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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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