problem4-51 - PROBLEM 4.51 KNOWN Square shape subjected to...

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PROBLEM 4.51 K NOWN: Square shape subjected to uniform surface temperature conditions. FIND: (a) Temperature at the four specified nodes; estimate the midpoint temperature T o , (b) Reducing the mesh size by a factor of 2, determine the corresponding nodal temperatures and compare results, and (c) For the finer grid, plot the 75, 150, and 250 ° C isotherms. SCHEMATIC: A SSUMPTIONS: (1) Steady-state, two-dimensional conduction, (2) Constant properties. ANALYSIS: (a) The finite-difference equation for each node follows from Eq. 4.29 for an interior point written in the form, T i = 1/4 T neighbors . Using the Gauss-Seidel iteration method, Section 4.5.2, the finite- difference equations for the four nodes are: ( ) k k 1 k 1 k 1 k 1 1 2 3 2 3 T 0.25 100 T T 50 0.25T 0.25T 37.5 = + + + = + + ( ) k k 1 k 1 k 1 2 4 1 1 4 T 0.25 100 200 T T 0.25T 0.25T 75.0 = + + + = + + k 1 ( ) k k 1 k 1 k 1 k 1 3 1 4 1 4 T 0.25 T T 300 50 0.25T 0.25T 87.5 = + + + = + + ( ) k k 1 k 1 k 1 k 1 4 2 3 2 3 T 0.25 T 200 300 T 0.25T 0.25T 125.0 = + + + = + + The iteration procedure using a hand calculator is implemented in the table below.
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