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.One valid set of Qoi's is Qo1 = 1.0, Qo2 = 1.0, Qo3 = 0.0, Qo4= 1.0, Qo5 = 1.0, Qo6 = 0.0, and Qo7 = 0.Finally, the pump head now becomeshp1 = A Q( o 1 + ∆ Q 1 − ∆ Q 3)2 + B Q( o 1 + ∆ Q 1 − ∆ Q 3)+ C when the pump curve is fittedwith a second-order polynomial.If desired, as an alternative either a linear or a higher-orderpolynomial could be chosen to describe the operating characteristics of this pump.Now let us revisit the network in Fig.4.10 that contains a BPV as a second illustrationof forming the ∆ Q-equations.In this analysis we can visualize the two sets of loops asshown in Fig.4.17.The ∆ Q loops ignore the presence of the BPV in this network, but the energy loops will be written for the modified network with the BPV converted into anartificial reservoir.The resulting ∆ Q-equations for this network appear as Eqs.4.31.InF 1 = K 1 Q( o 1 + ∆ Q 2) n 1 − hp 1 − K 2 Q( o 2 − ∆ Q 2 − ∆ Q 3) n 2 + 20 = 0F 2 = K 1 Q( o 1 + ∆ Q 2) n 1 − hp 1 + K 3 Q( o 3 − ∆ Q 3) n 3+ K' Q() n 4 +15 = 0(4.31)4o 4 − ∆ Q 1 − ∆ Q 3F 3 = K 9 Q( o 9 + ∆ Q 3) n 9 + K 6 Q( o 6 − ∆ Q 1) n 6 − K 7 Q( o 7 + ∆ Q 1) n 7− K'8 Q( o 8 + ∆ Q 1) n 8 + K Q() n 4 + 60 = 04o 4 − ∆ Q 1 − ∆ Q 3I∆Q200 m150 - 20002[1] 250 - 1200(2)140(1)0.015 m3/s00mDiameters in mmII(3)1P1Lengths in m-∆Q30.015 m3/s300HGL = 195 m0.02 m3/sm200 - 200070180 m[3][2](4)800 mmBPV200100 (5)150-(8)-∆15000.03 m3/sQ1(9)m135 m150(6)[5]60(7)[6]250 - 1000[4]m150 - 1500100 m150 - 1200 80 0.02 m3/sIII0.02 m3/sFigure 4.17 The network of Fig.4.10, modified for the ∆ Q-equation system.these equations hp 1 = A Q( o 1 + ∆ Q 2)2 + B Q( o 1 + ∆ Q 2)+ C.The initial flows that satisfythe junction continuity equations are chosen as Qo1 = 0.1, Qo2 = 0.0, Qo3 = 0.085,Qo4 = 0.015, Qo5 = 0.0, Qo6 = 0.0, Qo7 = 0.02, Qo8 = 0.05, and Qo9 = 0.02.The substitution of these values into Eqs.4.31 yields the final set of ∆ Q-equations.If large differences in ground elevation occur in a network, PRV's are often installed in asequence of pipes to prevent excessively large pressures in the lower part of the network.Such a series of PRV's may cause pressures in one subregion to be completely independentof the remainder of the network.Such isolation creates what are commonly called separatepressure zones.When separate pressure zones are created, it is normally better to form sub-networks and analyze each one separately, starting with the subnetwork at the lower eleva-tion.The solution from the isolated lower subnetwork can then be used to determine thedemands at the nodes of the next higher network, and so on.© 2000 by CRC Press LLCThe 10-pipe, 6-node network in Fig.4.18 contains three PRV's in pipes 4, 5, and 7, respectively; it typifies such a situation.In this network the three PRV's cause thepressures at nodes 4 and 6 to be independent of pressures in the remainder of thenetwork.The best analysis, therefore, would begin by studying separately the subnetworkthat is composed of pipes 5, 4, 7, and 8 downstream from the PRV's.In thissubnetwork the PRV's are modeled as three constant-head reservoirs.The values of Q4,Q5, and Q7 from the solution of the subnetwork are next added to the other demands todetermine the demands at nodes 3, 2, and 5, respectively, in organizing the remainder ofthe network for analysis.400'500''0000'P1(9)103.5 ft3/s10-(10)-"[5][1]6" - 2000'10400' [2] 6" - 1000'10" 400'400'(1)'(6)' 0 0500700'(PRV)532(PRV)-(HGL) = 150'(3)'136"(HGL) = 150'(7)1(2)7 (4)- 2000'1000'All pipesP326"--e = 0.002"8"8"[3](PRV)2[4]6" - 800'(5)6" - 3300'350'60'400'60'(8)[6](HGL) = 150'1.0 ft3/s20.5 ft3/s0.5 ft3/sPump CharacteristicsPump 1 Pump 2QHQHft3/sft.ft3/sft.1.51101.515.02.51042.012.03.5923.07.5Figure 4.18 A network with two pressure zones.While it is generally not difficult to determine by visual examination of a map of thepiping system whether PRV's isolate a portion of a network into a separate pressure zone,in computer programs a simple test is needed to identify this situation.Such a test can bebased on the fact that no series of connected pipes exists between any of the artificialreservoirs created by the PRV's and any of the other reservoirs and source pumps.That noconnection exists in the network example can be seen by resketching the network, asshown in Fig.4.19.As a consequence, if pseudo loops between artificial reservoirs or source pumps cannot be found by a computer program that uses its own internal loop-finding algorithm, then the PRV's isolate a subnetwork into a pressure zone that isseparate from the remainder of the network.One difficulty with this kind of test, whichrelies on the inability to find paths which connect all supply sources, is that errors in thenetwork input data or an ill-defined network itself can also cause this test to be satisfied;network computer programs are supposed to identify such input errors and terminate if anysuch errors are found.Thus it is desirable to have an independent verification, i.e [ Pobierz całość w formacie PDF ]