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.5)tQ1h 1in which the area A = A(h) is in general a function of the tank configuration, and thedischarge Q = Q(h) from the steady-flow work-energy principle, Eqs.7.1-7.3.Onecommon additional notational simplification is to let© 2000 by CRC Press LLC11 / 2=L1 + KE + f(7.6)CD so the discharge can be written in the form of the standard orifice equationQ = CAp 2 gh(7.7)Now we apply these equations to determine the time that is required to drain partially atank of constant cross-sectional area A = Ao.This situation includes the common case ofa cylindrical tank of fixed diameter with a vertical centerline, and it also includes tankshaving square and other cross-sectional shapes.In this case the time to drain the tank fromlevel h1 to level h2 ishh∆22−t = t1 / 22 − t 1 = −Adh∫=− Aohdh∫= − 2 Aoh[ 2 − h 1] (7.8)hCA1p2 ghCAp 2 g h 1CAp 2 gin which the removal of C from within the integral is only permissible when f isconstant.If the top and bottom of this fraction are multiplied by the common factorh[ 2 + h 1], an interesting practical interpretation of this result is obtained:A ()∆ t =o h 1 − h 2(7.9)1 CA[] = VolumeAverage Qp2 gh 1 + h 22In words, the elapsed time is the ratio of the tank volume that is emptied to the average ofthe discharges that occur at the beginning and end of the time period, a result that can aidcomputations and is intuitively appealing.For this result to be valid, however, the cross-sectional area and also the friction factor that is a part of C must remain constantthroughout the draining process.If either of the foregoing restrictions does not hold, the integral in Eqs.7.5 and 7.8 willnot simplify as it did in Eq.7.8.For example, if the cylindrical tank is laid on its side,then A(h) no longer is constant.It is then possible (but not very practical) to evaluatethe resulting expression as an elliptic integral (Byrd and Friedman, 1971), but it isnormally more convenient just to evaluate Eq.7.5 by use of some numerical integrationprocedure; the Trapezoidal rule or the more accurate Simpson's rule (Press et al., 1992) arejust two of many possibilities.Closed-form solutions are also known to exist for certainarea variations A(h) with a vertical centerline, specifically the cone, pyramid and parabo-loid, but the form of these solutions is algebraically more complex and of limited utility.The flow defined in Fig.7.3 can be made more general by allowing a nonzero constantinflow Qo at the top of the tank.We will again write the outflow from the pipe in theform of Eq.7.7.At first glance there appear to be two inflow cases, one with Qo > Qand the water surface in the tank rises, and the other with Qo < Q and the water surfacefalls.Such turns out not to be the case, for an individual consideration of each case leadsto the restatement of Eq.7.4 for both possibilities asAdh = (Qo − Q)dt(7.10)If we again assume that A = Ao and f are constants, then Eqs.7.7 and 7.10 lead to© 2000 by CRC Press LLCdt =A⋅dh(7.11)Ca 2 gQo− hCa 2 gWith integration between the same limits as in Eq.7.8, we obtain∆Qt =2 AQo − Q 2 1 − Q 2 − Qo ln(7.12)Ca 2 g()2Q o − Q 1 after some care in integration and several lines of algebra.This result, however, is onlyvalid if Qo is outside the discharge interval ( Q2, Q1); otherwise Eq.7.12 will lead tothe logarithm of a negative number.The cause of this behavior is not difficult tounderstand.During the outflow process the discharge Q takes on all values between Q1and Q2.If Qo were one of these intermediate values, then an equilibrium betweeninflow and outflow in the tank would occur at that discharge, the unbalanced driving forcefor the transient would cease, and the process would not continue on to state 2.Moreover,if the inflow were to match either Q1 or Q2, then Eq.7.12 predicts that the timeinterval that is required to reach the end state is infinite (i.e., a steady equilibrium is neverquite reached, according to this representation of the flow).One additional generalization that can be useful is to allow the inflow to be Qo(t), atime-varying inflow.A re-organization of Eq.7.10 yieldsdh = Qo(t) − Q(h) = F(t, h)(7.13)dtA(h)in which F(t, h) is simply a shorthand, functional representation of the formula thatprecedes it.Only a little effort is needed to convince oneself that this equation can not beintegrated directly as a quadrature.Press et al.(1992) present a chapter on variousalternatives in integrating ordinary differential equations, and others have written entirebooks; Appendix A on numerical methods presents the fourth-order Runge-Kutta formula(Section A.4.2) as one reliable way to solve this kind of problem.The formula in theappendix is described in terms of the variables y(x) which replace the variables h(t) here.The concept of quasi-steady flow can be applied to a variety of system configurations,including some which are much more extensive than the cases discussed here, so long as itis correct to assume that no large accelerations are present in the transient.In fact, theextended-time simulations in Chapter 6 to determine long-term variations in networkdemand are quasi-steady flow applications
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