problem4-54 - PROBLEM 4.54 KNOWN Flue of square cross...

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PROBLEM 4.54 KNOWN: Flue of square cross section with prescribed geometry, thermal conductivity and nner and outer surface temperatures. i FIND: Heat loss per unit length from the flue, q . SCHEMATIC: ASSUMPTIONS: (1) Steady-state, two-dimensional conduction, (2) Constant properties, (3) o internal generation. N ANALYSIS: Taking advantage of symmetry, the nodal network using the suggested 75mm grid spacing is shown above. To obtain the heat rate, we first need to determine the unknown temperatures T 1 , T 2 , T 3 and T 4 . Recognizing that these nodes may be treated as interior odes, the nodal equations from Eq. 4.29 are n (T 2 + 25 + T 2 + 350) - 4T 1 = 0 (T 1 + 25 + T 3 + 350) - 4T 2 = 0 (T 2 + 25 + T 4 + 350) - 4T 3 = 0 (T 3 + 25 + 25 + T 3 ) - 4T 4 = 0. The Gauss-Seidel iteration method is convenient for this system of equations and following he procedures of Section 4.5.2, they are rewritten as, t k k-1 1 2 k k k-1 2 1 3 k k k-1 3 2 4 k k 4 3 T 0.50 T 93.75 T 0.25 T 0.25 T 93.75 T 0.25 T 0.25 T 93.75 T 0.50 T 12.5.
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