Heat Convection
Latif M. Jiji Heat Convection With 206 Figures and 16 Tables
Prof. Latif M. Jiji City University of New York School of Engineering Dept. of Mechanical Engineering Convent Avenue at 138th Street 10031 New York, NY USA E-Mail: jiji@ccny.cuny.edu Library of Congress Control Number: 2005937166 ISBN-10 3-540-30692-7 Springer Berlin Heidelberg New York ISBN-13 978-3-540-30692-4 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science + Business Media springer.com Springer-Verlag Berlin Heidelberg 2006 Printed in The Netherlands The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover Design: Erich Kirchner, Heidelberg Cover Image: Microchannel convection, courtesy of Fluent Inc. Production: SPI Publisher Services, Pondicherry Printed on acid free paper 30/3100/as 5 4 3 2 1 0
To my sister Sophie and brother Fouad for their enduring love and affection
PREFACE Why have I chosen to write a book on convection heat transfer when several already exist? Although I appreciate the available publications, in recent years I have not used a text book to teach our graduate course in convection. Instead, I have relied on my own notes, not because existing textbooks are unsatisfactory, but because I preferred to select and organize the subject matter to cover the most basic and essential topics and to strike a balance between physical description and mathematical requirements. As I developed my material, I began to distribute lecture notes to students, abandon blackboard use, and rely instead on PowerPoint presentations. I found that PowerPoint lecturing works most effectively when the presented material follows a textbook very closely, thus eliminating the need for students to take notes. Time saved by this format is used to raise questions, engage students, and gauge their comprehension of the subject. This book evolved out of my success with this approach. This book is designed to: Provide students with the fundamentals and tools needed to model, analyze, and solve a wide range of engineering applications involving convection heat transfer. Present a comprehensive introduction to the important new topic of convection in microchannels. Present textbook material in an efficient and concise manner to be covered in its entirety in a one semester graduate course. Liberate students from the task of copying material from the blackboard and free the instructor from the need to prepare extensive notes. Drill students in a systematic problem solving methodology with emphasis on thought process, logic, reasoning, and verification. Take advantage of internet technology to teach the course online by posting ancillary teaching materials and solutions to assigned problems.
viii Hard as it is to leave out any of the topics usually covered in classic texts, cuts have been made so that the remaining materials can be taught in one semester. To illustrate the application of principles and the construction of solutions, examples have been carefully selected, and the approach to solutions follows an orderly method used throughout. To provide consistency in the logic leading to solutions, I have prepared all solutions myself. This book owes a great deal to published literature on heat transfer. As I developed my notes, I used examples and problems taken from published work on the subject. As I did not always record references in my early years of teaching, I have tried to eliminate any that I knew were not my own. I would like to express regret if a few have been unintentionally included. Latif M. Jiji New York, New York January 2006
CONTENTS Preface vii CHAPTER 1: BASIC CONCEPTS 1 1.1 Convection Heat Transfer 1 1.2 Important Factors in Convection Heat Transfer 1 1.3 Focal Point in Convection Heat Transfer 2 1.4 The Continuum and Thermodynamic Equilibrium Concepts 2 1.5 Fourier s Law of Conduction 3 1.6 Newton s Law of Cooling 5 1.7 The Heat Transfer Coefficient h 6 1.8 Radiation: Stefan-Boltzmann Law 8 1.9 Differential Formulation of Basic Laws 8 1.10 Mathematical Background 9 1.11 Units 12 1.12 Problem Solving Format 13 REFERENCES 17 PROBLEMS 18 CHAPTER 2: DIFFERENTIAL FORMULATION OF THE BASIC LAWS 21 2.1 Introduction 21 2.2 Flow Generation 21 2.3 Laminar vs. Turbulent Flow 22 2.4 Conservation of Mass: The Continuity Equation 22
x Contents 2.4.1 Cartesian Coordinates 22 2.4.2. Cylindrical Coordinates 24 2.4.3 Spherical Coordinates 25 2.5 Conservation of Momentum: The Navier-Stokes Equations of Motion 27 2.5.1 Cartesian Coordinates 27 2.5.2 Cylindrical Coordinates 32 2.5.3 Spherical Coordinates 33 2.6 Conservation of Energy: The Energy Equation 37 2.6.1 Formulation: Cartesian Coordinates 37 2.6.2 Simplified form of the Energy Equation 40 2.6.3 Cylindrical Coordinates 41 2.6.4 Spherical Coordinates 42 2.7 Solution to the Temperature Distribution 45 2.8 The Boussinesq Approximation 46 2.9 Boundary Conditions 48 2.10 Non-dimensional Form of the Governing Equations: Dynamic and Thermal Similarity Parameters 51 2.10.1 Dimensionless Variables 52 2.10.2 Dimensionless Form of Continuity 52 2.10.3 Dimensionless Form of the Navier-Stokes Equations of 53 Motion 2.10.4 Dimensionless Form of the Energy Equation 53 2.10.5 Significance of the Governing Parameters 54 2.10.6 Heat Transfer Coefficient: The Nusselt Number 55 2.11 Scale Analysis 59 REFERENCES 61 PROBLEMS 62 CHAPTER 3: EXACT ONE-DIMENSIONAL SOLUTIONS 69 3.1 Introduction 69 3.2 Simplification of the Governing Equations 69 3.3 Exact Solutions 71 3.3.1 Couette Flow 71 3.3.2 Poiseuille Flow 77 3.3.3 Rotating Flow 86 REFERENCES 93 PROBLEMS 94
Contents xi CHAPTER 4: BOUNDARY LAYER FLOW: APPLICATION TO EXTERNAL FLOW 99 4.1 Introduction 99 4.2 The Boundary Layer Concept: Simplification of the 99 Governing Equations 4.2.1 Qualitative Description 99 4.2.2 The Governing Equations 101 4.2.3 Mathematical Simplification 101 4.2.4 Simplification of the Momentum Equations 101 4.2.5 Simplification of the Energy Equation 109 4.3 Summary of Boundary Layer Equations for Steady Laminar 114 Flow 4.4 Solutions: External Flow 115 4.4.1Laminar Boundary Layer Flow over Semi-infinite Flat Plate: Uniform Surface Temperature 116 4.4.2 Applications: Blasius Solution, Pohlhausen s Solution, and Scaling 131 4.4.3 Laminar Boundary Layer Flow over Semi-infinite Flat Plate: Variable Surface Temperature 140 4.4.4 Laminar Boundary Layer Flow over a Wedge: Uniform Surface Temperature 143 REFERENCES 149 PROBLEMS 150 CHAPTER 5: APPROXIMATE SOLUTIONS: THE INTEGRAL METHOD 161 5.1 Introduction 161 5.2 Differential vs. Integral Formulation 161 5.3 Integral Method Approximation: Mathematical Simplification 162 5.4 Procedure 162 5.5 Accuracy of the Integral Method 163 5.6 Integral Formulation of the Basic Laws 163 5.6.1 Conservation of Mass 163 5.6.2 Conservation of Momentum 165 5.6.3 Conservation of Energy 168 5.7 Integral Solutions 170 5.7.1 Flow Field Solution: Uniform Flow over a Semi-infinite Plate 170 5.7.2 Temperature Solution and Nusselt Number: Flow over a Semi-infinite Plate 173
xii Contents 5.7.3 Uniform Surface Flux 185 REFERENCES 193 PROBLEMS 194 CHAPTER 6: HEAT TRANSFER IN CHANNEL FLOW 203 6.1 Introduction 203 6.2 Hydrodynamic and Thermal Regions: General Features 204 6.2.1 Flow Field 205 6.2.2 Temperature Field 205 6.3 Hydrodynamic and Thermal Entrance Lengths 206 6.3.1 Scale Analysis 206 6.3.2 Analytic and Numerical Solutions: Laminar Flow 207 6.4 Channels with Uniform Surface Heat Flux 212 6.5 Channels with Uniform Surface Temperature 218 6.6 Determination of Heat Transfer Coefficient h(x) and Nusselt Number Nu D 224 6.6.1 Scale Analysis 224. 6.6.2 Basic Considerations for the Analytical Determination of Heat Flux, Heat Transfer Coefficient and Nusselt Number 226 6.7 Heat Transfer Coefficient in the Fully Developed Temperature Region 229 6.7.1 Definition of Fully Developed Temperature Profile 229 6.7.2 Heat Transfer Coefficient and Nusselt Number 230 6.7.3 Fully Developed Region for Tubes at Uniform Surface Flux 231 6.7.4 Fully Developed Region for Tubes at Uniform Surface Temperature 236 6.7.5 Nusselt Number for Laminar Fully Developed Velocity and Temperature in Channels of Various Cross-Sections 237 6.8 Thermal Entrance Region: Laminar Flow Through Tubes 242 6.8.1 Uniform Surface Temperature: Graetz Solution 242 6.8.2 Uniform Surface Heat Flux 252 REFERENCES 254 PROBLEMS 255 CHAPTER 7: FREE CONVECTION 259 7.1 Introduction 259
Contents xiii 7.2 Features and Parameters of Free Convection 259 7.3 Governing Equations 261 7.3.1 Boundary Conditions 262 7.4 Laminar Free Convection over a Vertical Plate: Uniform Surface Temperature 263 7.4.1 Assumptions 263 7.4.2 Governing Equations 263 7.4.3 Boundary Conditions 264 7.4.4 Similarity Transformation 264 7.4.5 Solution 267 7.4.6 Heat Transfer Coefficient and Nusselt Number 267 7.5 Laminar Free Convection over a Vertical Plate: Uniform Surface Heat Flux 274 7.6 Inclined Plates 279 7.7 Integral Method 279 7.7.1 Integral Formulation of Conservation of Momentum 279 7.7.2 Integral Formulation of Conservation of Energy 282 7.7.3 Integral Solution 283 7.7.4 Comparison with Exact Solution for Nusselt Number 288 REFERENCES 289 PROBLEMS 290 CHAPTER 8: CORRELATION EQUATIONS: FORCED AND FREE CONVECTION 293 8.1 Introduction 293 8.2 Experimental Determination of Heat Transfer Coefficient h 294 8.3 Limitations and Accuracy of Correlation Equations 295 8.4 Procedure for Selecting and Applying Correlation Equations 295 8.5 External Forced Convection Correlations 296 8.5.1 Uniform Flow over a Flat Plate: Transition to Turbulent Flow 296 8.5.2 External Flow Normal to a Cylinder 302 8.5.3 External Flow over a Sphere 303 8.6 Internal Forced Convection Correlations 303 8.6.1 Entrance Region: Laminar Flow Through Tubes at Uniform Surface Temperature 303 8.6.2 Fully Developed Velocity and Temperature in Tubes: Turbulent Flow 309 8.6.3 Non-circular Channels: Turbulent Flow 310 8.7 Free Convection Correlations 311 8.7.1 External Free Convection Correlations 311 8.7.2 Free Convection in Enclosures 319
x iv Contents 8.8 Other Correlations 328 REFERENCES 329 PROBLEMS 331 CHAPTER 9: CONVECTION IN MICROCHANNELS 343 9.1 Introduction 343 9.1.1 Continuum and Thermodynamic Hypothesis 343 9.1.2 Surface Forces 343 9.1.3 Chapter Scope 345 9.2 Basic Considerations 345 9.2.1 Mean Free Path 345 9.2.2 Why Microchannels? 346 9.2.3 Classification 347 9.2.4 Macro and Microchannels 348 9.2.5 Gases vs. Liquids 349 9.3 General Features 349 9.3.1 Flow Rate 350 9.3.2 Friction Factor 350 9.3.3 Transition to Turbulent Flow 352 9.3.4 Nusselt number 352 9.4 Governing Equations 352 9.4.1 Compressibility 353 9.4.2 Axial Conduction 353 9.4.3 Dissipation 353 9.5 Velocity Slip and Temperature Jump Boundary Conditions 354 9.6 Analytic Solutions: Slip Flow 356 9.6.1 Assumptions 356 9.6.2 Couette Flow with Viscous Dissipation: Parallel Plates with Surface Convection 356 9.6.3 Fully Developed Poiseuille Channel Flow: Uniform Surface Flux 368 9.6.4 Fully Developed Poiseuille Channel Flow: Uniform Surface Temperature 386 9.6.5 Fully Developed Poiseuille Flow in Microtubes: Uniform Surface Flux 391 9.6.6 Fully Developed Poiseuille Flow in Microtubes: Uniform Surface Temperature 402 REFERENCES 404 PROBLEMS 406
Contents xv APPENDIX A Conservation of Energy: The Energy Equation 413 APPENDIX B Pohlhausen s Solution 422 APPENDIX C Laminar Boundary Layer Flow over Semiinfinite Plate: Variable Surface Temperature 426 APEENDIC D Properties of Dry Air at Atmospheric Pressure 429 APPENDIX E Properties of Saturated Water 430 INDEX 431