2nd Edition
Heat Conduction Using Green's Functions
Since its publication more than 15 years ago, Heat Conduction Using Green’s Functions has become the consummate heat conduction treatise from the perspective of Green’s functions—and the newly revised Second Edition is poised to take its place. Based on the authors’ own research and classroom experience with the material, this book organizes the solution of heat conduction and diffusion problems through the use of Green’s functions, making these valuable principles more accessible. As in the first edition, this book applies extensive tables of Green’s functions and related integrals, and all chapters have been updated and revised for the second edition, many extensively.
Details how to access the accompanying Green’s Function Library site, a useful web-searchable collection of GFs based on the appendices in this book
The book reflects the authors’ conviction that although Green’s functions were discovered in the nineteenth century, they remain directly relevant to 21st-century engineers and scientists. It chronicles the authors’ continued search for new GFs and novel ways to apply them to heat conduction.
New features of this latest edition—
- Expands the introduction to Green’s functions, both steady and unsteady
- Adds a section on the Dirac Delta Function
- Includes a discussion of the eigenfunction expansion method, as well as sections on the convergence speed of series solutions, and the importance of alternate GF
- Adds a section on intrinsic verification, an important new tool for obtaining correct numerical values from analytical solutions
A main goal of the first edition was to make GFs more accessible. To facilitate this objective, one of the authors has created a companion Internet site called the Green’s Function Library, a web-searchable collection of GFs. Based on the appendices in this book, this library is organized by differential equation, geometry, and boundary condition. Each GF is also identified and cataloged according to a GF numbering system. The library also contains explanatory material, references, and links to related sites, all of which supplement the value of Heat Conduction Using Green’s Functions, Second Edition as a powerful tool for understanding.
Introduction to Green’s Functions
Heat Flux and Temperature
Differential Energy Equation
Boundary and Initial Conditions
Integral Energy Equation
Dirac Delta Function
Steady Heat Conduction in One Dimension
GF in the Infinite One-Dimensional Body
Temperature in an Infinite One-Dimensional Body
Two Interpretations of Green’s Functions
Temperature in Semi-Infinite Bodies
Flat Plates
Properties Common to Transient Green’s Functions
Heterogeneous Bodies
Anisotropic Bodies
Transformations
Non-Fourier Heat Conduction
Numbering System in Heat Conduction
Geometry and Boundary Condition Numbering System
Boundary Condition Modifiers
Initial Temperature Distribution
Interface Descriptors
Numbering System for g(x, t)
Examples of Numbering System
Advantages of Numbering System
Derivation of the Green’s Function Solution Equation
Derivation of the One-Dimensional Green’s Function Solution Equation
General Form of the Green’s Function Solution Equation
Alternative Green’s Function Solution Equation
Fin Term m2T
Steady Heat Conduction
Moving Solids
Methods for Obtaining Green’s Functions
Method of Images
Laplace Transform Method
Method Of Separation of Variables
Product Solution for Transient GF
Method of Eigenfunction Expansions
Steady Green’s Functions
Improvement of Convergence and Intrinsic Verification
Identifying Convergence Problems
Strategies to Improve Series Convergence
Intrinsic Verification
Rectangular Coordinates
One-Dimensional Green’s Functions Solution Equation
Semi-Infinite One-Dimensional Bodies
Flat Plates: Small-Cotime Green’s Functions
Flat Plates: Large-Cotime Green’s Functions
Flat Plates: The Nonhomogeneous Boundary
Two-Dimensional Rectangular Bodies
Two-Dimensional Semi-Infinite Bodies
Steady State
Cylindrical Coordinates
Relations for Radial Heat Flow
Infinite Body
Separation of Variables for Radial Heat Flow
Long Solid Cylinder
Hollow Cylinder
Infinite Body with a Circular Hole
Thin Shells, T = T (φ, t)
Limiting Cases for 2D and 3D Geometries
Cylinders with T = T (r, z, t )
Disk Heat Source on a Semi-Infinite Body
Bodies with T = T (r, φ, t )
Steady State
Radial Heat Flow in Spherical Coordinates
Green’s Function Equation for Radial Spherical Heat Flow
Infinite Body
Separation of Variables for Radial Heat Flow in Spheres
Temperature in Solid Spheres
Temperature in Hollow Spheres
Temperature in an Infinite Region Outside a Spherical Cavity
Steady State
Steady-Periodic Heat Conduction
Steady-Periodic Relations
One-Dimensional GF
One-Dimensional Temperature
Layered Bodies
Two- and Three-Dimensional Cartesian Bodies
Two-Dimensional Bodies in Cylindrical Coordinates
Cylinder with T = T (r, φ, z,ω)
Galerkin-Based Green’s Functions and Solutions
Green’s Functions and Green’s Function Solution Method
Alternative form of the Green’s Function Solution
Basis Functions and Simple Matrix Operations
Fins and Fin Effect
Conclusions
Applications of the Galerkin-Based Green’s Functions
Basis Functions in some Complex Geometries
Heterogeneous Solids
Steady-State Conduction
Fluid Flow in Ducts
Conclusion
Unsteady Surface Element Method
Duhamel’s Theorem and Green’s Function Method
Unsteady Surface Element Formulations
Approximate Analytical Solution (Single Element)
Examples
Problems
References
Appendices
Index
Biography
Kevin D. Cole received his M.S. in aerospace engineering and mechanics from the University of Minnesota in 1979 and his Ph.D. in mechanical engineering from Michigan State University in 1986. Dr. Cole has held several positions in academia and industry and is currently Associate Professor of Mechanical Engineering at the University of Nebraska—Lincoln. Dr. Cole is active in writing and reviewing in the areas of heat conduction and thermal measurements. He is the creator of the Green’s Function Library internet site.
James V. Beck received his S.M. in mechanical engineering from MIT in 1957 and his Ph.D. from Michigan State University in 1964. Dr. Beck is currently Professor Emeritus of Mechanical Engineering at Michigan State University. He has been honored with the MSU Distinguished Faculty Award and the ASME Heat Transfer Memorial Award. He is the originator of the Inverse Problems Symposium and is the inventor, with Professor Litkouhi of the numbering system for heat conduction solutions. Dr. Beck has contributed to the field of heat transfer with numerous refereed journal articles and books.
A. Haji-Sheikh received his M.S. in M.E., M.A. in Math from the University of Michigan and a Ph.D. in 1965 from the University of Minnesota. In 1966, he joined the Department of Mechanical Engineering at the University of Texas at Arlington, and is currently Professor and member of the Distinguished Scholars Academy. His contributions to heat conduction include the floating random walk in Monte Carlo method, Green’s function in two-ste models, inverse problems, and Galerkin-based integral methods. He is a registered PE in the state of Texas, a Fellow of ASME, a recipient of the ASME Memorial Award in Science.
Bahman Litkouhi received his M.S. and Ph.D. from Michigan State University and is presently Professor and Graduate Program Director of the Mechanical Engineering Department at Manhattan College. Dr. Litkouhi is a registered professional engineer in the state of New York and a member of the American Society of Mechanical Engineers. He has authored several technical publications in heat transfer and has served as an industria consultant.
