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sample-final-sol

sample-final-sol - ME 608 Numerical Methods in Heat Mass...

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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 0 E = 10 , a 0 W = 0 , a 0 P = 30 , b 0 = 6000 Cell 2 : a 0 E = 10 , a 0 W = 10 , a 0 P = 20 , b 0 = 0 Cell 4 : a 0 E = 19 . 801 , a 0 W = 10 , a 0 P = 29 . 801 , b 0 = 0 Cell 5 : a 0 E = 1000 , a 0 W = 19 . 801 , a 0 P = 1019 . 801 , b 0 = 0 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 = 0 , a 1 P = 30 , b 1 = r 0 1 + r 0 2 Cell 2 : a 1 E = 19 . 801 , a 1 W = 10 , a 1 P = 29 . 801 , b 1 = r 0 3 + r 0 4 Cell 3 : a 1 E = 1000 , a 1 W = 19 . 801 , a 1 P = 1019 . 801 , b 1 = r 0 5 + r 0 6 Here, r 0 1 = ( b 0 + a 0 E T 0 E + a 0 W T 0 W a 0 P T 0 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 = 0 , 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 = 0 , 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 V-cycle with ν 1 = 0 and ν 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, T 9 = 316 . 8609, T 10 = 316 . 8609, T 11 = 383 . 4163, T 12 = 383 . 4163. 1
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2. The discrete equations for level 0 are given below. Cell 1 : a 0 E = 10 , a 0 W = 0 , a 0 N = 10 , a 0 S = 0 , a 0 P = 30 , b 0 = 7000 Cell 2 : a 0 E = 0 , a 0 W = 10 , a 0 N = 10 , a 0 S = 0 , a 0 P = 50 , b 0 = 12 , 000 Cell 3 : a 0 E = 0 , a 0 W = 10 , a 0 N = 0 , a 0 S = 10 , a 0 P = 80 , b 0 = 17 , 000 Cell 4 : a 0 E = 10 , a 0 W = 0 , a 0 N = 0 , a 0 S = 10 , a 0 P = 50 , b 0 = 12 , 000 The discrete equations at level 1 are given below for the mesh numbering shown in Fig. 2.
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