Heat Conduction Using Green's Functions (Second Edition) by Beck James V., Cole Kevin D., Haji Sheikh A., Litkouhi Bahman
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Heat Conduction Using Green's Functions (Second Edition) written by
Beck James V. and
Cole Kevin D. and
Haji Sheikh A. and
Litkouhi Bahman .
The purpose of this book is to simplify and organize the solution of heat conduction
and diffusion problems and to make them more accessible. This is accomplished
using the method of Green’s functions, together with extensive tables of Green’s
functions and related integrals. The tables of Green’s functions were first compiled
as a supplement to a first-year graduate course in heat conduction taught at Michigan
State University. The book was originally envisioned as a reference volume, but it
has grown into a heat conduction treatise from a Green’s function perspective.
There is enough material for a one-semester course in analytical heat conduction
and diffusion. There are worked examples and student problems to aid in teaching.
Because of the emphasis on Green’s functions, some traditional topics such as Fourier
series and Laplace transform methods are treated somewhat briefly; this material could
be supplemented according to the interest of the instructor. The book can also be used
as a supplementary text in courses on heat conduction, boundary value problems, or
partial differential equations of the diffusion type.
We hope the book will be used as a reference for practicing engineers, applied
mathematicians, physicists, geologists, and others. In many cases, a heat conduction
or diffusion solution may be assembled from tabulated Green’s functions rather than
derived. The book contains the most extensive set of Green’s functions and related
integrals that is currently available for heat conduction and diffusion.
The book is organized on a geometric basis because each Green’s function is associated
with a unique geometry. For each of the three coordinate systems—Cartesian,
cylindrical, and spherical—there is a separate appendix of Green’s functions named
Appendix X, Appendix R, and Appendix RS, respectively. Each of the Green’s functions
listed is identified by a unique alphanumeric character that begins with either
X, R, or RS to denote the x, r, or the spherical r coordinate, respectively. It is important
for the reader to know something about this numbering system to use the tables
of Green’s functions.Amore detailed numbering system, which covers both Green’s
functions and temperature solutions, is discussed in Chapter 2.We find the numbering
system very helpful in identifying exactly which solution is under discussion, and all
of the solutions discussed in the text are listed in Appendix N indexed according to
the numbering system.
The level of treatment is intended for senior and first-year graduate students in engineering and mathematics.We have emphasized solution of problems rather than theorems and proofs, which are generally omitted.Aprerequisite is an undergraduate course in ordinary differential equations. A previous introduction to the method of separation of variables for partial differential equations is also important.
Heat Conduction Using Green's Functions (Second Edition) written by
Beck James V. and
Cole Kevin D. and
Haji Sheikh A. and
Litkouhi Bahman
cover the following topics.
Preface to the First Edition
Preface to the Second Edition
Authors
Nomenclature
1. Introduction to Green’s Functions
1.1 Introduction
1.1.1 Advantage of the Green’s Function Method
1.1.2 Scope of This Chapter
1.2 Heat Flux and Temperature
1.3 Differential Energy Equation
1.4 Boundary and Initial Conditions
1.5 Integral Energy Equation
1.6 Dirac Delta Function
1.7 Steady Heat Conduction in One Dimension
1.7.1 Solution by Integration
1.7.2 Solution by Green’s Function
1.8 GF in the Infinite One-Dimensional Body
1.8.1 Auxiliary Problem for G
1.8.2 Laplace Transform, Brief Facts
1.8.3 Derivation of the GF
1.9 Temperature in an Infinite One-Dimensional Body
1.9.1 Green’s Function Solution Equation
1.9.2 Fundamental Heat Conduction Solution
1.10 Two Interpretations of Green’s Functions
1.11 Temperature in Semi-Infinite Bodies
1.11.1 Boundary Condition of the First Kind
1.11.2 Boundary Condition of the Second Kind
1.12 Flat Plates
1.12.1 Temperature for Flat Plates
1.12.2 Auxiliary Problem for Flat Plates
1.13 Properties Common to Transient Green’s Functions
1.14 Heterogeneous Bodies
1.15 Anisotropic Bodies
1.16 Transformations
1.16.1 Orthotropic Bodies
1.16.2 Moving Solids
1.16.3 Fin Term
1.17 Non-Fourier Heat Conduction
Problems
References
2. Numbering System in Heat Conduction
2.1 Introduction
2.2 Geometry and Boundary Condition Numbering System
2.3 Boundary Condition Modifiers
2.4 Initial Temperature Distribution
2.5 Interface Descriptors
2.6 Numbering System for g(x, t)
2.7 Examples of Numbering System
2.8 Advantages of Numbering System
2.8.1 Data Base in Transient Heat Conduction
2.8.2 Algebra for Linear Cases
Problems
References
3. Derivation of the Green’s Function Solution Equation
3.1 Introduction
3.2 Derivation of the One-Dimensional Green’s Function Solution Equation
3.3 General Form of the Green’s Function Solution Equation
3.3.1 Temperature Problem
3.3.2 Derivation of the Green’s Function Solution Equation
3.4 Alternative Green’s Function Solution Equation
3.5 Fin Term m2T
3.5.1 Transient Fin Problems
3.5.2 Steady Fin Problems in One Dimension
3.6 Steady Heat Conduction
3.6.1 Relationship between Steady and Transient Green’s Functions
3.6.2 Steady Green’s Function Solution Equation
3.7 Moving Solids
3.7.1 Introduction
3.7.2 Three-Dimensional Formulation
Problems
References
4. Methods for Obtaining Green’s Functions
4.1 Introduction
4.2 Method of Images
4.3 Laplace Transform Method
4.3.1 Definition 106
4.3.2 Temperature Example
4.3.3 Derivation of Green’s Functions
4.4 Method of Separation of Variables
4.4.1 Plate with Temperature Fixed at Both Sides (X11)
4.5 Product Solution for Transient GF
4.5.1 Rectangular Coordinates
4.5.2 Cylindrical Coordinates
4.6 Method of Eigenfunction Expansions
4.7 Steady Green’s Functions
4.7.1 Integration of the Auxiliary Equation: The Source Solutions
4.7.2 Pseudo-Green’s Function for Insulated Boundaries
4.7.3 Limit Method
Problems
References
5. Improvement of Convergence and Intrinsic Verification
5.1 Introduction
5.1.1 Problems Considered in This Chapter
5.1.2 Two Basic Functions
5.1.3 Convergence of the GF
5.2 Identifying Convergence Problems
5.2.1 Convergence Criterion
5.2.2 Monitor the Number of Terms
5.2.3 Slower Convergence of the Derivative
5.3 Strategies to Improve Series Convergence
5.3.1 Replacement of Steady-State Series
5.3.2 Alternate GF Solution Equation
5.3.3 Time Partitioning
5.4 Intrinsic Verification
5.4.1 Intrinsic Verification by Complementary Transients
5.4.2 Complementary Transient and 1D Solution
5.4.3 Intrinsic Verification by Alternate Series
Expansion
5.4.4 Time-Partitioning Intrinsic Verification
Problems
References
6. Rectangular Coordinates
6.1 Introduction
6.2 One-Dimensional Green’s Functions Solution Equation
6.3 Semi-Infinite One-Dimensional Bodies
6.3.1 Initial Conditions
6.3.2 Boundary Conditions
6.3.3 Volume Energy Generation
6.4 Flat Plates: Small-Cotime Green’s Functions
6.4.1 Initial Conditions
6.4.2 Volume Energy Generation
6.5 Flat Plates: Large-Cotime Green’s Functions
6.5.1 Initial Conditions
6.5.2 Plane Heat Source
6.5.3 Volume Energy Generation
6.6 Flat Plates: The Nonhomogeneous Boundary
6.7 Two-Dimensional Rectangular Bodies
6.8 Two-Dimensional Semi-Infinite Bodies
6.8.1 Integral Expression for the Temperature
6.8.2 Special Cases
6.8.3 Series Expression for the Temperature
6.8.4 Application to the Strip Heat Source
6.8.5 Discussion
6.9 Steady State
Problems
References
7. Cylindrical Coordinates
7.1 Introduction
7.2 Relations for Radial Heat Flow
7.3 Infinite Body
7.3.1 The R00 Green’s Function
7.3.2 Derivation of the R00 Green’s Function
7.3.3 Approximations for the R00 Green’s Function
7.3.4 Temperatures from Initial Conditions
7.4 Separation of Variables for Radial Heat Flow
7.5 Long Solid Cylinder
7.5.1 Initial Conditions
7.5.2 Boundary Conditions
7.5.3 Volume Energy Generation
7.6 Hollow Cylinder
7.7 Infinite Body with a Circular Hole
7.8 Thin Shells, T = T (f, t)
7.9 Limiting Cases for 2D and 3D Geometries
7.9.1 Fourier Number
7.9.2 Aspect Ratio
7.9.3 Nonuniform Heating
7.10 Cylinders with T = T (r, z, t)
7.11 Disk Heat Source on a Semi-Infinite Body
7.11.1 Integral Expression for the Temperature
7.11.2 Closed-Form Expressions for the Temperature
7.11.3 Series Expression for the Surface Temperature at Large Times
7.11.4 Average Temperature
7.12 Bodies with T = T (r, f, t)
7.13 Steady State
Problems
References288
8. Radial Heat Flow in Spherical Coordinates
8.1 Introduction
8.2 Green’s Function Equation for Radial Spherical Heat Flow
8.3 Infinite Body
8.3.1 Derivation of the RS00 Green’s Function294
8.4 Separation of Variables for Radial Heat Flow in Spheres
8.5 Temperature in Solid Spheres
8.6 Temperature in Hollow Spheres
8.7 Temperature in an Infinite Region outside a Spherical Cavity
8.8 Steady State
Problems
9. Steady-Periodic Heat Conduction
9.1 Introduction
9.2 Steady-Periodic Relations
9.3 One-Dimensional GF
9.3.1 One-Dimensional GF in Cartesian Coordinates
9.3.2 One-Dimensional GF in Cylindrical Coordinates
9.3.3 One-Dimensional GF in Spherical Coordinates
9.4 One-Dimensional Temperature
9.5 Layered Bodies
9.6 Two- and Three-Dimensional Cartesian Bodies
9.6.1 Rectangles and Slabs
9.6.2 Infinite and Semi-Infinite Bodies
9.6.3 Rectangular Parallelepiped
9.7 Two-Dimensional Bodies in Cylindrical Coordinates
9.7.1 GF with Eigenfunctions along r
9.7.2 GF with Eigenfunctions along z
9.7.3 Axisymmetric Half-Space
9.8 Cylinder with T = T (r, f,z, ?)
9.8.1 GF with Eigenfunctions along z
9.8.2 GF with Eigenfunctions along r
Problems
References
10. Galerkin-Based Green’s Functions and Solutions
10.1 Introduction
10.2 Green’s Functions and Green’s Function Solution Method
10.2.1 Galerkin-Based Integral Method
10.2.2 Numerical Calculation of Eigenvalues
10.2.3 Nonhomogeneous Solution
10.2.4 Green’s Functions Expression
10.2.5 Properties of Green’s Functions
10.2.6 Green’s Function Solution Equation
10.3 Alternative Form of the Green’s Function Solution
10.4 Basis Functions and Simple Matrix Operations
10.4.1 One-Dimensional Bodies
10.4.2 Matrices A and B for One-Dimensional Problems
10.4.3 Matrix Operations When N = 1 and N = 2
10.5 Fins and Fin Effect
10.6 Conclusions
Problems
Note 1: Mathematical Identities
Note 2: A Mathematica Program for Determination of Temperature in Example 10.2
References
11. Applications of the Galerkin-Based Green’s Functions
11.1 Introduction
11.2 Basis Functions in Some Complex Geometries
11.2.1 Boundary Conditions of the First Kind
11.2.2 Boundary Conditions of the Second Kind
11.2.3 Boundary Conditions of the Third Kind
11.3 Heterogeneous Solids
11.4 Steady-State Conduction
11.5 Fluid Flow in Ducts
11.6 Conclusion
Problems
References
12. Unsteady Surface Element Method
12.1 Introduction
12.2 Duhamel’s Theorem and Green’s Function Method
12.2.1 Derivation of Duhamel’s Theorem for Time- and Space-Variable Boundary Conditions
12.2.2 Relation to the Green’s Function Method
12.3 Unsteady Surface Element Formulations
12.3.1 Surface Element Discretization
12.3.2 Green’s Function Form of the USE Equations
12.3.3 Time Integration of the USE Equations
12.3.4 Flux-Based USE Equations for Bodies in Contact
12.3.5 Numerical Solution of the USE Equations for Bodies in Contact
12.3.6 Influence Functions
12.4 Approximate Analytical Solution (Single Element)
12.5 Examples
Problems
Note 1: Derivation of Equations 12.65a and 12.65b
References
Appendix
B Bessel Functions
D Dirac Delta Function
E Error Function and Related Functions
F Functions And Series
i Integrals
L Laplace Transform Method
P Properties of Selected Materials
R Green’s Functions for Radial-Cylindrical Coordinates (r)
RF Green’s Functions for Cylindrical Coordinates (r, f)
F Cylindrical Polar Coordinate, f Thin Shell Case
RS Green’s Functions for Radial Spherical Geometries
X Green’s Functions: Rectangular Coordinates
Index of Solutions by Number System
Author Index
subject Index
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