Thermodynamics HW Solutions 497

Thermodynamics HW Solutions 497 - Chapter 5 Numerical...

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Chapter 5 Numerical Methods in Heat Conduction Review Problems 5-106 Starting with an energy balance on a volume element, the steady three-dimensional finite difference equation for a general interior node in rectangular coordinates for T ( x, y, z ) for the case of constant thermal conductivity and uniform heat generation is to be obtained. Analysis We consider a volume element of size centered about a general interior node ( m , n, r ) in a region in which heat is generated at a constant rate of and the thermal conductivity k is variable . 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 ΔΔΔ xy ×× z 0 g & g Δ x Δ y Δ z n +1 n r +1 r m +1 m -1 g o Δ x Δ y Δ z &&& & & & & QQQ Q Q Q G E t cond, left cond, top cond, right cond, bottom cond, front cond, back element element ++ + + + + = Δ Δ 0 = for the steady case. Again assuming the temperatures between the adjacent nodes to vary linearly, the energy balance relation above becomes 0 ) ( ) ( + ) ( ) ( + ) ( ) ( + ) ( 0 , , 1 , , , , 1 , , , , , 1 , , , , , 1 , , , 1 , , , , , 1 = Δ × Δ
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This note was uploaded on 01/22/2012 for the course PHY 4803 taught by Professor Dr.danielarenas during the Fall '10 term at UNF.

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