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Numericalheatransfer - cen58933_ch05.qxd 11:41 AM Page 265...

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NUMERICAL METHODS IN HEAT CONDUCTION S o far we have mostly considered relatively simple heat conduction prob- lems involving simple geometries with simple boundary conditions be- cause only such simple problems can be solved analytically. But many problems encountered in practice involve complicated geometries with com- plex boundary conditions or variable properties and cannot be solved ana- lytically. In such cases, sufficiently accurate approximate solutions can be obtained by computers using a numerical method. Analytical solution methods such as those presented in Chapter 2 are based on solving the governing differential equation together with the boundary con- ditions. They result in solution functions for the temperature at every point in the medium. Numerical methods, on the other hand, are based on replacing the differential equation by a set of n algebraic equations for the unknown temperatures at n selected points in the medium, and the simultaneous solu- tion of these equations results in the temperature values at those discrete points. There are several ways of obtaining the numerical formulation of a heat conduction problem, such as the finite difference method, the finite element method, the boundary element method, and the energy balance (or control volume) method. Each method has its own advantages and disadvantages, and each is used in practice. In this chapter we will use primarily the energy bal- ance approach since it is based on the familiar energy balances on control vol- umes instead of heavy mathematical formulations, and thus it gives a better physical feel for the problem. Besides, it results in the same set of algebraic equations as the finite difference method. In this chapter, the numerical for- mulation and solution of heat conduction problems are demonstrated for both steady and transient cases in various geometries. 265 CHAPTER 5 CONTENTS 5–1 Why Numerical Methods 266 5–2 Finite Difference Formulation of Differential Equations 269 5–3 One-Dimensional Steady Heat Conduction 272 5–4 Two-Dimensional Steady Heat Conduction 282 5–5 Transient Heat Conduction 291 Topic of Special Interest: Controlling the Numerical Error 309 cen58933_ch05.qxd 9/4/2002 11:41 AM Page 265
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5–1 WHY NUMERICAL METHODS? The ready availability of high-speed computers and easy-to-use powerful soft- ware packages has had a major impact on engineering education and practice in recent years. Engineers in the past had to rely on analytical skills to solve significant engineering problems, and thus they had to undergo a rigorous training in mathematics. Today’s engineers, on the other hand, have access to a tremendous amount of computation power under their fingertips, and they mostly need to understand the physical nature of the problem and interpret the results. But they also need to understand how calculations are performed by the computers to develop an awareness of the processes involved and the lim- itations, while avoiding any possible pitfalls.
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