Numerical Method (Special Cases)

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Presentation transcript:

Numerical Method (Special Cases) For all the special cases discussed in the following, the derivation will be based on the standard nodal point configuration as shown to the right. m,n m,n+1 m,n-1 m+1, n m-1,n q1 q2 q4 q3 Symmetric case: symmetrical relative to the A-A axis. In this case, Tm-1,n=Tm+1,n Therefore the standard nodal equation can be written as A axis A-A

Special cases (cont.) m,n+1 m-1,n m+1, n m,n m,n-1 Insulated surface case: If the axis A-A is an insulated wall, therefore there is no heat transfer across A-A. Also, the surface area for q1 and q3 is only half of their original value. Write the energy balance equation (q2=0): A m,n m,n+1 m,n-1 m+1, n m-1,n q1 q2 q4 q3 Insulated surface A

Special cases (cont.) With internal generation G=gV where g is the power generated per unit volume (W/m3). Based on the energy balance concept:

Special cases (cont.) Radiation heat exchange with respect to the surrounding (assume no convection, no generation to simplify the derivation). Given surface emissivity e, surrounding temperature Tsurr. qrad Tsurr From energy balance concept: q2+q3+q4=qrad m,n m-1,n m+1,n + + = q2 q4 q3 m,n-1 Non-linear term, can solve using the iteration method