Heat Conduction

PREFACE
CONTENTS
1 BASIC CONCEPTS
	1.1 Examples of Conduction Problems
	1.2 Focal Point in Conduction Heat Transfer
	1.3 Fourier\'s Law of Conduction
	1.4 Conservation of Energy: Differential Formulation of the Heat Conduction Equation in Rectangular Coordinates
	1.5 The Heat Conduction Equation in Cylindrical and Spherical Coordinates
	1.6 Boundary Conditions
		1.6.1 Surface Convection: Newton\'s Law of Cooling
		1.6.2 Surface Radiation: Stefan-Boltzmann Law
		1.6.3 Examples of Boundary Conditions
	1.7 Problem Solving Format
	1.8 Units
	REFERENCES
	PROBLEMS
2 ONE-DIMENSIONAL STEADY-STATE CONDUCTION
	2.1 Examples of One-Dimensional Conduction
	2.2 Extended Surfaces: Fins
		2.2.1 The Function of Fins
		2.2.2 Types of Fins
		2.2.3 Heat Transfer and Temperature Distribution in Fins
		2.2.4 The Fin Approximation
		2.2.5 The Fin Heat Equation: Convection at Surface
		2.2.6 Determination of dAs / dx
		2.2.7 Boundary Conditions
		2.2.8 Determination of Fin Heat Transfer Rate f
		2.2.9 Steady-State Applications: Constant Cross-Sectional Area Fins with Surface Convection
		2.2.10 Corrected Length c
		2.2.11 Fin Efficiency f
		2.2.12 Moving Fins
		2.2.13 Application of Moving Fins
		2.2.14 Variable Area Fins
	2.3 Bessel Differential Equations and Bessel Functions
		2.3.1 General Form of Bessel Equations
		2.3.2 Solutions: Bessel Functions
		2.3.3 Forms of Bessel Functions
		2.3.4 Special Closed-form Bessel Functions: n = (odd integer) / 2
		2.3.5 Special Relations for n = 1, 2, 3, ….
		2.3.6 Derivatives and Integrals of Bessel Functions [2,3]
		2.3.7 Tabulation and Graphical Representation of Selected Bessel Functions
	2.4 Equidimensional (Euler) Equation
	2.5 Graphically Presented Solutions to Fin Heat Transfer Rate qf
	REFERENCES
	PROBLEMS
3 TWO-DIMENSIONAL STEADY-STATE CONDUCTION
	3.1 The Heat Conduction Equation
	3.2 Method of Solution and Limitations
	3.3 Homogeneous Differential Equations and Boundary Conditions
	3.4 Sturm-Liouville Boundary-Value Problem: Orthogonality [1]
	3.5 Procedure for the Application of Separation of Variables Method
	3.6 Cartesian Coordinates: Examples
	3.7 Cylindrical Coordinates: Examples
	3.8 Integrals of Bessel Functions
	3.9 Nonhomogeneous Differential Equations
	3.10 Nonhomogeneous Boundary Conditions: The Method of Superposition
	REFERENCES
	PROBLEMS
4 TRANSIENT CONDUCTION
	4.1 Simplified Model: Lumped-Capacity Method
		4.1.1 Criterion for Neglecting Spatial Temperature Variation
		4.1.2 Lumped-Capacity Analysis
	4.2 Transient Conduction in Plates
	4.3 Nonhomogeneous Equations and Boundary Conditions
	4.4 Transient Conduction in Cylinders
	4.5 Transient Conduction in Spheres
	4.6 Time-Dependent Boundary Conditions: Duhamel’s Superposition Integral
		4.6.1 Formulation of Duhamel’s Integral [1]
		4.6.2 Extension to Discontinuous Boundary Conditions
		4.6.3 Applications
	4.7 Conduction in Semi-Infinite Regions: The Similarity Transformation Method
	REFERENCES
	PROBLEMS
5 CONDUCTION IN POROUS MEDIA
	5.1 Examples of Conduction in Porous Media
	5.2 Simplified Heat Transfer Model
		5.2.1 Porosity
		5.2.2 Heat Conduction Equation: Cartesian Coordinates
		5.2.3 Boundary Conditions
		5.2.4 Heat Conduction Equation: Cylindrical Coordinates
	5.3 Applications
	REFERENCE
	PROBLEMS
6 CONDUCTION WITH PHASE CHANGE: MOVING BOUNDARY PROBLEMS
	6.1 Introduction
	6.2 The Heat Equations
	6.3 Moving Interface Boundary Conditions
	6.4 Nonlinearity of the Interface Energy Equation
	6.5 Nondimensional Form of the Governing Equations: Governing Parameters
	6.6 Simplified Model: Quasi-Steady Approximation
	6.7 Exact Solutions
		6.7.1 Stefan’s Solution
		6.7.2 Neumann’s Solution: Solidification of Semi-Infinite Region
		6.7.3 Neumann’s Solution: Melting of Semi-Infinite Region
	6.8 Effect of Density Change on the Liquid Phase
	6.9 Radial Conduction with Phase Change
	6.10 Phase Change in Finite Regions
	REFERENCES
	PROBLEMS
7 NONLINEAR CONDUCTION PROBLEMS
	7.1 Introduction
	7.2 Sources of Nonlinearity
		7.2.1 Nonlinear Differential Equations
		7.2.2 Nonlinear Boundary Conditions
	7.3 Taylor Series Method
	7.4 Kirchhoff Transformation
		7.4.1 Transformation of Differential Equations
		7.4.2 Transformation of Boundary Conditions
	7.5 Boltzmann Transformation
	7.6 Combining Boltzmann and Kirchhoff Transformations
	7.7 Exact Solutions
	REFERENCES
	PROBLEMS
8 APPROXIMATE SOLUTIONS: THE INTEGRAL METHOD
	8.1 Integral Method Approximation: Mathematical Simplification
	8.2 Procedure
	8.3 Accuracy of the Integral Method
	8.4 Application to Cartesian Coordinates
	8.5 Application to Cylindrical Coordinates
	8.6 Nonlinear Problems [5]
	8.7 Energy Generation
	REFERENCES
	PROBLEMS
9 PERTURBATION SOLUTIONS
	9.1 Introduction
	9.2 Solution Procedure
	9.3 Examples of Perturbation Problems in Conduction
	9.4 Perturbation Solutions: Examples
	9.5 Useful Expansions
	REFERENCES
	PROBLEMS
10 HEAT TRANSFER IN LIVING TISSUE
	10.1 Introduction
	10.2 Vascular Architecture and Blood Flow
	10.3 Blood Temperature Variation
	10.4 Mathematical Modeling of Vessels-Tissue Heat Transfer
		10.4.1 Pennes Bioheat Equation [1]
		10.4.2 Chen-Holmes Equation [5]
		10.4.3 Three-Temperature Model for Peripheral Tissue [7]
		10.4.4 Weinbaum-Jiji Simplified Bioheat Equation for Peripheral Tissue [8]
		10.4.5 The s-Vessel Tissue Cylinder Model [16]
	REFERENCES
	PROBLEMS
11 NUMERICAL SOLUTIONS USING MATLAB
	11.1 Introduction
		11.1.1 Categories of Heat Conduction Problems from a Mathematical Perspective
		11.1.2 Purpose and Scope of this Chapter
	11.2 Numerical Solution to Differential Equations using Finite Difference Methods
	11.3 Numerical Solution to Differential Equations using Finite Element Methods
	11.4 IVP Problems
		11.4.1 ODE Solvers in MATLAB
		11.4.2 Examples with ODE Solvers in MATLAB
	11.5 1D-BVP Problems
		11.5.1 BVP Solvers in MATLAB
		11.5.2 1D Steady Conduction in Multiple Domains
		11.5.3 1D Steady Radial Conduction in Solid Cylinders and Spheres
		11.5.4 1D Steady Conduction in a Semi-Infinite Domain
		11.5.5 1D Steady Conduction with an Unknown Parameter
		11.5.6 Limitations of MATLAB’s BVP Solver
	11.6 1D-PDE Problems
		11.6.1 PDE Solver in MATLAB
		11.6.2 1D Transient Conduction in a Finite-Length Rectangular Slab
		11.6.3 1D Transient Conduction in Cylindrical and Spherical Domains
		11.6.4 Stefan and Neumann’s Solutions
		11.6.5 Limitations of MATLAB’s PDE Solver
	11.7 Multidimensional BVP and PDE Problems
		11.7.1 Multidimensional BVP and PDE Solver in MATLAB
		11.7.2 2D Steady Conduction in a Finite-Length Rectangular Slab
		11.7.3 2D Steady Axisymmetric Conduction in a Cylinder
		11.7.4 Limitations of the Partial Differential Equation Toolbox
	REFERENCES
	PROBLEMS
12 MICROSCALE CONDUCTION
	12.1 Introduction
		12.1.1 Categories of Microscale Phenomena
		12.1.2 Purpose and Scope of this Chapter
	12.2 Understanding the Essential Physics of Thermal Conductivity using the Kinetic Theory of Gases
		12.2.1 Derivation of Fourier’s Law and an Expression for the Thermal Conductivity
	12.3 Energy Carriers
		12.3.1 Ideal Gases: Heat is Conducted by Gas Molecules
		12.3.2 Metals: Heat is Conducted by Electrons
		12.3.3 Electrical Insulators and Semiconductors: Heat is Conductedby Phonons (Sound Waves)
		12.3.4 Radiation: Heat is Carried by Photons (Light Waves)
	12.4 Thermal Conductivity Reduction by Boundary Scattering: The Classical Size Effect
		12.4.1 Accounting for Multiple Scattering Mechanisms: Matthiessen’s rule
		12.4.2 Boundary Scattering for Heat Flow Parallel to Boundaries
		12.4.3 Boundary Scattering for Heat Flow Perpendicular to Boundaries
	12.5 Closing Thoughts
	REFERENCES
	PROBLEMS
APPENDIX A ORDINARY DIFFERENTIAL EQUATIONS
	A.1 Second-Order Ordinary Differential Equations with Constant Coefficients
	A.2 First-Order Ordinary Differential Equations with Variable Coefficients
	REFERENCES
APPENDIX B INTEGRALS OF BESSEL FUNCTIONS
APPENDIX C VALUES OF BESSEL FUNCTIONS
APPENDIX D FUNDAMENTAL PHYSICAL CONSTANTS AND MATERIAL PROPERTIES
	D.1 Fundamental Physical Constants
	D.2 Unit Conversions
	D.3 Properties of Helium Gas
	D.4 Properties of Copper at 300 K
	D.5 Properties of Fused Silica (Amorphous Silicon Dioxide) at 300 K
	D.6 Properties of Silicon
	D.7 Measured Thermal Conductivity of a 56-nm-Diameter Silicon Nanowire at Selected Temperatures [5]
	D.8 Calculated Thermal Conductivity of Single-Walled Carbon Nanotubes, Selected Values [6]
	REFERENCES
APPENDIX E INTRODUCTION TO MATLAB
	E.1 Basics of MATLAB
		E.1.1 Syntax for Useful Functions
		E.1.2 Matrices and Matrix Operations
	E.2 Loops and Conditional Statements
		E.2.1 Loops
		E.2.2 Conditional Statements
	E.3 Finding Roots of an Equation
	E.4 Differentiation and Integration
		E.4.1 Differentiation
		E.4.2 Integration
	E.5 Plotting
INDEX