Computational Methods for Heat and Mass Transfer: 1st Edition (Hardback) book cover

Computational Methods for Heat and Mass Transfer

1st Edition

By Pradip Majumdar, Pradip Majumdar

CRC Press

744 pages | 324 B/W Illus.

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Hardback: 9781560329947
pub: 2005-09-28
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The advent of high-speed computers has encouraged a growing demand for newly graduated engineers to possess the basic skills of computational methods for heat and mass transfer and fluid dynamics. Computational fluid dynamics and heat transfer, as well as finite element codes, are standard tools in the computer-aided design and analysis of processes and products involving coupled transport and multi-physic phenomena. This textbook introduces the fundamentals of two important computational techniques for solving heat and mass transfer and fluid flow problems: finite difference and finite element methods.

The objective of the book is to help the students thoroughly understand the basic concepts and procedures of fluid dynamics, heat and mass transfer and implement computational methodology into a computer code and solve more complex problems on their own. Theory and practice are combined in a simple and straightforward manner. Classic problems in heat transfer, mass transfer, and fluid flows are solved and illustrated through step-by-step derivations and numerous figures. End-of-chapter problems are provided at the end of every chapter for extra practice and homework assignments.

The book is divided into three parts: Part One contains a review of basic equations of heat transfer, mass transfer and fluid dynamics; concepts of numerical approximations and errors; numerical solution techniques for systems of linear algebraic equations; and numerical integrations and quadrature formulas. (The last two topics are included primarily for students who have had no prior course on numerical analysis). Part Two introduces the finite difference/control volume method. Part Three presents the finite element method.

As an introductory text, this book is appropriate for senior undergraduate and first-year graduate level courses. Students taking independent study can use the text as a comprehensive reference guide. Others who will find it a useful resource include practicing engineers and scientists developing their own codes and using commercial codes for the analysis and design of processes and products involving heat and mass transfer and fluid dynamics.


""This extensive work… contains more than an adequate amount of information for two courses at this level. …The classical principles and methods are presented in a clear and detailed manner, and the many examples are helpful in explaining the methods."."

–Choice, March 2006

Table of Contents



Part I: Basic Equations and Numerical Analysis

1. Review of Basic Laws and Equations

1.1 Basic Equations

1.2 Fluid Flows

1.2.1 Fluid Properties

1.2.2 Basic Equations in Integral Forms

1.2.3 Differential Analysis of Fluid Motion

1.2.4 Boundary Conditions for Flow Field

1.3 Heat Transfer

1.3.1 Basic Modes and Transport Rate Equation

1.3.2 The First Law of Thermodynamics and Heat Equations

1.3.3 Thermal Initial and Boundary Conditions

1.4 Mass Transfer

1.4.1 Basic Modes and Transport Rate Equation

1.4.2 Conservation of Mass Species and Mass Concentration Equation

1.4.3 Initial and Boundary Conditions for Mass Transfer

1.5 Mathematical Classification of Governing Equation


2. Approximations and Errors

2.1 Truncation Error

2.2 Round off Error

2.2.1 Significant Figures or Digits

2.2.2 Computers Number System

2.2.3 Machine Epsilon

2.3 Error Definitions

2.4 Approximate Error

2.5 Convergence Criteria


3. Numerical Solution of Systems of Equations

3.1 Mathematical Background

3.1.1 Representation of the System of Equations

3.1.2 The Cramer's Rule and the Elimination of Unknowns

3.2 Direct Methods

3.2.1 Gauss Elimination

3.2.2 Gauss-Jordon Elimination Method

3.2.3 Decomposition of Factorization Methods

3.2.4 Banded Systems

3.2.5 Error Equations and Iterative Refinement

3.3 Iterative Methods

3.3.1 Jacobi Method

3.3.2 Gauss-Seidel Method

3.3.3 Convergence Criterion for Iterative Methods

3.3.4 Successive Over-Relaxation (SOR) method

3.3.5 Conjugate Gradient Method

3.3.6 Pre-conditioned Conjugate Gradient Method


4. Numerical Integration

4.1 Newton - Cotes Integration Formulas

4.1.1 The Trapezoidal Rule

4.1.2 Simpson's Integration Formula

4.1.3 Summary of Newton-Cotes Integration Formulas

4.2 Romberg Integration

4.3 Gauss Quadrature

4.3.1 Two-point Gauss-Legendre Formula

4.3.2 Higher-Points Gauss-Legendre Formulas

4.4 Multi-Dimensional Numerical Integration


Part II: Finite Difference - Control Volume Method

5. Basics of Finite Difference and Control Volume Method

5.1 Introduction and Basic steps in Finite Difference Method

5.2 Discretization of the Domain

5.3 Discretization of the Mathematical Model

5.3.1 The Taylor Series Method

5.3.2 Control Volume Method

5.4 One-dimensional Steady State Diffusion

5.5 Variable Source Term

5.6 Boundary Conditions

5.7 Grid Size Distribution

5.8 Non-uniform Transport Property

5.9 Nonlinearity

5.10 Linearization of a Variable Source Term


6. Multi-Dimensional Problems

6.1 Two-dimensional Steady State Problems

6.2 Boundary Conditions

6.2.1 Corner Boundary Nodes

6.3 Irregular Geometries

6.4 Three Dimensional Steady State Problems


7. Diffusion Equation

7.1 Time Discretization Procedure

7.2 Explicit Scheme

7.2.1 Discretization by Control Volume Approach

7.2.2 Finite Difference Equations by Taylor Series Expansions

7.2.3 Stability Consideration

7.2.4 Other Explicit Scheme

7.2.5 Boundary Conditions

7.3 Implicit Scheme

7.3.1 Discretization Equation by Control Volume Approach

7.3.2 Finite Difference Equation by Taylor Series Expansion

7.3.3 A General Formulation of Fully-Implicit Scheme for One-dimensional Problems

7.3.4 A General Formulation of Fully-Implicit Scheme for Two-dimensional Problems

7.3.5 Solution Methods for Two-dimensional Implicit Scheme

7.3.6 Boundary Conditions for Implicit Scheme

7.4 Crank-Nicolson Scheme

7.4.1 Solution Method for Crank-Nicolson Scheme

7.5 Splitting Method

7.5.1 ADI Method

7.5.2 ADE Method


8. Finite Difference-Control Volume Method: Convection Heat Transfer

8.1 Spatial Discretization using Control Volume Method

8.1.1 Central Difference Scheme

8.1.2 Upwind Scheme

8.1.3 Exponential Scheme

8.1.4 Hybrid Scheme

8.1.5 Power Law Scheme

8.1.6 Generalized Convection-Diffusion Scheme

8.2 Discretization of a General Transport Equations

8.2.1 One-dimensional Unsteady State Problem

8.2.2 Two-dimensional Unsteady State Problem

8.2.3 Three-dimensional Unsteady State Problem

8.3 Solution of Flow Field

8.3.1 Stream Function/Vorticity-Based Method

8.3.2 Direct Solution with the Primitive Variable


Part III: Finite Element Method

9. Introduction and Basic Steps in Finite Element Method

9.1 Comparison of Finite Difference/Control Volume Method and Finite Element Method

9.2 Basic Steps in Finite Element Methods

9.3 Integral Formulation

9.3.1 Variational Formulations

9.3.2 Method of Weighted Residuals

9.4 Variational Methods

9.4.1 The Rayleigh-Ritz Variational Method

9.4.2 Weighted Residual Variational Methods


10. Element Shape Functions

10.1 One-dimensional Element

10.1.1 One-dimensional Linear Element

10.1.2 One-dimensional Quadratic Line Element

10.1.3 One-dimensional Cubic Element

10.2 Two-dimensional Element

10.2.1 Linear Triangular Element

10.2.2 Quadratic Triangular Element

10.2.3 Two-dimensional Quadrilateral Element

10.3 Three-dimensional Element

10.3.1 Three-dimensional Tetrahedron Element

10.3.2 Three-dimensional Hexahedron Element


11. Finite Element Method: One-dimensional Steady State Problems

11.1 Finite Element Formulation using Galerkin Method

Discretization of the Solution Domain

Selection of Approximation Solution Function

Formation of Integral Statement of the Problem

Formation of Element Characteristics Equation

Assembly of Element Equations to Form the Global System

Implementation of Boundary Conditions

Solution of Global System

11.2 Finite Element Formulation using Variational Approach

Formation of the Integral Statement of Variational Form

Formation of Element Characteristics Equation

11.3 Boundary Conditions

11.3.1 Boundary Conditions of the First Kind or Constant Surface Flux

11.3.2 Mixed Boundary Conditions

11.3.3 Variable Source Term

11.3.4 Axisymmetric Problems


12. Finite Element Method: Multi-dimensional Steady State Problems

12.1 Two-dimensional Steady State Diffusion Equation

Mesh Generation or Discretization of Solution Domain

Element and node Numbering

Selection of Approximation Solution Function

Formulation of an Integral Statement using Galerkin Approach

Formation of Element Characteristics Equations

Assembly of Element Equations and Formation of Global System

12.2 Two-dimensional Axisymmetric Problems

Selection of Approximate Solution Function

Formation of an Integral Statement

Formation of Element Characteristics Equations

Implementation of Boundary Conditions

Axisymmetric Element

12.3 Three-dimensional Problems

12.4 FE Formulation Using Variational Approach

12.5 Point Source


13. Finite Element Method: Unsteady State Problems

13.1 Discretization Scheme

13.2 One-dimensional Unsteady State Problems

13.2.1 Semi-discrete Finite Element Formulations

13.2.2 Time Approximations

13.2.3 Stability Considerations

13.3 Two-dimensional Unsteady State Diffusion Equation

13.4 Three-dimensional Unsteady State Diffusion Equation


14. Finite Element Method: Convection Heat Transfer

14.1 Classification of Finite Elements Methods for Convection Problems

14.2 Velocity-Pressure or Mixed Method

14.2.1 One dimensional Convection-Diffusion Problem

14.2.2 Two-dimensional Viscous Incompressible Flow

14.2.3 Unsteady Two-dimensional Viscous Incompressible Flow

14.2.4 Unsteady Two-dimensional Viscous Incompressible Flow

14.2.5 Convection Problems

14.3 Solution Methods

14.3.1 Steady State Problems

14.3.2 Unsteady State Problem


Appendix A. Review of Vectors and Matrices

Appendix B. Integral Theorems



About the Series

Series in Computational and Physical Processes in Mechanics and Thermal Sciences

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Subject Categories

BISAC Subject Codes/Headings:
SCIENCE / Mechanics / Dynamics / Thermodynamics