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Numerical Method (Special Cases)

m,n+1. m-1,n. m+1, n. m,n. m,n-1. Numerical Method (Special Cases). A. 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. q 1. q 2. q 4. q 3.

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Numerical Method (Special Cases)

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  1. m,n+1 m-1,n m+1, n m,n m,n-1 Numerical Method (Special Cases) A 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. 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

  2. m,n+1 m-1,n m+1, n m,n m,n-1 Special cases (cont.) • 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 q1 q2 q4 q3 Insulated surface A

  3. 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:

  4. 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 + = + q4 q2 q3 m,n-1 Non-linear term, can solve using the iteration method

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