598 Pages 100 B/W Illustrations
    by CRC Press

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    Finite Difference Methods in Heat Transfer, Second Edition focuses on finite difference methods and their application to the solution of heat transfer problems. Such methods are based on the discretization of governing equations, initial and boundary conditions, which then replace a continuous partial differential problem by a system of algebraic equations. Finite difference methods are a versatile tool for scientists and for engineers. This updated book serves university students taking graduate-level coursework in heat transfer, as well as being an important reference for researchers and engineering.


  • Provides a self-contained approach in finite difference methods for students and professionals
  • Covers the use of finite difference methods in convective, conductive, and radiative heat transfer
  • Presents numerical solution techniques to elliptic, parabolic, and hyperbolic problems
  • Includes hybrid analytical–numerical approaches
  • Basic Relations

    Classification of Second-Order Partial Differential Equations

    Parabolic Systems

    Elliptic Systems

    Hyperbolic Systems

    Systems of Equations

    Boundary Conditions

    Uniqueness of the Solution Problems

    Discrete Approximation of Derivatives

    Taylor Series Formulation

    Finite Difference Operators

    Control-Volume Approach

    Application of Control-Volume Approach

    Boundary Conditions

    Errors Involved in Numerical Solutions Problems

    Methods of Solving Sets of Algebraic Equations

    Reduction to Algebraic Equations

    Direct Methods

    Iterative Methods

    Nonlinear Systems Problems

    One-Dimensional Steady-State Systems

    Diffusive Systems

    Diffusive-Convective System

    Diffusive-Convective System with Flow Problems

    One-Dimensional Parabolic Systems

    Simple Explicit Method

    Simple Implicit Method

    Crank-Nicolson Method

    Combined Method

    Cylindrical and Spherical Symmetry

    A Summary of Finite-Difference Schemes Problems

    Multidimensional Parabolic Systems

    Simple Explicit Method

    (i) Two-Dimensional Diffusion

    (ii) Two-Dimensional Steady Laminar Boundary Layer Flow

    (iii) Two-Dimensional Transient Convection-Diffusion

    Combined Method

    (i) Three-Dimensional Diffusion

    Alternating Direction Implicit (ADI) Method

    Alternating Direction Explicit (ADE) Method

    (i) One-Dimensional Diffusion

    (ii) Two-Dimensional Diffusion

    Modified Upwind Method

    (i) Transient Forced Convection Inside Ducts for Step Change in Fluid Inlet


    Pressure-Velocity Coupling Problems

    Elliptic Systems

    Steady-State Diffusion

    Velocity Field for Incompressible, Constant Property, Two-Dimensional Flow

    Vorticity – Stream Function Formulation


    Hyperbolic Systems

    Hyperbolic Convection (Wave) Equation

    Hyperbolic Heat Conduction Equation

    System of Vector Equations Problems

    Nonlinear Diffusion

    Lagging Properties by One Time Step

    Use of Three-Time Level Implicit Scheme


    Method of False Transients for Solving Steady-State Diffusion

    Simultaneous Conduction and Radiation in Participating Media – Diffusion


    Three-Dimensional Simultaneous Conduction and Radiation in Participating Media


    Phase Change Problems

    Mathematical Formulation of Phase Change Problems

    Variable Time Step Approach for Single-Phase Solidification

    Variable Time Step Approach for Two-Phase Solidification

    Enthalpy Method

    Phase Change Problems with Natural Convection


    Numerical Grid Generation

    Coordinate Transformation Relations

    Basic Ideas in Simple Transformations

    Basic Ideas in Numerical Grid Generation and Mapping

    Boundary Value Problem of Numerical Grid Generation

    Finite Difference Representation of Boundary Value Problem of Numerical Grid Generation

    Steady State Heat Conduction in Irregular Geometry

    Laminar Forced Convection in Irregular Channels

    Laminar Free Convection in Irregular Enclosures


    Hybrid Numerical-Analytic Solutions

    The Classical (CITT) and the Generalized Integral Transform (GITT) Techniques

    GITT with Partial Transformation

    Unified Integral Transforms (UNIT) Algorithm

    Applications in Heat Conduction

    Applications in Heat Convection




    Appendix I Discretization Formulae



    Helcio Rangel Barreto Orlande was born in Rio de Janeiro on March 9, 1965. He obtained his B.S. in Mechanical Engineering from the Federal University of Rio de Janeiro (UFRJ) in 1987 and his M.S. in Mechanical Engineering from the same University in 1989. After obtaining his Ph.D. in Mechanical Engineering in 1993 from North Carolina State University, he joined the Department of Mechanical Engineering of UFRJ, where he was the department head during 2006 and 2007. His research areas of interest include the solution of inverse heat and mass transfer problems, as well as the use of numerical, analytical and hybrid numerical-analytical methods of solution of direct heat and mass transfer problems. He is the co-author of 4 books and more than 280 papers in major journals and conferences. He is a member of the Scientific Council of the International Centre for Heat and Mass Transfer and a Delegate in the Assembly for International Heat Transfer Conferences. He serves as an Associate Editor for the journals Heat Transfer Engineering, Inverse Problems in Science and Engineering and High Temperatures – High Pressures.

    Marcelo J. Colaço is an Associate Professor in the Department of Mechanical Engineering at the Federal University of Rio de Janeiro - UFRJ, Brazil. He received his Ph.D. from UFRJ in 2001. He then spent 15 months as a postdoctoral fellow at the University of Texas at Arlington working on optimization algorithms, inverse problems in heat transfer, and electro-magneto-hydrodynamics including solidification. Afterwards, he spent one year performing research at UFRJ/COPPE on a prestigious CNPq grant as an Instructor and a researcher. From there, he joined Brazilian Military Institute of Engineering where he was teaching and performing research for five years in numerical algorithms for analysis of MHD flows, EHD flows, solidification problems, optimization algorithms utilizing response surfaces, and fuel research. For the past years, he has been teaching and performing research in Brazil and helping to build a large and unique fuels and lubricants research center at the UFRJ. He is the co-author of some book-chapters, and more than 200 papers in major journals and conferences. He was the recipient of the Young Scientist Award, given by state of Rio de Janeiro, in 2007 and 2009 and the Scientist Award in 2013 and 2015. Prof. Colaço is also member of the Scientific Council of the International Centre for Heat and Mass Transfer.