Thermodynamics HW Solutions 469

Thermodynamics HW Solutions 469 - Chapter 5 Numerical...

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Chapter 5 Numerical Methods in Heat Conduction 5-79 Starting with an energy balance on a volume element, the two-dimensional transient implicit finite difference equation for a general interior node in rectangular coordinates for for the case of constant thermal conductivity and no heat generation is to be obtained. Txyt (,,) Analysis (See Figure 5-49 in the text). We consider a rectangular region in which heat conduction is significant in the x and y directions, and consider a unit depth of Δ z = 1 in the z direction. There is no heat generation in the medium, and the thermal conductivity k of the medium is constant. Now we divide the x-y plane of the region into a rectangular mesh of nodal points which are spaced Δ x and Δ y apart in the x and y directions, respectively, and consider a general interior node ( m , n ) whose coordinates are x m x and . Noting that the volume element centered about the general interior node ( m , n ) involves heat conduction from four sides (right, left, top, and bottom) and expressing them at previous time step i , the transient implicit finite difference formulation for a general interior node can be expressed
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