# Mathematical Principles of Heat Transfer

K. N. Shukla
Karunya Institute of Technology and Sciences Coimbatore-641114, India

## Description

This book presents an investigative account of Mathematical Principles of Heat Transfer. It is concerned with three aspects of heat transfer analysis: theoretical development of conservation equations, analytical and numerical techniques of the solution, and the physical processes involved in the three basic modes of heat transfer, namely conduction, convection, and radiation. A concept of mathematical modeling is developed through the use of differential equations. In doing so, the well-posed boundary value problems are constructed and the solutions are attempted.

© 2005

## Table of Contents:

Chapter 1 Basic Concepts of Heat Transfer

1.1 Basic Modes of Heat Transfer

1.1.2 Temperature gradient

1.3.1 Thermal conductivities of gases

1.3.2 Thermal conductivities of liquids

1.3.3 Thermal conductivities of solids

1.6 Heat Transfer with Change of Phase

Chapter 2 Conservation Equations

2.1 General Conservation Equation

2.2 Equation of Continuity

Chapter 3 Similarity Theory and the Generalized Variables

3.1 Boundary Value Problem in Generalized Variables: A Mathematical Presentation

3.2 Method of Dimensionality

Chapter 4 Mathematical Methods for Boundary Value Problems

4.1 Separation of Variables

4.2 Integral Transform Method

4.3.1 Laplace transform-fundamental properties

4.3.2 Inversion theorem for Laplace transform

4.6 Finite Integral Fourier and Hankel Transforms

4.7 Limiting Cases of Laplace Transform

4.8 Green's Function for the Solution of Heat Conduction

4.9 Approximate Methods in the Solution of Heat Transfer Problems

4.9.5 Least-squares method

Chapter 5 Numerical Methods in Heat Transfer

5.1 Finite Difference Method

5.2 Gauss Elimination Method

5.3 Gauss-Seidel Method of Iteration

5.4 Successive Overrelaxation Method

5.5 Derivative Type of Boundary Conditions

5.6 Stability: Analytical Treatment

5.7 Convergence: Analytical Treatment

5.9 Two-Dimensional Problem of Heat Conduction

5.9.1 Locally one-dimensional method

Chapter 6 Steady-State Heat Conduction

6.1 Heat Transfer in a Slab

6.1.1 Dirichlet boundary conditions

6.1.2 Temperature-dependent thermal conductivity

6.1.4 Newtonian boundary conditions

6.1.5 Mixed boundary conditions

6.2 Heat Transfer through a Cylindrical Wall

6.2.1 Dirichlet boundary conditions

6.2.2 Temperature-dependent thermal conductivity

6.2.3 Composite cylindrical wall

6.2.4 Newtonian boundary conditions

6.2.5 Mixed boundary conditions

6.3 Heat Transfer through a Spherical Wall

6.3.1 Dirichlet boundary conditions

6.3.2 Temperature-dependent thermal conductivity

6.4 Critical Thickness of Insulation

6.5 Heat Conduction through a Thin Rod

6.6.2 Rectangular profile

6.6.4 Rectangular fin of minimum weight

6.6.5 Efficiency of the fin

6.6.6 Longitudinal fin of triangular profile

6.7 Steady Heat Flow in a Rectangle

6.7.1 Conjugate functions

Chapter 7 Transient Heat Conduction

7.5 Periodic Variation of Ambient Temperature

Chapter 8 Heat Conduction with Phase Change

8.1 Statement of Problem and Existence of Solution

8.2 State-of-the-Art Solution Technique

8.3 Solidification of a Semi-Infinite Liquid

8.6 Dynamics of Melt Growth and Axisymmetric Melting

9.1 Hydrodynamics and Thermal Boundary Layers

9.2 Similarity Solution for Boundary Layers

9.2.1 Skin friction and heat transfer

9.2.2 Laminar boundary layers by the integral method

9.3 Similarity Solution for Boundary Layers, u = cxm

9.4 Finite Difference Solution

9.4.1 Transient solution for a pipe flow

9.5.1 Natural convection from an isothermal vertical plate

Chapter 10 Diffusive Processes

10.1 Conservation Equations

10.2 Isothermal Ternary System

10.3 Nonisothermal Binary System

10.4 Thermohaline Convection

10.4.1 Free boundaries with specified solute concentration and temperature

10.4.2 Diffusive convection

10.6 Diffusion Processes in Material Processing and Microgravity Environments

Chapter 11 Radiation Heat Transfer

11.4 Environmental Radiation

11.5 Radiant Interchange between Surfaces Separated by a Nonparticipating Medium

11.6.1 Evaluation of integral

11.6.2 Contour integral representation

11.7 Electrical Network Analog for an Enclosure

11.8 Enclosures with Diffuse Gray Surfaces

11.9 Enclosure with Specularly Reflecting Surfaces

11.9.1 Solution for radiative transfer

11.10 Radiative Transfer in a Plane Layer

11.12 Radiation with Conduction

11.12.3 Optically thick limit: The diffusion approximation

B.1 Configuration factor for some common surfaces