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THEORETICAL,NUMERICAL

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THEORETICAL,NUMERICALthe

    THEORETICAL,NUMERICAL

20

    ;JournalofHydrodynamics,

    ;ChinaOceanPress,Beijing

    ;Ser.B,5(2003),2028

    ;

    ;PrintedinChina

    ;THEORETICAL,NUMERICALANDEXPERIMENTALSTUDYOFWATER ;HAMMERINPIPESYSTEMWITHCOLUMNSURGECHAMBER

    ;zhouZe.xuan,TanSoonKeat

    ;ENV.NTUEnvironmentalEngineeringResearchCentre,SchoolofCivilandStructuralEngineering,

    ;NanyangTechnologicalUniversity,Singapore639798,email:czxzhou@ntu.edu.sg ;(ReceivedJan.11,2002)

    ;ABSTRACT:Anewkindofgoverningequationsforwater ;hammerbasedontheelasticcolumntheorywasproposedand ;adoptedtoanalysewaterhammerphenomenoninthepipesys

    ;temwithaverticalcolumnsurgechamberandwaterlevelfluc

    ;tuationinthesurgechamberduringpressuretransient.The ;wrongnessexistingintheclassicalgoverningequationsforwa

    ;terhammerwasanalysed.Atypicalreservoirvalvepipesys

    ;tamwaschosenasanexampletoverifythenewgoverninge

    ;quationsnumericallyandexperimentally.Thefinitedifference ;methodbasedonthemethodofcharacteristicswasusedto ;solvenumericallythenonlinearcharacteristicequations.The ;temporalevolutionsoftransientvolumefluxandheadandof ;waterlevelfluctuationforvarioussurgechamberconfigura

    ;tionswereworkedout,assumingthattheairinthesurge ;chamberarecompressible.Therelevantexperimentwascon

    ;ductedtoverifythenewgoverningequationsandnumerical ;method.Thenumericalandexperimentalresultsshowthat ;thenewgoverningequationsarevalidandtheconventionalas

    ;sumptionthatthepressureheadatthebaseofasurgecham

    ;berequalsthatofthestaticheadaboveitduringpressuretran

    ;sientisnotalwaysvalid.Thesurgechambergenerallyreises ;theperiodoftransientpressurewaveinpipesystem,reduces ;themaximumpressureenvelopeandliftstheminimumenve

    ;lopesubstantially.Thewaterlevelfluctuationinthesurge ;chamberwasnumericallyandexperimentallyobserved.In

    ;creasingthesizeofthesurgechamberand/ordecreasingthe ;initialairpressureinthesurgechamberenhancetheeffective

    ;nessofthesurgechamberinsuppressingpressurewave. ;KEYWORDS:waterhammer,verticalcolumnsurgecham ;ber,pipesystem

    ;1.ITRoDUCTIoN

    ;Waterhammerreferstoanon??linearphenom?? ;enonwherepressuresurgesoccurfollowingasud- ;denchangeinflowvelocityinapipesystem.The ;changeinflowvelocitymaybeinducedbyvalve ;operation,pumpstartuporshut-down,fluidin

    ;jectionordischarge,thesuddencessationofener

    ;gYsupplierdevicescausedbypowerfailureorpipe ;movementundercertainexternalactionsuchasim- ;pactandearthquake.etc.Theseinducenonlinear

    ;pressuresurgesorwavespropagationthroughout ;thepipesystematacousticspeed.Thesurgepres‟

    ;surecouldhavedangerouslyhighamplitudeand ;maygeneratenoiseandfatiguethatcoulddamage ;thepipesystem.Thus,waterhammerphenome

    ;nonhasarousedmanyresearchinterests[-. ;Theresearchonwaterhammerprotectionis ;prominentcurrently.EsselmanandKupinski ;performedtherootcauseanalysisofwaterhammer

    ;eventsandproposedpossiblepreventionmethods. ;GastouniotisandGray.proposedtocontrolwater ;hammerusingEBARsupports.Verhoveneta1.[73 ;conductedacomparativestudyofwaterhammer ;protectionusingairvesselsorsurgechambers.Lai ;andHau[.]investigatedtheeffectofgasvoidson ;thewaterhammerphenomenoninapipenetwork. ;Thegaswasassumednon??condensableintheirin?? ;vestagation.Thisassumptionisnotvalidingeneral ;asgasisusuallycompressibleandthestateequation ;ofgasshouldbeusedE.Bruceeta1.

    ;[]studiedthe

    ;effectofone??wayandtwo?-waysurgechamberson ;waterhammerinpipesystemcausedbypumppow

    ;er-failure.Thesurgechambereffectonsuppress

    ;ingpressurewavehasbeenfoundbyconsidering ;theheadlOSSrateattheentrytothesurgechamber ;constant.Thewaterlevelfluctuationinthesurge ;chamber,howeverhasnotbeenstudied.

    ;Sofartheclassicalgoverningequationsfor ;waterhammerhavebeenwidelyadoptedinthere. ;searchofwaterhammerL.Theclassicalequa.

    ;tions,howevercannotbeusedtodescribesteady ;stateofpipesystematall,althoughgeneratingsat

    ;isfactoryresultsduetothespecialMethodOf ;Characteristics(M0C).Therefore,thoseclassical ;equationsarenottheoreticallyidea1.Inthisstudy. ;r

    ;

    ;theauthorsfocusonthederivationofnewgover- ;ningequationsforwaterhammerandstudythe ;effectofsurgechamberonwaterhammerphenom- ;enoninapipesystemnumericallyandexperimen- ;tally.Thewaterlevelfluctuationinthesurge ;chamberisalsoincludedintheanalysisofwater ;hammerphenomenon.

    ;Fig.1Illustrationofhydraulicsystem ;…r

    ;Fig.2Illustrationofsurgechamber

    ;21

    ;13P+

    ;

    ;ap)+1(O

    ;d

    ;A

    ;

    ;+”a

    ;d

    ;A)oqu

    ;

    ;0(2)

    ;inwhichpisthewaterdensity,Pisthepressure,u ;isthecross-sectionalaveragevelocity,fisthewall ;frictionfactor.Distheinnerdiameterofpipe,gis ;thelocalgravityconstant,0istheangleatwhich ;thepipeisinclinedwiththehorizontallineandAis ;thecross.sectioncurrentareaofpipe. ;Accordingtothenaturesofpipeflow,itisde- ;finedthat

    ;lV.

    ;.+gH(,f)

    ;.oNLINEARMAI‟HEMAI‟ICALMoDELAND

    ;EQUATIONS

    ;Thereservoir-valvepipesystemtobeconsid.- ;eredisshowninFig.1,identicaltothepipesystem ;usedbyWylieandStreeter[]ifthesurgechamber

    ;illustratedinFig.2,asaprotectionaccessorya- ;gainstdamagebywaterhammeronpipe,pumps, ;valvesandtheothers,isremoved.Theoriginof ;theCartesianco.ordinatesystem(,Y)issetat ;thepointo(theexitfromthereservoir)andthe. ;axiscoincideswiththelongitudinalaxisofpipe1. ;ValvesAandBareopeninitiallyandCandDare ;closedtokeepgasfromfleeingoutofthechamber ;top.Afterasteadyflowhasbeenestablished, ;valveA,theoperatingvalve,issuddenlyclosed. ;Anonlineartransientflowisdevelopedandfades ;withtime[„.,,.Foralongpipe.theequationof

    ;motionandequationofcontinuitybasedonthe ;one-dimensionalunsteadyflowtheoryE.]are ;Op+”au+au+n+l”l0(1)

    ;(3)

    ;inwhichPisthelocalambientairpressure,is ;thestandardwaterdensityandHistherelative ;pressureheadofwaterinsidepipe.

    ;BasedonEqs.(1).(3)andthedefinitionand ;thedefinitionofbulkmodulusofelasticityofwa. ;ter.thefollowingclassicalwaterhammergover. ;ningequationsE,.,‟,.]comeout:

    ;aH+

    ;

    ;3u3u+g

    ;sggsinU+l”l0(4)+”++”l”l—o(4

    ;+”3Hiaz3t0一十十一u(5)

    ;inwhichaisthewavespeedofpressurewave. ;Obviously,theaboveequationscannotgener- ;atetheoreticallyanysteadystateoflargeuniform ;pipesystematallduetowallfrictionandgravita- ;tion.Asoneknows,thesteadyvelocityoflargeu- ;niformpipesyspemisthespecifiedsteadylineve- ;locity,aconstantdeterminedbysomeflowre- ;quirement.

    ;Thispointmeansthattheclassicalgoverning ;equationsarewrongintheory.Butthoseequations ;cangenerateanysteadystateofanypipesystem ;numericallyandstandtoexperimenttests~?0,‟,.

    ;Intheinvestigationofthenumericalmethods ;basedontheMOC.itisfoundthattheMOCtoler. ;atesthewrognessintheclassicalgoverningequa- ;tionsduetothelargeacousticspeedinhydraulic

    ;projects,about1000m/s.IntheMOC,thefactor, ;thatmergestheequationofcontinuityintothee. ;quationofmotion,is?g/a.Itisnaturalthatthe ;verysmallfactormakesthewrognessoftheequa. ;tionofcontinuityunimportantinnumericalsimula. ;tion?Resultantlyitismeaningfultogivenewgov. ;

    ;22

    ;erningequationsforwaterhammev.Nowtheau_ ;thorstrytoderiveanewkindofgoverningequa- ;tionstillbasedontheelasticcolumntheory. ;Thebulkmodulusofelasticityofwaterin ;termsofpressure[„.‟‟.]isdefinedas

    ;Aop

    ;inwhich?meansthesmallvariationand

    ;watermodulus.

    ;(6)

    ;Kisthe

    ;Accordingtothelinearshelltheory[„.‟‟9]

    ;onehas

    ;AA==CAADAp

    ;1O

    ;a

    ;p__

    ;z

    ;__zo

    ;]-

    ;OUo

    ;+‰+gSin+

    ;f

    ;.

    ;.

    ;1Opo+‰)+.OUo

    ;“.

    ;(10)

    ;

    ;0(12)

    ;whereDrepresentstheinitialpipeinnerdiame- ;(7)ter?

    ;whereCA(112)/Ee.Here,Eandare

    ;Youg‟smodulusandPoisson‟sratioofwallmateri.

    ;alrespectively,C12isthestresscorrelationcoeffi- ;cientandeistheabsolutewallthickness.Thestress ;correlationcoefficientdependsonpipesupports

    ;andtherelativethicknessofpipewal1.e/D. ;Inthedefinitionofbulkmodulusofelasticity ;andformulaofpipecross.sectionalcurrentarea ;variation,itisseenthattwotime.sequentialstates ;ofthepipesystemowninhydarulicprojectthatthe ;changeofwaterdensityisnegligiblysmal1.The ;variationoflargepipeindiameterduringwater ;hammerisverysmallcomparedwithpipediame. ;ter.Andthelargepipeisuniformindiameter. ;Consequently,itisnaturalthat

    ;„Ilhewavespeedformula,Eq.(10),means

    ;thattheacousticspeed,thepropagatingspeedof ;pressurewave,isaconstantonlyiflargeuniform ;pipeisrelativelystiff.Eqs.(11)and(12)areiden- ;ticaltotheclassicalgoverningequationsofwater ;1

    ;hammerexceptforthelastterminEq.(12),

    ;ap

    ;‰,whichjustensuresthenewgoverningequa-

    ;tionstokeepwellanysteadystateintheory,which ;makestheclassicalgoverningequationstheoreti- ;callyinvalid.

    ;Attheupstreamend,i.e.,thereservoir,the ;variationoftheheadofthereservoirHsduring ;transientissmallandcanbeconsideredtobezero. ;Inthecasewheretheexitandentrylossissmallin ;comparisontotheheadofthereservoir,thenthe ;pressureheadattheentrytothehorizontalpipe ;neartheexitfromreservoiris

    ;HtHtyHsconstant

    ;Atthedownstreamend,i.e.,atvalveA

    ;Q()(cA.).rv厂万

    ;(13)

    ;(14)

    ;fromWylieandStreeter.Here,?Histhein.

    ;stantaneousdropinheadacrossvalveA;(CG). ;representstheproductofthedischargecoefficient ;ofvalveA,Coandtheeffectivefullyopeningarea ;ofvalveA;ristheratiooftheopeningareatothe ;totalcurrentareaofvalveA.theclosurerelation. ;ship.

    ;WylieandStreeter[]providedatypicalvalve ;closurerelationship,

    ;

;

    ;f(1一,/),t?E0,t)

    ;I,?Et0tLt,o.)I,?,?)

    ;wheretisthetimeatwhichvalveAisfullyclosed ;andEistheexponentoftheclosurecurve.Obvi- ;ouslytherearetwostagesduringtheclosureof ;valveA.

    ;Forthecomparabilityoftherelevantexperi

    ;ments,valveAisalsoconsideredanordinaryin- ;dustrialvalvewidelyusedwhoseclosurerelation- ;shipisgivenbytheequation.

    ;f10.2t/t

    ;l1.41.8t/,一

    ;1o.2o.2t/,【

    ;o,

    ;t?EO,0.25t)

    ;,?[0.25G,o?(

    ;16)

    ;,t?EO.75t,t)

    ;t?Et,co)

    ;Here,theclosurerelationshipincludes ;TheinstantaneousheaddropAH

    ;AHHI一一HI

    ;fourstages.

    ;(17)

    ;whereHIandH.Irepresenttheheadatthe

    ;leftandtherightsidesofvalveA,respectively. ;TherightsideofvalveAisexposedtothelocalen. ;vironmentsoHI0duetoEq.(3).

    ;BeforevalveAisclosed,thesteadyflowana1. ;ysisyieldsthefollowingtwoequationsfortheini- ;tialconditions:

    ;Qo(z)

    ;H.(z)H.Q:(z)

    ;(18)

    ;23

    ;sumedthattheairinthesurgechamberiscompres_ ;sibleanditsmassisconstantunderthecondition ;thattheairpressureismoderateandvalvesCandD ;botharetightlyclosed.Fromthisassumption,one ;obtainstheairstateequationasfollows[?: ;PViiconst(2O)

    ;wheretheexponent,zdependsonthethermody? ;namicprocessfollowedbythegasinsurgechamber

    ;andvariesfrom1to1.4inhydraulicprojects.The ;watervolumedeterminestheairvolume.

    ;Theair.waterinterfaceinthesurgechamberis ;balancedbeforevalveAisclosedsotheformulaof ;theinitialairpressureis

    ;P.P.+g(HH+a.)(21)

    ;inwhichHistheheighofthetopofthesurge ;chamberanda.theinitiallengthofaircolumnin ;thesteadystatethatindicatestheinitialpressureof ;theair.

    ;AftervalveAbeginsclosing,thewaterlevelin ;thesurgechamberwillfluctuate.Ifwateriscon? ;sideredincompressibleanditsinertiaisignored, ;thentheairvolumeis.Here,Visthe

    ;volumeofthesurgechamberandisthevolume ;ofwaterinthesurgechambergivenby

    ;V(+t.)V()?V(.)(22)

    ;inwhicht.istheobservedtimeand‟+‟means

    ;thatwaterfromthemaintransportpipeentersthe ;(19)surgechamber.Correspondingly,thepressureat ;theentrytothesurgechamberis

    ;inwhichR.fL/ZgDA.andListhetotallength

    ;ofthemaintransportpipe.

    ;Thetwolocalconnectionsexistbetweenthe ;horizontalpipes,verticalpipeandsurgechamber. ;Consideringthatthesizeofthelocalconnectionsis ;muchsmallerthanthecharacteristiclengthofthe ;pipesystemandthelocalheadlOSSiSverysmall ;comparedwiththereservoirhead,oneassumes ;thatthepointcontinuousconditionsareeffective ;atthetwolocalconnections~?.

    ;Thestaticflowexistsintheverticalpipeand ;surgechamberinthesteadystate.Thepressure ;continuousconditionisdefinedatthewater.airin. ;terfaceasthesurfacetensionisneglected.Itisas. ;P.Pi+g(23)

    ;whereyistheheightofwaterinthesurgecham- ;ber.

    ;3.NUMERICALMETHoDANDCoMPUTA-

    ;TIoNALDATA

    ;3.1Numericalmethod

    ;Basedonthedefinitionsofaveragevolume ;fluxandrelativepressureheadandMOC,thefo1. ;1owingandC-equationsarederivedfromEqs.

;(13)and(14):

    ;

    ;24

    ;+B+.1Qo1

    ;“.—

    ;dH

    ;i

    ;0

    ;dx.

    ;

    ;+

    ;

    ;B….lQodtdtgDlZA...

    ;“.—

    ;dH

    ;i

    ;

    ;0

    ;dx

    ;一一

    ;agatevalve,worksasaswitchthatdetermines ;whetherthesurgechamberisinstalledornot.Ta- ;ble4showsthestandardsizeofthesurgechamber. ;r9d,Amongtheoperatingparametersofthetypical ;valveofferedbyWylieandStreeterL,theclosing ;timeis2.1sandtheclosureexponentequals1.5. ;(24b)

    ;(25a)

    ;(25b)

    ;whereQisthevolumefluxandBa/gA.

    ;Eqs.(24)and(25)arethenonlinearcharac. ;teristicequationsofthenewgoverningequations ;forwaterhammer.Thetwounknownparameters ;areheadandvolumefluxandtheirvariationsare ;criticaltodesignandmaintenanceofhydraulic ;pipesystem.

    ;FDMisadoptedtosolvethecharacteristice. ;quations(24)and(25).Themeshforthepipe ;flowsatisfiestheCourantstabilitycriteriatoen. ;surethestabilityandconvergencefornumerical ;simulation.Thedifferenceschemealongtheposi. ;tiveandnegativecharacteristiclinesisthetempo. ;ralforwardingdifferenceonthetemporalderva. ;tivesinEqs.(24)and(25)andthevaluesofthe

    ;nonlinearfrictionalitemsatthecurrentstepare ;replacedwiththevaluesatthepreviousstep.This ;approximationisefficientonlyiffQ/4DA1

    ;?

    ;.

    ;Herezfltisthetimesteplengthandthesub. ;scriptfdenotesthecharacteristicparametersofthe ;pipesystem.Itwillbeexpoundedinthenextsub- ;sectionthatthesystemconsideredsatisfiesthecri. ;terion.Thefirst.orderaccuracyisusedintheeva1. ;uationofthepreviouspressuregradientatmostof ;computationalnodes.AtvalveAandtheentryto ;thesurgechamber,thesecond.orderaccuracyisa. ;doptedtoevaluatethepreviouspressuregradient. ;3.2Computationaldata

    ;ThedevicesysteminFig.1isextractedfroma ;pumpingstation.Table1indicatesitsspecifiedpa. ;rameters.ItsworkingconditionsareshowninTa. ;ble2.Table3givesthespecifiedparametersof ;valveAofferedbythepumpingstation.ValveB, ;Table1Theparametersofthehydraulicsystem ;Table2Theworkingconditionsofthesystem ;Table3TheoperatingparametersofvalveA ;Valvedischarge

    ;constant

    ;(CaAG).

    ;Valveclosing

    ;time

    ;t

    ;0.009m20s

    ;Table4Theparametersofsurgechamber

    ;intheauthors‟numericalcomputationbased

    ;onthespecifieddataandexperimentaldata.itis ;certainthat,?tQ/4DfAf?1.79627×10where

    ;thesubscriptlvariesintheset{1,2,3}.Obviously, ;thecriterionmentionedaboveissatisfiedinthe ;systemadopted.Asaresult,theaccuracyofthe ;firstorderintegrationusedinthedifference ;schemeisneverindoubt[.

    ;

    ;4.EXPERIMENT

    ;Theexperimentisconductedinthepumping ;stationduringdaytime.Thethermometershows ;thattheenvironmenttemperatureis30.C.Thelo.

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