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Unformatted text preview: ME 608 Numerical Methods in Heat, Mass, and Momentum Transfer Sample Final Exam Problems April 25, 2006 Instructor: J. Murthy 1. (a) For the fine mesh, the discrete equations for representative points are given below. Cell 1 : a E = 10 , a W = , a P = 30 , b = 6000 Cell 2 : a E = 10 , a W = 10 , a P = 20 , b = Cell 4 : a E = 19 . 801 , a W = 10 , a P = 29 . 801 , b = Cell 5 : a E = 1000 , a W = 19 . 801 , a P = 1019 . 801 , b = Other points can be discretized analogously. For level 1, using the grid numbering shown in Fig. 1, we have the following discrete equations for representative cells: Cell 1 : a 1 E = 10 , a 1 W = , a 1 P = 30 , b 1 = r 1 + r 2 Cell 2 : a 1 E = 19 . 801 , a 1 W = 10 , a 1 P = 29 . 801 , b 1 = r 3 + r 4 Cell 3 : a 1 E = 1000 , a 1 W = 19 . 801 , a 1 P = 1019 . 801 , b 1 = r 5 + r 6 Here, r 1 = ( b + a E T E + a W T W a P T P ) for cell 1 at the finest level and so on. Simlarly, for level 2, using the grid numbering shown in Fig. 1, we have the following discrete equations for represen tative cells: Cell 1 : a 2 E = 19 . 801 , a 2 W = , a 2 P = 39 . 801 , b 2 = r 1 1 + r 1 2 Cell 2 : a 2 E = 19 . 801 , a 2 W = 19 . 801 , a 2 P = 39 . 602 , b 2 = r 1 3 + r 1 4 Cell 3 : a 2 E = , a 2 W = 19 . 801 , a 2 P = 39 . 801 , b 2 = r 1 5 + r 1 6 Here, r 1 1 = ( b 1 + a 1 E T 1 E + a 1 W T 1 W a 1 P T 1 P ) for cell 1 at level 1 and so on. 1 2 3 4 5 6 7 8 9 10 11 12 Level 0 Level 1 Level 2 400K 300K 1 2 3 4 5 6 1 2 3 Figure 1: Computational Domain for Problem 1 (b) The solution after one Vcycle with 1 = 0 and 2 = 1 is given by T 1 = 300, T 2 = 300, T 3 = 300, T 4 = 300, T 5 = 300, T 6 = 300, T 7 = 300 . 9754, T 8 = 300 . 9754,...
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 Fall '10
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