Exercicio 07 - Redes 2 2012-2 Gab

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