HEAT CONDUCTION – BASIC RESEARCH Edited by Vyacheslav S. Vikhrenko Heat Conduction – Basic Research Edited by Vyacheslav S. Vikhrenko Published by InTech Janeza Trdine 9, 51000 Rijeka, Croatia Copyright © 2011 InTech All chapters are Open Access distributed under the Creative Commons Attribution 3.0 license, which allows users to download, copy and build upon published articles even for commercial purposes, as long as the author and publisher are properly credited, which ensures maximum dissemination and a wider impact of our publications. After this work has been published by InTech, authors have the right to republish it, in whole or part, in any publication of which they are the author, and to make other personal use of the work. Any republication, referencing or personal use of the work must explicitly identify the original source. 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Used under license from Shutterstock.com First published November, 2011 Printed in Croatia A free online edition of this book is available at www.intechopen.com Additional hard copies can be obtained from orders@intechweb.org Heat Conduction – Basic Research, Edited by Vyacheslav S. Vikhrenko p. cm. ISBN 978-953-307-404-7 free online editions of InTech Books and Journals can be found at www.intechopen.com Contents Preface IX Part 1 Inverse Heat Conduction Problems 1 Chapter 1 Inverse Heat Conduction Problems 3 Krzysztof Grysa Chapter 2 Assessment of Various Methods in Solving Inverse Heat Conduction Problems 37 M. S. Gadala and S. Vakili Chapter 3 Identifiability of Piecewise Constant Conductivity 63 Semion Gutman and Junhong Ha Chapter 4 Experimental and Numerical Studies of Evaporation Local Heat Transfer in Free Jet 87 Hasna Louahlia Gualous Part 2 Non-Fourier and Nonlinear Heat Conduction, Time Varying Heat Sorces 109 Chapter 5 Exact Travelling Wave Solutions for Generalized Forms of the Nonlinear Heat Conduction Equation 111 Mohammad Mehdi Kabir Najafi Chapter 6 Heat Conduction Problems of Thermosensitive Solids under Complex Heat Exchange 131 Roman M. Kushnir and Vasyl S. Popovych Chapter 7 Can a Lorentz Invariant Equation Describe Thermal Energy Propagation Problems? 155 Ferenc Márkus Chapter 8 Time Varying Heat Conduction in Solids 177 Ernesto Marín Moares VI Contents Part 3 Coupling Between Heat Transfer and Electromagnetic or Mechanical Excitations 203 Chapter 9 Heat Transfer and Reconnection Diffusion in Turbulent Magnetized Plasmas 205 A. Lazarian Chapter 10 Energy Transfer in Pyroelectric Material 229 Xiaoguang Yuan and Fengpeng Yang Chapter 11 Steady-State Heat Transfer and Thermo-Elastic Analysis of Inhomogeneous Semi-Infinite Solids 249 Yuriy Tokovyy and Chien-Ching Ma Chapter 12 Self-Similar Hydrodynamics with Heat Conduction 269 Masakatsu Murakami Part 4 Numerical Methods 293 Chapter 13 Particle Transport Monte Carlo Method for Heat Conduction Problems 295 Nam Zin Cho Chapter 14 Meshless Heat Conduction Analysis by Triple-Reciprocity Boundary Element Method 325 Yoshihiro Ochiai Preface Heat conduction is a fundamental phenomenon encountered in many industrial and biological processes as well as in everyday life. Economizing of energy consumption in different heating and cooling processes or ensuring temperature limitations for proper device operation requires the knowledge of heat conduction physics and mathematics. The fundamentals of heat conduction were formulated by J. Fourier in his outstanding manuscript Théorie de la Propagation de la Chaleur dans les Solides presented to the Institut de France in 1807 and in the monograph ThéorieAnalytique de la Chaleur (1822). The two century evolution of the heat conduction theory resulted in a wide range of methods and problems that have been solved or have to be solved for successful development of the world community. The content of this book covers several up-to-date approaches in the heat conduction theory such as inverse heat conduction problems, non-linear and non-classic heat conduction equations, coupled thermal and electromagnetic or mechanical effects and numerical methods for solving heat conduction equations as well. The book is comprised of 14 chapters divided in four sections. In the first section inverse heat conduction problems are discuss. The section is started with a review containing classification of inverse heat conduction problems alongside with the methods for their solution. The genetic algorithm, neural network and particle swarm optimization techniques, and the Marching Algorithm are considered in the next two chapters. In Chapter 4 the inverse heat conduction problem is used for evaluating from experimental data the local heat transfer coefficient for jet impingement with plane surface. The first two chapter of the second section are devoted to construction of analytical solutions of nonlinear heat conduction problems when nonlinear terms are included in the heat conduction equation (Chapter 5) or the nonlinearity appears through boundary conditions and/or temperature dependence of the heat conduction equation coefficients (Chapter 6). In the last two chapters of this section wavelike solutions are attained due to construction of a hyperbolic heat conduction equation (Chapter 7) or because of time varying boundary conditions (Chapter 8). X Preface The third section is devoted to combined effects of heat conduction and electromagnetic interactions in plasmas (Chapter 9) or pyroelectric material (Chapter 10), elastic deformations (Chapter 11) and hydrodynamics (Chapter 12). Two chapters in the last section are dedicated to numerical methods for solving heat conduction problems, namely the particle transport Monte Carlo method (Chapter 13) and a meshless version of the boundary element method (Chapter 14). Dr. Prof. Vyacheslav S. Vikhrenko Belarusian State Technological University, Belarus [...]... solving the inverse heat conduction problems Many analytical and semi-analytical approaches have been developed for solving heat conduction problems Explicit analytical solutions are limited to simple geometries, but are very efficient computationally and are of fundamental importance for investigating basic properties of inverse heat conduction problems Exact solutions of the inverse heat conduction problems... Part 1 Inverse Heat Conduction Problems 1 Inverse Heat Conduction Problems Krzysztof Grysa Kielce University of Technology Poland 1 Introduction In the heat conduction problems if the heat flux and/or temperature histories at the surface of a solid body are known as functions of time, then the temperature distribution can be found This is termed as a direct problem However in many heat transfer situations,... expressed in dimensionless form as follows: 10 Heat Conduction – Basic Research 2T ξ , T ξ , ξ , (0, f ] , , (17) where ξ stands for dimensionless spatial location and τ = k/c denotes dimensionless time (Fourier number) In further consideration we will use notation x =( x, y, z) and t for dimensionless coordinates For dimensionless heat conduction equation in 1D the set of T-functions... 12 Heat Conduction – Basic Research T T0 x for x (0, l ) and t = 0 For further analysis it is assumed that q(t) is not known Instead, some measured temperature histories are given at interior locations: T x j , tk Ui , k , x j j 1, , J 0, l , tk k 1, ,K 0, t f The heat flux is more difficult to calculate accurately than the surface temperature When knowing the heat. .. j q and 0 J T x j j 1 Tcon w 2 Yj T x j j (12) 8 Heat Conduction – Basic Research Equations (12) involve two sensitivity coefficients which can be evaluated from (10), T x j / q x j / k and T x j / Tcon 1 , j = 1,2,…,J , (Beck et al., 1985) Solving the system of equations (12) for the unknown heat flux gives J 2 J 2 J J w j w j x jYj ... scattered noisy measurements Yi( k ) , i 1, 2, , M , k 1, 2, , J i It is worth to mention that with reconstructed T and T / n on SR (0, t f ) it is easy to identify heat transfer coefficient, hc , on SR 14 Heat Conduction – Basic Research The fundamental solution of (20)1 in Rd is given by F x,t x2 H t exp 4t 4 t d /2 1 (29) where H(t) is the Heaviside function Assuming... A leads to values of measurements, the next n rows – to values of the right-hand side of the initial condition and, of course, time variable is then equal to zero, the next p rows leads to values of the right-hand side of the Dirichlet condition and the last q rows - to values of the right-hand side of Neumann condition 16 Heat Conduction – Basic Research The solvability of the system (31) depends... based on singular value decomposition of the N N matrix A can be expressed as N 2 i 1 i i 2 w T bvi i (40) 18 Heat Conduction – Basic Research The determination of a suitable value of the regularization parameter 2 is crucial and is still under intensive research Recently the L-curve criterion is frequently used to choose a good regularization parameter, (Hansen, 1992; Hansen & O’Leary,... conditions (3) to (6) describes an initial-boundary value problem for transient heat conduction In the case of stationary problem the equation (2) becomes a Poisson equation or – when the source function Q is equal to zero – a Laplace equation v Broadly speaking, inverse problems may be subdivided into the following categories: inverse conduction, inverse convection, inverse radiation and inverse phase change... given function and Tik known from e.g measurements As examples of such problems can be presented papers (Reinhardt et al., 2007; Soti et al., 2007; Ciałkowski & Grysa, 2010) and many others 6 Heat Conduction – Basic Research 3.2 Initial value determination inverse problems In this case an initial condition is not known, i.e in the condition (6) the function T0 is not known In order to find the initial . HEAT CONDUCTION – BASIC RESEARCH Edited by Vyacheslav S. Vikhrenko Heat Conduction – Basic Research Edited by Vyacheslav S. Vikhrenko. Inverse Heat Conduction Problems 1 Inverse Heat Conduction Problems Krzysztof Grysa Kielce University of Technology Poland 1. Introduction In the heat conduction problems if the heat flux. book covers several up-to-date approaches in the heat conduction theory such as inverse heat conduction problems, non-linear and non-classic heat conduction equations, coupled thermal and electromagnetic