinges on the ends of the plate fins, flows down the width of the heat exchanger, drains into a collection trough at the bottom, and is recycled. For an apex angle of 50°, the loss of deluge water on the interior face of the bundle was found to be minimal. If the ambient temperature exceeds approximately 13 °C and if the tower is at full load, the tower is augmented. Cooling air is supplied by four 4.88 m diameter axial flow fans with controllable blade pitches. The fan system is able to supply the design airflow rate through each heat exchanger set operated alone. Plywood panels or plastic sheets placed in front of the set in use block off all airflow during tests with individual heat exchanger sets.
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In addition to the previous configuration, other wet/dry cooling system arrangements may be considered for application in power generating plants according to Larinoff, Von Cleve, and Smith. A parallel-connected dry/wet cooling system, employing a divided waterbox condenser and separate dry and wet towers, is shown in Figure 1.3.9. At maximum ambient temperatures, both the dry and wet towers operate at full capacity. Lefevre has suggested that the dry- and the wet-cooling tower be connected in series or in parallel with a conventional surface condenser. This would require expensive corrosion resistant tubing throughout the condenser and in the dry-cooling tower, unless the open wet-cooling tower is replaced by an evaporative surface cooler.
Fig. 1.3.9 Parallel Connected Dry/Wet Cooling with Divided Waterbox Condenser
Figure 1.3.10 shows a dry/wet cooling tower system arrangement in which a direct air-cooled condenser is connected in parallel with a wet-cooling tower circuit equipped with a surface condenser. During normal operation, all the turbine exhaust steam is condensed in the air-cooled condenser according to Fay. At high ambient temperatures, the wet-cooling tower coupled to a surface condenser improves the cooling capacity of the system. The wet-cooling tower can be replaced by an evaporative cooler. Precooling by humidification or adiabatic cooling of inlet air prior to its entering a dry tower can be accomplished by spraying water into the air or passing it through wet tower fill (Fig. 1.3.11). This is done in order to
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enhance the performance of the air-cooled heat exchanger during hot weather operation. Evaporation of a portion of the secondary cooling water flowing through the spray or fill section cools the air to a temperature near its wetbulb temperature.
Fig. 1.3.10 Air-Cooled Condenser in Parallel with Wet-Cooling Tower
Fig. 1.3.11 Dry Tower with Adiabatic Cooling of Inlet Air
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The plume, saturated discharge air, rising from a wet-cooling tower is further cooled as a result of a natural temperature lapse and mixing with ambient air. This cooling causes the water vapor in the plume to further condense to form small water droplets that make up the visible plume. On rising, the evaporation of droplets from the edge of the plume further increases the tempo and amount of cooling and droplet formation. The plume may become denser than the atmospheric air, and as it moves downwind, it edges towards the ground. At this stage the plume can become a visibility hazard, especially if there are highways or airports nearby. Icing of roads and landing strips may occur. Corrosion of steel and other structures may also be a problem, while visible pollution and shading caused by the plume may be unacceptable. In many of the regions where these problems occur, hybrid cooling systems become an option because of their ability to control or inhibit plume formation. In these towers, a combination of evaporative and dry-cooling results in a lower humidity ratio for the air leaving the tower. By minimizing the height of these structures, their visual impact is also reduced. An increasing number of wet/dry or hybrid cooling systems are being constructed to reduce visible plumes. Some of the early hybrid systems include a 150 MWe unit at the BASF petrochemical plant in Ludwigshafen described by Kolcott, a 245 MW cooling tower at the Harvard power plant commissioned in 1977 in the United States, and others described by Bouton and Isles. The hybrid cooling system at the 420 MWe Altbach/Deizisau power plant was commissioned in 1985 according to Alt and Mäule. The system, which rejects 558 MW, consists of a concrete cooling tower with a base diameter of 70 m and a height of 45 m. As shown in Figure 1.3.12, water flows through the dry elements in one pass from the bottom to the top prior to being distributed through the wet section—a conventional type. The airflow arrangement is parallel with dry air blown horizontally into the wet airstream emerging from the next section. In order to mix the two airstreams, channels of different lengths are provided for the dry air.
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Fig. 1.3.12 Altbach Hybrid Cooling Tower
The tower can be operated in five main modes:
40
1.
Wet operation at full fan capacity. The wet section of the tower is designed to dissipate 100% of the heat load, and it can operate without the dry section at any heat load condition. The sliding gates behind the dry heat exchangers must be closed in order to prevent the recirculation of the plumes through the openings of the fans of the dry section. The formation of visible plumes above the tower is dependent on the temperature and humidity conditions of the atmosphere at that time.
2.
Wet operation at reduced fan capacity. This mode of operation can be used at low heat load, particularly in winter. The plume above the cooling tower will be visible.
3.
Integrated operation at full fan capacity of both the wet and dry sections. At design conditions of drybulb temperature of 10 °C and relative humidity of 70.8%, the visible plume is eliminated.
4.
Integrated operation at reduced fan capacity. This mode is used during low heat load periods when the visible plume must be eliminated.
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5.
Dry-cooling operation. At an ambient air temperature of –15 °C with the heat load not exceeding 35%, the tower can be operated using the dry section only. The wet section of the tower is then bypassed by having the water flow directly from the heat exchangers to the water basin via a bypass line.
The much larger hybrid cooling tower at the 1300 MWe Neckarwestheim power plant (GKN) in Germany is located amongst beautiful vineyards where an unsightly cooling tower emitting a plume that would cast shadows over the area was unacceptable according to Alt. The resultant hybrid tower has a base diameter of 160 m and a height of 51 m. During wet operation, the required air is supplied by 44 fans requiring a power input of 8.14 MW, while an additional 44 fans requiring 11.22 MW are required during the hybrid operating mode. The tower can reject 2500 MW heat. A schematic drawing of the tower is shown in Figure 1.3.13.
Fig. 1.3.13 GKN Hybrid Cooling Tower
Under certain circumstances, it may even be justified to retrofit an existing wet-cooling tower with a dry section to control plume formation, such as the case at the Killingholme combined cycle power plant according to Vodicka.
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1.4 Conservation Equations The fundamental conservation equations of mass, momentum, and energy form the basis for solving fluid flow and heat transfer problems. The general equation of continuity, conservation of mass, for steady flow through a control volume or duct between sections 1 and 2 (Fig. 1.4.1) is
m = ∫ ρvdA = ∫ ρvdA A1 A2
(1.4.1)
where ρ = the fluid density v = the velocity normal to the cross-sectional area A. The values of ρ for some fluids are given in appendix A.
Fig. 1.4.1 Control Volume
Thermodynamics is the science of the relationship between heat, work, and the properties of systems. The first law of thermodynamics states that, for steady flow, the external work done on any system plus the thermal energy transferred into or out of the system is equal to the change of energy of the system (conservation of energy). By applying this law to the flow through the control volume shown in Figure 1.4.1 one obtains according to Shapiro
P+Q = m[(i2+αe2v22/2+gz2)–(i1+αe1v21/2+gz1)]
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(1.4.2)
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where P = power or rate of work Q = rate of heat input into the fluid
The enthalpy of a fluid is defined as
i = u+p/q
(1.4.3)
where u = internal energy of the fluid
Upon substitution of Equation 1.4.3 into Equation 1.4.2, we find
P+Q = m[(u2+p2/q2+αe2v22/2+gz2)–(u1+p1/q1+αe1v12/2+gz1)] (1.4.4)
In Equations 1.4.2 and 1.4.4, the αev2/2 term represents the kinetic energy of the fluid. Since the velocity may vary across a cross section, a kinetic energy velocity distribution correction factor or kinetic energy coefficient is defined as
αe =
1 m
∫(v2/2)dm/(vm2 /2) = ∫v3dA/(Avm3 )
A
(1.4.5)
A
if the density remains constant over the section. The mean velocity at any cross section is defined as
∫
vm = vdA/A A
(1.4.6)
For a uniform velocity distribution, αe equals unity.
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The potential energy of the fluid may change, depending on its elevation, z, above any arbitrary datum plane.
General features of isentropic flow Consider the one-dimensional isentropic (frictionless adiabatic) flow of a fluid through a passage of varying cross sections. For the particular case where the cross-sectional area is infinite, the fluid velocity is zero, and the pressure at this state, pt, is called the isentropic stagnation pressure or the total pressure. The value of the stagnation enthalpy, it, is independent of whether or not entropy changes occur, since it has the same value for all states that are adiabatically reachable from it. Equation 1.4.2 can be simplified where the control surfaces extend between a stagnation section 1 and another section 2 at the same elevation in the duct for isentropic, one-dimensional, uniform flow with no work interaction, i.e.
it1 = i2+v22/2
(1.4.7)
For a fluid, the change in enthalpy can also be written as
Di = cpDT
(1.4.8)
where cp = specific heat of the fluid at constant pressure The value of cp for some fluids is given in appendix A.
Equation 1.4.7 can be written as
cp(Tt1–T2) = v22/2
44
(1.4.9)
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and for a gas
cp – cv = R
(1.4.10)
where cv = specific heat at constant volume R = gas constant Since
c p /c v = γ
(1.4.11)
it follows that
c p = γR/(γ–1)
(1.4.12)
For a perfect gas, the relationship between its pressure, density, and temperature is given by
p = ρRT
(1.4.13)
If a perfect gas undergoes an isentropic change, the following relationship holds:
p/ρ γ = constant
(1.4.14)
Substitute Equations 1.4.12, 1.4.13, and 1.4.14 into Equation 1.4.9, and find γ
[( ) ]
p t1 = p 2 1+
γ–1
ρ 2 v 22
γ
2p 2
γ–1
(1.4.15a)
or
γ
[( ) ]
p 2 = p t1 1–
γ–1 q t1 v 22 γ
2p t1
γ–1
(1.4.15b)
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In many practical problems, compressibility effects are small but cannot always be ignored. If the second term in the square brackets is smaller than unity, Equation 1.4.15 can be expanded using the binomial theorem. This gives the following formulae for low speed isentropic flow if the higher order terms are neglected.
or
p t1 ≈ p 2 +q 2 v 22/2
(1.4.16a)
p t1 ≈ p 2 +q t1 v 22/2
(1.4.16b)
The preferred form of the expression for the stagnation pressure is Equation 1.4.16(a). Since the stagnation pressure for a given isentropic flow has a particular value, the following relation is applicable between any two cross sections of a duct. Section 1 need not have an infinite cross-sectional area.
p 1 + q 1 v 12/2 = p 2 + q 2 v 22/2
(1.4.17)
Using the same procedure, the following relation is applicable where flow occurs isentropically between two sections at different elevations, where no work interaction occurs, or the velocity distribution is uniform:
p 1 +α e1 ρ 1 v 12/2+q 1 gz 1 = p 2 +α e2 q 2 v 22/2+q 2 gz 2
(1.4.18)
Or if the velocity distribution is uniform:
p 1 +ρ 1 v 12/2+ρ 1 gz 1 = p 2 +ρ 2 v 22/2+ρ 2 gz 2
(1.4.19)
For the purely hydrostatic case, Equation 1.4.19 becomes
p 1 – p 2 = g(ρ 2 z 2 – ρ 1 z 1 )
(1.4.20)
More accurate expressions for the hydrostatic pressure differential between two elevations in the atmosphere are given by equations in chapter 9. If work is done isentropically on the control volume, find
(p 1 +α e1 ρ 1 v 21 /2+ρ 1 gz 1 ) – (p 2 +α e2 q 2 v 22/2+ρ 2 gz 2 ) = –P isen /V (1.4.21)
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where V = volume flow rate through the control volume
If the velocity distribution is uniform at sections 1 and 2, this equation becomes
(p 1 +ρ 1 v 12/2+ρ 1 gz 1 ) – (p 2 +ρ 2 v 22 /2+ρ 2 gz 2 ) = –P isen /V
(1.4.22)
Momentum theorem Real flows in ducts are usually not isentropic, due to frictional effects. Consider the steady incompressible fluid flow through the elementary control volume in a duct (Fig. 1.4.2).
Fig. 1.4.2 Elementary Control Volume in a Duct
According to the continuity Equation 1.4.1, at any section in the duct
m=
∫ ρvda = ρv
mA
= constant
A
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or, if written in differential form,
dm = d (ρv A) = 0 m dx dx
(1.4.23)
From Newton’s second law of motion, the net force due to frictional or other body forces, dF, acting on the fluid within the elementary control volume is equal to the difference in momentum between the outgoing and incoming flow during steady state conditions, i.e.,
d dF dp dx – A = dx dx dx
(
∫ρv dA 2
A
)
dx =
d (α m ρv m2 A)dx dx
d d = ρv m A (α m v m )dx + α m v m (ρv m A)dx dx dx
(1.4.24)
where the momentum velocity distribution correction factor is defined as
∫
α m = v 2 dA/(v m2 A)
(1.4.25)
A
Substitute Equation 1.4.23 into Equation 1.4.24, and integrate the latter between sections 1 and 2 to find 2
∫ dF/A = (p
2
+ α m2 ρv 22/2) – (p 1 + α m1 ρv 12/2)
(1.4.26)
1
If the velocity distribution at any cross section of the duct is uniform, αm = 1 and Equation 1.4.26 becomes 2
∫ dF/A = (p
1
48
2
+ ρv 22/2) – (p 1 + ρv21 /2)
(1.4.27)
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References Alt, W., “Concept Design, Construction and Commissioning of the First Hybrid Cooling Tower for a Heating Power Station with a Net Power Generating Unit Capacity of 420 MW,” 5th International Association for Hydraulics Research Cooling Tower Workshop, Monterey, California, October 1986. Alt, W., “Design and Operation of Hybrid Cooling Towers,” Vereinigung der Grosskraftwerksbetreiber Kraftwerkstechnik, vol. 1, 1987. Alt, W., and H. Damjakob, “Grosse Hybrid- und Trockenkühltürme an den Beispielen GKN II und KW Kendal,” Vorträge Vereinigung der GrosskraftwerksbetreiberFachtagung Kühltürme 1991, Vereinigung der GrosskraftwerksbetreiberKraftwerkstechnik, Essen, 1991. Alt, W., and R. Mäule, “Hybridkühltürme im Wirtschaftlichen Vergleich zu Nass- und Trockenkühltürmen,” Vereinigung der Grosskraftwerksbetreiber Kraftwerkstechnik, 8: 763–768, 1987. Bartz, J. A., and J. S. Maulbetsch, “Are Dry-Cooled Power Plants a Feasible Alternative?” Mechanical Engineering, 103: 34–41, October 1981. Berliner, P., Kühltürme, Springer-Verlag, Berlin, 1975. Bodás, J., and Z. Szabó, “A Wet/Dry Cooling System for a 600 MW Combined Cycle Power Plant,” 5th International Association for Hydraulics Research Cooling Tower Workshop, Monterey, California, 1986. Bodás, J., Z. Szabó, and L. Awerbach, “Combined Wet/Dry Dry Cooling Plants ‘System Heller’ of EGI,” American Society of Mechanical Engineers Winter Annual Meeting, Chicago, 1988. Bouton, F., and M. Monjoie, “Testing Procedure for Wet/Dry Plume Abatement Cooling Towers,” 9th International Association for Hydraulics Research Cooling Tower and Spraying Pond Symposium, von Karman Institute, Brussels, September 1994. Burger, R., Cooling Tower Technology Maintenance, Upgrading and Rebuilding, The Fairmont Press Inc., Lilburn, GA, 1994. Buxmann, J., and H. Heeren, “Oberflächenkondensator für Trockenkühlung,” Vereinigung der Grosskraftwerksbetreiber Kraftwerkstechnik, 5: 301–306, May 1974. Chapelain, B., “Cooling Towers and Thermal Release,” 9th International Association for Hydraulics Research Cooling Tower and Spraying Pond Symposium, von Karman Institute, Brussels, September 1994. Cheremisinoff, N. P., and P. N. Cheremisinoff, Cooling Towers: Selection, Design and Practice, Ann Arbor Science Publishers, Inc., Ann Arbor, Michigan, 1981. Christopher, P. J., and V. T. Forster, “Rugeley Dry Cooling Tower System,” Proceedings, Institution of Mechanical Engineers, part 1, no. 11, 184: 197–222, 1969–1970.
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Lefevre, M. R., “Heat Transfer Technology, Atmospheric Cooling Equipment Water Consumption,” Heat Transfer-Niagara Falls, American Institute of Chemical Engineers Symposium Series, 80–236: 361–370, 1984. Lindahl, P. A., and R. W. Jameson, “Plume Abatement and Water Conservation with the Wet/Dry Cooling Tower,” Electric Power Research Institute Cooling Towers and Advanced Cooling Systems Conference, St. Petersburg, Florida, August/September, 1994. Ludvig, L., “Heller Type Dry Cooling System for the 8 x 250 MWe Shahid Rajai TPP in Iran,” 9th International Association for Hydraulics Research Cooling Tower and Spraying Pond Symposium, von Karman Institute, Brussels, September 1994. March, F., and F. Schulenberg, Air-Cooled Condenser for a 160 MW Steam Power Station, Planning of the Plant and Experience Gained in Two Years of Operation, GEA Bochum, 1972. Mathews, R. J., “Air Cooling in Chemical Plants,” Chemical Engineering Progress, vol. 55, no. 5, May 1959. Mäule, R., G. Ernst, and G. Bräuning, “Results of Experiments on the Performance and the Emission of the Hybrid Cooling Tower at the Altbach-Deizisau Plant at Neckarwerke,” 5th International Association for Hydraulics Research Cooling Tower Workshop, Monterey, California, October 1986. Maze, R. W., “Air vs. Water Cooling,” The Oil and Gas Journal, 74–78, November 1974. McHale, C. E., G. E. Jablonka, J. A. Bartz, and D. J. Webster, “New Developments in Dry Cooling of Power Plants,” Combustion, 28–36, May 1980. McKelvey, K. K., The Industrial Cooling Tower, Elsevier Publishing Co., Amsterdam, 1959. McQuiston, F. C., and J. D. Parker, Heating Ventilating and Air Conditioning, 4th ed., John Wiley and Sons. Inc., New York, 1994. Mikovics, A., “Water Cooled Condensers in Dry Cooling Systems,” Symposium on Dry Cooling Towers, Tehran, 1991. Mitchell, R. D., Survey of Water-Conserving Heat Rejection Systems, Electric Power Research Institute, Final Report GS-6252, Project 1260–59, March 1989. Moore, F. K., “On the Minimum Size of Large Dry Cooling Towers with Combined Mechanical and Natural Draft,” Transactions of the American Society of Mechanical Engineers, Journal of Heat Transfer, 383–389, August 1973. Nagel, P., “New Developments in Aircooled Steam Condensers and Dry Cooling Towers,” 9th International Association for Hydraulics Research Cooling Tower and Spraying Pond Symposium, von Karman Institute, Brussels, 1994. Nowosad, R. M., A Review of the Development and Application of Industrial Dry/Wet Cooling, Council for Scientific and Industrial Research Report O/DPT1, South Africa, December 1988.
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Petzel, H. K., “6 Years Operational Experience of Cooling Tower Discharge of Desulphurized Flue Gas,” 6th International Association for Hydraulics Research Cooling Tower Workshop, Pisa, Italy, 1988. Plank, R., Handbuch der Kältetechnik, Springer-Verlag, Berlin, 1988. Rathje U. J., and H.-J. Pflaumbaum, “Die Generation 2000 luftgekühlter Abdampfkondensatoren,” Vereinigung der Grosskraftwerksbetreiber Kraftwerkstechnik, vol. 79, no. 1, 1996. Scherf, O., “Die luftgekühlte Kondensationsanlage im 150-MW-Block des PreussagKraftwerkes in Ibbenbüren,” Energie und Technik, 7: 260–264, 1969. Schrey, H. G., “Cost Evaluation of Dry Cooling Systems for Power Stations,” Proceedings, 10th International Association for Hydraulics Research Cooling Tower and Spraying Pond Symposium, 289–301, Tehran, Iran, 1996. Schulenberg, F., “Der Luftkondensator für den 365-MW-Block in Wyoming,” Sammelband Vereinigung der Grosskraftwerksbetreiber-Konferenz, 1977. Shapiro, A. H., The Dynamics and Thermodynamics of Compressible Fluid Flow, vol. 1, Ronald Press, New York 1953. Smith, E. C., and M. W. Larinoff, “Alternative Arrangements and Design for Wet/Dry Cooling Towers,” Power Engineering, 58–61, May 1976. Smith E. C., and M. W. Larinoff, “Alternative Arrangements and Designs Wet/Dry Cooling Towers for Power Plant Applications,” Combustion, 23–27, May 1977. Smith E. C., and M. W. Larinoff, “Analyzing Wet/Dry Cooling Towers,” Power, 78–80, May 1976. Spilko, J., “Dry Cooling Enhances Syria’s Teshrin Plant Operation,” Power Engineering International, February 1994. Steele, B. L., Selection of Plant Cooling Source(s), 75-IPWR-6, American Society of Mechanical Engineers, 1975. Stupic, D. M., “Innovative Application of Film Fill in Large Industrial Crossflow Cooling Towers,” Journal of the Cooling Tower Institute, 17–1: 76–81, 1996. Surface, M. O., “System Designs for Dry Cooling Towers,” Power Engineering, September 1977. Szabo, Z., “Why Use ‘Heller System’? Circuitry, Characteristics and Special Features,” Symposium on Dry Cooling Towers, Teheran 1991. Szabó, Z., and I. Szentgyörgyi, “Main Design Features and Operation Results of the Heller-Type Wet/Dry Cooling System Serving the Trakya CCPP,” 6th International Association for Hydraulics Research Cooling Tower Workshop, Pisa, Italy, October 1988.
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Techet, K., and W. Schug, “GEA’s and Balcke-Dürr’s Contribution in the Basic and Detail Engineering of the World’s First Large Size Natural Draft Dry Cooling Towers with Integrated Flue Gas Duct in ENEL’s Trino-Vercellese Combined Cycle Power Plant in Italy,” Proceedings, 10th International Association for Hydraulics Research Cooling Tower and Spraying Pond Symposium, 155–167, Tehran, Iran, 1996. Trage, B., and F. J. Hintzen, “Design and Construction of Indirect Dry Cooling Units,” Proceedings, Vereinigung der Grosskraftwerksbetreiber Conference, “South Africa 1987,” 1: 70–79, 1987. Trage, B., and F. J. Hintzen, “Planung und Bau von Anlagen mit indirekter Trockenkühlung,” Vereinigung der Grosskraftwerksbetreiber Kraftwerkstechnik, 2: 183–189, 1989. Trage, B., A. J. Ham, and Th. C. Vicary, “The Natural Draught, Indirect Dry Cooling System for the 6 x 686 MWe Kendal Power Station,” RSA, 90-JPGC/Pwr-25, American Society of Mechanical Engineers Joint Power Conference, Boston, 1990. U.S. Army Material Command, Engineering Design Handbook, Military Vehicle Power Plant Cooling, AMCP-706-361, Virginia, 1975. Van der Walt, N. T., L. A. West, T. J. Sheer, and D. Kuball, “The Design and Operation of a Dry Cooling Tower System for a 200 MW Turbo-Generator at Grootvlei Power Station, South Africa,” 9th World Energy Conference, Detroit, Michigan, USA, 1974. Vereinigung der Grosskraftwerksbetreiber Technisch-Wissenschaftliche Berichte, Wärmekraftwerke, 5, Kühlturm und Umwelt, Essen, 1975. Vodicka, V., and D. Blanck, “Killingholme Refit Reduces Plume Formation,” Modern Power Systems, 49–53, July, 1996. Von Cleve, H. H., “Comparison of Different Combinations of Wet and Dry Cooling Towers,” 75-WA/Pwr-10, American Society of Mechanical Engineers, 1975. Von Cleve, H. H., “Die Luftgekühlte Kondensationsanlage des 4000 MW-Kraftwerks Matimba/Südafrika,” Vereinigung der Grosskraftwerksbetreiber Kraftwerkstechnik 64, 4, April 1984. Von Cleve, H. H., “Dry Cooling Tower for the 300 MW THTR at Schmehausen, Germany,” American Institute of Chemical Engineers Symposium Series, no. 174, 74: 288–292, 1978. Willa, J. L., “How to Improve the Thermal Performance of Power Plant Cooling Towers,” Electric Power Research Institute Cooling Towers and Advanced Cooling Systems Conference, St. Petersburg, Florida, August/September, 1994.
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2 Fluid Mechanics
2.0 Introduction Fluid mechanics is the study of the behavior of fluids at rest, fluid statics, and in motion, fluid dynamics, and of the properties of fluids insofar as they affect the fluid motion. A fluid may be either a gas or a liquid. The molecules of a gas are much farther apart than those of a liquid. Hence a gas is compressible while a liquid is relatively incompressible. A vapor is a gas whose temperature and pressure are such that it is very near the liquid phase. Steam is considered a vapor because its state is normally not far from that of water. A gas may be defined as a highly superheated vapor, that is, its state is far removed from the liquid phase. Air is considered a gas because its state is normally very far from that of liquid air. The volume of a gas or vapor is greatly affected by changes in pressure or temperature or both. It is usually necessary to take into account the changes in volume and temperature in dealing with gases or vapors. Whenever significant temperature or phase changes are involved in dealing with vapors and gases, the subject is largely dependent on heat phenomena, thermodynamics.
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The objective of this chapter is to introduce the reader to certain terminology, concepts, laws, and equations that are directly applicable to the design of air-cooled heat exchangers and cooling towers.
2.1 Viscous Flow Consider the flow of a fluid over a flat plate (Fig. 2.1.1).
Figure 2.1.1 Boundary Layer Development along a Flat Plate
Beginning at the leading edge of the plate, a region develops where the influence of viscous forces is felt. These viscous forces are described in terms of a shear stress, τ, between the fluid layers. If this stress is assumed to be proportional to the normal velocity gradient, the defining equation for viscosity, known as Newton’s equation of viscosity, is
τ = –µ
dv dy
(2.1.1)
The constant of proportionality, µ, is called the dynamic viscosity. The values of µ for some fluids are given in appendix A.
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The velocity or hydrodynamic boundary layer is the region of flow that develops from the leading edge of the plate in which the effects of viscosity are observed. The y-position where the boundary layer ends is arbitrarily chosen at a point where the velocity becomes 99% of the free stream value. The boundary layer thickness, δ, is defined as the distance between this point and the plate. Initially, the boundary layer development is laminar. At some critical distance from the leading edge, small disturbances in the flow begin to become amplified and a transition process takes place until the flow becomes turbulent. This depends on the flow field and fluid properties. The physical mechanism of viscosity is one of momentum exchange. In the laminar portion of the boundary layer, molecules move from one lamina to another and carry momentum corresponding to the velocity of the flow. There is a net momentum transport from regions of high velocity to regions of low velocity, which creates a force in the direction of flow. This force may be expressed in terms of the viscous shear stress as given by Equation 2.1.1. The rate at which the momentum transfer takes place is dependent on the rate at which the molecules move across the fluid layers. In a gas, the molecules would move about with some average speed proportional to the square root of the absolute temperature since we identify temperature with the mean kinetic energy of a molecule in the kinetic theory of gases. The faster the molecules move, the more momentum they will transport. Hence we should expect the viscosity of a gas to be approximately proportional to the square root of temperature, and this expectation is corroborated fairly well by experiment. The laminar velocity profile is approximately parabolic in shape. The transition from laminar to turbulent flow occurs typically when
qv ∞ x l
=
v ∞x m
≥3.2x10 5
where v∞ = free stream velocity x
= distance from the leading edge of the plate
ν
= µ/ρ, the kinematic viscosity of the fluid
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This particular dimensionless group or ratio of inertial force to viscous force is called the Reynolds number after the British scientist-engineer who first did extensive research on flow in the late 1800s.
Re x =
ρv ∞ x µ
(2.1.2)
Although the critical Reynolds number for transition on a flat plate is usually taken as 3.2 x 105 for most analytical purposes, the critical value in a practical situation is strongly dependent on the surface roughness conditions and the turbulence level of the free stream. The normal range for the beginning of transition is between 3.2 x 105 and 106. With very large disturbances present in the flow, transition may begin with Reynolds numbers as low as 105. For flows which are very free from fluctuations, it may not start until Rex = 2 x 106 or more. In reality, the transition process covers a range of Reynolds numbers. Completed transition and fully developed turbulent flow usually is observed at Reynolds numbers twice the value at which transition began. A qualitative picture of the turbulent flow process may be obtained by imagining macroscopic chunks of fluid transporting momentum instead of microscopic transport on the basis of individual molecules. The turbulent boundary layer is more complex than the laminar boundary layer because the nature of the flow in the former changes with distance from the plate surface. The zone adjacent to the wall is a layer of fluid, which, because of the stabilizing effect of the wall, remains laminar even though most of the flow in the boundary layer is turbulent. This very thin layer is called the laminar sublayer, and the velocity distribution in this layer is related to the shear stress and viscosity using Newton’s viscosity law. The flow zone outside the laminar sublayer is turbulent. The turbulence alters the flow regime so much that the shear stress, as given by τ = - µ dv/dy, is not significant. The mixing action of turbulence causes small fluid masses to be swept back and forth in a direction transverse to the mean flow direction. As a small mass of fluid is swept from a low-velocity zone next to the sublayer into a relatively high-velocity zone farther out in the stream, the mass has a retarding effect on the high-velocity stream. This mass of fluid, through an exchange of momentum, creates the effect of a retarding shear stress applied to a high-velocity stream. A small mass of fluid originates farther out in the boundary layer in a high-velocity flow zone and is swept into a region of
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relatively low velocity. This has an effect on the low-velocity fluid much like shear stress augmenting the flow velocity. In other words, the mass of fluid with relatively higher momentum will tend to accelerate the lower velocity fluid in the region into which it moves. Although the process described previously is a momentum-exchange phenomenon, it has the same effect as a shear stress applied to the fluid. In turbulent flow, these stresses are termed apparent shear stresses or Reynolds stresses. The turbulent velocity profile has a nearly linear portion in the sublayer and a relatively flat profile outside this region. Consider the flow in a tube shown in Figure 2.1.2. A boundary layer develops at the entrance. Eventually the boundary layer fills the entire tube, and the flow is said to be fully developed. If the flow is laminar, a parabolic velocity profile is experienced as illustrated in Figure 2.1.2a. When the flow is turbulent, a somewhat blunter profile is observed (Fig. 2.1.2b). In a tube, the Reynolds number based on the mean fluid velocity and the tube diameter is again used as a criterion for laminar and turbulent flow. For Red = ρvd/µ ≤ 2300, the flow is usually observed to be laminar, whereas for Red ≥ 10,000, it is turbulent.
Fig. 2.1.2 Velocity Profiles in a Tube: (a) Laminar Flow (b) Turbulent Flow
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Again, a range of Reynolds numbers for transition may be observed depending on the roughness of the pipe and smoothness of the flow. The generally accepted range for transition, also referred to as the critical region, is 2000 < Red < 4000. Laminar flow has been maintained up to Reynolds numbers of 25,000 in carefully controlled laboratory conditions. The mass flow rate or continuity relationship for one-dimensional flow in a tube is
m = ρvA
(2.1.3)
where m = mass rate of flow v = mean velocity A = cross-sectional area of the tube
The mass flux or mass velocity is defined as
G = m/A = ρv
(2.1.4)
so the Reynolds number may be written as
Re d = Gd/µ
(2.1.5)
Similar flow patterns are observed in ducts that do not have a circular cross section. In those cases, it is convenient to define the following equivalent or hydraulic diameter for calculating the Reynolds number:
de =
4 x cross-sectional flow area wetted perimeter
(2.1.6)
This particular grouping of terms is used because it yields the value of the physical diameter when applied to a circular cross section.
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2.2 Flow in Ducts Real flows in ducts experience boundary stresses due to frictional effects, which cause a pressure drop to occur between any two cross sections. Typically, a force due to such frictional effects acts on the incompressible fluid in the elementary control volume shown in Figure 1.4.2, i.e., dF = - τappPedx. Here, τapp is the apparent shear stress at the fluid-wall interface, and Pe is the wetted perimeter of the duct according to Shames. If this force is substituted into Equation 1.4.27 for flow in a round duct or pipe of length L and fixed diameter d (v1 = v2), find 2
L
∫ –τ 1
app P e dx
/A=
∫ –τ
app (πd)dx
/ (πd 2 /4) = –4τ app L / d = p 2 – p 1
0
(2.2.1)
or
∆p = p 1 – p 2 = 4τ app L / d
Dimensional analysis shows that, for fully developed pipe flow, the frictional pressure drop, ∆p, between any two sections is generally related to the pipe geometry and fluid properties in the following way: ∆p = Function ρv 2 / 2
(
ρvd L ε , , l d d
)
The quantity ρv2/2 is known as the dynamic pressure. The term L/d considers the geometry of the pipe and ε/d is a measure of the roughness of the pipe surface. Based on this analysis, the pipe friction equation, also commonly referred to as the Darcy-Weisbach equation for pressure drop in a circular pipe, is obtained according to Weisbach, i.e.,
∆p = f D (L/d)(ρv 2 /2)
(2.2.2)
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The number of velocity heads, v2/2, lost for a given pressure drop, ∆p, is expressed by the product of the Darcy friction factor fD and the geometric factor L/d. The friction factor under consideration corresponds to fully developed velocity profiles, both laminar and turbulent. These are encountered only after 25 or more diameters downstream of a pipe inlet. By equating Equations 2.2.1 and 2.2.2, find
4τ appL/d = f D (L/d)(ρv 2 /2) or
f D = 4τ app / (ρv 2 /2)
Other definitions of the friction factor appear in the literature. In some cases, the right side of this equation is divided by a factor of 4, giving a friction factor f = fD/4, also referred to as the Fanning friction factor. The friction factor is a function of Re, the cross-sectional shape of the duct and, in the turbulent flow regime, the relative roughness of the duct surface. Equations 2.2.1 and 2.2.2 are also applicable to ducts other than circular pipes, in which case d is replaced by de.
Laminar flow An extensive summary of Fanning friction factors for laminar flow in a variety of ducts is presented by Shah. Using the Hagen-Poiseuille solution for fully developed laminar flow in a circular duct or pipe, f = 16/Re = fD/4 according to Shames. The friction factor is independent of the roughness of the surface in the case of laminar flow. The results for other duct shapes are shown in Figure 2.2.1. For the rectangular duct, the friction curve in Figure 2.2.1 may be expressed as
[
fRe = 24 1–1.3553
( ) ( )
+ 0.9564
62
b a
b a
+ 1.9467
4
– 0.2537
2
( ) ( )] b a
b a
– 1.7012
5
( ) b a
3
(2.2.3)
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Fig. 2.2.1 Friction Factors for Fully Developed Flow
In certain applications where very viscous fluids are to be cooled, e.g., oil coolers, a twisted tape may be inserted into a circular heat exchanger pipe or tube. Smooth heat exchanger pipes are usually referred to as tubes. The tape is inserted to establish swirl flow, thereby increasing the heat transfer coefficient and is twisted around the longitudinal axis shown in Figure 2.2.2.
Fig. 2.2.2 A Tube with a Twisted Tape Insert
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Using Hong and Bergles, the friction factor for fully developed flow in a tube with such an insert is given by
f = 45.9 / Re where Re = m/[µ(πd/4-tt )] m = mass flow rate tt = tape thickness A more recent equation for the friction factor was derived numerically by Du Plessis and Kröger:
f = a1
[ 1 + { Re/( 70(P/d) ) } ] 1.3
1.5
0.333
(2.2.4)
where a1 = a2/[Re(15.767 – 0.14706 tt/d)] a2 = Ats d2/(a3a42) a3 = 2P2 (a6 – 1) /p – dtt a4 = 4a3/a5 a5 = 2d – 2tt + πd/a6 a6 = [1 + (πd/2P)2 ] 0.5 Ats = πd2/4
This equation is valid for 50 ≤ Re ≤ 2000 and for P/d ≥ 2. In the case of hydrodynamically developing flow in a duct from an initial uniform velocity distribution, an apparent Fanning friction factor is defined. The factor takes into account both the skin friction and the change in momentum rate caused by a change in the shape of the velocity profile in the hydrodynamic entrance region. In a long duct, the apparent friction factor may be expressed in terms of an incremental pressure drop number K∞ as
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f appRe = f Re + K ∞ d e Re/4L
(2.2.5)
For a circular duct or pipe,
f appRe=16+0.313dRe/L for L/ (dRe)≥0.06
(2.2.6)
Du Plessis proposes the following general correlation which can be applied to developing laminar flow in ducts of various cross sections:
[
{
f a pp Re= (fRe) n + 3.44/(L/(d e Re)) 0 .5
}
n
]
1 /n
(2.2.7)
where f Re = the value for fully developed flow n = an exponent dependent on the duct geometry This correlation agrees well with similar ones by Shah. For concentric annular ducts having inner and outer radii of ri and ro, values of n are listed in Table 2.2.1.
Table 2.2.1 Values of Exponent for Annular Ducts
The case ri /ro = 0 corresponds to a pipe while ri /ro = 1 can be used for parallel plates.
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For rectangular ducts, fRe is determined using Equation 2.2.3. The values for n are listed in Table 2.2.2.
Table 2.2.2 Values of Exponent for Rectangular Ducts
For isosceles triangular ducts having apex angles of 2θ degrees, the values of n are listed in Table 2.2.3.
Table 2.2.3 Values of Exponent for Isosceles Triangular Ducts
In general, the hydraulic entry length Lhy = x/(deRe) is the dimensionless length required for the centerline velocity to attain 99% of its fully developed value. Values for Lhy and K∞ for different duct sections are listed in Table 3.2.1. When heat is transferred to or from the fluid, all physical properties are evaluated at the mean fluid temperature. The latter is also referred to as the bulk or mixing cup temperature according to Holman. For those problems involving large temperature differences between the fluid and the duct wall, Shah introduced corrections to provide for the temperature dependence of the fluid properties. In the case of gases, the friction factor evaluated at the bulk mean temperature is multiplied by one of the following factors:
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(Tw/T) for 1 < (Tw/T) < 3 (heating)
(2.2.8a)
(Tw/T)0.81 for 0.5 < (Tw/T) < 1 (cooling)
(2.2.8b)
For liquids, the friction factor evaluated at the bulk mean temperature is multiplied by one of the following factors to obtain the correct value.
(µw/µ)0.58 for µw/µ < 1 (heating)
(2.2.8c)
(µw/µ)0.54 for µw/µ > 1 (cooling)
(2.2.8d)
The subscript w refers to the mean value of the duct wall temperature and the temperatures are in degrees Kelvin. These relationships are also applicable to developing flow.
Example 2.2.1 Air at a pressure of p = 101,025 N/m2 and a temperature of T = 16.87 °C flows uniformly into a rectangular duct with a
= 50 mm
b
= 3.5 mm
at a rate of m
= 6.403 x 10-4 kg/s
the duct length is L = 200 mm Determine the pressure differential between the inlet and the outlet of the duct.
Solution Using the perfect gas law given by Equation 1.4.13, the density of the air at the specified conditions can be expressed as
ρ =
p RT
=
101025 287.08x(273.15+16.87))
= 1.2134 kg/ m 3
where the gas constant for air is R = 287.08 J/kgK.
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Using Equation A.1.3, the dynamic viscosity of the air at 16.87 °C or (273.15 + 16.87) = 290.02 K is
µ = 2 . 2 8 7 9 7 3 x 1 0 – 6 + 6 . 2 5 9 7 9 3 x 10 – 8 T – 3 . 1 3 1 9 5 6 x 1 0 –1 1 T 2 + 8 . 1 5 0 3 8 x 1 0 –1 5 T 3 = 2 . 2 8 7 9 7 3 x 1 0 –6+ 6 . 2 5 9 7 9 3 x 1 0 –8 x 2 9 0 . 0 2 – 3 . 1 3 1 9 5 6 x 1 0 –11x 2 9 0 . 0 2 2 + 8 . 1 5 0 3 8 x 1 0 – 1 5 x290.02 3 = 1 . 8 0 0 7 x 1 0 – 5 kg/ s m
The mean air speed in the duct follows from Equation 2.1.3, i.e.,
v = m/(ρab) = 6.403 x 10-4/(1.2134 x 0.05 x 0.0035) = 3.015 m/s Using Equation 2.1.6, the hydraulic diameter of the duct is d e=
4ab 2(a +b )
=
4x 0 . 0 5 x 0 . 0 0 3 5 2(0. 0 5 + 0 . 0 0 3 5 )
= 0.006542 m
The Reynolds number for the air flowing in the duct is, using Equation 2.1.5,
Re = ρv de /µ = 1.2134 x 3.015 x 0.006542/(1.8007 x 10-5) = 1329.1 It follows from Equation 2.2.3 that for duct flow
[
fRe=24 1–1.3553(0.0035/0.05)+1.9467(0.0035/0.05)2–1.7012(0.0035/0.05)3
]
+0.9564(0.0035/0.05)4–0.2537(0.0035/0.05)5 =21.938
For b/a = 0.0035/0.05 = 0.07, find n ≈ 2.3 from Table 2.2.2. Substitute the values for fRe and n into Equation 2.2.7 to find
[
f app Re= (21.938) 2.3 + { 3.44/(0.2/0.006542x1329.1) 0.5 }
or fapp = 30.16/1329.1 = 0.02269
68
2.3
]
1/2.3
=30.16
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Using Equation 2.2.2, the pressure drop between the inlet and the outlet of the duct is
∆p=4f app
L
ρv 2
de
2
( )
=4x0.02269
(
0.2 0.006542
)(
1.2134x3.015 2 2
)
=15.3 N/m 2
Turbulent flow With fully developed turbulent flow in ducts, the friction loss depends on flow conditions as characterized by the Reynolds number and on the nature of the duct wall surface. The quantity, ε, having the dimension of length is introduced as a measure of the surface roughness. From dimensional analysis, it follows that the friction factor is a function of the Reynolds number and the relative roughness ε/d. The graphical representation of this relationship is known as the Moody diagram and is presented in Figure 2.2.3 according to Moody.
Fig. 2.2.3 Friction Factors for Pipe Flow
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As shown, the laminar friction factor for pipe flow is a single straight line and is not influenced by the relative roughness. Reynolds numbers in the range from 2000 to 4000 lie in a critical region where flow can be either laminar or turbulent. For Reynolds numbers larger than those in the critical region, turbulent flow exists. The two regions into which the turbulent zone is divided, transition and complete turbulence, categorize the state of the viscous sublayer as influenced by roughness. Based on Nikuradse’s data, the following implicit relationship is applicable to turbulent flow in smooth pipes 1/ f D0.5 = 0.86 n (Re f
0.5 D
) - 0.8
(2.2.9)
or according to Filonenko, -2
f D = (1.82log10 Re - 1.64 )
(2.2.10)
Inspection indicates, for high Reynolds numbers and relative roughness, the friction factor becomes independent of the Reynolds number in the region of complete turbulence. Then f D = [1.14 - 0.86 n (ε /d)]
-2
(2.2.11)
Transition between this region and the smooth wall friction factor is represented by an empirical implicit transition function developed by Colebrook.
ε /d 2.51 = log 10 + 0.5 0.5 fD 3.7 Re f D 1
-2
(2.2.12)
For purposes of computation, the following explicit relationship from Benedict is of value:
fD
ε /d 5.74 = 0.25 log 10 + 0.9 3.7 Re
-2
(2.2.13)
Haaland recommends an equation that yields results comparable to the implicit Colebrook equation:
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For ε/d > 10-4 fD
6.9 ε /d 1.11 = 0.3086 log 10 + Re 3.7
-2
(2.2.14)
For situations where ε/d is very small, Haaland proposes 7.73 ε /d 3.33 + f D = 2.7778 log10 Re 3.75
-2
(2.2.15)
The curves in the turbulent and transitional zones in Figure 2.2.3 were drawn employing Equation 2.2.11 and the Haaland relations, respectively. An approximate indication of the relative roughness of typical pipe surfaces encountered in practice is shown in Figure 2.2.4 according to Kirschmer.
Fig. 2.2.4 Surface Roughness in Pipes
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Dimensional analysis does not relate the performance of ducts having circular and noncircular cross sections. The fully turbulent friction factor for noncircular cross sections (annular spaces, rectangular and triangular ducts, etc.) may be evaluated from the data for circular pipes. This applies if the pipe diameter is replaced by an equivalent diameter, also referred to as the hydraulic diameter, defined by Equation 2.1.6. The equivalent or hydraulic diameter for an annulus of inner and outer diameter di and do is
de =
4(π /4)(d o2 - d i2 ) = d o- d i π (d o + d i )
(2.2.16)
For a rectangular duct having sides a and b, it is
d e = 2 ab /(a+ b)
(2.2.17)
Launder and Ying show, for a rectangular duct, the secondary velocity distribution gives rise to an increase in the friction factor of about 10%. Even so, their full theory slightly underestimates the measurements of Hartnett et al. According to White, the friction factor for turbulent flow between parallel plates as given by the following equation is higher than the value that would be obtained if the equivalent diameter were substituted into the equation for pipe flow.
1/f 0.5 D = 2 log10( Ref
) - 1.19
0.5 D
(2.2.18)
where Re is based on the hydraulic diameter which is equal to twice the distance between the plates. For developing turbulent flow near the entrance of a duct, the friction factor is considerably higher than for fully developed flow according to Deissler. When heat transfer occurs in turbulent duct flow, changes in thermophysical properties should be considered. This effect is taken into consideration by multiplying the friction factor, evaluated at the bulk temperature of the fluid, by one of the following appropriate correction factors from Petukhov:
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(7 – µ/µw)/6 for (µw/µ) < 1 (heating)
(2.2.19a)
(µw/µ)0.24 for (µw/µ) > 1 (cooling)
(2.2.19b)
for 1.3 < Pr < 10 and where µw is evaluated at the duct wall temperature. For air and hydrogen—temperature in degrees Kelvin
(Tw /T )[-0.6+ 5.6( Re w ρ w / ρ )
] (heating)
-0.38
(Tw /T )[-0.6+ 0.79( Re w ρ w / ρ )
-0.11
] (cooling)
(2.2.20a)
(2.2.20b)
Example 2.2.2 Calculate the approximate mean Darcy friction factor when air •
at a pressure of pa = 1.013 x 105 N/m2
•
a bulk temperature of T = 93.33 °C
•
flows at a speed of 6.096 m/s through a smooth pipe having an inside diameter of 25.4 mm
•
Tw = 426.67 °C is the inside pipe wall temperature
•
µaw = 3.355 x 10-5 kg/ms is the dynamic viscosity of air at 426.67 °C
Solution Evaluate the Reynolds number of the airstream at bulk temperature. Using Equation A.1.1, the density of the air at the bulk temperature of (273.15 + 93.33) = 366.48 K is
ρa =
pa 287.08 T
5
=
1.013 x 10
287.08 x (273.15 + 93.33)
= 0.9628 kg/m3
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The dynamic viscosity of the air at this temperature follows from Equation A.1.3.
µa = 2.287973 x 10-6 + 6.259793 x 10-8 x 366.48 - 3.131956 x 10-11 x 366.482 + 8.15038 x 10-15 x 366.483 = 2.14236 x 10-5 kg/ms Thus, Re =
ρ a vd µa
=
0.9628 x 6.096 x 0.0254 = 6958.61 -5 2.14236 x 10
The flow is turbulent, and the friction factor for the smooth tube may be determined at the bulk temperature using Equation 2.2.10, i.e., -2
f D = (1.82 log10 6958.61- 1.64) = 0.03489
This factor must be corrected by multiplying it by Equation 2.2.20a, which includes the Reynolds number of the air evaluated at the wall temperature Tw = 426.67 °C or (273.15 + 426.67) = 699.82 K The air density at this temperature is 5
ρ aw =
1.013 x 10 3 = 0.5042 kg/m 287.08 x 699.82
With this density and the specified dynamic viscosity of the air evaluated at the pipe wall temperature, find the Reynolds number
Re w =
0.5042 x 6.096 x 0.0254 = 2326.96 3.355 x 10 5
The corrected friction factor is thus
f Dc = f D (Tw /T)
[-0.6+ 5.6( Re w ρ aw / ρ a)-0.38 ] - 0.38
= 0.03489 (699.82/ 366.48 )[-0.6+ 5.6(2326.96x 0.5042/0.9628)
74
]
= 0.03019
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Transition laminar-turbulent flow In the critical zone where transition from laminar to turbulent flow takes place, the friction factor is uncertain, and there is an uncertainty in pressure drop estimates if the Reynolds number falls in this range, i.e., 2000 ≤ Re ≤ 4000. A single correlating equation that covers the entire range from the laminar through the critical region to turbulent flow in smooth pipes or tubes is proposed by Churchill: 10 Re 1 fD = 8 + 2.21n 0.5 8 10 7 Re 20 + Re 36500
-0.2
(2.2.21)
A more comprehensive equation including the effect of surface roughness is also presented by Churchill. 12 1 fD = 8 8 + Re 1.5 (a1 + a2)
0.0833
(2.2.22)
where 1 a1 = 2.457 n 0.9 (7 /Re) + 0.27ε /d
16
and a2 =
[
(37530) Re
]
16
2.3 Losses in Duct Systems As a result of the frictional resistance experienced during flow in a horizontal duct, the mechanical energy (the p/ρ + αev 2/2 terms in Equation 1.4.4) between any two sections of the duct is reduced, i.e., converted to thermal energy. Similar reductions or losses in mechanical energy may occur at inlets, outlets, abrupt changes in duct cross-sectional area, valves, bends, and other appurtenances.
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A dimensionless loss coefficient, in general, can be defined between two cross sections in a horizontal duct as
K =
( p /ρ 1
1
+ α e1 v12 / 2 )-( p2 / ρ 2 + α e 2 v 22 / 2) 2 v /2
(2.3.1)
where v is usually based on conditions at either section 1 or section 2. Since most loss coefficients are determined experimentally, it is important to specify the velocity on which a loss coefficient for a particular duct element is based, i.e., inlet, outlet, or some mean condition. If the flow is incompressible and the velocity distribution at sections 1 and 2 is uniform, as is approximately the case in turbulent flow, the kinetic energy coefficient is αe ≈ 1, and Equation 2.3.1 can be written as
K =
(p + ρ v /2) - (p + ρ v /2) 1
2 1
2
2 2
ρ v /2 2
(2.3.2)
or K = ( p t1 - p t2 ) / ( ρ v 2 / 2 )
where pt1 = total pressure at section 1 pt2 = total pressure at section 2 K is also referred to as the total pressure loss coefficient
A static loss coefficient is sometimes defined as 2 K s = ( p1 - p2 )/( ρ v / 2 )
(2.3.3)
This loss coefficient is equivalent to K if there is no change in velocity between sections 1 and 2.
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For frictional, fully developed, incompressible flow in a pipe of constant diameter, the pressure drop is given by the empirical Darcy-Weisbach Equation 2.2.2. A loss coefficient, using Equation 2.3.2, may be expressed as
Kf = ∆p/(ρv2/2) = fDL/d
(2.3.4)
The region of influence for a component can be determined experimentally. Attach straight pipes to the exit and entry of the component to attain fully developed conditions upstream and downstream. Consider the pressure distribution along a pipeline containing a bend for incompressible turbulent flow shown in Figure 2.3.1. The variations in static pressure, present across a section within the bend, extend for a diameter or two into the straight pipes upstream or downstream. The constant pressure gradient associated with fully developed flow in a straight pipe is not reestablished until fifty or more diameters downstream from the bend.
Fig. 2.3.1 Pressure Distribution in Horizontal Pipeline Containing a Bend
For most other pipework components, the variations in static pressure at a cross section are far less marked than for bends, and the fully developed velocity profile is recovered more quickly. For example, the flow recovers after about five diameters downstream of a sudden enlargement according to Hall.
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When the flow upstream and downstream of a component is fully developed, the component is said to be free of interference effects, since its performance is independent of the flow beyond the regions of fully developed flow. In the calculation of system pressure losses, it may be necessary to approximate the real situation by using data obtained from tests in interference-free flow. This approach should not lead to large errors, provided that a spacer length of at least 5 diameters or, in the case of bends, 10 diameters separates one component from another. In many practical systems, interference exists between the components of the pipe system. This occurs when the regions of influence of two components overlap. According to ESDI, the pressure loss through combinations may be higher or lower than the sum of the losses of the components in interference-free flow. There are so many possible combinations of components that, at present, the prediction of the performance of a system with interferences cannot be made with any accuracy except in special circumstances. If the interference in the system is confined to the interactions between a few of the components for which data are available, the principles outlined for interference-free flow can be applied by considering these components as single entities. In some circumstances, it may be possible to deduce, by broad physical arguments, that the interference effects will not be important. Consequently, the performance of the system can be estimated quite accurately using data obtained under interference-free conditions. Alternatively, there may be one or two particularly large sources of pressure loss in the system, and a much lower order of accuracy is acceptable for the other components—if their magnitudes are known with reasonable accuracy. However, these situations must be regarded as special cases. With the present state of knowledge, it is necessary to resort to model tests or computer analysis if an accurate assessment is required of the performance of a duct flow system in which interferences occur. Extensive data for loss coefficients of different components in pipe and duct systems is presented in the literature in works by Miller, Holms, Crane, Idelchik, and Fried.
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Abrupt contractions and expansions Head or mechanical energy losses occur at abrupt changes in duct flow cross-sectional areas. Consider incompressible flow in the duct shown in Figure 2.3.2, which includes a contraction at the inlet and a sudden enlargement at the outlet.
Fig. 2.3.2 Static Pressure Distribution in a Duct
From Equation 2.3.2 for uniform velocity distributions, the total inlet pressure drop due to a reduction in the flow area resulting in an acceleration of the flow and a loss due to separation of the boundary layer can be expressed as
( p + ρ v / 2) - ( p 1
2 1
+ ρ v 22/ 2) = K c ρ v 2 / 2 2
2
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The static pressure drop is p1 - p2 = ( ρ v 22 / 2)[(1- σ 21 ) + K c ] 2
(2.3.5)
where σ21 = A2 /A1 Kc
= the contraction coefficient containing the irreversible loss
At the outlet of the duct, there will be a rise in static pressure owing to the increase in flow area, whereas a loss will occur due to boundary layer separation and momentum changes following the abrupt expansion. The resultant change in pressure is 2 )] p3 - p4 = ( ρ v32 / 2)[K e - (1- σ 34
(2.3.6)
where Ke
= expansion coefficient
σ34 = A3/A4
The loss coefficients Kc and Ke refer to the kinetic energy of the flow in the smaller cross-sectional area. For highly turbulent flow, Kays expressed these coefficients using the following two equations: 2
K c = 1- 2 /σc + 1/σ c = (1- 1/σc ) 2
(2.3.7)
and Ke
80
= (1 – σ34)2
(2.3.8)
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Equations 2.3.7 and 2.3.8 apply to square-edged abrupt changes in cross section for single tubes or for tube bundles. The contraction ratio σc = Ac /A2 , shown in Figure 2.3.3 for two-dimensional and three-dimensional circular contractions, is usually determined experimentally according to Weisbach. The minimum area of the jet between sections 1 and 2, Ac, is referred to as the vena contracta.
Fig. 2.3.3 Contraction Ratio for Round Tubes and Parallel Plates
The curves shown in Figure 2.3.3 are approximated by the following empirical relations for round tubes σc = 0.61375 + 0.13318 σ21 - 0.26095 σ 221 + 0.51146 σ 321
(2.3.9)
and for parallel plates according to Rouse.
σ c = 0.6144517+ 0.04566493 σ 21 - 0.336651σ 221 + 0.4082743 σ 321
(2.3.10) + 2.672041 σ - 5.963169 σ + 3.558944 σ 4 21
5 21
6 21
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It is possible to reduce the contraction loss coefficient for a tube significantly by rounding off the inlet edge. This is illustrated in Fig. 2.3.4 and Table 2.3.1, according to Fried.
Fig. 2.3.4, Table 2.3.1 Contraction Loss Coefficient for Rounded Inlet
For tubes penetrating into a manifold, Fried shows a higher loss coefficient is applicable (Fig. 2.3.5 and Table 2.3.2).
Fig. 2.3.5, Table 2.3.2 Contraction Loss Coefficient for Penetrating Tube
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Example 2.3.1 A heat exchanger bundle consists of tubes having an inside diameter of 22.09 mm and a total length of 15,024 mm. The tubes are arranged in a staggered pattern and are welded into tube sheets (Fig. 2.3.6). Water at 52.5 °C flows through each tube at a rate of 0.4015 kg/s. The tube inlet has a square edge. Determine the difference in static pressure between the headers for a smooth tube and for the case where ε/di = 10-3.
Fig. 2.3.6 Tube Dimensions and Layout
Solution The thermophysical properties of water are listed in appendix A. Evaluate properties at 52.5 °C (325.65 K). Density of water from Equation A.4.1:
ρw = (1.49343 x 10–3 – 3.7164 x 10–6 x 325.65 + 7.09782 x 10–9 x 325.652 – 1.90321 x 10–20 x 325.656)–1 = 986.9767 kg/m3
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Dynamic viscosity of water from Equation A.4.3:
µ w = 2.414 x 10–5 x 10247.8/(325.65 – 140) = 5.21804 x 10–4 kg/ms The Reynolds number for the water flowing in the tube is Re =
ρw v di 4m 4 x 0.4015 x10 7 = = = 44,349.9 π di µw π x 22.09 x 5.21804 µw
The flow in the tube is turbulent, e.g., transition zone. The mean velocity of the water in the tube is determined from
v =
4m 4 x 0.4015 = = 1.06144 m/s qw p d i2 986.9767 x π (0.02209)2
The frictional pressure drop may be determined using Equation 2.2.2. L ρ v2 ∆ pf = f D w di 2
For a smooth tube, the friction factor follows from Equation 2.2.10:
fD = (1.82 log10 44349.9 – 1.64)–2 = 0.021516 The frictional pressure drop is thus 2
15024 986.9767 x 1. 06144 2 = 8136.15 N/ m ∆ pf = 0.021516 2 22.09
For the rough pipe, it follows from Equation 2.2.14 that fD = 0.3086/[log10 {6.9/44349.9 + (10-3/3.7)1.11 }]2 = 0.0241234 The resultant frictional pressure drop is 2
15024 986.9767 x 1. 06144 2 ∆ pf = 0.0241234 = 9122.12 N/ m 2 22.09
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For the particular tube layout, the area ratio for the entering water stream is
σ = π x 22.092/(4 x 58 x 50.22) = 0.13158 whereas the jet contraction ratio is, using Equation 2.3.9,
σc = 0.61375 + 0.13318 x 0.13158 – 0.26095 x 0.131582 + 0.51146 x 0.131583 = 0.628
For turbulent flow, the inlet contraction loss coefficient may be approximated by Equation 2.3.7, i.e.,
Kc = 1 – 2/0.628 + 1/0.6282 = 0.351 The static pressure drop at the inlet to the tube follows from Equation 2.3.5:
∆pi = 0.5 x 986.9767 x 1.061442 [(1 – 0.131582) + 0.351] = 741.5 N/m2 The outlet expansion loss coefficient is approximated by Equation 2.3.8:
Ke = (1 – 0.13158)2 = 0.7542 The static pressure drop at the outlet of the tube follows from Equation 2.3.6:
∆pe = 0.5 x 986.9767 x 1.061442 [0.7542 – (1 – 0.131582)] = –127.0 N/m2 For a smooth tube, the total static pressure differential between the headers is
∆p = 8136.15 + 741.5 – 127.0 = 8751 N/m2 For the rough tube, the pressure drop is
∆p = 9122.12 + 741.5 – 127.0 = 9737 N/m2
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Reducers and diffusers When the duct flow area is reduced gradually (Figs. 2.3.7a and b), the number of velocity heads lost is very small. Based on the smaller flow area, a loss coefficient of Kred = 0.04 or less is commonly quoted.
b
a
c
Fig. 2.3.7 Reducers (a) (b) (c)
For the conical reducer shown in Figure 2.3.7(c), the loss coefficient based on the smaller area may be obtained from the following equation according to Fried:
K red = (- 0.0125 σ 21 + 0.0224 σ 21 - 0.00723 σ 21 + 0.00444 σ 21 - 0.00745 ) 4
3
2
x (8 θ - 4π θ - 20 θ ) 3
2
where σ21 = A2/A1 θ is in radians
The loss coefficient is based on the velocity at 2.
86
(2.3.11)
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Whenever it is necessary to increase the flow area of a pipe gradually, the conical diffuser shown in Figure 2.3.8 may be used.
Fig. 2.3.8 Conical Diffuser
Ideally, in the absence of losses, the total pressure remains constant, i.e., 2
2
p2id + ρ v 2 /2 = p1 + ρ v 1 /2
and the pressure recovery is p2id - p1 = ρ (v1 - v 2) / 2 = ρ v1 (1- σ 12) / 2 2
2
2
2
(2.3.12)
where σ12 = A1/A2 id
= ideal conditions
In practical diffusers, only a part of this pressure recovery is possible, and a diffuser efficiency is defined as
η dif =
p2 - p1 p2 - p1 = 2 p2id - p1 ρ v12(1- σ 12 )/2
(2.3.13)
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For relatively small expansion angles, the diffuser efficiencies have been determined by Patterson, whose results are shown in Figure 2.3.9. The loss coefficient of a diffuser with uniform inlet and outlet flow is
K dif =
pt1 - pt 2 ρv /2 2 1
=
(p
1
+ ρ v12/ 2) -( p2 + ρ v 22/ 2) 2 ρ v1 / 2
(2.3.14)
Fig. 2.3.9 Conical Diffuser Efficiencies
Substitute Equation 2.3.13 into Equation 2.3.14 and find K dif = (1- ηdif)(1- σ12) 2
(2.3.15)
For practical applications, Daly suggests it may be convenient to employ Figure 2.3.10.
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Fig. 2.3.10 Losses in Duct Diffusers
The losses for open-outlet diffusers are shown in Figure 2.3.11. A uniform approach velocity, such as a venturi nozzle flow meter, allows more rapid expansion and lower loss, shown by the broken lines. Extensive data on flat and conical diffusers is presented by Runstadler et al.
Fig. 2.3.11 Losses in Open Outlet Diffusers
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Three-leg junctions Pressure loss data for dividing flows through planar three-leg junctions are cited in various references such as Engineering Sciences Data Unit (ESDU), Verein Deutscher Ingenieure-Wärmeatlas, the General Electric Fluid Flow Data Book, and Miller. The total pressure differences across pairs of inlet and outlet legs of a junction (Fig. 2.3.12) are calculated from
pt3 – pt1 = K31ρv32/2 + fD3L3ρv32/2d3 + fD1L1ρv12/2d1
(2.3.16)
pt3 – pt2 = K32ρv32/2 + fD3L3ρv32/2d3 + fD2L2ρv22/2d2
(2.3.17)
and
Fig. 2.3.12 Variation of Total Pressure in the Vicinity of a Junction
The last two terms on the right side of Equations 2.3.16 and 2.3.17 are the straight-pipe friction losses over lengths L3, L1, and L2. The loss coefficient for a 90° junction, Kj90 = K31, with Re3 ≥ 2 x 105 between leg 3 and leg 1 is given in Figure 2.3.13 for square corners (r31 = r12 = 0) as a function of A1/A3 and V1/V3.
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Fig. 2.3.13 Loss Coefficient for a 90° Junction with Square Corners
With rounded corners, the loss coefficient may be reduced, i.e.,
V1 / V3 K31r = K j90 r = K j 90- 0.9 A1 / A 3
2
0.5
V /V r 31 – 0.26 1 3 d 1 A1/ A 3
2
0.5
r 12 d1
(2.3.18)
for r12/d1 < 0.15 and r31/d1 < 0.15, and
V1/ V3 K 31r = K j 90 r = K j 90- 0.45 A1/ A 3
2
(2.3.19)
for r12/d1 > 0.15 and r31/d1 > 0.15.
The information strictly applies to junctions where the inlet flow is fully developed and where there is a long downstream duct length. However, in practice it can be applied without significant error when there are 15 or more equivalent diameters upstream and at least 4 diameters downstream of the junction. It is possible to reduce the loss coefficient by installing suitable guide vanes.
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The loss coefficient between leg 3 and 2 can be assumed to be unaffected by the geometry of leg 1 and is given in Figure 2.3.14 as a function of the flow ratio only. There is no significant change in the loss coefficient, K32, due to rounding of the junction corners.
Fig. 2.3.14 Loss Coefficient K32 for a 90° Junction
In the case of a square-edged T-junction, (KT = K31, the loss coefficient is shown in Figure 2.3.15.
Fig. 2.3.15 Loss Coefficient for a T-Junction with Square Corners
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Curved ducts or bends A few common types of curved ducts or bends are shown in Figure 2.3.16. Pressure loss data for flow through such bends are available from ESDU.
a
b
c
Fig. 2.3.16 Bends (a) Circular-Arc Bend (b) Single Miter Bend (c) Composite Miter Bend
The pressure loss due to a square bend and, in particular, a miter bend may be reduced by fitting guide vanes. It is common practice to use a number of guide vanes in a cascade in a miter bend. The vane geometry is not fixed by the bend geometry, and a number of designs exist. One example of such a miter bend is shown in Figure 2.3.17.
Fig. 2.3.17 Miter Bend with Cascade and Circular-Arc Guide Vanes
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Insufficient systematic data exist to provide detailed information on pressure losses in miter bends with cascades shown in Figure 2.3.18. The probable range of the loss coefficient, Kbm, is between 0.15 and 0.4, the lower values requiring very careful construction for their achievement. These values compare with a loss coefficient equal to approximately 1.1 for a similar bend without a cascade.
Fig. 2.3.18 Miter Bends Kbm = 0.28, Kbm = 0.25, Kbm = 0.4 (Jorgensen)
2.4 Manifolds The design of manifolds for the distribution or division of a fluid stream into several branching streams is of importance in the design of different types of heat exchangers. The same is true for the formation of a single main stream by the collection or confluence of several smaller streams. A manifold basically consists of a main channel, a header, to which several smaller conduits, tubes, or laterals are attached at right angles. Manifolds commonly used in flow distribution systems can be classified in the following five categories (Fig. 2.4.1).
94
•
simple distributing or dividing
•
collecting or combining
•
cocurrent flow or parallel flow
•
countercurrent flow or reverse flow
•
mixed flow or combined flow
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The cocurrent or countercurrent flow configurations are also referred to as Z or U type heat exchangers.
Fig. 2.4.1 Types of Manifolds
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Often, the objective of the design is to provide equal flow rates through the branches or laterals. This can be achieved if the cross-sectional area of the header is designed such that the fluid velocity and the pressure in the header remain constant. The variations in fluid pressure in the header are due to frictional effects and changes in momentum. The frictional loss is always in the direction of the flow. The momentum changes cause an increase in pressure in the direction of flow in the distributing header and a decrease in pressure in the direction of flow for the collecting header. By applying the energy equation to the flow in the header, Miller and Hudson quantified the flow and pressure distribution in the header. Enger and Levy, Keller, Acrivos et al., Markland and Bassiouny, and Martin based their continuous mathematical model on local momentum balance considerations in the header. Discrete mathematical models based on local momentum balances are derived by Kubo and Euda, Majumdar and Datta, and Majumdar. Bajura and Bajura and Jones derived continuous mathematical models based on integral momentum balances of the fluid in the header. They integrated the momentum equation in vector form over a control volume to quantify the flow and pressure distribution. A discrete mathematical model based on an integral momentum balance is due to Nujens. To find the approximate flow distribution and pressure change in a manifold, consider a section of a header of uniform cross section, Ah, and wetted perimeter, Peh, fitted with closely spaced laterals shown in Figure 2.4.2.
Fig. 2.4.2 Header Control Volume (a) Dividing Header (b) Combining Header
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By applying a mass balance to an incompressible fluid in an elementary control volume, Ah∆x, in a header, find ρAh(dvh/dx)∆x + ρvA = 0
(2.4.1)
where the subscripts h and refer to the header and the laterals, for a header of length Lh and n laterals, ∆x = Lh/n. The approximate momentum balance applicable to the control volume may be written as
ph A h - [ ph+ (dph / dx) ∆ x] Ah - c τ h Peh ∆ x = α mhρ [v h + (dvh /dx) δx ] A h + α m ρ v A v h 2
(2.4.2)
2 α mhρ v h A h
where the factor c = 1 for vh > 0 and c = -1 for vh < 0. The momentum correction factors, αmh and αm provide for nonuniformities in local header and lateral inlet velocity distributions. Using Equations 2.2.1 and 2.2.2, the header shear stress can be expressed in terms of the Darcy friction factor, i.e.,
τ h = f D ρ v h2 /8
(2.4.3)
Substitute Equations 2.4.1 and 2.4.3 into Equation 2.4.2, and simplify to find
2 dph dvh c f D v h + αmh ρv h + =0 dx dx 2d e
(2.4.4)
where the overall momentum correction factor αmh = 2αmh - αm and de = 4Ah /Peh. Bajura et al. graphically present some values for overall momentum correction factors.
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If a lateral is located between a dividing and a combining header, the static pressure difference between these headers can be expressed as 2 2 2 2 2 phd - phc= [K i - αed v hd/ v i + K ]ρv i / 2 + (K o+ α ecv hc / v o) ρ v o /2
(2.4.5)
In this equation, Ki and Ko are the lateral inlet and outlet loss coefficients while αed and αec are the header energy correction factors. In a conventional single phase heat exchanger, the lateral velocity vi = vo; however, in a condenser, the inlet velocity is greater than the outlet velocity. Bassiouny and Martin, after making some simplifying assumptions, present solutions for the manifold momentum equations for single phase flow in U or Z type configurations. The more complicated problem of flow distribution in an air-cooled steam condenser was analyzed by Zipfel and Kröger. While the lateral inlet loss in a single-phase heat exchanger is usually negligible, this is not the case in a condenser. They show, under certain circumstances, backflow of vapor can occur in some of the laterals.
2.5 Drag Drag is defined as the force component, parallel to the relative approach velocity exerted on the body by the moving fluid. Mathematically it can be expressed as
FD = CD A ρ v2/2 where A
= the characteristic projected area normal to the flow
ρv2/2 = the dynamic pressure of the main stream CD
98
= the drag coefficient
(2.5.1)
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The drag coefficient is found by dimensional analysis to be a function of the geometrical configuration of the immersed body, the Reynolds number, the turbulence characteristics of the incoming free stream, and the surface roughness of the body. Experimental data on drag coefficients versus Reynolds numbers for several different two-dimensional bodies are plotted in Figure 2.5.1.
Fig. 2.5.1 Coefficient of Drag for Two-Dimensional Bodies
A summary of results on forces associated with flow across circular cylinders is reported by the Engineering Sciences Data Unit (ESDU). As shown in Figure 2.5.1, the drag coefficient remains almost constant at a value of 1.2 for 104 < Re < 2 x 105 in the case of infinitely long cylinders. The drag coefficient is not significantly affected by surface roughness or free-stream turbulence for Re < 3 x 104. Above this value, these effects do become important according to ESDU.
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If the cylinder axis is rotated through an angle θD with respect to the normal approach flow direction, the drag coefficient may be calculated approximately from CDθ = CD (cos θD)3
(2.5.2)
for 0° < θD < 45°. Empirical equations based on experimental data have been developed by Hoerner for predicting the drag coefficient about elliptical sections for different Reynolds numbers, i.e., CD = 2.656 (1 + a/d) Re–0.5 + 1.1 (d/a)
(2.5.3)
for 103 < Re < 106 and where a and d are the dimensions of the major and the minor axes. This relationship is plotted in Figure 2.5.1. If the elliptical section is inclined relative to the flow, the drag coefficient is corrected in the same way as the circular cylinder, i.e. Equation 2.5.2. The drag coefficient for an infinitely long square section is CD = 2 for Re > 104 (Fig. 2.5.1). An extensive study on square and rectangular sections is presented by the ESDU. The drag coefficients for other two-dimensional structural shapes are listed in Table 2.5.1 according to Simiu.
Table 2.5.1 Two-Dimensional Drag Coefficients for Structural Shapes
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The drag coefficient for all objects with sharp corners is independent of the Reynolds number because the separation points are fixed by the geometry of the object. A more detailed list of drag coefficients is presented by Sachs. Data for estimating the mean fluid forces acting on lattice frameworks are presented in ESDU Item Number 75011. According to Turton and Levenspiel, the drag coefficient for a sphere is given by CD = 24(1 + 0.173 Re0.657)/Re + 0.413/(1 + 16300 Re-1.09)
(2.5.4)
for Re ≤ 200000.
2.6 Flow through Screens or Gauzes A screen may be defined as a regular assemblage of elements forming a pervious sheet, which is relatively thin, in the direction of flow through the screen. Screens of various types may be installed in systems to:
•
remove foreign objects from the fluid stream
•
protect equipment (e.g., against hailstones)
•
reduce fouling or clogging in heat exchangers
•
smooth flow
•
produce turbulence
In such cases, the prediction of the total pressure loss caused by the screen is of interest. An expression for the loss coefficient across a plane screen has been deduced by Cornell for a compressible fluid. Since the losses across screens
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employed in air-cooled heat exchangers are normally small, the loss coefficient can be expressed approximately in terms of the pressure difference across the screen and mean, free, stream velocity, i.e.,
Ks = 2(pt1 – pt2)/ρv2 = 2(p1 - p2)/ρv2
(2.6.1)
A round-wire screen or gauze of square mesh shown in Figure 2.6.1 is usually specified by the mesh, defined by the number of openings per unit length, 1/Ps, and by the diameter, ds, of the wires.
Fig. 2.6.1 Geometry of a Square-Woven Screen
An example of standard mesh data is listed in Table 2.6.1.
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Table 2.6.1 Standard Mesh Data
The porosity of the screen is defined as
βs = area of holes/total area = (1 – ds /Ps)2
(2.6.2)
According to Simmons, the following equation holds for a screen placed at right angles to an airstream at velocities above 10 m/s under ambient conditions:
Ks = (1 – βs)/βs2
(2.6.3)
This equation can be recommended for application in the case of most screens of practical interest where screen Reynolds numbers exceed 300.
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The screen Reynolds number is defined as Res = ρvds /(βs µ)
(2.6.4)
According to Wieghardt, the loss coefficient may be expressed as Ks = 6(1 – βs)βs–2 Res–0.333
(2.6.5)
for 60 < Res < 1000. In Figure 2.6.2, Equations 2.6.3 and 2.6.5 are compared with the experimental results obtained by various investigators. Since the screen geometry can have a significant influence on the loss coefficient, specific tests should be performed when these losses are of importance.
Fig. 2.6.2 Screen Loss Coefficient
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For cases where ds/Ps 100 kg/m2s, the Chisholm correlation should be employed.
•
For µ/µg > 1000 and G < 100 kg/m2s, the Martinelli correlation should be used.
The pressure gradient due to acceleration or deceleration effects may be expressed as 2 2 dpm d (1- x ) = G2 + x dz dz (1- α ) ρ α ρg
120
(2.7.16)
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where the term in brackets is the local effective specific volume and x is the vapor mass fraction. The void fraction, α, can be determined using Premoli’s correlation. The static or geodetic pressure gradient is given by
[
]
dps = α ρ g + (1 - α ) ρ g sin ϕ dz
(2.7.17)
where ϕ is the angle of the tube with respect to the horizontal plane with downward flow.
The homogeneous and separated flow models tend to give an inadequate representation of many real two-phase flows, and the detailed physics of the flow, including the flow pattern, are important. By employing flow pattern related frictional pressure drop correlations, it is possible to predict, stepwise, the pressure gradient along the tube according to Olujic. A systematic and practical approach for determining changes in pressure in two-phase flows is presented by Carey. Correlations on counterflow pressure drops are presented by a few researchers such as Feind, Dukler, Bharathan, and Stephan. The pressure drop for nonadiabatic flow is discussed in more detail in chapter 3.
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References Acrivos, A., B. D. Babcock, and R. L. Pigford, “Flow Distribution in Manifolds,” Chemical Engineering Science, 10:112–124, 1959. American Petroleum Institute (API) Standard 661, Air-Cooled Heat Exchangers for General Refinery Services, API, Washington, 1978. Andrews, J., A. Akbarzadeh, and I. Sauciuc, ed., Heat Pipe Technology, 156–163, Pergamon Press, 1997. Bajura, R. A., “A Model for Flow Distribution Manifolds,” Transactions of the American Society of Mechanical Engineers, Journal of Engineering for Power, 70-Pwr-3, 7–14, 1971. Bajura, R. A., and E. H. Jones, “Flow Distribution Manifolds,” Transactions of the American Society of Mechanical Engineers, Journal of Fluid Engineering, 98:656–666, 1976. Bajura, R. A., V. F. Le Rose, and L. E. Williams, “Fluid Distribution in Combining, Dividing and Reverse Flow Manifolds,” American Society of Mechanical Engineers, 73-PWR-1, 1973. Baker, O., “Simultaneous Flow of Oil and Gas,” Oil and Gas Journal, 53:185–195, 1954. Bankoff, S. G., and S. C. Lee, “A Critical Review of the Flooding Literature,” Multiphase Science and Technology, 2:95–180, 1986. Barnea, D., “A Unified Model for Predicting Flow Pattern Transitions for the Whole Range of Pipe Inclinations,” International Journal of Multiphase Flow, 13-1:1–12, 1987. Barnea, D., and N. Brauner, “Holdup of the Liquid Slug in Two-Phase Intermittent Flow,” International Journal of Multiphase Flow, 11-1:43–49, 1985. Baroczy, C. J., “A Systematic Correlation for Two-Phase Pressure Drop,” Chemical Engineering Progress Symposium Series 62, 232–249, 1965. Bassiouny, M. K., and H. Martin, “Flow Distribution and Pressure Drop in Plate Heat Exchangers-I,” Chemical Engineering Science, 39-4:693–700, 1984. Bassiouny, M. K., and H. Martin, “Flow Distribution and Pressure Drop in Plate Heat Exchangers-II,” Chemical Engineering Science, 39-4:701–704, 1984. Bell, K. J., J. Taborek, and F. Fenoglio, “Interpretation of Horizontal Intube Condensation Heat Transfer Correlation with a Two-Phase Flow Regime Map,” Chemical Engineering Progress Symposium Series, 66-102:159–165, 1970. Benedict, R. P., Fundamentals of Pipe Flow, John Wiley, New York, 1980. Bharathan, D. and Wallis, G. B., “Air-Water Countercurrent Annular Flow,” International Journal of Multiphase Flow, 9-4:349–366, 1983. Breber, G., Intube Condensation, Heat Transfer Equipment Design, ed. R. K. Shah, , E. C. Subbarao, and R. A. Mashelkar, Hemisphere Publishing Co., New York, 1988.
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Hartnett, H. P., J. C. Y. Koh, and S. T. McComas, “A Comparison of Predicted and Measured Friction Factors for Turbulent Flow through Rectangular Ducts,” Transactions of the American Society of Mechanical Engineers, Series C, 84-1:82–88, 1962. Hewitt, G. F., Flow Regimes, Handbook of Multiphase Systems, ed. G. Hetsroni, 1982. Hewitt, G. F., and S. Jayanti, “To Churn or Not to Churn,” International Journal of Multiphase Flow, 19-3:527–529, 1993. Hewitt, G. F., and D. N. Roberts, Studies of Two-Phase Flow Patterns by Simultaneous Flash and X-Ray Photography, AERE-M2159, Association of Environmental and Resource Economists, 1969. Hoerner, S. F., Fluid Dynamic Drag, Hoerner, New Jersey, 1965. Holman, J. P., Heat Transfer, McGraw-Hill Book Co., New York, 1986. Holms, E., Handbook of Industrial Pipework Engineering, McGraw-Hill Book Co., U.K., 1973. Hong, S. W., and A. E. Bergles, “Augmentation of Laminar Flow Heat Transfer in Tubes by Means of Twisted-Tape Inserts,” American Society of Mechanical Engineers, 75-HT-44, 1975. Hudson, H. E., R. B. Uhler, and R. W. Bailey, “Dividing-Flow Manifolds with Square Edged Laterals,” Proceedings of American Society of Mechanical Engineers, Journal of Environmental Engineering, vol. 105, 1979. Idelchik, I. E., Handbook of Hydraulic Resistance, Hemisphere Publishing Co., Washington, 1986. James, P. W. et al., “Developments in the Modelling of Horizontal Annular Two-Phase Flow,” International Journal of Multiphase Flow, 13-2:173–198, 1987. Jayanti, S., G. F. Hewitt, D. E. F. Low, and E. Hervieu, “Observation of Flooding in the Taylor Bubble of Co-Current Upwards Slug Flow,” International Journal of Multiphase Flow, 19-3:531–534, 1993. Jorgensen, R., Fan Engineering, Buffalo Forge Co., Buffalo, N.Y., 1961. Kays, W. M., “Loss Coefficients for Abrupt Changes in Flow Cross Section with Low Reynolds Number Flow in Single and Multiple-Tube Systems,” Transactions of the American Society of Mechanical Engineers, 72-8:1067–1074, 1950. Keller, J. D., “The Manifold Problem,” Transactions of the American Society of Mechanical Engineers, 71:77–85, 1949. Kirschmer, O., “Kritische Betrachtungen zur Frage der Rohrreibung,” Zeit, Verein Deutscher Ingenieure, 94:785, 1952. Kubo, T., and T. Ueda, “On the Characteristics of Dividing Flow and Confluent Flow in Headers,” Bulletin of Japan Society of Mechanical Engineers, 12-52:802–809, 1969. Launder, B. E., and W. M. Ying, “Prediction of Flow and Heat Transfer in Ducts of Square Cross Sections,” Proceedings of the Institute of Mechanical Engineers, 187:455–461, 1973.
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Launder, B. E., and W. M. Ying, “Secondary Flow in Ducts of Square Cross Section,” Journal of Fluid Mechanics, 54-2:289–295, 1972 Lin, P. Y., and T. J. Hanratty, “Effect of Pipe Diameter on Flow Patterns for Air-Water Flow in Horizontal Pipes,” International Journal of Multiphase Flow, 13-4:549–563, 1987. Lockhart, R. W., and R. C. Martinelli, “Proposed Correlation of Data for Isothermal Two-Phase Two-Component Flow in Pipes,” Chemical Engineering Progress, 45-1:39–48, 1949. Majumdar, A. K., “Mathematical Modelling of Flow in Dividing and Combining Flow Manifolds,” Applied Mathematical Modelling, 4:424–431, 1980. Mao, Z. S., and A. E. Dukler, “The Myth of Churn Flow,” International Journal of Multiphase Flow, 19-2:377–383, 1993. Markland, E., “Analysis of Flow from Pipe Manifolds,” Engineering, 189:150–151, 1959. Martinelli, R. C., and D. B. Nelson, “Prediction of Pressure Drop During Forced Circulation Boiling of Water,” Transactions of the American Society of Mechanical Engineers, 70:695–702, 1948. McQuillan, K. W., and P. B. Whalley, “A Comparison between Flooding Correlations and Experimental Flooding Data for Gas-Liquid Flow in Vertical Tubes,” Chemical Engineering Science, 40-8:1425–1441, 1985. McQuillan, K. W., and P. B. Whalley, “Flow Patterns in Vertical Two-Phase Flow,” International Journal of Multiphase Flow, 11-2:1425–1440, 1985. Miller, D. S., Internal Flow: A Guide to Losses in Pipe and Duct Systems, British Hydraulic Research Association, Cranfield, 1971. Miller, D. S., Internal Flow Systems, 2d ed., BHRA Fluid Engineering Series, British Hydraulic Research Association, Cranfield, 1990. Moody, L. F., “Friction Factors for Pipe Flow,” Transactions of the American Society of Mechanical Engineers, 66:671, 1944. Mukherjee, H., and J. P. Brill, “Empirical Equations to Predict Flow Patterns in TwoPhase Inclined Flow,” International Journal of Multiphase Flow, 11-3:299–315, 1985. Nikuradse, J., “Gesetzmässigkeiten der turbulenten Strömung in glatten Rohren,” Verein Deutscher Ingenieure Forschungsh, vol. 356, 1932. Nujens, P. G. J. M., A Discrete Model for Flow Distribution Manifolds, CSIR Report, Chemical Engineering Research Group, Pretoria, 1983. Olujic, Z., “A Method for Predicting Friction Pressure Drop of Two-Phase Gas-Liquid Flow in Horizontal Pipes,” American Institute of Chemical Engineers 75th Annual Meeting, Washington, 1983. Oschinowo, T., and M. E. Charles, “Vertical Two-Phase Flow,” Canadian Journal of Chemical Engineering, 52:25–35, 1974. Patterson, G. N., “Modern Diffuser Design,” Aircraft Engineering, 10:267, 1938.
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Petukhov, B. S., Heat Transfer and Friction in Turbulent Pipe Flow with Variable Physical Properties, Advances in Heat Transfer, vol. 6, ed. J. P. Hartnett and T. F. Irvine, Academic Press, New York, 503–564, 1970. Premoli, A., D. Di Francesco, and A. Prina, “An Empirical Correlation for Evaluating Two-Phase Mixture Density Under Adiabatic Conditions,” European Two-Phase Flow Group Meeting, Milan, 1970. Rouse, H., Elementary Mechanics of Fluids, John Wiley & Sons, London, 1946. Runstadler, P. W., F. X. Dolan, and R. C. Dean, Diffuser Data Book, Creare Inc., Hanover, New Hampshire, 1975. Sachs, F., Wind Forces in Engineering, Pergamon, New York, 1972. Scott, D. S., Properties of Co-Current Gas-Liquid Flow, Advances in Chemical Engineering, 4:199–277, Academic Press, New York, 1963. Shah, R. K., Fully Developed Laminar Flow Forced Convection in Channels, Low Reynolds Number Flow Heat Exchangers, ed. S. Kakac, R. K. Shah, and A. E. Bergles, Hemisphere Publishing Corp., Washington, 1983. Shah, R. K., and A. L. London, Laminar Flow Forced Convection in Ducts, Academic Press, New York, 1978. Shames, I. H., Mechanics of Fluids, McGraw-Hill Book Co., New York, 1962. Simiu, E., and R. H. Scanlan, Wind Effects on Structures: An Introduction to Wind Engineering, J. Wiley, Washington, 1977. Simmons, L. F. G., Measurements of the Aerodynamic Forces Acting on Porous Screens, R & M, no. 2276, 1945. Spedding, P. L., and V. T. Nguyen, “Regime Maps for Air-Water Two-Phase Flow,” Chemical Engineering Science, 35:779–793, 1980. Spedding, P. L. et al., “Two-Phase Co-Current Flow in Inclined Pipe,” International Journal of Heat and Mass Transfer, 41:4205–4228, 1998. Stanislav, J. F., S. Kokal, and M. K. Nicholson, “Intermittent Gas-Liquid Flow in Upward Inclined Pipes,” International Journal of Heat and Mass Transfer, 12-3:325–335, 1986. Stephan, M., “Untersuchungen zur gegenstrombegrenzung in vertikalen GasFlüssigkeits-Strömungen,” Doctoral thesis, Thermodynamik Technische Universität, München, August 1990. Stephan, M., and F. Mayinger, “Experimental and Analytical Study of Countercurrent Flow Limitations in Vertical Gas/Liquid Flow,” Chemical Engineering and Technology, 15:51–62, 1992. Taitel, Y., and B. Barnea, “Counter-Current Gas-Liquid Vertical Flow, Model for Flow Pattern and Pressure Drop,” International Journal of Multiphase Flow, 9-6:637– 647, 1983.
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Taitel, Y., and A. E. Dukler, “A Model for Predicting Flow Regime Transition in Horizontal and Near Horizontal Gas-Liquid Flow,” American Institute of Chemical Engineers Journal, 22-1:47–55, 1976. Taitel, Y., D. Barnea, and A. E. Dukler, “Modelling Flow Pattern Transition for Steady Upward Gas-Liquid Flow in Vertical Tubes,” American Institute of Chemical Engineers Journal, 26-3:345–354, 1980. Tien, C. L., and C. P. Liu, Survey on Vertical Two-Phase Counter-Current Flooding, Report NP-984, Electric Power Research Institute, February 1987. Turton, R., and O. Levenspiel, “A Short Note on the Drag Correlation for Spheres,” Power Technology, 47:83–86, 1986. Verein Deutscher Ingenieure-Wärmeatlas, Verein Deutscher Ingenieure, Düsseldorf, 1977. Wallis, G. B., Flooding Velocities for Air and Water in Vertical Tubes, AEEW-R123, 1961. Wallis, G. B., D. C. de Sieyes, R. J. Rosselli, and J. Lacombe, Countercurrent Annular Flow Regimes for Steam and Subcooled Water in a Vertical Tube, Research Project 443-2, EPRI NP-1336, Electric Power Research Institute, California, January 1980. Weisbach, J., Die Experimental-Hydraulik, J. S. Engelhardt, Freiberg, 1855. Whalley, P. B., Boiling, Condensation, and Gas-Liquid Flow, Clarendon Press, Oxford, 1987. Whalley, P. B., and K. W. McQuillan, “Flooding in Two-Phase Flow: The Effect of Tube Length and Artificial Wave Injection,” Physicochemical Hydrodynamics, 6-1:3–21, 1985. Whalley, P. B., and G. F. Hewitt, Multiphase Flow and Pressure Drop, Heat Exchanger Design Handbook, 2:2.3.2–2.3.11, Hemisphere, Washington D.C. 1983, 1980. White, F. M., Viscous Fluid Flow, McGraw-Hill Book Co., Singapore, 1991. Wieghardt, K. E. G., “On the Resistance of Screens,” The Aeronautical Quarterly, 4:186–192, February 1953. Zapke, A., and D. G. Kröger, “Countercurrent Gas-Liquid Flow in Inclined and Vertical Ducts. Part 1: Flow Patterns, Pressure Drop Characteristics and Flooding,” International Journal of Multiphase Flow, 26:1439–1455, 2000. Zapke, A., and D. G. Kröger, “Countercurrent Gas-Liquid Flow in Inclined and Vertical Ducts. Part 2: The Validity of the Froude-Ohnesorge Number Correlation for Flooding,” International Journal of Multiphase Flow, 26:1457–1468, 2000. Zapke, A., and D. G. Kröger, “The Effect of Fluid Properties on Flooding in Vertical and Inclined Rectangular Ducts and Tubes,” Proceedings, American Society of Mechanical Engineers Fluids Engineering Division Summer Meeting, FED239:527–532, San Diego, July 1996. Zapke, A., and D. G. Kröger, “The Influence of Fluid Properties and Inlet Geometry on Flooding in Vertical and Inclined Tubes,” International Journal of Multiphase Flow, 22-3:461–472, 1996.
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Zapke, A., and D. G. Kröger, “Pressure Drop during Gas-Liquid Countercurrent Flow in Inclined Rectangular Ducts,” Proceedings, 5th International Heat Pipe Symposium, Melbourne, November 1996. Zipfel, T., and D. G. Kröger, “A Design Method for Single-Finned-Tube-Row Air-Cooled Steam Condenser to Avoid Back Flow of Steam into Any Finned Tube,” South African Institution of Mechanical Engineering R + D Journal, 13:68–75, 1997. Zipfel, T., and D. G. Kröger, “Flow Loss Coefficients at Air-Cooled Condenser Finned Tube Inlets,” South African Institution of Mechanical Engineering R + D Journal, 13:62–67, 1997.
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3 Heat Transfer
3.0 Introduction Engineering thermal science includes thermodynamics and heat transfer. Thermodynamic analyses consider only systems in equilibrium. The role of heat transfer is to supplement thermodynamics with additional laws that allow the prediction of rates of energy transfer. These laws are based on different modes of heat transfer, namely conduction, radiation, and heat transfer by convection. In this chapter, many of the heat transfer problems encountered in aircooled heat exchangers and cooling towers are addressed. Specific solutions are presented for application in later chapters.
3.1 Modes of Heat Transfer The mechanism by which heat is transferred in physical equipment is quite complex; however, there appear to be two rather basic and distinct types of heat transfer processes—conduction and radiation. Conduction is the transfer by molecular motion of heat between one part of a body to another part of the same body or by one body and another in physical contact with it. In fluids, heat is conducted by nearly elastic collisions of the molecules or by an energy diffusion process. Theory of heat
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conduction in solids distinguishes between conductors and nonconductors or dielectrics of electricity. In a dielectric, heat is conducted by lattice waves caused by atomic motion; in electrical conductors, free electrons behaving almost like gas molecules contribute additionally to heat conduction. Radiation, or more precisely thermal radiation, is a phenomenon identical to the emission of light and is significant across the entire range of wave lengths from zero to infinity. Frequently, the processes of conduction and radiation occur simultaneously, even within solid bodies. However, in many engineering problems, the heat transferred by one of the modes is negligible compared with the other and can be assumed with good approximation to involve only one of the processes. Heat transferred between a flowing fluid and its bounding surface is often referred to as heat transfer by convection. In actual fact, the word convection applies to the fluid motion while the mechanisms of heat transfer anywhere in the fluid are only conduction and radiation.
Conduction When a temperature gradient exists within a homogeneous substance, there is an energy transfer from the high temperature region to the low temperature region. Heat is transferred by conduction, and the heat transfer rate per unit area is proportional to the normal temperature gradient, i.e.,
q =
Q dT ∝ A dx
where the heat flux, q, is the ratio of the heat transfer rate, Q, through the area A, and dT/dx is the temperature gradient in the direction of heat flow. When the proportionality coefficient is inserted,
Q = - kA
132
dT dx
(3.1.1)
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The positive coefficient, k, is called the thermal conductivity of the material, and the minus sign is inserted so the second principle of thermodynamics will be satisfied, i.e., heat must flow downhill on the temperature scale. Equation 3.1.1 is called Fourier’s law of heat conduction. Thermal conductivities of various fluids are given in appendix A. Consider the problem of one-dimensional steady-state conduction in a plane wall of homogeneous material having constant thermal conductivity with each face held at a constant uniform temperature (Fig. 3.1.1a). For the element of thickness, dx, the following energy balance is applicable:
Qx = Qx + dx Using Equation 3.1.1, this can be written as
-kA
or
dT dT dT d = -kA + - k A dx dx dx dx dx
(3.1.2)
d dT - k A = 0 dx dx
Fig. 3.1.1 One-Dimensional Heat Conduction (a) (b)
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With T = T1 at x = x1 and T = T2 at x = x2, the integration of this relation, assuming a constant value for k, yields the following temperature distribution:
T =
(T2 - T1) (x - x1) (x2 - x1)
+ T1
(3.1.3)
The heat transfer rate may now be determined using Equation 3.1.1, i.e.,
Q =- kA
dT k A (T2 - T1) =dx (x 2 - x1)
(3.1.4)
This equation can also be written as
Q =
T1 - T2 = ( x2 - x1) / (kA)
thermal potential difference thermal resistance
(3.1.5)
These principles are readily extended to the case of a composite plane wall (Fig. 3.1.1b). The steady state heat transfer rate entering the left face is the same as that leaving the right face. Thus,
Q=
T1 - T2 T2 - T3 and Q = (x 3 - x 2 )/(kb A ) x 1)/(ka A )
(x 2 -
(3.1.6)
Eliminate T2 in these relations, and find
Q =
(T1 - T3)
(3.1.7)
(x2 - x1) /(k a A) + (x3 - x 2) /(k b A)
Equations 3.1.5 and 3.1.7 illustrate the analogy between conductive heat transfer and electrical current flow, an analogy rooted in the similarity between Fourier’s and Ohm’s laws. It is convenient to express Fourier’s law as
conductive heat transfer =
134
overall temperature difference summation of thermal resistance
=
∆ T overall ∑ R th
(3.1.8)
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where Rth = the thermal resistances of the various materials
In the case of cylindrical coordinates, Fourier’s law in the radial direction is written as
Q = -k A
dT dr
(3.1.9)
Consider a long cylinder of inside radius ri and outside radius ro and length L. If the inside surface of the cylinder is maintained at a temperature Ti and the outside surface at To (Fig. 3.1.2a), it may be assumed the heat flows in the radial direction.
Fig. 3.1.2 Radial Heat Conduction (a) (b)
For the elementary cylindrical control volume of thickness dr, the following energy balance is applicable if steady state conditions are assumed:
Qr = Qr
+ dr
(3.1.10)
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With Equation 3.1.9, this becomes
- 2π rLk
dT dT d dT = - 2 π rLk + - 2 π rLk dr dr dr dr dr
(3.1.11)
or
d dT - 2 π rLk =0 dr dr With T = Ti at r = ri and T = To at r = ro, the integration of this relation, for a constant value of k, yields the following temperature distribution
T = Ti + (To – Ti) n (r/ri)/n (ro/ri)
(3.1.12)
The temperature distribution in the cylinder is a logarithmic function of the radius. From this relationship and Equation 3.1.9, the radial heat transfer rate is given by
Q = 2πLk(Ti – To)/n (ro/ri)
(3.1.13)
For the two-layered cylinder shown in Figure 3.1.2(b), find the heat transfer rate
Q =
136
2 π L (Ti - To) n (r1 / r i) / k a + n (r o / r1) / k b
(3.1.14)
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Convection Convection is a process involving the mass movement of fluids. When a temperature difference produces a density difference resulting in the mass movement, the process is called free or natural convection. When a pump or other similar device causes the mass motion to take place, the process is called forced convection. The heat transferred between a fluid and solid surface when the fluid flows either by free or by forced convection is usually referred to as heat transfer by convection. In actual fact, the fundamental mechanisms of heat transfer in such flows are those of conduction and radiation according to Rohsenow. Consider the heated plate shown in Figure 3.1.3. The temperature of the plate is Tp, and the temperature of the fluid is T∞. Just as the hydrodynamic boundary layer was defined as the region of the flow where viscous forces are felt, a temperature or thermal boundary layer may be defined as the region where temperature gradients are present in the flow. Analogous to the hydrodynamic case, the thermal boundary layer thickness, δT, is defined as that distance from the plate where Tp – T = 0.99(Tp – T∞). The thermal boundary layer is not necessarily the same thickness as the velocity boundary layer.
Fig. 3.1.3 Convection Heat Transfer from a Plate
Since the velocity of the flow at the plate is zero as a result of viscous action, heat is being transferred only by conduction at that point. However, the temperature gradient is dependent on the rate at which the fluid carries
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the heat away. To express the overall effect of convection, a quantity, h, called the convection heat transfer coefficient is defined by the expression
Q = h A (Tp – T∞)
(3.1.15a)
Q = h A (Tw – Tm)
(3.1.15b)
In the case of duct flow,
where Tw = wall temperature Tm = bulk mean fluid temperature This equation is known as Newton’s law of cooling. Flow in a two-dimensional duct is designated as fully developed thermally when the dimensionless temperature distribution, as expressed in brackets in the following, is invariant at a cross section, i.e., independent of x.
d Twm - T = 0 dx Twm - Tm
(3.1.16)
where Twm = peripheral mean wall temperature Tm
= bulk mean fluid temperature
The ratio of the convective conductance or heat transfer coefficient, h, to the molecular thermal conductance, k/L, for flow over a surface, or k/de for duct flow, is defined as a Nusselt number, i.e.,
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Nu =
hL for a plate k
(3.1.17a)
Nu =
hde for a duct k
(3.1.17b)
or
The Nusselt number is constant for fully developed thermal and hydrodynamic laminar flow. It is dependent on x/(deRePr) for developing laminar temperature profiles and for developing laminar velocity as well as temperature profiles. The ratio of momentum diffusivity to thermal diffusivity of the fluid is known as the Prandtl number, i.e.,
Pr = v/α = µcp/k
(3.1.18)
The product of the Reynolds and Prandtl numbers is known as the Péclet number, i.e.,
Pe = RePr
(3.1.19)
The ratio of Nusselt number to Péclet number is referred to as the Stanton number, i.e.,
St =
Nu h = RePr Gcp
(3.1.20)
Overall heat transfer coefficient It is often convenient to express the heat transfer rate for a conductiveconvective problem in terms of an overall heat transfer coefficient U.
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Consider the plane wall or plate having a surface area, A, which is exposed to a hot fluid, h, on one side and a cold fluid, c, on the other side (Fig. 3.1.4a). The heat transfer rate through the plate is given by Q = hh A (Th - T1) = kA (T1 - T2) /L = h c A (T2 - Tc )
(3.1.21)
or, written in terms of the thermal resistances,
Q =
(T - T ) (T2 - Tc ) Th - T1 = 1 2 = 1/( hh A) L/(kA) 1/(hc A)
(3.1.22)
Fig. 3.1.4 Conductive-Convective Geometries (a) (b)
Upon eliminating the plate surface temperatures T1 and T2 in this relation, find
Q =
Th - Tc = UA(Th - Tc ) 1/(hh A) + L/(kA) + 1/(hc A)
(3.1.23)
This equation defines the overall heat transfer coefficient
U =
140
1 1/ hh + L/k + 1/ hc
(3.1.24)
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In the case of a long cylinder or tube, the radial heat flux through the wall exposed to two fluids at different temperatures (Fig. 3.1.4b) is given by
Q =
Ti - To 1/(hi A i) + n(ro / ri) /(2π kL) + 1/(ho A o)
(3.1.25)
where L = length of the cylinder i
= the inside of the cylinder
o
= the outside of the cylinder
The overall heat transfer coefficient may be based on either the inside or the outside area of the cylinder, i.e.,
Ui =
1 1/ h i + A i n(ro / ri) /(2π kL) + A i /( A o ho )
(3.1.26)
1 A o /( A i hi) + A o n( r o / r i ) /(2π kL) + 1/( ho)
(3.1.27)
or
Uo =
In most practical heat exchangers, fouling of one form or another will occur according to Somerscales and Garrett-Price. Fouling tends to introduce an additional thermal resistance and reduce the performance of the heat exchanger. If the fouling resistance is specified, it can be incorporated in the evaluation of the overall heat transfer coefficient. Notwithstanding the great amount of research that has been done on fouling, there is very little that can be translated into improved practical design applications according to Taborek and Palen. In view of this problem, the initial heat exchanger design should assume clean operating conditions, unless detailed information on fouling is specified or is available. Subsequent further analyses may be done to determine the implications of various degrees of fouling.
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3.2 Heat Transfer in Ducts The heat transfer during the flow of a particular fluid in ducts is determined by the nature of the flow, i.e., laminar, transitional, or turbulent. Furthermore, the flow may be fully developed hydrodynamically and thermally, or it may develop in one form or another. In the case of laminar flow, the duct geometry and the thermal boundary conditions have a significant effect on the heat transfer rate. Numerous analytical and numerical solutions have been found for the heat transfer coefficient during laminar flow in various duct geometries. In the region of transitional and turbulent flow, correlations based on experimental measurements are employed to predict heat transfer rates.
Laminar flow The heat transfer coefficient during laminar flow in a duct, Re ≤ 2300, is dependent on a large number of parameters. The complexity of the problem is well illustrated by the extensive summary of results and correlations presented by Shah and London. Different temperature and/or heat flux conditions may occur at the inside wall of the duct. Of these, the most commonly encountered are the conditions of constant axial and peripheral temperature. This is approximated in certain condensers, evaporators, and liquid-to-gas heat exchangers with high liquid flows as well as cases of constant heat flux. A few useful equations will be presented here. According to Hausen, the mean Nusselt number for hydrodynamic fully developed flow in a round tube at a constant wall temperature is
NuT = 3.66 +
0.0668 Re Pr d/L 1 + 0.04 (Re Pr d/L) 0.667
(3.2.1)
Schlünder proposes the following correlation for these conditions: 0.333
N uT = (3.663 + 4.2Re Pr d/L)
142
(3.2.2)
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A more recent relationship was proposed by Gnielinski:
[
}]
{
3 0.333
3 3 Nu T = 3.66 + 0.7 + 1.615(Re Pr d/L)0.333- 0.7
(3.2.3)
When the inlet velocity distribution to the tube is uniform, Kays recommends the following equation:
NuT = 3.66 +
0.104 Re Pr d/L 0.8 1+ 0.016 (Re Pr d/L)
(3.2.4)
or, according to Churchill and Ozoe,
Nu T =
0.5
1.2732
π Re Pr d 0.25 4L
(3.2.5)
[1+ (Pr/0.0468) ] 0.667
Gnielinski proposes:
[
{
3
}
NuT = 3.66 3+ 0.7 3+ 1.615(Re Pr d/L )0.333- 0.7
(3.2.6)
{
} ]
+ 2 (Re Pr d/L)3/ (1 + 22 Pr )
0.5 0.333
Under certain operating conditions, the heat flux between the tube wall and the fluid may be constant. When this is the case and the inlet velocity distribution is fully developed, Shah finds the mean Nusselt number to be
Nuq = 4.364 + 0.0722 Re Pr d /L for L/(Re Pr d) > 0.03
(3.2.7)
= 1.953 (Re Pr d/L)0.333 for L/(Re Pr d) ≤ 0.03
(3.2.8)
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The duct geometry has a significant influence on the heat transfer coefficient and the friction factor. This is illustrated in Table 3.2.1, which lists Nusselt numbers and friction solutions for fully developed flow for several ducts at constant wall temperature. It also includes the case where heat flux is constant in the axial direction while temperature is constant peripherally according to Shah.
Table 3.2.1 Solutions for Heat Transfer and Friction for Fully Developed Flow in Ducts
For a fully developed velocity profile entering between two parallel plates at a constant temperature, Shah finds the following equations are applicable:
NuT = 7.541 + 0.0235 Re Pr de /L for L/(Re Pr de) > 0.006
(3.2.9)
= 1.849(Re Pr de /L)0.333 + 0.6 for 0.0005 < L/(Re Pr de) ≤ 0.006
(3.2.10)
In the case of uniform heat flux,
144
Nuq = 8.235 + 0.0364 Re Pr de /L for L/(Re Pr de) ≥ 0.01
(3.2.11)
= 2.236 (Re Pr de /L)0.333 + 0.9 for 0.001 < L/(Re Pr de) ≤ 0.01
(3.2.12)
= 2.236 (Re Pr de /L)0.333 for L/(Re Pr de) ≤ 0.001
(3.2.13)
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When the inlet velocity between the parallel plates is uniform and the wall temperature is constant, the Nusselt number is given by Stephan as
NuT = 7.55 +
0.024 (Re Pr d e /L )1.14 1+ 0.0358 (Re Pr d e /L )
0.64
Pr
0.17
(3.2.14)
The Nusselt numbers for fully developed flow in rectangular ducts are shown in Figure 3.2.1. Curves through these points are given by the following equations from Shah: NuT = 7.541 [1 – 2.610 b/a + 4.970 (b/a)2 –5.119 (b/a)3 (3.2.15) + 2.702
(b/a)4–
0.548
(b/a)5]
for a constant wall temperature. For a uniform heat flux,
Nuq = 8.235 [1 – 2.0421 b/a + 3.0853 (b/a)2 – 2.4765 (b/a)3 (3.2.16) + 1.0578
(b/a)4
– 0.1861
(b/a)5]
Fig. 3.2.1 Heat Transfer for Fully Developed Flow through Rectangular Ducts
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With these equations, it is possible to determine the limiting values for any rectangular duct. When L/d is less than 0.0048Re in tubes and when L/de is less than 0.0021Re in ducts of a rectangular cross section for both constant wall temperature conditions and uniform heat flux conditions,
Nu =
Re Pr d e 1 n 0.5 0.167 4L (Re Pr de /L ) } 1 - 2.654 /{Pr
(3.2.17)
Kreith has found this equation to be useful for ordinary liquids and gases. Jamil, Cheng, and Jamil and Zarling analyzed the problem of fully developed flow in noncircular ducts shown in Table 3.2.2 for the case of uniform heat flux. These results are compared to those of a rectangular duct in Figure 3.2.2.
Table 3.2.2 Solutions for Heat Transfer and Friction for Fully Developed Flow in Ducts
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Fig. 3.2.2 Heat Transfer for Fully Developed Flow in Ducts
Various investigators according to Shah have analyzed the problem of fully developed flow in elliptical ducts at constant wall temperature as well as for the case of a uniform heat flux. The results are listed in Table 3.2.3. Abdel-Wahed et al. reported the results of experimental studies.
Table 3.2.3 Solutions for Heat Transfer and Friction for Fully Developed Flow in Elliptical Ducts
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Hydrodynamic- and thermal-developed flows at the inlets to heat exchanger ducts or across interrupted surfaces generally lead to higher heat transfer coefficients compared to fully developed flow. Solutions for many of these problems appear in the literature according to Kakac. Free-convection effects may become important at very low Reynolds numbers, at high temperature differences, or if the passage geometry has a large hydraulic diameter. For horizontal tubes, free convection sets up secondary flows at a cross section and aids the convection process. Hence, the heat transfer coefficient for the combined convection is higher than for the pure forced convection. Metais and Eckert have classified free, mixed, and forced convection regimes (Fig. 3.2.3a) for horizontal tubes with the axially constant wall temperature boundary condition. The limits of the forced- and mixed-convection regimes are defined so that free convection effects contribute only about 10% to the heat flux; therefore, Figure 3.2.3(a) may be used as a guide to determine whether or not free convection is important.
Fig. 3.2.3 Free, Forced, and Mixed Convection (10-2 < Pr d/L < 1) (a) Horizontal Tube (b) Vertical Tube
Among others, Brown and Thomas and Oliver present heat transfer equations for laminar combined free and forced convection inside horizontal tubes having a constant wall temperature. According to Oliver,
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NuT = 1.75 [Gz + 0.0083 (GrPr)0.75]0.333 (µ/µw)0.14
(3.2.18)
where the Graetz number is defined as Gz = Re Pr d/L the Grashof number is Gr = gρ2d3β (Tw - T)/µ2
All of the fluid properties are evaluated at the fluid bulk mean temperature except µw, which is evaluated at the wall temperature. The effect of superimposed free convection for vertical tubes, unlike horizontal tubes, is dependent upon the flow direction and on whether or not the fluid is heated or cooled. The flow regime chart of Metais and Eckert for vertical tubes (Fig. 3.2.3b) provides guidelines to determine the significance of the superimposed free convection. Laminar flow generally results in relatively low heat transfer coefficients. Variable fluid property effects, which are predominantly due to viscosity and density variation, tend to increase heat transfer coefficients. The need for more effective heat transfer systems has stimulated interest in techniques to augment or enhance heat transfer. The various techniques to augment heat transfer inside tubes are generally classified as passive or active. With passive techniques, no external energy other than pump work is required to produce the augmentation. These techniques are further classified as
•
surface roughness
•
internal extended surface
•
displaced promoters
•
swirl flow
•
additives
•
compound techniques—when more than one of these techniques are employed
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With active techniques, external energy is required to produce the augmentation. These techniques include:
•
mechanical aids
•
heated surface vibration
•
fluid pulsation
•
electrostatic fields
•
suction or injection
Surveys and evaluations of augmentation techniques appear in the literature from Bergles. Hong and Bergles present an equation for fully developed laminar flow in a circular uniformly heated tube with a twisted tape insert shown in Figure 2.2.2.
Nu = 5.172[1 + 0.00548 Pr0.7 (Re d/P)1.25]0.5
(3.2.19)
where Nu = hd/k Re = Gd/µ G = m/(πd2/4 - ttd)
Other correlations are presented by Manglik and Bergles and Agarwal and Rao. A more complex correlation for thermally developing flow was presented by Du Plessis and Kröger. Considerable test data are available for internally finned tubes according to both Shah and Carnavos. The following correlations of Watkinson et al. are applicable in the case of laminar flow. Spirally finned tubes are
Nu = 19.2 Re0.26 (Pr de /L)0.333 (Lf /Pf)0.5 / ϕ
150
(3.2.20)
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where ϕ = 2.25 (1 + 0.01 Gr0.333)/log10 Re
(3.2.21)
and Lf is the average distance between fins. In this equation, the Grashof number is defined as
Gr = gρ2 d3 β (Tw – T) / µ2 Straight-finned tubes are
Nu = 2.43 Re0.46 (Pr de/L)0.333 (1/nf)0.5/ϕ
(3.2.22)
These correlations are based on data for oil in horizontal tubes having an approximately uniform temperature. Other data from Marner for both water and ethylene glycol in both steam- and electric-heated tubes are in approximate agreement with these correlations. A configuration consisting of a combination of internal fins and spiral tapes may be considered to improve the effective heat transfer coefficient when cooling viscous media according to both Van Rooyen and Jesch. In some of the previous equations, it is assumed the fluid properties are constant and they are evaluated at the bulk mean temperature. According to Shah, no corrections for temperature-dependent property effects are required when the previous heat transfer equations are applied to gases and if temperature differences are not excessive. For liquids, where only the viscosity is strongly temperature dependent, the following relationship is adequate for both heating and cooling:
µ Nu = Nucp µw
0.14
(3.2.23)
Bergles presents surveys of analytical solutions and experimental studies evaluating the influence variable transport properties may have on the heat transfer rate in laminar duct flows.
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Example 3.2.1 Oil flows at a rate of mo = 0.11094 kg/s in a 1 m long horizontal tube having an inside diameter of di = 13.5 mm. The flow entering the tube is hydrodynamically fully developed. The tube wall is at a constant temperature of Tw = 99.105 °C. The mean temperature of the oil in the tube is To = 44.1958 °C. Determine the Nusselt number applicable to this flow. The thermophysical properties of the oil are: Volume coefficient of expansion
βo = 7.47 x 10-4 K-1
Density
ρo = 880.5 - 0.613T kg/m3
Thermal conductivity
ko = 0.13428 - 7.185 x 10-5 T W/mK
Specific heat
cpo = 3.642 T + 1809 J/kg K
Dynamic viscosity in the range: 5 °C < T < 50 °C, µo = exp[-0.3948 (n T)2 + 1.2709 (n T) - 2.8523] kg/ms 50 °C ≤ T ≤ 100 °C, lo = 38.876/T1.9325 kg/ms where
T is in degrees Centigrade
Solution Evaluate the thermophysical properties of the oil at To = 44.1958 °C, i.e.,
ρo = 880.5 - 0.613 x 44.1958 = 853.408 kg/m3 ko = 0.13428 - 7.185 x 10-5 x 44.1958 = 0.1311 W/mK cpo = 3.642 x 44.1958 + 1809 = 1969.961 J/kg K
µo = exp [-0.3948 (n 44.1958)2 + 1.2709 (n 44.1958) - 2.8523] = 0.02462 kg/sm At a wall temperature of Tw = 99.105 °C, find the dynamic viscosity
µow = 38.876/(99.105)1.9325 = 0.0053979 kg/ms
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For the given oil mass flow rate, find the Reynolds number
Re o =
4 mo 4 x 0.11094 = = 424.99 π di µ 6 π x 13.5 x 10 -3 x 0.02462
i.e., the flow is laminar. The Prandtl number of the oil at its mean temperature is
Pro = µo cpo / k o = 0.02462 x 1969.961 / 0.1311 = 369.9499 To evaluate the influence free convection effects may have on the Nusselt number, determine the Grashof number for the flow inside the tube. 2
3
2
Gro = g ρo d i βo ( T w - T o ) / µo 2
3
-4
2
= 9.8 x 853. 408 x 0. 0135 x 7.47 x 10 (99.105 - 44.1958) / 0. 02462 = 1188.3 Thus GroProdi/L = 1188.3 x 369.9499 x 0.0135 = 5934.8 In Figure 3.2.3(a), flow conditions are close to the dividing line between forced convection laminar flow and mixed convection laminar flow for Reo = 424.99 and GroProdi/L = 5934.8. Equations ignoring free convection effects and those taking into consideration free convection are applicable. From Equation 3.2.1 and Equation 3.2.23, which correct for temperaturedependent properties,
-3 0.02462 0.0668 x 424.99 x 369.9499 x 13.5 x 10 NuT = 3.66 + -3 0.667 1 + 0.04 (424.99 x 369.9499 x 13.5 x 10 ) 0.0053979
0.14
= 27.528
Using Equations 3.2.2 and 3.2.23, find NuT = (3.663 + 4.2 x 424.99 x 369.9499 x 13.5 x 10-3)0.333 (0.02462/0.0053979)0.14 = 25.69
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From Equations 3.2.3 and 3.2.23,
[
{
} 3]
NuT = 3. 66 3+ 0. 7 3+ 1.615 (424.99 x 369.9499 x 0.0135) 0.333- 0.7
0.333
x (0.02462 / 0.0053979 )0.14 = 24.7135 Equation 3.2.18 can also be applied in this case. NuT = 1.75 [424.99 x 369.9499 x 13.5/1000 + 0.0083 (1188.3 x 369.9499)0.75]0.333 (0.02462/0.0053979)0.14 = 28.346 This latter value is preferred since it makes provision for free convection effects, which tend to increase the effective heat transfer coefficient.
Note A series of heat transfer tests were conducted with this particular oil flowing through the 1 m tube at different rates. The tube wall temperature was maintained at approximately 100 °C by condensing steam on its outside surface. The results of the tests are shown in Figure 3.2.4 and compared to Equations 3.2.1 and 3.2.18. It is noted in this simplified presentation (Gro and Pro are essentially constant) Equation 3.2.1 correlates the data well at higher Reynolds numbers. At the lower values, free convection effects clearly become more significant, and the data in this region is better correlated by Equation 3.2.18.
Fig. 3.2.4 Nusselt Number during Laminar Flow
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Turbulent flow In 1930, Dittus and Boelter summarized available heat transfer data for turbulent flow in tubes and plotted them in a single graph. Their work was critically evaluated by Hsu and Winterton. The various experiments described in thirteen papers were done with fluids, which covered a wide range of viscosities. The fluids were heated or cooled. The resultant equations, which have since found wide application in the design of heat transfer equipment, are
Nu = 0.0243 Re0.8 Pr0.4
(3.2.24)
Nu = 0.0265 Re0.8 Pr0.3
(3.2.25)
for heating, and
for cooling. The thermophysical properties are evaluated at the arithmetic mean bulk temperature. An improved equation considering flow development was proposed by Hausen in 1959 and modified by him in 1974:
d 0.667 N u = 0.0235 (Re 0.8 - 230) (1.8 Pr 0.3- 0.8) 1 + L
(3.2.26)
For fully developed turbulent flow, Petukhov developed an equation that was subsequently modified:
Nu =
( f D / 8) Re Pr 1.07+12.7( f D / 8)
0.5
(Pr 0.667- 1)
(3.2.27)
All properties are evaluated at the bulk temperature of the fluid. For smooth tubes, the recommended friction factor is, using Equation 2.2.10,
fD = (1.82 log10 Re – 1.64)–2
(3.2.28)
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Equation 3.2.27 is applicable in the following ranges:
•
0.5 < Pr < 200 for 6% accuracy
•
0.5 < Pr < 2000 for 10% accuracy
•
104 < Re < 5 x 106
•
0 < µ/µw < 40
On the basis of the earlier work of Hausen and Petukhov as well as on most of the available experimental data, Gnielinski proposes the following equation:
Nu =
( f D / 8) (Re -1000)Pr [1 + (d/L)0.67 ] 0.5
1 +12.7( f D / 8) (Pr
0.67
(3.2.29)
- 1)
Equation 3.2.29 is valid for the ranges, 2300 < Re < 106, 0.5 < Pr < 104, and 0 < d /L < 1. Care should be taken when applying this equation at low Reynolds numbers and to very short tubes. All thermophysical properties are evaluated at the bulk mean temperature of the fluid. When the temperature difference between the fluid and the duct or tube wall is large, variations in the properties of the fluid may be taken into consideration by multiplying the right side of the previous heat transfer equations by a correction factor. Sieder and Tate proposed a viscosity ratio correction factor (µ/µw)0.14 for heating and cooling of liquids. Büyükalaca and Jackson found the exponent of the ratio to be a function of the Reynolds number, i.e., (0.048 + 2.6 x 10-6Re) for Re > 20,000 when heating water in a tube. Petukhov found the correction factor to be (Pr/Prw)0.11 when heating a liquid and (Pr/Pw)0.25 when cooling a liquid. For gases in the range of 0.1 < (Pr/Prw) < 10, multiply by (T/Tw)0.45 when heating. No correction is required when cooling a gas. The temperatures T and Tw are in degrees Kelvin. Cain et al. investigated the heat transfer characteristics of flow in elliptical ducts and found existing equations to be in good agreement with experimental results for Re > 20,000.
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Transitional flow The transitional region between laminar and turbulent flow has not been extensively examined, and methods for determining the extent of this region and obtaining heat transfer coefficients may be unreliable. For round tubes, Hausen proposed the following equation:
Nu = 0.116 (Re0.667 – 125) Pr0.333 [1 + (d/L)0.667]
(3.2.30)
for 2100 < Re < 10,000 Barrow and Roberts recommend a relation for elliptical ducts in the range of 4000 < Re < 20,000, i.e.,
Nu = 0.00165 Re1.06 Pr0.4
(3.2.31)
Metais and Eckert attempt to establish certain criteria that define the limits of different transfer regimes in transitional flow shown in the modified plot in Figure 3.2.5.
Fig. 3.2.5 Modified Metais and Eckert Plot
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The heat transfer coefficients for the cooling of oil at a constant tube wall temperature were determined by Rogers for the ranges 550 < Re < 2800 and 132 < Pr < 642. With the transfer regions divided as given in Figure 3.2.5, correlating equations have been obtained for each regime.
Mixed turbulence: Nu = 0.0184(Re0.8 – 213.9)(1.8 Pr0.333 – 0.8)[1 + (d/L)0.67](µ/µw)0.14
(3.2.32)
Upper transitional: Nu = 0.0277(Re0.8 – 272.5)(1.8 Pr0.333 – 0.8)[1 + (d/L)0.67](µ/µw)0.14
(3.2.33)
Middle transitional: Nu = 0.0176(Re0.8 – 106.1)(1.8 Pr0.333 – 0.8)[1 + (d/L)0.67](µ/µw)0.14
(3.2.34)
Mixed laminar: Nu = 0.002 Re1.42 Pr0.333 (µ/µw)0.14
(3.2.35)
No suitable equation has been found to correlate the data in the lower transitional regime. A single correlating equation that covers the entire range from the laminar through the transitional to the turbulent regime in smooth tubes is proposed by Churchill. The behavior for all Reynolds and Prandtl numbers for developed or developing conditions is represented by
2 exp {(2200 -Re) / 365} 1 Nu 10 = Nu10 + + 0.5 Nuc2 Nu + 0.02793Re Pr f D o 0.8 0.833 (1+ Pr )
158
-5
(3.2.36)
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where Nu = Nusselt number for laminar flow Nuc = Nusselt number at Re = 2100 Nuo = the value the Nusselt number approaches as Re Pr approaches zero
The latter Nusselt number is approximately 4.8 for flow at a constant wall temperature and 6.3 for a uniform heat flux. For flow in a tube at a constant wall temperature, Nu may be approximated by
Re Pr d/L 2.67 NuT = 3.657 1 + 7.60
0.125
(3.2.37)
and, at a uniform heat flux, by
2
Re Pr d/L Nuq = 4.364 1 + 7.3
0.167
(3.2.38)
The value for Nuc is obtained by substituting Re = 2100 into Equation 3.2.37 or Equation 3.2.38. For a smooth tube, the friction factor, fD, can be determined using Equation 2.2.21. Taborek notes that data in the transition region can be plotted approximately as a straight line on a linear scale between the limits 2000 ≤ Re ≤ 8000. Using a value of the Nusselt number for laminar and turbulent flow, Nu and Nut, at these boundary Reynolds numbers, the linear proration results in
Nu = (1.333 - Re / 6000)Nu + (Re / 6000 - 0.333)Nut
(3.2.39)
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Example 3.2.2 In Example 2.3.1, if heat is removed from the water at a low uniform rate (constant heat flux), determine the Nusselt number for the case where the tube wall is smooth. Compare the values obtained using Equations 3.2.25, 3.2.26, 3.2.27, 3.2.29, and 3.2.36.
Solution Since Rew = 44349.9 > 2300, the flow is turbulent. In addition to the thermophysical properties of water at 52.5 °C (325.65 K) evaluated in Example 2.3.1, find the specific heat of water from Equation A.4.2 cpw = 8.15599 x 103 – 2.80627 x 10 x 325.65 + 5.11283 x 10–2 x 325.652 – 2.17582 x 10–13 x 325.656 = 4179.926 J/kgK and the thermal conductivity of water from Equation A.4.4 kw = –6.14255 x 10–1 + 6.9962 x 10–3 x 325.65 –1.01075 x 10–5 x 325.652 + 4.74737 x 10–12 x 325.654 = 0.64556 W/mK The Prandtl number is Prw = lw cpw/kw = 5.21804 x 10–4 x 4179.926/0.64556 = 3.379 For cooling, the Dittus and Boelter Equation 3.2.25 gives, Nu = 0.0265 x 44349.90.8 x 3.3790.3 = 199.25 The Hausen Equation 3.2.26 for small differences in wall and fluid temperature gives Nu = 0.0235 (44349.90.8 – 230)(1.8 x 3.3790.3–0.8) [1 + (22.09/15024)0.667] = 212.96
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Using the Petukhov Equation 3.2.27,
Nu =
0.021516 x 44349.9 x 3.379 / 8 1.07 + 12.7(0.021516 / 8 )0.5(3. 379
0.667
- 1)
= 212.68
Gnielinski’s Equation 3.2.29 gives
Nu =
[
] = 217.94
(0.021516 / 8)(44349.9 - 1000)3.379 1+ (22.09 / 15024)0.67 0.5
1 + 12.7(0.021516 / 8 ) (3. 379
0.67
- 1)
Applying Churchill’s equation, it follows from Equation 3.2.38 that
44349.9 x 3.379 x 22.09 2 Nuq = 4.364 1 + 7.3 x 15024
0.167
= 13.621
Furthermore,
2100 x 3.379 x 22.09 2 Nuc = 4.364 1 + 7.3 x 15024
0.167
= 5.255
The friction factor follows from Equation 2.2.21, i.e., -0.2 1 44349.9 10 fD = 8 + 2.21 n = 0.02137 7 8 10 44349.9 20 0.5 + 36500 44349.9
Substitute these values into Equation 3.2.36, and find
exp {(2200 - 44349.9) / 365)} Nu10 = 13. 62110 + 2 5.255 2 1 + 0.5 0.02793 x 44349.9 x 3.379 x 0. 02137 6.3 + 0.833 (1 + 3. 3790.8 )
-5
or Nu = 214.59
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3.3 Extended Surfaces In the design and construction of various types of heat transfer equipment, shapes such as cylinders, bars, and plates are used to implement the flow of heat between source and sink. They provide heat-absorbing or heat-rejecting surfaces, and each is known as a prime or base surface. When the prime surface is extended by appendages intimately connected with it, the additional surface is known as an extended surface. The elements used to extend prime surfaces are referred to as fins.
Fin efficiency or effectiveness Consider a one-dimensional plate fin of uniform thickness, tf, exposed to the surrounding fluid at a temperature, T∞ (Fig. 3.3.1). The temperature at the root of the fin is Tr.
Fig. 3.3.1 Schematic Drawing of a Longitudinal Fin of Rectangular Profile
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Neglecting radiation effects, apply an energy balance to an elementary length of the fin, and find the difference between the heat conducted into the control volume and that conducted out is equal to the heat transferred by convection to the surrounding fluid
d dT k f L f tf dx = 2 h (L f + t f ) dx (T - T∞ ) dx dx
(3.3.1)
or, assuming constant thermal conductivity kf,
d 2T dx
2
-
2 h(L f + t f ) k f Lf tf
(T - T ∞) = 0
(3.3.2)
This equation assumes the heat transfer coefficient is uniform and substantial temperature gradients occur only in the x-direction. The latter assumption will be satisfied if the fin is thin. This is a reasonable approximation in practice. Assume, that tf 2300, but this has a negligible influence on dpvf in practical air-cooled condenser tubes. The change in pressure due to the effects of acceleration can be expressed directly in terms of the momentum of the vapor entering the tube, i.e., 3
[
]
∆ p vm = ∫ G 2v d (1 - x ) / {(1 - x )ρ c}+ x 2 / (α ρ v ) ≈ - ρ v v v22 2
2
(3.4.59)
This equation assumes the velocity distribution of the entering vapor is uniform and no vapor flows out of the tube. In addition to the frictional and momentum effects, the loss at the inlet to the tube is significant in a condenser tube. Using Equation 2.3.5, 2 ) + K c] p v1 - p v 2 = ( ρ v vv22 / 2)[ (1 - σ21
(3.4.60)
If all steam entering the tube condenses, the change in pressure between the inlet and the outlet manifolds can be expressed in dimensionless form by adding Equations 3.4.58, 3.4.59, and 3.4.60 and dividing by ρvv2v2/2, i.e.,
Pv1 - Pv4 0.3164 L a1 a2 2 = (K c - σ 21 - 1) + Re v2.75 Re v1.75 2 + 2 2 2 Re v 2 d e 2.75 1.75 ρ v v v2 / 2
(3.4.61)
The frictional pressure differential between the inlet and any other section of the duct is given by Rev ρ v 2 dz ∆ p vf = p v2 - pvf = ∫ f De v v Rev2 2 d e
(3.4.62)
2
0.1582 µ v L a1 a2 2.75 1.75 (Re 2.75 (Re v1.75 = v 2 - Re v ) + 2 - Re v ) 3 1.75 ρ v d e Re v 2 2.75
The pressure differential due to accelerational effects is
∆ p vm = ρ v (v 2v - v 2v 2 )
(3.4.63)
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To find the mean static pressure in the tube, integrate the sum of Equation 3.4.62 and Equation 3.4.63 over the entire tube length, and find L
∫ [( p v2 − p vf ) + ρ v (v 2v − v 2v2)] 0
dz L
or
p vm ≈ pv 2-
2
0.1582 µ v L 3 ρv de Re v 2
2 1.75 2 (0.267 a1 Re v2.75 2 + 0.364 a 2 Re v 2 )+ ρv v v 2 3
(3.4.64)
The temperature of saturated steam can be expressed approximately in terms of the saturation pressure as
T v = 5149.6889682 /[ n (1.020472843 x 10 11/ pv )]
(3.4.65)
for 3500 N/m2 ≤ pv ≤ 75000 N/m2 and where Tv is in degrees Kelvin.
If the steam condensing inside the duct remains saturated, there will be a change in temperature in addition to the pressure drop. The mean steam temperature can be determined approximately by substituting the value of pvm into Equation 3.4.65. The previous procedure can also be followed to determine the pressure change during condensation in a tube. In that case, the coefficients in Equation 3.4.53 are, according to Groenewald,
a1 =1.0046 +1.719 x 10-3 Re vn - 9.7746 x 10 -6 Re 2vn a2 = 574.3115 + 24.2891 Re vn +1.8515 Re 2vn where Revn = ρvvv2d2/(4µL)
An additional resistance due to condensate surface waviness during annular tube flow can be determined from a presentation by Bergelin.
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For condensers used in the petrochemical industry, the vapor inlet velocity to the condenser tubes is usually limited to 30 m/s. This is considerably lower than the velocity encountered in low-pressure steam condensers according to Rose. Depending on the nature of the flow in the former case, other equations for determining the pressure drop may be appropriate, e.g., Baroczy.
3.5 Heat Exchangers The principles of heat transfer are applied to the design of heat transfer equipment. Depending on the particular application, the geometry and performance characteristics of heat exchangers may differ significantly.
Logarithmic mean temperature difference Consider a simple heat exchanger consisting of two concentric pipes (Fig. 3.5.1). One fluid flows on the inside of the smaller pipe while the other fluid flows in the annular region between the two pipes.
Fig. 3.5.1 Double Pipe Heat Exchanger
The fluids may flow in the same direction (Fig. 3.5.1), which is referred to as parallel flow, or they may flow in opposite directions or counterflow. Examples of the fluid temperature profiles are shown in Figure 3.5.2.
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Fig. 3.5.2 Temperature Profiles (a) Parallel Flow (b) Counterflow
In order to determine how much heat is transferred from the hot to the cold fluid, an expression for the mean temperature difference between the two streams must be determined. For the parallel flow heat exchanger (Fig. 3.5.2a), the heat transferred through an element of tube area dA is expressed as
dQ = U (Th – Tc)dA
192
(3.5.1)
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where U = the overall heat transfer coefficient referred to the tube area h = the hot fluid c = the cold fluid
For purposes of this analysis, the overall heat transfer coefficient is assumed to be constant. In certain practical cases, this assumption is a reasonable approximation. Where significant entrance effects and changes in physical properties occur, a numerical step-by-step integration of Equation 3.5.1 may be necessary. The heat transfer can also be expressed as
dQ = - m h cph d T h = m c cpc dT c
(3.5.2)
From Equation 3.5.2 it follows that
1 1 dT h - dT c = - dQ + m c m c cpc h ph
(3.5.3)
By substituting Equation 3.5.1 into Equation 3.5.3, find
1 d( Th - Tc) 1 dA = - U + (T h - T c) m h cph m c cpc
(3.5.4)
If constant specific heats are assumed, this equation may be integrated between the ends of the heat exchanger:
T h2 - T c2 ln = - UA T h1 - T c1
1 1 m c + m c c pc h ph
(3.5.5)
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Integration of Equation 3.5.2 between the same limits gives
Q = - m h cph (T h 2 - T h1) = m c cpc (T c2 - T c1)
(3.5.6)
Substitute the values of mcp from Equation 3.5.6 into Equation 3.5.5, and find
(T h 2 - T c2) - (T h1 - T c1) = UA ∆ Tm n [(T h 2 - T c 2) /( T h1 - T c1)]
(3.5.7)
(T h 2 - T c2) - (T h1 - T c1) ∆ T 2 - ∆ T1 = n [(T h 2 - T c2) /(T h1 - T c1)] n (∆ T 2 / ∆ T1)
(3.5.8)
Q = UA
where
∆ T m =
This is known as the logarithmic mean temperature difference (LMTD). Equation 3.5.8 is also applicable in the case of counterflow conditions or when the temperature of one of the fluids is constant during condensation or boiling. The mean temperature of the hot fluid for the parallel flow heat exchanger can be found by integrating Equation 3.5.4 over only a part of the heat exchanger area Ax from section 1 to give
1 1 Thx = Tcx + (Th1 - Tc1) exp - UA x + m h cph m c cpc
(3.5.9)
Integrate Equation 3.5.2 between the same limits.
- m h cph (T hx - T h1) = m c cpc (T cx - T c1) or
Tcx = Tc1 - mh cph (Thx - Th1) /( mc cpc)
194
(3.5.10)
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Substitute Equation 3.5.10 into Equation 3.5.9, and find
T c1 + T hx =
1 m h c ph T h1 1 + (T h1 - T c1) exp - UA x + m c cpc m h cph m c cpc
(3.5.11)
[1 + m h cph /(m c cpc)]
The mean temperature of the hot fluid is obtained by integrating Equation 3.5.11 over the entire heat exchanger, i.e., A
T hm =
1 T dA A ∫0 hx (3.5.12)
m h cph T h1 (T - T ) exp - UA 1 + 1 -1 h1 c1 m c m c ccp h ph m c cpc = (1 + m h cph / m c cpc ) UA(1 + m h cph / m c cpc )(1 / m h cph + 1 / m c cpc ) T c1 +
The mean temperature of the cold fluid is found to be
T h1 + T cm =
m c cpcT c1 m h cph
1 + m c cpc / m h c ph
+
1 1 - 1 (T h1 - T c1) exp - UA + m h cph m c cpc
(3.5.13)
UA(1 + m h cph / m c cpc)(1 / m h cph + 1 / m c cpc)
The mean temperatures for a counterflow heat exchanger are
Tc1T hm =
m h cph T h1 m c cpc
1 1 - 1 (T h1 - T c1) exp- UA m h cph m c cpc
(1 - m h cph / m c cpc) UA(1 + m h cph / m c cpc)(1 / m h cph -1 / m c cpc)
(3.5.14)
and
1 1 - 1 (T h1 - T c1) exp- UA m h cph m c cpc (3.5.15) T cm = + 1 - m c ccp / m h cph UA(1 - m h cph / m c cpc)(1 / m h cph - 1 / m c cpc) T h1 -
m c cpc T c1 m h cph
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Crossflow air-cooled heat exchangers find many applications in practice. An example of such an exchanger is shown in Figure 3.5.3(a) where the fluid flows inside the tubes while cooling air flows across the tube bundles. In this exchanger, the hot fluid, which is confined to the separate tubes, is said to be unmixed. The airstream, which can move about freely as it flows through the bundle, is said to be mixed. Unmixed conditions also occur when both streams are confined to specific channels, as in the finned tube heat exchanger shown in Figure 3.5.3(b).
Fig. 3.5.3 Crossflow Heat Exchangers (a) (b)
For crossflow heat exchangers having mixed and unmixed flow, the mathematical derivation of an expression for the mean temperature difference becomes quite complex according to Bowman, Taborek, and Roetzel. The usual procedure is to modify the simple counterflow LMTD by a correction factor, FT, determined for a particular arrangement. The product of FT and ∆Tm is called the corrected mean temperature difference. The heat transfer equation then takes the form
Q = UAFT ∆Tm
(3.5.16)
The approximate temperature correction factor for an unmixed single-pass crossflow heat exchanger from Tucker is shown in Figure 3.5.4. Methods for evaluating FT for different heat exchangers are presented in appendix B.
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Fig. 3.5.4 Correction Factor for Single-Pass Crossflow Heat Exchanger, Both Fluids Unmixed
Equation 3.5.7 is useful when all terminal temperatures for the evaluation of the logarithmic mean temperature difference are known. There are, however, numerous occasions when the fluid outlet temperatures are not known. This type of problem is encountered in the selection of a heat exchanger or when the unit has been tested at one flow rate but service conditions require different flow rates for one or both fluids. The outlet temperatures and the rate of heat transfer can then be found only by an iterative procedure. To avoid this problem, the effectiveness-NTU (Number of Transfer Units) method presented in the following section may be a more appropriate approach.
Effectiveness-NTU method The effectiveness-NTU method of heat exchanger analysis is valuable when variables other than the stream temperatures are specified. Furthermore,
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this method makes it possible to compare various types of heat exchangers so that the one best suited to accomplishing a particular heat transfer objective can be selected. The effectiveness of a heat exchanger, e, is defined as the ratio of the actual rate of heat transfer in a given heat exchanger to the maximum possible rate of heat transfer. The maximum rate would be obtained in a counterflow heat exchanger of infinite heat transfer area where the outlet temperature of the colder fluid equals the inlet temperature of the hotter fluid. This is in the absence of heat losses to the environment, when mc cpc < mh cph or the outlet temperature of the warmer fluid equals the inlet temperature of the colder one when mh cph < mc cpc. Depending on which of the heat capacity rates is smaller, the effectiveness for the parallel flow exchanger may be expressed as
e = Ch (T h1 - T h 2) /[Cmin(T h1 - T c1)]
(3.5.17)
e = Cc (T c2 - T c1) /[Cmin(T h1 - T c1)]
(3.5.18)
or
where the heat capacity rates are defined as Ch = mh cph and Cc = mc cpc, and Cmin is the smaller of these values. The method of deriving an expression for the effectiveness of a heat exchanger is illustrated by applying it to a parallel flow arrangement. Rewrite Equation 3.5.5 with Equation 3.5.17 and 3.5.18 to obtain
n (1 - eCmin / Ch - eCmin / Cc) = - UA(1 / Ch +1 / Cc) or
1 - e (Cmin / Ch + Cmin / Cc) = exp [- UA (1 / Ch + 1 / Cc)]
(3.5.19)
Solving for e yields
e =
198
1 - exp [- UA(1 / Ch + 1 / Cc)] (Cmin / Ch + Cmin / Cc)
(3.5.20)
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In general for parallel flow,
e =
1 - exp[- UA / Cmin (1 + Cmin / Cmax)] 1 + Cmin / Cmax
(3.5.21)
This derivation illustrates how the effectiveness for a given flow arrangement can be expressed in terms of two dimensionless parameters— the heat capacity rate ratio Cmin/Cmax and the ratio of the overall conductance to the smaller heat capacity rate UA/Cmin. The latter of the two parameters is called the number of transfer units or NTU. The number of heat transfer units is a measure of the heat transfer size of the exchanger. The larger the value of NTU, the closer the heat exchanger approaches its thermodynamic limit. In the case of condensation where the fluid temperature stays constant or the fluid acts as if it had infinite specific heat, Cmin/Cmax approaches zero, and the effectiveness relation for all heat exchanger arrangements becomes
e = 1 – exp (–UA/Cmin)
(3.5.22)
By analyses similar to the one presented here for parallel flow, the effectiveness may be evaluated for most flow arrangements of practical interest. The effectiveness of a crossflow heat exchanger with both streams unmixed is shown in Figure 3.5.5.
Fig. 3.5.5 Effectiveness for Crossflow Exchanger with Fluids Unmixed
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Kays summarizes effectiveness relations for a number of different types of heat exchangers in Table 3.5.1. Relations for other geometries are presented in the literature from ESDU. It should be noted that the relations for effectiveness listed in Table 3.5.1 are true only for a constant value of the overall heat transfer coefficient throughout the heat exchanger.
N = NTU = UA / Cmin ; C = Cmin / Cmax
Table 3.5.1 Heat Exchanger Effectiveness Relations
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For a multipass overall-counterflow arrangement with the process fluid mixed between passes, the effectiveness is given by Kays and London as
e=
[{(1 - e C) / (1 - e )} - 1]/[{(1 - e C)/ (1 - e )} - C] np
p
p
np
p
p
(3.5.23)
where np = the number of identical passes in the overall-counterflow arrangement ep = the effectiveness of each pass, a function of N/np and the basic flow configuration of the pass The individual passes can be any one of the basic flow arrangements. Although Equation 3.5.23 was deduced for the case of fluid mixing between passes, it can also be employed to predict the effectiveness for unmixed flows to a satisfactory degree of accuracy for practical heat exchangers. Unmixed flow implies that temperature differences within the fluid in at least one direction normal to the flow can exist but that no heat flux occurs owing to these differences. In the case of complete mixing, all the fluid in any given plane normal to the flow has the same temperature. Many practical flow situations may not satisfy any of the previous cases, and it may be necessary to analyze the performance of consecutive tubes or sections of tubes. Uncertainties in the effectiveness-NTU calculations for crossflow heat exchangers are evaluated by Di Giovanni and Webb.
Example 3.5.1 Oil at 100 °C enters a new air-cooled, single-pass, crossflow heat exchanger at a rate of mo = 2 kg/s. The oil is distributed uniformly through 40 tubes (Fig. 3.5.6). A 0.2 mm thick spiral tape insert with a 30 mm pitch is located inside each tube to improve the effective heat transfer coefficient on the oil side.
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Fig. 3.5.6 Air-Cooled Oil Cooler
Air at 50 °C and a pressure of 105 N/m2 approaches the finned surface at a uniform speed of 3 m/s. The aluminum-plate fins are 0.24 mm thick and are mechanically bonded (neglect thermal contact resistance) at a 3 mm pitch to copper tubes having an effective length of 726 mm and an outside diameter do = 10 mm. The air-side heat transfer coefficient is ha = 45.6 W/m2 K. Find the rate of heat transfer from the oil to the air. The thermophysical properties of the oil are: Density
ρo = 880.5 - 0.613 T kg/m3
Thermal conductivity
ko = 0.13428 - 7.185 x 10-5 T W/mK
Specific heat
cpo = 3.642T + 1809 J/kgK
Dynamic viscosity
µo = 38.876/T1.9325 kg/ms
where T is in °C. The thermal conductivity of aluminum is ka = 204 W/mK and copper is kcu = 364 W/mK.
Solution To determine the approximate heat transfer coefficient on the inside of the tubes, the properties of oil are initially evaluated at 100 °C, i.e.,
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ρo = 880.5 - 0.613 x 100 = 819.2 kg/m3 ko = 0.13428 - 7.185 x 10-5 x 100 = 0.1271 W/mK cpo = 3.642 x 100 + 1809 = 2173.2 J/kgK
µo = 38.876/1001.9325 = 0.005305 kg/ms Pro = µocpo/ko = 0.005305 x 2173.2/0.1271 = 90.707 The effective cross-sectional flow area through a tube is given by
A ts= π di2 / 4 - t t di = ( π x 9 2/ 4 - 0.2 x 9 )x 10 -6 = 61.817 x 10 -6m 2
The Reynolds number inside the tube can be expressed as Re o = m odi /(40 A ts µo)= 2 x 9 x 10 -3/ (40 x 61.817 x 10 -6 x 0.005305) = 1372.2
Although the oil is being cooled, the oil-side heat transfer coefficient can be approximated by Equation 3.2.19.
ho =
0.1271 x 5.172 0.7 1372.2 x 9 1 + 0.00548 x 90. 707 x 30 9 x 10-3
1.25 0.5
2 = 1130 W/ m K
To find the efficiency of the plate fins, the equivalent diameter is first determined using Equation 3.3.14. From Figure 3.5.6,. find L1 = L2 = 10 mm. Thus
0.5
dfe 10 10 = 2.56 - 0.2 = 2.28973 dr 10 10
Substitute this value into Equation 3.3.13, and find
ϕ = (2.28973 - 1)[1 + 0.35 n (2.28973)]= 1.66369 Furthermore,
b = [2 h a / ( k f t f )] = [2 x 45.6 / (204 x 0.24 x 10-3)] = 43.16 0.5
0.5
where kf = ka
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The fin efficiency is determined using Equation 3.3.12. ηf = tanh(43.16 x 0.01 x 1.66369 / 2)/(43.16 x 0.01 x 1.66369 / 2) = 0.959 Consider a control volume about one tube and between two fins, and employ Equation 3.1.26 with kt = kcu to find,
UA i =
1
1 / (h o A i )+ n (do / di ) / (2 π k t Pf )+ 1/[h a( ηf A f + A re)]
106 103 x n(10 / 9) = + 1130π x 9 x 3 2π x 364 x 3
+
106
45.6{0.959(20 x 20 - π x 10 2 / 4 )x 2 + π x 10(3 - 0.24)}
-1
-1
= (10.43297 + 0.01536 + 31.18272 ) = 0.024 W/K
From the terms in the previous equation, it is noted that the major thermal resistance of this particular heat exchanger is on the air side (75% of total resistance) while the resistance due to the copper tube wall is negligible. The greatest potential for improving the design of this heat exchanger lies in enhancing the heat transfer on the air side. For the entire heat exchanger UA = 0.024 x 726 x 40/3 = 232.32 W/K The density of the air entering the heat exchanger, using Equation A.1.1, is ρa = 105 / {287.08 x (273.15 + 50)]=1.077936 kg/ m3
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Using Equation A.1.2, the specific heat of the entering air is
cpa = 1.045356 x 10 3- 0.3161783 x 323.15 + 7.083814 x 10-4 x 323.15 2 - 2.705209 x 10 -7 x 323.15 3= 1008 J/kgK
The air mass flow rate through the face area or frontal area, Afr, is
m a = ρa v a A fr = 1.077936 x 3 x 0.726 x 0.8 = 1.8782 kg/s The approximate heat capacity rate for the airstream is Ca = macpa = 1.8782 x 1008 = 1893.23 W/K and for the oil stream, it is Co = mocpo = 2 x 2173.2 = 4346.4 W/K Since Ca < Co, it follows from Table 3.5.1 with Cmin = Ca and Cmax = Co that the effectiveness of the heat exchanger is 0.22 0.78 1.8782 x 1008 232.32 232.32 e =1 - exp exp - 1 2 x 2173.2 1.8782 x 1008 1.8782 x 1008
1.8782 x 1008 / = 0.111 2 x 2173.2 The theoretical maximum heat transfer rate from the oil to the air is m a c pa (T oi - T ai ). Multiply the latter value by the effectiveness to obtain the actual heat transfer rate, i.e.,
Q = emacpa (Toi - Tai) = 0.111 x 1.8782 x 1008 (100 - 50) = 10507 W The mean oil outlet temperature can be found as follows: Q = mocpo(Toi - Too) or Too = Toi - Q/(mocpo) = 100 - 10507/(2 x 2173.2) = 97.58 °C
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The mean outlet temperature of the airstream follows from Tao = Tai + Q/(macpa) = 50 + 10507/(1.8782 x 1008) = 55.55 °C The temperature on the inside of the tube wall, Tti, can be found from Q = hoAti(Tom - Tti) or Tti = Tom - Q/(hoAti) = (Too + Toi)/2 - Q/(hoAti) = (100 + 97.58)/2 - 10507/[1130(π x 9 x 10-3 x 0.726 x 40)] = 87.466 °C where Ati = the inside area of all tubes Tom = the mean oil temperature The dynamic viscosity of the oil at this temperature is
µow = 38.876/87.4661.9325 = 0.006872 kg/ms The dynamic viscosity at the mean oil temperature, i.e., Tom = (Toi + Too)/2 = (100 + 97.58)/2 = 98.79 °C, is
µom = 38.876/98.791.9325 = 0.005431 kg/ms Using Equation 3.2.23, the corrected heat transfer coefficient on the inside of the tube is hom = ho(µom/µow)0.14 = 1130(0.005431/0.006872)0.14 = 1093.38 W/m2k Due to the relatively small change in air-side properties, the heat transfer coefficient can be assumed to remain unchanged. By repeating the previous procedure with the improved heat transfer coefficient on the oil side and by evaluating all properties at the mean temperatures of the fluids, an improved value of the heat transfer rate can be obtained, i.e., Q = 10,413 W.
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Note If they contain more than one row of tubes, single-pass (process fluid) aircooled crossflow heat exchangers in which viscous fluids such as oils are cooled must be designed carefully in order to avoid uneven oil flow through the tubes. In a single-pass multirow oil cooler, oil flowing through the tubes facing the entering cooling airstream would be cooled more effectively than oil in the rows of downstream tubes. The cooler oil is more viscous and will flow more slowly than the hotter oil for a given pressure differential between the headers, which reduces the effectiveness of the heat exchanger according to Jesch. It is also important that the airstream flows uniformly through the heat exchanger to avoid maldistribution of oil flow in the tubes. Other factors that may affect the performance of heat exchangers in which the flow is laminar are addressed by Rohsenow.
Example 3.5.2 Superheated refrigerant 134a enters an air-cooled condenser tube (Fig. 3.5.7) at a rate of mr = 0.03 kg/s and at a temperature of Tvi = 60 °C. The refrigerant condenses at a pressure of pvs = 1.13 x 106 N/m2 and a saturation vapor temperature of Tvs = 44 °C. The serpentine copper tube (thermal conductivity kt = 364 W/mK) has an outside diameter do = 10 mm, an inside diameter di = 8 mm, and consists of 40 effective lengths of 726 mm each pitched at Pt = 20 mm.
Fig. 3.5.7 Air-Cooled Condenser
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Aluminum-plate fins (thermal conductivity kf = 204 W/mK) having a thickness of 0.24 mm are mechanically bonded to the tubes (assume thermal contact resistance is negligible) at a pitch of Pf = 3 mm. Air at Tai = 20 °C and a pressure of pa = 105 N/m2 approaches the finned surface at a uniform speed of va = 3 m/s. The air-side heat transfer coefficient is ha = 45.6 W/m2K. Fouling can be ignored. Determine the rate of heat transfer from refrigerant to air and the temperature of the refrigerant at the condenser outlet if changes in its pressure are neglected and all thermophysical properties are evaluated at 44 °C. The thermophysical properties of refrigerant 134a at 44 °C from the ASHRAE Handbook of Fundamentals are as follows: Saturated liquid (condensate). Specific heat
cpc = 1525 J/kg K
Dynamic viscosity
µc = 169.8 x 10-6 kg/ms
Thermal conductivity
kc = 0.0731 W/m K
Critical pressure
pcr = 4.056 x 106 N/m2
Saturated vapor Specific heat
cpvs = 1156 J/kg K
Dynamic viscosity
µvs = 13.38 x 10-6 kg/ms
Thermal conductivity
kvs = 0.0161 W/m K
The latent heat of vaporization
ifg = 1.589 x 105 J/kg.
Solution As shown in Figure 3.4.4 for the case of high heat flux, superheated refrigerant vapor entering a condenser tube is usually cooled initially without condensation occurring. Where the surface temperature at the inside of the tube corresponds to the saturation temperature of the vapor, condensation will commence. The flow pattern will tend to become one of shear-driven annularcondensate flow with a superheated vapor core. Ultimately, the tube will be filled with liquid, which will be subcooled. To find the location where condensation commences, the heat transfer coefficient must be obtained in the region where the vapor is superheated. For superheated refrigerant 134a, the vapor Reynolds number is given by
Re v =
208
ρ v vv d i m d 0.03 x 4 = 356 850 ≈ r i = Ats µ vs π x 8 x 10 -3 x 13.38 x 10 -6 µv
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The approximate Prandtl number of the superheated vapor is
Prv ≈
µ vs cpvs 13.38 x 10 -6 x 1156 = = 0.961 k vs 0.0161
When cooling a fluid during turbulent flow inside a smooth tube, the heat transfer coefficient may be determined using Equation 3.2.25
hv =
=
0.3 N u v k v 0.0265 Re 0.8 v Prv k v = di di
0.0265 x 356 850 0.8 x 0. 9610.3 x 0.0161 = 1 458.1 W/ m2 K 8 x 10- 3
Consider an elementary control volume of arbitrary length, say Pf, of the condenser tube at the point where the inside wall surface temperature becomes 44 °C. To find the rate of heat transfer from this surface to the outside airstream, find the relevant product of overall heat transfer coefficient and corresponding area, i.e.,
n (do / di ) 1 (UA )ai = + 2 π k t Pf h a(ηf A f + A re)
-1
-1
n(10 / 8) = + 31.18272 = 32.04 x 10- 3 W / K -3 2 π x 364 x 3 x 10
where 1/[ha(ηf Af + Are)] = 31.18272 K/W was evaluated in Example 3.5.1. The overall heat transfer coefficient based on the inside surface area of the control volume is
Uai = 32.04 x 10- 3 / (π x 8 x 10- 3 x 3 x 10- 3 )= 424.94 W/ m2 K It is noted that this heat transfer coefficient is considerably smaller than hv, which means the heat transfer rate is determined primarily by conditions on the air side, i.e., Uai.
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If it is assumed that the inside surface of the tube wall is at a uniform temperature of 44 °C over the distance Pf, the heat transfer rate can be expressed in terms of the effectiveness as given by Equation 3.5.22. To apply this equation, determine the density of the ambient air using Equation A.1.1, i.e., ρa = 105/[287.08 x (273.15 + 20)] = 1.1882 kg/m3 and the specific heat using Equation A.1.2, i.e.,
cpa = 1.045356 x 103 - 3.161783 x 10- 1 x 295.15 + 7.083814 x 10- 4 (295.15 )2 3
- 2.705209 x 10- 7 (295.15 ) = 1006.79 J/kg K The heat capacity rate for the airstream flowing through the elementary control volume between two fins is given by
Cmincv = macv cpa = ρa v a Pf Pt cpa = 1.1882 x 3 x 3 x 10-3 x 20 x 10-3 x 1006.79
= 0.2153 W/K Substitute the values for (UA)ai and Cmincv into Equation 3.5.22, and find
ecvai = 1 - exp (- 32.04 x 10- 3 / 0.2153)= 0.1383 The heat transfer rate for the elementary control volume can be expressed as
Qcv = ecvai Cmincv(44 - Tai) = 0.1383 x 0.2153(44 - 20) = 0.7146 W The product of the overall heat transfer coefficient between the vapor and the airstream and the area for the elementary control volume is given by
1 n (d o / di) 1 (UA) cv = + + 2 + ( ) π π h d P k P h A A η t f a re v i f f f
-1
1 n(10 / 8) + = - 3 + 31.18272 -3 -3 π 2 x 364 x 3 x 10 1458.1 x π x 8 x 10 x 3 x 10 = 24.8 x 10- 3 W / K
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With Cmincv = 0.2153 W/K and Cmaxcv ≈ mrcpvs = 0.03 x 1156 = 34.68 W/K such that Ccv = Cmincv/Cmaxcv = 0.2153/34.68 = 6.208 x 10-3, find the effectiveness for a crossflow heat exchanger from Table 3.5.1 in which both streams are unmixed, i.e., 0.78 24.8 x 10- 3 0.22 24.8 x 10- 3 exp - 6.208 x 10- 3 ecv =1- exp - 1 / (6.208 x 10- 3) = 0.10874 0.2153 0.2153
The heat transfer rate for the elementary control volume can be expressed in terms of this effectiveness, i.e.,
Qcv = 0.7146 = ecv Cmincv (T vo - T ai )= 0.10874 x 0.2153 (T vo - 20) = 0.02341 (T vo - 20) or
Tvo = 0.7146/0.02341 + 20 = 50.52 °C In cooling the superheated vapor from 60 °C to 50.52 °C, its enthalpy is reduced by
Qv = mrcpv (Tvi - Tvo) ≈ mrcpvs (Tvi - Tvo) = 0.03 x 1156 (60 - 50.52) = 328.8 W This heat is transferred to the airstream. The overall heat transfer between the superheated vapor and the airstream is
Pf 1 n (d o / di ) + (UA)v = + π π 2 k L h A + A L h d L ( ) η t v a re v i v v f f
-1
1 n (10 / 8) + 31.18272 x 3 x 10 -3 = Lv + -3 2 π x 364 1458.1 x π x 8 x 10
-1
= 8.269 L v where Lv is the length of tube preceding the cross section where condensation begins.
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The heat capacity ratio in the region of superheated vapor is
Cv =
Cminv m r cpvs 0.03 x 1156 0.4832 = = ≈ Cmaxv ρava Pt Lv cpa 1.1882 x 3 x 20 x 10-3 x 1006.79 L v Lv
The assumption that mrcpvs < ρavaPtLvcpa is true only if Lv > 0.4832 m. In this region, both fluid streams are unmixed and result in an effectiveness that, according to Table 3.5.1, is given by
8.267 L 0.22 0.4832 8.269 L v 0.78 L v v e v = 1 - exp - 1 exp L v 0.03 x 1156 0.4832 0.03 x 1156 -0.22 = 1 - exp [1.51 L1.22 v {exp (- 0.1579 L v )- 1}]
The heat transfer rate in that section of the tube where no condensation occurs is
Qv ≈ ev m r cpvs(T vi - T ai) - 0.22 )- 1}] ]0.03 x1156 (60 - 20) = 328.8 W = [1 - exp[1.51 L1.22 v {exp (- 0.1579 L v
By following an iterative procedure, find Lv = 1.223 m. Since Lv > 0.4832 m, the assumption is that Cminv = mrcpv was correct. After Lv = 1.223 m, condensation of the vapor occurs, and the local condensation heat transfer coefficient is given by Equation 3.4.41. If changes in vapor pressure can be neglected, the rate of condensation in an air-cooled condenser will be relatively uniform (the quality varies linearly), and Equation 3.4.42 can be employed to determine the mean condensation heat transfer coefficient, i.e., 0.8
hc =
0.023 x 0.0731 0.03 x 8 x 10-3 x 4 -3 2 -6 -6 8 x 10 π x 8 x 10 x 169.8 x 10
[
]= 4 985 W / m 2 K
0.38
x 0.55 + 2.09 / (1.13 x 106 / 4.056 x 106)
212
169.8 x 10- 6 x 1525 0.0731
0.4
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With this value, find the product of overall heat transfer coefficient and the corresponding area in the region of condensing vapor.
Pf 1 n (d o / di ) + (UA)c = + h c π di L c 2 π k t L c h a(ηf A f + A re)L c
-1
1 n (10 / 8) + 31.18272 x 3 x 10- 3 = Lc + -3 2π x 364 4985 π x 8 x 10
-1
= L c [13.236 x 10- 3 + 0.0976 x 10- 3 + 93.548 x 10- 3]
-1
= 9.84 L c
where Lc is the length of tube over which condensation occurs. Note that the first term in the previous equation is considerably smaller than the last term, i.e., the thermal resistance on the condensing side is relatively small compared to that on the air side. The assumption of a mean condensing heat transfer coefficient in this analysis is an acceptable approximation. If this had not been the case, better accuracy could have been achieved by analyzing consecutive relatively short lengths of the condensing tube according to Li. The effectiveness in the region of condensation is, using Equation 3.5.22,
ec = 1 - exp [- 9.84 L c / (ρ a v a Pt L c cpa)]
= 1 - exp [- 9.84 / (1.1882 x 3 x 20 x 10- 3 x 1006.79 )] = 0.1281 The heat transfer rate in the condensing region is
Qc = ec ρa va Pt Lc cpa (44 - 20) = 0.1281 x 1.1882 x 3 x 20 x 10-3 x 1006.79 x 24 Lc = 220.67 Lc
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This heat transfer rate is equal to the change in enthalpy between the superheated vapor entering and the saturated liquid leaving the condensing section, i.e.,
Qc = mr[ifg + cpvs (Tvo-44)] = 0.03[1.589 x 105 + 1156(50.52 - 44)] = 4993.1 W Thus
Lc = Qc/220.67 = 4993.1/220.67 = 22.627 m After condensing, the refrigerant is subcooled in the remaining length of the tube, i.e.,
L = (40 x 0.726) - Lv - Lc = 29.04 - 1.223 - 22.627 = 5.19 m The Reynolds number of the liquid in this region is
Re=
4 mr 4 x 0.03 = = 28 119.3 π di µc π x 8 x 10- 3 x 169.8 x 10- 6
The flow is turbulent, and Equation 3.2.25 can be employed to find the heat transfer coefficient between the liquid and the tube wall. 0.3
h =
169.8 x 10 - 6 x 1525 0.0265 x 0.0731 = 1282.52 W/ m2 K (28 119.3)0.8 -3 0.0731 8 x 10
The product of overall heat transfer coefficient and area is
Pf 1 n (do / di ) + (UA) = + 2 π k t L h a (ηf A f + A re)L e h π di L
-1
3 x 10 -3 x 31.18272 1 n (10 / 8) + = + -3 5.19 1282.52 x π x 8 x 10 x 5.19 2 π x 364 x 5.19 = 41.63 W / K
214
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The minimum heat capacity rate is
Cmin = mrcpc = 0.03 x 1525 = 45.75 W/K while
Cmax = ρa va Pt L cpa = 1.1882 x 3 x 20 x 10-3 x 5.19 x 1006.79 = 372.518 W/K and
C = Cmin/Cmax = 45.75/372.518 = 0.1228 For a crossflow heat exchanger where both fluids are unmixed, find the effectiveness from Table 3.5.1 0.78 41.63 0.22 41.63 e = 1 - exp exp - 0.1228 - 1 / 0.1228 = 0.577 45.75 45.75
The heat transfer rate in the subcooling region is
Q = eCmin (44 - 20) = 0.577 x 45.75 x 24 = 633.5 W The outlet temperature of the condensate or liquid can be determined from the following relation:
Q = mrcpc (44 - To) = 633.5 W or To = 44 - 633.5/(0.03 x 1525) = 30.15 °C The rate of heat transfer from the refrigerant to the air is
Qv + Qc + Q = 328.8 + 4993.1 + 633.5 = 5955.4 W
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References Abdel-Wahed, R. M., A. E. Attia, and M. A. Hifni, “Experiments on Laminar Flow and Heat Transfer in an Elliptical Duct,” International Journal of Heat and Mass Transfer, 27-12, 1984. Agarwal, S. K., and M. R. Rao, “Heat Transfer Augmentation for the Flow of a Viscous Liquid in Circular Tubes Using Twisted Tape Inserts,” International Journal of Heat and Mass Transfer, 39-17:3547–3557, November 1996. Akers, W. W., H. A. Deans, and O. K. Crosser, Condensing Heat Transfer within Horizontal Tubes, Chemical Engineering Progress Symposium Series, 55-29:171, 1958. Altman, M., F. W. Stanle and R. H. Norris, “Local Heat Transfer and Pressure Drop for Refrigerant-22 Condensing in Horizontal Tubes,” Third National Heat Transfer Conference, Chemical Engineering Progress Symposium Series, 56-30:157, 1960. American Society of Heating, Refrigeration and Air Conditioning (ASHRAE), Handbook of Fundamentals, SI Edition, Atlanta, 1993. Baroczy, C. J., A Systematic Correlation for Two-Phase Pressure Drop, Chemical Engineering Progress Symposium Series, 62-64:232–249, 1966. Barrow, H., and A. Roberts, “Flow and Heat Transfer in Elliptical Ducts,” 4th International Heat Transfer Conference, Versailles, September 1970. Bergelin, O. P., P. K. Kegel, F. G. Carpenter, and C. Gazley, “Co-Current Gas-Liquid Flow, Flow in Vertical Tubes,” American Society of Mechanical Engineers Heat Transfer and Fluid Mechanics Institute, Berkely, California, 19–28, June 1949. Bergles, A. E., and S. D. Joshi, Augmentation Techniques for Low Reynolds Number In-Tube Flow, Low Reynolds Number Flow Heat Exchangers, ed. S. Kakac, R. K. Shah, and A. E. Bergles, Hemisphere Publishing Corp., Washington, 1983. Bergles, A. E., “Enhancement of Heat Tranfer, Heat Transfer 1978,” Proceedings, Sixth International Heat Transfer Conference, 6:89–108, Hemisphere Publishing Corp., Washington, 1978. Bergles, A. E., Experimental Verification of Analyses and Correlation of the Effects of Temperature-Dependent Fluid Properties on Laminar Heat Transfer, Low Reynolds Number Flow Heat Exchangers, ed. S. Kakac, R. K. Shah, and A. E. Bergles, Hemisphere Publishing Corp., Washington, 1983. Bergles, A. E., Experimental Verification of Analyses and Correlation of the Effects of Temperature-Dependent Fluid Properties on Laminar Heat Transfer, Heat Exchanger Sourcebook, ed. J. W. Palen, Hemisphere Publishing Corp., Washington, 1986. Bergles, A. E., Prediction of the Effects of Temperature Dependent Fluid Properties on Laminar Heat Transfer, Low Reynolds Number Flow Heat Exchangers, ed. S. Kakac, R. K. Shah, and A. E. Bergles, Hemisphere Publishing Corp., Washington, 1983. Blangetti, F., R. Krebs, and E. U. Schlünder, “Condensation in Vertical Tubes—Experimental Results and Modelling,” Chemical Engineering Fundamentals, 1-2:20–63, 1982.
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Bowman, R. A., A. C. Mueller, and W. M. Nagle, “Mean Temperature Difference in Design,” Transactions of the American Society of Mechanical Engineers, 62:283–294, 1940. Boyko, L. D., and G. N. Kruzhilin, “Heat Transfer and Hydraulic Resistance During Condensation of Steam in a Horizontal Tube and in a Bundle of Tubes,” International Journal of Heat Mass Transfer, 10:361–373, 1967. Brauer, H., “Wärmeübergang bei der Filmkonsation reiner Dämpfe an lotrechten Wänden,” Forschung auf den Gebiete des Ingenieurswesen, 24:105–117, 1958. Breber, G., Intube Condensation, Heat Transfer Equipment Design, ed. R. K. Shah, E. C. Subbarao, and R. A. Mashelkar, Hemisphere Publishing Corp., New York, 1988. Breber, G., J. W. Palen, and J. Taborek, “Prediction of Horizontal Tubeside Condensation of Pure Components Using Flow Regime Criteria,” Transactions of the American Society of Mechanical Engineers, 102:471–476, 1980. Bronstein, I. N., and K. A. Semendjajew, Taschenbuch der Mathematik, ed. G. Grosche, V. Ziegler, and D. Ziegler, Verlag Harri Deutsch, Thun und Frankfurt/Main, 1990. Brown, A. R., and M. A. Thomas, “Combined Free and Forced Convection Heat Transfer for Laminar Flow in Horizontal Tubes,” Journal of Mechanical Engineering. Science, 7:440–448, 1965. Büyükalaca, O., and J. D. Jackson, “The Correction to Take Account of Variable Property Effects on Turbulent Forced Convection to Water in a Pipe,” International Journal of Heat Mass Transfer, nos. 4–5, 41:665–669, 1998. Cain, D., A. Roberts, and H. Barrow, “An Experimental Investigation of Turbulent Flow and Heat Transfer in Elliptical Ducts,” Wärme und Stoffübertragung, 2:101–107, 1973. Carey, V. P., Liquid-Vapor Phase-Change Phenomena, Hemisphere Publishing Corp., Washington, 1992. Carnavos, T. C., Some Recent Developments in Augmental Heat Elements, Heat Exchangers: Design and Theory Sourcebook, ed. N. H. Afgan and E. U. Schlünder, 441–489, Scripta Book Co., Washington, 1974. Carpenter, E. F., and A. P. Colburn, “The Effect of Vapor Velocity on Condensation Inside Tubes,” Proceedings, Institute of Mechanical Engineers and the American Society of Mechanical Engineers General Discussion on Heat Transfer, 20–26, London, 1951. Chato, J. C., “Laminar Condensation Inside Horizontal and Inclined Tubes,” American Society of Heating Refrigerating and Air-Conditioning Journal, 52–60, 1962. Chawla, J. M., “Wärmeübergang in Durchströmten Kondensator-Rohren,” KältetechnikKlimatisierung, 24:233–240, 1972. Chen, I. Y., and G. Kocamustafaogullari, “Condensation Heat Transfer Studies for Stratified, Co-Current Two-Phase Flow in Horizontal Tubes,” International Journal of Heat Mass Transfer, 30-6:1133–1148, 1987. Chen, S. L., F. M. Gerner, and C. L. Tien, “General Film Condensation Correlations,” Experimental Heat Transfer, 1:93–107, 1987.
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Cheng, K. C., and M. Jamil, “Laminar Flow and Heat Transfer in Circular Ducts with Diametrically Opposite Flat Sides and Ducts of Multiple Connected Cross Sections,” Canadian Journal of Chemical Engineering, 48:333–334, 1970. Chun, M. H., and K. T. Kim, “Assessment of New and Existing Correlations for Laminar and Turbulent Film Condensation on a Vertical Surface,” International Comm. Heat Mass Transfer, 17:431–441, 1990. Churchill, S. W., “Comprehensive Correlating Equations for Heat Mass and Momentum Transfer in Fully Developed Flow in Smooth Tubes,” Industrial and Engineering Chemistry, Fundamentals, vol. 16, no. 1, 1977. Churchill, S. W., and H. Ozoe, “Correlations for Laminar Forced Convection in Flow Over an Isothermal Flat Plate and in Developing and Fully Developed Flow in an Isothermal Tube,” Journal of Heat Transfer, 95:416–419, 1973. Di Giovanni, M. A., and R. L. Webb, “Uncertainty in Effectiveness-NTU Calculations for Crossflow Heat Exchangers,” Heat Transfer Engineering, 10-3:61–70, 1989. Dittus, F. W., and L. M. K. Boelter, “Heat Transfer in Automobile Radiators of the Tubular Type,” University of California Publications in Engineering, 2:443–461, 1930. Du Plessis, J. P., and D. G. Kröger, “Heat Transfer Correlation for Thermally Developing Laminar Flow in a Smooth Tube with a Twisted-Tape Insert,” International Journal of Heat Mass Transfer, 30-3:509–515, 1987. Engineering Sciences Data Unit, Effectiveness-NTU Relationships for the Design and Performance Evaluation of Multi-Pass Crossflow Heat Exchangers, ESDU Item No. 87020, London 1987. Fieg, G. P., and W. Roetzel, “Calculation of Laminar Film Condensation in/on Inclined Elliptical Tubes,” International Journal Heat Mass Transfer, 37-4:619–624, 1994. Fürst, J., “Kondensation in geneigten ovalen Rohren,” Fortschritt - Berichte Verein Deutscher Ingenieure, vol. 19, no. 36, Düsseldorf, 1989. Garrett-Price, B., S. A. Smith, and R. L. Watts et al., Fouling of Heat Exchangers: Characteristics, Costs, Prevention, Control and Removal, Noyes Publications, Park Ridge, New Jersey, 1985. Gnielinski, V., Forschung Ingenieurswesen, vol. 41, no. 1, 1975. Gnielinski, V., “Zur Wärmeübertragung bei laminarer Rohrströmung und konstanter Wandtemperatur,” Chem. - Ing. - Tech., 61-2:161–163, 1989. Groenewald, W., “Heat Transfer and Pressure Change in an Inclined Air-Cooled Flattened Tube During Condensation of Steam,” Master’s Thesis, Department of Engineering, University of Stellenbosch, Stellenbosch, 1993. Groenewald, W., and D. G. Kröger, “Effect of Mass Transfer on Turbulent Friction During Condensation Inside Ducts,” International Journal of Heat Mass Transfer, 38-18:3385–3392, 1995. Hausen, H., “Extended Equation for Heat Transfer in Tubes at Turbulent Flow,” Wärme und Stoffübertragung, 7:222–225, 1974.
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Hausen, H., “Neue Gleichungen für die Wärmeübertragung bei freier oder erzwungener Konvektion,” Allgemeinë Wärmetechnik, 9:75–79, 1959. Hausen, H., Z. “Verein Deitscher Ingenieure,” Fortschrit-Berichte, vol. 4, no. 91, 1943. Hewitt, G. F., “Two-Phase Flow, Two-Phase Heat Transfer in Steady and Transient States,” Course presented at Stanford University, Stanford, California, 1979. Holman, J. P., Heat Transfer, Mc Graw Hill Book Co., New York, 1986. Hong, S. W., and A. E. Bergles, “Augmentation of Laminar Flow Heat Transfer by Means of Twisted-Tape Inserts,” Journal of Heat Transfer, 98:251–256, 1976. Hsu, S. T., Engineering Heat Transfer, D. van Nostrand Co. Inc., Princeton, 1962. Hu, X., and A. M. Jacobi, “Local Heat Transfer Behaviour and Its Impact on a SingleRow, Annularly Finned Tube Heat Exchanger,” Transactions of the American Society of Mechanical Engineers, Journal of Heat Transfer, 115:66–74, February 1993. Jamil, M., “Laminar Forced Convection in Non-Circular Ducts,” Master’s thesis, Department of Mechanical Engineering, University of Alberta, Edmonton, Alberta, 1967. Jesch, J., J. Lutcha, and V. Michal, “The Use of Air Coolers for Viscous Media,” International Chemical Engineering, 19-4:680–688, 1979. Kakac, S., and Y. Yener, Laminar Forced Convection in Combined Entrance Region of Ducts, Low Reynods Number Flow Heat Exchangers, eds. S. Kakac, R. K. Shah, and A. E. Bergles, Hemisphere Publishing Corporation, Washington, 1983. Kays, W. M., “Numerical Solution for Laminar Flow–Heat Transfer in Circular Tubes,” Transactions of the American Society of Mechanical Engineers, 77:1265–1274, 1955. Kays, W. M., and London, A. L., Compact Heat Exchangers, Mc Graw-Hill Book Co., New York, 1984. Kern, D. Q., and A. D. Kraus, Extended Surface Heat Transfer, McGraw Hill Book Co., New York, 1972. Kosky, P. G., and F. W. Staub, “Local Condensing Heat Transfer Coefficients in the Annular Flow Regime,” American Institute of Chemical Engineers Journal, 17:1037–1043, 1971. Kreith, F., Basic Heat Transfer, Harper and Row Publishers, New York, 1980. Kröger, D. G., “Laminar Condensation Heat Transfer Inside Inclined Tubes,” American Institute of Chemical Engineers Symposium Series, 73-164:256–260, 1977. Li, P-W., and W-Q. Tao, “An Experimental Investigation on Forced-Convection Condensation of HFC-134a Inside Horizontal Tube,” Proceedings, 9th International Symposium on Transport Phenomena in Thermal-Fluids Engineering, 2:1317–1322, Singapore, 1996. Manglik, R. M., and A. E. Bergles, An Analysis of Laminar Flow Heat Transfer in Uniform Temperature Circular Tubes with Tape Inserts, Technical Report ERI Project 1744, Department of Mechanical Engineering, Iowa State University, Ames, 1986.
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Manglik, R. M., and A. E. Bergles, “Heat Transfer and Pressure Drop Correlation for Twisted-Tape Inserts in Isothermal Tubes: Part 1—Laminar Flow,” Journal of Heat Transfer, 115:881–889, 1993. Marner, W. J., and A. E.Bergles, “Augmentation of Tubeside Laminar Flow Heat Transfer by Means of Twisted-Tape Inserts, Static-Mixer Inserts and Internally Finned Tubes,” Heat Transfer, 1978, 2:583–588, Hemisphere Publishing Corp., Washington, 1978. Metais, B., and E. R. G. Eckert, “Forced, Mixed and Free Convection Regimes,” Journal of Heat Transfer, 86:295–296, 1964. Murthy, V. N., and P. K. Sarma, “Condensation Heat Transfer Inside Horizontal tubes,” The Canadian Journal of Chemical Engineering, 50:547–549, 1972. Nusselt, W., “Die Oberflächen Kondensation des Wasserdampfes,” Zeitschrift, Verein Deutscher Ingenieure, 60:541–546, 569, 1916. Owen, R. G., R. S. Sardesai, and D. Butterworth, “Two-Phase Pressure Drop for Condensation Inside a Horizontal Tube,” Symposium on Advancement in Heat Exchangers, International Centre for Heat and Mass Transfer, Dubrovnik, 1981. Palen, J. W., “On the Road to Understanding Heat Exchangers: A Few Stops Along the Way,” Heat Transfer Engineering, 17-2:41–53, 1996. Palen, J. W., G. Breber, and J. Taborek, “Prediction of Flow Regimes in Horizontal Tubeside Condensation,” Heat Transfer Engineering, 1-2:47–57, 1979. Parr, P. H., “The Water Film on Evaporating and Condensing Tubes,” The Engineer, 113:559–561, 1921. Petukhov, B. S., “Heat Transfer and Friction in Turbulent Pipe Flow with Variable Physical Properties,” Advances in Heat Transfer, vol. 6, ed. J. P. Hartnett and T. F. Irvine, Academic Press, New York, 504–564, 1970. Rich, D. G., “The Efficiency and Thermal Resistance of Angular and Rectangular Fins,” Proceedings, International Heat Transfer Conference, 1966. Roetzel, W., Berechnung von Wärmeübertragern, VDI-Wärmeatlas, Cal–Ca31, Verein Deutscher Ingenieure Verlag GmbH, Düsseldorf, 1984. Roetzel, W., “Laminare Filmkondensation an einer senkrechten ebenen Wand mit örtlich veränderlicher Oberflächentemperatur,” Wärme und Stoffübertragung, 27:173–175, 1992. Rogers, D. G., Experimental Heat Transfer Coefficients for the Cooling of Oil in Horizontal Internal Forced Convective Transitional Flow, Report CENG 371, Council for Industrial and Scientific Research, South Africa, 1981. Rohsenow, W. M., Handbook of Heat Transfer, Mc Graw Hill Book Co., New York, 1973. Rohsenow, W. M., Why Laminar Flow Heat Exchangers Can Perform Poorly, in Heat Exchangers, Thermal-Hydraulic Fundamentals and Design, ed. S. Kakac, A. E. Bergles, and F. Mayinger, 1057–1071, McGraw Hill, New York, 1981.
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Rohsenow, W. M., and H. Y. Choi, Heat Mass and Momentum Transfer, Prentice Hall Inc., Englewood Cliffs, New Jersey, 1961. Rose, J. C., D. J. Wilson, and G. H. Cowan, The Performance of Air-Cooled Heat Exchangers, Part II. Air-Cooled Heat Exchanger Performance Specification, Heat Transfer and Fluid Flow Service, Harwell, 1973. Roth, J. E., “Wärmeübertragung bei der Kondensation in geneigten ovalen Rohren,” Fortschritt-Berichte Verein Deutscher Ingenieure, vol. 6, no. 145, Düsseldorf, 1984. Sardesai, R. S., R. G. Owen, and D. J. Pulling, “Pressure Drop for Condensation of a Pure Vapor in Downflow in a Vertical Tube,” Proceedings of the 7th International Heat Transfer Conference, Paper Cs 23, 5:139–145, München, 1982. Schlünder, E.-U., Einführung in die Wärme- und Stoffübertragung, Vieweg, Braunschweig, 1972. Schmidt, T. E., “La Production Calorifique des Surfaces Munies Dailettes,” Annexe Du Bulletin De L’Institut International Du Froid, Annex G-5 1945–1946. Schulenberg, F., “Wärmeübergang und Druckverlust bei der Kondensation von Kältemitteldämpfen in Luftgekühlten Kondensatoren,” Kältetechnik-Klimatisierung, 3:75–81, 1970. Schulenberg, F. J., “Wärmeübergang und Druckänderung bei der Kondensation von Strömenden Dampf in geneigten Rohren,” Doctoral thesis, Universität Stuttgart, 1969. Shah, M. M., “A General Correlation for Heat Transfer During Film Condensation Inside Pipes,” International Journal of Heat Mass Transfer, 22:547–556, 1979. Shah, R. K., Compact Heat Exchangers, in Heat Exchangers, Thermal-Hydraulic Fundamentals and Design, ed. S. Kakac, A. E. Bergles, and F. Mayinger, 111–151, McGraw Hill Book Co., New York, 1981. Shah, R. K., Fully Developed Laminar Flow Forced Convection in Channels, Low Reynolds Number Flow Heat Exchangers, ed. S. Kakac, R. K. Shah, and A. E. Bergles, Hemisphere Publishing Corp., Washington, 1983. Shah, R. K., “Thermal Entry Length Solutions for the Circular Tube and Parallel Plates,” Proceedings of the 3rd National Heat Mass Transfer Conference, Indian Institute of Technolgy, Bombay, vol. 1, HMT-11-75, 1975. Shah, R. K., and A. L. London, Laminar Flow Forced Convection in Ducts, Academic Press, New York, 1978. Soliman, M., J. R. Schuster, and P. J. Berenson, “A General Heat Transfer Correlation for Annular Flow Condensation,” Transactions of the American Society of Mechanical Engineers, Journal of Heat Transfer, 90:267–276, May, 1968. Somerscales, E. F. C., and J. G. Knudsen, Fouling of Heat Transfer Equipment, Hemisphere Publishing Corp., Washington, 1981. Stephan, K., “Wärmeübergang und Druckabfall bei nicht ausgebildeter Laminarströmung in Rohren und in Ebenen Spalten,” Chem. -Ing. -Techn., 31:773–778, 1959.
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Taborek, J., “Design Method for Tube-Side Laminar and Transition Flow Regime with Effects of Natural Convection,” Open Forum Session, 9th International Heat Transfer Conference, Jerusalem, 1990. Taborek, J., Mean Temperature Difference, Heat Exchanger Design Handbook, vol. 1, Hemisphere Publishing Corp., Washington, 1983. Taborek, J., T. Aoki, R. B. Ritter, J. W. Palen, and J. G. Knudsen, “Fouling—The Major Unresolved Problem in Heat Transfer,” Chemical Engineering Progress, 68-2:59–67, 68-7:69–78, 1972. Traviss, D. P., W. M. Rohsenow, and A. B. Baron, “Forced Convection Condensation in Tubes: A Heat Transfer Correlation for Condenser Design,” Transactions of the American Society of Heating Refrigerating and Air-Conditioning, vol. 30, part 1, 157–165, 1973. Tucker, A. S., “The LMTD Correction Factor for Single-Pass Crossflow Heat Exchangers with Both Fluids Unmixed,” Transactions of the American Society of Mechanical Engineers, Journal of Heat Transfer, 118:488–490, May 1996. Van Rooyen, R. S., and D. G. Kröger, “Laminar Flow Heat Transfer in Internally Finned Tubes with Twisted Tape Inserts,” Proceedings of the 6th International Heat Transfer Conference, Paper FC(a)-16, Toronto, August 1978. Walt, J. v.d., and D. G. Kröger, “Heat Transfer During Film Condensation of Saturated and Superheated Freon-12,” Progress in Heat and Mass Transfer, 6:75–97, 1972. Watkinson, A. P., D. C. Miletti, and G. R. Kubanek, “Heat Transfer and Pressure Drop of Internally Finned Tubes in Laminar Oil Flow,” American Society of Mechanical Engineers, 75-HT-41, 1975. Whalley, P. B., B. J. Azzopardi, G. F. Hewitt, and R. G. Owen, “A Physical Model for TwoPhase Flows with Thermodynamic and Hydrodynamic Non-Equilibrium,” Proceedings of the 7th International Heat Transfer Conference, Paper Cs 29, München, 1982. Winterton, R. H. S., “Where Did the Dittus and Boelter Equation Come From?” International Journal of Heat Mass Transfer, nos. 4–5, 41:809–810, 1998. Zabronsky, H., “Temperature Distribution and Efficiency of a Heat Exchanger Using Square Fins on Round Tubes,” Transactions of the American Society of Mechanical Engineers, Journal of Applied Mechanics, vol. 77, 1955. Zarling, J. P., “Application of the Schwarz-Newmann Technique to Fully Developed Laminar Heat Transfer in Non-Circular Ducts,” American Society of Mechanical Engineers, 76-WA/HT-49, 1976. Zeller, M., and M. Grewe, “Verallgemeinerte Näherungsgleichung für den Wirkungsgrad von Rippen auf kreisförmigen und elliptischen Kernrohren,” Wärme und Stoffübertragung, 29:379–382, 1994.
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4 Mass Transfer and Evaporative Cooling 4.0 Introduction Psychrometry is the study of the properties of mixtures of air and water vapor. The subject is important in cooling system practice because atmospheric air is not completely dry but is a mixture of air and water vapor. In certain cooling systems, water is added to the air-water-vapor mixture. Psychrometric principals will be applied in this and later chapters to evaluate the performance of cooling towers, evaporative coolers, and systems incorporating adiabatic precooling of the air. Since there is usually both a heat and mass transfer process between the air and some wetted surface, information on mass transfer is presented. Because of the analogy that exists between momentum, heat, and mass transfer, there is a similarity in the equations employed.
4.1 Mass Transfer When a mixture of gases or liquids is contained such that a concentration gradient of one or more of the components exists across the system, there will be mass transfer on a microscopic level. This is a result of
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diffusion from regions of high concentration to regions of low concentration. Mass transfer can occur on a molecular basis. In turbulent flow systems, accelerated diffusion rates can occur as a result of rapid-eddy mixing processes just as these mixing processes created increased heat transfer and viscous action in turbulent flow. Consider the system shown in Figure 4.1.1. A thin partition separates two gases—a and b. When the partition is removed, the two gases diffuse through one another until equilibrium is established and the concentration of the gases is uniform throughout the container. The diffusion rate is given by Fick’s law of diffusion, which states that the mass flux of a constituent per unit area is proportional to the concentration gradient. Thus,
m dc =-D A dx
(4.1.1)
where the constant of proportionality, D, is called the diffusion coefficient and is measured in m2/s. The concentration, c, is the mass of a constituent per unit volume. Notice the similarity between Equation 4.1.1, Equation 3.1.1 (Fourier’s law of heat conduction), and Equation 2.1.1 (Newton’s equation of viscosity). The diffusion equation describes the transport of mass, the equation of viscosity describes the transport of momentum, and the conduction equation describes the transport of energy.
Fig. 4.1.1 Diffusion of Components
To understand the physical mechanism of diffusion, consider the imaginary plane shown by the dashed line in Figure 4.1.2.
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Fig. 4.1.2 Sketch Showing Diffusion Dependence on Concentration Profile
The concentration of constituent, b, is greater on the left side of this plane than on the right side. A higher concentration means there are more molecules per unit volume. If the system is a gas or a liquid, the molecules move about in a random fashion, and the higher the concentration, the more molecules will cross a given plane per unit of time. On the average, there are more molecules moving from left to right across the plane than in the opposite direction. This results in a net mass transfer from the region of high concentration to the region of low concentration. The fact that the molecules collide with each other influences the diffusion process strongly. In a mixture of gases, there is a decided difference between a collision of like molecules and a collision of unlike molecules. The collision of like molecules does not appreciably alter the basic molecular movement because it does not make any difference which of the two identical molecules crosses the plane. The collision of two unlike molecules, say molecules a and b, might result in molecule b crossing the plane instead of molecule a. In general, when the colliding molecules have different masses, the mass transfer is influenced by the collision. Using the kinetic theory of gases, it is possible to predict the diffusion rates for some systems by taking into account the collision mechanisms and molecular weights of the constituent gases. In gases, the diffusion rates are dependent on the molecular speed. Since the temperature indicates the average molecular speed, we should expect a dependence of the diffusion coefficient on temperature. Gilliland has proposed a semi-empirical equation for the diffusion coefficients in gases:
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[
]
D = 0.04357 T 1.5(1 / M a +1 / M b) / p (V a0.333+ V b0.333) , m 2 /s 0.5
2
(4.1.2)
where Va and Vb = the molecular volumes of gases a and b Ma and Mb = the molecular masses T = temperature in degrees Kelvin p = the total system pressure For air: Va = 29.9 and Ma = 28.97 For water vapor: Vv = 18.8 and Mv = 18.016.
Equation 4.1.2 offers a convenient expression for calculating the diffusion coefficient for various compounds and mixtures. It should not be used as a substitute for experimental values of the diffusion coefficient when they are available for a particular system. Calculation of momentum and heat transfer rates at a solid-fluid interface by the appropriate rate equation requires a knowledge of the velocity and the temperature profiles within the boundary layer. Mass transfer at an interface is determined by the concentration boundary-layer profile. Figure 4.1.3 shows a fluid mixture flowing over a flat plate with freestream velocity and concentration designated by v∞ and cb∞. If the plate surface is maintained at a concentration cbo > cb∞, mass, mb, diffuses from the surface into the fluid stream. A concentration boundary layer grows from the leading edge in the same way that a velocity or a thermal boundary layer grows. The thickness of the concentration boundary-layer, δc, is defined as the distance from the plate where (cbo – cb) = 0.99 (cbo – cb∞).
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Fig. 4.1.3 Velocity and Concentration Boundary Layer on a Flat Plate
By analogy with heat transfer, a mass transfer coefficient hD is defined by
m b = h D A (cbo - cb∞)
(4.1.3)
A dimensionless mass transfer number, called the Sherwood number, corresponding to the Nusselt number for heat transfer may be defined for a plate as
Sh = hD L/D
(4.1.4a)
Sh = hD de/D
(4.1.4b)
or for a duct as
The relative rates of growth of the velocity and concentration boundary layers are determined by the Schmidt number of the fluid, i.e., Sc = µ /( ρ D) = m /D
(4.1.5)
The Schmidt number is analogous to the Prandtl number in heat transfer.
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If a temperature difference exists between the plate and the free stream in addition to the concentration difference, a thermal boundary layer will grow concurrently with the velocity and the concentration boundary layers. The relative rates of growth of the thermal and concentration boundary layers are determined by the Lewis number.
Le = k/ ( ρ cp D ) = α/D = Sc/Pr
(4.1.6)
The temperature and concentration profiles will coincide when Le = 1. Similarities between the governing equations of heat, mass, and momentum transfer suggest that empirical correlations for mass transfer would be similar to those for momentum and heat transfer.
According to Holman, the analogy for local mass transfer for laminar pipe flow over a smooth flat plate becomes
h Dx Sc 0.667/ v∞ = 0.332 Re -x0.5
(4.1.7)
h Dx Sc 0.667/ v ∞ = 0.0296 Re -x0.2
(4.1.8)
or for turbulent flow,
while for turbulent pipe flow,
h D Sc 0.667/ v m = f D / 8
(4.1.9)
Gilliland presents an equation found in Holman’s 1986 book for the vaporization of liquids into air inside circular columns where the liquid wets the surface and the air is forced through the column.
Sh = h D d/D = 0.023 Re 0.83Sc 0.44 This equation is valid for 2000 < Re < 35,000 and 0.6 < Sc < 2.5.
228
(4.1.10)
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Ranz and Marshall present a semiempirical correlation for the mass transfer coefficient from a single spherical water droplet suspended in an airstream.
Sh = h D d/D = 2 + 0.6 Re 0.5 Sc 0.33
(4.1.11)
for 2 ≤ Re ≤ 800.
When evaluating the mass transfer coefficients for an air-water-vapor mixture, the thermophysical properties of the mixture are determined using the equations listed in appendix A. In air-water-vapor systems, the concentration of the water vapor in the air is often expressed in terms of the humidity ratio, w, which is defined as the ratio of the mass of water vapor per unit mass of dry air. According to the ASHRAE Handbook of Fundamentals and Stoecker, when moist air is considered to be a mixture of independent perfect gases, dry air, and water vapor, each is assumed to obey the perfect gas equation of state, and it follows that
w=
kg of water vapor p v V/ R v T R pv = = kg of dry air pa V/RT R v ( p - p v) 0.622 p v 287.08 pv = = ( p - pv) 461.52 ( p - p v)
(4.1.12)
where pv = partial pressure of the water vapor pa = partial pressure of the air p = pa + pv = total mixture pressure V = volume occupied by the air-vapor mixture
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The gas constant for air is R = 287.08 J/kg K; for water vapor, it is Rv = 461.52 J/kg K. For air saturated with water vapor, the partial pressure of the water vapor can be determined using Equation A.2.1. Equation A.3.5 may also be employed to determine the humidity ratio. If the plate in Figure 4.1.3 were to be replaced by a water surface exposed to an airstream, the mass transfer coefficient would be defined by the following equation:
m w = h d A (w swo - w ∞)
(4.1.13)
At the water-air interface, the humidity ratio corresponding to water saturation conditions is, using Equation 4.1.12,
w swo= 0.622 p vs / ( p - p vs)
(4.1.14)
Equate Equation 4.1.3 and Equation 4.1.13 to find
h d = h D (cbo - cb∞)/ (w swo - w ∞)
(4.1.15)
Note the species or constituent concentration, cb, is equivalent to the partial density of the substance, ρb. Introducing the perfect gas law, Equation 4.1.15, for an air-water-vapor system yields Equation 4.1.16. The subscript b refers to constituent or component b, which is the water vapor in this case.
hd =
hD ( pvso - pv ∞)/ (w swo - w ∞) RvT
(4.1.16)
From Equation 4.1.12, it follows that
p v = pw/ (w + 0.622 )
230
(4.1.17)
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Substitute Equation 4.1.17 into Equation 4.1.16, and find
hd =
w swo w∞ R v T (w swo - w ∞) (w swo + 0.622 ) (w ∞ + 0.622 ) hD p
(4.1.18)
This equation is not only limited to flow over flat plates but can be extended to other geometries where water is exposed to an airstream, i.e.,
hd =
w sw w w w R v T (w sw - ) (w sw + 0.622 ) ( + 0.622 ) hD p
(4.1.19)
Poppe derived the following expression for the mass transfer coefficient under isothermal conditions:
hd =
hD p
R v T (w sw - w )
n [(w sw + 0.622 )/ (w + 0.622 )]
(4.1.20)
It is sometimes convenient to express the mass transfer coefficient in terms of the relative humidity. The latter is defined as the ratio of the mole fraction of water vapor in a given moist air sample to the mole fraction in an air sample saturated at the same temperature and pressure from the ASHRAE Handbook of Fundamentals. For perfect gas relationships, another expression for the relative humidity is
ϕ = p v / p vs
(4.1.21)
Using Equation 4.1.21 and Equation 4.1.19, according to Hayashi, the following can be shown for isothermal conditions,
hd =
h D p p vs ϕ p vs 1 - 1 R v T p p
(4.1.22)
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The relationship between the humidity ratio and the relative humidity is, using Equations 4.1.12 and 4.1.21,
w = 0.622 ϕ pvs/(p - ϕ pvs)
(4.1.23)
Psychrometric properties of air-water-vapor mixtures can be presented either mathematically as equations or graphically. Psychrometric charts are a useful and widely accepted tool for design and analysis of processes of heat and mass exchange involving moist air according to Stoecker. Unfortunately, no universal chart exists that is suitable for all applications. This is because:
•
A given chart is only valid for the particular barometric pressure for which it was drawn.
•
Depending on the application, different ranges of temperature and moisture content of the air may be required to maintain the necessary clarity and accuracy of the chart according to Johannsen.
The conventional psychrometric chart has a vertical co-ordinate for humidity ratio and an inclined coordinate for enthalpy—both with linear scales (Fig. 4.1.4). The abscissa shows the drybulb temperature.
Fig. 4.1.4 Psychrometric Chart Coordinates
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The enthalpy of a mixture of dry air and water vapor is the sum of the enthalpy of the dry air and the enthalpy of the water vapor. Enthalpy values are always based on some reference value, and the zero value of dry air is usually chosen as air at 0 °C. The zero value of water vapor is saturated liquid water at 0 °C, the same reference value that is used for steam. An equation for the approximate enthalpy of a mixture of air and water vapor is ima = cpaT + w(ifgwo + cpvT), J/kg dry air
(4.1.24)
where the specific heats are evaluated at T/2 °C, and the latent heat of vaporization, ifgwo, is obtained from Equation A.4.5 at 0 °C. This equation is listed in appendix A as Equation A.3.6(b) together with Equation A.3.6(a), which expresses the enthalpy per kg of air-water-vapor mixture. In certain processes, it is useful to employ the equation based on dry air since the latter does not change. If the mass of the mixture were used as the basis, an iterative method of solving the problem would usually be required since the mass of the mixture changes during the process. It is evident from the previous equation for enthalpy that the isotherms will not be parallel lines. The maximum temperature isotherm is usually chosen as vertical. The saturation curve is obtained with the aid of Equation A.2.1. Lines of constant relative humidity are found with the aid of Equations 4.1.21 and A.2.1. Lines of constant wetbulb temperature are determined by means of Equation A.3.5. A detailed psychrometric chart for moist air at a barometric pressure of 101,325 N/m2 is shown in Figure 4.1.5.
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Fig. 4.1.5 Psychrometric Chart
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Example 4.1.1 Determine the relative humidity for ambient air with a drybulb temperature Ta= 288.6 K and a wetbulb temperature Twb = 284.2 K at an atmospheric pressure of pa= 84100 N/m2.
Solution The pressure of saturated water vapor at Twb = 284.2 K may be determined using Equation A.2.1 where
z = 10.79586(1 - 273.16 / 284.2) + 5.02808 og10 (273.16 / 284.2)
+ 1.50474 x 10 -4[1 - 10 -8.29692{(284.2/273.16)-1}]
+ 4.2873 x 10 -4[10 4.76955(1-273.16/284.2)-1] + 2.786118312 = 3.119284 to give
p vwb= 103.119284 = 1316.0854 N/ m2 The humidity ratio follows from Equation A.3.5). 0.62509 p vwb 2501.6 - 2.3263(Twb - 273.15 ) w = 2501.6 +1.8577(Ta - 273.15)- 4.184(Twb - 273.15) pa - 1.005 vwb 1.00416 (Ta - Twb) _ 2501.6 +1.8577 (Ta - 273.15)- 4.184(Twb - 273.15 ) 2501.6 - 2.3263(284.2 - 273.15) = 2501.6 + 1.8577(288.6 - 273.15) - 4.184(284.2 - 273.15) 0.62509 x 1316.0854 x 84100 - 1.005 x 1316.0854 1.00416(288.6 - 284.2 ) - = 0.008127 kg/kg ( ) ( ) 2501.6 + 1.8577 288.6 273.15 4.184 284.2 273.15
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The saturation pressure of water vapor at Ta = 288.6 K follows from Equation A.2.1) where
z = 10.79586(1 - 273.16 / 288.6 )+ 5.02808 og10 (273.16 / 288.6) = 1.50474 x 10 -4 [1 - 10 -8.29692{(288.6/273.16)-1}] + 4.2873 x 10 -4 [10 4.76955(1-273.16/288.6) -1]+ 2.786118312 = 3.24407 to give
p vs = 103.24407= 1754.157 N/ m 2 From Equation 4.1.23, find
ϕ=
pa w 84100 x 0.008127 = = 0.61834 (0.622 + w) p vs (0.622 + 0.008127)1754.157.157
or the relative humidity is 61.834%.
4.2 Heat and Mass Transfer in Wet-Cooling Towers Consider an elementary control volume in the fill or packing of a counterflow wet-cooling tower (Fig. 4.2.1). Evaporation of the downward flowing water occurs at the air-water interface where the air is saturated with water vapor. The vapor subsequently diffuses into the free stream air, which has a lesser vapor concentration.
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Fig. 4.2.1 Control Volume for Derivation of Governing Equations for Counterflow Fill
It will be assumed that the interface water temperature, Ts, is the same as the bulk water temperature, Tw. The effect of this assumption on the transfer process has been investigated by a number of researchers including Baker, Webb, and Marseille. Air and water properties at any horizontal cross section are assumed to be constant, and the area dA for heat and mass transfer is identical.
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A mass balance for the control volume yields,
dm dw m a (1 + w) + m w + w dz = m a 1 + w + dz + m w dz dz or
dm w dw = ma dz dz
(4.2.1)
where ma = mass flow rate of the air constituent
An energy balance for the control volume yields
dm dT di m a ima + m w + w dz cpw T w + w dz = m a ima + ma dz + m w cpw T w (4.2.2) dz dz dz
where Tw is in °C.
Neglecting second order terms, Equation 4.2.2 simplifies to
m w cpw
dT w dm w di + cpw T w = m a ma dz dz dz
(4.2.3)
where ima refers to the enthalpy of the air-water vapor mixture per unit mass of dry air, which, using Equation A.3.6b, is expressed as
ima = cpa T a + w (ifgwo+ cpv T a) and where ifgwo is evaluated at 0 °C and cpa and cpv at Ta/2 °C.
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(4.2.4)
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Substitute Equation 4.2.1 into Equation 4.2.3 to find
dT w m a 1 dima dw = - Tw dz m w cpw dz dz
(4.2.5)
The total enthalpy transfer at the air-water interface consists of an enthalpy transfer associated with the mass transfer due to the difference in vapor concentration and the heat transfer due to the difference in temperature. Accordingly, one has dQ = dQm + dQc
(4.2.6)
where the subscripts m and c refer to the enthalpies associated with mass transfer and convective heat transfer. The mass transfer at the interface is expressed by
dm w dz = h d (w sw - w)dA dz
(4.2.7)
where wsw is the saturation humidity ratio of air evaluated at the local bulk water temperature, Tw. The enthalpy transfer is
dQm = iv
dm w dz = i v h d ( w sw - w) dA dz
(4.2.8)
The enthalpy of the water vapor, iv, at the bulk water temperature, Tw, is given by iv = ifgwo + cpv Tw where Tw is in °C, and cpv is evaluated at Tw/2 °C.
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The convective transfer of sensible heat at the interface is given by
dQc = h(T w - T a) dA
(4.2.9)
The enthalpy of the saturated air evaluated at the local bulk water temperature is given by imasw = cpa Tw + wsw (ifgwo + cpv Tw) = cpa Tw + wsw iv which may be rewritten as
imasw = cpa T w + wi v + (w sw - w) iv
(4.2.10)
where cpa is evaluated at Tw/2 °C. Subtract Equation 4.2.4 from Equation 4.2.10. The resultant equation can be simplified if the small differences in the specific heats, which are evaluated at different temperatures, are ignored, i.e.,
imasw - ima ≈ (cpa + w c pv) (T w - T a) + (w sw - w) i v or
T w - T a = [ (imasw - ima)- (w sw - w )iv ]/ cpma
(4.2.11)
where cpma = cpa + w cpv,
Substitute Equations 4.2.8, 4.2.9, and 4.2.11 into Equation 4.2.6 to find upon re-arrangement
h dQ = h d (imasw - ima)+ 1 - h iv (w sw - w ) dA cpma h d cpma h d
(4.2.12)
where h/(cpma hd) = Lef, which is known as the Lewis factor and is an indication of the relative rates of heat and mass transfer in an evaporative process.
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Bosnjakovic proposed the following equation to express the Lewis factor for air-water vapor systems:
w + 0.622 w + 0.622 L ef = 0. 8660.667 sw - 1 / n sw w + 0.622 w + 0.622
(4.2.13)
Noting the enthalpy transfer must be equal to the enthalpy change of the airstream, one has from Equation 4.2.12
dima 1 dQ h d dA = = [ Lef (imasw - i ma) + (1 - Lef ) iv ( w sw - w) ] dz m a dz m a dz
(4.2.14)
For a one-dimensional model of the cooling tower fill where the available area for heat and mass transfer is the same at any horizontal section through the fill, the transfer area for a section dz deep is usually expressed as dA = afi Afr dz
(4.2.15)
where afi = the wetted area divided by the volume of the fill or area density Afr = the frontal area or face area
Substitute Equation 4.2.15 into Equation 4.2.14, and find
di ma h d afi A fr = [Lef (imasw - ima)+ (1 - Lef )iv (w sw - w )] dz ma
(4.2.16)
When the ambient humidity is high enough, the air becomes saturated with water vapor prior to its exit from the fill. In this case, the previous equations fail to describe the evaporative process in the fill. Since the temperature of the saturated air at the interface is still higher than the temperature of the now saturated free stream air, a potential for heat and mass transfer will still exist. The excess water vapor transferred to the free stream air will condense as a mist.
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Assume that the heat and mass transfer coefficients for the mist zone is the same as those for unsaturated air as is proposed by Bourillot and Poppe and Rögener. The evaporation rate in the mist zone depends on the difference in moisture content of the saturated air at the interface, at the local bulk water temperature, and the moisture content of the free stream air, thus
dm w = h d afi A fr [w sw - w sa ] dz
(4.2.17)
where wsa is the humidity ratio of saturated air at temperature Ta. Since the excess water vapor will condense, the enthalpy of supersaturated air is expressed by
i ss = cpa T a + w sa (ifgwo + cpv T a) + (w - w sa) cpw T a
(4.2.18)
Proceeding along the same lines as in the case of unsaturated air, find
di ma h d afi A fr [Lef (imasw - iss) + (1 - Lef ) iv ( wsw - wsa )+Lef (w - wsa ) cpw Tw] (4.2.19) = dz ma
In addition to the assumption stated earlier, Merkel assumes the Lewis factor is equal to unity and the evaporation loss is negligible. Introducing these two assumptions, the governing Equations 4.2.16 and 4.2.5 simplify to
di ma h d afi A fr = (imasw - ima) dz ma
(4.2.20)
dT w m a 1 dima = dz m w cpw dz
(4.2.21)
and
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With only the previous equations, it is impossible to calculate the state of the air leaving the fill, since at least two properties must be known in order to achieve this. Hence, the exit air temperature, essential to calculating the airflow rate through a natural draft tower, is unknown. Merkel assumes that the air leaving the fill is saturated with water vapor, which enables him to determine the temperature and density of the air and the draft. In many practical cases, this assumption will yield reasonable results. Traditionally, Equations 4.2.20 and 4.2.21 are combined to yield upon integration,
h d A h d afi A fr L fi h d afi L fi = = = Gw mw mw
Twi
cpw dT w = Me ( i -i ) wo masw ma
∫ T
(4.2.22)
commonly referred to as Merkel’s equation. The nondimensional coefficient of performance or transfer characteristic hdafiLfi/Gw, is known as the Merkel number. In this equation, Lfi is the height of the fill or the air travel distance (ATD) and Gw = mw/Afr. In the literature, the notation frequently used for the Merkel number is KaV/L where K = hd a = afi V = AfrLfi L = mw The implications of some of the approximations made in deducing Equation 4.2.22 are evaluated by Lefevre and Ibrahim et al. More detailed models are due to Poppe, Bourillot, Sutherland, Webb, Feltzin and Benton, and Poppe and Rögener. Poppe and Rögener do not make the simplifying assumptions of Merkel. Their more rigorous approach is as follows:
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Substitute Equations 4.2.7 and 4.2.14 into Equation 4.2.3, and find with cpwdTw = diw
m w cpw d T w = m w d i w = h d dA [Le f (i masw- ima) + (1 - Le f ) i v ( w sw - w) - cpw T w ( w sw - w) ]
(4.2.23)
Equation 4.2.5 can be rearranged to give
dima di ma dw mw mw = = dT w T w cpw dT w m a T w T w di w m a T w
(4.2.24)
Substitute Equations 4.2.14 and 4.2.23 into Equation 4.2.24, and find upon rearrangement dw cpw m w ( w sw - w) / m a (4.2.25) = ( ) + ( 1)[( d T w imasw ima Le f imasw- i ma) - (w sw - w) i v] - ( w sw - w) cpw T w
Upon substitution of Equation 4.2.25 into Equation 4.2.24, find dima m = cpw w dT w ma cpw T w ( w sw - w) x 1 + (imasw- ima) + (Le f - 1) [(imasw- ima) - ( w sw - w) i v] - (w sw - w) cpw T w
(4.2.26)
Combine Equations 4.2.1. and 4.2.7 to find
h d dA =
m a dw ( w sw - w)
(4.2.27)
Divide both sides of Equation 4.2.27 by mw, introduce dTw/dTw to the right side of Equation 4.2.27, and integrate to find the Merkel number according to Poppe, i.e.,
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hd A = ∫ m a dw /dT w dT w = MeP w mw w m w sw
(4.2.28)
Upon substitution of Equation 4.2.25 into Equation 4.2.28 and differentiation of the latter with respect to water temperature, find
dMeP cpw = dTw (imasw- ima) + (L ef - 1)[(imasw- ima) - ( wsw - w) iv ] - ( wsw - w) cpw Tw
(4.2.29)
To evaluate the change in the ratio of mw /ma as the air flows upward through the fill, consider the elementary control volume in the fill (Fig. 4.2.2).
Fig. 4.2.2. Control Volume in Fill
A mass balance in terms of the inlet water mass flow rate, mwi, applicable to the control volume is
mwi = mw + ma ( w o - w)
(4.2.30)
Upon re-arrangement of Equation 4.2.30,
m w m wi 1 - m a ( w - w) o = m a m a m wi
(4.2.31)
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The air outlet conditions can now be obtained from Equations 4.2.13, 4.2.25, 4.2.26, and 4.2.31 in terms of enthalpy and humidity ratio. The previous equations are applicable if the air is unsaturated. For supersaturated air, the enthalpy iss, as given by Equation 4.2.18, is employed in the calculations instead of ima. Equation 4.2.7 is not valid in the case of supersaturated air. The mass transfer at the interface of the water and supersaturated air is given by
dmw dz = hd (wsw - w sa) dA dz
(4.2.32)
The Bosnjakovic equation for the Lewis factor for supersaturated air is expressed as
w + 0.622 w + 0.622 Lef = 0. 8660.667 sw - 1 / n sw w + 0.622 w sa + 0.622 sa
(4.2.33)
By following the same procedures as for unsaturated air but employing Equations 4.2.18, 4.2.32, and 4.2.33, find for supersaturated air
dw/ dT w = cpw m w (w sw - w sa) / m a /[(imasw - iss) + (L ef - 1){(imasw - iss) - ( w sw - w sa) i v + (w - w sa) cpw T w}
(4.2.34)
+ (w - w sw) cpw T w ]
The enthalpy gradient is given by
di ma / dT w = (cpw m w / m a)[1 + cpw T w ( w sw - w sa) /{(imasw - iss) + (L ef - 1){(imasw - iss) - ( w sw - w sa) i v + (w - w sa) cpw T w} (4.2.35) + (w - w sw) cpw T w}]
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while
dMep / dTw = cpw / [(i masw - iss) + (L ef - 1){(imasw - iss) - ( wsw - wsa) i v + (w - w sa) cpw Tw } (4.2.36) + (w - w sw ) cpw T w ] The outlet air conditions in terms of enthalpy and humidity ratio can now be determined using Equations 4.2.31, 4.2.33, 4.2.34, and 4.2.35.
Example 4.2.1 A performance test on a counterflow cooling tower fill material (water flows downward and air flows upward) shows that Twi = 39.67 °C (312.82 K) for an inlet water temperature Two = 27.77 °C (300.92 K) the outlet water temperature The enthalpies of saturated air corresponding to these temperatures are imaswi = 163546.9337 J/kg dry air imaswo = 88710.7054 J/kg dry air Based on measured air inlet conditions, the air enthalpy is imai = 25291.87496 J/kg dry air while the air outlet enthalpy on top of the fill is imao = 73379.5976 J/kg dry air The specific heat of water at a mean temperature of (Twi + Two)/2 = (39.67 + 27.77)/2 = 33.72 °C (306.87 K) is cpwm = 4177.40244 J/kg
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Determine the approximate Merkel number for this fill using Equation 4.2.22. It may be assumed that 1/(imasw - ima) is linear.
Solution Using Equation 4.2.22, T wi
Me =
T wo
=
cpwm (T wi - T wo ) c pw d T w 1 1 + ≈ ( i i ) 2 ( i i ) masw - i ma ) maswi mao maswo mai
∫ (i
4177.40244(39.67 - 27.77) 2
+
1 (88710.7054 - 25291.87496)
1 = 0.6676 (163546.9337 - 73379.5976)
Note As shown in Figure 4.2.3, the relation 1/(imasw - ima) is nonlinear between the temperature limits of integration, i.e., Twi = 312.82 K and Two = 300.92 K.
Fig. 4.2.3 Enthalpy of Air
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The value obtained for the Merkel number in this example is only an approximation. It will be shown in Example 4.3.1 that an improved method of integration will give a better value for the Merkel number of 0.68468.
4.3 Fills or Packs Cooling tower fills or packs have been developed from the simple timber or bamboo splash bar to the modern vacuum formed or injection molded plastic fills now in common use according to Mirsky. Fills should be structurally strong, chemically inactive, fire resistant, resistant to fouling and erosion, and have a low airflow resistance. A critical comparative study is presented by Monjoie. His main objective is to identify the deficiencies of seven different plastic fill materials. Among others, properties concerning forming, assembly, fire, chemical, thermal, recycling, and environmental impact are evaluated. Examples of some plastic fills are shown in Figure 4.3.1.
Fig. 4.3.1 Plastic Fills and Spray Nozzles (1) Film (2) Trickle Grid (3) Film (4) Splash (5) Spray Nozzles
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Further examples of fills are shown in Figure 4.3.2 from Lowe and Figure 4.3.3 from Johnson. Many other configurations find application in practice according to Dumitru. PVC can be used as fill material up to a water temperature of about 50 °C, chlorinated PVC (CPVC) up to 65 °C, while Burger finds higher temperatures require polypropylene or stainless steel.
Fig. 4.3.2 Fills (a) (b) and (h) Splash, (c) (d) (e) (f) (g) and (i) Film
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Fig. 4.3.3 (a) Fills
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Fig. 4.3.3 (b) Fills
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In some applications, fill materials that can be recycled may be preferred. Many of the deficiencies of plastic fills may be avoided by employing stainless steel fills, an example is shown in Figure 4.3.4.
Figure 4.3.4 Expanded Metal Fill
The water distribution system for a counterflow wet-cooling tower usually consists of a piping manifold that serves to support and supply an array of low pressure spray nozzles. A few examples are shown in Figure 4.3.1. In large natural draft cooling towers, nozzles require relatively low pressures of between 5000 to 15,000 N/m2. Medium pressure nozzles requiring 20,000 to 100,000 N/m2 and producing smaller droplets are employed in mechanical draft industrial cooling towers according to Thacker. A most comprehensive source of nozzle information is presented by Lefebvre. Nozzles should be arranged so the distribution of water entering the fill is as uniform as possible. Kranc found non-uniformity of flow occurred where nozzles produced circular spray patterns with radial variation overlap patterns of the sprays from adjacent nozzles. He also found non-uniform flow of water may be corrected partially in certain fills. Improved distribution may also be achieved by employing nozzles that give almost square spray patterns. In a counterflow cooling tower, the entire cooling region including the spray, fill, and rain zones are influenced by the characteristics of the droplets introduced through the spray nozzles. Up to 15% of the cooling may occur in the spray zone above the fill. The spray may
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be directed downward or upward. In the case of the latter, the longer droplet residence time improves the transfer process in the spray zone. The spray produced in a cooling tower depends on the type of nozzle employed according to both Scriven and Bellagamba. The lightest drops (less than about 0.3 mm in diameter) are carried upward by the air to the droplet eliminators where most are collected and returned downward to the fill in the form of larger drops.
Splash fill. The splash type fill or pack is designed to break the mass of water falling through the cooling tower into a large number of drops. The water surface area exposed to cooling air increases as well as the amount of heat transferred to the surrounding air by conduction, convection, radiation, and evaporation. As water falls through the fill, droplets collide with successive layers of splash bars, which cause redistribution of water and heat due to the formation of fresh droplets. As a further benefit, the retention time of water falling through the tower is prolonged by contact with the fill, extending the period during which the water is exposed to cooling air. The disadvantage of splash fill is, by its very nature, a large volume is required to break up the water flow, which in turn necessitates large towers. There is a natural tendency for free falling droplets to agglomerate. Splash bars are arranged in layers some 200 to 600 mm apart, frequently resulting in fill heights of perhaps 5 to 8 m in the largest towers. The effectiveness of a particular splash type fill is governed by its ability to form droplets. Its efficiency also depends upon its airside flow resistance and, to a lesser extent, on economical use of material. Treated timber is used due to its availability, structural strength, relative cheapness, and long working life under most conditions. Injection molded plastic fills have been in service for many years. Splash fills tend to produce more carryover than other types of fill, particularly if high air rates are used. Efficient spray eliminators are employed to overcome this potential disadvantage. However, the increased air resistance of the complete tower demands additional draft or fan capacity and additional running costs. The inherent disadvantages of splash fills have created a demand for the development of film type fills, which are more compact and preferred. The latter require less material and water pumping power due to the lower fill height.
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Trickle pack. Trickle packs or grids are much finer than splash packs and are made up of plastic or metal grids onto which the water is sprayed. It runs down the grid rather than splashing. This type of fill has been introduced in recent years with the advances in plastic injection molding. Because of the much finer mesh than the splash type fill, they tend to clog more easily and have a greater pressure drop.
Film fill. Although the purpose is to produce a large water surface area, film fills are different from splash fills because this is achieved by allowing the water to spread in a thin layer over a large area of fill rather than forming droplets. This reduces the problem of carryover of water droplets into the atmosphere and allows higher air velocities to be used. Fill types may be placed in several categories, the simplest is the timber grid. This consists of a series of closely spaced slats placed in tightly packed layers, each layer at right angles to the previous layer. This arrangement provides good water distribution over a large area, but air resistance is high. Thin timber sections have low structural strength and limited resistance to chemical attack and distortion. Corrugated or flat asbestos sheeting was used in the past, but resin impregnated cardboard, metal, and more effective plastics are preferred now. A theoretical examination by Kelly suggests a fill consisting of a series of close parallel vertical film surfaces would give good transfer with low pressure drop. Various manufacturers have designed packs along these lines but found in practice they were not reliable. The packs have a tendency towards uneven water distribution, which reduces heat transfer effectiveness. Fouling is more of a problem in this fill than in the splash type.
Extended film fills. Although problems were encountered with early thermoplastic film type fills, these difficulties have been largely overcome. There are now a wide variety of pressed or vacuum formed fills available. These vary in design but have high transfer characteristics, low weight, acceptable strength, and adequate durability. The problems of water and air distribution have been reduced by the development of geometrical designs, which incorporate interconnected channels and secondary profiles. Both
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these features improve water and air distribution and encourage greater mixing of the layer of saturated air. The air layer forms adjacent to both the water layer and the bulk of air traveling through the fill, and it further improves performance. A large variety of proprietary cooling tower fills are produced by commercial manufacturers. The results of studies on the performance characteristics of fills have been reported in the literature by authors such as Lowe, the Cooling Tower Institute (CTI), Kelly, Cale, Fulkerson, and Johnson. Some of the more recent studies reported by Johnson have evaluated test facilities and methods of data evaluation. Thermal and pressure drop data obtained in one facility does not always agree with that obtained in another facility. Some of the reasons for these discrepancies are:
•
distorted flow patterns
•
test facility edge effects
•
influences due to the type of spray nozzle and spray or rain zone according to Fulkerson
•
different test temperatures and pressures
•
changes in fill wetting patterns (the degree of wetting of the fill surface may change with time)
•
errors in measurement
In a large modern test facility, the cross section of the fill may have dimensions up to 7 m x 7 m for counterflow and 5 m x 10 m for crossflow and most of the previously mentioned problems can be greatly reduced according to Fabre and Caytan. It is important to elaborate on the method employed in evaluating the test data, i.e., Merkel, Poppe, or others. Where the Merkel method is employed, it is convenient to present the transfer and pressure drop characteristics per meter of fill depth as follows: -b d
h d afi A fr / m w = h d afi / G w = ad (G w / G a)
(4.3.1)
and
K fi1 = ap (G w / G a)+ bp
256
(4.3.2)
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where mean mass flow rates through the fill are given by Gw = mw/Afr and Ga = ma/Afr. The subscript 1 refers to one meter height of the fill or air travel distance. In fill performance characteristics presented in the literature, some correlations may be expressed in terms of Gav based on inlet or mean conditions through the fill. If not clearly defined, this may lead to errors, especially when evaluating the draft equation in the case of a natural draft wet-cooling tower. Other forms for approximating the previous characteristics empirically are as follows
h d afi / G w = ad G bwdG cad
(4.3.3)
-b
(4.3.4)
and
h d afi / G w = ad (G w / G a ) d L cfid or b
b
K fi1 = ap Gw paG a pb
(4.3.5)
where the values of ad, ap, bd, and bp are determined experimentally in each case. Obviously, these simple correlations cannot consider all variables, resulting in considerable scatter of test data which may lead to less reliable cooling system designs. Ideally, fill performance tests for a particular cooling tower should be conducted under conditions similar to those specified for the tower design operating point according to Kloppers. The performance characteristics of a few fills are listed in Table 4.3.1 from Lowe.
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Table 4.3.1 Data for Counterflow Fills (Merkel’s Theory)
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Table 4.3.1 Data for Counterflow Fills (Merkel’s Theory) - continued
A more recent publication on counterflow and crossflow fill performance data is presented by Johnson et al. Four methods or codes for determining the transfer coefficients were evaluated, i.e., FACTS assessed by Benton, TEFERI by Bourillot, VERA 2D by Majumdar, and ESC by Zivi and by Baker. They note the value of the mass transfer coefficient calculated by these methods varied as much as 10% from each other. They also stress the importance of using the same method to predict cooling tower performance as was used to derive the characteristics of a particular fill.
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The publication includes: •
complete data listing of the calculated mass transfer coefficients
•
resulting correlations and confidence limits associated with the correlations
•
statistical summary of the capability of the codes to predict large scale cooling tower performance
Some of the results are shown in Tables 4.3.2 (a) and 4.3.2 (b) taken from Johnson. Dreyer presents a mathematical model to predict the performance characteristics of splash fill material and lists extensive experimental performance data.
Table 4.3.2 (a): Data for Crossflow Fills (Merkel’s Theory)
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Table 4.3.2 (b) Data for Counterflow Fills (Merkel’s Theory)
Combinations of different types of fill may be installed in a cooling tower to achieve a desired performance or to enhance the performance and reduce fouling in an existing tower according to both Gösi and Phelps. Fouling of fills due to biological growths can reduce the performance of the fill when installed in a cooling tower according to Aull, Combaz, Gill, Monjoie, Newton, and Puckorius. These include:
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•
algae and bacteria
•
colloidal materials transported in the recirculating water
•
airborne dirt or particles
•
silt or suspended solids in the make-up water
•
scaling due to dissolved materials carried in solution
Monjoie, Mortensen report certain fills tend to be more susceptible to fouling than others. When selecting a particular fill for a cooling system, it is important not only to consider initial performance characteristics and cost but also the long term structural performance and fouling characteristics. These can have significant cost implications on plant performance or output.
Example 4.3.1 The performance characteristics of an expanded metal (stainless steel) fill of the type shown in Figure 4.3.4 are determined during counterflow in a large insulated test facility. To minimize edge effects, the actual measurements are limited to a section of the fill having a frontal area of 1.5 m x 1.5 m in the middle of the test section. Hot water is sprayed uniformly across the fill and flows downward while air is forced to flow uniformly upward through the Lfi = 1.878 m high fill. During the particular test the following measurements are made: Atmospheric pressure
262
pa = 101712.27 N/m2
Air inlet temperature
Tai = 9.7 °C (282.85 K)
Air inlet temperature (wetbulb)
Twb = 8.23 °C (281.38 K)
Dry air mass flow rate
ma = 4.134 kg/s
Static pressure drop across fill
∆pfi = 4.5 N/m2
Water inlet temperature
Twi = 39.67 °C (312.82 K)
Water outlet temperature
Two = 27.77 °C (300.92 K)
Water mass flow rate (in)
mw = 3.999 kg/s
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Determine the Merkel number and loss coefficient per unit height of the fill. It may be assumed that mw ≈ mwm.
Solution Different numerical integration methods may be considered to approximate the Merkel integral Equation 4.2.22. These methods vary in both accuracy and computational effort. One of the more frequently used approaches is from the mathematician Chebyshev according to Fröberg, Cale, and British Standard 4485. Chebyshev’s method uses values of the integrand at predetermined values within the integration interval, selected so that the sum of these values multiplied by the interval times a constant gives the desired approximate integral. In its four-point form, the approximate formula is b
∫a f (x) dx
≈
(b- a) [ f ( x1) + f ( x 2) + f ( x 3) + f ( x 4)] 4
The values of f(x) have to be evaluated at values of x, which are 0.102673, 0.406204, 0.593796, and 0.897327 of the interval (b-a). These values can be rounded off to the nearest tenth and still give good results. Thus f(x1), f(x2), f(x3), and f(x4) have to be evaluated at the following values of x: f(x1) = value of f(x) at x = a + 0.1 (b-a) f(x2) = value of f(x) at x = a + 0.4 (b-a) f(x3) = value of f(x) at x = a + 0.6 (b-a) f(x4) = value of f(x) at x = a + 0.9 (b-a) Application of the Chebyshev method to the Merkel equation gives: T
h d afi A fr L fi h d afi L fi wi cpw dT w = = ∫ mw Gw T i masw - i ma wo
≈
≈
(T wi - T wo) cpw1 + cpw2 + cpw3 + cpw4 4
∆ i(1)
∆ i(2)
∆ i(3)
∆ i(4)
cpwm(T wi - T wo) 1 1 1 1 + + + 4 ∆ i(1) ∆ i(2) ∆ i(3) ∆ i(4)
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In this equation, the specific heat of the water can be evaluated using Equation A.4.2 at the mean water temperature, i.e.,
(Twi + Two)/2 = (312.82 + 300.92)/2 = 306.87 K cpwm = 8.15599 x 103 - 2.80627 x 10 x 306.87 + 5.11283 x 10-2 x 306.872 - 2.17582 x 10-13 x 306.876 = 4177.40244 J/kg The enthalpy differentials, ∆i, are dependent on the following intermediate temperatures: Tw(1) = Two + 0.1(Twi - Two) = 27.77 + 0.1(39.67 - 27.77) = 28.96 °C Tw(2) = Two + 0.4(Twi - Two) = 27.77 + 0.4(39.67 - 27.77) = 32.53 °C Tw(3) = Two + 0.6(Twi - Two) = 27.77 + 0.6(39.67 - 27.77) = 34.91 °C Tw(4) = Two + 0.9(Twi - Two) = 27.77 + 0.9(39.67 - 27.77) = 38.48 °C The bracketed subscript numbers refer to the intervals in the Chebyshev integral. To find the enthalpy per unit mass of the entering dry air given by Equation A.3.6b, certain thermophysical properties have to be evaluated at (Tai + 273.15)/2 = (282.85 + 273.15)/2 = 278 K: Specific heat of dry air from Equation A.1.2 cpai = 1045.356 - 3.161783 x 10-1 x 278 + 7.083814 x 10-4(278)2 - 2.705209 x 10-7(278)3 = 1006.39285 J/kg K Specific heat of water vapor from Equation A.2.2 cpvi = 1360.5 + 2.31334 x 278 - 2.46784 x 10-10 x (278)5 + 5.91332 x 10-13(278)6 = 1866.79847 J/kg K Pressure of water vapor from Equation A.2.1 evaluated at Twb = 281.38 K with
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z = 10.79586 (1 - 273.16 / 281.38) + 5.02808 log 10(273.16 / 281.38) + 1.50474 x 10 -4[1 - 10 -8.29692(281.38/273.16-1)] + 4.2873 x 10 -4[10 4.76955(1-273.16/281.38)-1]+ 2.786118312 = 3.036985
to give pvwbi = 103.036985 = 1088.89333 N/m2 Humidity ratio from Equation A.3.5
wi =
2501.6 - 2.3263 (281.38 - 273.15) 2501.6 + 1.8577 (282.85 - 273.15) - 4.184 (281.38 - 273.15) x
-
0.62509 x 1088.89333 101712.27 - 1.005 x 1088.89333
1.00416 (282.85 - 281.38) = 0.00616336 kg/kg 2501.6 + 1.8577(282.85 - 273.15) - 4.184 (281.38 - 273.15)
Latent heat at 273.15 K follows from Equation A.4.5
ifgwo = 3.4831814 x 106 - 5.8627703 x 103 x 273.15 + 12.139568 x 273.152 - 1.40290431 x 10-2 x 273.153 = 2501598.53 J/kg
Upon substitution of the previous properties into Equation A.3.6b, find the enthalpy of the entering air.
imai = 1006.39285(282.85 - 273.15) + 0.00616336[2501598.53 + 1866.79847(282.85 - 273.15)] = 25291.87496 J/kg dry air To find the enthalpy of saturated air at the local bulk water temperature Tw(1) = 28.96 °C (302.11 K), the following properties must be determined at (Tw(1) + 273.15)/2 = (302.11 + 273.15)/2 = 287.63 K:
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Specific heat of dry air from Equation A.1.2 cpa(1) = 1045.356 - 3.161783 x 10-1 x 287.63 + 7.083814 x 10-4(287.63)2 - 2.705209 x 10-7(287.63)3 = 1006.5815 J/kg K
Specific heat of water vapor from Equation A.2.2 cpv(1) = 1360.5 + 2.31334 x 287.63 - 2.46784 x 10-10 x (287.63)5 + 5.91332 x 10-13 x (287.63)6 = 1874.8913 J/kg K Pressure of water vapor at 302.11 K from Equation A.2.1 with
z = 10.79586 (1 - 273.16 / 302.11) + 5.02808 log10 (273.16 / 302.11) + 1.50474 x 10 -4[1 - 10 -8.29692(302.11/273.16-1) ] + 4.2873 x 10 -4x [10 4.76955(1-273.16/302.11) -1]+ 2.786118312 = 3.6016042 to give pv(1) = 103.6016042 = 3995.8041 N/m2 Humidity ratio for saturated air at 302.11 K from Equation A.3.5 ws(1) = 0.62509 x 3995.8041/(101712.27 - 1.005 x 3995.8041) = 0.0255663 kg/kg dry air The enthalpy of saturated air evaluated at the local bulk water temperature Tw(1) = 302.11 K follows from Equation A.3.6b imasw(1) = 1006.5815 x (302.11 - 273.15) + 0.0255663 [2501598.53 + 1874.8913 (302.11 - 273.15)] = 94495.37386 J/kg dry air Using Equation 4.2.21, the enthalpy of the air at the location where Tw(1) = 302.11 K is ima(1) = mwcpwm (Tw(1) - Two)/ma + imai = 3.999 x 4177.40244 x (28.96 - 27.77)/4.134 + 25291.87496 = 30100.6472 J/kg dry air
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With the previous enthalpies, find Di(1) = (imasw(1) - ima(1)) = 94495.37386 - 30100.6472 = 64394.727 J/kg dry air The values of the other differentials are found to be ∆i(2) = 69298.4667 J/kg dry air ∆i(3) = 74438.2259 J/kg dry air ∆i(4) = 85477.1871 J/kg dry air With these values, the Merkel equation becomes
h d afi L fi 4177.40244 (39.67 - 27.77) ≈ Gw 4 1 1 1 1 x + + + = 0.68468 64394.727 69298.4667 74438.2259 85477.1871 Per unit height of fill with Lfi = 1.878 m, find
h d afi 0.68468 = = 0.3646 m-1 Gw 1.878 at an air flux of Ga = ma/(1.5 x 1.5) = 4.134/2.25 = 1.837 kg/sm2. The measured static pressure drop across the fill is due to frictional and form drag resistance in addition to the acceleration of the air due to heating and mass transfer. The buoyancy due to the difference in density of the air in the fill and that in the manometer tube external to the test section will tend to counteract these effects. 2 ) - ( ρ ava - ρ avm)gL fi ∆ pfi = ∆ pfd + (ρ avo v 2avo - ρ avi v avi
where subscript
fd
refers to frictional and drag effects
ρava
= the density of the ambient air, in this case it is equal to the density of the air entering the fill, i.e., ρavi
density of the air leaving the fill = ρavo mean density ρavm
= 2/(1/ρavi + 1/ρavo)
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The second term on the right side of the equation represents the momentum change experienced by the airstream while the third term considers buoyancy effects. This equation assumes the porosity of the particular fill, defined as the ratio of the free flow area at a cross section to the frontal cross-sectional area of the film pack, is unity. For fills having a porosity of less than unity, this must be considered in the evaluation of ∆pfi described in chapter 5. In the absence of momentum changes, a loss coefficient determined by frictional and drag effects can be defined, i.e.,
K fd =
2 2 )+ (ρ avi - ρ avm)gL fi] 2∆ pfd 2 [∆ pfi - (ρ avo v avo - ρ avi v avi = 2 2 ρv ρv
In practice, the reference conditions chosen for the denominator in this equation may differ. Some examples follow.
Illustration 1. The loss coefficient for a particular fill can be defined in terms of the mean air-vapor flow rate and its density through the fill, i.e.,
K fd = 2 [∆ pfi - (ρ avo vavo - ρ avi v avi)+ (ρ avi - ρ avm)gL fi]ρ avmA fr / m avm 2
2
2
2
where mavm = ρavm vavm Afr To find the mean density of the air-vapor through the fill, certain thermophysical properties must be determined. Using Equation A.3.1, the density of the inlet air at Tai = 9.7 °C is
ρavi = (1 + wi )[1 - wi / (wi + 0.62198)][ pa / (287.08 Tai)] 0.00616336 101712.27 3 = (1 + 0.00616336) 1 = 1.248 kg /m 0.00616336 + 0.62198 287.08(9.7 + 273.15)
The inlet air-vapor mass flow rate is
mavi = ma + wima = 4.134 + 0.00616336 x 4.134 = 4.1595 kg/s
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The velocity of the air entering the fill is
vavi = mavi/(ρavi Afr) = 4.1595/(1.248 x 1.5 x 1.5) = 1.481 m/s Using the conservation of energy
ma (imao - imai) = mw cpwm (Twi - Two) Re-arrange this equation to find an expression for the enthalpy of the outlet air.
imao = mw cpwm (Twi - Two)/ma + imai = 3.999 x 4177.40244 (39.67 - 27.77)/4.134 + 25291.875 = 73379.5976 J/kg dry air Since it is assumed that the air leaving the fill (outlet air) is saturated, it follows from Equation A.3.6b that
imao = imaos = cpao Tao + wso (ifgwo + cpvo Tao) = 73379.5976 J/kg dry air The thermophysical properties cpao, wso, and cpvo are all functions of Tao, and the latter can be determined from this equation by following an iterative procedure. Find Tao = 24.278 °C (297.428 K). To confirm that this value is correct, evaluate the previously mentioned thermophysical properties. The mean specific heat of dry air is evaluated at 24.278/2 °C or 285.289 K using Equation A.1.2. cpao = 1.045356 x 103 - 3.161783 x 10-1 x 285.289 + 7.083814 x 10-4 x 285.2892 - 2.705209 x 10-7 x 285.2893 = 1006.5274 J/kg K The specific heat of water vapor at this temperature follows from Equation A.2.2. cpvo = 1.3605 x 103 + 2.31334 x 285.289 - 2.46784 x 10-10 x 285.2895 + 5.91332 x 10-13 x 285.2896 = 1872.9051 J/kg K
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The pressure of saturated water vapor at Tao = 24.278 °C (297.428 K) is given by Equation A.2.1.
z = 10.79586 (1 - 273.16 / 297.428) + 5.02808 log10 (273.16 / 297.428) + 1.50474 x 10 -4[1 - 10 -8.29692{(297.428/273.16)-1}] + 4.2873 x 10 -4[10 4.76955(1-273.16/297.428) -1]+ 2.786118312 = 3.4819
Thus pvo = 103.4819 = 3032.9647 N/m2 The humidity ratio of the saturated air at this temperature is given by Equation A.3.5.
w so =
0.62509 x 3032.9647 = 0.019215 kg/kg dry air 101712.27 - 1.005 x 3032.9647
Substitute these properties into Equation A.3.6b, and find
imaos = 1006.5274 x 24.278 + 0.019215 (2501598.5334 + 1872.9051 x 24.278) = 73379.597 J/kg dry air This value is in agreement with the value obtained for imao, which means that the given value for Tao = 24.278 °C is correct. The density of the outlet air at this temperature is, using Equation A.3.1,
0.019215 101712.27 ρ avo= (1 + 0.019215) 1 0.019215 + 0.62198 287.08 (24.278 + 273.15) = 1.1777 kg /m3
The outlet air mass flow rate is
mavo = ma + wso ma = 4.134 + 0.019215 x 4.134 = 4.2134 kg/s
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The velocity of the air leaving the fill is
vavo = mavo/(ρavo Afr) = 4.2134/(1.1777 x 1.5 x 1.5) = 1.59 m/s The arithmetic mean air mass flow rate through the fill is
m avm =
m avi + m avo 4.1595 + 4.2134 = = 4.1865 kg/s 2 2
The harmonic mean density of the air through the fill is -1
-1
1 1 1 1 = 2 + + ρ avm = 2 = 1.2118 kg / m3 1.248 1.1777 ρ ρ avi avo Substitute these values into the equation defining the loss coefficient, i.e., 2 2 2 )+ (ρ avi - ρ avm)gL fi] ρ avmA fr2 / m avm K fdm = 2 [∆ pfi - (ρ avov avo - ρ aviv avi
= 2[4.5 - (1.1777 x 1. 592 - 1.248 x 1. 4812)+ (1.248 - 1.2118) x 9.8 x 1.878]1.2118 x (1.5 x 1.5) / 4.18652 = 3.449 2
or per unit height of the fill Kfdm1 = Kfdm/Lfi = 3.449/1.878 = 1.8365 m-1
Illustration 2. Another definition of the loss coefficient is as follows:
K fdam1 = 2 ρam ∆ pfd A fr2 / (ma2L fi ) where ρam is the density of the dry air based on the arithmetic mean temperature through the fill, i.e.,
ρ am =
2 pa
=
2 x 101712.27
R (T ai + T ao) 287.08(282.85 + 297.428)
= 1.221 kg / m3
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With this value, find
K fdam1 = 2 x 1.221 x [4.5 - (1.1777 x1. 592 - 1.248 x 1. 4812)+ (1.248 - 1.2118)x 9.8 x1.878] 2
(1.5 x 1.5 ) / (4.1342 x 1.878) = 1.8975 m-1
This definition of the loss coefficient is similar to that proposed by Johnson.
Illustration 3. A third definition of the loss coefficient due to friction and drag effects per unit height of fill is in terms of the dry air mass flow rate and its density at the inlet to the fill, i.e.,
K fda1= 2 ρ ai ∆ pfd A 2fr / (m a2 L fi) The density of the dry inlet air is given by
ρ ai ≈ pa / (RT ai)= 101712.27 / (287.08 x 282.85) = 1.2526 kg / m3
With this value, find the loss coefficient per unit depth of fill.
K fda1= 2 x 1.2526 x[4.5 - (1.1777 x 1. 592 - 1.248 x 1. 4812)+ (1.248 - 1.2118)x 9.8 x 1.878] 2
(1.5 x 1.5 ) / (4.1342 x 1.878) = 1.9466 m-1
Any of the previous definitions for the loss coefficient are permissible, although their values may differ measurably (Kfda1 = 1.9466 is 6% larger than Kfdm1 = 1.8365). It is very important when fill performance characteristics are presented that their definitions be clearly specified.
Note When designing a cooling tower incorporating a counterflow fill installation, the actual fill loss coefficient is obtained from
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2 2 K fi =[( pi+ ρ aviv 2avi / 2)-(po + ρ avov avo / 2)]/(ρ avmv avm / 2)
2 2 =[∆ pfd+ (ρ avov avo - ρ aviv 2avi)+(ρ aviv 2avi / 2 - ρ avov avo / 2)]/(ρ avmv 2avm / 2)
2 = K fd + (ρ avov avo / 2 - ρ aviv 2avi / 2 )/ (ρ avmv 2avm / 2)
2 = K fd + (G avo / ρ avo- G 2avi / ρ avi)/ (G 2avm / ρ avm)
where 1/ρavm = 0.5 (1/ρavi + 1/ρavo) Gavm = (Gavi + Gavo)/2 The approximate mean air and water temperatures in the fill are 17 °C and 34 °C. By conducting additional tests with these different flow rates, it is possible to correlate the resultant transfer characteristics of the particular fill by an empirical relation similar to Equation 4.3.3. The following expression correlates data obtained for this particular fill shown in Figure 4.3.5:
h d afi / G w = 0.2692 G -w0.094G 0.6023 a
Fig. 4.3.5 Fill Performance Characteristics—Merkel Assumptions with Chebyshev Integration Based on 1.878 m and 2.504 m Fill Heights
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The fill loss coefficient per unit fill height may be correlated by Equation 4.3.5. In this case, the applicable correlation is
K fdm1 = 1.9277 G1.2752 G -a1.0356 w Further tests were conducted on a fill consisting of eight layers giving a total height of 8 x 313 = 2504 mm. The performance correlations are:
h d a fi /G w = 0.25575 G w- 0.094G a0.6023
Kfdm1 = 1.851G 1.2752 G -a1.0356 w The previous correlations are shown in Figure 4.3.5 for Gw = 1.8 kg/sm2. Fill height clearly influences performance.
4.4 Effectiveness-NTU Method Applied to Evaporative System Jaber and Webb developed the equations necessary to apply the Effectiveness-NTU method to counterflow or crossflow cooling towers. The approach is useful in crossflow and simplifies the method of solution when compared to a more conventional numerical procedure. Consider the equation for the enthalpy transfer in an evaporative process given by Equation 4.2.12, which may be written as
dQ = h d [Lef (imasw - ima) + (1 - Lef ) i v ( w sw - w) ] dA
(4.4.1)
With the assumption of Merkel that the Lewis factor is equal to unity, Equation 4.4.1 reduces to
dQ = h d (imasw - ima) dA
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(4.4.2)
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where (imasw - ima) is the enthalpy driving potential used by the Effectiveness-NTU method in the case of evaporative cooling. For the control volume shown in Figure 4.2.1, it follows from Equations 4.2.14 and 4.2.21 that
dQ = mwcpwdTw = madima
(4.4.3)
It is convenient to relate dQ to the slope of the saturated air enthalpy (imasw) water temperature (Tw) curve. Equation 4.4.3 is written as
dQ = m w cpw dimasw/(di masw/d T w ) = m a di ma
(4.4.4)
from which it follows that
di masw = dQ(di masw /d T w) /( m w cpw)
(4.4.5)
It follows from Equation 4.4.3 that dima = dQ/ma. Subtract this relation from Equation 4.4.5, and find
di masw - dima = d(imasw - i ma) = dQ[(di masw /d Tw) /( m w cpw) - 1 / m a]
(4.4.6)
From Equations 4.4.6 and 4.4.2, it follows that
(di d(imasw - ima) /d T w) 1 dA = h d masw m a (i masw - i ma) m w cpw
(4.4.7)
This equation, applicable in an evaporative system, will correspond to the heat exchanger design determined in Equation 3.5.4 if one defines the air capacity rate (cold fluid) as ma and the water capacity rate (hot fluid) as mwcpw/(dimasw/dTw). The maximum theoretical amount of enthalpy that can be transferred, is Qmax = (minimum capacity rate) x (imaswi - imai), where imaswi is the saturated air enthalpy at the water inlet condition and imai denotes air inlet enthalpy. There are two possible cases to be considered.
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Case 1 mwcpw/(dimasw/dTw) < ma where consistent with heat exchanger design terminology Cemin = mwcpw/ (dimasw/dTw) and Cemax = ma. The evaporative capacity rate ratio for this particular case is given by
Ce = Cemin / Cemax = m w cpw /[(di masw/ dT w ) m a]
Substitute Ce into Equation 4.4.7 to find
d(imasw - ima) h d (di masw/ dT w )(1 - Ce) dA = (i masw - ima) m w cpw
(4.4.8)
Integration of Equation 4.4.8 between the entering and leaving air states, iai and iao, gives
(imaswo- imai)/(imaswi - imao) = exp [- NTUe (1 - Ce)]
(4.4.9)
where imaswo and imaswi refer to the saturated air enthalpy at the water outlet and inlet conditions. The analogous definition for NTU in this particular evaporative system or wet-cooling tower is
NTUe = h d A( dimasw / dTw ) /(m w cpw)
(4.4.10)
where A is the total wetted transfer area. The heat exchange effectiveness is defined as
ee = Q/ Qmax
276
(4.4.11)
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Integration of Equation 4.4.3 between inlet and outlet conditions gives
Q = m w cpw (T wi - Two) = m a (imao - imai )
(4.4.12)
The maximum enthalpy transfer rate can be expressed approximately as
Qmax ≈ m w cpw (imaswi - imai) /(di masw / dTw ) = m a Ce (imaswi - imai)
(4.4.13)
where the gradient of the saturated air enthalpy-temperature curve over the control volume is
dimasw imaswi -imaswo ≈ dT w T wi - T wo
(4.4.14)
It follows from Equations 4.4.11, 4.4.12, 4.4.13, and 4.4.14 that
ee = (imaswi - imaswo) /(imaswi - i mai)
(4.4.15)
and from Equations 4.4.11, 4.4.12, and 4.4.13 that
Ce ee = (imao - imai) /(imaswi - i mai )
(4.4.16)
From Equations 4.4.15 and 4.4.16, it follows that
(ee - 1)/ (ee Ce -1)= (imaswo- i mai)/ (imaswi - imao)
(4.4.17)
Equating Equations 4.4.9 and 4.4.17 gives the Effectiveness-NTU equation for a counterflow evaporative system or cooling tower
ee =
1 - exp[ - NTU e (1 - Ce)] 1 - Ce exp[ - NTU e (1 - Ce)]
(4.4.18)
which is similar to the Effectiveness-NTU expression for a counterflow heat exchanger listed in Table 3.5.1.
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Case 2 ma < mwcpw/ (dimasw/dTw) In this case, Ce = ma (dimasw/dTw)/(mwcpw) Follow a procedure similar to that of Case 1, and again find the effectiveness given by Equation 4.4.18. The Effectiveness-NTU method is subject to approximations involved in linearizing the imasw versus Tw curve as a straight line. The accuracy of the method can be increased by breaking up the design into a number of increments. An analytical method was developed by Berman to improve the approximation of the imasw versus Tw curve as a straight line. He proposed a correction factor k given by
λ = (imaswo+ i maswi - 2imasw ) / 4
(4.4.19)
where imasw denotes the enthalpy of the saturated air at the mean water temperature Twm = (Twi + Two)/2. This factor is used to obtain a more correct value for Qmax as follows:
Qmax = Cemin(i maswi - λ - imai)
(4.4.20)
The use of the correction factor gives a two-increment design. The Effectiveness-NTU equations described previously show the Effectiveness-NTU equations for a counterflow heat exchanger can be applied to a counterflow cooling tower. The equations for a crossflow heat exchanger can be applied to a crossflow cooling tower. Jaber and Webb recommend the use of the unmixed/unmixed Effectiveness-NTU equation as listed in Table 3.5.1. A more refined analysis for counterflow cooling towers is presented by El-Dessouky et al.
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Example 4.4.1 Determine the Merkel number or transfer characteristic of the fill described in Example 4.3.1, employing the Effectiveness-NTU method as applicable to evaporative systems.
Solution To obtain the Merkel transfer characteristic, the effectiveness ee = Q/Qmax of the evaporative system must be known. The heat transfer rate is given by Equation 4.4.12. Q = mwcpw (Twi - Two) ≈ mwcpwm (Twi - Two) = 3.999 x 4177.40244 (39.67 - 27.77) = 198794.645 W where cpwm was determined in Example 4.3.1. To find Qmax, the enthalpies of saturated air at the water inlet and outlet temperatures are required. To evaluate the latter, determine the specific heat of dry air at (273.15 + Two/2) = (273.15 + 27.77/2) = 287.035 K using Equation A.1.2. cpao = 1.045356 x 103 - 3.161783 x 10-1 x 287.035 + 7.083814 x 10-4 x 287.0352 - 2.705209 x 10-7 x 287.0353 = 1006.5672 J/kg K Using Equation A.2.2, the specific heat of saturated water vapor at 287.035 K is cpvo = 1.3605 x 103 + 2.31334 x 287.035 - 2.46784 x 10-10 x 287.0355 + 5.91332 x 10-13 x 287.0356 = 1874.3847 J/kg K The pressure of saturated vapor at Two = (273.15 + 27.77) = 300.92 K follows from Equation A.2.1, where
z = 10.79586 (1 - 273.16 / 300.92) + 5.02808 log10 (273.16 / 300.92) + 1.50474 x 10 -4[1 - 10-8.29692(300.92/273.16-1)] + 4.2873 x 10 -4[104.76955(1-273.16/300.92) -1]+ 2.786118312 = 3.5716
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such that psvo = 10z = 103.5716 = 3728.8246 N/m2 The humidity ratio for saturated air at Two = 27.77 °C is, using Equation A.3.5,
0.62509 x 3728.8246 w so = = 0.0237927 kg/kg dry air 101712.27 1.005 x 3728.8246 The enthalpy of saturated air at this temperature is given by Equation A.3.6b. With the previous values and the latent heat at 0 °C obtained in Example 4.3.1, find imaswo = 1006.5672 x 27.77 + 0.0237927 (2501598.53 + 1874.3847 x 27.77) = 88710.7054 J/kg dry air The enthalpy of saturated air at Twi = 39.67 °C is found to be imaswi = 163546.9337 J/kg dry air while the enthalpy of saturated air at the mean water temperature, Twm = (Twi + Two) = (39.67 + 27.77)/2 = 33.72 °C, is imaswm = 121000.7923 J/kg dry air With these values, find the approximate gradient of the saturated air enthalpy (imasw) and water temperature (Tw) curve using Equation 4.4.14.
dimasw 163546.9337 - 88710.7054 ≈ = 6288.7587 J/kg K dT w 39.67 - 27.77 In order to determine the relevant case described in section 4.4, find
m w cpw m w cpwm 3.999 x 4177.40244 = = = 2.6564 6288.7587 dimasw/dTw dimasw / dT w Since Cemin = mwcpwm/(dimasw/dTw) = 2.6564 < Cemax = ma = 4.134, find the capacity ratio for Case 1.
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Ce = Cemin/Cemax = mwcpwm/(madimasw/dTw) = 2.6564/4.134 = 0.6426 To find Qmax, evaluate the correction factor λ using Equation 4.4.19.
λ = (88710.7054 + 163546.9337 - 2 x 121000.7923)/4 = 2564.0136 J/kg With these values, find Qmax using Equation 4.4.20. Qmax = 2.6564(163546.9337 - 2564.0136 - 25291.87496) = 360449.1272 J/s where imai was determined in Example 4.3.1. The effectiveness given by Equation 4.4.11 is ee = 198794.645/360449.1272 = 0.55152 However, in a counterflow evaporative system, the effectiveness can be expressed in terms of NTUe shown in Equation 4.4.18.
ee = 0.55152 =
1 - exp [- NTUe (1 - 0.6426)] 1 - 0.6426 exp [- NTUe (1 - 0.6426)]
Solve, and find NTUe = 1.0193. With this value, find from Equation 4.4.10 that
h d A (di masw / dT w ) = NTUe = 1.0193 m w cpwm or the Merkel transfer characteristic is
hd A mw
=
h d afi L fi Gw
=
NTU e x cpwm 1.0193 x 4177.40244 = = 0.67709 6288.7587 (dimasw /dT w )
This value of the transfer characteristic compares very well with the value obtained in Example 4.3.1, i.e., 0.68468.
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4.5 Closed Circuit Evaporative Cooler In a closed-circuit evaporative cooler, an example is shown in Figure 1.3.1, cooling a process fluid flowing in tubes is achieved by spraying water onto the tubes, deluging them, and forming a film of water that flows downward under the action of gravity. As the water flows down the surface of the tube, it is evaporated by air flowing over it and results in cooling the process fluid. One of the earliest useful analytical treatments of closed circuit evaporative coolers was by Parker and Treybal. The method was derived before low cost computing facilities were generally available and used Merkel’s approximation for the heat-mass transfer process. One of the most significant features of this work is that the variations in the recirculating water temperature as it flowed through the bundle were taken into consideration. In addition, the enthalpy of the saturated air was assumed to be a linear function of temperature, making it possible to integrate the simultaneous differential equations over the height of the coil. Mizushina et al. using a similar approach to Parker and Treybal integrated their equations numerically using a computer. At the same time, they carried out some useful experiments to determine the applicable heat and mass transfer coefficients in a smooth tube bundle having a triangular tube layout. Perez-Blanco and Bird did an analysis on the performance of a rather idealized vertical counterflow evaporative cooling unit but used the correct thermodynamic equations without any approximations. Kreid et al. presented an approximate method of analyzing deluged heat exchangers with fins using an effective overall heat transfer coefficient based on the logarithmic mean enthalpy difference. They demonstrated a practical method to predict heat transfer rates within 5% of the actual values. Leidenfrost and Korenic presented a rigorous analysis of finned tube evaporative condensers, which could be applied to crossflow or counterflow devices. They did not make use of Merkel’s approximation of a Lewis factor of unity, and their analysis can even accommodate partially dry heat exchangers. They proposed the use of a stepwise integration process using a graphic method originally derived by Bosnjakovic. The process was used to determine the exit state of the air and water leaving an element, which amounts to a computerization of that method. Some useful software was developed by Webb to approximate the performance of various types of evaporative cooling devices using a unified approach for the air side. The prediction accuracy is stated to be on the order
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of 3% of the manufacturer’s data for several devices. However, the evaporative cooler program is limited to vertical counterflow equipment. Erens and Dreyer present a general approach to the performance evaluation of evaporative coolers based on the methods of Poppe and Bourillot. Hellmann predicts the thermal performance of evaporative coolers by solving a system of coupled nonlinear differential equations. In view of the complexity of this latter approach, correlations have been developed for a new design method. This method allows the estimation of errors introduced by simple analytical solutions as functions of dimensionless operating parameters. Consider an elementary control volume of a counterflow tube evaporative cooler (Fig. 4.5.1.b). Water is sprayed over the tubes to deluge them and then falls through an upward flowing airstream. A waterfilm forms on the smooth or finned tube outside surface. In order to derive the governing equations for the evaporative cooler, the following assumptions are made:
1.
The waterfilm throughout the cooler is at a constant mean film temperature. Since the deluge or cooling water is recirculated (Fig. 1.3.1) its inlet temperature is equal to its outlet temperature. As shown by Mizushina, the deviation in water temperature as it flows through the tube bank or bundle is often relatively small. However, in certain cases, this assumption is not acceptable according to Finlay.
2.
The air/water interface area is approximately the same as the outer surface area of the tube bundle, i.e., the waterfilm on the tubes is very thin such that the area exposed to the airstream dAa is the same as the outside surface area of the tubes.
3.
The deluge water is evenly distributed between the different tube rows, and the outside tube surface area is totally wetted.
4.
The Lewis factor is equal to unity, and the evaporation loss is negligible.
Fig. 4.5.1 Control Volume of Tube Evaporative Cooler (a) Crossflow (b) Counterflow
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Following the same procedure as in section 4.2, the governing equations for the evaporative cooler are:
di ma = h d (imasw - ima)dA a / m a
(4.5.1)
dT w = (m a dima + m p cpp dTp )/( m w cpw)
(4.5.2)
Equation 4.5.1 corresponds to Equation 4.2.20, and Equation 4.5.2 corresponds to Equation 4.2.21. A third variable, the process fluid temperature Tp, is present for the evaporative cooler, and a third governing equation is necessary.
dT p = - U a (T p - T w )dA a /( m p cpp)
(4.5.3)
Ua is the overall heat transfer coefficient between the process fluid inside the tubes and the deluge water film on the outside.
1 A A + a + R a R n Ua = h w ef A p h p n A n
-1
(4.5.4)
In this equation,
•
hw is the heat transfer coefficient between the waterfilm and the tube outer surface
•
hp is the heat transfer coefficient on the inside of the tube
•
Rn includes all other resistances (tube wall, fouling, contact resistance if the outer surface is finned, etc.) between the waterfilm and the process fluid
•
ef is the effectiveness of a finned surface.
By assuming that the deluge water film has a constant mean temperature, Twm, throughout the cooler, Equation 4.5.2 is eliminated. Integrate Equation 4.5.1 between imai and imao.
imao = imaswm- (imaswm- imai) exp (- NTUa )
284
(4.5.5)
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where NTUa = Aahd/ma Equation 4.5.3 can be integrated between the inlet and outlet process fluid temperatures to give
T po = T wm + (T pi - T wm) exp(- NTUp)
(4.5.6)
where
NTUp = A a Ua /( m p cpp) The heat transfer rate of the evaporative cooler is given by the following equation.
Q = m a (imao - imai ) ≈ m p cpp (Tpi - Tpo)
(4.5.7)
Substitute Equations 4.5.5 and 4.5.6 into Equation 4.5.7, and simplify.
m a [imaswm - (imaswm - imai) exp(- NTUa) - imai] = m p cpp [Tpi - Twm - (Tpi - Twm ) exp(- NTUp)] or
T wm = T pi -
m a (i maswm - i mai) [1- exp (- NTUa)] m p cpp [1 - exp (- NTUp)]
(4.5.8)
From Equation 4.5.8, the mean deluge water-film temperature can be determined iteratively. The outlet conditions can be determined from Equations 4.5.5 and 4.5.6. The outlet air is assumed to be saturated with water vapor. Numerous correlations are found in the literature for the mass transfer coefficient, hd, between the cooling water flowing downward over banks of horizontal tubes and the upward flowing airstream.
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Parker and Treybal tested tube banks consisting of 19 mm outside tubes arranged on a 2do triangular pitch, and found
h d = 0.04935 [(1 + w )(m a / A c)]
0.905 ≈ 0.04935 (m avm / A c) , kg/ sm 2
0.905
(4.5.9)
where Ac is the minimum cross-sectional airflow area between the tubes. The air-vapor mass velocity was in the range of 0.68 < (mavm/Ac) < 5 kg/sm2. Mizushina et al. tested bundles having 12–40 mm outside diameter tubes arranged on a 2do triangular pitch, and found 0.9 0.15 -1.6 h d = 5.5439 x 10 -8 Re avm Re wm do , kg / sm 2
(4.5.10)
This equation is valid for 1.2 x 103 < Reavm = mavmdo/(Acµavm) < 1.4 x 104 and 50 < Rewm = mwmdo/(Acµwm) = 4 Γm/µwm < 240 The deluge water mass flow rate per unit length over half of a cooling tube can be expressed as Γm = mwmdo/(2Afr). Nitsu et al. tested banks of both plain and finned tubes and found that the mass transfer coefficient was in all cases independent of water flow rate above a critical Γm/do value of 0.7. The correlations obtained were for:
1.
Plain tubes, 16 mm outside diameter with P/do = 2.38 and Pt/do = 2.34 hd = 0.076(mavm/Ac)0.8, kg/sm2
for 1.5 ≤ (mavm/Ac) ≤ 5
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(4.5.11)
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2. Finned tubes, 42.6 mm fin diameter, having a 16 mm outside diameter and arranged with P/do = 2.38 and Pt/do = 2.34 and 1.5 < (mavm/Ac) < 5. hd = 0.0135(mavm/Ac)1.25, kg/sm2
(4.5.12)
for a fin spacing of 11 mm, and hd = 0.0112(mavm/Ac)1.25, kg/sm2
(4.5.13)
for a fin spacing of 6.1 mm.
Dreyer and Erens studied a crossflow evaporative cooler having 38.1 mm outside diameter tubes arranged in a 2do mm triangular pattern. Based on the Merkel method of analysis, they find 0.64 0.2 h d = 5.5749 x 10 -5Re avm Re wm , kg / sm2
(4.5.14)
for 2500 < Reavm = mavmdo/(ρavµavmAc) 350,000, the assumed value of Cn = 0.994 is correct.
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The density of the air at the heat exchanger using Equation A.3.1 is
0.0085 99695 ρ av = (1 + 0.0085)1 = 1.1647 kg / m3 0.0085 + 0.622 287.08 x 296.65
The characteristic pressure drop number follows from Equation 5.4.16, 2 = 1.1647 x 188.5 /(1.82 x 10-5 )2= 6.624362 x 1011m-2 Ey iso = ρ av ∆ pa / µ av
and the characteristic flow parameter follows from Equation 5.4.11. Ryiso = mav/µav Afr = 2.1129/(1.82 x 10–5 x 0.4875) = 2.38075 x 105 m-1 To find the friction factor, determine the flow per unit area through the minimum flow area in the bundle, i.e., Ac= Afr – ntr Lt [df tf + (Pf –tf) dr] /Pf = 0.4785 – 16.5 x 0.5 [0.0572 x 0.0005 + (0.0028 – 0.0005)0.0276]/0.0028 = 0.2072 m2 The mass velocity through this area is given by Gc = mav/Ac = 2.1129/0.2072 = 10.198 kg/m2s Using Equation 5.4.2, the Euler number based on the minimum flow area is Eu = 1.1647 x 188.5/(10.198)2 = 2.111 and the loss coefficient is, using Equation 5.4.3, K = 2 x 1.1647 x 188.5/10.1982 = 4.22 Based on the free stream velocity, the value of this loss coefficient is K = 2 ρav ∆pa /(mav/Afr )
2
= 2 x 1.1647 x 188.5/(2.1129/0.4785)2 = 22.52
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The isothermal loss coefficient for this bundle based on free stream conditions was also determined at other airflow rates, and the results are plotted in Figure 5.6.6. The data may be correlated by the following empirical equation for the range tested: K = 1383.94795 Ry–0.332458
Example 5.4.4 Repeat examples 5.4.1 and 5.4.2, assume the air to be dry (w = 0), and find the values for Qa, Ny, and K.
Solution Qa = 33989 W Ny = 2.43 x 105 m–1 for Ry = 227999 m–1 K = 22.52 for Ry = 236452 m–1 In all cases, the deviation from the original value is less than 1%.
5.5 Heat Transfer and Pressure Drop Correlations Although numerous heat transfer and pressure drop correlations for flow through bundles of finned tubes have been reported in the literature, all have their limitations. Great care should be taken when applying them to a particular design. In the following equations, all thermophysical properties are evaluated at the mean air temperature unless specified otherwise.
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Heat transfer correlations for staggered circular finned tubes Some of the early empirical relations applicable to bundles of staggered circular finned tubes are proposed by Jameson, Katz and Young, Kutateladze and Borishaniskii, and Schmidt. The heat transfer correlations proposed by Briggs and Young are based on a wide range of data. Their general equation for six-row, equilaterally arranged, finned tube bundles can be recommended: 0.2
0.1134
2(Pf - t f ) Pf - t f hd r = 0.134 Pr0.33 Re 0.681 k df - d r t f
(5.5.1)
where Re = Gc dr/µ
This equation is valid within the following limits:
1000 < Re < 18000 11.13 mm < dr < 40.89 mm 1.42 mm < (df – dr)/2 < 16.57 mm 0.33 mm < tf < 2.02 mm 1.30 mm < Pf < 4.06 mm 24.49 mm < Pt < 111 mm
Brauer investigated the heat transfer and pressure drop characteristics of round and elliptical finned tubes for staggered and in-line arrangement but did not develop any new empirical equations to correlate his data. Schulenberg proposed a correlation in which the exponent of the Reynolds number is a function of the finned surface area. In 1966, Vampola proposed an equation based on an extensive study of different finned tubes. For more than three tube rows,
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-0.2
P -d h de = 0.251 Re 0.67 t r k dr
-0.2
Pt - d r +1 Pf - t f
0.4
Pt - d r Pd - d r
(5.5.2)
when (Pt – dr)/(Pd – dr) > 1. For all other values of the latter term, it is replaced by unity in the previous equation. The diagonal pitch is given by 2
0.5
Pd = [(Pt / 2 ) + P l2 ]
The Reynolds number is defined as Re = Gc de/µ where
de =
A re d r + A f (A f / 2 n f ) A f + A re
0.5
All values are calculated for a one-meter length of tube. Af is the fin surface area, and Are is the exposed root area. This equation is valid within the following limits: 1000 < Re < 10000 10.67 mm < dr < 26.01 mm 5.20 mm < (df – dr)/2 < 9.70 mm 0.25 mm < tf < 0.70 mm 2.28 mm < Pf < 5.92 mm 20.32 mm < P < 52.40 mm 24.78 mm < Pt < 49.55 mm 16.20 mm < de < 34.00 mm 0.48 mm < (Pt – dr)/dr < 1.64 4.34 < (Pt – dr)/(Pf – tf) + 1 < 25.2 0.45 < (Pt – dr)/(Pd – dr) < 2.50 Further correlations were proposed by Zozulya et al. and Kuntysh. In 1974, Mirkovi´c proposed the following correlation for an eight-row tube bundle:
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0.1
0.15
P -d d hd et = 0.224 Pr 0.33Re 0.662 t r r k d r P - d r
2(Pf - t f ) d -d f r
0.25
(5.5.3)
where Re = Gc det/µ det = A1/[π(df – dr + Pf)] A1 is the area exposed to the airstream by one fin and its exposed root area.
This equation is valid within the following limits:
3000 < Re < 56,000 25.40 mm < dr < 50.80 mm 9.53 mm < (df – dr)/2 < 15.88 mm 1.27 mm < tf < 2.03 mm 4.23 mm < Pf < 8.47 mm 60 mm < P < 80 mm 100 mm < Pt < 120 mm Other correlations include those of Zhukauskas, Elmahdy and Biggs, Hofmann, Biery, Gianolio and Cuti, Brandt and Wehle, ESDU, and Nir. Weierman correlated performance data for finned tubes and presented his findings in graphic form. Ganguli et al. propose the following correlation for three or more rows of finned tubes:
hd r -0.15 = 0.38 Re 0.6 Pr 0.333(A/Ar ) k
(5.5.4)
where Re = Gcdr /µ
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If no fins are present, the ratio of the total air-side area to the root area, Ar, is given by A/Ar = (Af + Are)/Ar = [(df2 - dr2)/2 + df tft + dr(Pf - tfr)]/(drPf) This correlation is valid for:
11.176 mm < dr < 50.8 mm 5.842 mm < (df - dr)/2 < 19.05 mm 2.3 mm < Pf < 3.629 mm 0.254 mm < tf < 0.559 mm 27.432 mm < Pt < 98.552 mm 1800 < Re < 100000 1 < (Af + Are)/Ar ≤ 50
The previous correlations are applicable to bundles having three or more rows of finned tubes and are based on tests performed in wind tunnels. In the tests, the turbulence levels of the incoming airstream are low, as in the case of induced-draft heat exchangers. These equations should be multiplied by a row correction factor, Fr, when the bundle consists of fewer tube rows. Information in the literature for bundles less than six rows deep is inconsistent. Brauer found heat transfer stability by the second row, Ward and Young by the fourth to sixth row, and Mirkovi´c by the eighth row. Eckels and Rabas also found stability by the fourth row. Ward and Young found the row-effect to be a function of the air velocity, vc, through the smallest cross section of the finned tube bundle (Fig. 5.5.1). According to Gionolio and Cuti, this trend is correlated approximately by
h nr = h 6 (1 + vc / n 2r )-0.14
(5.5.5)
where h6 is the mean heat transfer coefficient for a six-row bundle and nr is the number of tube rows in the flow direction.
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Fig. 5.5.1 Tube Row Correction Factor
Because of this row-effect, the heat transfer performance characteristics of individual tube rows in a bundle should be determined in wind tunnel tests for accurate design of heat exchangers such as induced-draft condensers. Where the turbulence levels of the incoming airstream are high, other correction factors are recommended (see section 5.9). The previous equations are given because they represent a wide spectrum of data and have been applied in the design of practical systems. From this extensive yet incomplete summary of references, it is obvious no single correlation exists that can predict accurate heat transfer characteristics of circular finned tubes arranged in a staggered pattern. Extensive surveys of finned tube performance characteristics are presented by PFR Engineering Systems, Inc., Stasiulevicius and Srinska, and Zukauskas and Ulinskas. Where uncertainties exist or where the limits of applicability are exceeded, the results of different equations should be compared for a particular design. When a more sophisticated analysis is required, laboratory tests must be conducted on a bundle of the particular finned tubes to be employed.
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Pressure drop correlations for staggered circular finned tubes Numerous correlations with which the static pressure drop through bundles of staggered circular finned tubes can be predicted are reported in the literature. Gunter and Shaw, Jameson, and Ward and Young presented some of the earlier correlations. An equation from Robinson and Briggs is frequently used for tubes arranged in a staggered or equilateral pattern -0.927
Eu =
ρ∆ p P = 18.93 n r Re -0.316 t 2 Gc dr
0.515
Pt Pd
(5.5.6)
for nr tube rows and where Re = Gc dr /l and the diagonal pitch can be found from
Pd =
[(P / 2) + P ] 2
t
2
0.5
This equation is valid for: 2000 < Re < 50,000 18.64 mm < dr < 40.89 mm 39.68 mm < df < 69.85 mm 10.52 mm < (df – dr)/2 < 14.48 mm 42.85 mm < Pt < 114.3 mm 37.11 mm < P < 98.99 mm 2.31 mm < Pf < 2.82 mm 1.8 < Pt /dr < 4.6 Vampola proposes the following empirical relation for the pressure drop
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-0.9
Eu =
ρ∆ p P -d = 0.7315 n r Re -0.245 t r G c2 dr
0.7
Pt - d r +1 Pf - t f
0.9
de dr
(5.5.7)
where the definition of Re and the limits of applicability of this equation are similar to those of Equation 5.5.2. Based on a series of tests conducted by Mirkovi´c,
Eu =
ρ∆ p P -d = 3.96 n r Re -0.31 t r G c2 dr
0.14
dr P - d r
0.18
df - d r 2( - t ) Pf f
0.2
(5.5.8)
where Re = Gc deh /µ
d eh = 4 [P Pt Pf - π (df2 t f + Pf d 2r - d 2r t f )]/A1 A1 = the area exposed to the airstream by one fin and its exposed root area The limits of applicability of this equation are similar to those of Equation 5.5.3 with 1600 < Re < 31000. Other correlations are proposed by Hofmann, Schack, Kast and Eberhard, Nir, and ESDU. Ganguli et al. propose the following equation
Eu = ρ∆ p / G c2 = 2 n r [1 + 2 exp{- (Pt - df ) / (4 d r)}/{1 + (Pt - df ) / d r}]
[
x 0.021 + 13.6 (d f - d r) / Re (Pf - t f ) + 0.25246{(df - d r) / Re (Pf - t f )}
]
0.2
(5.5.9)
where Re = Gc dr /µ Equation 5.5.9 is valid across the range 2.5 < (df - dr )/[2(Pf - tf )] < 12.5
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Example 5.5.1 For the finned tube dimensions and operating conditions as specified in Example 5.4.2, determine the air-side heat transfer coefficient for the four-row bundle by employing Equations 5.5.1 and 5.5.4.
Solution Consider Equation 5.5.1 to evaluate h. The given specifications satisfy conditions where this equation is applicable, i.e., 1000 < Re = 14575 < 18,000 11.13 mm < dr = 27.6 mm < 40.89 mm 1.42 mm < (df – dr)/2 = (57.2 – 27.6)/2 = 14.8 mm < 16.57 mm 0.33 mm < tf = 0.5 mm < 2.02 1.30 mm < Pf = 2.8 mm < 4.06 24.49 mm < Pt = 58 mm < 111 mm Furthermore, Pr = 0.71218, and k = 2.67 x 10–2 W/mK. Substitute these values into Equation 5.5.1 to find
h=
0.134 x 2.67 x 10-2 0.33 0.681 2(2.8 - 0.5) (0.71218 ) (14575 ) (57.2 - 27.6) 0.0276 2.8 - 0.5 x 0.5
0.2
0.1134
= 65 W/ m2 K
Note that Equation 5.5.1 is applicable to bundles having six tube rows. As shown in Figure 5.5.1, the heat transfer coefficient for a four-row bundle will be less than that for a six-row bundle. The calculated heat transfer coefficient is much less than the experimentally determined value of 72.17 W/m2 K.
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For this finned tube, the ratio of the air-side area to the root area is (Af + Are)/Ar = [(57.22 - 27.62)/2 + 57.2 x 0.25 + 27.6(2.8 - 0.75)]/(27.6 x 2.8) = 17.15735 Equation 5.5.4 is applicable since 11.176 mm < dr = 27.6 mm < 50.8 mm 5.842 mm < (df - dr)/2 = 14.8 mm < 19.05 mm 2.3 mm < Pf = 2.8 mm < 3.629 mm 0.254 mm < tf = 0.5 mm < 0.559 mm 27.432 mm < Pt = 58 mm < 98.552 mm 1800 < Re = 14575 < 100,000 1 < (Af + Are)/Ar = 17.15735 ≤ 50 Find
h=
0.38 x 2.67 x 10-2 x 14575 0.6 x 0. 71218 0.333x 17.15735- 0.15 = 67.4987 W/ m2 K 0.0276
This value is closer to the measured value of 72.17 W/m2K.
Example 5.5.2 For conditions specified in Example 5.4.3, determine the pressure drop across the bundle using Equations 5.5.6 and 5.5.9.
Solution Consider Equation 5.5.6 to evaluate the pressure drop. The given specifications satisfy conditions where this equation is applicable, i.e.,
2000 < Re =
G c d r 10.198 x 0.0276 = = 15461 < 50000 1.82 x 10 -5 µ
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18.64 mm < dr = 27.6 mm < 40.89 mm 10.52 mm < (df – dr)/2 = (57.2 – 27.6)/2 = 14.8 mm ≈ 14.48 mm 42.85 mm < Pt = 58 mm < 114.3 mm 37.11 mm < P = 50.22 mm < 98.99 mm 2.31 mm < Pf = 2.8 mm < 2.82 mm 1.8 < Pt /dr = 58/27.6 = 2.10 < 4.6 Furthermore,
Pd =
[(P / 2) + P ] = [(58 / 2 ) + 50. 22 ] 2
t
2 0.5
2 0.5
2
= 57.99 mm
Substitute these values into Equation 5.5.6, and find for the four-row bundle -0.927
∆p =
10.198 2 x 18.93 x 4 x 15461-0.316 58 1.1647 27.6
58 57.99
0.515
= 161.2 N/ m2
This value is somewhat less than the 188.5 N/m2 measured. Equation 5.5.9 is applicable since 2.5 < (df - dr)/[2(Pf - tf)] = (57.2 - 27.6)/[2(2.8 - 0.5)] = 6.4348 < 12.5. For the four-row bundle,
∆p=
2 x 4 x 10.198 2 [1 + 2 exp {- (58 - 57.2) /(4 x 27.6)} 1.1647
/ {1 + (58 - 57.2) / 27.6}][0.021 + 13.6 (57.2 - 27.6) /{15461 (2.8 - 0.5)}
+ 0.25246 [(57.2 - 27.6) /{15461 x (2.8 - 0.5)}]
] = 195.57 N/ m
0.2
2
This value compares with the measured pressure differential of 188.5 N/m2.
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5.6 Oblique Flow through Heat Exchangers In large air-cooled systems, heat exchanger bundles are often arranged in the form of A-frames, deltas, or V-arrays to conserve space. The general flow pattern through two finned tube bundles of an A-frame or a V- or A-frame array heat exchanger system is sketched in Figure 5.6.1.
Fig. 5.6.1 General Picture of the Flow-Through Inclined Finned Tube Bundles (a) A-Frame (b) A-Frame Array
On passing through the bundles in the V- or A-frame array, the streamlines will emerge in a direction almost perpendicular to the downstream face. This occurs if the loss coefficient of the finned tube bundle is high. The streamlines will converge into a jet with severe separation at the downstream corners of the bundle face to form distorted velocity profiles. Because the downstream streamlines are curved, the pressure is not uniform along the downstream face of the bundle, and this in turn causes the velocity through the bundle to vary with position. The result is the upstream flow may approach the bundle with greater obliquity than it would if this flow were uniform. The obliquity of the incoming flow to the bundle front face and the jet formed at the downstream face may be regarded as two interrelated effects. This causes a considerable loss, contributes to overall aerodynamic losses, and will affect the performance of the cooling system.
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Oblique flow analysis The inlet pressure loss for oblique or inclined flow through a finned tube heat exchanger bundle may be analyzed by a simplified model proposed by Moore and Torrence. Consider a set of parallel thin sheets of negligible thickness shown in Figure 5.6.2. Incoming flow streamlines approach the upstream face with an angle θ, and the flow is constrained to leave normally. The entering flow tends to separate on the thin sheet leading edge upon entering, and a jet is formed. The velocity of the flow is assumed uniform over the cross section at 1. After turning, the flow evolves into a parallel flow at 2. Since the flow is potential between 1 and 2, the velocity at 2 can be assumed to be uniform. It is also assumed the static pressure within the initial separation region is constant and is equal to the upstream static pressure p1. By Bernoulli’s equation, the velocity at the minimum cross section must be equal to the upstream velocity v1. The flow then mixes downstream from this minimum cross section and attains the velocity v3 and static pressure p3. The drop in total pressure difference between positions 2 and 3 is given by the following equation:
p t1 - pt 3 = ∆ pit = p1 - p3 + ρ (v12 - v 32) / 2
(5.6.1)
Fig. 5.6.2 Behavior of the Flow at the Entrance of a Parallel Thin Sheet Bundle
Applying the momentum equation between 2 and 3, one obtains p1 – p3 = ρv3(v3 – v1)
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(5.6.2)
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Substitution of Equation 5.6.2 into Equation 5.6.1 yields
∆ pit = ρ v 32 - ρ v 3 v1 + ρ v12 / 2 - ρ v 32 / 2 which can be written as 2
∆ pit = ρ v 32 ( v1 / v 3 - 1 ) / 2
(5.6.3)
Since v1 /v3 = 1/sin θ by continuity, i.e., the contraction coefficient is equal to sin θ, and by introducing the definition,
Ki h = ∆ pit / ( q v 32 / 2)
(5.6.4)
the previous equation may be written as
1 - 1 Ki h = sin θ
2
(5.6.5)
Mohandes proposes the following modified expression for the loss coefficient based on experimental observations with bundles of finned tubes or sheets having a finite thickness
1 - 1 Ki h = K 0.5 ci + sin θ
2
(5.6.6)
where Kci is the entrance contraction loss coefficient for the normal flow condition based on the normal approach free stream velocity. For turbulent flow between flat sheets,
K ci = K c / σ 221 where Kc is given by Equation 2.3.7.
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The loss coefficient for the complete bundle is
K heθ = Ki θ + K f + K ei = ∆ pt /( ρ v 2i / 2)
(5.6.7)
where Kf = the friction expansion losses Kei = the outlet expansion losses vi = the mean normal inlet velocity Substitute Equation 5.6.6 into Equation 5.6.7, and find 2
1 K heθ = K ci0.5 + - 1 + K f + K ei sin θ (5.6.8)
1 1 = K he + - 1 - 1 + 2 K ci0.5 sin θ sin θ where Khe is the loss coefficient for normal flow through the bundle and includes the contraction and expansion losses under these conditions. In the case of a single inclined bundle located in a duct of constant cross section or in an array of V- or A-frames, the stream leaving the bundle in a normal direction is redirected (Fig. 5.6.3).
Fig. 5.6.3 Sketch of the Co-Ordinate System and Flow Process
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The total pressure drop after the inclined bundle has two causes. First is the turbulent decay of the jet near the centerline, which occurs directly downstream in the “V” region from position 1 to position 2. The second is due to the final mixing process between 2 and ∞, which is responsible for smoothing out the velocity profile according to Moore. The downstream loss coefficient is given by
K d = (p t1- pt ∞ )/ ( ρ 1 v i2 / 2)
(5.6.9)
where pt1 = the mean total pressure immediately downstream of the bundle vi = the mean velocity normal to the bundle at this point ρ1 = the corresponding density The value of this coefficient for different angles is shown in Figure 5.6.4 and may be expressed in terms of the following empirical relation according to Kotzé K d = exp(5.488405 - 0.2131209 θ + 3.533265 x 10-3 θ 2 - 0.2901016 x 10-4 θ 3 )
(5.6.10)
where the apex semi-angle θ is given in degrees.
Fig. 5.6.4 Overall Downstream Losses
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According to Kotzé, the jetting loss alone may be expressed as: Kdj = exp(5.32262 - 0.22888 θ + 4.193055 x 10-3θ 2 - 4.082383 x 10-5θ 3 )
(5.6.11)
The total loss coefficient for the bundle with inlet and downstream losses due to oblique flow is obtained by the addition of Equations 5.6.8 and 5.6.9.
1 1 Khe θ = Khe + - 1 - 1 + 2 K 0.5 ci + K d sin θ m sin θ m
(5.6.12)
where θm = the mean flow incidence angle
Due to the flow distortion downstream of the bundle, the actual mean flow incidence angle will not be uniform along the bundle face and is smaller than θ (Fig. 5.6.5). This effect should be taken into account when evaluating the inlet loss in Equation 5.6.12. The curve shown in Figure 5.6.5 is represented by the following empirical relation: θm = 0.0019 θ2 + 0.9133 θ – 3.1558
(5.6.13)
Fig. 5.6.5 Mean Flow Incidence Angle as Function of the Semi-Apex Angle
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From Equation 5.6.12, the change in total pressure is 2
∆ pt θ =
0.5 m ρ A fr
1 1 - 1 - 1 + 2 K 0.5 K he + ci + K d (5.6.14) sin θ m sin θ m
When heat is transferred to the airstream, the total pressure drop across an inclined heat exchanger bundle is
2 1 1 1 m K he 1 1 + + ∆ pt θ = 0.5 ρ sin θ - 1 sin θ - 1 A 2 ρ ρ fr m m o i i
(5.6.15)
Kd + 2 K 0.5 ci }+ ρo where Khe is obtained from Equation 5.4.22 or Equation 5.4.23. The corresponding loss coefficient is
Khe θ = Khe +
2 ρi K d 2 ρo 1 1 - 1 - 1 + 2 K ci0.5 + ( ρo + ρi ) sin θ m sin θ m ( ρ o + ρi )
(5.6.16)
The loss coefficient given by Equation 5.6.12 may be determined directly in an experimental test and expressed in terms of the flow parameter Ry. In the case of non-isothermal conditions, the resultant loss coefficient is approximately
Kheθ = aK Ry bKh + h
2 ρi - ρo σ 2 ρi + ρo
(5.6.17)
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Oblique flow experiments Meyer and Kröger tested a range of finned tubes and determined the tube geometry had an influence on the inlet loss under oblique flow conditions. Kotzé et al. compared Equations 5.6.8 and 5.6.12 to isothermal experimental results obtained on single-inclined three-, four- and eight-tube row bundles. Details of their finned tubes and tube arrangement in the test bundle are given in Example 5.4.1. As shown in Figure 5.6.6, good agreement between theory and experiment is obtained for the eight-row bundle, which has a high flow resistance. Some discrepancy is observed when Equation 5.6.12 is applied to bundles where the outlet flow is directed by the duct walls. However, Equation 5.6.8 also predicts the pressure drop accurately for the remaining bundles where the flow approaches obliquely but leaves the bundle normally.
Fig. 5.6.6 Pressure Drop across Heat Exchanger
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When the total pressure drop across an inclined bundle is determined experimentally in isothermal conditions, losses under non-isothermal conditions must be approximated since inlet and outlet losses are not known. They are dependent on the respective densities.
Fig. 5.6.7 Heat Transfer in Three- and Four-Row Heat Exchangers
The heat transfer characteristics of the single three- and four-row heat exchanger bundles are shown in Figure 5.6.7. There is a small increase in heat transfer rate compared to normal flow conditions when the flow enters obliquely but leaves the bundle in a normal direction.
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However, a reduction is observed where the flow enters and leaves the bundle at 30°. Experimental results obtained by Becker are in agreement with this trend. In the case of the four-row V-bundle arrangement, the heat transfer coefficient is similar to that obtained under normal flow conditions. The results of the isothermal pressure drop measurements across a tworow plate-finned heat exchanger are shown in Figure 5.6.8. Based on a curve fitted through the normal flow data, the losses for other arrangements are predicted. Good agreement is obtained between theory and experiment up to inclination angles of 25°. The heat transfer results are shown in Figure 5.6.9. Where the inlet flow only is inclined, there is an increase in the heat transfer rate as observed in the round-tube bundles. There is also a slightly higher heat transfer rate compared to the normal flow conditions when both the inlet and the outlet flows are inclined.
Fig. 5.6.8 Pressure Drop across Two-Row Plate-Finned Heat Exchanger
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Fig. 5.6.9 Heat Transfer in Two-Row Plate-Finned Heat Exchanger
The enhanced heat transfer may be due to eddy shedding and boundary layer development near the leading edge of the tilted fins. Samie and Sparrow have observed significant increases in the Nusselt number during flow across a single yawed finned tube. Somerton et al. observed similar trends in the case of inclined radiator cores. Additional test results are reported by Fisher and Bucher and Monheit and Freim. In view of their method of testing, care should be taken when interpreting their pressure drop data. Monheit and Freim find there is no measurable effect of the angle of inclination on the thermal performance of banks of finned tubes. Van Aarde and Kröger obtained test results for flow losses in heat exchanger arrangements shown in Figure 5.6.10. They incorporated process fluid ducts and walkways similar to those found in air-cooled condensers and defined a total loss coefficient that includes kinetic energy losses at the outlet of an A-frame array, i.e.,
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K θ t = [( p1 + ρ v12 / 2 ) - pa] / ρ v i2 / 2 = Ki θ + K f + K e + K dj + K o
1 1 = K he + - 1 - 1 + 2 K ci0.5 + K dj + K o sin θ m sin θ m
(5.6.18)
or for non-isothermal flow
Kθ t = K he +
2 ρ2 1 2 ρ1 - ρ 2 1 + - 1 - 1 + 2 K 0.5 ci σ 2min ρ 1 + ρ 2 (ρ 1 + ρ 2) sin θ m sin θ m
(5.6.19) +(K dj + K o) 2 ρ 1 / ( ρ 1 + ρ 2)
θm is given by Equation 5.6.13, while Khe is obtained from Equation 5.4.23.
Fig. 5.6.10 Section of an Array of A-Frames
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The inlet contraction loss coefficient based on the normal upstream velocity for turbulent flow between plates follows from Equation 2.3.7
K ci = [ (1 - 1 / σ c )/ σ ]
2
(5.6.20)
where σ is the ratio of the flow area between the fins at their leading edge to the corresponding area immediately upstream of the fins. For plate fins, σc is given by Equation 2.3.10. The value of Kci is 0.05 for typical industrial round finned tube bundles. The jetting loss coefficient is expressed by the following relation 2 0.4 L L L 28 L K dj = - 2.89188 w + 2.93291 w t b L t L s L s θ Lt 0.5
L L + exp (2.36987 + 5.8601 x 10-2 θ - 3.3797 x 10-3 θ 2) s t L b L r
2
(5.6.21)
where θ is in degrees. The outlet loss coefficient is given by 2 3 L L L K o = - 2.89188 w + 2.93291 w s L t L b Lt
(5.6.22) 2
d d + 1.9874 - 3.02783 s + 2.0187 s 2 L b 2 L b
Lt L s
2
These equations are valid for Khe ≥ 30 semi-apex angles of 20° ≤ θ ≤ 35° 0 ≤ ds/(2Lb) ≤ 0.17886 0 ≤ (Lw /Lt) ≤ 0.09033
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Equation 5.6.19 is conservative at low air velocities through the heat exchanger. When applying Equations 5.6.12, 5.6.16, 5.6.18, and 5.6.19 to tubes where the fins do not cover the entire effective frontal area (Fig. 5.6.11b), the second term on the right side of the equations must be multiplied by the square of the area ratio, i.e., [Pt /(Pt – d)]2. In practice, this tends to give an inlet loss coefficient higher than measured.
Fig. 5.6.11 Finned Tubes (a) (b)
Due to losses, the airflow distribution through the heat exchanger bundles is not uniform (Fig. 5.6.12). The velocities of measured bundles follow trends predicted by Moore.
Fig. 5.6.12 Dimensionless Velocity Distribution through Bundle
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Various methods have been proposed to reduce the additional pressure loss during flow through inclined bundles. These include the introduction of guide vanes or airfoils described by both Moore and Fischer or the use of formed-plate fins reported by Phillips (Fig. 5.6.13). The latter not only reduce the total pressure drop but also improve the heat transfer because of more uniform flow through the bundle.
Fig. 5.6.13 Formed Plate Fins
5.7 Corrosion, Erosion, and Fouling A finned tube heat exchanger may be erected in an area where the environment is corrosive or harsh. Corrosive atmospheres may contain gases including sulphur dioxide, chlorine compounds, carbon monoxide, carbon dioxide, and NOx. These impurities combined with moisture, rain, hailstones, snow, ice, and other airborne inorganic materials such as dust, flyash, coal dust, etc. encourage the corrosion of finned surfaces in a heat exchanger. Physical damage during erection, poor maintenance, and ineffective cleaning accelerate this trend.
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Corrosion problems have been experienced by a few large air-cooled heat exchangers in different parts of the world. The natural draft Ibbenbüren tower in Germany was fitted with aluminum finned tubes. Irregular cleaning, poor maintenance, and polluted air of the industrial environment have been cited as the major causes for the deterioration of the finned tubes. The dry Rugeley tower in England was fitted with the same type of all-aluminum tubes as the Ibbenbüren tower. Intermittent operation, high moisture content, and extreme chlorine concentrations in the atmosphere caused serious corrosion of the aluminum. Some corrosion has been observed on the surface of the unprotected steel core tube in the Grootvlei 6 tower in South Africa. Galvanized surfaces may be attacked by atmospheres containing sulfides, especially when these collect in suspensions on the fin according to Schulenberg. Many laboratory and field tests have been conducted to determine the characteristics of different types of finned tubes subjected to corrosive atmospheres. An extensive survey of materials and corrosion performance in dry-cooling applications was conducted by Battelle’s Pacific Northwest Laboratories. The experience gained through many years of operation of industrial plants is summarized in the survey and some of the major causes of corrosion are identified. For example, the service life of galvanized coatings subjected to different types of atmospheric environments is shown in Figure 5.7.1.
Fig. 5.7.1 Service Life or Zinc Coatings versus Zinc Thickness
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Other air-side corrosion tests were conducted on six different types of finned tubes employed in dry-cooling towers in the period from 1977 to 1983 by the Vereinigung der Grosskraftwerksbetreiber (VGB). VGB collaborated with three utilities and five producers of finned tubes, including Alesa Alusuisse, Zürich; Balcke-Dürr, Ratingen, Germany; GEA, Bochum, Germany; Linde, Munich; and Transelektro/EGI, Budapest according to VGB-TW 504 and Süthoff. In a paper on heat exchanger materials for dry-cooling towers, Bodás refers to galvanized structural surfaces that show signs of corrosion after 5 to 6 years of exposure to a polluted industrial atmosphere. Palfalvi also refers to the corrosion of galvanized surfaces in the polluted environment at the Gargarin power station. In Gargarin, the average thickness of the zinc layer on the heat exchanger support frames and louvers was significantly reduced after more than 12 years of operation. Bodás noted that the fin efficiency would be reduced if a similar reduction in zinc thickness were to occur on a galvanized finned surface. The observed corrosion appears to occur primarily on the cold outside surfaces of the bundle frames. The VGB study concluded that galvanized (50 to 80 µm zinc) steel surfaces offered satisfactory corrosion resistance. Where the protective zinc coating was locally damaged, adequate cathodic protection was maintained. The loss in zinc was found to be much less on surfaces operated under continuous warm conditions in a cooling tower than in samples tested at ambient temperatures. The VGB study also found that unprotected aluminum was not likely to offer adequate corrosion resistance when installed in a cooling tower for a period of 25 to 30 years; however, coated aluminum surfaces were acceptable. Where the coating is damaged, corrosion tends to occur primarily near the inlet of the heat exchanger bundle. Aluminum finned tubes have been installed in numerous dry-cooling towers, and good performance characteristics have been maintained for many years according to Bódás. Finned tubes protected by an electrophoretic coating reported by Höfling are claimed by Fischer to have enhanced performance characteristics. Extruded bimetallic finned tubes shown in Figure 5.1.1 ensure protection of the steel core tube except at the ends where additional protection is required after welding. In extreme cases, the fins should be coated since they may suffer intergranular corrosion due to substantial deformation during the manufacturing process as reported by the Battelle Pacific Northwest Laboratories.
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Wrapped-on aluminum finned tubes (Fig. 5.1.3) were installed at the Grootvlei cooling towers in South Africa. In Grootvlei 5, the outside surface of the steel core tube was galvanized, and no corrosion problems were experienced after 15 years of operation. In the case of Grootvlei 6, where the core tube was not galvanized, corrosion occurred on the upper tubes and reduced the effective overall heat transfer coefficient by approximately 10% according to Ham. Although some corrosion has occurred in a few air-cooled heat exchangers exposed to harsh environments, it is unlikely similar problems will be experienced in future plants. Practice has shown both galvanized steel and chemically oxidized aluminum surfaces are suitable for use in air-cooled heat exchangers and dry-cooling towers with a specified service life of more than 25 years. These can be treated with the modified Bauer-Vogel process (MBV) or other protection according to Bódás. Continuous operation at temperatures higher than ambient tend to reduce galvanic corrosion and the removal of zinc. Regular cleaning further extends the service life. Bódás also suggests, where the environment is corrosive, it may be advisable to erect a corrosion test stand at the site for several years before selecting a heat transfer surface. Such a unit can also be used to monitor fouling characteristics. Corroded finned surfaces and fin deformation reported by Paikert cause the air-side flow resistance to increase. Erosion of finned surfaces has been observed in areas where sandstorms are common. Adjustable louvers may be required in such cases to protect the heat exchangers. Erosion is not uncommon in mechanical draft systems operating in dry, sandy areas. Fouling of the finned surface of an air-cooled heat exchanger tends to increase the thermal and the flow resistance on the air side, resulting in a net reduction in the heat transfer rate. Although the effect of dirt deposits on the overall heat transfer coefficient may be small, the reductions in airflow rate and mean temperature difference may reduce performance significantly. An increased pressure drop of 20% caused by fouling has been reported by Rose while Preece et al. observed a 52% increase in pressure drop due to corrosion of fins. Russell suggests pressure drop increases of up to 40% were not unrealistic in practical designs. A similar trend is observed by Cowell in vehicle radiators or in most air-cooled heat exchangers. Clearly, exact figures depend on the heat exchanger type and the type and magnitude of the fouling layer. Laboratory experiments conducted by Hunn showed increases in pumping power of up to 45% are required to maintain the same rate of heat transfer. This occurred when different samples of dry and wet coal dust were introduced into the airstream flowing through finned tube geometries characteristic of dry-cooling
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tower applications. The fouling layer thickness increased to the point where 48% percent blockage of the freeflow area occurred. Some uncertainty exists concerning the effect of fouling on the performance of practical systems. One of the reasons for this uncertainty is the diverse nature of the material that causes fouling. These range from normal atmospheric dust to organic matter, products of corrosion, soot, and insects. Fouling tends to be a transient phenomenon. Based on the deposition model from Kern and Seaton, Bemrose and Bott postulate the following relationship for the friction factor on the finned side as a measure of fouling at a time τ: f = fo [1 + a(1 – e–bs)]
(5.7.1)
where fo = the friction factor corresponding to the original clean surface at τ = 0 a = constant b = constant Fouling tests were conducted by Bemrose and Bott in a test rig containing one- to four-row bundles of spiral-wound finned tubes. Calcium dust with a particle median diameter of 14 µm was introduced into the airstream for a range of air velocities. The increase in friction factor followed the predicted trend given by Equation 5.7.1. As observed in practical heat exchangers subjected to dust laden airstreams, more foulant was seen to deposit on the front and rear faces than in the middle of the bank of tubes. The trend predicted by Equation 5.7.1 is not necessarily representative of all types of fouling, and increases in flow resistance following quite a different function have been observed. Since most fouling in air-cooled systems occurs at the inlet to the heat exchanger or at the first two tube rows in a multi-row tube bundle, cleaning is done best from the outlet side using jets of compressed air or water. Rinsing or low-pressure jets are adequate for lightweight, slightly fouled fins. Pressures of 50 bar to 200 bar are not uncommon in the case of galvanized steel tubes. At some sites, soap is added to the water jet, and in extreme cases, fly ash or
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fine sand may be introduced, but care must be taken not to erode the protective coating or the fin itself. Steam cleaning is also used in some extreme cases. Aircraft jet engines have been used to remove particulates from finned surfaces in dry-cooling towers in the USSR according to Neal. At Grootvlei, a gradual buildup of impurities during a number of years formed a tough layer on the finned surface that could not be removed completely by the previously mentioned methods. In practice, evaluate the potential for fouling in a particular area by exposing various types of finned tube heat exchanger bundles to a stream of ambient air for an extended period. In addition to pollutants generated in industrial areas, both local flora and fauna may lead to fouling. During the flowering season or in the fall, fouling may be more pronounced— particularly during windy periods. The presence of seeds and other plant matter, especially during harvest time, may further encourage fouling. Alternating periods of dust and rain will cause a buildup of material on the finned surface. Flies, butterflies, moths, locusts, and other insects, especially during breeding and hatching periods, tend to clog up the system. Unless essential, all lights in the vicinity of the finned surfaces and the ducts leading to them should be extinguished to avoid attracting insects at night. Warm heat exchanger surfaces tend to attract insects, which may be followed by birds. The result is an accumulation of fluff and feathers on the inlet side of the finned surface. The orientation of the plant should take into account the location of potential sources of pollutants such as coal or ash as well as wind direction. Oil leaking from poorly maintained gearboxes or other lubricated devices is another fouling hazard. The frequency of cleaning is determined by the type and rate of fouling that occurs in a particular application. Semiannual or annual cleaning is sufficient to control fouling at many locations according to Schulenberg. The installation of a suitable screen may be of value in the control of certain types of fouling. Screens are also installed in areas where damage by hailstones may occur. Fouling on the inside surface of finned-tube air-cooled heat exchangers is as varied and complex as that occurring on the air side. Considerable information on this type of fouling is available in the literature from authors such as Somerscales and Schnell.
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5.8 Thermal Contact and Gap Resistance Finned tubes used in air-cooled heat exchangers may be classified under four broad headings descriptive of the nature of the junction between tube and fins. •
integral fins
•
embedded fins
•
soldered, galvanized or brazed fins
•
interference-fit fins
Examples of interference-fit fins are edge tension and footed tensionwound helical fins, helical-extruded finned muff, and expanded-plate fins. The finned tube types rely on residual radial stresses of the finning operation to maintain intermetallic heat transfer contact between them. The tube wall has a lesser diameter and the fin base a greater diameter than either would have in the absence of the contact pressure. The sum of both displacements is the interference. Most interference-fit finned tubes are bimetallic, aluminum being the most common fin material. At elevated temperatures, the fins tend to expand away from the tube walls of lesser thermal expansion coefficients with a possible loss of thermal contact. This occurs even though the fins in aircooled equipment are at a temperature lower than that of the tube. The imperfect metal-to-metal contact results in a resistance to heat transfer known as contact resistance. A gap exists between the surfaces in poor quality finned tubes or at high temperatures when almost no metal-to-metal contact exists and no contact pressure is exerted. The resistance to heat transfer is referred to as the gap resistance. Figure 5.8.1 defines various dimensions of an extruded finned tube under these conditions.
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Fig. 5.8.1 Extruded Bimetallic Finned Tube
Analysis of contact and gap resistance To analyze the contact and gap resistance problem, Gardner and Carnavos considered an annular disk subjected to a uniform radial load and heat flux. They derived the following expression for the radial gap, g. The gap is between the fin base and the tube wall upon elastic relaxation from a state of residual stress brought about by the finning operation
g =
R ia do R a / ef -α t (α f - α t)(T i - T p) - α f 1 R + R 2 R + R ta cga cga ta
x (T i - T a) + a (pc - pcp )
408
]
(5.8.1)
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where Ti = the temperature of the fluid inside the tube Tp = the fin and tube temperature during production Ta = the ambient fluid or air bulk temperature pc = the contact pressure between the fin base and the tube pcp = the contact pressure during original production α = the thermal expansion coefficient Furthermore,
a=
1 Ef
2 d 2f + d 2o t d 2 + (d o - 2 t t ) 2 2 + m f + f o2 2 -m t d d E P d ( d 2 t ) t f o f o o t
(5.8.2)
where E = the elastic modules m = Poisson’s ratio The total heat transfer resistance in the absence of any gap resistance is Rta = Ra/ef + Ria
(5.8.3)
where the fin effectiveness using Equation 3.3.11 is ef = 1 – Af (1 – ηf)/Aa and Ra is the thermal resistance on the outside or finned side. Ria includes the resistances due to the fluid film inside the tube as well as the tube wall. All refer to the total outside surface area. The sum of any contact and gap resistance based on the outside surface area is Rcga = Rca + Rga
(5.8.4)
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Negative values of the gap, g, are not possible. Only zero or positive values have significance. When g = 0, fins and core tube are in contact with no gap resistance, and the contact pressure may be obtained from Equation 5.8.1.
pc = pcp -
1 [(α f - α t) (T i - T p) a (5.8.5)
R a / ef α t R ia - α f 1 (T i - T a) R ta + R ca R ta + R ca The subject of thermal metal-to-metal contact resistance has been studied by numerous investigators according to Lemczyk. Contact correlations developed by Shlykov and Ganin and more recently by Yovanovich appear to be the most reliable. The former presents a simple mathematical model of parallel resistance between the places of actual contact of the two surfaces and the voids existing between them. For two dissimilar materials, they derived the following expression for contact resistance:
Rc =
1 ka pc k f k t 4 + 4.2 x 10 s t bp k f + k t ε
Based on the outside surface area in the case of a finned tube,
R ca ≈
df2 - do2 k p k k 2 do Pf a + 4.2 x10 4 c f t s t bp k f + k t ε
(5.8.6)
where ε = the arithmetic mean value of the height of the micro roughness of the surfaces in contact ka = the thermal conductivity of gases or air entrapped among roughness asperities between the surfaces
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bp = a function of the cold workability of the metal value for aluminum = 3 value for copper = 5 st = the yield stress of the tube material equal to the ultimate stress According to Gardner and Carnavos, the initial contact pressure, pcp, is estimated to be 0.67 of the yield stress of the fin material as originally cold worked. In the extreme case of fully annealed aluminum, this value is approximately 24.1 x 106 N/m2. Tests performed by Young and Briggs suggest the value is a good approximation. Smith, Gunther, and Victory performed a series of strain gauge tests to determine initial contact pressures on commercial tubes available at that stage. They obtained values of nearly 30 x 106 N/m2. Even higher values are possible for good quality tubes according to Coetzee. Under certain conditions, the contact pressure, pc, may become zero. In this case, using Equation 5.8.6, the contact resistance becomes
R ca = ε (d 2f - d 2o) /(2 k a d o Pf )
(5.8.7)
Any further increase in the inside fluid temperature, Ti, will result in the formation of a gap between the tube wall surface and the base of the finned section. The thermal resistance of the gap alone, based on the outside surface area, is
R ga = g (d f2 - d o2) /(2 k a d o Pf )
(5.8.8)
The total thermal resistance under these conditions is obtained by the addition of Equations 5.8.7 and 5.8.8
R cga = (d f2 - d 2o) (ε + g) /(2 k a do Pf )
(5.8.9)
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Substitute g from Equation 5.8.1 into Equation 5.8.9, and find
d 2 - d 2 2ε R cga = f o + (α f - at) (T i - T p) 4 k a Pf d o αt R ia R /e - αf 1 - a f (T i - T a) - apcp R ta + R cga R ta + R cga or
(d2r - d2o) 4 R ta k a Pf - 2e - (α - α ) (T - T ) f t i p 4 k a Pf (df2 - do2) do
R 2cga + R cga
+ a f (T i - T a) + a pcp + R ta [(df2- do2 ) / (4 k a Pf )] (5.8.10)
[
]
x α f (T i - T a) - (α f - α t) (T i - T p) + a pcp - 2 ε / d o
-
[(d2f - d2o) /(4 k a Pf )](T i - T a)[ α f R a / ef + α t R ia]= 0
In general, this equation gives satisfactory results if the value of pcp is known. Kulkarni and Young graphically present the contact and gap resistances for specific cases based on these equations. The previous analysis is applied to a particular finned tube under given operating conditions shown in Figure 5.8.2 from Coetzee. Initially, the thermal resistance increases gradually until g = 0 followed by a rapid increase. From the results, it is obvious the initial contact pressure will have a significant influence on the contact and gap resistances between the aluminum fin and the steel tube. Limited experimental results are in agreement with this trend.
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Fig. 5.8.2 Contact and Gap Resistance as a Function of Production Contact Pressure Pcp
If the heat transfer coefficient on the outside surface of the finned tube is increased while other conditions remain unchanged, the contact and gap resistances tend to decrease (Fig. 5.8.3). An initial contact pressure of 30 x 106 N/m2 is assumed.
Fig. 5.8.3 Contact and Gap Resistance as a Function of Outside Heat Transfer Coefficient ha
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An increase in thermal resistance can be expected when the fluid temperature on the outside surface is increased if other parameters are the same. This is illustrated in Figure 5.8.4.
Fig. 5.8.4 Contact and Gap Resistance as a Function of Ambient Temperature Ta
Example 5.8.1 Air at a temperature of 33.4915 °C flows across an extruded bimetallic finned tube shown in Figure 5.8.2. Saturated steam at 100.7 °C condenses inside the tube. The air-side heat transfer coefficient is ha = 86.2 W/m2 K. The condensation heat transfer coefficient inside the tube is hi = 10099.21 W/m2 K. The elastic module of the aluminum fin material is 6.945665 x 1010 N/m2, and that of the steel tube is 1.940773 x 1011 N/m2. The thermal expansion coefficients are 2.36516 x 10–5 K–1 and 1.156774 x 10–5 K–1. The respective thermal conductivities are 227.7142 W/mK and 52.42969 W/mK. The Poisson ratio is 0.334 for aluminum and 0.292 for steel. At a production or fabrication temperature of 21.11 °C, the contact pressure is 38.2 x 106 N/m2. The arithmetic mean value of the height of the micro roughness of the surfaces in contact is 10–6 m. The yield stress of the steel tube material is 5.4 x 108
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N/m2. Determine the thermal contact resistance between the fin root and the core tube, based on the bond area.
Solution The approximate surface area of a single fin is found from
Af =
π 2 2 π (df - d r ) = (57. 52 - 27. 742 )10-6 = 3.985 x 10-3 m2 2 2
The exposed root area is approximately
A r = π d r (Pf - t f ) = π x 27.74(2.79 - 0.4) x 10-6 = 0.2083 x 10-3 m2 By adding the previous values, the total effective air-side area around one fin is found: Aa = 4.1933 x 10-3 m2 The area inside the tube is
A i = π di Pf = π x 21.5 x 2.79 x 10-6 = 0.18845 x 10-3 m2 The thermal resistance due to the fluid film inside the tube as well as the tube wall resistance, all referred to the total outside surface area, is given by
1 n (do / di) A i A a R ia = (1 / h i + R t ) A a / A i = + 2 π k t Pf A i hi
For a given heat transfer coefficient, the fin efficiency may be calculated. Using Equation 3.3.13,
57.5 57.5 ϕ= - 1 1 + 0.35 n = 1.346515 27.74 27.74 Furthermore, 0.5
b = [2 x 86.2 /(227.7142 x 0.4 x 10-3) ] = 43.505441
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The fin efficiency follows from Equation 3.3.12.
ηf =
tanh (43.505441 x 27.74 x 1.346515 / 2000) = 0.8258 (43.505441 x 27.74 x 1.346515 / 2000)
Using Equation 3.3.11, the surface effectiveness is
ef = 1 - 3.985 (1 - 0.8258) / 4.1933 = 0.8345 From Equation 5.8.3, it follows that
R ta = 1 /( h a ef ) + R ia = 1 /(86.2 x 0.8345) + 2.9638 x 10-3 = 1.6865 x 10-2 m2 K/W Using Equation 5.8.2,
a=
+
1 57. 52 + 25. 42 + 0.334 10 6.945665 x 10 57.5 2 - 25. 42
25. 42 + (25.4 - 2 x 1.95 )2 0.4 2 - 0.292 11 10 1.940733 x x 2.79 25.4 - (25.4 - 2 x 1.95 )
= 3.044423 x 10-11 m 2 /N The desired contact pressure is found by solving Equations 5.8.5 and 5.8.6 simultaneously. This is done by following an iterative procedure. Assume an arbitrary value of Rca = 4.152 x 10-4 m2 K/W, substitute this into Equation 5.8.5, and find
pc = 38.2 x 106 -
1011 [(2.36516 - 1.156774)10-5 (100.7 - 21.11) 3.044423
1 - 2.36516 x 10-5 1 -2 -4 86.2 x 0.8345 (1.6865 x 10 + 4.152 x 10 ) -
416
1.156774 x 10-5 x 2.9638 x 10-3 (100.7 - 33.4915) = 1.2438 x 107 N/ m 2 -2 -4 1.6865 x 10 + 4.152 x 10
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The thermal conductivity of the air entrapped between the surface asperities at the bond interface is evaluated using Equation A.1.4 at 100.7 °C and found to be 3.148 x 10–2 W/mK. Substitute the previous values of pc and ka into Equation 5.8.6, and find
R ca =
x
1 2 x 25.4 x 2.79
(57. 52 - 25. 42) 3.148 x 10-2 1.2438 x 107 227.7142 x 52.42969 + 4.2 x 104 -6 8 10 5.4 x 10 x 3 227.7142 + 52.42969
= 4.152 x 10-4 m 2 K/W Based on the surface area, the approximate contact resistance is
R c = R ca x 2 do Pf /(d 2f - d2o) = 4.152 x 10-4 x 2 x 25.4 x 2.79 /(57. 52 - 25. 42)
= 2.21 x 10-5 m2 K/W
Measuring contact and gap resistance Gardener and Carnavos present the results of experimental studies on contact and gap resistance. The heat transfer rate is measured in a four-row bundle of finned tubes installed in a wind tunnel. Because of the major thermal resistance on the air side, the testing method is reliable only in a laboratory environment where good quality tubes with a small contact resistance are tested. It is an expensive test method. Young and Briggs tested single-finned tubes of the extruded muff type in a concentric pipe heat exchanger. The test fluids were recirculating streams in axial counterflow. The fluid through the tube was Mobiltherm 600, and the fluid through the annular space between the shell and the finned tube was Mobiltherm Light. Heat was provided to the tube stream by an electrical resistance heater and discharged from the shell stream through a water-cooled heat exchanger. One
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reason for using a heat transfer oil rather than air on the finned side of the tube was to reduce the heat transfer resistance on that side, making the contact or gap resistance a large fraction of the overall resistance. Smith, Gunter, and Victory tested single extruded muff and footed tensionwound tubes in an apparatus developed originally for industrial manufacturing control purposes. Air is blown across the finned surface while steam condenses inside the tube. It is unlikely this particular type of facility can guarantee repeatable quantitative results when good quality tubes are being tested in an industrial environment. This is true because of the relatively high thermal resistance on the air side according to Coetzee. Other studies include those of Young et al., Dart, Eckels, Christensen and Fernandes, Sheffield et al., and Critoph et al. Electrical methods to determine the contact resistance have been considered but found to be impractical. In another indirect method of testing, the core tube is withdrawn mechanically from a specified length of finned tubing. The force required and the thermal contact resistance are supposed to be related. This is not necessarily true since the core tube tends to be very sensitive to the mechanical and geometric characteristics of the contact surfaces according to Ernest. In view of the limitations of the previously described test equipment, Coetzee and Kröger recommend a test method for good quality extruded bimetallic finned tubes. They suggest having saturated steam condense on the outside finned surface at approximately 100 °C and hot water flow inside the tube. The method is accurate and can be applied in practice by measuring the difference between the steam and the water temperatures as well as the condensate flow rate. The thermal contact resistance will increase in finned tubes subjected to thermal cycles or excursions. Although no theoretical method is available for predicting the performance degradation under such conditions, experimental studies have been carried out. Smith et al. used a multipurpose cycling apparatus capable of operation with the heating medium always flowing in the tubes and steady or intermittent airflow or water spray on the finned side. In the case of air cycling between 24 °C and 47 °C, the airflow was continued for 1.5 minutes and the total cycle time was approximately 5 minutes. For water cycling between 47 °C and 115 °C, the spray time was 45 seconds and the total cycle time was 5 to 6 minutes. The finned tubes were 6.1 m long and were the extruded muff and L-footed tension-wound types. A minimum of 500 cycles were run on each tube. The results of these tests are shown in Figure 5.8.5 from Smith.
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Fig. 5.8.5 Increased Resistance Due to Thermal Cycling
In properly galvanized steel-finned tubes (Fig. 5.8.6a), the thermal contact resistance should be negligible. However, poor temperature control or impurities in the zinc bath may result in the formation of cavities, poor penetration, and an irregular coating thickness (Fig. 5.8.6b).
a
b
Fig. 5.8.6 Galvanized Finned Tube (a) Good Quality (b) Poor Quality
Taborek evaluates accepted industrial practices and notes the lack of progress in the standardization of test procedures for determining the thermal contact resistance in finned tubes.
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5.9 Free Stream Turbulence The air-side heat transfer and pressure drop characteristics of heat exchangers are obtained experimentally in wind tunnels in which the air upstream of the heat exchanger has a low turbulence level of 1–3%. Where the heat exchanger consists of bundles of plain or finned tubes, the heat transfer coefficient in the first rows is lower than further downstream due to the higher levels of turbulence that exist in the wake stream. In real aircooled heat exchangers, the level of turbulence in the inlet airstream may be much higher. This is due to the absence of smooth inlets or the presence of flow-control devices, turning vanes, louvers, bar grids, grills, or screens. In a forced-draft unit, the wake of the fan introduces a degree of turbulence. Baines and Peterson experimentally investigated the establishment and decay of turbulence downstream from biplane lattices or screens. They compared the decay to theoretical expressions derived by other investigators and found the approximate expression of Frenkiel most closely represents the true decay, i.e., Tu = 1.12 (ds /xs)
5/7
(5.9.1)
where xs = the downstream distance from the screen ds = the wire diameter Experiments indicate a distance of 5–10 times the wire pitch or mesh length downstream from any screen is necessary to ensure good flow establishment. The maximum intensity of turbulence is reached within 2 or 3 mesh lengths from the screen. Equation 5.9.1 can be used in practical designs to predict the turbulence downstream from a screen if the Reynolds number based on the free stream velocity (Re = ρvds/µ) exceeds 100. In the case of a bar grid, vortices separate from the cylindrical bars and move downstream in the fluid. At larger distances from the grid, it is possible to discern a regular pattern of vortices, which move alternatively clockwise and counter-clockwise. This is known as the von Kármán vortex street. This
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vortex street moves in front of the grid with a velocity smaller than the free flow stream velocity. The fluctuations are oriented in one plane. They create fluctuations only along the direction of free stream movement perpendicular to the cylindrical bars of the grid. Correlations of turbulence intensity are shown in Figure 5.9.1. These were measured by Fourie and Kröger as a function of Reynolds number and grid position together with the frequency of vortex shedding and based on the measurements of Blenk et al.
Fig. 5.9.1 Measured Turbulence Intensity and Corresponding Frequency of Vortex Shedding behind Bar Grid
Stephan and Traub determined the influence of turbulence intensity on heat transfer and pressure drop in bundles consisting of in-line and staggered plain tube banks. The tube diameter was 28 mm, and the longitudinal pitchto-diameter ratio was maintained at 1.54. The transversal ratio varied between 1.54 and 3.07. The air-side Reynolds number ranged from 2 x 104 to 1.5 x 105 . It was based on the tube diameter and the mean velocity through the minimum flow area. Different turbulence intensities were generated with the aid of wire screens having wire diameters from 1–2.5 mm, pitches from 3.5–12 mm, and a biplanar grid consisting of a 6 mm wide bar at a pitch of 30 mm. It was possible to generate turbulence levels of 3.3–7% with the screens and 25% with the grid 90 mm downstream of the screens or the grid. In general, their turbulence measurements were in agreement with the values predicted by Equation 5.9.1.
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The correlation of heat transfer tests for bundles consisting of one to five rows of in-line plain tubes is shown in Figure 5.9.2 in the form of a reduced Nusselt number, Nu/Nuo. Nuo denotes the Nusselt number for the same bundle without a grid as a function of the inlet turbulence level.
Fig. 5.9.2 Increased Heat Transfer in an In-Line Tube Bundle
The enhancement of the mean heat transfer coefficient for bundles with five rows is 8%. It is 9.5% with four rows, 13.5% with three rows, and 42% for the single row at a turbulence intensity of 25%. The drag coefficients or friction factors for these bundles are found to be almost independent of the free stream turbulence level over a wide range of Reynolds numbers. A decrease in flow resistance is observed for the single row bundle only for very high turbulence levels and high Reynolds numbers. This is due to a downstream shift in the boundary-layer separation point. Zozulya et al. studied the influence of free stream turbulence on staggered finned tubes. The outside tube diameter was 22 mm with fins having a diameter of 38 mm, a thickness of 0.8 mm, and an interfin spacing of 3.8 mm. The transversal tube pitch was 53 mm and 30 mm in the longitudinal direction. The turbulence level of the incoming airstream was adjusted by installing orthogonal steel strips 6, 12, and 25 mm wide at
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pitches of 12, 24, and 45 mm. Figure 5.9.3 shows an increase in the flow turbulence to 22–25% having the greatest effect on the heat transfer coefficient for the first row with an increase of 20% at vc = 5 m/s and 35% at vc = 15 m/s. There is no change in the fourth row.
Fig. 5.9.3 Change in Heat Transfer Due to Turbulence
Tests were conducted by Fourie and Kröger on bundles of staggered finned tubes having root diameters of 27.2 mm. The mean fin thickness was 0.37 mm at a 2.54 mm pitch. Tube pitches were 55 mm in the longitudinal direction and 66.7 mm in the transversal direction. Free stream air speeds were varied from 2 m/s to 10 m/s. This resulted in Reynolds numbers, based on the tube root diameter and the minimum flow area in the bundle, of 6500 and 32,500. Square screens and a bar grid were installed at different positions upstream of the bundles. The screens had wire diameters of 0.5 mm (fine), 1.5 mm (medium), and 3 mm (coarse) at pitches of 1.85, 6.3, and 16 mm respectively. The bar grid consisted of 12.7 mm bars with a pitch of 25.4 mm. The results of their heat transfer tests with the coarse screen located in different positions in front of one-, two-, and three-row bundles are shown in Figure 5.9.4. The largest increase in heat transfer coefficient is observed when the screen is located close to the bundle at the higher Reynolds number. No corresponding change in pressure differential across the bundle was observed. Less enhancement was achieved with the finer screens.
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Fig. 5.9.4 Change of Heat Transfer with Coarse Screen Upstream of Finned-Tube Bundle
As shown in Figure 5.9.5, a maximum enhancement of 30% was observed in the single-row bundle with the bar grid located 50 mm upstream. For this particular case, a reduction in pressure drop was observed (Fig. 5.9.6).
Fig. 5.9.5 Change of Heat Transfer with Bar Grid Upstream of Finned Tube Bundle
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Fig. 5.9.6 Change of Pressure Differential with Bar Grid Upstream
The turbulence levels for the various tests can be determined with the aid of Equation 5.9.1 or from Figure 5.9.1 for the bar grid. No measurable change in heat transfer was observed for bundles having more than three tube rows. Nirmalan and Junkhan studied the influence of free stream turbulence intensity on five configurations of plate- and louvered-fin, single-pass, and crossflow heat exchanger surfaces. The geometric details of these are given in Table 5.9.1.
Table 5.9.1 Heat exchanger bundle geometries.
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Streamwise turbulence intensities of up to 16% were obtained with three planar grids 6.4, 12.7, and 19 mm wide at pitches of 25, 50, and 75 mm. Turbulence intensities generated by these grids were measured and found to be in good agreement with the equation of Baines and Peterson. Free stream air velocities ranged from 3 m/s to 15.3 m/s and resulted in Reynolds numbers based on conditions at the minimum free flow area ranging from 900 to 10,000 according to Junkhan. Exchangers 1 and 2 presented similar responses to the effects of inlet turbulence intensity. The Stanton number ratio versus Reynolds number curves indicates some augmentation. The maximum improvement (Fig. 5.9.7a) is about 6% for exchanger 1.
Fig. 5.9.7 Stanton Number Increase (a) (b)
This small increase in performance suggests that only the finned region near the inlet was affected by turbulence. Further downstream, the wake flow generated by the staggered tube arrangement produces mixing unaffected by upstream turbulence. In exchanger 2, the louvered fins and high fin density promote mixing. Fin boundary layers, unaffected by turbulence, are maintained.Figures 5.9.7(b), 5.9.8(a), and 5.9.8(b) show more enhancement is achieved in exchangers 3, 4, and 5 due to less effective mixing in the wake of the in-line tube arrangement. With the lower fin density, the resultant thicker boundary layer is more affected by free stream turbulence.
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Fig. 5.9.8 Stanton Number Increase (a) (b)
Gianolio and Cuti describe performance tests conducted on a pilot plant simulating an air-cooled heat exchanger operated in both the induced-draft and forced-draft modes. Under forced-draft conditions, the velocity of the airstream approaching the finned tube heat exchanger bundle has a tangential component in addition to the axial component. According to Kotb, the oblique approach velocity and the turbulence in the fan wake tend to improve the heat transfer coefficient. This is particularly true on the upstream tube rows compared to a sixrow bundle (Fig. 5.9.9). The enhancement is very high and not typical of all forced-draft heat exchangers.
Fig. 5.9.9 Comparison of Heat Transfer Coefficient for Induced and Forced Flow
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The degree of turbulence will also be influenced by other disturbances, such as support structures, walkways, screens, etc. located upstream and downstream of the fan. In such forced-draft systems, a row correction factor of unity is recommended for all tube rows by Huber. No general equation exists to quantify the enhancement in heat transfer for specific heat exchanger bundles due to free stream turbulence at different Reynolds numbers. However, available experimental data suggests free stream turbulence may enhance heat transfer in thin heat exchanger cores or bundles when little active internal mixing occurs due to the geometric design of finned surfaces. In addition to the level of turbulence, the nature of the turbulence in the approach airstream undoubtedly affects the degree of enhancement achieved. The influence on the pressure drop is small in most cases.
5.10 Non-Uniform Flow and Temperature Distribution An extensive review of the causes and effects of non-uniform or maldistributed flow and temperature distribution on the performance of different types of heat exchangers is presented by Mueller and Chiou and Kitto and Robertson. The causes of maldistribution in air-cooled heat exchangers as indicated by London, Mondt, Bell, and Shah and by Sparrow include:
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design of headers
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design of inlet and outlet duct systems
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upstream and downstream appurtenances
•
manufacturing tolerances of flow passages through the heat exchanger
•
variations in fluid viscosity
•
thermo-acoustic oscillations
•
fouling or blockage
•
corrosion
•
recirculation
•
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An early study by McDonald and Eng showed the effect of tube-side maldistribution in a crossflow heat exchanger was small and resulted in a maximum thermal performance reduction of about 4%. Chiou and Solar et al. evaluated the effect of non-uniform flow and/or temperature distribution on the thermal performance of crossflow heat exchangers. They used oneand/or two-dimensional finite difference techniques to solve the governing differential equation. Using numerical integration, Dobryakov et al. and Fagan analyzed the effects of airflow maldistribution on heat exchanger performance. Rabas extended their integral approach to both one- and two-dimensional nonuniform flow and temperature distributions found in air-cooled condensers. Beiler and Kröger evaluate the thermal performance reduction of forced draft air-cooled heat exchangers. They take into consideration maldistributions in airflow and temperature as well as variable performance characteristics of individual tube rows. They concluded the reduction in performance due to maldistribution of flow should be less than 2% in a well-designed industrial air-cooled heat exchanger. This is confirmed by Meyer and Kröger who show the approximate effectiveness of a forced-draft air-cooled heat exchanger can be expressed in terms of the kinetic-energy, air-velocity, distribution-correction factor of the heat exchanger, i.e., enu = Qnu/Q ≈ 1.05 - 0.05 αe
(5.10.1)
Typical values of αe are given by Equation 6.4.5. The air-side flow distribution in an induced- draft heat exchanger is more uniform than in a forced-draft unit. Maldistribution of the airflow due to space limitations and the proximity of the fan to the heat exchanger can significantly reduce the effectiveness in certain cases. Examples of highly distorted airflows through vehicle radiators where air is blown by axial-flow fans are found in practice according to Koffman. Due to the complex nature of such flows, the performance characteristics of these fan/heat exchanger systems are determined experimentally. Distortions of airflow patterns during windy periods can dramatically reduce the performance of air-cooled heat exchangers and cooling towers. These effects are evaluated in chapter 9.
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References Air-Conditioning and Refrigeration Institute, Standard for Forced Circulation Air-Cooling and Air-Heating Coils, Standard 410, ARI, Arlington, Virginia, 1972. American Society of Heating, Refrigeration and Air Conditioning, Methods for Testing Forced Circulation Air Cooling and Air Heating Coils, Standard 33-78, ASHRAE, New York, 1978. Baines, W. D., and E. G. Peterson, “An Investigation of Flow Through Screens,” Transactions of the American Society of Mechanical Engineers, 73:467–480, 1951. Battelle Pacific Northwest Laboratories, A Survey of Materials and Corrosion Performance in Dry Cooling Applications, BNWL-1958, UC-12, Richland, Washington, 1976. Battelle Pacific Northwest Laboratories, Aluminum Alloy Performance Under Dry Cooling Tower Conditions, Richland, Washington, 1977. Becker, N., “Druckverlust und Wärmeübertragung von Rippenrohrbündeln für Naturzug-Trockenkühltürme,” Doctoral thesis, Department of Engineering, Technische Hochschule, Aachen, 1984. Beiler, M. G., and D. G. Kröger, “Thermal Performance Reduction in Air-Cooled Heat Exchangers Due to Nonuniform Flow and Temperature Distributions,” Heat Transfer Engineering, vol. 17, no. 1, 1996. Bell, K. J., and W. H. Kegler, Analysis of Bypass Flow Effects in Tube Banks and Heat Exchangers, American Institute of Chemical Engineers Symposium Series, vol. 74, no. 174. Bemrose, C. R., and T. R. Bott, Correlations for Gas-Side Fouling of Finned Tubes, 1st U.K. National Conference on Heat Transfer, 1:357–367, Pergamon Press, London, 1984. Bemrose, C. R., and T. R. Bott, Studies of Gas-Side Fouling in Heat Exchangers, Progress in the Prevention of Fouling in Industrial Plant Conference, University of Nottingham, 2–20, April, 1981. Biery, J. C., “Prediction of Heat Transfer Coefficients in Gas Flow Normal to Finned and Smooth Tube Banks,” Transactions of the American Society of Mechanical Engineers Journal of Heat Transfer, 103:705–714, 1981. Blenk, H., D. Fuchs, and L. Liebert, “Über die Messung von Wirbelfrequenzen,” Luftfahrtforschung, 12:38–41, 1935. Bódás, J., “Guidelines in Selecting Heat Exchanger Materials for Dry Cooling Plants of Large Capacity Steam Turbines,” 6th International Association for Hydraulics Research Cooling Tower Workshop, Pisa, 1988. Bódás, J., “Heat Exchanger Materials for Dry Cooling Plants,” Symposium on Dry Cooling Towers, Tehran, 1991. Bonger, R., “New Developments in Aircooled Steam Condensing,” Electric Power Research Institute Cooling Towers and Advanced Cooling Systems Conference, St. Petersburg Beach, Florida, August 1994.
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Moore, F. K., and K. E. Torrence, Air Flow in Dry Natural-Draught Cooling Towers Subject to Wind, Cornell Energy Report, Final Report to ERDA, Ithaca, New York, 1977. Moore, F. K., and J. R. Ristorcelli, “Turbulent Flow and Pressure Losses Behind Oblique High-Drag Heat Exchangers,” International Journal of Heat and Mass Transfer, 22:1175–1186, 1979. Mueller, A. C., and J. P. Chiou, “Review of Various Types of Flow Maldistribution in Heat Exchangers,” ASME HDT-Vol. 75, 24th National Heat Transfer Conference, ed. J. B. Kitto and L. M. Robertson, New York, 1987. Mueller, A. C., and J. P. Chiou, “Review of Various Types of Flow Maldistribution in Heat Exchangers,” Heat Transfer Engineering, vol. 9, no. 2, 1988. Nagel, P., “New Developments in Aircooled Steam Condensers and Dry Cooling Towers,” 9th International Association for Hydraulics Research Cooling Tower and Spraying Pond Symposium, von Karman Institute, Brussels, September 1994. Neal, J. W., and W. F. Savage, Report of the Visit of the United States of America Delegation of the U.S.–U.S.S.R. Co-ordinating Committee on Scientific and Technical Co-Operation in the Field of Thermal Power Plant Heat Rejection Systems to the Union of Soviet Socialistic Republics, DOE/ET-0078, U.S. Dept. of Energy, 1979. Nir, A., “Heat Transfer and Friction Factor Correlations for Crossflow Over Staggered Finned Tube Banks,” Heat Transfer Engineering, 12-1:43–58, 1991. Nirmalan, V., and G. H. Junkhan, “Effects of Turbulence Parameters on Heat Transfer of Cross Flow Heat Exchangers,” Proceedings, 7th International Heat Transfer Conference, 6:233–238, München, 1982. Olsson, C. O., and B. Sundén, “Heat Transfer and Pressure Drop Characteristics of Ten Radiator Tubes,” International Journal of Heat and Mass Transfer, 39-15:3211–3220, 1996. Paikert, P., “Werkstoffauswahl für luftgekühlte Wärmeaustauscher,” Chemie Ingenieur Technik, 61-8:590–596, 1989. Palfalvi, G., “Experience with Aluminium Air Coolers in Power Plants,” 6th International Association for Hydraulics Research Cooling Tower Workshop, Pisa, 1988. Phillips, N. V., Gloeilampenfabrieken, Eindhoven, Patent No. 7314929, Netherlands, 1973. Pollack, H., Wärme und strömungstechnische Untersuchungen an Kunststoffwärmeaustauschern für Trockenkühltürme, Report No. BMFT-FB-T 81-004, Bundesministerium für Forschung und Technologie, Bonn, 1991. Preece, R. J., J. Lis, and J. A. Hitchcock, “Comparative Performance Characteristics of Some Extended Surfaces for Air Cooler Applications,” Conference on Air Coolers, Institute of Mechanical Engineers, London, 41–60, 24 September 1970. Rabas, T. J., The Effect of Nonuniform Inlet Flow and Temperature Distributions on the Thermal Performance of Air-Cooled Condensers, ASME Report HTD-Vol. 75, Maldistribution of Flow and its Effect on Heat Exchanger Performance, ed. J. B. Kitto and J. M. Robertson, American Society of Mechanical Engineers, 1987.
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Reay, D. A., “Heat Transfer Enhancement—A Review of Techniques and Their Possible Impact on Energy Efficiency in the U.K.,” Heat Recovery Systems and CHP, 11-1:1–40, 1991. Robinson, K. K., and D. E. Briggs, Pressure Drop of Air Flowing Across Triangular Pitch Banks of Finned Tubes, Chemical Engineering Progress Symposium Series, 6264:177–184, 1966. Rose, J. C., “Some Problems Associated with the Operation and Testing of Air-Cooled Heat Exchangers,” Conference on Air-Coolers, Institute of Mechanical Engineers, London, 77–87, 24 September 1970. Rozenman, T., S. K. Momoh, and J. M. Pundyk, Heat Transfer and Pressure Drop Characteristics of Dry Tower Extended Surfaces, Part 1, PFR Report No. BNWL-PFR-7100, PFR Engineering Systems, Inc., California, 1976. Russell, C. M. B., “The Effect on the Thermal Performance of External Corrosion of Fin Tubes,” Nat. Heat Transfer Conference, San Diego, California, August 1979. Samie, F., and M. Sparrow, “Heat Transfer from a Yawed Finned Tube,” Transactions of the American Society of Mechanical Engineers Journal of Heat Transfer, 108:479–482, 1986. Schack, K., “Berechnung des Druckverlustes in Querstrom von Rippenrohrbündeln,” Chemie Ingenieur Technik, 51:986–987, 1979. Schmidt, T. E., “Der Wärmeübergang an Rippenrohre und Berechnung von Rohrbündelnwärmeaustauschern,” Kältetechnik, vol. 15, no. 4, part 1 and part 11, 370–378, 1963. Schnell, H., Verschmutzung, Handbuch der Kältetechnik, Wärmeaustauscher, ed. R. Plank, 394–410, Springer-Verlag, Berlin, 1988. Schulenberg, F. J., “Finned Elliptical Tubes and Their Application in Air-Cooled Heat Exchangers,” Transactions of the American Society of Mechanical Engineers, Journal of Engineering for Industry, 179–190, May 1966. Schulenberg, F. J., “Wahl der Bezugslänge zur Darstellung von Wärmübergang und Druckverlust in Wärmetauschern,” Chemie Ingenieur Technik, vol. 37, 1965. Shah, R. K., and A. L. London, “Effects of Nonuniform Passages on Compact Heat Exchanger Performances,” Transactions of the American Society of Mechanical Engineers, Journal of Engineering for Power, 102:653–659, July 1980. Sheffield, J. W., M. Abu-Ebid, and H. J. Sauer, “Finned Tube Contact Conductance: Empirical Correlation of Thermal Conductance,” American Society of Heating, Refrigeration and Air Conditioning Transactions, vol. 91, pt. 2, 1985. Shlykov, Y. P., and Y. A. Ganin, “Thermal Resistance of Metallic Contacts,” International Journal of Heat and Mass Transfer, 7:921–929, 1964. Smith, C. E., A. Y. Gunter, and S. P. Victory, “Evaluation of Fin Tube Performance under Steady-State and Thermal Cycling Conditions,” Chemical Engineering Progress, vol. 62, no. 7, 1966.
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Solar, A. I., K. P. Singh, and T. L. Ng, “Effect of Nonuniform Inlet Flow and Air Cooler Heat Exchange Performance,” Proceedings, American Society of Mechanical Engineers—Japan Society of Mechanical Engineers Thermal Engineering Joint Conference, 1:537–542, 1983. Somerscales, E. F. C., and J. G. Knudsen, Fouling of Heat Transfer Equipment, Hemisphere Publishing Co., Washington, 1981. Somerton, C., R. D. Bridges, R. W. Callihan, and C. Taff, Heat Transfer Analysis of an Inclined Radiator, 85-WA/HT-24, American Society of Mechanical Engineers, 1985. Sparrow, E. M., and R. Ruiz, “Effect of Blockage-Induced Flow Maldistribution on the Heat Transfer and Pressure Drop in a Tube Bank,” Transactions of the American Society of Mechanical Engineers, Journal of Heat Transfer, 104:691–699, November 1982. Stasiulevicius, J., and A. Srinska, Heat Transfer of Finned Tube Bundles in Crossflow, Hemisphere Publishing Corp., New York, 1988. Stephan, K., and D. Traub, “Influence of Turbulence Intensity on Heat Transfer and Pressure Drop in Compact Heat Exchangers,” Proceedings, 8th International Heat Transfer Conference, San Francisco, California, 2739–2744, August 1986. Süthoff, Th., and H.-H. Reichel, “Vergleichende Korrosionsversuche an Trockenkühlelementen für Trockenkühltürme,” Vereinigung der Grosskraftwerksbetreiber Kraftwerkstechnik 65, 9:835–844, September 1985. Taborek, J., “Bond Resistance and Design Temperatures for High-Finned Tubes—A Reappraisal,” Heat Transfer Engineering, 8-2:26–34, 1987. Vampola, J., “Heat Transfer and Pressure Drop in Flow of Gases Across Finned Tube Banks,” Strojirenstvi, 7:501–507, 1966. Van Aarde, D. J., and D. G. Kröger, “Flow Losses Through an Array of A-Frame Heat Exchangers,” Heat Transfer Engineering, 14-1:43–51, 1993. Vergleichende Korrosionsversuche an Trockenkühl-Elementen für Trockenkühltürme, VGB-TW 504, Vereinigung der Grosskraftwerksbetreiber Kraftwerkstechnik GMBH, Essen, 1984. Ward, D. J., and E. H. Young, Heat Transfer and Pressure Drop of Air in Forced Convection Across Triangular Pitch Banks of Finned Tubes, Chemical Engineering Progress Symposium Series, 55-29:37–44, 1959. Webb, R. L., Principles of Enhanced Heat Transfer, John Wiley & Sons, 1994. Weierman, C., “Correlations Ease the Selection of Finned Tubes,” The Oil and Gas Journal, 74-36:94–100, September 1976. Wile, D. D., “Air Flow Measurement in the Laboratory,” ASRE, Refrigerating Engineering, 515–521, June 1947. Young, E. H., and D. E. Briggs, “Bond Resistance of Bimetallic Finned Tubes,” Chemical Engineering Progress, 61-7:71–78, 1965.
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Young, E. H., J. R. Fleming, and W. F. Conroy, Development of an Apparatus for Measurement of Low Bond Resistance in Finned and Bare Duplex Tubes, University of Michigan, Engineering Research Institute Project 1592, Report 48, 1957. Yovanovich, M. M., “New Contact and Gap Conductance Correlations for Conforming Rough Surfaces,” 81-1164, American Institute of Aeronautics and Astronautics 16th Thermophysics Conference, Palo Alto, California, June 1981. Zhukauskas, A. A., “Investigation of Heat Transfer in Different Arrangements of Heat Exchanger Surfaces,” Teploenergetika, 21-5:24–29, 1974. Zozulya, N. V., B. L. Kalinin, and A. A. Khavin, “Influence of the Layout of a Bank of Finned Aluminium Tubes on Heat Transfer,” Teploenergetika, vol. 17, no. 6, 1970. Zozulya, N. V., Yu. P. Vorobyev, and A. A. Khavin, “Effect of Flow Turbulization on Heat Transfer in a Finned-Tube Bundle,” Heat Transfer-Soviet Research, 5-1:154–156, 1973. Zukauskas, A., and R. Ulinskas, Heat Transfer in Tube Banks in Crossflow, Hemisphere Publishing Corp., New York, 1988.
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Appendix A Properties of Fluids
A.1 The Thermophysical Properties of Dry Air from 220K to 380K at Standard Atmospheric Pressure (101325 N/m2) Density ρa = pa/(287.08 T), kg/m3
(A.1.1)
Specific heat from General Electric Heat Transfer and Fluid Flow Data Book cpa = 1.045356 x 103 – 3.161783 x 10–1 T + 7.083814 x 10–4 T2 – 2.705209 x 10–7 T3, J/kgK
(A.1.2)
Dynamic viscosity from General Electric Heat Transfer and Fluid Flow Data Book µa = 2.287973 x 10–6 + 6.259793 x 10–8 T – 3.131956 x 10–11 T2 + 8.15038 x 10–15 T3, kg/sm
(A.1.3)
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Thermal conductivity ka = – 4.937787 x 10–4 + 1.018087 x 10–4 T – 4.627937 x 10–8 T2
(A.1.4)
+ 1.250603 x 10–11 T3, W/mK Table A.1 The Thermophysical Properties of Dry Air at Standard Atmospheric Pressure
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Figure A.1 The Thermophysical Properties of Dry Air at Standard Atmospheric Pressure (101325 N/m2)
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A.2 The Thermophysical Properties of Saturated Water Vapor from 273.15K to 380K Vapor pressure according to Goff pv = 10z, N/m2
(A.2.1)
z = 10.79586(1 – 273.16/T) + 5.02808 log10(273.16/T) + 1.50474 x 10–4 [1 – 10–8.29692{(T/273.16) –1}] + 4.2873 x 10–4[104.76955(1 – 273.16/T) –1] + 2.786118312 Specific heat cpv = 1.3605 x 103 + 2.31334T – 2.46784 x 10–10T5
(A.2.2)
+ 5.91332 x 10–13T6, J/kgK Dynamic viscosity µv = 2.562435 x 10–6 + 1.816683 x 10–8T + 2.579066 x 10–11T2 (A.2.3) – 1.067299 x 10–14T3, kg/ms Thermal conductivity from General Electric Heat Transfer and Fluid Flow Data Book kv = 1.3046 x 10–2 – 3.756191 x 10–5T + 2.217964 x 10–7T2 – 1.111562 x 10–10T3, W/mK
444
(A.2.4)
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Vapor density from the U.K. Steam Tables in SI Units 1970 ρv = – 4.062329056 + 0.10277044T – 9.76300388 x 10–4T2 +4.475240795 x 10–6T3 – 1.004596894 x 10–8T4 + 8.9154895 x 10–12T5, kg/m3
(A.2.5)
Temperature 2
-1
T = 164.630366 + 1.832295 x 10-3 p v + 4.27215 x 10-10 p v + 3.738954 x 10 3 p v
(A.2.6) -2
- 7.01204 x 105 pv + 16.161488 ᐉn p v - 1.437169 x 10-4 p v ᐉn p v , K Table A.2 The Thermophysical Properties of Saturated Water Vapor
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Figure A.2 The Thermophysical Properties of Saturated Water Vapor
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A.3 The Thermophysical Properties of Mixtures of Air and Water Vapor Density from ASHRAE Handbook of Fundamentals
ρ av = (1 + w) [1 - w/(w + 0.62198)] pabs/(287.08 T), kg air- vapor / m3 (A.3.1) Specific heat according to Faires cpav = (cpa + wcpv)/(1 + w), J/K kg air-vapor
(A.3.2a)
or the specific heat of the air-vapor mixture per unit mass of dry air cpma = (cpa + wcpv), J/K kg dry air
(A.3.2b)
Dynamic viscosity according to Godridge 0.5 0.5 0.5 µ av = (X a µ a M 0.5 a + X v µ v M v ) /( X a M a + X v M v ), kg/ms
(A.3.3)
where Ma = 28.97 kg/mole Mv = 18.016 kg/mole Xa = 1/(1 + 1.608 w) Xv = w/(w + 0.622) Thermal conductivity according to Lehmann 0.33 0.33 0.33 k av = (X a k a M 0.33 a + X v k v M v ) /( X a M a + X v M v ), W/mK1
(A.3.4)
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Humidity ratio according to Johannsen
0.62509 pvwb 2501.6 - 2.3263(Twb - 273.15) w = 2501.6 + 1.8577(T 273.15) 4.184( 273.15) T wb pabs - 1.005 pvwb (A.3.5)
1.00416 (T - Twb ) , kg/kg dry air - 2501.6 + 1.8577(T 273.15) 4.184( 273.15) T wb Enthalpy (A.3.6a) iav = [cpa (T – 273.15) + w{ifgwo + cpv (T - 273.15)}]/(1+w), J/kg air-vapor or the enthalpy of the air-vapor mixture per unit mass of dry air
ima = cpa (T - 273.15) + w [ifgwo + cpv (T - 273.15)], J/kg dry air
(A.3.6b)
where the specific heats are evaluated at (T + 273.15)/2 and the latent heat, ifgwo, is evaluated at 273.15 K according to equation (A.4.5), i.e., ifgwo = 2.5016 x 106 J/kg
A.4 The Thermophysical Properties of Saturated Water Liquid from 273.15k to 380k Density ρw= (1.49343 x 10–3 – 3.7164 x 10–6T + 7.09782 x 10–9T2 – 1.90321 x 10–20T6)–1, kg/m3
448
(A.4.1)
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Specific heat cpw = 8.15599 x 103 – 2.80627 x 10T + 5.11283 x 10–2T2 – 2.17582 x
10–13T6,
(A.4.2)
J/kgK
Dynamic viscosity from General Electric Heat Transfer and Fluid Flow Data Book µw = 2.414 x 10–5 x 10247.8/(T – 140), kg/ms
(A.4.3)
Thermal conductivity kw = – 6.14255 x 10–1 + 6.9962 x 10–3T – 1.01075 x 10–5T2 (A.4.4) + 4.74737 x 10–12T4, W/mK Latent heat of vaporization ifgw = 3.4831814 x 106 – 5.8627703 x 103T + 12.139568T2 (A.4.5) – 1.40290431 x 10–2T3, J/kg Critical pressure pwc = 22.09 x 106, N/m2
(A.4.6)
Surface tension from U.K. Steam Tables in SI Units 1970 σw = 5.148103 x 10–2 + 3.998714 x 10–4T – 1.4721869 x 10–6T2 + 1.21405335 x
10–9T3,
(A.4.7)
N/m
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Table A.3 The Thermophysical Properties of Saturated Water Liquid
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Figure A.3 The Thermophysical Properties of Saturated Water Liquid
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A.5 The Thermophysical Properties of Saturated Ammonia Vapor Vapor pressure (230K to 395K) according to Raznjevic pammv = 1.992448 x 106 – 57.56814 x 103T + 0.5640265 x 103T2 – 2.337352T3 + 3.54143 x 10–3T4, N/m2
(A.5.1)
Density (260K to 390K) according to Raznjevic ρammv = – 6.018936 x 102 + 5.361048T – 1.187296 x 10–2T2 – 1.161479 x 10–5T3 + 4.739058 x 10–8T4, kg/m3
(A.5.2)
Specific heat (230K to 325K) from ASHRAE Handbook of Fundamentals cpammv = – 2.7761190256 x 104 + 3.39116449 x 102T – 1.3055687T2 – 1.728649 x 10–3T3; J/kgK
(A.5.3)
Dynamic viscosity (240K to 370K) from ASHRAE Handbook of Fundamentals µammv = – 2.748011 x 10–5 + 2.82526 x 10–7T – 5.201831 x 10–10T2 – 6.061761 x 10–13T3 + 2.12607 x 10–15T4, kg/sm
(A.5.4)
Thermal conductivity (245K to 395K) from ASHRAE Handbook of Fundamentals kammv = – 0.1390216 + 1.35238 x 10–3T – 2.532035 x 10–6T2 – 4.884341 x 10–9T3 + 1.418657 x 10–11T4, W/mK
452
(A.5.6)
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Table A.4 The Thermophysical Properties of Saturated Ammonia Vapor
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Figure A.4 The Thermophysical Properties of Saturated Ammonia Vapor
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A.6 The Thermophysical Properties of Saturated Ammonia Liquid from 200K to 405K Density according to Yaws
ρ amm = 2.312 x 102 x 0. 2471[-(1 - T /405.5)
0.285714
], kg / m3
(A.6.1)
Specific heat (200K to 375K) according to Yaws cpamm = – 2.497276939 x 103 + 7.7813907 x 10T – 3.006252 x 10–1T2 + 4.06714 x 10–4T3, J/kgK
(A.6.2)
Dynamic viscosity according to Yaws
µ amm = 0.001 x 10(-8.591+876.4/T0.02681+T-3.612x10
-5T 2 ) , kg/sm
(A.6.3)
Thermal conductivity (200K to 375K) according to Yaws kamm = 1.068229 – 1.576908 x 10–3T – 1.228884 x 10–6T2, W/mK
(A.6.4)
Latent heat of vaporization according to Yaws 0.38
ifgamm = 1.370758 x 10 6[(405.55 - T) /(165.83) ]
, J/kg2
(A.6.5)
Critical pressure pammc = 11.28 x 106, N/m2
(A.6.6)
Surface tension according to Yaws 1.1548
σ = 0.0366[(405.55 - T ) / 177.4 ]
, N/m
(A.6.7)
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Table A.5 The Thermophysical Properties of Saturated Ammonia Liquid
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Figure A.5 The Thermophysical Properties of Saturated Ammonia Liquid
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References ASHRAE, Handbook of Fundamentals, American Society of Heating, Refrigeration and Air Conditioning Engineers, Inc., 1972. Faires, V. M., and C. M. Simmang, Thermodynamics, 6th ed., Macmillan Publishing Co. Inc., 1978. General Electric, Heat Transfer and Fluid Flow Data Book, General Electric Co., Corporate Research Division, New York, 1982. Godridge, A.M., British Coal Utilisation Research Association Monthly, vol. 18 no. 1, 1954. Goff, J.A., Saturation Pressure of Water on the New Kelvin Scale, Humidity and Moisture Measurement and Control in Science and Industry, ed., A. Wexler and W. H. Wildhack, Reinhold Publishing Co., New York, 1965. Johannsen, A., “Plotting Psychrometric Charts by Computer,” The South African Mechanical Engineer, 32:154–162, July 1982. Lehmann, H., Chemical Technology, 9:530, 1957. Popiel, C. O. and J. Wojtkowiak, “Simple Formulas for Thermophysical Properties of Liquid Water for Heat Transfer Calculations (from 0°C to 150°C),” Heat Transfer Engineering, 19-3:87–101, 1998. Raznjevic, K., Handbook of Thermodynamic Tables and Charts, McGraw-Hill Book Co., New York, 1976. United Kingdom Committee on the Properties of Steam, U.K. Steam Tables in SI Units 1970, Edward Arnold Ltd., London, 1970. Yaws, C. L., Physical Properties, Chemical Engineering Publ., McGraw-Hill Book Co., New York, 1977.
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Appendix B Temperature Correction Factor
To determine the performance characteristics of heat exchangers, it is essential to determine the mean temperature difference between fluids accurately. Since this difference depends on the geometry and the flow pattern through the heat exchanger, simple analytic solutions are not always possible. For computational purposes, the method proposed by Roetzel is of value. Consider the heat exchanger shown in Figure B.1.
Figure B.1 Schematic of Heat Exchanger
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In such an exchanger, the heat transfer rate is given by Q = UA∆Tm
(B.1)
where UA is the conductance of the exchanger U is the overall heat transfer coefficient and is assumed to be constant Q is the heat and flows from the hot fluid, subscript h, to the cold fluid, subscript c According to Equation B.1, the mean temperature difference between the two streams, ∆Tm, may be expressed as ∆Tm = Q/UA or, if made dimensionless with respect to the largest temperature difference,
ϕ=
Q ∆Tm = T hi - T ci UA( T hi - T ci)
(B.2)
Dimensionless temperature changes of the two streams may be defined as
ϕh = and
ϕc =
T hi - T ho Q = T hi - T ci m h cph (T hi - T ci)
(B.3)
T co - T ci Q = T hi - T ci m c cpc (T hi - T ci)
(B.4)
where m = the mass flow rate cp = the specific heat of the fluid
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In the case of counterflow, a dimensionless mean temperature difference can be expressed in terms of the logarithmic mean temperature difference, i.e.,
ϕcf =
ϕ h - ϕc D Tᐉm = (T hi - T ci) ᐉn[(1- ϕc) /(1 - ϕ h)]
(B.5)
According to Roetzel, a temperature correction factor can be expressed as
FT =
4 4 ϕ k ϕ = 1 − R R a i,k (1 − ϕcf ) sin 2i arctan h ϕcf ϕc i =1 k =1
(B.6)
Tables B.1 to B.10 present the sixteen values of the empirical constant ai,k for ten different heat exchanger geometries. Table B.1 Crossflow with One Tube Row
Table B.2 Crossflow with Two Tube Rows and One Pass
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Table B.3 Crossflow with Three Tube Rows and One Pass
Table B.4 Crossflow with Four Tube Rows and One Pass
Table B.5 Crossflow with Two Tube Rows and Two Tube Passes
Table B.6 Crossflow with Three Tube Rows and Three Tube Passes
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Table B.7 Crossflow with Four Tube Rows and Four Tube Passes
Table B.8 Crossflow with Four Tubes and Two Passes Through Two Tubes
Table B.9 Crossflow with Both Streams Unmixed
Table B.10 Crossflow with Both Streams Mixed
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References Roetzel, W., “Berechnung von Wärmeübertragern,” Verein Deutscher IngenieureWärmeatlas, Ca1–Ca31, VDI-Verlag GmbH, Düsseldorf, 1984. Roetzel, W., and J. Neubert, “Calculation of Mean Temperature Difference in AirCooled Cross-Flow Heat Exchangers,” Journal of Heat Transfer, Transactions of the American Society of Mechanical Engineers, 101:511–513, Augusut 1979. Roetzel, W., and J. Fürst, “Schnelle Berechnung der mittleren Temperaturdifferenz in luftgekühlten Kreuzstrom Wärmeaustauschern,” Wärme und Stoffübertragung, 14:131–136, 190. Spang, B., and W. Roetzel, “Neue Näherungsgleichung zur einheitlichen Berechnung von Wärmeübertragern,” Heat and Mass Transfer, 30:415–422, Springer Verlag, 1995.
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Appendix C Conversion Factors
Acceleration 1 cm/s2 = 1.000 x 10–2 m/s2 1 m/h2 = 7.716 x 10–8 m/s2 1 ft/s2 = 0.3048 m/s2 1 ft/h2 = 2.352 x 10–8 m/s2
Area 1 ha = 104 m2 1 acre = 4.047 x 103 m2 1 cm2 = 1.000 x 10–4 m2 1 ft2 = 9.290 x 10–2 m2 1 in2 = 6.452 x 10–4 m2 1 yard2 = 0.8361 m2 1 square mile = 2.59 x 106 m2
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AIR-COOLED HEAT EXCHANGERS AND COOLING TOWERS
Density 1 g/cm3 = 1000 kg/m3 1 1b/ft3 = 16.02 kg/m3 1 kg/ft3 = 35.31 kg/m3
Energy, work, heat 1 cal = 4.187 J 1 kcal = 4187 J 1 Btu = 1055 J 1 erg = 1.000 x 10–7 J 1 kWh = 3.600 x 106 J 1 ft pdl = 4.214 x 10–2 J 1 ft lbf = 1.356 J 1 Chu = 1899 J 1 therm = 1.055 x 108 J
Force 1 dyne = 1.000 x 10–5 N 1 kgf = 9.807 N 1 pdl = 0.1383 N 1 lbf = 4.448 N
Heat flux 1 cal/s cm2 = 4.187 x 104 W/m2 1 kcal/h m2 = 1.163 W/m2 1 Btu/h ft2 = 3.155 W/m2 1 Chu/h ft2 = 5.678 W/m2 1 kcal/h ft2 = 12.52 W/m2
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CONVERSION FACTORS
Heat transfer coefficient 1 cal/s cm2 °C = 4.187 x 104 W/m2 K 1 kcal/h m2 °C = 1.163 W/m2 K 1 Btu/h ft2 °F = 5.678 W/m2 K 1 Chu/h ft2 °C = 5.678 W/m2 K 1 kcal/h ft2 °C = 12.52 W/m2 K
Length 1 cm = 1.000 x 10–2 m 1 ft = 0.3048 m 1 micron = 1.000 x 10–6 m 1 in = 2.540 x 10–2 m 1 yard = 0.9144 m 1 mile = 1609 m
Mass 1 g = 1.000 x 10–3 kg 1 lb = 0.4536 kg 1 tonne = 1000 kg 1 grain = 6.480 x 10–5 kg 1 oz = 2.835 x 10–2 kg 1 ton (long) = 1016 kg 1 ton (short) = 907 kg
Mass flow rate 1 g/s = 1.000 x 10–3 kg/s 1 kg/h = 2.778 x 10–4 kg/s 1 lb/s = 0.4536 kg/s
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1 tonne/h = 0.2778 kg/s 1 lb/h = 1.260 x 10–4 kg/s 1 ton (long)/h = 0.2822 kg/s
Mass flux 1 g/s cm2 = 10.00 kg/s m2 1 kg/h m2 = 2.778 x 10–4 kg/s m2 1 lb/s ft2 = 4.882 kg/s m2 1 lb/h ft2 = 1.356 x 10–3 kg/s m2 1 kg/h ft2 = 2.990 x 10–3 kg/s m2
Mass transfer coefficient 1 lb/h ft2 = 1.3385 x 108 kg/sm2 1 g/s cm2 = 9.869 x 10–5 kg/sm2 1 kg/h m2 = 2.7415 x 10–9 kg/sm2
Power 1 cal/s = 4.187 W 1 kcal/h = 1.163 W 1 Btu/s = 1055 W 1 erg/s = 1.000 x 10–7 W 1 hp (metric) = 735.5 W 1 hp (British) = 745.7 W 1 ft pdl/s = 4.214 x 10–2 W 1 ft lbf/s = 1.356 W 1 Btu/h = 0.2931 W 1 Chu/h = 0.5275 W 1 ton refrigeration = 3517 W
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CONVERSION FACTORS
Pressure 1 dyne/cm2 = 0.100 N/m2 1 kgf/m2 = 9.807 N/m2 1 pdl/ft2 = 1.488 N/m2 1 standard atm = 1.0133 x 105 N/m2 1 bar = 1.000 x 105 N/m2 1 kgf/cm2 (1 at) = 9.807 x 104 N/m2 1 lbf/ft2 = 47.88 N/m2 1 lbf/in2 = 6895 N/m2 1 mm water = 9.807 N/m2 1 in water = 249.1 N/m2 1 ft water = 2989 N/m2 1 mm Hg = 133.3 N/m2 1 in Hg = 3387 N/m2
Specific enthalpy or latent heat 1 cal/g = 4187 J/kg 1 Btu/lb = 2326 J/kg 1 Chu/lb = 4187 J/kg 1 kcal/kg = 4187 J/kg
Specific heat capacity 1 cal/g °C = 4187 J/kgK 1 Btu/lb °F = 4187 J/kgK
Specific volume 1 cm3/g = 1.000 x 10–3 m3/kg 1 ft3/lb = 6.243 x 10–2 m3/kg 1 ft3/kg = 2.832 x 10–2 m3/kg
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Surface tension 1 dyne/cm = 1.000 x 10–3 N/m 1 pdl/ft = 0.4536 N/m 1 lbf/ft = 14.59 N/m
Temperature difference 1 deg F = 0.556 K
Thermal conductivity 1 cal/s cm °C = 4187 W/m K 1 kcal/h m °C = 1.163 W/m K 1 Btu/h ft °F = 1.731 W/m K
Time 1 h = 3600 s 1 day = 8.640 x 104 s 1 year = 3.156 x 107 s
Velocity 1 cm/s = 1.000 x 10–2 m/s 1 m/h = 2.778 x 10–4 m/s 1 ft/s = 0.3048 m/s 1 ft/h = 8.467 x 10–5 m/s 1 mile/h = 0.4470 m/s 1 knot = 0.5148 m/s
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CONVERSION FACTORS
Viscosity (dynamic) 1 g/cm s (poise) = 0.1000 kg/m s (N s/m2) 1 kg/m h = 2.778 x 10–4 kg/ms 1 lb/ft s = 1.488 kg/m s 1 lb/ft h = 4.134 x 10–4 kg/m s
Viscosity (kinematic) 1 cm2/s (stoke) = 1.000 x 10-4m2/s 1 ft2/s = 0.0929 m2/s
Volume 1 yard3 = 0.7646 m3 1 cm3 = 1.000 x 10–6 m3 1 ft3 = 2.832 x 10–2 m3 1 litre = 1.000 x 10–3 m3 1 in3 = 1.639 x 10–5 m3 1 UK gal = 4.546 x 10–3 m3 1 US gal = 3.785 x 10–3 m3
Volumetric flow 1 cm3/s = 1.000 x 10–6 m3/s 1 m3/h = 2.778 x 10–4 m3/s 1 ft3/s = 2.832 x 10–2 m3/s 1 ft3/h = 7.866 x 10–6 m3/s 1 UK gal/min = 7.577 x 10–5 m3/s 1 US gal/min = 6.309 x 10–5 m3/s
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Appendix D This is the Table of Contents for Volume II.
6
Fans Introduction
Meteorological Effects
Test Facilities and Procedures
Introduction
Presentation of Data and Results
Atmosphere
Tip Clearance
Effect of Wind on Cooling Towers
Fan System
7
Natural Draft Cooling Towers
9
Effect of Wind on Air-cooled Heat Exchangers Recirculation and Interference Inversions
Introduction Dry-cooling Tower Wet-cooling Tower Inlet Losses Cold Inflow
8
Mechanical Draft Coolers
10 Cooling System Selection and Optimization Introduction Power Generation Cooling System Optimization
Introduction Air-cooled Heat Exchangers and Cooling Towers Noncondensables Inlet Losses Recirculation
473
INDEX
Index Terms
Links
A Abatement angle Abrupt contractions and expansions Acceleration
316 79–85 119
181
189
267
304
359
28
37–38
313
1
44
Accu-pac fill, see Fill (accu-pac) Active heat transfer technique
149
Additive
149
Adiabatic cooling Adiabatic flow Adiabatic frictional pressure gradient
181
Adiabatic gas-liquid flow
186
Adiabatic two-phase flow, see Two-phase flow (adiabatic) Aerodynamic design A-frame
310 16
19
36
387
390
397–398
A-frame air-cooled condenser, see Condenser (A-Frame air-cooled) Agglomeration
300
Air capacity rate
275
Air-cooled condenser tube, see Condenser tube (air-cooled) Air-cooled condenser, see Condenser (air-cooled) This page has been reformatted by Knovel to provide easier navigation.
Index Terms
Links
Air-cooled crossflow heat exchanger, see Heat exchanger (air-cooled crossflow) Air-cooled finned tube condenser, see Condenser, air-cooled finned tube Air-cooled finned tube, see Finned tube (air-cooled) Air-cooled heat exchanger (forced draft), see Heat exchanger (air-cooled forced draft) Air-cooled heat exchanger (induced draft), see Heat exchanger (air-cooled induced draft) Air-cooled heat exchanger (rectangular), see Heat exchanger (air-cooled) Air-cooled heat exchanger (V-configuration), see Heat exchanger (air-cooled) Air-cooled heat exchanger (vertical), see Heat exchanger (air-cooled) Air-cooled heat exchanger, see Heat exchanger (air-cooled) Air-Cooled Heat Exchangers and Cooling Towers Introduction
1–54 1–2
Cooling Towers
2–12
Mechanical draft
4–6
Natural draft
6–12
This page has been reformatted by Knovel to provide easier navigation.
Index Terms
Links
Air-Cooled Heat Exchangers and Cooling Towers (Cont.) Air-Cooled Heat Exchangers
12–26
Mechanical draft
13–19
Natural draft
20–26
Dry/Wet and Wet/Dry Cooling Systems
27–41
Conservation Equations
42–48
General features of isentropic flow
44–47
Momentum Theorem
47–48
References;
49–54
Air-cooled oil cooler Air-cooled refrigerant condenser
202 15
Air-cooled steam condenser, see Condenser (air-cooled steam) Air-cooled system
2
11
16
74
267
344–345
363
373–374
25 Air cooler, see Heat exchanger (air-cooled) Air density Air discharge
317
Air distribution
256
Air enthalpy
247–248
Air flow
13
14
250
Airflow
30
39
300
317 Airflow rate
243
373
Air flow rate
289
341
This page has been reformatted by Knovel to provide easier navigation.
Index Terms
Links
Airflow resistance
249
Airfoil
401
Air layer
256
Air mixer
339
Air recirculation (hot plume)
301
13
14
Air resistance
254
255
Air-side capacity ratio
355
Air-side flow resistance
154
Air-side heat transfer surface
313
Air-side heat transfer
360
Air-side pressure loss
289
Air speed
312
Airstream velocity
339
Airstream
Air temperature
332
4
39
283
285
304
309
313
317–318
339
294
315
340
409 Air travel distance
243
Air vapor
291
Air-vapor flow rate
268
Air-vapor mass velocity
286
Air velocity
269
Air-water counterflow
340
Air-water interface
236
Air/water interface
283
Air-water vapor system
241
Alga
262
Algae control
309
Altbach/Deizisau Power Plant
257
399 239
39
This page has been reformatted by Knovel to provide easier navigation.
405
Index Terms Aluminum
Links 333
407
39
267
291
301
313–316
319
Aluminum plate fin, see Fin (aluminum plate) Aluminum plate fin-tube heat exchanger, see Heat exchanger (Aluminum plate fin-tube) Aluminum skived-fin-tube heat exchanger, see Heat Exchanger (Aluminum skived-fin-tube) Aluminum slotted plate finned tube, see Finned tube (slotted aluminum plate) Ambient air Ambient airstream
4
Ambient humidity
241
Ambient temperature
27
Ambient wetbulb temperature, see Wetbulb temperature American tower plastic fill, see Fill (American tower plastic) Ammonia
34
Ammonia condenser
35
Ammonia heat pump
35
Ammonia heat transport system
33
Ammonia phase-change system
33
Ammonia pressure
35
Ammonia vapor
35
Analysis of contact and gap resistance Annular-condensate flow
408–417 208
This page has been reformatted by Knovel to provide easier navigation.
Index Terms
Links
Annular duct, see Circular duct Annular finned tube, see Finned tube (annular) Annular flow
107–113
Annular flow pattern
179–182
Anti-recirculation fence
190
13
Apparent shear stress, see Reynolds stress Arc-welding
334
Arid climate
18
Armenia
24
27
Asbestos-free fiber cement fill, see Fill (asbestos-free fiber cement) Asbestos louver Asbestos sheet Atmospheric pressure Attachment point
259 258–259 235
262
362
372
79
Augmentation condenser system, see Dry/wet cooling system (augmentation condenser system) Auxiliary cooler
31
Axial component
427
Axial flow
7
Axial flow fan, see fan (axial floor) see fan (axial flow) Axial pressure gradient
181
Axial shear force
179
This page has been reformatted by Knovel to provide easier navigation.
350
Index Terms
Links
Axial temperature
142–144
Axial velocity vector
301–302
Axial velocity
181
B Bacteria
262
Baker flow Pattern map
112
Bakersfield, California
33
Barometric pressure
232–234
Base surface, see Prime surface Bend geometry
93
Bent triangular projection
334
Bernoulli’s equation
388
Bessel function (modified)
165
Binomial theorem
46
Biological growth
261
Bird
405
Blowdown disposal
10
Blowdown
9
Blower
4
Boiler feed water system
30
Boiler
20
10
Bosnjakovic equation
246
Boundary condition
148
165
310
312
56
58–59
79–80
226–227
343
426
Boundary layer Boundary layer development
397
Boundary layer separation
422
Boundary layer thickness
57
This page has been reformatted by Knovel to provide easier navigation.
170
Index Terms Boundary stress Brass tube
Links 61 337
Brazed fin, see Fin (galvanized) Brazil Bubble flow Bubble flow pattern
25 107–109 180
Bulk mean temperature, see Mixing cup temperature Bulk water temperature
237
239
265
267
268
266 Buoyancy
170
C Cable cooling tower, see Cooling tower (cable) Candiota cooling tower
25
Capacitive-cooling system, see Dry/wet cooling system (Capacitive-cooling system) Capacity rate
275
Capacity ratio
280
Capital cost
7
Carbon dioxide
401
Carbon monoxide
401
Cathodic protection
403
12
Cellular drift eliminator pack, see Drift eliminator pack (cellular) Characteristic flow parameter
358
370
This page has been reformatted by Knovel to provide easier navigation.
375
Index Terms
Links
Characteristic heat transfer and pressure drop parameter
357–376
Characteristic heat transfer parameter
357–358
Characteristic pressure drop parameter
359
Chebyshev integral
264
Chemically oxidized aluminum
404
China
24
Chisholm correlation
120
Chlorinated polyvinyl chloride
250
Chlorine
401–402
Churchill’s equation
155
160
Churn flow
107
109
Churn flow pattern
180
Circuit water
31
Circular-arc bend, see Curved duct (circular-arc bend) Circular duct
65
72
Circular fin, see Fin (circular) Circular tower
306
Circular tube
150
Circulating water Cleaning Climbing film flow
3 401–402 114
Closed circuit
10
Closed circuit cooling plant
15
Closed circuit evaporative cooler Coating
405–406
282–298 403
This page has been reformatted by Knovel to provide easier navigation.
Index Terms
Links
Cocurrent downward flow pattern, see Flow pattern (Cocurrent downward) Cocurrent flow
107–109
114
120
362
371–372
180 Cocurrent flow manifold, see Manifold (cocurrent flow) Cocurrent gas-liquid flow
186
Cocurrent upward flow pattern, see Flow pattern (Cocurrent upward) Coefficient of discharge
364
Coefficient of performance
243
Colburn j factor
347
Colburn j factor and friction factor Collar (zinc)
347–357 23
Collecting manifold, see Manifold (collecting) Collision (molecule)
225
Colloidal material
262
Combined flow manifold, see Manifold (mixed flow) Combining header, see Header (combining) Compact heat exchanger
12
Complete turbulence
70
Composite miter bend, see Curved duct (composite miter bend) Compressibility
55
Compressibility effect
46
Compressible fluid
340
Concentration boundary layer
228
This page has been reformatted by Knovel to provide easier navigation.
Index Terms
Links
Concentration boundary-layer profile Concentration gradient Concentration profile Condensate Condensate film Condensate flow rate
226 223–224 228 20
38
169–176 171
174
179
182 Condensate pump Condensate wave formation
38 186
Condensation correlations
179–185
Condensation heat transfer
172
Condensation heat transfer coefficient
182
Condensation
39
169–191
199
208
212–214
241
314–315
418
Condensation in duct
172–175
Condensation on cylinder
175–179
Condenser (A-Frame air-cooled)
14
Condenser (air-cooled finned tube)
16
Condenser (air-cooled) Condenser (air-cooled steam)
14–19
34
38
429
98
172–173
336 Condenser (counterflow)
15
Condenser (induced draft)
381
Condenser (low pressure steam)
169
191
Condenser (surface)
24
25
Condenser (water box)
37
This page has been reformatted by Knovel to provide easier navigation.
37 188
Index Terms Condenser (water-cooled) Condenser tube (air-cooled)
Links 35 207
212
Conduction
131–136
163
Conduction (radial)
135–136
Conduction equation
254
224
Conductive heat transfer, see Conduction Conical diffuser, see Diffuser (conical) Conical reducer, see Reducer Conservation of energy
42
Conservation of mass
42
Constant of proportionality
224
Contact pressure
407–416
Contact resistance
407–419
Continuity relationship Continuous phase
269
60 299
Continuous plate fin, see Fin (continuous plate) Contraction
343
Contraction coefficient
389
Contraction loss
390
Contraction loss coefficient
82
Contraction (pipe diameter)
79
Contraction ratio
81
Control volume
245
Convection
344
85
399
132
137–139
254
Convection (forced)
137
148
153
Convection heat transfer coefficient
138
Convection (natural)
137
148–149
153–154
This page has been reformatted by Knovel to provide easier navigation.
Index Terms Convective heat transfer
Links 3
132
163
239 Convective transfer
240
Cooldrop fill, see Fill (cooldrop) Cooling Cooling capacity Cooling delta Cooling system (closed cycle) Cooling tower (cable)
282
314
23
314
31–32 2 26
Cooling tower (Counterflow wet-cooling) Cooling tower Cooling tower (counterflow) Cooling tower (crossflow) Cooling tower (dry)
236
253
299
247
272–278
281
7
274
278
24–26
39
313
2–12
403–406 Cooling tower evaporation loss
9
Cooling tower (fan-assisted crossflow)
8
Cooling tower fill, see Fill or packs Cooling tower (forced draft)
4
Cooling tower (hyperbolic concrete natural draft dry-cooling)
20
Cooling tower (indirect dry)
20
25
Cooling tower induced draft (mechanical draft counterflow)
5
Cooling tower (induced draft Mechanical draft crossflow)
5
This page has been reformatted by Knovel to provide easier navigation.
Index Terms Cooling tower (induced draft)
Links 5
33
306
Cooling tower (induced draft single-cell circular) Cooling tower (mechanical draft)
6 4–6
317
Cooling tower (natural draft counterflow)
7
299–300
Cooling tower (natural draft crossflow)
7
8
Cooling tower (natural draft fan-assisted)
7
Cooling tower (natural draft) Cooling tower (natural draft wet) Cooling tower (natural draft wet-cooling)
6–12
243
253
257
299–300
402
276
309
2
27
285
290
291
19 257
Cooling tower pack, see Fill or packs Cooling tower (Parallel path airflow hybrid)
317–318
Cooling tower performance
259–262
Cooling tower (wet)
41 314–319
Cooling tube
286
Cooling (turbulent flow)
155
Cooling water Copper Correction factor Corrosion
333 196–197
278–281
428
39
314
319
332
403
401–406 Corrosion control
309
Corrosion, Erosion, and fouling
401–406
Corrosion protection
401–406
Corrosion resistance
329
This page has been reformatted by Knovel to provide easier navigation.
Index Terms
Links
Corrosion resistant finned tube, see Finned tube (corrosion resistant) Cofrosion resistant smooth tube, see Smooth tube (corrosion resistant) Corrosion resistant tubing
37
Corrosive environment
401
Cost
254
262
329–330
334 Countercurrent flow
114
180
Countercurrent flow manifold, see Manifold (countercurrent flow) Countercurrent two-phase flow
109
Counterflow drift eliminator, see Drift eliminator (counterflow) Counterflow fill, see Fill (counterflow) Counterflow
Counterflow temperature profile
6
191
193
237
256
262
274
278
281–283
299
306–307
340
368
417
192
Counterflow tube evaporative cooler, see Evaporative cooler (counterflow tube) Counterflow wet-cooling tower, see Cooling tower (Counterflow wet-cooling) This page has been reformatted by Knovel to provide easier navigation.
Index Terms
Links
CPVC, see Chlorinated polyvinyl chloride Critical pressure Critical zone Crossflow
183
208
75 6
256
274
282
287–289
299
368 Crossflow air-cooled heat exchanger, see Heat exchanger (crossflow air-cooled) Crossflow cooling tower, see Cooling tower (crossflow) Crossflow cooling tower drift eliminator, see Drift eliminator (Crossflow cooling tower) Crossflow evaporative cooler, see Evaporative cooler (crossflow) Crossflow fill, see Fill (Crossflow) Crossflow Heat exchanger, see Heat exchanger (crossflow) Crossflow radiator, see Radiator (crossflow) Crosswind
14
Cumulus cloud
319
Cupronickel
333
Curved duct
93
Curved duct (circular-arc bend)
93
Curved duct (composite miter bend)
93
Curved duct (miter bend)
93
300
This page has been reformatted by Knovel to provide easier navigation.
Index Terms
Links
Curved duct (miter bend with cascade)
94
Curved duct (single miter bend)
93
Curved duct (square bend)
93
Curved ducts or bends
93–94
Cylindrical polar coordinate
300
Cylindrical-vortex generator
317
D Damage
410–406
Darcy friction factor
62
73
Darcy-Weisbach equation
61
77
Datong Power Plant
24
Delta
97
387
Delta configuration
30
Deluge enhancement
28
Deluge system, see Dry/wet cooling system (deluge system) Deluge water temperature Deluge water Deluging pump Density
291–292
297
31
36
290–291
295
30 149
202
210
267
270
344–345
356
359
366–367
391 Desulphurized flue gas, see Flue gas (desulphurized) Diameter
283–286
361
This page has been reformatted by Knovel to provide easier navigation.
Index Terms Dielectric
Links 132
Differential equation
48
Diffuser (open-outlet)
89
Diffuser (conical)
86
Diffuser efficiency
87–89
Diffusion coefficient
224–226
Diffusion rate
224–225
Dilution line
316–319
Dimensionless loss coefficient Dimensionless mass transfer number
89
236
76 227
Dimpled fin, see Fin (dimpled) Direct contact spray condenser Dirt
29 262
Discharge coefficient, see Coefficient of discharge Dispersed bubble flow, see Bubble Flow Dispersed phase
299
Displaced promoter
149
Dissolved solid
9
11
Distributing flow manifold, see Manifold (distributing) Distribution-correction factor
429
Disturbance wave, see Wave (disturbance) Dittus and Boelter equation
155
160
Dividing header, see Header (dividing)
This page has been reformatted by Knovel to provide easier navigation.
Index Terms
Links
Doron V-bar fill, see Fill (Doron V-bar) Double L-fin, see Fin (double-L) Downstream
387
Draft equation
257
Drag
98–101
Drag coefficient
98–101
105
303
422 Drag effect Drag equation Drag force Drag resistance Drift
267–268
344
98 5
303–304
267 9
Drift elimination
310
Drift eliminator
254
Drift eliminator (counterflow)
310
309 300
Drift eliminator (crossflow cooling tower)
310
Drift eliminator efficiency
312
Drift eliminator pack (cellular)
310
Drift eliminator (sinusoidal wave shape) Drift loss Drive system (fan)
309 9 14
Droplet
299
Droplet collection efficiency
310
Droplet eliminator, see Drift eliminator Droplet formation
309
254
This page has been reformatted by Knovel to provide easier navigation.
309–312
Index Terms
Links
Dropwise condensation
169
Dry air
233
238
264
266–269
279
294
363
365
373
Dry-cooling
24
Dry-cooling system (see Air-cooled system) Dry-cooling tower, see Cooling tower (dry) Dry heat exchanger, see Heat exchanger (dry) Dry/wet and wet/dry cooling system
2
27–41
Dry/wet cooling system (augmentation condenser system)
35
Dry/wet cooling system (Capacitive–cooling system)
35
Dry/wet cooling system (deluge system)
35
Dry/wet cooling system (Parallel-connected) Drybulb temperature
37 12
33
40
232
235
291
313–315
318
339
Dryness factor
106
Durability (fill)
255
Dust
401
405–406
61
98
Dynamic pressure
This page has been reformatted by Knovel to provide easier navigation.
Index Terms Dynamic viscosity
Links 56
68
74
84
112
152
202
206
208
296
351
353
363–367
E Earthquake hazard
24
Ecodyne T-bar fill, see Fill (Ecodyne T-bar) Eddy shedding
397
Edge effects
25
Edge tension
407
Edge tension-wound helical fin, see Fin (edge tension-wound helical) Effective surface area Effectiveness—NTU method
294 197–215
Effectiveness—NTU Method applied to Evaporative System
274–281
E-Fin, see Finned tube (extruded) Eigenvalue
302
Elastic module
409
Electrical conducter
132
Electric-heated tube
151
Electrostatic field
150
Eliminator wall
309
414
Eliminator, see Drift eliminator Elliptical duct
147
156
This page has been reformatted by Knovel to provide easier navigation.
157
Index Terms
Links
Elliptical nozzle
339
Elliptical steel tube
369
372
341
367
332–335
Embedded fin, see Fin (embedded) Enclosed area between dilution line and saturation curve
316–319
Energy balance
238
Energy equation
299
Energy recovery turbine, Turbine (energy recovery) Energy transfer
132
England
402
Enthalpy
43–44
211
214
233–234
238–240
242–247
264–269
275–280
282
291–294
318
341
Enthalpy gradient
246
Enthalpy transfer
239
Enthalpy transfer rate
277
Entrance contraction loss coefficient
343
Environmental consideration
2
Environmental study
10
Equation of continuity
42
Equation of viscosity
224
Equilibrium
224
Equipment protection
101
Escape velocity Euler number Evaporation rate
274
10
13 346–347
372
242
This page has been reformatted by Knovel to provide easier navigation.
375
Index Terms Evaporation
Evaporative capacity rate ratio Evaporative condenser Evaporative condenser (finned tube) Evaporative cooler (closed circuit)
Links 3
9
27
35
39
236
240
241
254
274–298
313–315
276 35 282 28
282–298
Evaporative cooler (counterflow tube)
283
Evaporative cooler (crossflow)
287
Evaporative cooler (finned tube)
289
Evaporative cooler performance
282–283
295
35
275
2
4
Evaporative cooling Evaporative cooling system (cooling tower) Excursion
418
Exhaust
309
Exhaust steam Exhaust steam (turbine) Exit air temperature
3
33
16
20
298
37
243
Expanded-plate fin, see Fin (expanded-plate) Expansion angle Expansion loss
88 343
Expansion loss coefficient (outlet)
85
Expansion (pipe diameter)
79
344
Extended film fill, see Fill (extended film) Extended Surfaces
162–168
Extruded bimetallic finned tube, see Finned tube (extruded bimetallic) This page has been reformatted by Knovel to provide easier navigation.
390
Index Terms
Links
Extruded finned tube, see Finned tube (extruded)
F Falling film flow Fan
109 3
12–13
38
40–41 Fan-assisted natural draft cooling tower, see Cooling tower (natural draft fan-assisted) Fan (axial floor)
7
Fan (axial flow)
3
16
19
30
36
429
Fan guard Fan installation cost Fanning friction factor Fauna
105 7 62
64
405
Federal Republic of Germany, see Germany feedwater
3
Fick’s Law of diffusion
224
Fill (accu-pac)
252
Fill (American tower plastic)
251–252
Fill (asbestos-free fiber cement)
252
Fill characteristic
249
Fill (cooldrop)
251
Fill configuration
260
Fill (counterflow)
251
252
258–259
261
272
310
7
251
260
Fill (crossflow)
255
This page has been reformatted by Knovel to provide easier navigation.
Index Terms
Links
Fill (Doron V-bar)
251
Fill (Ecodyne T-bar)
251
Fill (extended film)
255
Fill (film)
3
Fill height
257
Fill loss coefficient
255
271–272
274
Fill (Marley bar)
251
252
Fill (metal)
253
Fill or packs
249–274
Fill performance
271
273
Fill (plastic)
249
254
Fill (splash)
254
260
Fill (stainless steel)
253
262
Fill strength
255
Fill (thermoplastic film)
255
Fill thickness
301
Fill (trickle pack)
255
Fill (wood lath)
251
Fill weight
255
Film
249
Film condensation
309
169–172
Film fill, see Fill (film) Film flow theory
114
Film pack
309
Film temperature
283
Film thickness (Film condensation)
170
172
Film (water)
295
313
162–167
282
Fin
401 This page has been reformatted by Knovel to provide easier navigation.
329–337
Index Terms Fin (aluminum plate) Fin (aluminum)
Links 202–203
208
19
22–25
331
162–168
203–204
352
403
409
416
412 Fin base
407–408
Fin corrosion
401–406
Fin (circular)
168
Fin (continuous plate)
335
Fin density
426
Fin (dimpled)
337
Fin (double-L) Fin (edge tension-wound helical) Fin efficiency Fin efficiency or effectiveness
331–332 407
162–167
Fin (embedded)
407
Fin (expanded-plate)
407
Fin (flat-plate)
335
349
Fin (footed tension-wound helical)
407
418
Fin (formed plate)
401
Fin (galvanized steel)
402
403
Fin (galvanized)
404
407
Fin (G-helically wound AI)
332
Fin height
352
Fin (helical)
330
Fin (helical-extruded muff)
407
Fin (helically wound wire-loop)
334
Fin (hexagonal)
168
Fin (I)
330
Fin (integral)
407
Fin (interference fit)
407
333
This page has been reformatted by Knovel to provide easier navigation.
Index Terms Fin (IW) Fin (L)
Links 330 331–33
Fin (longitudinal)
162
Fin (louvered C-T)
337
Fin (louvered)
337
Fin material
402
Fin (perforated plate)
332 165 425
30–31
337
Fin pitch
349–350
361
Fin plate
36
Fin (plate)
336
Fin (punched)
334
Fin (radial)
165
167
Fin root
164
168
334
361–362
370
415
Fin (segmented)
334
Fin shape
330
Fin (slotted)
333
Fin (slotted plate)
335
Fin spacing
287
Fin (steel-plate)
335
Fin (studded)
334
Fin surface area
370
Fin surface temperature
356
Fin (tension wound)
330–332
Fin thickness
349–350
Fin tip Fin tube
425
361
361
164 35
36
Fin (wavy)
330
337
Fin (wire-loop extended)
334
Finite difference
429
This page has been reformatted by Knovel to provide easier navigation.
Index Terms
Links
Finned elliptical tube
172
Finned flattened tube
172
Finned surface area
358
Finned surfaces Finned tube
330–337 16–27
41
150–151
283–289
330
332
341
349
357
360
376–389
403
414
418
19
25–26
407 Finned tube (air-cooled)
185
Finned tube (aluminum)
402–403
Finned tube (annular)
334
Finned tube (bimetallic)
403
Finned tube (corrosion resistant) Finned tube (elliptical)
27 377
Finned tube evaporative condenser, see Evaporative condenser (finned tube) Finned tube evaporative cooler, see Evaporative cooler (finned tube) Finned tube (extruded bimetallic) Finned tube (extruded muff)
330 417–418
Finned tube (extruded)
333
Finned tube (forgo-type)
335
Finned tube (galvanized elliptical)
17
Finned tube (galvanized plate)
330
Finned tube (galvanized steel)
419
Finned tube (grooved)
333
Finned tube heat exchanger, see Heat exchanger (finned tube) Finned tube heat exchanger bundle
13–14
This page has been reformatted by Knovel to provide easier navigation.
Index Terms
Links
Finned tube (helically wound bimetallic)
330
Finned tube (helically wound galvanized steel)
330
Finned tube (radial)
334
Finned tube (rectangular)
334
Finned tube (segmented)
334
Finned tube (serrated) Finned tube (skived fin) Finned tube (slotted aluminum plate)
330 21–22
Finned tube (spiral-wound)
405
Finned tube (square)
334
Finned tube (staggered)
423
Finned tube (staggered circular)
377–386
Finned tube (wavy-flattened)
336
Finned tube (wrapped-on aluminum)
404
Finned tube (yawed)
397
Fin-tube bond
334
Fire resistant
249
Flat-plate fin, see Fin (flat-plate) Flattened tube
118
Flat tube
337
Flooding
113–119
Flooding rate
115
Flora
405
Flow acceleration
343
Flow condition
69
Flow distribution
94
Flow field
57
Flow in ducts Flow instability/wave growth theory
336 312
96
61–75 114
This page has been reformatted by Knovel to provide easier navigation.
Index Terms
Links
Flow loss
397
Flow parameter
360
393
60
106–112
117
121
179
180
256
300
310
13–14
32
34
47
96
106
107
197
262
268
270–273
291
294
312
350
354
355
362–364
394
404–405
Flow pattern
387 Flow rate
374 Flow regime, see Flow pattern Flow resistance
330 422
Flow through Screens or Gauzes Flow velocity Flue gas (desulphurized) Flue gas stack Fluid bulk mean temperature Fluid dynamics
101–105 57
388
8 26 149
151
55
Fluid film
409
415
Fluid flow
42
191
Fluid Mechanics
55–129
Introduction
55–56
Viscous Flow
56–60
Flow in Ducts
61–75
Fluid mixing
201
This page has been reformatted by Knovel to provide easier navigation.
Index Terms
Links
Fluid motion
55
Fluid pressure
96
Fluid pulsation
150
Fluid statics
55
Fluid velocity
44
Fog
316
Fogging
316
96
Footed tension-wound helical fin, see Fin (footed tension-wound helical) Forced convection, see Convection (forced) Forced-draft
4
Forced flow
16
Foreign object removal
420
427
141
255
261
319
334
404–406
101
Forgo-type finned tube, see Finned tube (forgo-type) Formed-plate fin, see Fin (formed plate) Fouling Fouling characteristic Fourier’s law of heat conduction
262 133–135
224
Free convection, see Convection (natural) Free mixing region
79
Free mixing stall region
79
Free stream air Free stream turbulence Free stream value
236
241–242
420–428 57
This page has been reformatted by Knovel to provide easier navigation.
Index Terms Friction
Links 48
61
96
119
181
185
267–268
296
359
62–75
84
97
144
155
187
296
344–348
354
368
375
405
90
96
390 Friction factor
422 Friction gradient
90
Friction loss
69
Friction multiplier Friction (skin)
119–120 64
Friction (surface)
344
Frictional effect
47
Frictional pressure differential
344
189
Frictional pressure drop, see Pressure drop Frictional pressure gradient
120
Frictional pressure
84
Frictionless adiabatic flow
44
Friedel correlation
120
Froude number
116
Fuel cost
181
11
G Gagarin Power Plant
21
Galvanized elliptical finned tube, see Finned tube (galvanized elliptical)
This page has been reformatted by Knovel to provide easier navigation.
Index Terms
Links
Galvanized fin, see Fin (galvanized) Galvanized plate finned tube, see Finned tube (galvanized plate) Galvanized steel
290
Galvanized steel-finned tube, see Finned tube (galvanized steel) Galvanizing process
167
Gamma function
177
Gap resistance
407–419
Gargarin
403
Gas constant
230
Gas expansion factor
340
364
374
Gas flow rate
107
108
109
110
114
115
116 Gas liquid interface Gas turbine Gas-liquid flow
114
115
29 106
Gauze, see Mesh General features of isentropic flow Geodetic pressure
44–47 185
Geodetic pressure gradient, see Static pressure gradient Geometric factor
62
Germany
21
26
402 G-fin, see Finned tube (grooved) Gillette, Wyoming
18
This page has been reformatted by Knovel to provide easier navigation.
41
Index Terms
Links
Gnielinski’s equation
156
Graetz number
149
Graphic method
282
Grashof number
149
Gravitational force
304
Gravity controlled flow
160
368
151
153
304
179–182
Gravity controlled flow (slug flow)
179
Gravity controlled flow (stratified flow)
179
Gravity controlled flow (wavy flow)
179
Gravity effect
110
Gravity
170
181
Great Britain
20
21
Grootvlei Power plant
23
25
404
406
Guide vane
91
93
Gyongyos, Hungary
21
402
Grooved fin, see Finned tube (grooved)
H Hagen-Poiseuille solution Harmonic mean density Harvard Power Plant
62 271 39
Hausen equation
155
Hazard
314
160
Head loss, see Mechanical energy loss HEADd, see Heller/EGI Advanced Dry/deluged (HEADd) Combined This page has been reformatted by Knovel to provide easier navigation.
401
Index Terms
Links
HEADd (Cont.) cooling system Header (combining)
96
Header (dividing)
96
Header energy correction factor
98
Header pressure
96
Header shear stress, see Shear stress Heat and Mass Transfer in Wet-Cooling Towers
236–249
Heat capacity rate
205
Heat capacity ratio
212
Heat capacity
209–210
215
198–199
Heat conduction, see Conduction Heat dissipation capacity
27
Heat exchange
232
Heat exchange effectiveness
276
Heat exchanger (air-cooled)
2
12–27
33
35
38
313–314
329
387
402–405
427 Heat exchanger (air-cooled crossflow)
341
Heat exchanger (air-cooled forced draft)
13
429
Heat exchanger (air-cooled induced draft)
13–14
429
Heat exchanger (air-cooled mechanic draft)
13–19
Heat exchanger (air-cooled natural draft)
20–26
Heat exchanger (Aluminum plate fin-tube)
35
Heat Exchanger (Aluminum skived-fin-tube)
35
This page has been reformatted by Knovel to provide easier navigation.
Index Terms Heat exchanger bundle
Links 19
25–26
83
314
339
343
200–201
278
345
196
200–201
210
215
278
355
362 Heat exchanger (counterflow) Heat exchanger (crossflow)
429 Heat exchanger (crossflow air-cooled)
196
Heat exchanger delta
21–24
Heat exchanger (dry)
33
201–207 40
282
196
401
198–199
200
132
142–143
146
159
172–174
177
208
408
402 Heat exchanger (finned tube)
20
Heat exchanger frontal area
358
Heat exchanger mass
329
Heat exchanger material
403
Heat exchanger (parallel flow)
194
Heat exchanger (plate-finned)
396
Heat exchanger pipe
63
Heat exchanger (skived-fin)
36
Heat exchanger volume
329
Heat exchanger (water-cooled)
417
Heat flow, see Heat transfer Heat flux
Heat load
40
Heat loss
341
Heat rejection
10
12
299
309
This page has been reformatted by Knovel to provide easier navigation.
34
Index Terms Heat rejection capacity Heat rejection rate Heat rejection system Heat sink
Links 27 297 11 1
Heat transfer
131–222
Heat Transfer
131–222
Introduction
28 236–249
131
Modes of Heat Transfer
131–141
Conduction
132–136
Convection
137–139
Overall heat transfer coefficient
139–141
Heat Transfer in Ducts
142–161
Laminar flow
142–154
Turbulent flow
155–156
Transitional flow
157–161
Extended Surfaces
162–168
Fin efficiency or effectiveness
162–167
Condensation
169–191
Film condensation
169–172
Condensation in duct
172–175
Condensation on cylinder
175–179
Condensation correlations
179–185
Pressure and temperature distribution inside duct Heat Exchangers
185–191 191–215
Logarithmic mean temperature difference
191–197
Effectiveness—NTU method
197–215
References
216–222
This page has been reformatted by Knovel to provide easier navigation.
329–439
Index Terms
Links
Heat transfer and pressure drop correlations
376–386
Heat transfer by convection, see Convective heat transfer Heat transfer characteristic Heat transfer coefficient
381 63
139–144
148–151
154
157–158
163
165
172–175
192
200
202–203
206
215
242
282
284
295–297
329
340–342
346
354–358
368
371
380
384
396
404
413–415
420
422–423
370
372
427 Heat transfer correlation
377–381
Heat transfer correlations for staggered circular finned tubes Heat transfer equation Heat transfer equipment Heat transfer in ducts
377–381 148–149 162 142–161
Heat transfer parameter
358
Heat transfer performance
381
Heat transfer rate
134
136
142
151
167
169
171–172
185
210–215
226
279
282
285
298
314
This page has been reformatted by Knovel to provide easier navigation.
Index Terms
Links
Heat transfer rate (Cont.)
Heat transfer resistance
350
355
404
417
418
Heat transfer surface
337–439
Heat transfer surface area
329–439
Heat transfer surfaces
329–439
Introduction
329
Finned surfaces
330–337
Test facilities and Procedures
337–340
Interpretation of Experimental Data Presentation of data
341–349 346–376
Colburn j factor and friction factor
347–357
Characteristic heat transfer and pressure drop parameter
357–376
Heat Transfer and Pressure drop correlations
376–386
Heat transfer correlations for staggered circular finned tubes
377–381
Pressure drop correlations for staggered circular finned tubes
382–386
Oblique Flow through heat exchangers
387–401
Oblique flow analysis
388–393
Oblique flow experiments
394–401
This page has been reformatted by Knovel to provide easier navigation.
396
Index Terms
Links
Heat transfer surfaces (Cont.) Corrosion, Erosion, and Fouling
401–406
Thermal contact and Gap Resistance
407–419
Analysis of contact and gap resistance
408–417
Measuring contact and gap resistance Free stream Turbulence
417–419 420–428
Non-Uniform Flow and Temperature Distribution References
428–429 430–439
Heat transport system
35
Heat-absorbing surface
162
Heated surface vibration
150
Heating
267
Heating (turbulent flow)
155
Heat-rejecting surface
162
Heat-transfer-enhanced steam condenser/ammonia reboiler
33
Helical-extruded finned muff, see Fin (helical-extruded muff) Helical fin, see Fin (helical) Helically wound aluminum G-fin, see Fin (G-helically wound aluminum) Helically wound bimetallic finned tube, see Finned tube (helically wound bimetallic) Helically wound galvanized steel This page has been reformatted by Knovel to provide easier navigation.
Index Terms
Links
Helically wound galvanized steel (Cont.) finned tube, see Finned tube (helically wound galvanized steel) Helically wound wire-loop fin, see Fin (helically wound wire-loop) Heller/EGI Advanced Dry/deluged (HEADd) Combined cooling system
29
Heller system
20
Heller-type finned-tube bundle
24
24
Hewitt and Roberts Flow Pattern map
113
Hexagonal fin, see Fin (hexagonal) High velocity zone
58
Homogeneous model for two-phase flow Horizontal tube
119
121
151
152
28
37
39–40
313
317
229–230
232
235
242
246–247
264–266
270
280
292
294
313–315
318
363
373
285
Horizontal two-phase flow, see Two-phase flow (horizontal) Humidification Humidity Humidity ratio
Hungary Hybrid cooling system
21 39–41
Hybrid evaporative cooler
27
Hybrid system
27
This page has been reformatted by Knovel to provide easier navigation.
Index Terms Hydraulic diameter
Links 72
118
148
349
352–353
357
137
228
Hydraulic entry length
66
Hydrodynamic boundary layer
57
Hydrodynamic laminar flow Hydrostatic pressure
139 46
Hyperbolic concrete natural draft dry-cooling tower, see Cooling Tower (hyperbolic concrete natural draft dry-cooling)
I Ibbenburen Power Plant
21
402
I-fin, see Fin (I) Ince B power plant fan-assisted draft tower
7
Incidence angle
392
Inclined flow
388
Inclined two-phase flow, see Two-phase flow (inclined) Incompressible flow
47
76–77
Incompressible fluid
61
97
Indirect cooling system
24
Induced-draft
4
16
Induced draft cooling tower, see Cooling tower (induced draft) Inertial force
58
Inertial impaction separator, see Drift elimator This page has been reformatted by Knovel to provide easier navigation.
301
427
Index Terms Inlet loss coefficient
Links 98
In-line tube
288
Insect
405
Integral fin, see Fin (integral) Integrally finned tube, see Finned tube (skived fin) Interface
226
Interface water temperature
237
Interfacial friction factor
186
Interfacial shear stress
181
Interfacial tension
106
Interference effect
78
Interference
110
407
Interference-fit fin, see Fin (interference fit) Interference-free flow
78
Internal energy
43
Internal extended surface Interpretation of experimental data Iran
149 341–349 24
Irrotational inlet flow Isentropic flow
32
302 44–47
Isentropic stagnation pressure
44
Isfahan Power Plant
32
Isosceles Triangular duct
66
Isotherm
233
372
Isothermal condition
231
346
Isothermal loss coefficient
360
376
Isothermal pressure drop
396
This page has been reformatted by Knovel to provide easier navigation.
395
Index Terms
Links
IW-fin, see Fin (IW)
J j factor, see Colburn j factor Jet contraction ratio
85
Johannesburg, South Africa
19
K Kendal Power Plant
25
Kern Station
33
Killingholme Combined cycle power plant
41
Kinematic viscosity
57
Kinetic energy coefficient
43
76
Kinetic energy
43
57
76
80
341
397
429 Kinetic theory of gases Kutateladze number
57
225
184
L Laminar boundary layer
58
Laminar convection
148
Laminar film condensation
172
185
57–60
62–69
75
139
142
149
188
228
Laminar flow
This page has been reformatted by Knovel to provide easier navigation.
Index Terms
Links
Laminar flow
62–69
Turbulent flow
69–74
Transition laminar-turbulent flow Losses in Duct Systems
75 75–94
Abrupt contractions and expansions
79–85
Reducers and diffusers
86–89
Three-leg junctions
90–92
Curved ducts or bends
93–94
Manifolds
94–98
Drag
98–101
Flow through Screens or Gauzes
101–105
Two-phase flow
106–121
Two-phase flow patterns
106–113
Flooding
113–119
Pressure drop in two-phase flow References Laminar flow (in ducts)
119–121 122–129 142–154
Laminar friction factor, see Friction factor Laminar velocity profile
57
Latent heat of vaporization
208
Latent heat
265
Lateral velocity
98
Lateral
96
Lattice wave Legislation
280 97
132 11
This page has been reformatted by Knovel to provide easier navigation.
Index Terms Lephalale (Ellisras) Lewis factor
Links 18 228
240–246
274
282–283 Lewis number, see Lewis factor L-fin, see Fin (L) Limiting condition for countercurrent flow, see Onset of Flooding Linear model Liquid flow rate
302 107–110
Liquid phase
106
Liquid-Vapor interface
170
List of symbols
XII–XVII
Greek Symbols
XIII–XIV
Dimensionless Groups
XIV–XV
Subscripts Local effective specific volume
116
XV–XVII 121
Logarithmic mean temperature difference
191–197
Longitudinal fin, see Fin (longitudinal) Loss coefficient (pressure)
Losses in Duct Systems
76–94
101–105
263
268
304
307–308
311–312
347
360
372
375
387
389–393
397–400
75–94
This page has been reformatted by Knovel to provide easier navigation.
Index Terms Louver
Links 7
24
30–31
301
317
404
9
10
27
30
262
Louvered C-T Fin, see Fin (louvered C-T) Louvered fin, see Fin (louvered) Low pressure spray nozzle, see Spray nozzle (low pressure) Low pressure steam condenser, see Condenser (low pressure steam) Low-pressure steam
184–185
Low-velocity zone
58
Ludwigshafen Petrochemical plant
39
M Maintenance Majuba Power Plant Make-up water
401–402 19
Maldistributed flow, see Non-uniform flow Manifold (cocurrent flow)
94–95
Manifold (collecting)
94–95
Manifold (countercurrent flow)
94–95
Manifold (distributing)
94–95
Manifold (mixed flow)
94
Manifold momentum equation
98
Manifold
94–98
340
Marley bar fill, see Fill (Marley bar) This page has been reformatted by Knovel to provide easier navigation.
Index Terms Martinelli correlation Mass balance
Links 120 9
238
245
Mass exchange
232
Mass flow rate
60
106
170
177
188
238
245
257
286
309
317
339
Mass flux, see Mass velocity Mass Transfer and Evaporative Cooling Introduction Mass transfer
223–327 223 223–236
Heat and mass transfer in wet-cooling towers Fill or packs
236–249 249–274
Effectiveness—NTU method applied to evaporative system
274–281
Closed circuit evaporative cooler
282–298
Rain zone
299–309
Drift eliminators
309–312
Spray and Adiabatic cooling
313–314
Visible plume abatement
314–319
Mixing line ratio
315–316
Plume abatement angle
316
Enclosed area between dilution line and saturation curve References
316–319 320–327
This page has been reformatted by Knovel to provide easier navigation.
Index Terms Mass transfer coefficient
Links 227
229–231
242
259–260
282
285–287
292
294
305
Mass transfer
2
181
223–319
Mass velocity
60
106
224
343
347
375
177–178
184
Matimba Power Plant
18–19
Matra Power Plant, see Gagarin Power Plant Maximum temperature isotherm
233
Mean condensation coefficient
183
Mean condensation heat transfer coefficient, see Mean heat transfer coefficient Mean flow direction Mean flow Mean fluid velocity
58 392 59
Mean heat flux
170
Mean heat transfer coefficient
170
Mean Specific heat
269
Mean static pressure, see Static pressure Mean steam temperature
190
Mean temperature difference
175
Mean temperature
339
Mean velocity Mean water temperature
178
43 264
273
280 Measurement error
256
This page has been reformatted by Knovel to provide easier navigation.
278
Index Terms
Links
Measuring contact and gap resistance
417–419
Mechanical draft (Air-Cooled Heat Exchangers) Mechanical draft (Cooling Towers)
13–19 4–6
Mechanical draft counterflow cooling tower, see Cooling tower (induced draft) Mechanical draft crossflow cooling tower, see Cooling tower (induced draft) Mechanical draft tower, see Cooling tower (mechanical draft) Mechanical draft
3
Mechanical energy loss
79
Mechanical energy
75
Mechanical strength
329
Medium pressure spray nozzle, see Spray nozzle (medium pressure) Merkel equation
243–244
248–249
256
260–261
263
267
274
279
281–282
287
305–308
Merkel integral equation, see Merkel equation Merkel method, see Merkel equation Merkel number, see Merkel equation This page has been reformatted by Knovel to provide easier navigation.
Index Terms
Links
Merkel transfer characteristic, see Merkel equation Merkel’s approximation, see Merkel equation Merkel’s theory, see Merkel equation Mesh
101–102
Metais and Eckert plot (modified)
255
420
109
110
157
Metal fill, see Fill (metal) Middle transitional flow
158
Mist flow pattern
179
Mist flow
107
Mist zone
242
Mist
309
313
Miter bend with cascade, see Curved duct (miter bend with cascade) Miter bend, see Curved duct (miter bend) Mixed airstream
196
Mixed convection
148
153
Mixed flow manifold, see Manifold (mixed flow) Mixed laminar flow
158
Mixed turbulent flow
158
Mixing cup temperature Mixing line ratio Mixing Modes of Heat Transfer Modified Bauer-Vogel process
66–67
72–74
315–319 224
315–319
131–141 404
This page has been reformatted by Knovel to provide easier navigation.
391
Index Terms
Links
Moisture content
232
Molecular mass
226
Molecular movement
225
Molecular thermal conductance
138
Molecular volume
226
Molecular weight
225
Momentum correction factor
402
223–236
Molecular speed
Momentum balance
242
96–97
119
97
Momentum diffusivity
139
Momentum effect
170
Momentum equation
388
Momentum exchange
57–59
Momentum theorem
47–48
106
Momentum transfer, see Momentum exchange Momentum
64
96
185
189
226
228
268
299
343
448 Mono-dispersed spray Moody diagram Muff
299 69 333
N Natural Convection, see Convection (natural) Natural draft (air-cooled heat exchangers), see Heat exchanger (air-cooled natural draft) This page has been reformatted by Knovel to provide easier navigation.
Index Terms
Links
Natural draft (cooling tower), see Cooling Tower (natural draft) Natural draft cooling tower, counterflow, see Cooling tower (natural draft counterflow) Natural draft cooling tower, crossflow, see Cooling tower (natural draft crossflow) Natural draft
3
Natural temperature lapse
39
Neckarwestheim Power Plant (GKN)
41
Net force
48
New Mexico
33
Newton’s equation of viscosity
56
Newton’s law of cooling Newton’s second law of motion Nitrogen oxide Noise restriction Non-isothermal flow
6
138 48 401 2 398
Non-uniform flow and temperature distribution
428–429
Non-uniform flow
428–429
Normal velocity gradient
56
Nozzle coefficient
339
Nozzle
363
NTU method
274–281
NTU (Number of transfer units)
199
Number of transfer units
199
This page has been reformatted by Knovel to provide easier navigation.
Index Terms Nusselt number
Links 138
139
142
143
145
152
153
154
159
160
227
296
397
422
346
357
O Oblique flow analysis
388–393
Oblique flow experiments
394–401
Oblique flow through heat exchangers
387–401
Oblique velocity
427
Ohm’s Law
134
Ohnesorge number
118
One-dimensional flow One-dimensional model Onset of flooding Open circuit
44
60
241 114–115 10
Open-outlet diffuser, see Diffuser (open-outlet) Operating cost
27
Outlet loss coefficient
98
Outlet pressure tap Overall heat transfer coefficient
339 139–141
P Pacific Gas and Electric Company
33
This page has been reformatted by Knovel to provide easier navigation.
Index Terms
Links
Parallel flow heat exchanger, see Heat exchanger (parallel flow) Parallel flow manifold, see Manifold (cocurrent flow) Parallel flow temperature profile
192
Parallel flow
191
198–199
356
388
260
Parallel path airflow hybrid cooling tower, see Cooling tower (parallel path airflow hybrid) Parallel resistance
410
Parallel thin sheet
388
Parallel-connected dry/wet cooling system, see Dry/wet cooling system (Parallel-connected) Partial density Partial pressure
230 229–230
Passive heat transfer technique
149
Peclet number
139
Perfect gas equation of state
45
67
229–230
45
229
231
22
201
256–259
298–299
309–310
336–337
387
426–429
Peripheral temperature
142
144
Perpendicular flow
260
Perfect gas law, see Perfect gas equation of state Perfect gas Perforated plate fin, see Fin (perforated plate) Performance
This page has been reformatted by Knovel to provide easier navigation.
Index Terms
Links
Petukhov equation
155
Phase boundary
179
Phase change
55
Pipe flow
61
160
69–70
Pipe friction equation, see Darcy-Weisbach equation Plain tube bank
421–422
Plain tube evaporative cooler, see Evaporative cooler Plant material
405
Plastic fill, see Fill (plastic) Plate fin, see Fin (plate) Plate vortex generator Plate-finned elliptical tube
317 25
Plug flow, see Slug flow Plume abatement angle Plume abatement Plume air
316 27 315
Plume condensation
315–316
Plume dilution
315–316
Plume evaporation
315–316
Plume formation Plume severity Plume
317–318
39
314–319
315–316 39–40
314–319
Poisson’s ratio
409
414
Polar coordinate
301
Pollution
401–403
This page has been reformatted by Knovel to provide easier navigation.
Index Terms
Links
Poly-dispersed spray
299
Polypropylene
250
Polyvinyl chloride
250
Porosity (fill)
268
Porosity
103
310 345
Positive gauge pressure, see Pressure (positive gauge) Potential energy Potential flow field Power (fan) Power consumption Prandtl number
44 302 5 14 139
153
158
160
209
227
295
351
353
366
367
Precooling
37
Preheater/peak cooler
31
Presentation of data Pressure (positive gauge) Pressure (total system)
32
346–376 20 226
Pressure and temperature distribution inside duct Pressure change Pressure differential (static)
185–191 119
185
85
Pressure differential
207
346
Pressure distribution
77
96
Pressure drop correlation for staggered circular finned tubes Pressure drop in two-phase flow Pressure drop parameter
382–386 119–121 372
This page has been reformatted by Knovel to provide easier navigation.
386
Index Terms Pressure drop
Pressure gradient
Links 14
20
61
62
64
69
77–80
84
85
90
93
101
116
119–121
179
181
184
186–190
255
256
267
280
289
290
299
304
310–312
318
334
339
343
344
346
350
356–360
362
372
375–386
388
391
393
401
404
420
421
424
428
77
120
121
233
234
Pressure loss coefficient, see Loss Coefficient (pressure) Pressure loss, see Pressure drop Pressure recovery
87
Pretreated water
31
Prime surface Production cost Psychromatic chart
162 10 232 313–319
Psychrometric model
317
Psychrometric property
232
Psychrometry
223
232
This page has been reformatted by Knovel to provide easier navigation.
Index Terms
Links
Pump
41
Pumping (water)
10
Pumping power
254
309
Punched fin, see Fin (punched) PVC, see Polyvinyl chloride
R Radial fan
339
Radial fin, see Fin (radial) Radial finned tube, see Finned tube (radial) Radial gap
408
Radial heat conduction, see Conduction (radial) Radial heat flux
141
Radial inflow
301
Radial velocity vector
301
Radiation hazard
302
24
Radiation
132
Radiator (crossflow)
349
254
Radiator
15
Rain zone
4
256
Razdan Power Plant
24
32
Reboiler
34
Recirculating water
282
Recirculation pump
34
Recirculation
5
79
This page has been reformatted by Knovel to provide easier navigation.
299–309
419
Index Terms
Links
Recooled water
4
20
Rectangular duct
66
72
145–146
Rectangular finned tube, see Finned tube (rectangular) Rectangular tower
307
Reduce clogging
101
Reduce fouling
101
Reducer Reducers and diffusers Reflux condensation Refrigerant Relative humidity
86 86–89 180 207–208
214–215
27
40
231–236
19
23
58–60
68
70
73–75
84
99–105
139
148
153
154
156
158
171
172
178
182
184
188
203
208
214
287
288
296
314 Reliability (test data) Republic of South Africa
256–257 18 25
Residence time
254
Residual radial stress
407
Residual water load
312
Resistance
312
Reverse flow manifold, see Manifold (countercurrent flow) Reynolds number
This page has been reformatted by Knovel to provide easier navigation.
Index Terms
Links
Reynolds number (Cont.)
Reynolds stress Rinsing Riser
305
339
347
348
352
354
357
362
364
368
372
374
378
420–428
59
61
405 7
Rosin-Rammler distribution function
308
Round tube bundle
396
Row correction factor
380
381
Row-effect
380
381
20
23
402
237
240–242
247
256
265
266
270
275–282
285
291
292
297
Rugeley Power Plant Running cost
7
S Sampling tube San Juan Power Plant Saturated air
339 33
309 Saturated discharge air, see Plume Saturated steam
414
418
Saturated water vapor
169
235
279 Saturation curve
233
315–319
This page has been reformatted by Knovel to provide easier navigation.
270
Index Terms
Links
Saturation humidity ratio
239
Saturation line
315
Saturation pressure
190
236
169–170
207–208
Saturation
230
239
241
Sauter mean diameter
309
Sauter mean value droplet diameter
309
Scale
262
405
420–423
Saturation temperature
Schmehausen Nuclear Plant Schmidt number Screen Second principle of thermodynamics
26 227 101–105 133
Segmented fin, see Fin (segmented) Segmented finned tube, see Finned tube (segmented) Self-generated free mixing stall region Self-generating free mixing region Separated flow model Separation point
79 79 119
121
79
Separator, see Drift eliminator Serpentine copper tube
207
Serrated finned tube, see Finned tube (serrated) Service life Shahid Rajai Power Plant Shear stress
402
403
24 56–59
97
108
170
172
182
This page has been reformatted by Knovel to provide easier navigation.
Index Terms
Links
Sherwood number
227
Silt
262
Single miter bend, see Curved duct (single miter bend) Single-cell circular induced draft cooling tower, see Cooling tower (induced draft) Single-pass multirow oil cooler
207
Sinusoidal wave shape eliminator, see Eliminator (sinusoidal wave shape) Siting
11
Skin friction, see Friction (skin) Skived-fin heat exchanger, see Heat exchanger (skived-fin) Slotted aluminum plate finned tube, see Finned tube (slotted aluminum plate) Slotted fin, see Fin (slotted) Slotted-plate fin, see Fin (slotted-plate) Slug flow (gravity-contolled flow), see Gravity controlled flow (slug flow) Slug flow pattern
180
Slug flow
107
109
110
Slug
107
109
110
Smooth tube bundle
282
Soldered fin, see Fin (galvanized) South Africa, see Republic of South Africa This page has been reformatted by Knovel to provide easier navigation.
Index Terms Spacer Specific heat
Splash bar
Links 334 45
152
199
202
205
208
210
233
240
247
264
266
279
291–295
350
353
365–367
2
7
254
258–259
249
Splash fill, see Fill (splash) Spray and adiabatic cooling Spray condenser (direct contact) Spray cooling Spray eliminator Spray nozzle (low pressure) Spray nozzle (medium pressure) spray nozzle
313–314 20 313–314 254 7
253
253 2
4
36
249
253
256
256
299
Spray pattern
253
Spray zone
254
Square bend, see Curved duct (square bend) Square finned tube, see Finned tube (square) Staggered circular finned tube, see Finned tube (staggered circular) Stainless steel fill, see Fill (stainless steel) Stainless steel
250
This page has been reformatted by Knovel to provide easier navigation.
Index Terms Stanton number Static loss coefficient
Links 139 76
Static pressure differential, see Pressure Differential (static) Static pressure drop Static pressure gradient Static pressure
Staton number ratio
85 121 77
79–85
98
190
339
343–344
350
356
360
382
388
426
Steady flow
42
Steady state
48
Steam cleaning
405
Steam exhaust pipe
16
Steam turbine
29
Steam-heated tube
267
31
151
Steel-plate fin, see Fin (steel-plate) storage reservoir (water)
10
Storage tank
35
Straight-pipe friction loss, see Friction loss (straight-pipe) Strain gauge test
411
Stratified flow (gravity-controlled flow), see Gravity controlled flow (stratified flow) Stratified flow
109
Stratified gas-liquid flow
186
Streamline
387
This page has been reformatted by Knovel to provide easier navigation.
Index Terms Structural performance
Links 262
Studded fin, See Fin (studded) Suction
150
Sulfide
402
Sulfur dioxide
401
sump
27
290
Superficial mass velocity, see Mass velocity Super-heated vapor
209–214
Supersaturated air
242
Supersaturated curve
315
Supersaturated discharge air
246
314–319
Surface condenser, see Condenser (surface) Surface friction, see Friction (surface) Surface roughness Surface tension
58
69
99
149
111
112
Surface wetting, see Wetting (surface) Suspended solid
262
T Tangential component
427
Temperature boundary layer, see Thermal boundary layer Temperature correction factor
368
Temperature difference
404
369
This page has been reformatted by Knovel to provide easier navigation.
71 176
Index Terms Temperature distribution
Links 136
138
170
298
339
428–429
191–192
226
228
299
303
Temperature gradient, see Thermal gradient Temperature profile Tension wound fin, see Fin (tension wound) Terminal velocity Teshrin Power Plant Test facilities and procedures
24 337–340
Test facility
256
Thermal boundary condition
142
Thermal boundary layer
137
228
133
153
160
163
166–167
202–208
290
295
351–353
361
365–367
410
414
417
Thermal conductivity
Thermal contact and gap resistance Thermal contact resistance
262
407–419 331
361
370
335
342
415 Thermal contact
330–332 407–419
Thermal cycle
418
Thermal diffusivity
139
Thermal discharge Thermal drop Thermal energy Thermal expansion coefficient
2 256 42
75
331
407
414 This page has been reformatted by Knovel to provide easier navigation.
409
Index Terms
Links
Thermal expansion
331
Thermal gradient
132
137
163
169 Thermal metal-to-metal contact resistance Thermal performance Thermal resistance
Thermocline Thermocouple Thermodynamics
410 397
429
134–135
140–141
169
204
213
296
342
404
409–418
42
55
131
133
282
35 340
Thermoplastic film fill, see Fill (thermoplastic film) Thermosyphon recirculation
34
Three-dimensional circular contraction Three-leg junctions
81 90–92
Tilted fin
397
Timber grid
255
Tower outlet
314–319
Trakya Power Station
29
Transfer characteristic
243
279
281
Transfer coefficient
208
298
308
Transition laminar-turbulent flow
75–78
Transition turbulence
70
Transition zone (fluid flow)
84
Transitional flow (in ducts)
157–161
Transitional flow
157–161
296
This page has been reformatted by Knovel to provide easier navigation.
Index Terms Transmission cost Transport of energy
Links 11 224
Transport of momentum, see Momentum Treated timber
254
Triangular projection
335
Trickle grid
249
308
Trickle pack fill, see Fill (trickle pack) Trickle pack
309
Tube bank
283
286
292
Tube pitch
362
421
423
Tube wall resistance
415
Tube wall
296
Turbine (energy recovery)
20
Turbine back pressure
27
407–408
Turbine exhaust steam, see Exhaust steam (turbine) Turbine outlet pressure Turbine
19 3
16
19
21
24
33
35 Turbogenerator
27
Turbulence
58
99–101
115
116
339
380–381
420–428 Turbulent boundary layer
58
Turbulent flow (in ducts)
155–156
This page has been reformatted by Knovel to provide easier navigation.
Index Terms
Links
Turbulent flow
55–77
80
84–85
155–156
188
209
214
224
228
354
389
399
420–428 Turbulent friction factor, see Friction factor Turbulent regime (Film condensation) Turbulent vapor flow Turkey
172 187 29
Two-dimensional circular contraction Two-phase flow (adiabatic) Two-phase flow (horizontal) Two-phase flow (inclined)
81 181
186
109–110
112
111
116
117
112
114
119
179–181
119 Two-phase flow (vertical)
107–109 117
Two-phase flow patterns
106–113
Two-phase flow
106–121 309
Two-phase mixture
34
U Uniform flow
44
Unmixed flow
201
Unmixed fluid
196
Unstable flow
107
Upper transitional flow
158
This page has been reformatted by Knovel to provide easier navigation.
Index Terms
Links
Utrillas Power Plant
17
Utrillas/Teruel, Spain
16
V Vane geometry
93
Vane
339
Vapor concentration
239
Vapor condensation
169
Vapor flow rate
179
Vapor inlet velocity
191
Vapor mass flow rate
182
Vapor mass fraction
121
Vapor phase
106
Vapor pressure Vapor quality
184
34
212
234
292
363
373
34
Vapor shear effect
172
Vapor shear
179
Vapor shear-controlled flow
179
Vapor temperature
185
Vapor velocity distortion
187
Vapor velocity profile
186
Vapor velocity
172
180
178
184
294
315
187–188 Vapor Vaporization Vapor-liquid separator
55 228 34
V-array
387
Vector diagram
302
This page has been reformatted by Knovel to provide easier navigation.
Index Terms
Links
Velocity boundary layer, see Hydrodynamic boundary layer Velocity distribution
79
339
400
Velocity factor
340
Velocity head
62
86
Velocity profile
59
62
64
77
343
387
81
343
57
149
391 Velocity vector
301–302
vena contracta
79
Venturi nozzle
89
Venturi Vertical cooling delta
339 30
Vertical two-phase flow, see Two-phase flow (vertical) Viscosity
56 155
Viscous flow
56–60
Viscous force
58
Viscous shear stress
57
Visibility (Plume)
40
Visible plume abatement
314–315
Visonta Power Plant, see Gagarin Power Plant Void fraction
106
Volume yield
238
von Karman vortex street Vortex shedding
420–421 421
This page has been reformatted by Knovel to provide easier navigation.
Index Terms
Links
W Wall temperature
144–148
152
154
158–159
173–174
176
184
206
210
296
332–333
345
Waste water
11
Water availability
12
27
Water box condenser, see Condenser (water box) Water capacity rate
275
Water conserving heat rejection system Water consumption Water cost Water density
11 35 2
10
12
32
286
289
295
300
341
20
405
83
Water distribution system
253
Water distribution
256
Water film, see Film (water) Water flow rate Water jet Water layer
256
Water loss
27
Water pump Water pumping
3
29
309
Water recirculation
30
Water requirement
10
Water return
10
This page has been reformatted by Knovel to provide easier navigation.
38
Index Terms Water rights Water saturation
Links 10 230
Water storage tank, see Storage tank Water supply Water temperature
10
11
4
10
237
245
247
250
275
282
283
287
340
350
5
229–230
233
235–236
239–243
264
266
269
285
291
314
363
365
373
367 Water treatment
10
Water use priority
11
Water vapor
Water velocity
354
Water viscosity
112
Water-air interface
230
Water-cooled condenser, see Condenser (water-cooled) Waterfilm Water-side capacity ratio Water-to-air heat exchanger Wave (disturbance)
283–287 355 31 115–116
Wavy fin, see Fin (wavy) Wavy flow (gravity-controlled flow), see Gravity controlled flow (wavy flow) This page has been reformatted by Knovel to provide easier navigation.
Index Terms Wavy regime (Film condensation)
Links 172
Wavy-finned flattened tube, see Finned tube (wavy-flattened) Weather condition
13
Wet/dry cooling system, see Dry/wet wet/dry cooling system Wetbulb temperature
Wet-cooling system
4
12
33
38
233
235
262
291–292
313–314
339
362
9
10
11
12 Wet-cooling tower, see Cooling tower (wet) Wetted area
276
Wetting (surface)
169
Wetting characteristic
106
Wetting pattern (fill)
256
Wetting
313
Wind tunnel
338
360
363
380
417
420
103
105
Windmill, see Anti-recirculation fence Wire gage Wire-loop extended fin, see Fin (wire-loop extended) Wispy annula flow pattern
180
Wood lath fill, see Fill (wood lath) Wood lath
309
This page has been reformatted by Knovel to provide easier navigation.
Index Terms Wood slat Wyodak Power Plant
Links 255 18
Y Yield stress
411
414
Z Zero absolute velocity
300
Zinc coating
167
Zinc loss
403
402–403
This page has been reformatted by Knovel to provide easier navigation.
