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.In a branched system there are by definition noloops, and thus NL = 0 for any branched system.In branched systems the number ofnodes is always one larger than the number of pipes, or NP = NJ - 1, unless a reservoir is© 2000 by CRC Press LLCshown at the end of one pipe and this is not considered to be a junction.Then NP = NJ.(This situation actually occurs.) Figures 4.1a and 4.1b depict a small branched networkand also a small looped network.[1][2][3][1][2][3](1)(2)(1)(2)(3)(5)[4](3)(4)[5][4][5][6](4)(5)(6)(7)[6](6)(8)(9)(10)[7][7](11)[8](12)[9]Figure 4.1 (a) A small branched system.(b) A small looped system.6 pipes, 7 nodes 12 pipes, 9 nodesIn the branched system the number of nodes is 7 and the number of pipes is 6 (one lessthan the number of nodes), whereas in the looped system there are 12 pipes and 9 nodes,i.e., the number of nodes is less than the number of pipes.For a looped network the number of loops (around which independent energy equationscan be written) is given byNL = NP − NJ(4.1)if the network contains two or more supply sources, orNL = NP − ( NJ − 1 ) = NP − NJ + 1(4.2)If the network contains fewer than two supply sources and the flow from the single sourceis determined by adding all of the other demands, then this source is shown as a negativedemand and the source is called a node.We note that this is the case in the small loopednetwork in 4.1.b, so we have NP = 12, NJ = 9 and NL = 12 - (9 - 1) = 4.Equation 4.2 also applies to a branched system with NL = NP - NJ + 1 = 0, since abranched system can have at most one supply source.Actually, every pipe system musthave at least one supply source, but sometimes the source is not shown since the dischargefrom this supply source is known, and the source is replaced by a negative demand, whichis a flow coming into this junction, equal to the sum of the other demands.When this isdone, the elevation of the energy line (or HGL or pressure) must be specified at a node sothe other HGL elevations can be determined.Energy loops that begin at one supplysource and end at another are called pseudo loops, i.e., these loops do not close onthemselves.The number of pseudo loops, which are numbered as part of NL, equal thenumber of supply sources minus one.In forming pseudo loops all supply sources must belocated at the end of a pseudo loop.It is generally possible to form more loops than areneeded to produce a set of independent equations.As each new loop is formed, see that atleast one pipe in the new loop is not a part of any prior loop; in this way the formation ofredundant loops can usually be avoided.For special devices, such as pressure reductionvalves, this rule of experience must be modified slightly, as will be described later.4.2 EQUATION SYSTEMS FOR STEADY FLOW IN NETWORKSThree different systems of equations can be developed for the solution of networkanalysis problems.These systems of equations are named after the variables that areregarded as the principal unknowns in that solution method.These systems of equationsare called the Q-equations (when the discharges in the pipes of the network are the© 2000 by CRC Press LLCprincipal unknowns), the H-equations (when the HGL-elevations, also simply called the heads H, at the nodes are the principal unknowns), and the ∆ Q-equations (whencorrective discharges, ∆ Q, are the principal unknowns).Each of these three systems ofequations will be studied separately.4.2.1.SYSTEM OF Q - E Q U A T I O N SThe analysis of flow in pipe networks is based on the continuity and work-energyprinciples.To satisfy continuity, the volumetric discharge into a junction must equal thevolumetric discharge from the junction.Thus at each of the NJ (or NJ - 1) junctions anequation of the form of Eq.4.3 is obtained:QJ j − Σ Qi = 0(4.3)In this equation QJj is the demand at the junction j, and each Qi is the discharge in oneof the pipes that join at junction j.These junction continuity equations are the firstportion of the Q-equations.The work-energy principle provides additional equationswhich must be satisfied.These equations are obtained by summing head losses along bothreal and pseudo loops to produce independent equations.There are NL of these equations,and they are of the form of Eq.4.4 or 4.5, depending upon whether the loop is a real loopor a pseudo loop, respectively, and they are the second portion of the Q-equations:Σ hfi = 0(4.4a)Σ hfi = ∆ WS(4.5a)When the head losses are expressed in terms of the exponential formula, then theseequations take the forms∑ K niQ= 0(4.4b)i∑ K niQ= ∆ WS(4.5b)iin which the summation includes the pipes that form the loop.If the direction of the flowshould oppose the direction that was assumed when the energy loop equations were written,such that Qi becomes negative, then there are two alternatives: One is to reverse the signin front of this term, i.e., correct the direction of the flow.The second, which is generallypreferred when writing a program to solve these equations, is to rewrite the equations asfollows:Ki Qi Qi∑n−1= 0(4.4c)Ki Qi Qi∑n−1= ∆ WS(4.5c)To illustrate the system of Q-equations, consider the small 5-pipe network shown inFig.4.2.Since no supply sources are shown for this network, only NJ - 1 junctioncontinuity equations are available
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