PS3_sol_S08 - Problem Set 3 (2.006, Spring 08) Problem 1...

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Problem Set 3 (2.006, Spring 08) Problem 1 (a) Neglect any gravitational head change. It is expected that flow in the bypass line flows from 2 to 1 since pressure at 2 is higher. Then there are two conditions that the configuration should satisfy. 1) mass conservation: outside pump bypass VV V =− && & 2) pressure matching: 2 21 30 750 pump PP hHV g ρ == = & and 2 2 bypass bypass L hf K gg D υ ⎛⎞ + ⎜⎟ ⎝⎠ Loss at the exit (K~1.0, see Fig. 6.21, White, 6 th ed.) should be added to the sum of minor losses. Therefore 0.2 0.2 0.2 1 1.6 KK K =++++ =+ , where control valve loss 12 0 V K = . 30 750 pump h V = & and ( ) () 2 2 9.806 / 2 1.5 1.6 0.1 bypass V bypass ms h gh L m fK d m + ++ where friction factor f is from Haaland’ formula (Eq. 6.49, White 6 th ) 2 1.11 1 6.9 / 1.8log Re 3.7 d f d ε = ⎡⎤ + ⎢⎥ ⎣⎦ , ( ) ( ) 3 1000 / 0.1 Re 0.001 d kg m m d Pa s ρυ μ bypass is iteratively calculated from given h and K V , and volumetric flow rate in bypass line is calculated by 2 4 bypass bypass d V π = & . Then calculate outside pump bypass V & . 2 bypass V & pump V & pump V & bypass V & 1
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Closed K V =1 K V =5 K V =10 K V =20 h (m) outside pump VV = && bypass V & outside V & bypass V & outside V & bypass V & outside V & bypass V & outside V & 0 0.2000 0.0000 0.2000 0.0000 0.2000 0.0000 0.2000 0.0000 0.2000 1 0.1966 0.0207 0.1760 0.0133 0.1833 0.0101 0.1865 0.0074 0.1892 2 0.1932 0.0293 0.1639 0.0188 0.1744 0.0143 0.1789 0.0105 0.1827 3 0.1897 0.0359 0.1538 0.0231 0.1667 0.0175 0.1722 0.0129 0.1768 4 0.1862 0.0415 0.1447 0.0266 0.1596 0.0202 0.1660 0.0149 0.1713 5 0.1826 0.0464 0.1362 0.0298 0.1528 0.0226 0.1600 0.0166 0.1659 6 0.1789 0.0508 0.1280 0.0326 0.1463 0.0248 0.1541 0.0182 0.1607 7 0.1751 0.0549 0.1202 0.0352 0.1399 0.0268 0.1484 0.0197 0.1554 8 0.1713 0.0587 0.1125 0.0377 0.1336 0.0286 0.1427 0.0211 0.1502 9 0.1673 0.0623 0.1050 0.0400 0.1274 0.0303 0.1370 0.0223 0.1450 10 0.1633 0.0657 0.0976 0.0421 0.1212 0.0320 0.1313 0.0235 0.1398 11 0.1592 0.0689 0.0903 0.0442 0.1150 0.0336 0.1256 0.0247 0.1345 12 0.1549 0.0720 0.0829 0.0462 0.1087 0.0351 0.1199 0.0258 0.1291 13 0.1506 0.0749 0.0756 0.0481 0.1025 0.0365 0.1141 0.0268 0.1237 14 0.1461 0.0778 0.0683 0.0499 0.0962 0.0379 0.1082 0.0279 0.1182 15 0.1414 0.0805 0.0609 0.0516 0.0898 0.0392 0.1022 0.0288 0.1126 16 0.1366 0.0832 0.0535 0.0533 0.0833 0.0405 0.0961 0.0298 0.1068 17 0.1317 0.0857 0.0459 0.0550 0.0767 0.0417 0.0899 0.0307 0.1010 18 0.1265 0.0882 0.0383 0.0566 0.0699 0.0429 0.0836 0.0316 0.0949 19 0.1211 0.0906 0.0305 0.0581 0.0630 0.0441 0.0770 0.0325 0.0886 20 0.1155 0.0930 0.0225 0.0596 0.0558 0.0453 0.0702 0.0333 0.0822 21 0.1095 0.0953 0.0142 0.0611 0.0484 0.0464 0.0632 0.0341 0.0754 22 0.1033 0.0976 0.0057 0.0625 0.0407 0.0475 0.0558 0.0349 0.0684 23 0.0966 0.0998 -0.0031 0.0640 0.0327 0.0485 0.0481 0.0357 0.0609 24 0.0894 0.1019 -0.0125 0.0653 0.0241 0.0496 0.0399 0.0365 0.0530 25 0.0816 0.1040 -0.0224 0.0667 0.0150 0.0506 0.0310 0.0372 0.0444 26 0.0730 0.1061 -0.0330 0.0680 0.0050 0.0516 0.0214 0.0380 0.0351 27 0.0632 0.1081 -0.0449 0.0693 -0.0061 0.0526 0.0106 0.0387 0.0245 28 0.0516 0.1101 -0.0584 0.0706 -0.0189 0.0536 -0.0019 0.0394 0.0122 29 0.0365 0.1120 -0.0755 0.0718 -0.0353 0.0545 -0.0180 0.0401 -0.0036 30 0.0000 0.1140 -0.1140 0.0731 -0.0731 0.0554 -0.0554 0.0408 -0.0408 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20 0 5 10 15 20 25 30 h ( m ) V outside (m 3 /s) Bypass Closed K V =20 K V =10 K V =5 K V =1 Pump operation curves are valid only when 0 outside V &
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(b) At h=0m, all the graphs fall onto 3 0.2 / outside Vm s = & . It is impossible to change flow rate by using the control valve.
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PS3_sol_S08 - Problem Set 3 (2.006, Spring 08) Problem 1...

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