Chapter 5 Numerical Methods in Heat Conduction 5-78Starting with an energy balance on a volume element, the two-dimensional transient explicitfinite 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 Δz = 1 in the z direction. There is no heat generation in the medium, and the thermal conductivity kof the medium is constant. Now we divide the x-y plane of the region into a rectangular meshof nodal points which are spaced Δxand Δyapart in the x and y directions, respectively, and consider a general interior node (m, n) whose coordinates are xm x=Δand . 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
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