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Dimensionless Numbers Definition Name Use Boussinesq number Natural convection External forced convection Forced convection Internal forced convection Natural convection Condensation C f = (1I2}pU2 Tw Skin friction coefficient Eckert number Friction factor Grashof number Jakob number U2 E c= - - - cp(Tw - T,) f = - dpldz pU212D Gr Ja = f3g(Tw - T,)p2/ 3 JJ.2 = cp(Tw h ,g T,) M =a U Mach number Nusselt number Peclet number Prandtl number Rayleigh number Heat flux Rayleigh number Reynolds number Stanton number Weber number High speed forced convection Convective heat transfer . 1>orous media flows Convective heat transfer Natural convection Natural convection Forced convection Forced convection Surface tension effects hi Nu = k UI Pe = - a Pr = JJ.cp k Ra ~ f3g(Tw - T,)13 va Ra q = f3gqw /4 v ak Re = Upl p. . S t = _ h_ PCpU W e=-u pU21 Where: a cp d p/dz D g h fg = Speed of sound = Specific heat = Pressure gradient = Hydraulic diameter = Gravitational acceleration = Latent heat = Thermal diffusivity = Bulk coefficient = Coefficient of viscosity = Kinematic viscosity k l qw Tw - Tf U = Coefficient of thennal conductivity = Characteristic dimension = Surface heat flux = Characteristic temperature difference = Characteristic forced velocity ex p f3 p. ." 7"w u = Density = Surface shear stress = Surface tension coefficient Forced Convection: Nu = function[Re. P rj Natural Convection: Nu = function[Gr. P rj / An Introduction to Convective Heat Transfer Analysis Patrick H. Oosthuizen Heat Transfer Laboratory Department o f Mechanical Engineering Queen s University and David Naylor Department o f Mechanical Engineering Ryerson Polytechnic University _ McGraw-Hili New York St. Louis San Francisco Auckland Bogota Caracas Lisbon London Madrid Mexico City Milan Montreal New Delhi San Juan Singapore Sydney Tokyo Toronto _weB ./ W eB/McGraw-Hill A Division ofTheMcGraw·HiUCompanies A N INTRODUCTIeN~TO CONVECTIVE HEAT TRANSFER ANALYSIS C opyright © 1999 b y T he M cGraw-Hill Companies, Inc. All rights reserved. Printed in the"United States o f America. E xcept as permitted under the U nited States C opyright A ct o f 1976, no part o f t his publication m ay be reproduced or distributed i n a ny form o r b y a ny means, o r stored in a d ata b ase o r retrieval system, without t he prior written permission o f t he publisher. This book is p rinteaon a cid-free paper. 1 2 3 4 5 6 7 8 9 0 BKM/BKM 9 3 2 1 0 9 8 ISBN 0-07-048201-2 V ice president and editorial director: Kevin T. Kane Publisher: Tom Casson S enior s ponsoringeditor: Debra Riegert M arketing manager: John T. Wa1!nemacher P roject manager: J im w beots . S enior production supervisor: Elizabeth LaManna D esigner: Kiera Cunningham S upplement coordinator: Linda Huenecke Compositor: Publication Services, Inc. Typeface: 10.5/12 Times Roman Printer: Book-mart Press Li~rary of Congress Cataloging-in-Publication D ata Oosthuizen, P. H. A n introduction to c onvective heat transfer analysis / P atrick H. Oosthuizen and David Naylor. p. cm. Includes bibliographical references a nd index. I SBN 0 -07-048201-2 1. H eat-Convection. I. Naylor, David, P h.D. II. Title. T J260.057 1999 6 21.402'2-dc21 9 8-5484 McGraw-Hill Series i n Mechanical Engineering CONSULTING EDITORS Jack P. Holman, S outhern M ethodist U niversity John R. Lloyd, M ichigan S tate U niversity Anderson: Computational Fluid Dynamics: The Basics with Applications Anderson: Modern Compressible Flow: With Historical Perspective Arora: Introduction to Optimum Design Borman and Ragland: Combustion Engineering Burton: Introduction to Dynamic Systems Analysis Culp: Principles o f Energy Conversion Dieter: Engineering Design: A Materials & Processing Approach Doebelin: Engineering Experimentation: Planning, Execution, Reporting Oriels: Linear Control Systems Engineering Edwards and McKee: Fundamentals o f Mechanical Component Design Gebhart: Heat Conduction and Mass Diffusion Gibson: Principles o f Composite Material Mechanics Hamrock: Fundamentals o f Fluid Film Lubrication Heywood: Internal Combustion Engine Fundamentals Hinze: Turbulence Histand and Alciatore: Introduction to Mechatronics a nd Measurement Systems Holman: Experimental Methods f or Engineers Howell and Buckius: Fundamentals o f Engineering Thermodynamics laluria: Design a nd Optimization o f Thermal Systems luvinall: Engineering Considerations o f Stress, Strain, a nd Strength Kays and Crawford: Convective Heat and Mass Transfer Kelly: Fundamentals o f Mechanical Vibrations Kimbrell: Kinematics Analysis and Synthesis Kreider a nd Rabl: Heating a nd Cooling o f Buildings Martin: Kinematics a nd Dynamics o f Machines Mattingly: Elements o f Gas Turbine Propulsion Modest: Radiative Heat Transfer Norton: Design o f Machinery Oosthuizen a nd Carscallen: Compressible Fluid Flow ~-'l-· Oosthuizen and Naylor: Introduction to Convective Heat Transfer Analysis' Phelan: Fundamentals o f Mechanical Design Reddy: A n Introduction to Finite Element Method Rosenberg a nd Karnopp: Introduction to Physical Systems Dynamics Schlichting: Boundary-Layer Theory Shames: i Mechanics o f Fluids Shigley: Kinematic Analysis o f Mechanisms Shigley a nd M ischke: Mechanical Engineering Design Shigley a nd Uicker: Theory o fMachines and Mechanisms Stiffler: Design with Microprocessors f or Me~hanical Engineers Stoecker a nd Jones: Refrigeration and A ir Conditioning Turns: A n Introduction to Combustion: Concepts a nd Applications IDlman: The Mechanical Design Process Wark: A dvanced Thermodynamics f or Engineers Wark a nd Richards: Thermodynamics White: Viscous Fluid Flow Zeid: C AD/CAM Theory a nd Practice /' To my loving wife Jane, f or h er support a nd encouragement. P. H. O. To my wife Kathryn, f or all h er love a nd support. D .N. / C ONTENTS Preface Nomenclature xiii xvii 1 Introduction 1.1 Convective Heat Transfer 1.2 Forced, Free, and Combined Convection 1.3 External and Internal Flows 1.4 The Convective Heat Transfer Coefficient' 1.5 Application o f Dimensional Analysis to Convection 1.6 Physical Interpretation o f the Dimensionless Numbers 1.7 Fluid Properties 1.8 Co.ncluding Remarks Problems References 1 1 4 5 5 11 23 26 26 27 29 31 31 32 35 41 46 49 59 61 69 71 2 The Equations of Convective Heat Transfer 2.1 Introduction 2.2 Continuity and Navier-Stokes Equations 2.3 The Energy Equation for Steady Flow 2.4 Similarity in Convective Heat Transfer 2.5 Vorticity and Temperat~e Fields , 2.6 The Equ~ol1s for Turbulent Convective Heat Transfer 2.7 Simplifying Assumptions Used in the Analysis o f Convection 2.8 2.9 2.10 2.11 T he Boundary Layer Equations for Laminar Flow T he Boundary Layer Equations for Turbulent Flow T he Boundary Lay~r Integral Eq~ations Concluding Remarks Problems References 80 80 82 v ii 3 Some Solutions for External Laminar Forced Convection 3.1 Introduction 3.2 Similarity Solution for Flow over a n Isothermal Plate 3.3 Similarity Solutions for Flow over Flat Plates with O ther T hermal Boundary Conditions 3.4 O ther Similarity Solutions 3.5 Integral Equation Solutions 3.6 Numerical Solution o f the Laminar Boundary Layer \ Equations 3.7 Viscous Dissipation Effects on Laminar Boundary Layer Flow over a Flat Plate 3.8 ~.- Hiffect o f Fluid Property Variations with Viscous Dissipation Effects on Laminar Boundary Layer Flow over a Flat Plate 3.9 Solutions to the Full Governing Equations 3.10 Concluding Remarks Problems References 83 83 83 98 106 114 123 140 149 150 152 152 155 157 157 158 169 179 189 191 201 212 219 220 220 225 4 Internal Laminar Flows 4.1 Introduction 4.2 Fully Developed Laminar Pipe Flow 4.3 F ully Developed Laminar Flow in a P lane Duct 4.4 Fully Developed Laminar Flow in Ducts with Other Cross-Sectional Shapes 4.5 P ipe Flow with a Developing Temperature Field 4.6 P lane D uct Flow with a Developing Temperature Field 4.7 L aminar P ipe Flow with Developing Velocity and Temperature Fields 4.8 L aminar Flow in a Plane Duct with Developing Velocity and Temperature Fields 4.9 Solutions to the Full Navier-Stokes Equations 4.10 Concluding Remarks Problems References 5 Introduction to Thrbulent Flows 5.1 Introduction 5.2 Governing Equations 5.3 Mixing Length Turbulence Models 5.4 More Advanced Turbulence Models 5.5 Analogy Solutions for Heat Transfer in Turbulent Flow 5.6 Near-Wall Region 5.7 Transition from Laminar to Turbulent Flow 5.8 Concluding Remarks Problems References 227 227 228 234 239 244 245 247 250 250 252 254 254 254 272 281 296 299 299 300 302 304 304 304 322 329 336 337 337 339 342 342 342 6 External Thrbulent Flows 6.1 Introduction 6.2 Analogy Solutions for Boundary Layer Flows 6.3 Integral Equation Solutions 6.4 Numerical Solution o f the Turbulent Boundary Layer Equations 6.5 Effects o f Dissipation on Turbulent Boundary Layer Flow O ver a F lat Plate 6.6 Solutions to the Full Turbulent Flow Equations 6.7 Concluding Remarks Problems References 7 I nternal Thrbulent Flows 7.1 Introduction 7.2 Analogy Solutions for Fully Developed Pipe Flow 7.3 Thermally Developing Pipe Flow 7.4 Developing Flow i n a Plane Duct 7.5 Solutions to the Full Governing Equations 7.6 Concluding Remarks Problems References 8 N atural Convection Introduction \ 8.2 Boussinesq Approxirhation '. 8.1 / 8.3 8.4 8.5 8.6 8.7 8.8 8.9 Governing Equations Similarity i n Free Convective Flows Boundary Layer Equations for Natural Convective Flows Similarity Solutions for Free Convective Laminar Boundary Layer Flows Numerical Solution o f the. Natural Convective Boundary L ayer Equations 344 345 349 354 365 366 385 403 407 416 416 420 426 426 426 430 431 442 446 446 449 455 464 466 474 477 477 480 487 487 488 Free Convective Flow through a Vertical Channel Natural Convective Heat Transfer across a Rectangular Enclosure 8.10 Horizontal Enclosures Heated from Below , 8.11-"-'fiitrbulent Natural Convective Flows 8.12 Concluding Remarks Problems References j 9 Combined Convection 9.1 9 .2 9.3 9 .4 Introduction Governing Parameters Governing Equations Laminar Boundary Layer Flow O ver an Isothermal Vertical Flat Plate Numerical Solution o f Boundary Layer Equations Combined Convection O ver a Horizontal Plate Solutions to the Full Governing Equations 9.5 9 .6 9 .7 9 .8 Correlation o f Heat Transfer Results for Mixed Convection 9.9 Effect o f Buoyancy Forces on Turbulent Flows 9.10 Internal Mixed Convective Flows 9.11 Fully Developed Mixed Convective Flow in a Vertical Plane Channel 9.12 Mixed Convective Flow in a Horizontal Duct 9.13 Concluding Remarks Problems References 10 Convective H eat T ransfer Through Porous Media 10.1 Introduction 10.2 A rea Averaged Velocity Darcy Flow Model Energy Equation 10.5 Boundary Layer Solutions for Two-Dimensional Forced Convective Heat Transfer 10.6 Fully Developed Duct Flow 10.7 Natural Convective Boundary Layer Flows 10.8 Natural Convection in Porous Media-Filled Enclosures 10.9 Stability o f Horizontal Porous Layers Heated from Below 10.10 Non-Darcy a nd O ther Effects 10.11 Concluding Remarks Problems References 10.3 10.4 490 495 498 521 526 531 540 545 547 547 550 555 555 558 570 574 579 585 586 597 600 600 602 11 Condensation 11.1 Introduction 11.2 Laminar Film Condensation on a Vertical Plate 11.3 Wavy and Turbulent Film Condensation on a Vertical Surface 11.4 Film Condensation on Horizontal Tubes 11.5 Effect o f Surface Shear Stress on Film Condensation on a Vertical Plate ~ 11.6 Effect o f Noncondensible Gases on Film Condensation 11.7 Improved Analyses o f Laminar Film Condensation 11.8 Nongravitational Condensation 11.9 Concluding Remarks Problems References Appendices A Properties o f Saturated Water B Properties o f A ir at Standard Atmospheric Pressure Index 606 607 609 P REFACE C onvective heat transfer occurs in almost all branches o f engineering and a knowledge o f t he methods used to model convective heat transfer is therefore required b y many practicing engineers. Most conventional introductory courses on heat transfer deal with some aspects o f convective heat transfer but the treatment is usually relatively superficial. For this reason, many engineering schools offer a course dealing with convective heat transfer at the senior undergraduate or graduate level. The purpose o f such courses is to expand and extend the coverage given in the basic course on heat transfer. The present book is intended to provide a clear presentation o f t he material covered in such courses. Completion o f undergraduate courses in basic heat transfer, thermodynamics, and fluid mechanics i s assumed. C overage The book provides a comprehensive coverage o f the subject giving a full discussion o f forced, natural, and mixed convection including some discussion o f turbulent natural and mixed convection. A comprehensive discussion o f convective heat transfer in porous media flows and o f condensation heat transfer is also provided. The book contains a large number o f worked examples that illustrate the use o f the derived results. All chapters in the book also contain an extensive set o f problems. Convective heat transfer has become a subject o f very wide extent and the se'lection o f material for inclusion in a book o f t he present type requires careful consideration. In this book, heat transfer during boiling and solidification have not been considered. This is not in any way meant to suggest that these topics are o f l esser importance than those included in the book. Rather, it is felt that the student needs a good grounding in the topics covered i n t he present book before approaching the analysis o f h eat transfer with boiling and solidification. The book thus lays the foundation for more advanced courses on specialized aspects o f convective heat transfer. Approach T he basic aim o f the book is to p resent a discussion o f some currently available methods for predicting convective heat transfer rates. The main emphasis is, therefore, on the prediction o f h eat transfer rates rather than on the presentation o f l arge amounts o f experimental data. Attention is given to both analytical and numerical methods o f analysis. Another aim o f t he book is to present a thorough discussion o f the foundations o f the subject i n a clear, easy to follow, student-oriented style. Because new methods o f analyzing convection are constantly being introduced, the emphasis in the book is on develQping a thorough understanding o f t he basic equations and o f t he processes involved i n t he analysis o f convective heat transfer, together with developing an understarlding o f the assumptions on which existing x iii methods o f analysis are based and on developing a clear understanding o f the limitations o f these methods. Even a t t he graduate level, some students have difficulty dealing with the process o f taking a complex real situation and, b y introducing 11 s eries o f carefully reasoned and documented assumptions, deriving a model o f t he real situation that is amenable to solution. I n this book, emphasis has, therefore, been placed wherever possible on this modeling process. The widespread use o f c omputer software for the analysis o f e ngineering problems has, i n m any ways, increased the need to understand the assumptions and the theory on which such analyses are based. S uch an understanding is required to interpret the computer results and to j udge w hether a particular piece o f software will give results that are o f adequate accuracy for the application being considered. T hhefore, while there is an extensive discussion o f numerical methods i n this book, the major emphasis is on developing an understanding o f the material and o f t he assumptions conventionally used in analyzing convective heat transfer. Compared, to avail&ble textbooks on the subject,. then, the present book is, it is hoped, distinguished o y its attempt to develop a thorough understanding o f the theory and o f the assumptions on which this theory is based and by the breadth o f its coverage. Solutions Manual A manual, written by the authors o f this textbook, that contains detailed solutions to .' all o f the problems in the book, is available. Each solution in this manual starts on a separate page thus making it easier to post solutions to individual problems. Software A number o f computer programs are discussed in this book. These are all based on relatively simple finite-difference procedures that are developed i n the book. While the numerical methods used are relatively simple, it is believed that i f the students gain a good understanding o f these methods and are exposed to the power o f e ven simple numerical solution procedures, they will have little difficulty in understanding and using more advanced numerical methods. Examples o f t he use o f the computer programs are included in the text. The computer programs described in this book can all be downloaded from the web site o f one o f t he authors (P. H. Oosthuizen) at Queen's University. For more information on these programs and on how they can be obtained, please contact this author via e-mail A CKNOWLEDGMENTS Thill book has been developed from notes prepared over many years for use in courses on convective heat transfer taught to advanced undergradmlte and graduate students at Queen's University. The input o f students into a professor's understanding o f a subject cannot b e overemphasized and the material in this book and the way in which it is presented have been greatly influenced by the undergraduate and graduate students that have taken these courses. T he authors wish to express their deep and sincere gratitude to all o f these students. Jane Paul undertook the tedious j ob o f preparing and checking much o f the text and her help and encouragement is gratefully acknowledged. The authors would like to t hank John Lloyd and John D. Anderson for their advice and encouragement during the early stages o f the development o f this book. They would also like to express their gratitude to the following reviewers for their contributions to the development o f this text: A. M. Kanury Oregon State University Afshin J. Ghajar Oklahoma State University John Lloyd Michigan State University Jamal Seyed- Yagoobi Texas A &M University D avid G. B riggs Rutgers University R amendra P. Roy A rizona State University Patrick H. Oosthuizen David Naylor ./ N OMENCLATURE The following is a list o f some o f the main symbols used in this book: A a Cf cp D Dh E e Fr G Gr Gr* g H h Ii hfg Ja K k L e M m Nu N ux P p p p' Pe Pr PrT Q q Ra Re r T T T' area speed o f sound shear stress coefficient constant pressure specific heat diameter o r drag force hydraulic diameter Eckert number wall roughness Froude number buoyancy force parameter Grashof number heat flux Grashof number gravitational acceleration rate o f enthalpy transport heat transfer coefficient average heat transfer coefficient latent heat Jakob number turbulence kinetic energy or permeability thermal conductivity length ...~ mixing length mass o r M ach numb~r o r rate o f momentum transport mass flow rate Nusselt number local Nusselt n umber dimensionless pressure or perimeter pressure time-averaged pressure fluctuating component o f pressure Peclet n umber Prandtl n umber turbulent Prandtl number h eat transfer rate heat transfer rate per unit area Rayleigh number Reynolds number radius temperature time-averaged temperature fluctuating component o f temperature xvii .' Tb Tf Ts Tw TWad t U U Ii U' UI Uc U* v v V V' w w W' x y Z a r 8 E {3 'Y 8T EH (J JL v 7T' P 'YJ aT w b ulk temperature fluid temperature saturation temperature (also Tiat) surface temperature adiabatic wall temperature time forced velocity or dimensionless velocity velocity component average velocity component fluctuating component o f velocity free-stream velocity (also uoo) center-line velocity friction velocity volume or dimensionless velocity velocity <;PIIlPonent average velocity c omponent fluctuating component o f velocity velocity component average velocity component fluctuating component o f velocit):' coordinate direction coordinate direction coordinate direction dhermal diffusivity volumetric expansion coefficient specific heat ratio mass flow rate p er u nit width velocity boundary layer dhickness o r condensate film dhickness dhermal boundary l ayer t hickness eddy viscosity eddy diffusivity dimensionless temperature or angle dynamic viscosity kinematic viscosity dimensionless parameter density similarity parameter or dimensionless coordinate normal stress shear stress molecular shear stress turbulent shear stress dissipation function angular coordinate or dimensionless temperature or porosity dimensionless stream function stream function dimensionless vorticity vorticity