Chapter 5 Numerical Methods in Heat Conduction 5-49 Starting with an energy balance on a volume element, the steady two-dimensional finite difference equation for a general interior node in rectangular coordinates for T(x, y) for the case of variable thermal conductivity and uniform heat generation is to be obtained. AnalysisWe consider a volume elementof size ΔΔxy××1 centered about a general interior node (m, n) in a region in which heat is generated at a constant rate of and the thermal conductivity k is variable (see Fig. 5-24 in the text).Assuming the direction of heat conduction to be towardsthe node under consideration at all surfaces, the energy balance on the volume element can be expressed as &g&&&&&QQQQGEtcond, leftcond, topcond, rightcond, bottomelementelement++++=ΔΔ0=for the steadycase. Again assuming the temperatures between the adjacent nodes to vary linearly and noting that the heat transfer area is
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