Thermodynamics HW Solutions 439

Thermodynamics HW Solutions 439 - Chapter 5 Numerical...

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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. Analysis We consider a volume element of size Δ Δ xy × × 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 towards the node under consideration at all surfaces, the energy balance on the volume element can be expressed as & g &&& & & QQQ Q G E t cond, left cond, top cond, right cond, bottom element element ++ + + = Δ Δ 0 = for the steady case. Again assuming the temperatures between the adjacent nodes to vary linearly and noting that the heat transfer area is
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