外文翻译三维注射成型流动模拟的研究.docx
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外文翻译三维注射成型流动模拟的研究
附录
NumericalFillingSimulationofInjectionMolding
UsingThree—DimensionalModel
Abstract:
Mostinjectionmoldedpartsarethree-dimensional,withcomplexgeometricalconfigurationsandthick/thinwallsections.A3Dsimulationmodelwillpredictmoreaccuratelythefillingprocessthana2.5Dmode1.Thispapergivesamathematicalmodelandnumericmethodbasedon3Dmodel,inwhichanequal-ordervelocity-pressureinterpolationmethodisemployedsuccessfully.Therelationbetweenvelocityandpressureisobtainedfromthediscretizedmomentumequationsinordertoderivethepressureequation.A3Dcontrolvolumeschemeisemployedtotracktheflowfront.Thevalidityofthemodelhasbeentestedthroughtheananalysisoftheflowincavity.
Keywords:
threedimension;equal-orderinterpolation;simulation;injectionmolding
1Introduction
Duringinjectionmolding,thetheologicalresponseofpolymermeltsisgenerallynon-Newtonianandnoisothermalwiththepositionofthemovingflowfront.Becauseoftheseinherentfactors,itisdifficulttoanalyzethefillingprocess.Therefore,simplificationsusuallyareused.Forexample,inmiddle-planetechniqueanddualdomaintechnique[1],becausethemostinjectionmoldedpartshavethecharacteristicofbeingthinbutgenerallyofcomplexshape,theHele-Shawapproximation[2]isusedwhileananalyzingtheflow,i.e..Thevariationsofvelocityandpressureinthegapwise(thickness)dimensionareneglected.Sothesetwotechniquesareboth2.5Dmoldfillingmodels,inwhichthefillingofamoldcavitybecomesa2Dprobleminflowdirectionanda1Dprobleminthicknessdirection.
However,becauseoftheuseoftheHele-Shawapproximation,theinformationthat2.5Dmodelscangenerateislimitedandincomplete.Thevariationinthegapwise(thickness)dimensionofthephysicalquantitieswiththeexceptionofthetemperature,whichissolvedbyfinitedifferencemethod,isneglected.Withthedevelopmentofmoldingtechniques,moldedpartswillhavemoreandmorecomplexgeometryandthedifferenceinthethicknesswillbemoreandmorenotable,sothechangeinthegapwise(thickness)dimensionofthephysicalquantitiescannotbeneglected.Inaddition,theflowsimulatedlooksunrealisticinasmuchasthemeltpolymerflowsonlyonsurfacesofcavity,whichappearsmoreobviouswhentheflowsimulationisdisplayedinamouldcavity.
3Dsimulationmodelhasbeenaresearchdirectionandhotspotinthescopeofsimulationforplasticinjectionmolding.In3Dsimulationmodel,velocityinthegapwise(thickness)dimensionisnotneglectedandthepressurevariesinthedirectionofpartthickness,and3Dfiniteelementsareusedtodiscretizethepartgeometry.Aftercalculating,completedataareobtained(notonlysurfacedatabutalsointernaldataareobtained).Therefore,a3Dsimulationmodelshouldbeabletogeneratecomplementaryandmoredetailedinformationrelatedtotheflowcharacteristicsandstressdistributionsinthinmoldedpartsthantheoneobtainedwhenusinga2.5Dmodel(basedontheHele-Shawapproximation).Ontheotherhand,a3Dmodelwillpredictmoreaccuratelythecharacteristicsofmoldedpartshavingthickwalledsectionssuchasencounteredingasassistedinjectionmolding.Severalflowbehaviorsattheflowfront.suchas“fountainflow”.which2.5Dmodelcannotpredict,canbepredictedby3Dmode1.Meanwhile,theflowsimulationlooksmorerealisticinasmuchastheoverallananalysisresultisdirectlydisplayedin3Dpartgeometryortransparentmouldcavity.
ThisPaperpresentsa3Dfiniteelementmodeltodealwiththethree—dimensionalflow,whichemploysanequa1-ordervelocity-pressureformulationmethod[3,4].Therelationbetweenvelocityandpressureisobtainedfromthediscretizedmomentumequations,thensubstitutedintothecontinuityequationtoderivepressureequation.A3Dcontrolvolumeschemeisemployedtotracktheflowfront.Thevalidityofthemodelhasbeentestedthroughtheanalysisoftheflowincavity.
2GoverningEquations
Thepressureofmeltisnotverybigduringfillingthecavity,inaddition,reasonablemoldstructurecanavoidoverbigpressure,sothemeltisconsideredincompressible.Inertiaandgravitationareneglectedascomparedtotheviscousforce.
Withtheaboveapproximation,thegoverningequations,expressedincartesiancoordinates,areasfollowing:
Momentumequations
Continuityequation
Energyequation
where,x,y,zarethreedimensionalcoordinatesandu,v,warethevelocitycomponentinthex,y,zdirections.P,T,ρandηdenotepressure,temperature,densityandviscostyrespectively.
Crossviscositymodelhasbeenusedforthesimulations:
where,n,γ,rarenon-Newtonianexponent,shearrateandmaterialconstantrespectively.Becausethereisnonotablechangeinthescopeoftemperatureofthemeltpolymerduringfilling,Anheniusmodel[5]forη0isemployedasfollowing:
whereB,Tb,βarematerialconstants.
3NumericalSimulationMethod
3.1Velocity-PressureRelation
Ina3Dmodel,sincethechangeofthephysicalquantitiesarenotneglectedinthegapwise(thickness)dimension,themomentumequationsaremuchmorecomplexthanthoseina2.5Dmode1.Itisimpossibletoobtainthevelocity—pressurerelationbyintegratingthemomentumequationsinthegapwisedimension,whichisdoneina2.5Dmodel.Themomentumequationsmustbefirstdiscretized,andthentherelationbetweenvelocityandpressureisderivedfromit.Inthispaper,themomentumequationsarediscretizedusingGalerkin’smethodwithbilinearvelocity-pressureformulation.Theelementequationsareassembledintheconventionalmannertoformthediscretizedglobalmomentumequationsandthevelocitymaybeexpressedasfollowing
where
thenodalpressurecoefficientsaredefinedas
where
representglobalvelocitycoefficientmatricesinthedirectionofx,y,zcoordinaterespectively.
denotethenodalpressurecoefficientsthedirectionofx,y,zcoordinaterespectively.Thenodalvaluesfor
areobtainedbyassemblingtheelement-by-elementcontributionsintheconventionalmanner.N,iselementinterpolationandimeansglobalnodenumberandj,isforanode,theamountofthenodesaroundit.
3.2PressureEquation
Substitutionofthevelocityexpressions
(2)intodiscretizedcontinuityequation,whichisdiscretizedusingGalerkinmethod,yieldselementequationforpressure:
Theelementpressureequationsareassembledtheconventionalmannertoformtheglobalpressureequations.
3.3BoundaryConditions
Incavitywall,theno-slipboundaryconditionsareemployed,e.g.
Onaninletboundary,
3.4VelocityUpdate
Afterthepressurefieldhasbeenobtained,thevelocityvaluesareupdatedusingnewpressurefieldbecausethevelocityfieldobtainedbysolvingmomentumequationsdoesnotsatisfycontinuityequation.Thevelocitiesareupdatedusingthefollowingrelations
Theoverallprocedureforfluidflowcalculationsisrelaxationiterative,asshowninFig.landthecalculationisstablewithoutpressureoscillation.
3.5TheTracingoftheFlowFronts
Theflowoffluidinthecavityisunsteadyandthepositionoftheflowfrontsvalueswithtime.Likein2.5Dmodel,inthispaper,thecontrolvolumemethodisemployedtotracethepositionoftheflowfrontsaftertheFAN(FlowAnalysisNetwork)[6].But3Dcontrolvolumeisaspecialvolumeandmorecomplexthanthe2Dcontrolvolume.
Itisrequiredthat3Dcontrolvolumesofallnodesfillthepartcavitywithoutgapandhollowspace.Two3DcontrolvolumesareshowninFig.2.
4ResultsandDiscussion
ThetestcavityanddimensionsareshowninFig.3(a).TheselectedmaterialisABS780fromKumbo.Theparametricconstantscorrespondingtothen,γ,B,Tbandβofthefive-constantCross-typeViscositymodelare0.2638,4.514×le4Pa,1.3198043×le-7Pa*S,1.12236×1e4K,0.000Pa-1.Injectiontemperatureis45℃,mouldtemperatureis250℃,injectionflowrateis44.82cu.cm/s.Themeshed3DmodelofcavityisshowninFig.3(b).
“Fountainflow”isatypicalflowphenomenonduringfilling.Whenthefluidisinjectedintoarelativelycoldermould,solidlayerisformedinthecavitywallsbecauseofthediffusioncooling,sotheshearnearthewallstakesplaceandiszerointhemiddleofcavity,andthefluidnearthewallsdeflectstomovetowardthewalls.Thefluidnearthecentermovesfasterthantheaverageacrossthethicknessandcatchesupwiththefrontsotheshapeoftheflowfrontisroundlikethefountain.Theroundshapeoftheflowfrontoftheexampleinseveralfillingtimespredictedbypresent3DmodelandshowninFig.4(a),conformstothetheoryandexperiments.Contrarily,theshapeoftheflowfrontpredictedby2.5DmodelandshowninFig.4(b)donotrevealthe“Fountainflow”.
TheflowfrontcomparisonatthefillingstageisillustratedinFig.5.Itshowsthatthepredictedresultsbasedonpresent3DmodelagreewellwiththatbasedonMoldflow3Dmode1.ThegatepressureisillustratedinFig.6,comparedwiththepredictionofMoldflow3Dmodel.Itshowsthatthepredictedgatepressureofpresent3DmodelismainlyinagreementwiththatbasedonMoldflow3Dmode1.Themajorreasonforthisdeviationisdifferenceindealingwiththemodelandmaterialparameters.
5Conclusions
Atheoreticalmodelandnumericalschemetosimulatethefillingstagebasedona3Dfiniteelementmodelarepresented.Acavityhasbeenemployedasexampletotestthevalidity.3Dnumeralsimulationofthefillingstageininjectionmouldingisadevelopmentdirectioninthescopeofsimulationforplasticinjectionmoldinginthefuture.Thelongtimecostisatpresentaproblemfor3Dfillingsimulation,butwiththedevelopmentofcomputerhardwareandimprovementinsimul