Latif M. Jiji. Heat Conduction. Third Edition ABC

Transcription

1 Heat Conduction

2 Latif M. Jiji Heat Conduction Third Edition ABC

3 Professor Latif M. Jiji Department of Mechanical Engineering Grove School of Engineering The City College of The City University of New York New York, New York USA ''Additional material to this book can be downloaded from '' ISBN e-isbn DOI / Library of Congress Control Number: Applied for c 2009 Springer-Verlag Berlin Heidelberg 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. Violations are liable to prosecution under the German Copyright Law. 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. Typesetting by the Author. Production: Scientific Publishing Services Pvt. Ltd., Chennai, India. Cover Design: WMX Design GmbH, Heidelberg. Printed in acid-free paper 30/3100/as springer.com

4 This book is dedicated to my wife Vera for opening many possibilities and providing balance in my life.

5 PREFACE This book is designed to: Provide students with the tools to model, analyze and solve a wide range of engineering applications involving conduction heat transfer. Introduce students to three topics not commonly covered in conduction heat transfer textbooks: perturbation methods, heat transfer in living tissue, and microscale conduction. Take advantage of the mathematical simplicity of onedimensional conduction to present and explore a variety of physical situations that are of practical interest. Present textbook material in an efficient and concise manner to be covered in its entirety in a one semester graduate course. Drill students in a systematic problem solving methodology with emphasis on thought process, logic, reasoning and verification. To accomplish these objectives requires judgment and balance in the selection of topics and the level of details. Mathematical techniques are presented in simplified fashion to be used as tools in obtaining solutions. Examples are carefully selected to illustrate the application of principles and the construction of solutions. Solutions follow an orderly approach which is used in all examples. To provide consistency in solutions logic, I have prepared solutions to all problems included in the first ten chapters myself. Instructors are urged to make them available electronically rather than posting them or presenting them in class in an abridged form. This edition adds a new chapter, Microscale Conduction. This is a new and emerging area in heat transfer. Very little is available on this subject as textbook material at an introductory level. Indeed the preparation of such a chapter is a challenging task. I am fortunate

6 viii PREFACE that Professor Chris Dames of the University of California, Riverside, agreed to take on this responsibility and prepared all the material for chapter 11. Now for the originality of the material in this book. Much that is here was inspired by publications on conduction. I would like to especially credit Conduction Heat Transfer by my friend Vedat S Arpaci. His book contains a wealth of interesting problems and applications. My original notes on conduction contained many examples and problems taken from the literature. Not having been careful in my early years about recording references, I tried to eliminate those that I knew were not my own. Nevertheless, a few may have been inadvertently included. ACKNOWLEDGMENTS First I would like to acknowledge the many teachers who directly or indirectly inspired and shaped my career. Among them I wish to single out Professors Ascher H. Shapiro of the Massachusetts Institute of Technology, Milton Van Dyke of Stanford University, D.W. Ver Planck of Carnegie Institute of Technology and Gordon J. Van Wylen and John A. Clark of the University of Michigan. I only wish that I had recognized their lasting contributions to my education decades earlier. Chapter 11 was carefully reviewed by Professor Gang Chen of the Massachusetts Institute of Technology. The chapter author Chris Dames and I are grateful for his technical comments which strengthened the chapter. My wife Vera read the entire manuscript and made constructive observations. I would like to thank her for being a supportive, patient and understanding partner throughout this project. Latif M. Jiji New York, New York March, 2009

7 CONTENTS Preface vii CHAPTER 1: BASIC CONCEPTS Examples of Conduction Problems Focal Point in Conduction Heat Transfer Fourier s Law of Conduction Conservation of Energy: Differential Formulation of the Heat Conduction in Rectangular Coordinates The Heat Conduction Equation in Cylindrical and Spherical Coordinates Boundary Conditions Surface Convection: Newton s Law of Cooling Surface Radiation: Stefan-Boltzmann Law Examples of Boundary Conditions 1.7 Problem Solving Format Units REFERENCES 17 PROBLEMS 18 CHAPTER 2: ONE-DIMENSIONAL STEADY-STATE CONDUCTION Examples of One-dimensional Conduction Extended Surfaces: Fins The Function of Fins Types of Fins Heat Transfer and Temperature Distribution in Fins The Fin Approximation The Fin Heat Equation: Convection at Surface Determination of da s / dx Boundary Conditions Determination of Fin Heat Transfer Rate q f Steady State Applications: Constant Area Fins with Surface Convection 41

8 x CONTENTS Corrected Length L c Fin Efficiency f Moving Fins Application of Moving Fins Variable Area Fins Bessel Differential Equations and Bessel Functions General Form of Bessel Equations Solutions: Bessel Functions Forms of Bessel Functions Special Closed-form Bessel Functions: n = odd integer/ Special Relations for n = 1, 2, 3, Derivatives and Integrals of Bessel Functions Tabulation and Graphical Representation of Selected Bessel Functions Equidimensional (Euler) Equation Graphically Presented Solutions to Fin Heat Transfer Rate q 59 f REFERENCES 60 PROBLEMS 61 CHAPTER 3: TWO-DIMESIONAL STEADY STATE CONDUCTION The Heat Conduction Equation Method of Solution and Limitations Homogeneous Differential Equations and Boundary Conditions Sturm-Liouville Boundary-Value Problem: Orthogonality Procedure for the Application of Separation of Variables Method Cartesian Coordinates: Examples Cylindrical Coordinates: Examples Integrals of Bessel Functions Non-homogeneous Differential Equations Non-homogeneous Boundary Conditions: The Method of Superposition 109 REFERENCES 111

9 CONTENTS xi PROBLEMS 111 CHAPTER 4: TRANSIENT CONDUCTION Simplified Model: Lumped-Capacity Method Criterion for Neglecting Spatial Temperature Variation Lumped-Capacity Analysis Transient Conduction in Plates Non-homogeneous Equations and Boundary Conditions Transient Conduction in Cylinders Transient Conduction in Spheres Time Dependent Boundary Conditions: Duhamel s Superposition Integral Formulation of Duhamel s Integral Extension to Discontinuous Boundary Conditions Applications Conduction in Semi-infinite Regions: The Similarity Transformation Method 150 REFERENCES 154 PROBLEMS 154 CHAPTER 5: POROUS MEDIA Examples of Conduction in Porous Media Simplified Heat Transfer Model Porosity Heat Conduction Equation: Cartesian Coordinates Boundary Conditions Heat Conduction Equation: Cylindrical Coordinates Applications 168 REFERENCES 174 PROBLEMS 175

10 xii CONTENTS CHAPTER 6: CONDUCTION WITH PHASE CHANGE: MOVING BOUNDARY PROBLEMS Introduction The Heat Equation Moving Interface Boundary Conditions Non-linearity of the Interface Energy Equation Non-dimensional Form of the Governing Equations: Governing Parameters Simplified Model: Quasi-Steady Approximation Exact Solutions Stefan s Solution Neumann s Solution: Solidification of Semi- Infinite Region Neumann s Solution: Melting of Semi-Infinite Region Effect of Density Change on the Liquid Phase Radial Conduction with Phase Change Phase Change in Finite Regions 209 REFERENCES 210 PROBLEMS 210 CHAPTER 7: NON-LINEAR CONDUCTION PROBLEMS Introduction Sources of Non-linearity Non-linear Differential Equations Non-linear Boundary Conditions Taylor Series Method Kirchhoff Transformation Transformation of Differential Equations Transformation of Boundary Conditions Boltzmann Transformation Combining Boltzmann and Kirchhoff Transformations Exact Solutions 227 REFERENCES 230 PROBLEMS 230

11 CONTENTS xiii CHAPTER 8: APPROXIMATE SOLUTIONS: THE INTEGRAL METHOD Integral Method Approximation: Mathematical Simplification Procedure Accuracy of the Integral Method Application to Cartesian Coordinates Application to Cylindrical Coordinates Non-linear Problems Energy Generation 260 REFERENCES 264 PROBLEMS 264 CHAPTER 9: PERTURBATION SOLUTIONS Introduction Solution Procedure Examples of Perturbation Problems in Conduction Perturbation Solutions: Examples Useful Expansions 296 REFERENCES 296 PROBLEMS 297 CHAPTER 10: HEAT TRANSFER IN LIVING TISSUE Introduction Vascular Architecture and Blood Flow Blood Temperature Variation Mathematical Modeling of Vessels-Tissue Heat Transfer Pennes Bioheat Equation Chen-Holmes Equation Three-Temperature Model for Peripheral Tissue Weinbaum-Jiji Simplified Bioheat Equation for Peripheral Tissue The s-vessel Tissue Cylinder Model 323 REFERENCES 332 PROBLEMS 334

12 xiv CONTENTS CHAPTER 11: MICROSCALE CONDUCTION Introduction Categories of Microscale Phenomena Purpose and Scope of this Chapter Understanding the Essential Physics of Thermal Conductivity Using the Kinetic Theory of Gases Determination of Fourier s Law and Expression for Thermal Conductivity Energy Carriers Ideal Gas: Heat is Conducted by Gas Molecules Metals: Heat is Conducted by Electrons Electrical Insulators and Semiconductors: Heat is Conducted by Phonons (Sound Waves) Radiation: Heat is Carried by Photons (Light Waves) Thermal Conductivity Reduction by Boundary Scattering: The Classical Size Effect Accounting for Multiple Scattering Mechanisms: Matthiessen s Rule Boundary Scattering for Heat Flow Parallel to Boundaries Boundary Scattering for Heat Flow Perpendicular to Boundaries Closing Thoughts 391 REFERENCES 394 PROBLEMS 397 APPENDIX A: APPENDIX B: APPENDIX C: APPENDIX D: Ordinary Differential Equations (1) Second Order Differential Equations with Constant Coefficients (2) First Order Ordinary Differential Equations with Variable Coefficients Integrals of Bessel Functions Values of Bessel Functions Fundamental Physical Constants and Material Properties D-1 Fundamental Physical Constants D-2 Unit conversions

13 CONTENTS xv D-3 Properties of Helium Gas D-4 Properties of Copper at 300 K D-5 Properties of Fused Silica (Amorphous Silicon Dioxide, SiO 2 ) at 300 K D-6 Properties of Silicon D-7 Measured Thermal Conductivity of a 56 nm Diameter Silicon Nanowire at Selected Temperatures D-8 Calculated Thermal Conductivity of Single-Walled Carbon Nanotubes, Selected Values INDEX 416