QUASI-STEADY FLOW IN HYDRAULIC SYSTEMS USING MOC. E. Benjamin Wylie University of Michigan Ann Arbor, Ml

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1 IPC QUASI-STEADY FLOW IN HYDRAULIC SYSTEMS USING MOC E. Benjmin Wylie University of Michign Ann Arbor, Ml ABSTRACT The Method of Chrcteristics (MOC) is utilized effectively in mny industries to clculte hydrulic trnsients in lmost ll pipeline systems. Recently it hs been suggested s suitble clcultion procedure for long term (extended time) qusi-stedy flow problems. It might serve s n lterntive pproch to using rigid wter-column (lumped inerti) clcultions, or to integrting over time with series of stedy stte clcultions, which re two procedures receiving recent ttention. This pper provides unique pproch within the MOC frmework, by modifying terms in the pipeline prtil differentil equtions. When crried through the MOC it leds to minor modifiction in trnsient code by introducing the two new prmeters. The method is developed nd pplied to network exmple herein. INTRODUCTION In liquid pipeline systems four ctegories of flows re generlly recognized with respect to stediness - three dynmic nd one stedy. For nlysis or design, specific models offer dvntges for ech type. Cbrer et l (1995), mke the following suggestion, in which the reference times re order of mgnitude only: ) Use n elstic model when there is sudden vrition (seconds or less) in pressure nd flow. b) Use rigid model when there is significnt vrition in the system's min vribles, perhps due to rpid consumption chnge where the liquid's ccelertion remins importnt (minutes). c) Use qusi-sttic model (or extended-period simultion) when there is slow vrition (hours) in the system's opertion, such s tnk level or consumption chnges. d) Use sttic or stedy-stte model when there is no temporl chnge in vribles. is not efficient computtionlly. Prcticlly, if one's interest is primrily in prticulr one of these domins the bove ctegoriztion nd use of models is most pproprite. However, there is some ppel to the concept of performing stedy nd unstedy clcultions within the sme code utilising the sme dt set, prticulrly in more complex hydrulic systems. Fox nd Keech (1975) nd Vrdy nd Chn (1983) presented method using MOC for the expressed purpose of chieving stedy stte in hydrulic system prior to performing trnsient study. Shimd (1986) presented definitive work on how to optimize prmeter selection to chieve stedy stte most efficiently in given simple hydrulic system. In ech method the wve speed is djusted in every pipeline in the system nd ech pipe is treted s one rech regrdless of length. Shimd (1988) further developed his method for more complex systems by introducing common pipeline length throughout the system, by vrying the wve speed in ech pipe to chieve common time step, nd djusting the friction fctor to correct the friction loss. This mens preprocessor is needed, or specific dt set or specil coding is needed to estblish stedy stte prior to inititing the code for the trnsient event. With proper implementtion, Shimd's two-step method provides n efficient convergence to blnced initil condition. Luvizotto et l (1997) nd Righetto & Porto (1997) present the wve-speed vrition method in MOC s possibility for use in studying systems with slow vrition in vribles where inertil effects re still importnt. There re number of resons for the interest in this topic, not the lest of which is the feet tht few systems ever relly operte in true stedy-stte mode. Other resons might include clibrtion of the hydrulic performnce using rel qusi-stedy dt, btch trcking in long petroleum-product systems, trcking contminnt trnsport in piping networks, opertion mode decisions to chieve certin desirble pressure distributions or vilbility of product for specil need, etc. The concepts introduced herein my be extended to include these pplictions, long with others. From theoreticl view it is possible to use n elstic dynmic model to study system vribles under ll conditions, but it currently Copyright 2000 by ASME

2 A recent publiction, Wylie (2000), offered slightly different pproch within the MOC frmework, which enbled the use of stndrd MOC system model to be effectively used in estblishing stedy stte. The method involved modifiction of terms in the pipeline prtil differentil equtions, nmely the use of two mulipliers, n inertil multiplier in the Eqution of Motion nd n elsticity multiplier in Continuity. When crried through the MOC to numericl method nd ultimtely to robust trnsient code it involves only minor modifiction in dt by dding two new prmeters. In this pper the sme method is shown to be potentilly ttrctive nd prcticl in the intermedite region between stedy stte nd elstic wter hmmer problems. In the next section the prmeters re introduced, their physicl significnce is explined nd the method is developed. It is then pplied to n exmple. INERTIAL AND ELASTICITY MULTIPLIERS An inertil multiplier,, nd n elsticity multiplier, /?, re introduced to the prtil differentil equtions of momentum nd continuity, respectively, Wylie nd Streeter (1993). The modified equtions re: H x + Q t + & =0 * 2gDA] ga (i) (2) in which H = p/pg + z = piezometric hed, p = gge pressure, p = mss density, z = elevtion bove reference dtum, Q - volumetric flow rte, = distnce, T = time, = wve speed in liquid-filled pipeline, D = pipe dimeter, A r = pipe cross-sectionl re, / = Drcy-Weisbch friction fctor, nd g = grvittionl ccelertion. Following stndrd MOC trnsformtion the equtions become: RN ±DH+A/3BDQ +F^-Q 2 D = 0 (3) the elsticity term in Eq. (2), the effective impednce (5B in Eq. (3), nd the effective wve propgtion velocity f}l in Eq. (4) re ltered proportionlly. When * 1 the inerti term in Eq. (1) nd the effective impednce in Eq. (3) re ltered proportionlly, while the effective wve propgtion velocity in Eq. (4) is ltered inversely. For illustrtion, if («= /}) < 1, Eq. (1) shows the system would respond dynmiclly with lower inerti, while Eq. (2) shows the pprent elsticity would be less, providing softer, more complint system. If ( = P) > 1, Eq. (1) shows the system would respond dynmiclly with higher inerti, while Eq. (2) shows the pprent elsticity would yield stiffer, less complint system. The equtions my be non-dimensionlized to: ±dh+pdq + R L q 2 dx = 0 N in which h = H/H s,q = QIQ 0,x = /L,t = T/L/, Q 0 = stedy stte flow, RL = H/H s friction prmeter, H/= stedy stte friction loss, nd H s = potentil surge R,= H s = BQ 0 JL Q 0 _H f 2D A r H, (7) (8) (9) (10) In this form it my be seen from Eqs. (7) tht the behvior of different pipelines is chrcterized by the prmeter RL when A = P = 1, Liou (1991). The pipeline dynmic response lso depends on the prticulr end conditions. When A nd J3 re differentfromunity the pipeline response depends on A, /?, RL, nd the end conditions. A stbility nlysis of the linerized form of Eqs. (8) nd Eqs. (7), withfirst-orderfriction,shows tht p d = ± dt (4) in which the pipeline of length L is divided into N reches, A = L/N,B = pipe chrcteristic impednce, nd R = resistnce coefficient, defined s B=g^r (5) fa fl R = 2gDA 2 r 2gDA 2 (6) rn It should be noted tht if = p = 1 the equtions reduce to the stndrd set normlly utilized for trnsient studies. Since it is the intent to ssign to nd /? vlues different from unity it is of interest to note the effect of other vlues. When /? * 1, N <p (II) the generlly recognized vlue when = ft = 1, Shimd (1988), Wylie nd Streeter (1993). Further, for resonble numericl ccurcy with second-order representtion of the friction term in Eqs. (7), RJ N f}, Wylie (1996). During stedy stteflowthe inertil (second) term in Eq. (1) is zero, nd the elsticity (second) term in Eq. (2) is lso zero since HT must be zero. Thus or fi my tke on ny vlue without impcting stedy stte conditions. If prticulr vlues of or fi enble numericl procedure to chieve stedy stte more redily from n rbitrry initil condition it would be dvntgeous. Shimd's method (nd its predecessors) ccomplished this by chnging the wve propgtion velocity directly. In terms of the or p multipliers, Shimd's first method my be chieved by setting = 1 nd llowing p to tke on vlues different from unity to provide the desired wve speed. For non-stedy conditions the inertil nd

3 elsticity terms hve clerly been modified if nd P re not equl to unity. However, when the rigid or qusi-sttic model is pproprite = 0 in Eq. (2), so ysmy tke on vlues different from 1 if there is computtionl dvntge. Similrly, if the inertil term is smll in Eq. (1) the multiplier my tke on vlues different from 1 s long s the term remins insignificnt with respect to the other two terms in the Eqution of Motion. The use of n MOC model for rigid nd/or qusi-stedy problems my be prcticl nd efficient by selecting nd p ppropritely, s discussed below. s impednce mtching of pipelinefrictionwith pipeline impednce. In the context of the nottion herein the dimensionless impednce, f}, would be set equl to the dimensionless friction prmeter, RL. Any combintion of nd p to yield RL my be used to estblish stedy stte, wheres vlues tht yield P> 1 re recommended for rigid nd qusi-stedy conditions. p=^- = R L (12) Numericl method The bsic equtions in MOC re modified by introducing nd p. This chnges the effective pipeline impednce to pb, in ccordnce with Eq. (3), nd the effective wve propgtion velocity in the pipeline to p/ in ccordnce with Eq. (4). Ech of nd p directly chnge the pipeline chrcteristic impednce proportionlly. However, smller increses the effective wve speed without chnging the pipe elsticity, while smller P decreses the effective wve speed without influencing the pipe inertnce. Most significntly for qusi-stedy flow vlue of the rtio /p greter thn 1 permits the use of lrger time step for the sme distnce intervl, in ccordnce with Eq. (4). The convergence efficiency in the rigid or qusi-stedy process is regulted by the numericl vlue of the effective chrcteristic impednce, /pb, just s it is the importnt prmeter controlling the reltive ese or difficulty in mrching towrds stedy stte in system. To begin, it is ssumed the hydrulic system hs been schemtized for numericl unstedy-flow solution by MOC. Tht is, the fetures importnt to dynmic nlysis hve been identified, nd the physicl properties of the skeletonized system re known. Pipelines in the system hve been divided into reches to llow common time step throughout the system for trnsient clcultions. Initil conditions, normlly stedy flow, re needed prior to inititing the unstedy event. To perform this step die new prmeters re set, nd stedy stte is ccomplished by stepping through the trnsient portion of the code using step size from Eq. (4), step size tht hs nothing to do with rel time increment. As the literture hs shown this process my be surprisingly efficient. Once this itertion process hs chieved stedy stte within specified tolernce, the code my proceed directly to the dynmic nlysis. If stedy stte is known initilly this step my be bypssed. The dynmic solution is next sought. For wter hmmer event the two prmeters re set to unity. This returns the numericl method to the correct unstedy flow equtions before inititing the desired trnsient event. For n event in which either the rigid or qusi-stedy conditions re pproprite nd /?must be selected judiciously to provide for n optiml solution. For most systems the optiml dimensionless impednce, p, is best used, nd the rtio / ys > 1 is recommended to increse the time step to vlue s lrge s possible to efficiently simulte the longer durtion trnsient. To estblish stedy stte in single pipeline system Shimd (1988) suggested tht reduced wve speed be selected, ', to yield, 'Q /(ga,) = H f. Wylie (2000) demonstrted tht this ws the sme Complex Systems In estblishing stedy stte from n rbitrry initil condition in complex pipe systems it is importnt tht resonble initil vlues be ssumed for pressure cross the system. Additionlly, n estimte for the vlue of p is needed prior to the first run. A procedure is suggested which borrows from single pipeline. A resonble velocity must be initilly ssumed in ech pipe to enble n pproximte clcultion of R L =p. Then the vlue for the system my be obtined by weighting the individul element vlues in proportion to the number of reches in the element. P = T-^-Rl, (13) y=1 I J=1 Here j = element number nd n = the number of elements in the system. The sme vlue of the product p will be most effective during the unstedy clcultion, however for the slow unstedy process, it is desirble to select nd p so the rtio /p is s lrge s fesible to yield lrge time step (refer to Eqs. (4) nd (8)). For low inerti (very slow) problems my be lrger thn in problems in which higher ccelertions exist. Figure 1, grph showing vs. P, my help in the selection nd visuliztion. The negtivelysloped digonls represent vlues of constnt dimensionless impednce, with p =1 highlighted. The positively-sloped digonls represent constnt vlues of /p, which dictte the size of the time step. The highlighted vlue, /p=\, shows condition in which the Cournt condition in MOC would be stisfied. With constnt impednce (p), should be s lrge s possible to llow s lrge time step s fesible while still yielding relible solution. Depending upon the ccelertion in the rigid cse my be s lrge s 3. Similrly, depending upon the slowness of the qusi-stedy cse my be of the order of 10. APPLICATION TO NETWORK Figure 2 shows 22-element system including pump with dischrge vlve, 16 nodes including 2 lrge reservoirs nd one smller reservoir with over-flow weir, 2 nodl flow demnds, nd 7 loops. The system represents modifiction to system first presented by Vrdy nd Chn (1983) nd lter used by Shimd (1986). The element numbers, lengths, dimeters, friction fctors, wve speeds, nd number of reches in ech line re provided in Tble 1. With = p=\, the necessry MOC time step is seconds. Initil conditions involved zero flow ssumed in ech element, nd nodl pressure heds ssumed to vry incrementlly

4 Tble 1. Dt ssocited with network, Figure 2 pipe L D f N Flow Upstr. no. m m m/s m A 3/s H, m pump DH= *Q-23.83*Q A vlve CDA = tnk,ar=20 m A 2,El.=9.75m,Coef = between the reservoir levels. The stedy stte pipeline flows nd nodl pressure heds s determined by the nlysis re lso provided in Tble 1. When the system is operting with the pump running nd the vlve open Eq. (13) yields vlue of fi « With this vlue, nd using = 1, P = pproximtely 500 itertions re needed to chieve stedy stte, wheres pproximtely 8,000 itertions would be needed to chieve stedy stte if = 1, p = 1. When the system is operting with the pump off nd the vlve closed Eq. (13) yields vlue of p « These two vlues of the dimensionless impednce hve significnce in the vrious operting modes of the exmple tht follows. In this exmple, the network system is operting in the stedy condition provided in Tble 1. An unstedy flow is imposed by incresing the flow out of nodes 6 nd 8, using the sinusoidl reltionship, Q n = &Q n (1. - cos(2;zt / period)), in which the period ws 12 hours, AQ s = 0.04 m 3 /s, nd AQ t = 0.2 m 3 /s. The time step for the elstic nlysis, = P = 1, ws seconds. Figure 3 shows results t nodes 6. 8, nd 16 for the elstic nlysis. Bsiclly the sme results were obtined using = 3 nd p = 0.03, for which P = /p = 100, nd At = 2.5 s. The use of other vlues of nd P to provide lrger time steps produced results shown in Figs. 4 nd 5. The most pronounced vrition occurred Tble 2. Number of itertions for 12 hours Cse P <x0 ct/p dt No , during the pek flows between 4 nd 8 hours. Figure 4 shows n expnded view of the pressure hed t nodes 6 nd 8, while Fig. 5 provides the pressure hed nd flow t node 16, the overflow weir. Three cses re listed in Tble 2, in which the lst column refers to the number of itertions to simulte the 12-hour trnsient. As stted bove the results from cse 1 gree with the correct numericl solution when = p = 1. The results from cse 3 with At = 60 s re probbly uncceptble, however, the results from cse 2, with At = 20 s, re similr to results produced by other methods nd re quite cceptble for mny pplictions. In cse 2, with At = 20 s, 2160 itertions were needed to simulte 12 hours. This is the sme order of mgnitude s might be used during simultion of typicl elstic trnsient event on this system, CONCLUSIONS The purpose of this pper ws to provide method to numericlly model the complete rnge of time responses for pure liquid flows in complex hydrulic system. This opportunity is mde possible by the introduction into MOC n inertil multiplier,, nd n elsticity multiplier, tht my tke on different numericl vlues depending on the type of flow. The ctegories of flow re stedy stte, qusi stedy, rigid or inelstic (lumped) unstedy, nd elstic unstedy. The thought is tht the user knows in dvnce which ctegory is nticipted, nd utilizes the model effectively for tht event At this stge it is unlikely the method could be too effective in hndling ll ctegories of flow in one event lthough it is conceptully possible. MOC is well known for its success in deling with elstic unstedy events. This vrition in the method offers n lterntive to simultneous solution procedures for stedy stte cses, to qusi-sttic models for extended-period, nd to rigid models for lumped-inerti events. The focus ws to present method tht ) requires only minor ltertions to n existing MOC code, nd minimiztion of dditionl dt requirements, b) provides wy to select the prmeters to mtch impednces so s to numericlly minimize the number of oscilltions in the fictitious strt-up event in estblishing stedy stte, nd c) demonstrtes the benefits of djusting inerti or elsticity to increse the elstic-model time-step by s much s 3 orders of mgnitude for slow unstedy motion, while fithfully trcking the temporl chnges in vribles. In ddition to the development of the method, the physicl significnce of the new prmeters ws explined, the interction of pipeline impednce with pipeline friction ws demonstrted, the method ws pplied to complex network, nd the method ws relted to other procedures.

5 REFERENCES Cbrer, E., Grci-Serr, J., nd Inglesis, P. L., 1995, "Modeling Wter Distribution Networks: From Stedy Stte to Wter Hmmer," E. Cbrer nd A. F. Vel, eds., Improving Efficiency nd Relibility in Wter Distribution Systems, Kluwer Acdemic Publishers, Fox, J. A., nd Keech, A. E., 1975, "Pipe network Anlysis - Novel Stedy Stte Technique," J. Inst. Wter Engineers & Science, 29, Liou, C. P., 1991, "Mximum Pressure Hed Due to Liner Vlve Closure," J. of Fluids Engineering, ASME, Vol. 113, Dec., Luvizotto Jr., E., Koelle, E., nd Andrde, J. G., 1997, "Mngement nd Control of Wter Pipeline System using the Elstic Model," 3 rd Int. Conf. On Wter Pipelines Systems, ed. R Chilton, BHR, Crnfield, UK, Righetto, A. M., nd Porto, R.M., 1997, "Anlysis of Lrge Urbn Wter Distribution Network by the MOC," 3 rd Int. Conf. On Wter Pipelines Systems, ed. R Chilton, BHR, Crnfield, UK, Shimd, M., 1986, "Advnces in Numericl Anlysis using MOC," 5 th Int. Conf. On Pressure Surges, British Hydromechnics Reserch Assoc., Shimd, M., 1988, "Time-Mrching Approch for Stedy Flows", J. Hydrulic Engineering, ASCE, Vol. 114, No. 11, Vrdy, A. E., nd Chn, L. L, 1983, "Rpidly Attenuted Wter Hmmer nd Steel Hmmer," Pper Al, 4 th Int. Conf. On Pressure Surges, BHRA, Univ. ofbth, U.K., Wylie, E. B., 1996, "Unstedy Internl Flows - Dimensionless Numbers & Time Constnts," 7 th Int. Conf. On Pressure Surges & Fluid Trnsients, BHR Group, Wylie, E. B., 2000, "Stedy Stte in Hydrulic Systems by Mrching Methods," 8 th Int. Conf On Pressure Surges,, BHR Group, The Hgue, Netherlnds, April, Wylie, E. B., nd Streeter, V. L., 1993, FLUID TRANSIENTS IN SYSTEMS, Prentice Hll, New Jersey. \ P=I / \ / \ / \ / \ r' / \ / %< w A x - f / ' / * / / / 11 V 7 V, / ir 8 \ x \ / '? < \ ' -.. Figure 1. vs. p showing dimensionless impednce nd time-step vrition.

6 Figure 2. Network, includes 19 pipelines, 1 pump with vlve, 2 flow demnds, nd 3 reservoirs. Time, hr Figure 3. Outflow t nodes 6, 8, nd 16; pressure hed t nodes 6, 8, nd 18.

7 6 Time,hr Figure 4. Expnded scle of pressure heds t nodes 6 nd 8 for 3 cses. Time,hr Figure 5. Expnded scle of pressure hed nd outflow t node 16 for 3 cses.