exercicio 07 - redes 2 2012-2 gab
TRANSCRIPT
modelo de planilha para o método de Hardy-Cross
solução inicial
trecho noi nof L (m) øi (mm) r Q0 (m³/s) Qi (m³/s)1 1 2 700 400 108 0.700 55.85 79.78 0.7002 2 3 990 300 620 0.250 47.72 190.903 2 4 600 300 3764 4 5 600 200 27085 5 6 500 150 91606 4 7 600 250 9137 7 8 620 150 113598 8 9 480 150 87949 5 8 500 150 9160
10 3 6 500 250 76111 6 9 450 200 2031
anel
DH0 (m) DH/Q
DQ (m³/s) SDH (m) SDH/Q DQ (m³/s)
S|DH|
iteração 1 ... iteração final
... Qi (m³/s)55.85 79.78 ... 0.700 55.85 79.78
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DHi (m) DH/Q DHi (m) DH/Q
SDH (m) SDH/Q DQ (m³/s) SDH (m) SDH/Q
método de Hardy-Cross
solução inicial
trecho noi nof L (m) øi (mm) r Q0 (m³/s) Qi (m³/s)1 1 2 700 400 108 0.194 5.20 26.80 0.1942 2 3 990 300 620 0.018 0.37 20.40 0.0403 2 4 600 300 376 0.149 11.10 74.52 0.1274 4 5 600 200 2708 0.040 7.02 175.54 0.0185 5 6 500 150 9160 0.017 4.88 286.94 -0.0056 4 7 600 250 913 0.090 10.62 117.97 0.0907 7 8 620 150 11359 0.058 58.57 1009.81 0.0588 8 9 480 150 8794 0.037 19.74 533.51 0.0379 5 8 500 150 9160 0.000 0.00 0.00 0.000
10 3 6 500 250 761 0.000 0.00 0.00 0.02211 6 9 450 200 2031 0.000 0.00 0.00 0.000
anel
1 2 10 -5 -4 -3 0.022 -22.64 557.392 4 9 -7 -6 -62.16 1303.32 0.0303 5 11 -8 -9 -14.86 820.45
99.66
DH0 (m) DH/Q
DQ (m³/s) SDH (m) SDH/Q DQ (m³/s)
S|DH|
n1NA=30,00 m
n2: 27 L/s n3: 18 L/s
n7:32 L/s
n4:19 L/s
n5: 23 L/s
n8:21 L/s
n6:17 L/s
n9:37 L/s
t1,194
t3,149
t2,18
t6,90
t7,58
t4,40
t8,37
t5,17
t10,0
t11,0t9,0
DQ1
DQ2 DQ3
Estimativa inicialde vazões: ti,Qo
iteração 1 iteração 2 iteração 3 iteração
Qi (m³/s) Qi (m³/s) Qi (m³/s)5.20 26.80 0.194 5.20 26.80 0.194 5.20 26.80 0.194 5.201.60 40.17 0.040 1.60 40.17 0.040 1.60 40.17 0.058 3.208.27 65.08 0.127 8.27 65.08 0.127 8.27 65.08 0.109 6.231.61 89.25 0.048 9.86 205.19 0.048 9.86 205.19 0.030 4.13-0.50 100.56 -0.005 -0.50 100.56 0.012 2.51 211.37 -0.006 -0.7410.62 117.97 0.060 5.01 83.57 0.060 5.01 83.57 0.060 5.0158.57 1009.81 0.028 15.21 543.54 0.028 15.21 543.54 0.028 15.2119.74 533.51 0.037 19.74 533.51 0.020 6.43 318.72 0.020 6.430.00 0.00 0.030 13.96 465.18 0.013 3.05 231.36 0.013 3.050.65 29.63 0.022 0.65 29.63 0.022 0.65 29.63 0.040 1.970.00 0.00 0.000 0.00 0.00 0.017 1.06 63.03 0.017 1.06
-7.13 324.69 -15.38 440.63 0.018 -18.38 551.44 -4.45-67.58 1217.03 3.60 1297.48 -7.31 1063.66 0.007 -13.04-20.24 634.08 0.017 -34.20 1099.26 -5.92 824.49 -9.1794.94 53.18 31.61 26.66
DHi (m) DH/Q DHi (m) DH/Q DHi (m) DH/Q DHi (m)
SDH (m) SDH/Q DQ (m³/s) SDH (m) SDH/Q DQ (m³/s) SDH (m) SDH/Q DQ (m³/s) SDH (m)
4 iteração 5 iteração 6 iteração 7
Qi (m³/s) Qi (m³/s) Qi (m³/s)26.80 0.194 5.20 26.80 0.194 5.20 26.80 0.194 5.20 26.8055.12 0.058 3.20 55.12 0.058 3.20 55.12 0.068 4.34 63.4757.14 0.109 6.23 57.14 0.109 6.23 57.14 0.099 5.17 52.45
137.63 0.037 6.11 164.73 0.037 6.11 164.73 0.027 3.31 124.29120.98 -0.006 -0.74 120.98 0.002 0.11 49.86 -0.008 -1.30 156.0283.57 0.053 3.97 75.11 0.053 3.97 75.11 0.053 3.97 75.11
543.54 0.021 8.87 424.29 0.021 8.87 424.29 0.021 8.87 424.29318.72 0.020 6.43 318.72 0.012 2.41 202.83 0.012 2.41 202.83231.36 0.020 6.75 333.23 0.012 2.54 212.62 0.012 2.54 212.6249.32 0.040 1.97 49.32 0.040 1.97 49.32 0.050 3.03 60.1063.03 0.017 1.06 63.03 0.025 2.23 88.72 0.025 2.23 88.72
420.18 -6.43 447.28 0.010 -7.29 376.16 0.19 456.32996.09 0.02 997.36 -4.19 876.74 0.005 -6.99 836.30734.09 0.008 -12.87 835.96 -2.61 554.02 -4.01 660.19
19.33 14.09 11.20
DH/Q DHi (m) DH/Q DHi (m) DH/Q DHi (m) DH/Q
SDH/Q DQ (m³/s) SDH (m) SDH/Q DQ (m³/s) SDH (m) SDH/Q DQ (m³/s) SDH (m) SDH/Q
iteração 8 iteração 9 iteração 10
Qi (m³/s) Qi (m³/s) Qi (m³/s) Qi (m³/s)0.194 5.20 26.80 0.194 5.20 26.80 0.194 5.20 26.80 0.1940.068 4.34 63.47 0.068 4.34 63.47 0.068 4.34 63.47 0.0710.099 5.17 52.45 0.099 5.17 52.45 0.099 5.17 52.45 0.0960.031 4.43 142.00 0.031 4.43 142.00 0.031 4.43 142.00 0.028-0.008 -1.30 156.02 -0.004 -0.30 80.07 -0.003 -0.19 64.95 -0.0060.048 3.37 69.62 0.048 3.37 69.62 0.048 3.37 69.62 0.0480.016 5.65 344.93 0.016 5.65 344.93 0.016 5.65 344.93 0.0160.012 2.41 202.83 0.007 0.99 134.99 0.007 0.80 121.96 0.0070.016 4.60 279.30 0.012 2.54 212.72 0.011 2.23 200.15 0.0110.050 3.03 60.10 0.050 3.03 60.10 0.050 3.03 60.10 0.0530.025 2.23 88.72 0.030 3.03 102.09 0.030 3.19 104.50 0.030
-0.92 474.03 -1.92 398.09 0.003 -2.03 382.960.00 835.85 -2.05 769.27 -2.37 756.70 0.002
0.005 -6.07 726.87 0.001 -0.81 529.87 -0.03 491.567.00 4.78 4.43
DHi (m) DH/Q DHi (m) DH/Q DHi (m) DH/Q
DQ (m³/s) SDH (m) SDH/Q DQ (m³/s) SDH (m) SDH/Q DQ (m³/s) SDH (m) SDH/Q DQ (m³/s)
iteração 11 iteração 12 iteração 13 iteração
Qi (m³/s) Qi (m³/s) Qi (m³/s)5.20 26.80 0.194 5.20 26.80 0.194 5.20 26.80 0.194 5.204.69 65.72 0.071 4.69 65.72 0.071 4.69 65.72 0.072 4.804.89 51.15 0.096 4.89 51.15 0.096 4.89 51.15 0.095 4.813.70 130.84 0.031 4.26 139.60 0.031 4.26 139.60 0.030 4.03-0.67 115.39 -0.006 -0.67 115.39 -0.004 -0.42 92.68 -0.005 -0.593.37 69.62 0.046 3.09 66.87 0.046 3.09 66.87 0.046 3.095.65 344.93 0.014 4.31 304.40 0.014 4.31 304.40 0.014 4.310.80 121.96 0.007 0.80 121.96 0.005 0.52 100.57 0.005 0.522.23 200.15 0.013 3.13 233.94 0.012 2.58 214.11 0.012 2.583.36 62.98 0.053 3.36 62.98 0.053 3.36 62.98 0.054 3.463.19 104.50 0.030 3.19 104.50 0.032 3.45 108.34 0.032 3.45
0.12 426.08 -0.44 434.84 0.001 -0.70 412.13 0.01-3.09 745.54 0.00 744.80 -0.55 724.98 0.001 -0.78-0.51 542.00 0.001 -1.41 575.79 -0.07 515.71 -0.243.72 1.85 1.32 1.04
DHi (m) DH/Q DHi (m) DH/Q DHi (m) DH/Q DHi (m)
SDH (m) SDH/Q DQ (m³/s) SDH (m) SDH/Q DQ (m³/s) SDH (m) SDH/Q DQ (m³/s) SDH (m)
14 iteração 15 iteração 16 iteração 17
Qi (m³/s) Qi (m³/s) Qi (m³/s)26.80 0.194 5.20 26.80 0.194 5.20 26.80 0.194 5.20 26.8066.44 0.072 4.80 66.44 0.072 4.80 66.44 0.073 4.83 66.6750.73 0.095 4.81 50.73 0.095 4.81 50.73 0.094 4.78 50.60
136.03 0.030 4.18 138.31 0.030 4.18 138.31 0.030 4.10 137.16108.51 -0.005 -0.59 108.51 -0.005 -0.49 100.26 -0.005 -0.55 105.3666.87 0.046 3.01 66.15 0.046 3.01 66.15 0.046 3.01 66.15
304.40 0.014 3.98 293.64 0.014 3.98 293.64 0.014 3.98 293.64100.57 0.005 0.52 100.57 0.005 0.44 92.60 0.005 0.44 92.60214.11 0.013 2.82 222.94 0.012 2.62 215.71 0.012 2.62 215.7163.90 0.054 3.46 63.90 0.054 3.46 63.90 0.055 3.50 64.20
108.34 0.032 3.45 108.34 0.032 3.54 109.74 0.032 3.54 109.74
425.61 -0.14 427.90 0.000 -0.23 419.65 0.00 423.99721.41 0.00 721.04 -0.20 713.80 0.000 -0.27 712.65531.55 0.000 -0.48 540.38 -0.01 518.30 -0.07 523.41
0.62 0.44 0.34
DH/Q DHi (m) DH/Q DHi (m) DH/Q DHi (m) DH/Q
SDH/Q DQ (m³/s) SDH (m) SDH/Q DQ (m³/s) SDH (m) SDH/Q DQ (m³/s) SDH (m) SDH/Q
iteração 18 iteração 19 iteração 20
Qi (m³/s) Qi (m³/s) Qi (m³/s) Qi (m³/s)0.194 5.20 26.80 0.194 5.20 26.80 0.194 5.20 26.80 0.1940.073 4.83 66.67 0.073 4.83 66.67 0.073 4.85 66.75 0.0730.094 4.78 50.60 0.094 4.78 50.60 0.094 4.77 50.55 0.0940.030 4.16 137.96 0.030 4.16 137.96 0.030 4.13 137.56 0.030-0.005 -0.55 105.36 -0.005 -0.52 102.74 -0.005 -0.54 104.51 -0.0050.045 2.99 65.89 0.045 2.99 65.89 0.045 2.99 65.89 0.0450.013 3.87 289.85 0.013 3.87 289.85 0.013 3.87 289.85 0.0130.005 0.44 92.60 0.005 0.41 90.04 0.005 0.41 90.04 0.0050.012 2.70 218.80 0.012 2.64 216.50 0.012 2.64 216.50 0.0120.055 3.50 64.20 0.055 3.50 64.20 0.055 3.51 64.30 0.0550.032 3.54 109.74 0.032 3.57 110.18 0.032 3.57 110.18 0.032
-0.05 424.79 0.000 -0.08 422.16 0.00 423.670.00 712.50 -0.06 710.20 0.000 -0.09 709.80
0.000 -0.15 526.50 0.00 519.46 -0.02 521.23 0.0000.20 0.14 0.11
21 iterações de Hardy-Cross para
resultar erro máximo de 0.05 m em
um anel...
12 iterações de Hardy-Cross para
resultar correção máxima de 1 L/s em
um anel...
DHi (m) DH/Q DHi (m) DH/Q DHi (m) DH/Q
DQ (m³/s) SDH (m) SDH/Q DQ (m³/s) SDH (m) SDH/Q DQ (m³/s) SDH (m) SDH/Q DQ (m³/s)
iteração 21
5.20 26.804.85 66.754.77 50.554.15 137.82-0.54 104.512.98 65.813.84 288.620.41 90.042.67 217.513.51 64.303.57 110.18
-0.02 423.930.00 709.75-0.05 522.240.06
21 iterações de Hardy-Cross para
resultar erro máximo de 0.05 m em
12 iterações de Hardy-Cross para
resultar correção máxima de 1 L/s em
DHi (m) DH/Q
SDH (m) SDH/Q
método de Hardy-Cross, variante fazendo a correção simultânea nos três anéis
solução inicial
trecho noi nof L (m) øi (mm) r Q0 (m³/s) Qi (m³/s)1 1 2 700 400 108 0.194 5.20 26.80 0.1942 2 3 990 300 620 0.018 0.37 20.40 0.0403 2 4 600 300 376 0.149 11.10 74.52 0.1274 4 5 600 200 2708 0.040 7.02 175.54 0.0445 5 6 500 150 9160 0.017 4.88 286.94 0.0056 4 7 600 250 913 0.090 10.62 117.97 0.0647 7 8 620 150 11359 0.058 58.57 1009.81 0.0328 8 9 480 150 8794 0.037 19.74 533.51 0.0279 5 8 500 150 9160 0.000 0.00 0.00 0.016
10 3 6 500 250 761 0.000 0.00 0.00 0.02211 6 9 450 200 2031 0.000 0.00 0.00 0.010
anel
1 2 10 -5 -4 -3 0.022 -22.64 557.39 0.0192 4 9 -7 -6 0.026 -62.16 1303.32 0.0063 5 11 -8 -9 0.010 -14.86 820.45 0.010
99.66
DH0 (m) DH/Q
DQ (m³/s) SDH (m) SDH/Q DQ (m³/s)
S|DH|
iteração 1 iteração 2 iteração 3 iteração
Qi (m³/s) Qi (m³/s) Qi (m³/s)5.20 26.80 0.194 5.20 26.80 0.194 5.20 26.80 0.194 5.201.60 40.17 0.059 3.29 55.84 0.065 3.99 61.02 0.069 4.368.27 65.08 0.108 6.14 56.75 0.102 5.47 53.84 0.098 5.168.32 189.73 0.031 4.34 140.75 0.031 4.32 140.41 0.030 4.230.48 98.63 -0.004 -0.40 91.33 -0.005 -0.60 109.55 -0.005 -0.565.69 88.55 0.058 4.75 81.55 0.052 3.83 73.89 0.049 3.45
19.74 612.64 0.026 13.55 515.44 0.020 8.09 406.76 0.017 6.0611.18 410.84 0.018 4.97 283.15 0.012 2.51 206.81 0.009 1.374.36 272.39 0.012 2.67 217.44 0.013 3.07 231.93 0.013 2.850.65 29.63 0.041 2.05 50.25 0.047 2.70 56.95 0.051 3.040.39 39.80 0.019 1.39 71.30 0.025 2.19 87.89 0.028 2.77
-14.81 423.24 0.006 -4.73 394.92 0.003 -2.51 421.77 0.002 -1.42-12.75 1163.31 0.006 -11.30 955.19 0.003 -4.54 852.99 0.002 -2.43-14.67 821.66 0.005 -6.66 663.22 0.003 -3.99 636.19 0.002 -2.0242.23 22.69 11.04 5.87
DHi (m) DH/Q DHi (m) DH/Q DHi (m) DH/Q DHi (m)
SDH (m) SDH/Q DQ (m³/s) SDH (m) SDH/Q DQ (m³/s) SDH (m) SDH/Q DQ (m³/s) SDH (m)
4 iteração 5 iteração 6 iteração 7
Qi (m³/s) Qi (m³/s) Qi (m³/s)26.80 0.194 5.20 26.80 0.194 5.20 26.80 0.194 5.20 26.8063.56 0.070 4.57 65.00 0.071 4.70 65.79 0.072 4.77 66.2352.39 0.097 4.98 51.57 0.096 4.89 51.11 0.095 4.83 50.85
139.10 0.030 4.19 138.48 0.030 4.17 138.14 0.030 4.16 137.96106.54 -0.005 -0.56 105.87 -0.005 -0.55 105.11 -0.005 -0.54 104.7070.39 0.047 3.24 68.35 0.046 3.12 67.22 0.046 3.06 66.58
356.19 0.015 5.01 326.38 0.014 4.47 309.62 0.014 4.18 300.12156.49 0.007 0.88 127.67 0.006 0.65 110.83 0.005 0.53 101.24224.23 0.013 2.77 221.25 0.012 2.72 219.28 0.012 2.69 218.2460.22 0.052 3.25 62.06 0.053 3.37 63.07 0.054 3.43 63.6497.99 0.030 3.12 103.45 0.031 3.32 106.52 0.032 3.44 108.23
421.80 0.001 -0.79 422.96 0.001 -0.44 423.22 0.000 -0.25 423.38789.90 0.001 -1.29 754.47 0.001 -0.71 734.26 0.000 -0.39 722.91585.25 0.001 -1.09 558.24 0.001 -0.59 541.73 0.000 -0.32 532.41
3.17 1.74 0.96
DH/Q DHi (m) DH/Q DHi (m) DH/Q DHi (m) DH/Q
SDH/Q DQ (m³/s) SDH (m) SDH/Q DQ (m³/s) SDH (m) SDH/Q DQ (m³/s) SDH (m) SDH/Q
iteração 8 iteração 9 iteração 10
Qi (m³/s) Qi (m³/s) Qi (m³/s) Qi (m³/s)0.194 5.20 26.80 0.194 5.20 26.80 0.194 5.20 26.80 0.1940.072 4.80 66.48 0.072 4.83 66.61 0.073 4.84 66.69 0.0730.095 4.80 50.71 0.095 4.79 50.63 0.094 4.78 50.58 0.0940.030 4.15 137.87 0.030 4.15 137.81 0.030 4.14 137.78 0.030-0.005 -0.54 104.44 -0.005 -0.54 104.30 -0.005 -0.54 104.22 -0.0050.046 3.02 66.22 0.045 3.00 66.02 0.045 2.99 65.91 0.0450.014 4.02 294.80 0.013 3.93 291.81 0.013 3.88 290.14 0.0130.005 0.47 95.79 0.005 0.44 92.73 0.005 0.42 91.00 0.0050.012 2.67 217.64 0.012 2.66 217.31 0.012 2.66 217.13 0.0120.054 3.47 63.96 0.054 3.49 64.13 0.055 3.50 64.23 0.0550.032 3.50 109.18 0.032 3.54 109.71 0.032 3.56 110.01 0.032
0.000 -0.14 423.45 0.000 -0.08 423.49 0.000 -0.04 423.510.000 -0.22 716.53 0.000 -0.12 712.96 0.000 -0.07 710.960.000 -0.18 527.06 0.000 -0.10 524.06 0.000 -0.06 522.37
0.53 0.30 0.16
11 iterações ou 33 correções de vazões
para resultar erro máximo de 0.05 m em
um anel...
será que ocorreria sempre essa
convergência fazendo todos os anéis
ao mesmo tempo ou foi sorte?
DHi (m) DH/Q DHi (m) DH/Q DHi (m) DH/Q
DQ (m³/s) SDH (m) SDH/Q DQ (m³/s) SDH (m) SDH/Q DQ (m³/s) SDH (m) SDH/Q DQ (m³/s)
iteração 11
5.20 26.804.84 66.734.77 50.564.14 137.77-0.54 104.182.98 65.853.85 289.210.41 90.042.66 217.033.51 64.293.57 110.18
-0.02 423.52-0.04 709.85-0.03 521.420.09
11 iterações ou 33 correções de vazões
para resultar erro máximo de 0.05 m em
será que ocorreria sempre essa
convergência fazendo todos os anéis
ao mesmo tempo ou foi sorte?
DHi (m) DH/Q
SDH (m) SDH/Q
utilizando o solver do Excel para resolver simultaneamente as correções dos três anéis
solução inicial final
trecho noi nof L (m) øi (mm) r Q0 (m³/s) Qi (m³/s)1 1 2 700 400 108 0.194 5.20 0.194 5.202 2 3 990 300 620 0.018 0.37 0.073 4.853 2 4 600 300 376 0.149 11.10 0.094 4.774 4 5 600 200 2708 0.040 7.02 0.030 4.145 5 6 500 150 9160 0.017 4.88 -0.005 -0.546 4 7 600 250 913 0.090 10.62 0.045 2.987 7 8 620 150 11359 0.058 58.57 0.013 3.828 8 9 480 150 8794 0.037 19.74 0.004 0.409 5 8 500 150 9160 0.000 0.00 0.012 2.65
10 3 6 500 250 761 0.000 0.00 0.055 3.5211 6 9 450 200 2031 0.000 0.00 0.033 3.59
anel
1 2 10 -5 -4 -3 0.055 -22.64 0.002 4 9 -7 -6 0.045 -62.16 0.003 5 11 -8 -9 0.033 -14.86 0.00
99.66 0.00
DH0 (m) DHi (m)
DQ (m³/s) SDH (m) DQ (m³/s) SDH (m)
S|DH|
trecho noi nof L (m) øi (mm) r1 1 2 700 400 1082 2 3 990 300 6203 2 4 600 300 3764 4 5 600 200 27085 5 6 500 150 91606 4 7 600 250 9137 7 8 620 150 113598 8 9 480 150 87949 5 8 500 150 9160
10 3 6 500 250 76111 6 9 450 200 2031
anel
1 2 10 -5 -4 -32 4 9 -7 -63 5 11 -8 -9
método matricial da linearização - quem se habilita a preparar esse exemplo de aplicação?
n1NA=30,00 m
n2: 27 L/s n3: 18 L/s
n7:32 L/s
n4:19 L/s
n5: 23 L/s
n8:21 L/s
n6:17 L/s
n9:37 L/s
t1,194
t3,149
t2,18
t6,90
t7,58
t4,40
t8,37
t5,17
t10,0
t11,0t9,0
DQ1
DQ2 DQ3
Estimativa inicialde vazões: ti,Qo
