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Unformatted text preview: Chapter 5 Numerical Methods in Heat Conduction Chapter 5 NUMERICAL METHODS IN HEAT CONDUCTION Why Numerical Methods 51C Analytical solution methods are limited to highly simplified problems in simple geometries . The geometry must be such that its entire surface can be described mathematically in a coordinate system by setting the variables equal to constants. Also, heat transfer problems can not be solved analytically if the thermal conditions are not sufficiently simple. For example, the consideration of the variation of thermal conductivity with temperature, the variation of the heat transfer coefficient over the surface, or the radiation heat transfer on the surfaces can make it impossible to obtain an analytical solution. Therefore, analytical solutions are limited to problems that are simple or can be simplified with reasonable approximations. 52C The analytical solutions are based on (1) driving the governing differential equation by performing an energy balance on a differential volume element, (2) expressing the boundary conditions in the proper mathematical form, and (3) solving the differential equation and applying the boundary conditions to determine the integration constants. The numerical solution methods are based on replacing the differential equations by algebraic equations . In the case of the popular finite difference method, this is done by replacing the derivatives by differences . The analytical methods are simple and they provide solution functions applicable to the entire medium, but they are limited to simple problems in simple geometries. The numerical methods are usually more involved and the solutions are obtained at a number of points, but they are applicable to any geometry subjected to any kind of thermal conditions. 53C The energy balance method is based on subdividing the medium into a sufficient number of volume elements, and then applying an energy balance on each element. The formal finite difference method is based on replacing derivatives by their finite difference approximations. For a specified nodal network, these two methods will result in the same set of equations. 54C In practice, we are most likely to use a software package to solve heat transfer problems even when analytical solutions are available since we can do parametric studies very easily and present the results graphically by the press of a button. Besides, once a person is used to solving problems numerically, it is very difficult to go back to solving differential equations by hand. 55C The experiments will most likely prove engineer B right since an approximate solution of a more realistic model is more accurate than the exact solution of a crude model of an actual problem....
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 Spring '11
 ENgel
 Heat Transfer

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