ASCE and Qinfen Zhang PhD Abstract The pressure variations associated with a filling undulating pipeline containing an entrapped air pocket are investigated both experimentally and numerically The influence of entrapped air on abnormal transient pre ID: 83740
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MathematicalModelFig.showstheconfigurationofthepipesystemenvisionedinthiswork,whichconsistsofasourcereservoir,anupstreamwatercol-umn,avalve,andadownstreamwatercolumnwithanentrappedairpocketandanair-waterinterface.istheupstream(gauge)headandisthecorrespondingabsolutepiezometichead.InFig.,andarethethreeidentifiedpipelengthsup-streamofthevalve,and,andarethethreeidentifiedpipelengthsdownstreamofthevalve;isthepipediameterandisconstantforallsections;aretheinitialvolumeandabsolutepressurehead(inmeters)attheairpocket,respectively;istheelevationofpipecenterlineattheair-waterinterface.Theinitialvoidfractionoftheairpocket,,istheinitialairvolumetototalvolumeofairandwater.Thefollowingassumptionsaremadeinthedevelopmentoftheone-dimensionalmathematicalmodel(Cabreraetal.1992etal.20021.Theairpocketoccupiestheentirecrosssection;2.Theair-waterinterfaceisperpendiculartothecenterlineofpipe(i.e.,theairpocketremainscylindricalinshape);3.Duringthefillingprocess,boththedissolutionandreleaseofairarenegligible;4.Thepipelineisfullyclosedandnowaterandairareleakedor5.Thewallfrictionfactorforsteadyflowisapplicableunderunsteadyflowcondition;6.Apolytropiclawisapplicablefortheairphaseandthepoly-tropicexponentisconstant;and7.Theinertiaofairpocketisnegligible,andthustheconstantpressureisthroughoutwithintheairpocket.GoverningEquationsApplyingthemassconservationandmomentumequationstothewaterphaseinthepiperesultsinthestandardgoverningequationsforwaterhammer Htþ a2gA Qx¼0ð1Þg Hxþ 1A Qtþ where=time;=distance;=head;=discharge;=wavespeed;=wallfriction;=pipescross-sectionalarea;andgravitationalacceleration(981mForthesolutionasawhole,thestandardmethodofcharacter-istics(MOC)isemployedtoanalyzethetransientflowinthefillingpipecontaininganentrappedairpocket(WylieandStreeter1993LeeandMartin1999Lee2005).Comparedtoacompleteelasticwatermodel,asimplermodelforthewatercolumnisproposed.ThegoverningequationsofthewaterphasearesolvedwiththeMOC,butthechangesofboththeheadandflowrateofatinywatercolumnneartheair-waterinterfaceareneglectedtomoreconven-ientlytracktheair-waterinterface.Thecomputationalsystemin-cludesthewater,airpocket,andair-waterinterfaceparts.ControlEquationsforWaterPhaseTheupstreamboundarycondition,takingintoaccountoftheminorlossattheinlet,canbecalculatedby Q2P2gA2j =minorlosscoefficientoftheinlet;forpositiveflow,5;fornegativeflow,1.Thestandardorificeequationisusedtorepresentthevalveboundarycondition(WylieandStreeter1993GoverningEquationforAirPhaseThegoverningequationfortheconfinedairpocketis,and=instantaneousabsolutepressure,volume,andlength,respectively,oftheentrappedairpocketattime=initiallengthoftheairpocketbeforethevalveisopened;=polytropicexponent.Fasttransientphenomenaareoftenassumedtobeadiabaticprocesseswith4.Moreover,Zhou)andLee()demonstratedthatthefirstpeakpressureofanentrappedairpocketinarapidlyfillinghorizontalpipeisbetterpredictedwithapolytropiccompressionexponent4ratherthanwith2and0.Therefore,fortherelativesmallairpocketvolumeandthefastresponseofthesystemtothefirstpressurepeak,apolytropicexponentof4wasemployed.TrackingtheAir-WaterInterfaceOneofthemostchallengingaspectsofthesimulationisthattheair-waterinterfacemoveswithtime.DirectlyapplyingtheMOCtosolvethemovinginterfaceboundarytendstocreateinterpolationproblems.Tocircumventthischallenge,akeyassumptionismade:thattheliquidinertiaandenergylossofatinywatercolumnadjacenttotheair-waterinterfacecanbeignored(namely,theheadandwaterflowratearebothconstantthroughoutthewatercolumn).Theiscontinuallychangedbythemovementoftheair-waterinterface,butthelengthextentofisrelatedtotheMOCgridlength.AsshowninFig.,aslongasissuf-ficientlysmall,thismodelwillgeneratenearlythesamecalculatednumericalresultsasthecompleteelasticwatermodel,whileavoid-inginterpolations.However,aninfinitesimalwillleadtorun-ningmistakesinthesimulationprogram.Therefore,arationallengthextentofiscrucialtoensurethecalculatedaccuracyandrealizationofthenumericalmodel,and0isadoptedhere. ValveReservoirDead End L Hr ZC Va0 Ha0 Fig.1.Definingsketchfortheoreticalanalysis Lw x Dtx CC+C+C-BAAPPPt+tt Air-water interface Valve Filling water Filling water Entrapped air t Fig.2.DefinitionsketchinplanewithfixedgridJOURNALOFHYDRAULICENGINEERING©ASCE/DECEMBER2011/ AsshowninFig.arelocationsoftheair-waterinter-faceattheknowntimeandtobedeterminedat.Duringthecalculation,thegridstepareconstant,butthemove-mentofair-waterinterfacecausesthechange.Consequently,solvingisthekeytotrackingtheinterfaceboundary.GoverningEquationofAir-WaterInterfaceWithanundulatingprofile,consideringtheelevationchangeoftheair-waterinterface,thecontinuityequationandmomentumequa-tioncanbewrittenas or where=distanceoftheair-waterinterfacefromtheupstream=waterflowrateattheair-waterinterface;=distanceandheadoftheinterfaceat,respectively;=continuityfunctionof,andwhentheprofileofpipesystemisgiven,therelationbetweenisalsoknown.Trackingthemovingboundaryallowsthelocationoftheair-waterinterfacetobeknownatanytime,andintegrationofEq.()canbeexpressedas ApplyingthemeanvaluetheoremofintegralsandthemethodofmeanfunctioninEq.(),theequationcanbeapproximatedas whichdefinesthenewlocationofthemovingboundary,andwhere=flowrateatpoints,respectively;andothernewtermsaredefinedinFig.Accordingtotheprecedingassumption,theinertiaandenergylossofareignored.ThecontinuityequationandmomentumequationareTheproposedanalyticalmodelismadeupofEqs.(NumericalSolutionFig.showsthecomputationalgridusedinsolvingthegoverningequations.Inthismodel,thetimestepandthegridstepbothconstantduringthesimulation,and0Bytakingaccountofthevalveopening,thesolutionschemeschangewiththewaterlengthbehindthevalve(thedistanceisbetweenthevalveandtheair-waterinterface).Theprimarydif-ferencescanbenumericallyassociatedwiththesolutionofthevalve:1.When,theflowrateandpressureheadattheup-streaminletcanbecomputedbycombiningEq.()andthecharacteristicequation.TheinternalnodescanbecalculatedwiththeMOC.Theflowrateandpressureheadatthevalvecanbeobtainedthroughtheorificeequationforthevalve.Duringthecalculationoftheinterfaceboundary,combiningcharacteristicandEqs.(),theNewtonmethodisem-ployedtosolve,and.Thepressureoftheentrappedairpocket,,canbeobtainedthroughthegaslaw.2.When0,itnolongermeetsthecondition.Now,theupstreaminletandthein-ternalnodesofwaterpartcanbecomputedbyusingthesamemethodasinthefirstscenario.However,thecalculationatthevalveandinterfaceboundaryneedstobecarriedoutbycom-biningcharacteristic,theorificeequationforthevalve,andEqs.().Thiscaseisrareandonlyoccursatthebeginningofthevalvemovementwhenthepipeupstreamofthevalveisinitiallyfullofwaterandthepipedownstreamofthevalveisinitiallyemptyorpartlyfilledwithasmallamountofwater(inthispaper,thecasewithairpassingthroughthevalveisnotconsidered).Theboundaryandinitialconditionsare:(1)theupstreamres-ervoirheadremainsconstantthroughoutthesimulation;(2)theinitialpressureheadoftheentrappedairpocketisassumedtobeatmospheric;(3)theinitialflowrateofthewaterisassumedtobezerotobeconsistentwiththeexperimentaltestsconducted(althoughthemodelisnotlimitedtothissituation);(4)theinitiallengthoftheairpocketcanbeconvertedfromthesameairvolumebasedonthebasicassumptionsofthemodel;(5)theinitialdimen-sionlessvalveopeningisassumedtobeasmallvalue(0.05isusedhere)toensurethatthesimulationprogramrunssmoothly,andthefullvalveopeningis1.0.Duringthecalculationofthewaterfillingprocess,becauseofthecompressionandexpansionoftheairpocket,conditionsof1mayoccur.Tovalidatetheassumptionof0,itisnecessarytoaddordeleteacomputednodeandthevariablesatthenewlyaddednodemustbeestimatedbyinterpolationbetweenthenearbynodes.ExperimentalStudiesTheexperimentinvestigatestheeffectsoftheinitialvoidfractionoftheentrappedairpocketonthepressureofafillingpipelinesystem,focusingoncaseswithrelativelysmallinitialvoidfractionswithinanundulatedpipeline.ExperimentProgramTheexperimentswereconductedattheHydraulicLaboratoryofHohaiUniversityusingtheexperimentalapparatusdepictedin.Thesystemconsistsofanupstreamreservoirwith5mcrosssection,agatevalve,aquarter-turnballvalve,awatervent,anda4.4445-m-longpipelinewitha9cminternaldiameterand0.5cmpipewallthickness.Thepipelinecontainsfiveparts:a125-cm-longhorizontalpipe(Pipe1),a73-cm-longverticalpipe(Pipe2),a121.45-cm-longhorizontalpipe(Pipe3),a100-cm-longverticalpipe(Pipe4),anda25-cm-longhorizontalpipe(Pipe5).ThehorizontalPipe1withthegatevalveismadeofPVCandtheremainingundulatedpipelineismadeoftransparentorganicglass.Intheexperiments,theairpocketisentrappedatthedeadendofthepipelinesystem.ThewaterventatthebottomhorizontalPipe3and Dead End Ball Valve 73 49 24 39.5 PT2PT3PT4Unit: cmPT: Pressure TransducerPG: Pressure Gauge PG Gate Valve 52 Water Vent Fig.3.Diagramofexperimentalapparatus/JOURNALOFHYDRAULICENGINEERING©ASCE/DECEMBER2011 avalveatthedeadendofPipe5areusedtoregulatetheinitiallocationoftheentrappedairpocket.Initially,thegatevalveisfullyopened,theballvalveisclosed,theairpocketisentrappedatthedownstreamend,andthewaterventisclosed.Themeasuringsystemconsistsofonepressuregaugeandfivepressuretransducers.Thepressuregaugeisinstalledimmediatelyupstreamoftheballvalvetomeasuretheinitialstaticpressurefromwhichtheinletpressureheadattheupstreamreservoircanalsobeobtained.Thelocationsof5pressuretransducersareshowninFig.:PT1islocatednearthedeadendintheairpocketandthuseffectivelymeasuresthevaryingpressurewithintheairpocket,PT2isinstalledatthelowerpartoftheelbownearthedeadend,PT3isatthemiddleoftheverticalpipe(Pipe4),andPT4andPT5arefixedalongthelowerhorizontalpipe(Pipe3)at24cmdownstreamoftheballvalveand10cmupstreamoftheballvalve,respectively.These5pressuretransducershavethesamescale,thatis,thefluidpressurescaleisfrom1to2.0MPa;responsefrequencyrangesfrom30to300kHz;operatingtemperaturesarepermittedwithintherangeof40and150ºC;linearityrangesfrom0.03to0.3%ofthefullrange;lagerrorisbetween0.03and0.3%ofthefullrange.Eachofthesetransducerswascalibratedinadvanceofthetests.Duringthemeasurement,thetransducerscollectat1,000Hz.Eachofthesetransducersisconnectedtoacarrierdemodulatorthatis,inturn,connectedtoadataacquisitionboardcontrolledbyapersonalcomputer.Thedataacquisitionsoftwareissettoautomaticallyinitiatedatacollection.Thepressureoscillationhistoryateachofthesetransducersisdisplayedonthecomputerscreenandthensavedtoafileforlateranalysis.Tofacilitatelatercalibrationofthemodel,thefrictionfactorthepipeandtheminorlosscoefficientforthevalvesandthepipeelbowareindirectlymeasuredundersteadyflowconditions.Thelocallosscoefficientforthefullyopenballvalveisfirstmeasuredintherangeoffrom0.1to0.13.Thefrictionfactor,,isdeterminedbasedontotallosses,rangingbetween0.045and0.05,and05isadoptedinthereportedcalculations.Pipefillingisachievedbymanuallyturningthequarter-turnballvalve,soinstantaneousvalveopeningisanidealcondition.Therecordofahigh-speeddigitalcamerashowsthatthevalveopeningtime(fromfullyclosedtofullyopened)rangesfrom0.05to0.1s;avalveopeningtimeof0.1sisassumedinsubsequentcalculations.ThespeedofthewaterhammerwavespeedisnecessaryfortheMOCsolution.Thewavespeedismeasuredundertherapidvalveclosuretestswithnoair.ThewavespeedofPVCandorganicglasspipingfilledwithwateris400mExperimentalResultsAssumingthatthecenterlineofthehorizontalpipe(Pipe5)isthedatumline,0.Theabsolutepiezometricheadattheinlet,between14.70and16.63m(correspondingly,isfrom4.37to6.30m).Theinitialelevationoftheair-waterinterface,,ranges15to045m.Thetotalvolumeofwaterandair,andinitialairvolumecanbereadfromthethree-dimensionalschematicoftheexperimentalsystem,whichisdrawnbyusingcomputer-aideddesign(CAD)software.Correspondingly,theinitialvoidfractionoftheentrappedairpocketvariesfrom0to8.02%.showsthepiezometricheadoscillationpatternsrecordedbytheTransducers1,2,and5(absolutehead),withdifferentvoidfractionsfortheentrappedairpocket.Comparingtheheadatthedifferenttransducers,itisfoundthattheheadatPT2isalwaysbasicallyclosetotheheadatPT1.Thisisbecausebothtransducersarelocatedintheactiveareaoftheairpocket.Whentheinitialvoidfractionofairpocketis6.18%,becausetheaircushioningeffectisbasicallynegligibleandthewaterimpactforceisdominant,themaximumheadatPT1andPT2arebothlargerthanatPT5,whichislocatedupstreamoftheballvalve.Whiletheinitialvoidfractionoftheairpocketiscloseto0,themaximumheadsofPT1andPT2decreaseandbothareclosetothatofPT5.Thisisbecausewhentheinitialvoidfractionoftheairpocketissmall,boththeaircushion-ingeffectandwaterimpactforcearealsosmall.AsshowninFig.,theexperimentalinvestigationindicatesthat,undertheconstantinletpressure,astheinitialvoidfractionoftheairpocket,,becomessmaller,themaximumpressureoftheairpocketincreasesatthebeginningbecausetheaircushioningeffectisdominantandgraduallyreduced;then,themaximumpres-sureofairpocketobtainsitshighestvaluebecausethewaterimpactforceisgraduallymoredominantwhentheinitialvoidfractionoftheairpocketreachesaspecificvalue(inthisexperiment,).Subsequently,becausetheeffectsofaircushioningandwaterimpactforcearebothreduced(practicallynegligible),themaximumpressureoftheairpocketdecreaseswiththedecrease.Meanwhile,thetimedurationtogetthemaximumpressurebecomesshorterwiththedecreaseoftheinitialvoidfractionoftheairpocket.Asaforementioned,Zhou(),Zhouetal.(),LeeandMartin(),andLee()concludedthatthemaximumpeakpressureofairpocketincreaseswiththedecreaseoftheinitialvoidfractionofairpocket,basedontheirexperimentswithahigherinitialvoidfractionofairpockets().Physically,thelargeairpocketentrappedintheconfinedpipesystemsissimilartoanaccumulatorthatpreventshighpressurefrombeinggenerated,andthecushioningeffectismoreobviouswiththelargerairpocket.Thus,Lee()andZhou()drewtheprecedingconclusionaboutthelargeairpocket.However,whentheinitialairpocketissmall,especiallywhenisclosetozero,thepeakpressureisalmostthesameasapurewaterhammerwithnoair;thewater 00.10.20.30.40.50.6 PT2 00.050.10.150.2Time (s) PT1 PT2 PT5 Fig.4.Headattransducers:(a)53m,voidfractionofair;(b)53m,voidfractionofairpocketJOURNALOFHYDRAULICENGINEERING©ASCE/DECEMBER2011/ impactforcebecomesgraduallydominantwiththeincreaseoftheairvoidfraction,whichisdemonstratedbythepreviouslydis-cussedexperimentalresults.Therefore,thecurrentexperimentalinvestigationsarecomplementarytotheprecedingones.Theinflu-enceoftheairpocketonthepeakpressurecanbesummarizedasfollows:underahigherinitialvoidfractionoftheairpocket,owingtothedecreaseoftheaircushioningeffect,themaximumpeakpressureoftheairpocketbecomesgreaterasdiminishes;whenreachesacertainsmallvalue,thesmallerspaceforthewatermovementleadstothedecreaseinthewaterimpactforce,andthusthemaximumpressuredecreasesasbecomessmaller.MathematicalModelVerificationComparedwiththecompleteelasticwatermodel,theproposedmodelisintroducedwithasimplificationtoavoidtheinterpola-tions.Thelengthrangeofisthekeyofthecurrentmodelandaffectsthecalculatedaccuracy.Lee()developedacom-pleteelasticwatermodelwithaninstantaneousopenvalveandahorizontalpipe(Leesmodel).TheparametervaluesusedinLeecalculations()wereasfollows:pipediameter026m,pipelength4m,waterdensity000kg,wavespeed600m025,airpolytropicexponentrecom-4,initialpressureofairpocket0×10Pa(102mHO).Thecasewith8m,,and01missimulatedwithboththecurrentmodelandLeeselasticmodel.Thelengthrangeofinthecurrentmodelisassumedtobe.Fig.showstheresultsofdifferentlengthrangesof(differentvalues),andthesmalleris,thecloserthecalculatedresultistoLeesmodel.Thecurrentmodelwith0canachievetheconsistentairpressureoscillationpatternwithLeeelasticmodel.Themathematicalmodeldevelopedherecanbealsoappliedforanundulatedpipelinesysteminwhichthevalveopeningtimeistakenintoaccount.Thewavespeed,frictionfactor,theminorlossesattheballvalve,andthevalveopeningtimemeasuredfromtheexperimentsareusedasinputparametersinthemodeling.Fig.comparesthecomputedandmeasuredpressureoscillationpatterns.Fig.comparesahorizontalpipecontaininganen-trappedairpocket.ThemeasureddataandrigidmodelresultareabstractedfromZhousdoctorialdissertation(Zhou2000whichconsidersthevariablelengthofwatercolumns.Fig.comparestheundulatedpipelinediscussedhere.Thesecompari-sonsdemonstratethattheproposedmodelprovidesimproved Time (s) =8.02% =6.18% =1.0% =0.07% H u 00.10.20.30.40246810 Measured at 1#PT Current Model H u=16.53m(b) Fig.5.Effectsofthevoidfractionofairpocketonthepressureoftheairpocket:(a)pressureoscillationpatternrecordedbyPT1;(b)changeofthemaximumpressureoftheairpocket Fig.6.ComparisonofthecurrentmodelandLeeselasticmodel Time (s) Measured by F.Zhou Rigid Model of F.Zhou(a) 00.511.5200.10.20.30.4 Measured at PT1 Current Model Rigid Model of F.Zhou(b) Fig.7.Comparisonsbetweencalculatedandexperimentalpressureoscillationpattern:(a)horizontalpipeline[Zhousexperiment((b)undulatedpipeline(currentexperiment,26m)/JOURNALOFHYDRAULICENGINEERING©ASCE/DECEMBER2011 (moreaccurate)predictionsforbothhorizontalandundulatedpipelines.5(b)showsthatastheinitialvoidfractionofairpocketin-creases,theresultsfrombothelasticandrigidmodelsaregenerallyconsistent,andbothareslightlyhigherthantheexperimentalpres-sures.However,thedifferencefrombothmodelsbecomesprogres-sivelymoresignificantwiththedecreaseintheinitialvoidfractionoftheairpocket.Theresultsfromthecurrentlyproposedmodeldemonstratethatthemaximumpeakpressureincreasesslowlyatbeginningandthendecreasesrapidlywiththedecreaseoftheinitialvoidfractionofairpocket,whichhasthesamelawasthemeasuredresult;whereastheresultsfromtherigidmodelofZhoushowsthatthemaximumpeakpressureproportionallyincreaseswiththedecreaseoftheinitialvoidfractionoftheairpocket.Thus,therigidmodeliseitherhighlyconservativeorentirelyunsuitableforpredictingthepressurewhensmallerinitialvoidfractionsareconsidered.comparedthecomputedandobservedpeakpressures.Tofurtherverifytheproposedmodelusingtheexperimentalmeasure-ment,thenumericalmodelingandexperimentarerepeatedbychangingtheinletpressureandinitialvoidfractionoftheairpocket,andasimplestatisticalanalysisisconductedbydefiningarelativeerror: Theaverageerrorisfoundtobe5.99%,themaximumerroris13.95%,andtheminimum0.78%.Theerrorislessthan10%for90%ofthetests.Thisfindingdemonstratesthatthecurrentmodelpracticallyquantifiespressureoscillation,especiallyforthe(usu-allymostsevere)firstcycleofoscillation.Theprecedingcomparisonsandanalysesindicatethatthepeakpressurepredictedbythecurrentlyproposedmodeltendstobeconservativefromadesignperspective,becauseitisslightlyhigherthantheexperimentalvalue.Comparedwiththemeasurements,theattenuationofthecalculatedpressurecurveafterthefirstpeakisslowerandthecycleislonger.Possiblereasonsforthesedifferen-cesinclude(1)usingthesteady-flowfrictionfactorfortheunsteadyflowwouldunderestimatethepressurewaveattenuationatmoder-ateandhighfrequencies(Chaudhry1985);(2)theairphaseistreatedasanadiabaticprocesswithaconstantpolytropicexponent4),whereasthetruedynamicinvolvesacomplicatedheattransferprocess(Graze1996);(3)theintegralityoftheentrappedairpocketthatisassumedthroughoutthetransientprocessmightactuallybebrokenupbythefillingwater,especiallyafterthefirstcompressionoftheairpocket;(4)theflowtransientsintheelbowpartoftheundulatedpipelinearecomplicated,andtheenergylossisalsodifficulttobequantify.Therefore,ifthefrictionfactorandcharacteristicsoftheairpocketweredescribedmoreaccurately,theproposedmodelwouldpresumablybeabletomoresuccessfullypredictforthepressureoscillationpattern;inpractice,suchasearchforafurtherimprovedmodelwouldseldombeeconomicallyConclusionsandDiscussionsTheobservationfromthesephysicalexperimentsrevealstheeffectoftheentrappedairpocketontheflowtransientsduringthepipe-linefilling.Keepingbothpipelengthandinletpressureconstant,themaximumpeakpressureoftheairpocketincreasesatfirstandthendecreaseswiththereductionofthevoidfractionofairpocket.Thehighestpressureoccursatasmallervoidfractionoftheairpocket(atinthepreviouslydiscussedcasestudy).ThepressurechangelawfoundinthispaperisnotincompatiblewiththeconclusionsmadebyZhou(),Zhouetal.(LeeandMartin(),andLee(),thatis,themaximumpeakpressureofairpocketincreaseswiththedecreaseoftheinitialvoidfractionofairpocket,buttheirexperimentswereconductedforahigherinitialvoidfractionoftheairpocket(notlessthan5.8%)inahorizontalpipe.Insum,thecurrentresearchisnotonlyconsistentwiththeconclusionofZhouandLee,butalsocomplementaryfortheinvestigationontheinfluenceofthevoidfractionoftheairpocketonthetransientpressureduringpipefilling.Theproposedmodelmakesareasonablesimplificationtocap-turetheair-waterinterface;theresultingsimulationsnotonlyholdthesameaccuracywiththecompleteelasticwatermodel,butalsoavoidtheinterpolationcomplexity.Inaddition,themodelconsidersmanypotentiallyimportantfactors,suchaselevationchangeofthepipelineandvalveopeningtime.Althoughsomeunavoidableas-sumptionsandlimitsofthenumericalmodelresultedindifferencesbetweenthecalculatedandmeasuredresults,themaximumpeakpressurefromtheproposedmodelisclosertoexperimentalmea-surementsthantherigidwatermodel.Forvariationsofthevoidfractionoftheairpocket,thepredictionofthecurrentmodelpro-videsthesamepatternofthemaximumpeakpressureasfromtheexperimentalobservation.However,therigidwatermodelcannoteasilypredictthepeakpressureandcanevenresultinanoppositepeakpressurevariationpatternwhenthevoidfractionoftheairpocketissufficientlysmall.AcknowledgmentsTheauthorsgratefullyacknowledgethefinancialsupportonthisresearchfromtheNationalNaturalScienceFoundationofChina(GrantNo.50979029).Thefollowingsymbolsareusedinthispaper:=sectionareaofpipe;=wavespeed;=nameofcharacteristicequations;=diameterofpipe;=frictioncoefficient;=gravitationalacceleration(981m=instantaneouspiezometrichead;=airpressurehead(absolutevalue);=initialairpressurehead(absolutevalue);=themaximumairpressurehead(absolutevalue); 051015202530Experimental Fig.8.ComparisonbetweencomputedandmeasuredmaximumJOURNALOFHYDRAULICENGINEERING©ASCE/DECEMBER2011/ =calculatedvalueofthemaximumabsoluteairpressure;=calculatedvalueofthemaximumabsoluteairpressure;=piezometricheadatnode;=upstreamwaterhead(relativevalue);=upstreamwaterhead(absolutevalue);=lengthofpipe;=lengthofairpocket;=initiallengthofairpocket;=thedistancebetweenvalveandair-waterinterface;=polytropicexponent;=flowrate;=flowrateatnode;=airvolume;=initialairvolume;=locationofair-waterinterface;=elevationofpipecenterofair-waterinterface;=initialvoidfractionofairpocket(initialairvolumetototalvolumeofairandwater);=distancefromair-waterinterfacetopointnearbytheinterface;=timestep;=gridlength;=relativeerrorofcalculation;=minorlosscoefficientoftheinlet;and=waterdensity.ReferencesCabrera,E.,Abreu,J.,Perez,R.,andVela,A.(1992).Influenceofliquidlengthvariationinhydraulictransients.J.Hydraul.Eng.,118(12),Chaudhry,M.H.(1985).Limitationofhydraulic-transientcomputation.21stIAHRCongress,Int.AssociationforHydraulicResearch,Madrid,Spain,132DeMartino,G.,Fontana,N.,andGiugni,M.(2008).Transientflowcausedbyairexpulsionthroughanorifice.J.Hydraul.Eng.,134(9),EnvironmentalHydraulicsGroup(EHG).(1996).HydraulictransientevaluationoftheCityofEdmontonseweragesystem,PhaseI.ResearchRep.,EHG,Ontario,Canada.Graze,H.R.(1996).Thermodynamicbehaviorofentrappedairinanairchamber.7thInt.Conf.onPressureSurges,BHRGroup,Bedfordshire,UK,549Guo,Q.Z.,andSong,C.C.S.(1990).SurginginurbanstormdrainageJ.Hydraul.Eng.,116(12),1523Lee,N.H.(2005).Effectofpressurizationandexpulsionofentrappedairinpipelines.Doctoraldissertation,GeorgiaInstituteofTechnology,Lee,N.H.,andMartin,C.S.(1999).Experimentalandanalyticalinves-tigationofentrappedairinahorizontalpipe.Proc.,3rdASME/JSMEJointFluidsEngineeringConf.,ASME,NewYork,1Liu,D.Y.,andSuo,L.S.(2004).Rigidmodelfortransientflowinpres-surizedpipesystemcontainingtrappedairmass.Adv.WaterSci.16(6),717(inChinese).Liu,D.Y.,andZhou,L.(2009).Numericalsimulationoftransientflowinpressurizedwaterpipelinewithtrappedairmass.2009Asia-PacificPowerandEnergyEngineeringConf.(Appeec),IEEEPowerandEnergySociety,NewYork,104Martin,C.S.(1976).Entrappedairinpipelines.Proc.,2ndInt.Conf.onPressureSurges,BritishHydromechanicsResearchAssociation,Bedford,UK,15Ocasio,J.A.(1976).Pressuresurgingassociatedwithpressurizationofpipelinescontainingentrappedair.SpecialM.S.ResearchRep.SchoolofCivilEngineering,GeorgiaInstituteofTechnology,Atlanta.Wylie,E.B.,andStreeter,V.L.(1993).Fluidtransientsinsystems,PrenticeHall,NewYork.Zhou,F.(2000).Effectsoftrappedaironflowtransientsinrapidlyfillingsewers.Doctoraldissertation,UniversityofAlberta,Alberta,Canada.Zhou,F.,Hicks,F.E.,andSteffler,P.M.(2002).Transientflowinarap-idlyfillinghorizontalpipecontainingtrappedair.J.Hydraul.Eng.128(6),625Zhou,L.,Liu,D.Y.,andOu,C.Q.(2011).SimulationofflowtransientsinawaterfillingpipecontainingentrappedairpocketwithVOFmodel.EngineeringApplicationsofComputationalFluidMechanics,5(1),/JOURNALOFHYDRAULICENGINEERING©ASCE/DECEMBER2011 TechnicalNoteInfluenceofEntrappedAirPocketsonHydraulicTransientsinWaterPipelinesLingZhou;DeyouLiu;BryanKarney,M.ASCE;andQinfenZhang,Ph.D.Abstract:Thepressurevariationsassociatedwithafillingundulatingpipelinecontaininganentrappedairpocketareinvestigatedbothexperimentallyandnumerically.Theinfluenceofentrappedaironabnormaltransientpressuresisoftenambiguousbecausethecompress-ibilityoftheairpocketpermitstheliquidflowtoacceleratebutalsopartlycushionsthesystem,withthebalanceofthesetendenciesbeing Professor,CollegeofWaterConservancy&HydropowerEngineering,HohaiUniv.,1XikangRd.,Nanjing,China210098.E-mail:Liudyhhuc@Professor,Dept.ofCivilEngineering,Univ.ofToronto,35St.George