Heat Conduction : Third Edition

This textbook presents the classical topics of conduction heat transfer and extends the coverage to include chapters on perturbation methods, heat transfer in living tissue, and microscale conduction. This makes the book unique among the many published textbook on conduction heat transfer. Other noteworthy features of the book are: The material is organized to provide students with the tools to model, analyze and solve a wide range of engineering applications involving conduction heat transfer. Mathematical techniques are presented in a clear and simplified fashion to be used as instruments in obtaining solutions. The simplicity of one-dimensional conduction is used to drill students in the role of boundary conditions and to explore a variety of physical conditions that are of practical interest. Examples are carefully selected to illustrate the application of principles and the construction of solutions. Students are trained to follow a systematic problem solving methodology with emphasis on thought process, logic, reasoning and verification. Solutions to all examples and end-of-chapter problems follow an orderly problems solving approach. Extensive training material is available on the web A solution manual for teachers is available on request Subject front-matter 1 Chapter 01 14 BASIC CONCEPTS 14 Examples of Conduction Problems 14 Focal Point in Conduction Heat Transfer 15 Fourier's Law of Conduction 15 Conservation of Energy: Differential Formulation of the Heat Conduction Equation in Rectangular Coordinates 18 The Heat Conduction Equation in Cylindrical and Spherical Coordinates 22 Boundary Conditions 23 Surface Convection: Newton's Law of Cooling 23 Surface Radiation: Stefan-Boltzmann Law 24 Examples of Boundary Conditions 25 Problem Solving Format 28 Units 29 REFERENCES 30 Chapter 02 37 ONE-DIMENSIONAL STEADY STATE CONDUCTION 37 Examples of One-dimensional Conduction 37 Extended Surfaces: Fins 47 The Function of Fins 47 Types of Fins 47 Heat Transfer and Temperature Distribution in Fins 48 The Fin Approximation 49 The Fin Heat Equation: Convection at Surface 50 Determination of $\frac{dA_{s}}{dx}$ 52 Boundary Conditions 53 Determination of Fin Heat Transfer Rate $q_{f}$ 53 Steady State Applications: Constant Area Fins with Surface Convection 54 Corrected Length $L_{c}$ 57 Fin Efficiency $\eta_{f}$ 57 Moving Fins 58 Application of Moving Fins 60 Variable Area Fins 62 Bessel Differential Equations and Bessel Functions 65 General Form of Bessel Equations 65 Solutions: Bessel Functions 65 Forms of Bessel Functions 67 Special Closed-form Bessel Functions:$n = \frac{odd integer}{2}$ 67 Special Relations for n = 1, 2, 3, .... 68 Derivatives and Integrals of Bessel Functions [2,3] 69 Tabulation and Graphical Representation of Selected Bessel Functions 69 Equidimensional (Euler) Equation 71 Graphically Presented Solutions to Fin Heat Transfer Rate [5] 72 REFERENCES 73 Chapter 03 85 TWO-DIMENSIONAL STEADY STATE CONDUCTION 85 The Heat Conduction Equation 85 Method of Solution and Limitations 86 Homogeneous Differential Equations and Boundary Conditions 86 Sturm-Liouville Boundary-Value Problem: Orthogonality [1] 87 Procedure for the Application of Separation of Variables Method 89 Cartesian Coordinates: Examples 96 Cylindrical Coordinates: Examples 110 Integrals of Bessel Functions 115 Non-homogeneous Differential Equations 116 Non-homogeneous Boundary Conditions: The Method of Superposition 122 REFERENCES 124 Chapter 04 132 TRANSIENT CONDUCTION 132 Simplified Model: Lumped-Capacity Method 132 Criterion for Neglecting Spatial Temperature Variation 132 Lumped-Capacity Analysis 134 Transient Conduction in Plates 137 Non-homogeneous Equations and Boundary Conditions 141 Transient Conduction in Cylinders 145 Transient Conduction in Spheres 151 Time Dependent Boundary Conditions: Duhamel’s Superposition Integral 154 Formulation of Duhamel’s Integral [1] 155 Extension to Discontinuous Boundary Conditions 157 Applications 158 Conduction in Semi-infinite Regions: The Similarity Method 163 REFERENCES 167 Chapter 05 176 CONDUCTION IN POROUS MEDIA 176 Examples of Conduction in Porous Media 176 Simplified Heat Transfer Model 177 Porosity 177 Heat Conduction Equation: Cartesian Coordinates 178 Boundary Conditions 180 Heat Conduction Equation: Cylindrical Coordinates 181 Applications 181 REFEENCES 187 Chapter 06 197 CONDUCTION WITH PHASE CHANGE: MOVING BOUNDARY PROBLEMS 197 Introduction 197 The Heat Equations 198 Moving Interface Boundary Conditions 198 Non-linearity of the Interface Energy Equation 201 Non-dimensional Form of the Governing Equations: Governing Parameters 202 Simplified Model: Quasi-Steady Approximation 203 Exact Solutions 210 Stefan’s Solution 210 Neumann’s Solution: Solidification of Semi-Infinite Region 213 Neumann’s Solution: Melting of Semi-infinite Region 216 Effect of Density Change on the Liquid Phase 217 Radial Conduction with Phase Change 218 Phase Change in Finite Regions 222 REFERENCES 223 Chapter 07 228 NON-LINEAR CONDUCTION PROBLEMS 228 Introduction 228 Sources of Non-linearity 228 Non-linear Differential Equations 228 Non-linear Boundary Conditions 229 Taylor Series Method 229 Kirchhoff Transformation 233 Transformation of Differential Equations 233 Transformation of Boundary Conditions 234 Boltzmann Transformation 237 Combining Boltzmann and Kirchhoff Transformations 239 Exact Solutions 240 REFERENCES 243 Chapter 08 249 APPROXIMATE SOLUTIONS: THE INTEGRAL METHOD 249 Integral Method Approximation: Mathematical Simplification 249 Procedure 249 Accuracy of the Integral Method 250 Application to Cartesian Coordinates 251 Application to Cylindrical Coordinates 259 Non-linear Problems [5] 264 Energy Generation 273 REFERENCES 277 Chapter 09 282 PERTURBATION SOLUTIONS 282 Introduction 282 Solution Procedure 283 Examples of Perturbation Problems in Conduction 284 Perturbation Solutions: Examples 286 Useful Expansions 309 REFERENCES 309 Chapter 10 315 Heat Transfer in Living Tissue 315 Introduction 315 Vascular Architecture and Blood Flow 315 Blood Temperature Variation 317 Mathematical Modeling of Vessels-Tissue Heat Transfer 318 Pennes Bioheat Equation [1] 318 Chen-Holmes Equation [5] 325 Three-Temperature Model for Peripheral Tissue [7] 326 Weinbaum-Jiji Simplified Bioheat Equation for Peripheral Tissue [8] 328 The $s$-Vessel Tissue Cylinder Model [16] 336 REFERENCES 345 Chapter 11 360 MICROSCALE CONDUCTION 360 Introduction 360 Categories of Microscale Phenomena 361 Purpose and Scope of this Chapter 363 Understanding the Essential Physics of Thermal Conductivity Using the Kinetic Theory of Gases 364 Derivation of Fourier’s Law and an Expression for the Thermal Conductivity 364 Energy Carriers 368 Ideal Gases: Heat is Conducted by Gas Molecules 368 Metals: Heat is Conducted by Electrons 372 Electrical Insulators and Semiconductors: Heat is Conducted by Phonons (Sound Waves) 374 Radiation: Heat is Carried by Photons (Light Waves) 385 Thermal Conductivity Reduction by Boundary Scattering: The Classical Size Effect 390 Accounting for Multiple Scattering Mechanisms: Matthiessen’s rule 390 Boundary Scattering for Heat Flow Parallel to Boundaries 392 Boundary Scattering for Heat Flow Perpendicular to Boundaries 400 Closing Thoughts 405 REFERENCES 408 back-matter 415 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 o- dimensional 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 textbook presents the classical topics of conduction heat transfer and extends the coverage to include chapters on perturbation methods, heat transfer in living tissue, and microscale conduction. This makes the book unique among the many published textbook on conduction heat transfer. Other noteworthy features of the book are: The material is organized to provide students with the tools to model, analyze and solve a wide range of engineering applications involving conduction heat transfer. Mathematical techniques are presented in a clear and simplified fashion to be used as instruments in obtaining solutions. The simplicity of one-dimensional conduction is used to drill students in the role of boundary conditions and to explore a variety of physical conditions that are of practical interest. Examples are carefully selected to illustrate the application of principles and the construction of solutions. Students are trained to follow a systematic problem solving methodology with emphasis on thought process, logic, reasoning and verification. Solutions to all examples and end-of-chapter problems follow an orderly problems solving approach. Extensive training material is available on the web The author provides an extensive solution manual for verifiable course instructors on request. Please send your request to heattextbook@gmail.com. Front Matter....Pages - BASIC CONCEPTS....Pages 1-23 ONE-DIMENSIONAL STEADY STATE CONDUCTION....Pages 24-71 TWO-DIMENSIONAL STEADY STATE CONDUCTION....Pages 72-118 TRANSIENT CONDUCTION....Pages 119-162 CONDUCTION IN POROUS MEDIA....Pages 163-183 CONDUCTION WITH PHASE CHANGE: MOVING BOUNDARY PROBLEMS....Pages 184-214 NON-LINEAR CONDUCTION PROBLEMS....Pages 215-235 APPROXIMATE SOLUTIONS: THE INTEGRAL METHOD....Pages 236-268 PERTURBATION SOLUTIONS....Pages 269-301 Heat Transfer in Living Tissue....Pages 302-346 MICROSCALE CONDUCTION....Pages 347-401 Back Matter....Pages - "This textbook presents the classical topics of conduction heat transfer and extends the coverage to include chapters on perturbation methods, heat transfer in living tissue, and microscale conduction. This makes the book unique among the many published textbooks on conduction heat transfer."--Jacket Presents the classical topics of conduction heat transfer and includes chapters on perturbation methods, heat transfer in living tissue, and microscale conduction. This book provides the tools to model, analyze and solve a wide range of engineering applications involving conduction heat transfer.