Pré-visualização parcial do texto
Baixe Kays- Conevective Heat and mass transfer e outras Notas de estudo em PDF para Engenharia Aeronáutica, somente na Docsity!
McGraw-Hill Series in Mechanical Engineering Jack P. Holman, Southern Methodist University Consulting Editor Barron: Cryogenic Systems Eckert: Introduction to Heat and Mass Transfer Eckert and Drake; Analysis of Heat and Mass Transfer Eckert and Drake: Heat and Mass Transfer Ham, Crane, and Rogers: Mechanics of Machinery Hartenberg and Denavit: Kinemaric Synthesis of Linkages Hinze: Turbulence Jacobsen and Ayre: Engineering Vibrations Juvinall: Engineering Considerations of Stress, Strain, and Strength Kays and Crawford: Convective Heat and Mass Transfer Lichty: Combustion Engine Process Martin: Kinematics and Dynamics of Machines Phelan: Dynamics of Machinery Phelan; Fundamentals of Mechanical Design Raven: Automatic Control Engineering Schenck: Theories of Engineering Experimentation Schlichting: Boundary-Layer Theory Shigley: Dynamic Analysis of Machines Shigley: Kinematic Analysis of Mechanics Shigley: Mechanical Engineering Design Shigley: Simulation of Mechanical Systems Shigley and Uicker: Theory of Machines and Mechanisms Stoecker: Refrigeration and Air Conditioning CONVECTIVE HEAT AND MASS TRANSFER Second Edition W.M. Kays Professor of Mechanical Engineering Dean of Engineering Stanford University M. E. Crawford Assistant Professor of Mechanical Engineering Massachusetts Institute of Technology McGraw-Hill Book Company New York St. Louis San Francisco Auckland Bogotá Hamburg Johannesburg London Madrid Mexico Montreal New Delhi Panama Paris São Paulo Singapore Sydney Tokyo Toronto This book was set in Times Roman. The editors were Frank J. Cera and J. W. Maisel; the production supervisor was Diane Renda. New drawings were done by Santype International Limited. R.R. Donnelley & Sons Company was printer and binder. CONVECTIVE HEAT AND MASS TRANSFER Copyright O 1980, 1966 by McGraw-Hill, Inc. Al rights reserved. Printed in the United States of America. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, withont the prior written permission of the publisher. 1234567890 DODO 89876543210 Library of Congress Cataloging in Publication Dat Kays, William Morrow, Convective heat and mass transfer. (McGraw-Hill series in mechanical engineering) Includes index. 1. Heat-Convection. 2. Mass transfer L Crawford, Michael E, joint author. IL Title. QCI7K37 1980 536.25 79-16282 ISBN 0-07-033457.9 To Alma Campbell Kays and Carol Ann Crawford viii CONTENTS Chapter 5 Chapter 6 Chapter 7 Chapter 8 The Integral Equations of the Boundary Layer 45 The Momentum Integral Equation / The Displacement and Momentum Thickness / Alternative Forms of the Momentum Integral Equation / The Energy Integral Equation / The Enthalpy and Conduction Thicknesses / Alternative Forms of the Energy Integral Equation / Problems / Reference Momentum Transfer: Laminar Flow Inside Tubes s8 Fully Developed Laminar Flow in Circular Tubes / Fully Developed Laminar Flow in Other Cross-Sectional Shape Tubes / The Laminar Hydrodynamic Entry Length / Problems / References Momentum Transfer: The Laminar External Boundary Layer 7 Similarity Solutions: The Laminar Incompressible Boundary Layer with Constant Properties and Constant Free-Stream Velocity / Similarity Solutions for the Laminar Incompressible Boundary Layer for to = Cx" / Similarity Solutions for the Laminar Incompressible Boundary Layer for vo £ 0/ Nonsimilar Momentum Boundary Layers / An Approximate Laminar Boundary-Layer Solution for Constant Free-Stream Velocity Developed from the Momentum Integral Equation / An Approximate Laminar Boundary-Layer Solution for Arbitrarily Varying Free-Stream Velocity over a Body of Revolution / Problems / References Heat Transfer: Laminar Flow inside Tubes 88 The Energy Differential Equations for Flow through a Circular Tube / The Circular Tube with Fully Developed Velocity and Temperature Profiles / The Concentric Circular-Tube Annulus with Fully Developed Velocity and Temperature Profiles, Asymmetric Heating / Solutions for Tubes of Noncircular Cross Section with Fully Developed Velocity and Temperature Profiles / Circular-Tube Thermal-Entry-Length Solutions / Thermal- Entry-Length Solutions for the Rectangular Tube and Annulus / The Effect of Axial Variation of the Chapter 9 Chapter 10 Chapter 11 CONTENTS ix Surface Temperature / The Effect of Axial Variation of Heat Flux / Combined Hydrodynamic and Thermal Entry Length / Problems / References Heat Transfer: The Laminar External Boundary Layer 133 Constant Free-Stream Velocity Flow along a Constant-Temperature Semi-Infinite Plate / Flow with Uo = Cx” along a Constant-Temperature Semi-Infinite Plate / Flow along a Constant-Temperature Semi- Infinite Plate with Injection or Suction / Nonsimilar Thermal Boundary Layers / Constant Free-Stream Velocity Flow along a Semi-Infinite Plate with Unheated Starting Length / Constant Free-Stream Velocity Flow along a Semi-Infinite Plate with Arbitrarily Specified Surface Temperature / Constant Free-Stream Velocity Flow along a Semi-Infinite Plate with Arbitrarily Specified Surface Heat Flux / Flow over a Constant-Temperature Body of Arbitrary Shape / Flow over a Body of Arbitrary Shape and Arbitrarily Specified Surface Temperature / Flow over Bodies with Boundary-Layer Separation / Problems / References Momentum Transfer: The Turbulent Momentum Boundary Layer 161 Transition of a Laminar Boundary Layer to a Turbulent Boundary Layer / The Qualitative Structure of the Turbulent Boundary Layer / The Concept of Eddy Diffusivity, Eddy Viscosity / The Prandtl Mixing-Length Theory / The Shear Stress Distribution Near a Wall / The Law of the Wall for the Case of p* = 00 and vj = 00 / An Approximate Solution to the Momentum Boundary Layer / Equilibrium Turbulent Boundary Layers / The Transpired Turbulent Boundary Layer / A Continuous Law of the Wall; the Van Driest Model / Summary of a Complete Mixing-Length Theory / The Effects of Surface Roughness / The Effects of Axial Pressure Gradient / High-Order Models of Turbulence / Problems / References Momentum Transfer: Turbulent Flow in Tubes 196 Fully Developed Flow in a Circular Tube / Other Flow Cross-Section Shapes / Effects of Surface Roughness / Problems / References X CONTENTS Chapter 12 Chapter 13 Chapter 14 Heat Transfer: The Turbulent Boundary Layer The Concept of Eddy Diffusivity for Heat Transfer, Eddy Conductivity / The Reynolds Analogy / À Law of the Wall for the Thermal Boundary Layer / À Heat-Transfer Solution for Constant Free-Stream Velocity and Surface Temperature along a Semi-Infinite Plate / Constant Free-Stream Velocity Flow along a Semi-Infinite Plate with Unheated Starting Length / Constant Free-Stream Velocity Flow along a Semi-Infinite Plate with Arbitrarily Specificd Surface Temperature / Constant Free-Stream Veloeity Flow along à Semi-Infinite Plate with Arbitrarily Specified Heat Flux / Axisymmetric Body with Arbitrarily Varying Free-Stream Velocity and Surfaçe Temperature / The Transpired Turbulent Boundary Layer / Film Cooling / The Turbulent Prandtl Number / Summary of a Complete Mixing-Length Theory / The Effects of Surface Roughness / Problems / References Heat Transfer: Turbulent Flow Inside Tubes Circular Tube with Fully Developed Velocity and Temperature Profiles, Constant Heat Rate, Prandtl Number near 1.00 / Circular Tube with Fully Developed Flow, Higher Prandtl Numbers / Very Low-Prandtl-Number Heat Transfer, Liquid Metals / Circular Tube, Fully Developed Profiles, Constant Surface Temperature / Effect of Peripheral Heat Flux Variation / Fully Developed Turbulent Flow between Parallel Planes and in Concentric Circular-Tube Annuli / Fully Developed Turbulent Flow in Other Tube Geometries / Experimental Correlation for Flow in Tubes / Thermal Entry Length for Turbulent Flow in a Circular Tube / Thermal Entry Length for Turbulent Flow between Parallel Planes / The Effects of Axial Variation of Surface Temperature and/or Heat Flux / Combined Hydrodynamic and Thermal Entry Length in a Circular Tube / The Influence of Surface Roughness / Problems / References The Influence of Temperature-Dependent Fluid Properties Laminar Flow in Tubes: Liquids / Laminar Flow in Tubes: Gases / Turbulent Flow in Tubes: Liquids / 204 236 275 Chapter 15 Chapter 16 Chapter 17 Chapter 18 CONTENTS xi Turbulent Flow in Tubes: Gases / The Laminar External Boundary Layer: Gases / The Turbulent External Boundary Layer: Gases / Problems / References Convective Heat Transfer at High Velocities 288 The Stagnation Enthalpy Equation / The High-Velocity Thermal Boundary Layer for Fluid with Pr = 1 / The Laminar Constant-Property Boundary Layer for Pr 1/The Laminar Boundary Layer for a Gas with Variable Properties / The Use of Reference Properties for High-Velocity Laminar Boundary-Layer Calculations / The Turbulent Boundary Layer [or a Gas with Variable Properties / Reference Properties for High-Velocity Turbulent Boundary Layer Calculations / Mach Number and large-Temperature-Difference Corrections for Variable Free-Stream Velocity and Variable Temperature Diflerences / Problems / References Free-Convection Boundary Layers a13 Boundary-Layer Equations for Free Convection / Similarity Solutions: Laminar Flow on a Constant-Temperature, Vertical, and Semi-Infinite Flat Plate / Similarity Solutions with Variable Surface Temperature / Similarity Solutions with Wall Suction or Blowing / Approximate Integral Solutions: Laminar Flow on a Constant-Temperature, Vertical, and Semi-Infinite Flat Plate / The Effect of Variable Properties / Free-Convection Flow Regimes / Turbulent Flow on a Vertical and Semi-Infinite Flat Plate / Heat-Transfer for Other Geometries / Mixed Free and Forced Convection / Problems / References Mass Transfer: Formulation ofa Simplified Theory 332 Definitions / The Differential Equations of the Concentration Boundary Layer / Boundary Conditions at the Interface / Definition of the Mass-Transfer Conductance and Driving Force / Problems Mass Transfer: Some Solutions to the Conserved-Property Equation 352 The Laminar Constant-Property Boundary Layer, Xiy PREFACE TO THE SECOND EDITION requires an understanding of both the basic processes and, in at least a general way, the consequences of particular sets of conditions. This understanding is difficult to obtain when the computer is relisd on exclusively. Equally im- portant is the fact that a very high percentage of engineering heat-transfer problems do not require the high precision and detail generally available from a computer solution, but they are problems for which quick, low-cost answers are essential. For such problems the computer-based finite-difference solution is elegant, but overkill. The point at which overkill occurs is moving inexor- ably away from the classical methods, but in the authors” view the distance to go is still large, and in any case is going to vary greatly with local conditions. Ttis a question of engineering judgment; an engineer must optimize not only the system being designed but also his or her own expenditure of time. Tn this second edition the authors have retained the basic objectives of the first edition, while at the same time modernizing it and shifting emphasis where appropriate, but they have also tried to provide a theoretical frame- work for finite-difference methods. The relevant differential equations are developed, and simple turbulent transport models applicable to finite- difference procedures are discussed. Since the literature abounds with refer- ences to computer programs and model developments, the authors have tried mainly to refer to survey articles in the discussions. As before, not everything can be covered. The topics chosen and the depth of coverage represent a personal judgment as to what is of first importance for a mechanical, aerospace, or nuclear engineering student at about the fifth- year level. A chapter on free convection has been added, and the discussion ofthe effects of surface roughness has been greatly expanded. Several chapters have been reorganized to completely separate laminar and turbulent flows, and the approach to turbulent transport processes has been drastically modified. Because of their continuing evolution, though, higher-order turbulence closure models are not discussed. Finally, it should be emphasized that only two-dimensional boundary-layer flows are treated, and only single- phase systems are considered. Finally, the second author would like to express his gratitude to Professor Kays for the honor and privilege of being asked to coauthor the second edition. Professor Kays's style of teaching, both in the classroom and other- wise, and his approach to the subject of heat transfer will be forever with me. I would also like to express my indebtedness to Professor R. J. Moffat at Stanford who taught me all I profess to know about experimental heat transfer and the art of written and oral communication. Lastly, 1 would like to express appreciation to two more colleagues, Professor À. L. London at Stanford and Professor J. L. Smith, Jr., at M.LT., who have taught me the closely allied field of thermodynamics and the general methodology behind engineering problem solving. W. M. Kays M. E. Crawford PREFACE TO THE FIRST EDITION Priorto World War II, convective heat and mass transfer were largely empiri- cal sciences, and engineering design was accomplished almost exclusively by the use of experimental data, generalized to some degree by dimensional analysis. During the past two decades great strides have been made in developing analytical methods of convection analysis, to the point where today experiment is assuming more its classical role of testing the validity of theoretical models. This is not to say that direct experimental data are not still of vital importance in engineering design, but there is no question that the area of complete dependence on direct experimental data has been greatly diminished. With this change our understanding of convection phenomena has been greatly enhanced, and we find ourselves in a position to handle, with confidence, problems for which experiment would be time consuming and expensive. This book has been prepared as a response to this trend. Tt is axiomatic that the engineering student must learn to reason from first principles so that she or he is not at a loss when faced by new problems. But time spent solving a complex problem from first principles is time wasted if the solution already exists. By their very nature analytic convection solu- tions often tend to be lengthy and difficult. Thus familiarity with, and an understanding of, some of the more important of the available analytic con- vection solutions should be an important part of the background of the heat- transfer engineer. One of the objectives of this book is to bring together in an easily usable form some of the many solutions to the boundary layer equa- tions. Although these are available in the heat transfer literature, they are not always readily accessible to the practicing engineer, for whom time is an im- portant consideration. The author feels that a study of these solutions, in a logical sequence, also provides the best way for a student to develop an understanding of convective heat and mass transfer. Thus it is hoped that this book will serve both as a classroom text and as a useful reference book for the engineer. This book is the outgrowth of a set of notes which the author has devel- oped over the past ten years to supplement lectures in the “convection” x XYI PREFACE TO THE FIRST EDITION portion ofa one-year course in heat transfer for first-year graduate students, The students in the course have been largely mechanical, nuclear, and aero- nautical engincers, interested in problems associated with thermal power systems and thermal environmental control. It is assumed that the student has a typical undergraduate background in applied thermodynamics, fluid mechanics, and heat transfer. Heat transfer, although not mandatory, is usually of considerable help in orienting the student's thinking and establishing a sense of need for a deeper study of the subject. In particular, some familiarity with the commonly employed empiri- calmethods ofcalculating convection heat-transfer rates is assumed, but only so that the student has an appreciation for the usefulness of a heat transfer coefficient and some grasp of the basic physics of the convection process. The choice of subject matter reflects quite frankly the author's own interests, and the depth to which each topic is pursued represents a com- promise made necessary by what can be practicably accomplished in approx- imately one semester (or perhaps two quarters). It will be found that the momentum boundary layer is heavily compressed with only sufficient material presented to support the heat and mass transfer sections. The student desiring to concentrate heavily in boundary layer theory will undoubtedly want to take a separate course on viscous fluid mechanics, for which adegquate texts exist. And, for that matter, there is certainly a great deal more to con- vective heat and mass transfer than is presented here, not only in the topics considered but also in those not even mentioned. In the latter category the reader may miss such topics as natural convection, heat exchanger theory, rotating surfaces, nonsteady flows, two-phase flows, boiling and conden- sation, non-Newtonian fluids, internally radiating gases, rarefied gases, magnetohydrodynamic flows, and coupling between heat and mass transfer. But this only suggests why second editions are usually bigger than first editions. Finally, I would like to acknowledge my indebtedness to some of my colleagues, without whose assistance, conscious or otherwise, this book could never have been written. First, Professor A. L. London taught me all that I profess to know about teaching, introduced me to heat transfer, and has been a constant source of help and inspiration. Professor W. C. Reynolds has worked with me on some of the research that is summarized in the book, and substantial parts of it are the result of his work alone. Several months spent with Professor D. B. Spalding at Imperial College in London were a rare Privilege, and his influence will be found throughout the book. But most specifically, Spalding's generalization of the convective mass transfer problem forms the entire basis for the last three chapters. Although available in Spalding's many papers, it is hoped that its inclusion here will encourage its more extensive use. Lastly I would like to express appreciation to Mr. R.J. Mofiat who read the manuscript and made many helpful suggestions. W.M. Kays TO THE INSTRUCTOR The authors have rather strong feelings about the kinds of home problems that a student in a senior- or graduate-level course should be asked to attack and about the manner in which problems should be presented. The notion that practice on a variety of problems is the effective route to proficiency in engineering analysis is not disputed. But the real world of engineering does not present itself as a series of neatly packaged and well-defined exercises. At some point, and the earlier the better, the student must face up to the analysis of problems that may require considerable time, possibly extensive outside study, and that are incompletely stated or specified so that individual judg- ment must be exercised. Above all, the student must be weaned away from the notion that the end product of an engineering analysis is merely a number that is either correct or incorrect. The difficulty with using lengthy and comprehensive problems in an engineering course is that the student's time is limited, relatively fewer problems can be assigned, and thus it is usually not possible to cover many aspects of the subject with this kind of practice. The instructor must be extremely careful in the choice of assignments if the problems are to be a meaningful supplement to the other methods of learning. A further difficulty is that such problems must be read and constructively criticized if full value is to be realized, which involves more work for the instructor than merely checking off answers. The problems at the ends of the chapters have been selected to provide a choice ofboth relatively short exercises and much more lengthy tasks, many of which might better be described as projects rather than problems. Most of them have been used by the authors in class, but only a fraction of them in any one class. Both the longer and shorter variety have been selected so that some represent engineering applications, whereas others are of a fundamental analysis variety designed to enhance understanding of the various develop- xvii LIST OF SYMBOLS fa English Jetter symbols area, surface area, m? Van Driest constant; see Eq. (10-33) flow cross-sectional area, m? see Eq. (8-39) transpiration parameter; see Eq. (10-32) mass-transfer driving force, defined by Eg. (17-34) transpiration parameter; see Eq. (10-26) heat-transfer transpiration parameter; see Eq. (12-33) see Eq. (8-33) specific heat at constant pressure, J/kg:K), m?/(s?-K) local friction coefficient, defined by Eq. (5-10) mean friction coefficient with respect to length apparent mean friction coefhicient, defined by Eq. (6-19) specific heat at constant volume, J/(kg-K), m?/(52-K) specific heat at constant pressure for component j of a mixture, J/(kg'K), m?2/(s?-K) inside diameter of a circular tube, m hydraulic diameter, 4r, = 44.L/A,m mass difusion coefficient for component ; in a multi- component mixture, m?/s outside diameter of a circular tube, m mass diflusion coefficient for a binary (two-component) mixture; note that 2, = 9, m?/s rate of energy transfer by convection across a control surface, J/s, m?kg/s* xxii LIST OF SYMBOLS = internal thermal and chemical energy, J/kg, m?/s? e F = resultant of all external forces acting on a control volume, N, m/kg/s? G = mass flux, or mass velocity; see Egs. (2-2) and (2-3), kg/m? G = a shape factor; see Eq. (10-25) é Go = mass velocity in the free stream, Pato » kg/m no G = mass fiux, or mass velocity vector V, at any point in the stream, kg/m? , Gs, G,, etc. = components of the mass flux vector, kg/m o Gar, ; = mass flux of component j transported by diffusion; see Eq. (3-12), kg/m? G, = see Eq. (8-34) . Gr, = local Grashof number, gpx(to — fo) Grt = modified Grashof number, gBasl/kv?) Grp = Grashof number based on diameter . Gr, = Grashof number based on L = surface area/surface peri- meter g = mass-transfer conductance, defined by Eq. (17-33), kg/(s:m?) 9; = enthalpy conductance, defined by Eq. (15-9) kg/(s:m?) g acceleration of gravity, m/s? g* = value of mass-transfer conductance for very small mass- transfer rate, Eq. (17-36), kg/(s:m?) H = boundary-layer shape factor, defined by Eq. (7-39) H, = “heat” ofcombustion, per unit of fuel mass, at a temperature fo, Jkg, mp? . h = heat-transfer coefficient, or convection conductance, Eq. (1-1) W/(m?-K), 1/ls-m?-K), kg/(8-K) 1a i = enthalpy, and enthalpy of a mixture, e + P/p, Jjkg. mis Í partial enthalpy of component j of a mixture, J, /ke, m?/s = stagnation enthalpy, | + u?/2, J/kg, m?js? . ix = reference enthalpy for evaluation of fluid properties, Eq. (15-42), Jkg K = acceleration parameter, sec Eg. (10-27) K, = equilibrium constant, sec Chap. 19 = thermal conductivity, Eq. (3-8), W/(m-K) J/(sm:K), mkg/(9º-K) = equivalent “sand grain” roughness, m kr = thermal diffusion ratio, Eq. (3-9) k, = eddy conductivity; see Eq. (12-3), W/(m-K) L = flow length of a tube, m LIST OF SYMBOLS xxiii Le, = Lewis number, 7,1, Pr/Sc, = mixing length, m M = a blowing rate parameter; see Eq. (12-39) M = Mach number, Eg. (15-32) m = mass, kg m = an exponent, see Egs. (7-22) and (7-23) my = mass concentration (mass fraction) of substance j in a mixture m = mass flow rate, kg/s m" = total mass flux (mass flow rate per unit of area) at surface or phase interface, kp/(m?-s) ; mass flux of substance j at surface or phase interface, kg/(m?:s) rate of creation of substance j, per unit of volume, by chemical reaction, kg/(m?-s) %W = molecular weight, kg/kmol M = mass fraction of element « in a mixture of compounds Ho,j mass fraction of element « in a compound substance j Nu = Nusselt number, hD/k, 4r,h/k, hD,/k, xh/k P = pressure, N/m?, Pa, kg/m -s? P, = partial pressure of substance a in à gas mixture, N/m? p* = nondimensional pressure gradient; Eg. (10-13) ? = conserved property of the second kind; see Eq. (17-21) Pr = Prandtl number, pc/k, u/T, v/a Pr, = turbulent Prandtl number; see discussion preceding Eq. (12-4) q = heat, energy in transit by virtue of a temperature gradient, J, m?-kg/s? 4 = heat-transfer rate, J/s, W, N-m/s, m?-kg/s? q” = heat flux vector, heat-transfer rate per unit of area, J/(s -m?), W/m?, kg/s? do = heat flux (heat-transfer rate per unit of arca) at surface or phase interface, J/(s-m?), Wjm?, kg/s? R = radius ofa body of revolution, Fig. 5-4; radius of cylinder or sphere, Fig. 9-2; m R = gas constant; see App. B; J/(kmol-K) R, = see Eq. (8-33) » = radial distance in cylindrical or spherical coordinates, m * = boundary-layer thickness ratio, A/ô r = mass ratio of oxident to fuel in a simple chemical reaction te = recovery factor, Eq. (15-21) EU 1 & h "7 XXYi LIST OF SYMBOLS = à d2 de Em Em ú vs thickness of a momentum boundary layer [for example, see Eq. (7-34)], m displacement thickness of momentum boundary layer, defined by Eq. (5-5), m momentum thickness of boundary layer, defined by Eq. (5-6), m shear thickness of boundary layer, defined by Eq. (7-38), m eddy diffusivity for heat transfer, Eq. (12-1), m?/s eddy diflusivity for momentum, Eq. (10-2), m?/s dependent variable in Blasius equation, u/uo = C(n), Eq. (7-8) similarity parameter, Eg. (7-11), independent variable in Blasius and other similarity solution equations film cooling eflectiveness; see Eg. (12-39) time, s nondimensional fluid temperature in a tube, (to—t)/ (to — to), à solution to Eg. (8-29) nondimensional fluid temperature in an external boundary layer, (to — (to — ta), for the case of a step change in surface temperature nondimensional fluid temperature in a high-velocity boun- dary layer, defined by Eg. (15-18) a temperature difference, see Eq. (9-23), K angular coordinate in a spherical coordinate system; see App. D mixing-length constant; see Eq. (10-7) a nondimensional, boundary-layer parameter defined by Eg. (7-41) a diffusion coefficient, or y;, kg/(m-s) sec Eq. (8-34) = dynamic viscosity coefficient, defined by Eq. (3-1), N-s/m?, Pas, kg/(m:s) eddy viscosity, Eq. (10-4), N-s/m? kinematic viscosity, 4/p, m?/s distance from beginning of tube or plate to point where heat transfer starts; also a dummy length variable; may be dimensional or nondimensional, according to context of use fluid density, mass per unit of volume, kg/m? Stefan-Boltzmann constant, see App. B; W/(m?-K*), kg/(s?-K?) eee se? ase8 ” aw ler] LIST OF SYMBOLS Xxvii normal stress on an element of fluid, see Egs. (3-4) to (3-6), N/m?, kg/(m-s?) shear stress; see Egs. (3-1) to (3-3); N/m?, kg/(m-s?) nondimensional temperature in an external boundary layer, (t — tol/to — to), à solution to Eg. (9-4) shear stress evaluted at wall surface, N/m?, kg/(m-s?) angular coordinate in cylindrical and spherical coordinate systems; sec App. D relative humidity, Eg. (19-6) stream function, defined by Eg. (7-9), m?/s absolute humidity, Eq. (19-5) Subscripts evaluated within the boundary layer or considered phase but at the surface refers to outer surface of an annulus outer surface of an annulus when outer surface alone is heated evaluated at the free-stream state evaluated at the tube entrance evaluated at the mixed mean state; also denotes other means refers to reference temperature, or properties evaluated at reference temperature evaluated at a particular point along a surface evaluated at “adiabatic” wall state evaluated at the surface but within the neighboring phase evaluated at the transferred substance state, Egs. (17-1) to (17-3) refers to a substance j, which is a component of a mixture refers to a particular chemical element a in a compound and/or mixture refers to inner surface of an annulus inner surface of an annulus when inner surface alone is heated refers to constant-property solution with all fluid properties introduced at either mixed mean or free-stream state refers to solution for constant heat rate per unit of tube length refers to solution for constant surface temperature xxvili LIST OF SYMBOLS (The meanings of other subscripts should be apparent from the context of their use.) Note: 'The dimensions of all quantities listed are given in standard SI base or derived units or both. Some quantities are customarily expressed in decimal multiples or submultiples of these units, and care should be taken to note this fact in making arithmetic calculations. Typical examples are pressure, usually expressed in kilopascals, kPa, specific heat capacity, kJ/(kg'K); enthalpy, MJ/kg or kJ/kg. CHAPTER ONE INTRODUCTION Among the tasks facing the engineer is the calculation of energy-transfer rates and mass-transfer rates at the interface between phases in a fluid system. Most often we arg concerned with transfer at a solid-fluid interface where the fluid may be visualized as moving relative to a stationary solid surface, but there are also important applications where the interface is between a liquid and a gas. Tf the fuids are everywhere at rest, the problem becomes one of either simple heat conduction where there are temperature gradients normal to the interface (which wil! be subsequently referred to as the surface) or simple mass diflusion where there are mass concentration gradients normal to the surface. However, if there is fluid motion, energy and mass are transported both by potential gradients (as in simple conduction) and by movement of the fluid itself. This complex of transport processes is usually referred to as convection. Thus the essential feature of a convective heat-transfer or a con- vective mass-transfer process is the transport of energy or mass to or from a surface by both molecular conduction processes and gross fluid movement. Popular usage forces us to use the term convection somewhat loosely. We will speak of the convective terms in our differential equations, as opposed to diflusive terms; here we refer to that part of the transport process attributable to the gross fluid motion alone. If the fluid motion involved in the process is induced by some external means (pump, blower, wind, vehicle motion, etc.), the process is generally called forced convection. If the fluid motion arises from external force fields, such as gravity, acting on density gradients induced by the transport process itself, we usually call the process free convection. ro CHAPTER TWO CONSERVATION PRINCIPLES The solution of a convection problem starts with the combination of an appropriate conservation principle and one or more flux laws. In this chap- ter we set down the ground rules that we will employ for application of the conservation principles and then introduce these principles and a suitable system of nomenclature. THE CONTROL VOLUME Consider a defined region in space across the boundaries of which mass, cnergy, and momentum may flow, within which changes of mass, energy, and momentum storage may take place, and on which external forces may act. Such a region is termed a control volume, and the boundary surfaces are called the control surface. The control volume comprises our region of inter- est in application of the various conservation principles discussed below. The volume may be finite in extent, or alt its dimensions may be infinitesimal, The complete definition of a control volume must include at least the implicit definition of some kind of coordinate system, for the control volume may be moving or stationary, and the coordinate system may be fixed on the control volume or elsewhere. A control volume across whose surface no matter passes during the process in question is sometimes referred to as a fixed mass system, or a simple thermodynamic system. 4 CONSERVATION PRINCIPLES 5 PRINCIPLE OF CONSERVATION OF MASS Let the term creation have the connotation of outflow minus inflow plus increase of storage; then the principle of conservation of mass, when applied to a control volume (Fig. 2-1), may be expressed as Rate of creation of mass = O (2-1) Ií, in a flow system, the relative velocity normal to the control surface is designated as V, m/s, the total mass flux crossing the surface is G=Vo kg/(s:m?) (2-2) where p = fluid density, kg/mº G = mass velocity or mass flux, kg/(s - m?) JÉ G is constant over a cross-sectional area A, the total rate ofmass flow across A is then m=AG=AVp kgs (2-3) THE MOMENTUM THEOREM The momentum of an element of matter may be defined as a quantity equal to the product of the mass of that element and its velocity. Since velocity is a vector quantity, momentum is also à vector quantity. The momentum theorem may be expressed either in vector form or, using a cartesian coor- dinate system, as separate equations for the x, y, and z directions, employing the components of momentum and external forces in each of these respective Mass Mass Storage) flow flow 1 out in Figure 2-1 Control volume for application of conservation of mass principle. e) 6 CONVECTIVE HEAT AND MASS TRANSFER directions. The momentum theorem, when applied to a control volume, as in Fig. 2-2, may be expressed vectorially as Rate of creation of momentum = É (2-4) where mY = momentum m = mass, kg Y = velocity, m/s, referred to an ineitial or nonaccelerating coor- dinate system 1 = momentum rate across control surface m = rate of mass flow across control surface, kg/s F = resultant of all external forces acting on control surface or volume, N Again the term creation has the same connotation as for Eg. (2-1), that is, outflow minus inflow plus increase of storage, Application of Eg. (2-4) involves a summation of the momentum flux terms over the entire control surface to evaluate the outflow and inflow terms, a summation of rates of changes of momentum over the control volume to evaluate the increase of storage term, and a summation of the external forces over the surface and/or volume to evaluate the resultant external force. Note that the velocities associated with the outflow and inflow terms are, in general, different from the velocities involved in the storage terms, and all must be referred to a nonaccelerating coordinate system. PRINCIPLE OF CONSERVATION OF ENERGY The principle of conservation of energy is a general expression of the first law of thermodynamics as applied to a control volume. Consider the energy that might cross the control surface and the changes of energy storage that Momentum flow in Momentum flow out (Change of Momentum Storage) F Figure 2-2 Control volume for application of momentum theorem. CONSERVATION PRINCIPLES 7 Energy low out Ee (including including E cat and work) (Change of Ercráy Slorage) Figure 2-3 Control volume for application of conservation of energy principle. might take place in the control volume shown in Fig, 2-3. The principle of conservation of energy can then be stated as Rate of creation of energy = O (2-5) Application of Eq. (2-5) involves the same Kind of summation as required for Egs. (2-1) and (2-4). Energy can cross the surface in the form of work or heat transfer, or it can cross as energy stored in any mass which flows across the surface, Some of the forms of energy frequently encountered in flow systems are Energy that can cross a control surface Mechanical (or electrical) work: W Flow work: Work done by cach unit mass of flowing fluid on the control volume, or by the control volume, as it flows across the control surface. In general, the flow work done by a unit mass is equal to the product of the normal ftuid stress and the fluid specific volume, both evaluated at the control surface. If velocity gradients are not large, the normal stress is essentially the simple thermodynamic pressure, and the unit flow work becomes merely Pv, where v is the specific volume of the fluid. Heat: q, energy transferred by virtue of a temperature gradient Energy stored in each unit mass as it crosses the control surface Internal thermal energy: e (including stored chemical energy) Kinetic energy: V?/2 Potential energy: zg, where z is elevation with respect to some datum in a gravity field of constant strength g Enthalpy: i, a thermodynamic property, the sum e + Pv 10 CONVECTIVE HEAT AND MASS TRANSFER = [a e w Figure 3-1 Stresses acting on element of fluid; components of fluid velocity. components of velocity in the x, y, z directions, again as shown in Fig. 3-1, the shear stresses then become dv Ou Tay E Tu = dE + aa (6-1) ôw O == dE + a) 62) Ou Ow Tu TwS (ões =) (3-3) The equations for the normal stresses in a viscous fluid are simply postulated here without development or further argument. They can be derived for a simple gas by the kinetic theory of gases but require certain FLUID STRESSES AND FLUX LAWS 11 critical assumptions for a more general development. In particular, the use of —3y as the coefficient of the second term appears to be strictly limited to the low-density gas regime.” 2 (ôu dp dw êu = pulsa) a (8-4) 2 fôu O Ow ôv G= pousa) + 6855) 2 (du O Ow ow am po Tugas a E) 656) P is the simple thermodynamic pressure under these assumptions and, for the majority of applications, is the only term of real significance. For a constant-density fluid it is seen later that the second term in each equation is always zero and that only when there are very large velocity gradients in the direction of the stress (last term) does o differ appreciably from P. Analysis of a normal shock wave is an example where all thê terms are of significance. Equations (3-1) through (3-6) can be compactly written by using car- tesian tensor notationt as 2 Ou Cj= (rajada) y Some representative data on viscosity coefficients of fluids are given im App. A. The viscosity coefficient varies somewhat with the temperature of a fluid. Generally for a liquid it is a decreasing function of temperature, whereas for a gas it is an increasing function. y has the dimensions Pas, or kg/(m-s). 67) FOURIER'S LAW OF HEAT CONDUCTION Fourier's law of heat conduction states that heat transfer by molecular interactions at any point in a solid or fluid is proportional in magnitude and coincident with the direction of the negative gradient of the temperature + In using tensor notation the indices in the subscripts can have the values 1, 2, 3, and they have the following meaning: x =X, =), X1=2 and =4 =, 4 =w. Repeated indices in a term of an equation mean to sum the term over the three values of k (summation convention). The Kronecker delta is 8,,=0, ij, and ô;= 1, i=j. In the symbol o; the subscript í is the direction of the outward normal to the surface on which the stress acts, and the subscript j indicates the direction of the stress itself. By symmetry Of the stress tensor, Fun nTn On = Op Eta ET Ma 0, 030, 12 CONVECTIVE HEAT AND MASS TRANSFER field. Conduction heat transfer is thus a vector quantity, and the basic Four- ier equation for the heat flux vector is f=-kvt (3-8) where k is the thermal conductivity of the conducting media and has the dimensions W/(m-K). Fourier's law is the simplest form of a general energy flux law and is strictly applicable only when the system is uniform in all respects except for the temperature gradient, that is, no mass concentration gradients or gra- dients in other intensive properties. Given a system with gradients of temperature, pressure, mass concentra- tion, magnetic field strength, and so on, there is no a priori justification for ignoring the possibility that each of these gradients might contribute to the energy flux. The simplest expression which could describe this relationship would be a linear combination of terms, one for cach of the existing potential gradients. Experience shows that there are, in fact, measurable “ coupled” effects, such as energy fluxes due to mass concentration gradients, and that this more general form of the linear rate equation is necessary under certain conditions. The relationships among the coefficients of this equation have been the subject of several investigations? * * in irreversible thermodynamics. These relationships fall into the domain of irreversible thermodynamics by the very nature of the diffusive transport process. One of the principal contributions ofirreversible thermodynamics has been to show that, for example, if energy is transported because of a concentration gradient, then mass will be trans- ported because of a temperature gradient and the coefficients of these two coupled interactions will be the same, provided that the driving potentials are appropriately chosen as a thermodynamically consistent set. Since we are primarily concerned here with temperature gradients and mass concentration gradients, only the coupling between these effects is discussed. An explicit relation for the energy flux can be derived for a low- density gas from kinetic theory. The following equation for the thermal flux in a binary gas mixture (two components of the gas only) is given by Baron” and follows from the development by Chapman and Cowling. RTk, mam, MR = —kVt+ (i +h+ Gu 1) (6.9) where 1 and 2 refer to the two components of the mixture, i is the enthalpy, m is the mass concentration, Wt is the mean molecular weight of the mixture, R is the universal gas constant, Gar, 1 is the mass diffusion flux of component 1, and ky is a “thermal diffusion ratio.” The effect incorporated in the last term is frequently known as the diffusion-thermo, or Dufour, effect. Information on ky is rather meager and largely derived from kinetic FLUID STRESSES AND FLUX LAWS 13 theory for gases, but calculations to establish the importance of this effectin typical boundary layer flows have been made.” In this book, however, we assume that this eflect is negligible in all the applications considered, although the transport of enthalpy by mass diffusion (first term in the first set of parentheses) is included where appropriate. With this exception, then, Fourier's law of heat conduction is assumed to represent the heat flux in all cases. Some representative data on the thermal conductivity k of fluids are given in App. A. The thermal conductivity of liquids is relatively indepen- dent of temperature, but for gases it is an increasing function of temperature similar to the viscosity coefficient. In some applications, the thermal conductivity occurs naturally in com- bination with specific heat. In these cases, it becomes convenient to define r=º kg/(sm) (3-10) where c is the specific heat of the fluid, J/(kg - K). (For gases, c will be understood to be the specific heat at constant pressure c,.) Thus if T is employed, Eq. (3-8) becomes f=-Tev (3-11) FICK'S LAW OF DIFFUSION Before introducing Fick's law it is useful to discuss some definitions, and to make a distinction between mass transfer by convection, or bulk fluid move- ment, and mass transfer by diffusion, which is primarily caused by concentra- tion gradients. We will define G = total mass flux vector, kg/(s - m?) Y=Gp = total velocity vector, m/s where p = mixture density, kg/m?. These terms have the same meaning whether the fluid has one compo- nent, multiple components, or multiple components with concentration gradients, If there are multiple components in the fluid, we will define the total mass flux vector for some component j as Gio, 5» kg/(s * m?) 
