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外文翻译英文版--反应注射成型过程中熔体流动前沿的PETROV-GALERKIN有限元分析.pdf

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外文翻译英文版--反应注射成型过程中熔体流动前沿的PETROV-GALERKIN有限元分析.pdf

PergamonComputersFluidsVol.24,No.1,pp.5562,1995Copyright01995ElsevierScienceLtd0045793094000204PrintedinGreatBritain.Allrightsreserved00457930/959.500.00PETROVGALERKINFINITEELEMENTANALYSISFORADVANCINGFLOWFRONTINREACTIONINJECTIONMOLDINGNITINR.ANTURKARFordResearchLaboratory,FordMotorCompany,P.O.Box2053,MD3198,Dearborn,MI481212053,U.S.A.Received4August1993inrevisedform4May1994AbstractAnumericalschemeforcomputingtheadvancementofaflowfrontandrelatedvelocity,pressure,confersionandtemperaturedistributionsduringmoldfillinginreactioninjectionmoldingRIMisdescribedinthiswork.IntheRIMprocess,theconvectivetermintheenergyequationisdominant.Therefore,thenumericalschemehasincorporatedaPetrovGalerkinfiniteelementmethodtosuppressspuriousoscillationsandtoimproveaccuracyofthecalculations.Theotherfeatureofthenumericalschemeisthattheflowfrontlocationsarecomputedsimultaneouslywithprimaryvariablesbyusingasurfaceparameterizationtechnique.Thenumericalresultscomparewellwiththereportedexperimentaldata.ImprovedaccuracyobtainedbythisnumericalschemeintheflowfrontregionisexpectedtoassistinthepredictionsofthefiberorientationsandthebubblegrowthinRIM,whicharedeterminedprimarilybytheflowfrontregion.I.INTRODUCTIONReactioninjectionmoldingRIMisawidelyusedprocesstomanufactureexteriorfasciasintheautomobileindustry.Inthisprocess,aprepolymerizedisocyanateandapolyol/aminemixturearemixedtogether,andinjectedintoamold,wherepolymerizationoccurs.Afountainfloweffectintheadvancingflowfrontregionduringthemoldfillingstageplaysanimportantroleindeterminingtheresidencetimeofthefluidelementsandincontrollingthefiberorientationsinthefinalproduct11.Anaccuratesimulationofthisflowfront,however,posesachallengingproblem.Evolvingflowdomainwithadvancingflowfrontrequiresupdatingofthenumericalgridsandpredictionofthemovingboundaryateverytimestep.Lowthermalconductivityofthematerial,highflowratesintheRIMprocess,andhighlyexothermicrapidreactionsresultinconvectiondominatedenergytransportequation,whichneedsaspecialnumericaltreatment.Besides,movingcontactlinesnearthewallsneedsuitableboundaryconditionsthatdonotintroducenumericalinstability.AnumericalschemethatincorporatesallthesecomplexfeaturesoftheRIMprocessisrequiredforaccuratepredictionsneartheflowfrontregion.Previousstudieseitherhavemadesimplifyingassumptionsregardingtheflowfrontregion2l,orhavenotcomparedtheirresultswiththeexperiments5,6.Inthispaper,wedescribeanumericalschemeindetail,whichwilladdresstheabovementionedcomplexities,and.presentthereleventresultsthathighlightthenumericalschemerefertoourearlierwork7forthedetaileddiscussionofthegoverningequationsandadditionalresults.Noaprioriassumptionsaremadeinthenumericalschemeregardingtheshapeofthenewfrontorthevelocitydistributionintheflowdomain.Afreesurfaceparameterizationtechniqueisused,inwhichtheshapeoftheflowfrontiscalculatedsimultaneouslywithotherfieldvariables,suchaspressure,velocitiesandconversion,byincorporatingkinematicboundaryconditionatthesurfaceoftheflowfrontasoneofthegoverningequations.AconventionalGalerkinfiniteelementtechniqueisnotoriousforitsnumericalinstabilityinconvectiondominatedtransportproblems8.Theresultingspuriousoscillationscanbeusuallyeliminatedbymeshrefinement.However,fortransientproblemdescribedhere,meshrefinementisanimpracticalandexpensivealternative.Theotheralternativesincludevariousupwindingschemes9121,amethodofcharacteristics6,13,141,andaGalerkin/leastsquarestechnique151.Althoughtheconservativemethods,suchasmethodsofcharacteristicsandGalerkin/leastsquarestechniquesaremoreaccurate,asimplePetrovGalerkinupwindingmethodiseasierto5556NITINR.ANTURKARimplementandcosteffective,particularlyforatransientprobleminvestigatedinthiswork.Therefore,suchaschemeisimplementedherefollowingAdornatoandBrown9tosuppressnumericalinstabilitywithoutresolvingtoextremelyrefinedmeshes.ThegoverningequationsarepresentedbrieflyinSection2,andthenumericalmethodisdescribedindetailinSection3.ThetypicalresultsofthemoldfillingstageoftheRIMprocessinatwodimensionalrectangularplaquearepresentedinSection4.Theresultsarealsocomparedwiththereportedexperimentaldata2,andwiththenumericalresultsobtainedbyusingconventionalGalerkinfiniteelementmethod.2.GOVERNINGEQUATIONSThelumpedkineticrateexpressionforpolymerizationreactionsinRIMis16,171riA,expE,/RTCr,1where,Ciistheisocyanateconcentration,Tthetemperature,Rthegaslawconstant,mtheorderofthereaction,E,theactivationenergyofthereaction,andA,therateconstant.Theviscositydependsontheconversionandtemperature,andisexpressedintheformofCastroMacoskoviscosityfunction2,X,TrlXIITA,expiBXi,2whereXistheisocyanateconversion,X,thegelconversion,andA,,,E,,AandBaretheconstants.Forconstantthermalpropertiesanddensityofthereactivemixture,andfornegligiblemoleculardiffusion,thedimensionlessgoverningequationsare,continuityequationv.vo3conservationofmomentumequationRev.VvpV.IvrcjGz7,,vVXDak.lXmolebalanceequation45conservationofenergyequationGzgvVTVTBrrcjVvDarc,lXmL.16where,visthevelocityvector,qtherateofstraintensor,tthetime,pthepressure,andk,isthedimensionlessrateconstant,definedasexpE,/Rl/Tl/T,.TheequationsaremadedimensionlessusingtheaveragevelocityV,halfofthethicknessofthemoldH,andthetemperatureT,andtheviscosityqOrX0,TT,attheinletofthemold.AllthedimensionlessgroupsandtheirdefinitionsarelistedinTable1.Theboundaryconditionsintermsofdimensionlessvariablesare1.atthewallsv,,,,0noslip,TT,,,,2.atthemidplaneaTjay0,Jay0,V,03.attheinletvfullydevelopedflow,T1,XX,4.atthecontactlinenPI20fullslip5.attheflowfrontn.PI20forcebalance,n.vah/at0kinematicconditionTable1.Dimensionlessgroupsingoverningequations,whereAH,istheheatofreaction,AT,,,theadiabatictemperaturerise,andC,,theinitialconcentrationofisocyanateGZGraetznumberVHpC,lkReReynoldsnumberHVlrloKviscosityratio41BrBrinkmannnumbertoVlkT,DaDamkohlernumberAH,HC/kT,A,expE,/RTTadbadiabatictemperatureriseAT,,,IT,Flowfrontadvancementinreactioninjectionmolding57wherea.,andvYarethecomponentsofthevelocityvectorv,IItheunitnormalvector,rtheextrastresstensor,hthelocationvectoroftheflowfrontandTwal,thedimensionlesstemperatureatthemoldwall.Thedetailsofincorporatingtheboundaryconditionsinthenumericalanalysisareexplainedinthenextsection.3.NUMERICALANALYSISInthefiniteelementformulationtheunknownvelocities,temperatureandconversionareexpandedintermsofthebiquadraticbasisfunctions4,thepressureintermsofthebilinearbasisfunctionsll/iandtheflowfrontshapehintermsofthequadraticbasisfunctions7wherelandqarethecoordinatesinisoparametrictransformation,definedasi1ilintheisoparametricdomain1L5,where,PeisthelocalelementPecletnumberVA/D,Atheelementsizeandc,sisthecubicpolynomial5/85lt1.Theindexi1correspondstothevertexnodes,andi258NITINR.ANTIJRKARcorrespondstothecentroidnodesintheelement.Thestandardonedimensionalconvectiondiffusionproblemhasexactsolutionatthenodesif25,9cPe2tanhPe/2l3/PecothPe/4X/PecothPe/4,1lac216/Pe4cothPe/4.1lbInatwodimensionalproblem,thetensorialproductofequations10and11providesthefunctioncintheweightingfunctionsdescribedinequation9.ThelocalPecletnumberiscomputedforeachthreenodegroupbasedontheaveragevelocitiesattherelevantboundariesinthetwodimensionalelement9.Therearesixsuchgroupsthreeinthexdirection,andthreeintheydirectionandthus,thereare12upwindingparametersE.ThecalculationsofthePecletnumberinvolvelineardistances,whichessentiallyneglectthecurvilinearsidesoftheelements.However,itisagoodapproximationsinceflowfrontisnotseverelydeformedinourproblem.Thediffusivitiesarel/GzfortheenergyequationandisK/Rforthemomentumequation.ThePetrovGalerkinweightedresidualequationsare,RV.vdlO,sRLIvRegv.VvWfdV12yPIKVWidVssn.pIlcfWdSO,13sBrrcjVvDak,lXWdV1sVT.VWdVsn.VTWdSO,15VSRIsn.vah/41dS0.16swhere,VistheflowdomainandStheflowboundary.Theboundarytermsappearintheenergyandmomentumequationsbecausedivergencetheoremisappliedtothehigherorderterms.TheresidualsR,,R,,R,,R,andR,correspondtothevariablesp,v,X,Tandh,respectively.ThePetrovGalerkinweightingfunctionsareusedonlyformomentumandenergyequationsduetothepresenceofconvectiontermsintheseequations.BeforeintegratingtheaboveequationsusinganinepointGaussianquadrature,theequationsaremappedintheisoparametricdomainreferto26fordetailsandtheboundaryconditionsareapplied.TheessentialboundaryconditionsforvandTatthewallsforv,TandXattheinletofthemoldandforvYatthemidplaneaxisofsymmetryareappliedbysubstitutingtheboundaryconditionsfortheequations.Thenaturalboundaryconditions,namelythesymmetryconditionsatthemidplane,thefullslipzerofrictionconditionatthecontactpoint,andthezeroforceatthefreesurfaceareimplementedbysubstitutingtheboundarytermsintheresidualequations.Thekinematicboundaryconditionattheflowfrontisincorporatedasthegoverningequationforpredictingtheflowfrontlocations.Theweakformofenergyequationisextendedtotheflowfrontboundarybyevaluatingtheboundaryterms,insteadofbyimposinganyunknownessentialornaturalboundaryconditions27.Suchfreeboundarycondition,asdenotedbyPapanastasiouetal.27,minimizestheenergyfunctionalamongallpossiblechoices,atleastforvarioustypesofcreepingflows,andhasbeensuccessfullyusedinseveralapplications,includingthosewithhighReynoldsnumbers.Thespatialdiscretizationreducesthetimedependentequations1216toasystemofordinarydifferentialequations,M2Rq0,17

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