Thermodynamics HW Solutions 498

Thermodynamics HW Solutions 498 - Chapter 5 Numerical...

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Chapter 5 Numerical Methods in Heat Conduction 5-107 Starting with an energy balance on a volume element, the three-dimensional transient explicit finite difference equation for a general interior node in rectangular coordinates for T ( x, y, z, t ) for the case of constant thermal conductivity k and no heat generation is to be obtained. Analysis We consider a rectangular region in which heat conduction is significant in the x and y directions. There is no heat generation in the medium, and the thermal conductivity k of the medium is constant. Now we divide the x-y-z region into a mesh of nodal points which are spaced Δ x, Δ y , and Δ z apart in the x , y, and z directions, respectively, and consider a general interior node ( m , n, r ) whose coordinates are x = m Δ x , y = n Δ y , are z = r Δ z . Noting that the volume element centered about the general interior node ( m , n, r ) involves heat conduction from six sides (right, left, front, rear, top, and bottom) and expressing them at previous time step i , the transient
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