Heat Convection

Latif M. Jiji

Heat Convection

Second Edition

ABC

Prof. Latif M. JijiDepartment of Mechanical EngineeringGrove School of EngineeringThe City College ofThe City University of New YorkNew York, New York 10031USAE-mail: jiji@ccny.cuny.edu

ISBN 978-3-642-02970-7 e-ISBN 978-3-642-02971-4

DOI 10.1007/978-3-642-02971-4

Library of Congress Control Number: Applied for

c© 2009 Springer-Verlag Berlin Heidelberg

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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 textbook 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.

vi

This edition adds two new chapters on turbulent convection. I am fortunate that Professor Glen E. Thorncroft of California Polytechnic State University, San Luis Obispo, California, agreed to take on this responsibility and prepared all the material for chapters 8 and 9.

Hard as it is to leave out any of the topics usually covered in classic texts, cuts have been made so that the remaining material 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. Detailed solution to all end of chapter problems follow the same format. They are prepared for posting electronically.

This work 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. Since 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 Department of Mechanical Engineering The City Collegeof the City University of New York New York, New York May 2009

ii

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 17PROBLEMS 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

Motion 53 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

Governing Equations 99

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 Flow 114

4.4 Solutions: External Flow 115 4.4.1 Laminar 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 193PROBLEMS 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 2126.5 Channels with Uniform Surface Temperature 2186.6 Determination of Heat Transfer Coefficient h(x) and Nusselt

Number DNu 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 2426.8.2 Uniform Surface Heat Flux 252

REFERENCES 254PROBLEMS 255

CHAPTER 7: FREE CONVECTION 259

7.1 Introduction 259 7.2 Features and Parameters of Free Convection 259 7.3 Governing Equations 261

Contents xiii

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 289PROBLEMS 290

CHAPTET 8: CONVECTION IN EXTERNAL TURBULENT FLOW

293

8.1

8.2

8.3

Introduction 8.1.1 Examples of Turbulent Flows

8.1.2 The Reynolds Number and the Onset of Turbulence 8.1.3 Eddies and Vorticity 8.1.4 Scales of Turbulence 8.1.5 Characteristics of Turbulence 8.1.6 Analytical Approach Conservation Equations for Turbulent Flow 8.2.1 Reynolds Decomposition 8.2.2 Conservation of Mass 8.2.3 Momentum Equations 8.2.4 Energy Equation 8.2.5 Summary of Governing Equations for Turbulent Flow Analysis of External Turbulent Flow 8.3.1 Turbulent Boundary Layer Equations 8.3.2 Reynolds Stress and Heat Flux 8.3.3 The Closure Problem of Turbulence 8.3.4 Eddy Diffusivity

293 294 296 297 299 302 302 304 304 307 308 310

310 311 311 314 315 317

xi Contents

8.4

8.5

Momentum Transfer in External Turbulent Flow 8.4.1 Modeling Eddy Diffusivity: Prandtl’s Mixing Length

Theory8.4.2 Universal Turbulent Velocity Profile 8.4.3 Approximate Solution for Momentum Transfer: Momentum Integral Method 8.4.4 Effect of Surface Roughness on Friction Factor Energy Transfer in External Turbulent Flow 8.5.1 Momentum and Heat Transfer Analogies 8.5.2 Validity of Analogies 8.5.3 Universal Turbulent Temperature Profile 8.5.4 Algebraic Method for Heat Transfer Coefficient 8.5.5 Integral Method for Heat Transfer Coefficient 8.5.6 Effect of Surface roughness on Heat Transfer

REFERENCES PROBLEMS

318

318 320 328

334 335 336 344 345 349 351 352

353 356

CHAPTERE 9: CONVECTION IN TURBULENT CHANNEL FLOW 361

9.1 9.2 9.3

9.4

9.5

9.6

9.79.89.9

Introduction Entry Length Governing Equations 9.3.1 Conservation Equations 9.3.2 Apparent Shear Stress and Heat Transfer 9.3.3 Mean Velocity and Temperature Universal Velocity Profile 9.4.1 Results from Flat Plate Flow 9.4.2 Development in Cylindrical Coordinates 9.4.3 Velocity Profile for the Entire Pipe Friction Factor for Pipe Flow 9.5.1 Blasius Correlation for Smooth Pipe 9.5.2 The 1/7th Power Law Velocity Profile 9.5.3 Prandtl’s Law for Smooth Pipe 9.5.4 Effect of Surface Roughness Momentum-Heat Transfer Analogies 9.6.1 Reynolds Analogy for Pipe Flow 9.6.2 Adapting Flat-Plate Analogies to Pipe Flow 9.6.3 Other Analogy-Based Correlations Algebraic Method Using Universal Temperature Profile Other Correlations for Smooth Pipe Heat Transfer in Rough Pipes

361 361 362 362 363 363 364 364 365 366 367 367 367369 371 372 374 374 376 377 379 380

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Contents xv

REFERENCES PROBLEMS

381 382

CHAPTER 10: CORRELATION EQUATIONS: FORCED AND FREE CONVECTION 387

10.1 Introduction 387 10.2 10.3 10.4 10.5

10.6

10.7

10.8

Experimental Determination of Heat Transfer Coefficient hLimitations and Accuracy of Correlation Equations Procedure for Selecting and Applying Correlation Equations External Forced Convection Correlations 10.5.1 Uniform Flow over a Flat Plate: Transition to

Turbulent Flow 10.5.2 External Flow Normal to a Cylinder 10.5.3 External Flow over a Sphere Internal Forced Convection Correlations 10.6.1 Entrance Region: Laminar Flow Through Tubes at

Uniform Surface Temperature 10.6.2 Fully Developed Velocity and Temperature in Tubes:

Turbulent Flow 10.6.3 Non-circular Channels: Turbulent Flow Free Convection Correlations 10.7.1 External Free Convection Correlations 10.7.2 Free Convection in Enclosures Other Correlations

REFERENCES PROBLEMS

388 389 389 390

390 396 397 397

397

403 404 405 405 413 422

423 425

CHAPTER 11: CONVECTION IN MICROCHANNELS 437

11.1 Introduction 437 11.1.1 Continuum and Thermodynamic Hypothesis 437 11.1.2 Surface Forces 437 1.1.3 Chapter Scope 439

11.2 Basic Considerations 439 11.2.1 Mean Free Path 439 11.2.2 Why Microchannels? 440 11.2.3 Classification 441 11.2.4 Macro and Microchannels 442 11.2.5 Gases vs. Liquids 442

11.3 General Features 442 11.3.1 Flow Rate 444 11.3.2 Friction Factor 444

xvi Contents

11.3.3 Transition to Turbulent Flow 446 11.3.4 Nusselt number 446

11.4 Governing Equations 446 11.4.1 Compressibility 447 11.4.2 Axial Conduction 447 11.4.3 Dissipation 447

11.5 Velocity Slip and Temperature Jump Boundary Conditions 448 11.6 Analytic Solutions: Slip Flow 450

11.6.1 Assumptions 450 11.6.2 Couette Flow with Viscous Dissipation: Parallel Plates with Surface Convection 450 11.6.3 Fully Developed Poiseuille Channel Flow: Uniform

Surface Flux 450 11.6.4 Fully Developed Poiseuille Channel Flow: Uniform Surface Temperature 450 11.6.5 Fully Developed Poiseuille Flow in Microtubes: Uniform Surface Flux 485 11.6.6 Fully Developed Poiseuille Flow in Microtubes: Uniform Surface Temperature 496

REFERENCES 498PROBLEMS 500

APPENDIX A Conservation of Energy: The Energy Equation 507

APPENDIX B Pohlhausen’s Solution 516

APPENDIX C Laminar Boundary Layer Flow over Semi-infinite Plate: Variable Surface Temperature 520

APPENDIX D The von Karman Momentum and Heat Transfer Analogy 523

APPENDIX E Turbulent Heat Transfer from a Flat Plate with Unheated Starting Length 527

APEENDIC F Properties of Dry Air at Atmospheric Pressure 536

APPENDIX G Properties of Saturated Water 537

INDEX 539