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Do celu tam się wysiada. Lec Stanisław Jerzy (pierw. de Tusch-Letz, 1909-1966)
A bogowie grają w kości i nie pytają wcale czy chcesz przyłączyć się do gry (. . . ) Bogowie kpią sobie z twojego poukładanego życia (. . . ) nie przejmują się zbytnio ani naszymi planami na przyszłość ani oczekiwaniami. Gdzieś we wszechświecie rzucają kości i przypadkiem wypada twoja kolej. I odtąd zwyciężyć lub przegrać - to tylko kwestia szczęścia. Borys Pasternak
Idąc po kurzych jajach nie podskakuj. Przysłowie szkockie
I Herkules nie poradzi przeciwko wielu.
Dialog półinteligentów równa się monologowi ćwierćinteligenta. Stanisław Jerzy Lec (pierw. de Tusch - Letz, 1909-1966)
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.Whendemands are changing throughout a large distribution system, large pressure changes willalter the system demands so the pressure wave is rapidly absorbed.In this case the need foran elastic analysis may be restricted to that pipe, or possibly to it and a few nearby pipes.Rigid column theory can be applied to situations in which the demands on an elasticpipe network change rather rapidly but not instantaneously, causing the inertial effects inaccelerating the liquid to have a significant effect on the pressure.Examples are foundduring the morning hours in a large city when additional pumps must accommodaterelatively rapid increases in demand, or whenever a major user may shut down rapidly.These changes in demand are not so rapid that elastic effects become significant, yet theeffect of accelerating the fluid, owing to the long pipelines that exist between the supplysources and the demand sites, can cause the pressures far downstream in a distributionsystem to be significantly different than would be the case if only fluid friction wereconsidered.If both inertial and elastic effects can be ignored, then a quasi-static or extended timesimulation would be valid for much of the operation of a water distribution system.12.2 RIGID-COLUMN UNSTEADY FLOW IN NETWORKS12.2.1.THE GOVERNING EQUATIONSIn the latter portion of Chapter 7 some unsteady flows in single pipes were studied.That theory assumed the liquid to be incompressible and the pipes to be rigid, thusignoring the elastic properties of the liquid and the pipe.Here this same rigid columntheory will be expanded to multiple-pipe systems.Here we ignore the convectiveacceleration term V∂ V/∂ s for reasons discussed in Section 8.5.2.In the analysis of steadyflows in networks it is also common practice to ignore the difference between the hydraulic© 2000 by CRC Press LLCgrade line and the energy line by assuming they are coincident.This simplification isconsistent with the deletion of the convective acceleration term, and it is standard practicein the application of rigid column theory, so long as velocities are low.Equation 8.59, the equation of motion, can be written asdV = − ∂ Hg− f V V(12.1)dt∂ s2 DSince ∂ H/∂ s is constant along a pipe, it can be expressed as ( Hj - Hi)/ Lk.The subscripts indicate that pipe k has an upstream node i and a downstream node j.Substituting one expression for the other in Eq.12.1 givesdVHfk =i − H jg− kVk Vk(12.2)dtLk2 Dkin which the subscript k has been added to the velocity V , the diameter D, and thefriction factor f, to show the equation applies to pipe k in the system.Usually it ismore convenient to use the discharge Q = VA as a dependent variable in place of V; thenEq.12.2 can be written asdQHfk =i − H jgAkQk Qkk−(12.3)dtLk2 Dk AkFor unsteady flows Eq.12.3 relates the time-varying discharge in pipe k, the frictionalloss, and the instantaneous heads at the end nodes of the pipe.If dQk/dt = 0 so the flowis steady, we recover from Eq.12.3 the Darcy-Weisbach equation itself.Thus Eq.12.3 isthe unsteady-flow analog of the Darcy-Weisbach equation, or an empirical equation such asthe Hazen-Williams formula, for the relation between the frictional head loss and thedischarge.The junction continuity equations must also be satisfied for unsteady flows.Therefore,in addition to the equations that can be written by applying Eq.12.3 to a network, NJ (orNJ - 1 if all external flows are specified) junction continuity equations must be written,one for each node, in the formQk∑ − QJi = 0(12.4)Here the summation includes all pipes that join at junction i, and QJ i is the demand atthis junction.In Eq.12.4 the discharge is positive if it flows into junction i andnegative if it flows from the junction.12.2.2.THREE-PIPE PROBLEMWe begin by describing how Eqs.12.3 and 12.4 can be used to model the unsteady flowin a small network.For this example we select the three-pipe network in Fig.12.1.Since all external flows are specified, there are NJ - 1, or 2, junction continuityequations for this network.These continuity equations areF 1 = Q 1 − Q 2 − QJ 2 = 0(12.5)F 2 = Q 2 + Q 3 − QJ 3 = 0These two equations require the negative demand QJ1 at node 1 to equal the sum of the© 2000 by CRC Press LLCH2QJ[2]2(1)HQJ11[1](2)(3)H3[3]QJ3Figure 12.1 Three-pipe network.positive demands QJ2 at node 2 and QJ3 at node 3.In addition, the following threeordinary differential equations apply, one for each pipe:dQf1 =HgA1 − H 21 Q 1 Q 11−dtL 12 D 1 A 1dQf2 =HgA2 − H 32 Q 2 Q 22−(12.6)dtL 22 D 2 A 2dQf3 =HgA1 − H 33 Q 3 Q 33−dtL 32 D 3 A 3In this network we assume that the head H1 is constant since the fluid is supplied from areservoir.Therefore, if the demands QJ2 and QJ3 are specified for all time, then the fivevariables Q1, Q2, Q3, H2, and H3 are the unknown variables in this network.To determine five unknown variables, we must have five independent equations.In thisproblem Eqs.12.5 and 12.6 satisfy this requirement.But how can this system ofequations be solved when some of the equations are ordinary differential equations (ODEs)rather than algebraic equations? If a solution can be found, it must then be appliedrepeatedly as time advances.Thus such a solution is far more than a single steady-flowsolution
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