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Heat Conduction.pdf
Title Page
PREFACE
CONTENTS
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 Equation 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
Problem Solving Format
Units
REFERENCES
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 $\frac {dA_{s}{dx}}$
Boundary Conditions
Determination of Fin Heat Transfer Rate $q_{f}$
Steady State Applications: Constant Area Fins with Surface Convection
Corrected Length $L_{c}$
Fin Efficiency $\eta_{f}$
Moving Fins
Application of Moving Fins
Variable Area Fins
Bessel Differential Equations and Bessel Functions
General Form of Bessel Equations
Solutions: Bessel Functions
Special Closed-form Bessel Functions:$n = \frac {odd integer}{2}$
Special Relations for n = 1, 2, 3, ….
Derivatives and Integrals of Bessel Functions [2,3]
Tabulation and Graphical Representation of Selected Bessel Functions
Equidimensional (Euler) Equation
Graphically Presented Solutions to Fin Heat Transfer Rate [5]
REFERENCES
TWO-DIMENSIONAL STEADY STATE CONDUCTION
The Heat Conduction Equation
Method of Solution and Limitations
Homogeneous Differential Equations and Boundary Conditions
Sturm-Liouville Boundary-Value Problem: Orthogonality [1]
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
REFERENCES
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 [1]
Extension to Discontinuous Boundary Conditions
Applications
Conduction in Semi-infinite Regions: The Similarity Method
REFERENCES
CONDUCTION IN 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
REFEENCES
CONDUCTION WITH PHASE CHANGE: MOVING BOUNDARY PROBLEMS
Introduction
The Heat Equations
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
REFERENCES
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
REFERENCES
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 [5]
Energy Generation
REFERENCES
PERTURBATION SOLUTIONS
Introduction
Solution Procedure
Examples of Perturbation Problems in Conduction
Perturbation Solutions: Examples
Useful Expansions
REFERENCES
Heat Transfer in Living Tissue
Introduction
Vascular Architecture and Blood Flow
Blood Temperature Variation
Mathematical Modeling of Vessels-Tissue Heat Transfer
Pennes Bioheat Equation [1]
Chen-Holmes Equation [5]
Three-Temperature Model for Peripheral Tissue [7]
Weinbaum-Jiji Simplified Bioheat Equation for Peripheral Tissue [8]
The $s$-Vessel Tissue Cylinder Model [16]
REFERENCES
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
Derivation of Fourier’s Law and an Expression for the Thermal Conductivity
Energy Carriers
Ideal Gases: 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
REFERENCES
APPENDIX A: ORDINARY DIFFERENTIAL EQUATIONS
APPENDIX B INTEGRALS OF BESSEL FUNCTIONS
APPENDIX C: Values of Bessel Functions
APPENDIX D FUNDAMENTAL PHYSICAL CONSTANTS AND MATERIAL PROPERTIES
INDEX
Heat Conduction
Latif M. Jiji
Heat Conduction
Third Edition
ABC
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 10031
USA
E-Mail: jiji@ccny.cuny.edu
''Additional material to this book can be downloaded from http://extra.springer.com ''
ISBN 978-3-642-01266-2
e-ISBN 978-3-642-01267-9
DOI 10.1007/978-3-642-01267-9
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 mate-
rial 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. Dupli-
cation 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 5 4 3 2 1 0
springer.com
This book is dedicated to my wife Vera for opening
many possibilities and providing balance in my life.
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 one-
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 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
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
CONTENTS
Preface vii
CHAPTER 1: BASIC CONCEPTS
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
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
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
Problem Solving Format
Units
REFERENCES
PROBLEMS
1
1
2
2
5
9
10
10
11
12
15
16
17
18
CHAPTER 2: ONE-DIMENSIONAL STEADY-STATE
CONDUCTION 24
2.1
2.2
Examples of One-dimensional Conduction
Extended Surfaces: Fins
2.2.1 The Function of Fins
2.2.2 Types of Fins
2.2.2 Heat Transfer and Temperature Distribution in
Fins
2.2.4 The Fin Approximation
2.2.5 The Fin Heat Equation: Convection at Surface
dAs /
2.2.6 Determination of
2.2.7 Boundary Conditions
2.2.8 Determination of Fin Heat Transfer Rate
2.2.9 Steady State Applications: Constant Area Fins
dx
with Surface Convection
fq
24
34
34
34
35
36
37
39
40
40
41
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