Tema29 - CHAPTER 29 29.1 First iteration: 7.500000 9.750000...

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CHAPTER 29 29.1 First iteration: 7.500000 2.250000 15.675000 9.750000 3.600000 20.782500 55.425000 62.707500 85.047000 Error: 100.000000 100.000000 100.000000 100.000000 100.000000 100.000000 100.000000 100.000000 100.000000 Second iteration: 9.600000 8.212501 20.563500 26.137500 34.632000 52.916250 68.068500 88.782750 85.500310 Error: 21.875000 72.602740 23.772710 62.697270 89.604990 60.725670 18.574660 29.369720 5.301830E-01 Seventh iteration: 25.013610 28.806340 33.932440 46.216590 56.257030 56.921290 78.575310 93.082440 87.501180 Error: 2.954020E-01 2.531316E-02 1.679560E-02 2.267362E-02 2.082395E-02 1.041445E-02 2.165254E-03 3.590016E-03 1.743838E-03 29.2 The fluxes for Prob. 29.1 can be calculated as qx= -9.325527E-02 -2.185114E-01 -5.192447E-01 -7.657973E-01 -2.622653E-01 1.532972E-01 -1.668020 -2.186839E-01 1.055520 qy= -1.132306 -1.378297 -1.394572 -1.312262 -1.574765 -1.312434 -2.542694 -2.296703 -2.280428 qn= 1.136140 1.395511 1.488101 1.519367 1.596454 1.321357 3.040984 2.307091 2.512862 theta= -94.708180 -99.008540 -110.421900 -120.266600 -99.455400 -83.337820 -123.265100 -95.439100 -65.162450
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29.3 The plate is redrawn below 100 o C 0 o C 75 o C 50 o C After 15 iterations of the Liebmann method, the result is 0 100 100 100 100 100 100 100 0 50 73.6954 82.3973 86.06219 87.7991 88.54443 88.19118 85.32617 75 50 62.3814 69.8296 74.0507 76.58772 78.18341 78.8869 78.10995 75 50 55.9987 60.4898 63.72554 66.32058 68.71677 71.06672 73.23512 75 50 51.1222 52.4078 54.04625 56.25934 59.3027 63.42793 68.75568 75 50 46.0804 43.9764 43.79945 45.37425 48.80563 54.57569 63.33804 75 50 39.2206 33.6217 31.80514 32.62971 35.95756 42.71618 54.995 75 50 27.1773 19.4897 17.16646 17.3681 19.66293 25.31308 38.86852 75 0 0 0 0 0 0 0 0 0 with percent approximate errors of 0 0 0 0 0 0 0 0 0 0 0.0030 0.0040 0.0043 0.0049 0.0070 0.0114 0.0120 0 0 0.0050 0.0062 0.0057 0.0055 0.0079 0.0120 0.0109 0 0 0.0062 0.0067 0.0036 0.0007 0.0007 0.0097 0.0241 0 0 0.0076 0.0066 0.0020 0.0106 0.0067 0.0164 0.0542 0 0 0.0106 0.0079 0.0033 0.0074 0.0077 0.0400 0.1005 0 0 0.0149 0.0099 0.0119 0.0327 0.0630 0.1192 0.2343 0 0 0.0136 0.0013 0.0302 0.1259 0.2194 0.2925 0.7119 0 0 0 0 0 0 0 0 0 0 29.4 The solution is identical to Prob. 29.3, except that now the top edge must be modeled. This means that the nodes along the top edge are simulated with equations of the form 4 2 0 1 1 1 T T T T i j i j i j i j , , , , - - - = - + - The resulting simulation (after 14 iterations) yields 50 50.38683 51.16385 52.6796 55.17802 58.7692 63.41846 68.9398 75 50 50.17211 50.76425 52.15054 54.58934 58.20129 62.96008 68.67918 75 50 49.51849 49.56564 50.58556 52.86931 56.56024 61.64839 67.93951 75 50 48.31607 47.39348 47.78093 49.79691 53.61405 59.2695 66.58047 75 50 46.33449 43.91569 43.37764 44.99165 48.94264 55.38806 64.29121 75 50 43.09381 38.56608 36.8614 37.93565 41.91332 49.21507 60.37012 75 50 37.46764 30.4051 27.61994 28.08718 31.71478 39.39338 53.1291 75 50 26.36368 17.98153 15.18654 15.20479 17.63115 23.73251 38.00928 75 0 0 0 0 0 0 0 0 0 with percent approximate errors of
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0 0.0584 0.1318 0.2034 0.2606 0.2828 0.2493 0.1529 0 0 0.0722 0.1603 0.2473 0.3173 0.3424 0.2983 0.1862 0 0 0.0854 0.1883 0.2937 0.3788 0.4077 0.3438 0.2096 0 0 0.0933 0.2121 0.3441 0.4464 0.4754 0.3972 0.2247 0 0 0.0930 0.2300 0.3913 0.5097 0.5328 0.4468 0.2605 0 0 0.0873 0.2469 0.4299 0.5474 0.5611 0.4237 0.2747 0 0 0.0913 0.2827 0.4995 0.5852 0.5525 0.3157 0.0477 0 0 0.1131 0.3612 0.7054 0.9164 0.7958 0.5085 0.6345 0 0 0 0 0 0 0 0 0 0 29.5 The solution is identical to Examples 29.1 and 29.3, except that now heat balances must be developed for the three interior nodes on the bottom edge. For example, using the control-volume approach, node 1,0 can be modeled as - - + - + - - = - - - = - 0 49 5 10 0 49 5 10 0 49 10 10 2 10 0 4 2 8163265 10 00 20 10 11 10 10 00 20 11 . ( ) . ( ) . ( ) ( ) . T T T T T T T T T T The resulting simulation yields (with a stopping criterion of 1% and a relaxation coefficient of 1.5) 87.5 100 100 100 75 75 79.91669 77.76103 70.67812 50 75 66.88654 60.34068 55.39378 50 75 52.26597 40.84576 40.26148 50 75 27.12079 10.54741 14.83802 50 The fluxes for the computed nodes can be computed as q x -0.06765 0.226345 0.680145 0.359153 0.281573 0.253347 0.836779 0.29411 -0.22428 1.579088 0.300928 -0.96659 q y -0.81128 -0.97165 -1.09285 -0.67744 -0.90442 -0.74521 -0.97426 -1.21994 -0.99362 -1.23211 -1.48462 -1.24575
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Tema29 - CHAPTER 29 29.1 First iteration: 7.500000 9.750000...

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