Thermodynamics HW Solutions 470

Thermodynamics HW Solutions 470 - Chapter 5 Numerical...

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Chapter 5 Numerical Methods in Heat Conduction 5-80 Starting with an energy balance on a disk volume element, the one-dimensional transient explicit finite difference equation for a general interior node for Tz in a cylinder whose side surface is insulated for the case of constant thermal conductivity with uniform heat generation is to be obtained. t (,) Analysis We consider transient one-dimensional heat conduction in the axial z direction in an insulated cylindrical rod of constant cross-sectional area A with constant heat generation and constant conductivity k with a mesh size of 0 g & z Δ in the z direction. Noting that the volume element of a general interior node m involves heat conduction from two sides and the volume of the element is z A V Δ = element , the transient explicit finite difference formulation for an interior node can be expressed as t T T xC A x A g x T T kA x T T kA i m i m i m i m i m i m Δ Δ = Δ + Δ Δ + + 1 0 1 1 + ρ & m -1 m m +1 Insulation Disk
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