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____________________CorrespondingauthorAlbanAgazzi,UniversitédeNantesLaboratoiredethermocinétiquedeNantes,LaChantrerie,rueChristianPauc,BP50609,44306Nantescedex3France,phone33240683171,fax33240683141emailalban.agazziunivnantes.frAMETHODOLOGYFORTHEDESIGNOFEFFECTIVECOOLINGSYSTEMININJECTIONMOULDINGA.Agazzi1,V.Sobotka1,R.LeGoff2,D.Garcia2,Y.Jarny11UniversitédeNantes,NantesAtlantiqueUniversités,LaboratoiredeThermocinétiquedeNantes,UMRCNRS6607,rueChristianPauc,BP50609,F44306NANTEScedex3,France2PôleEuropéendePlasturgie,2ruePierreetMarieCurie,F01100BELLIGNAT,FranceABSTRACTInthermoplasticinjectionmoulding,partqualityandcycletimedependstronglyonthecoolingstage.Numerousstrategieshavebeeninvestigatedinordertodeterminethecoolingconditionswhichminimizeundesireddefectssuchaswarpageanddifferentialshrinkage.Inthispaperweproposeamethodologyfortheoptimaldesignofthecoolingsystem.Basedongeometricalanalysis,thecoolinglineisdefinedbyusingconformalcoolingconcept.Itdefinesthelocationsofthecoolingchannels.Weonlyfocusonthedistributionandintensityofthefluidtemperaturealongthecoolinglinewhichisherefixed.Weformulatethedeterminationofthistemperaturedistribution,astheminimizationofanobjectivefunctioncomposedoftwoterms.Itisshownhowthistwoantagonisttermshavetobeweightedtomakethebestcompromise.Theexpectedresultisanimprovementofthepartqualityintermsofshrinkageandwarpage.KEYWORDSInverseproblem,heattransfer,injectionmoulding,coolingdesign1INTRODUCTIONInthefieldofplasticindustry,thermoplasticinjectionmouldingiswidelyused.Theprocessiscomposedoffouressentialstagesmouldcavityfilling,meltpacking,solidificationofthepartandejection.Aroundseventypercentofthetotaltimeoftheprocessisdedicatedtothecoolingofthepart.Moreoverthisphaseimpactsdirectlyonthequalityofthepart12.Asaconsequence,thepartmustbecooledasuniformlyaspossiblesothatundesireddefectssuchassinkmarks,warpage,shrinkage,thermalresidualstressesareminimized.Themostinfluentparameterstoachievetheseobjectivesarethecoolingtime,thenumber,thelocationandthesizeofthechannels,thetemperatureofthecoolantfluidandtheheattransfercoefficientbetweenthefluidandtheinnersurfaceofthechannels.Thecoolingsystemdesignwasprimarilybasedontheexperienceofthedesignerbutthedevelopmentofnewrapidprototypingprocessmakespossibletomanufactureverycomplexchannelshapeswhatmakesthisempiricalformermethodinadequate.Sothedesignofthecoolingsystemmustbeformulatedasanoptimizationproblem.1.1HEATTRANSFERANALYSISThestudyofheattransferconductionininjectiontoolsisanonlinearproblemduetothedependenceofparameterstothetemperature.Howeverthermophysicalparametersofthemouldsuchasthermalconductivityandheatcapacityremainconstantintheconsideredtemperaturerange.Inadditiontheeffectofpolymercrystallisationisoftenneglectedandthermalcontactresistancebetweenthemouldandthepartisconsideredmoreoftenasconstant.TheevolutionofthetemperaturefieldisobtainedbysolvingtheFouriersequationwithperiodicboundaryconditions.Thisevolutioncanbesplitintwopartsacyclicpartandanaveragetransitorypart.Thecyclicpartisoftenignoredbecausethedepthofthermalpenetrationdoesnotaffectsignificantlythetemperaturefield3.Manyauthorsusedanaveragecyclicanalysiswhichsimplifiesthecalculus,butthefluctuationsaroundtheaveragecanbecomprisedbetween15and403.Thecloserofthepartthechannelsare,thehigherthefluctuationsaroundtheaverageare.Henceinthatconfigurationitbecomesveryimportanttomodelthetransientheattransfereveninstationaryperiodicstate.Inthisstudy,theperiodictransientanalysisoftemperaturewillbepreferredtotheaveragecycletimeanalysis.Itshouldbenoticedthatinpracticethedesignofthecoolingsystemisthelaststepforthetooldesign.Neverthelesscoolingbeingofprimaryimportanceforthequalityofthepart,thethermaldesignshouldbeoneofthefirststagesofthedesignofthetools.DOI10.1007/s1228901006952©SpringerVerlagFrance2010IntJMaterForm2010Vol.3Suppl11613–1.2OPTIMIZATIONTECHNIQUESINMOULDINGIntheliterature,variousoptimizationprocedureshavebeenusedbutallfocusedonthesameobjectives.Tangetal.4usedanoptimizationprocesstoobtainauniformtemperaturedistributioninthepartwhichgivesthesmallestgradientandtheminimalcoolingtime.Huang5triedtoobtainuniformtemperaturedistributioninthepartandhighproductionefficiencyi.eaminimalcoolingtime.Lin6summarizedtheobjectivesofthemoulddesignerin3facts.Coolthepartthemostuniformly,achieveadesiredmouldtemperaturesothatthenextpartcanbeinjectedandminimizethecycletime.Theoptimalcoolingsystemconfigurationisacompromisebetweenuniformityandcycletime.Indeedthelongerthedistancebetweenthemouldsurfacecavityandthecoolingchannelsis,thehighertheuniformityofthetemperaturedistributionwillbe6.Inversely,theshorterthedistanceis,thefastertheheatisremovedfromthepolymer.Howevernonuniformtemperaturesatthemouldsurfacecanleadtodefectsinthepart.Thecontrolparameterstogettheseobjectivesarethenthelocationandthesizeofthechannels,thecoolantfluidflowrateandthefluidtemperature.Twokindsofmethodologyareemployed.Thefirstoneconsistsinfindingtheoptimallocationofthechannelsinordertominimizeanobjectivefunction47.Thesecondapproachisbasedonaconformalcoolingline.Lin6definesacoolinglinerepresentingtheenvelopofthepartwherethecoolingchannelsarelocated.Optimalconditionslocationonthecoolingandsizeofthechannelsaresearchedonthiscoolingline.Xuetal.8gofurtherandcutthepartincoolingcellsandperformtheoptimizationoneachcoolingcell.1.3COMPUTATIONALALGORITHMSTocomputethesolution,numericalmethodsareneeded.Theheattransferanalysisisperformedeitherbyboundaryelements7orfiniteelementsmethod4.Themainadvantageofthefirstoneisthatthenumberofunknownstobecomputedislowerthanwithfiniteelements.Onlytheboundariesoftheproblemaremeshedhencethetimespenttocomputethesolutionisshorterthanwithfiniteelements.Howeverthismethodonlyprovidesresultsontheboundariesoftheproblem.Inthisstudyafiniteelementmethodispreferredbecausetemperatureshistoryinsidethepartisneededtoformulatetheoptimalproblem.TocomputeoptimalparameterswhichminimizetheobjectivefunctionTangetal.4usethePowellsconjugatedirectionsearchmethod.Matheyetal.7usetheSequentialQuadraticProgrammingwhichisamethodbasedongradients.Itcanbefoundnotonlydeterministicmethodsbutalsoevolutionarymethods.Huangetal.5useageneticalgorithmtoreachthesolution.Thislastkindofalgorithmisverytimeconsumingbecauseittriesalotofrangeofsolution.Inpracticetimespentformoulddesignmustbeminimizedhenceadeterministicmethodconjugategradientwhichreachesanacceptablelocalsolutionmorerapidlyispreferred.2METHODOLOGY2.1GOALSThemethodologydescribedinthispaperisappliedtooptimizethecoolingsystemdesignofaTshapedpartFigure1.ThisshapeisencounteredinmanypaperssocomparisoncaneasilybedoneinparticularlywithTangetal.4.Figure1HalfTshapedgeometryBasedonamorphologicalanalysisofthepart,twosurfaces1Γand3ΓareintroducedrespectivelyastheerosionandthedilationcoolinglineofthepartFigure1.Theboundaryconditionoftheheatconductionproblemalongthecoolingline3Γisathirdkindconditionwithinfinitetemperaturesfixedasfluidtemperatures.Theoptimizationconsistsinfindingthesefluidtemperatures.Usingacoolinglinepreventstochoosethenumberandsizeofcoolingchannelsbeforeoptimizationiscarriedout.Thisrepresentsanimportantadvantageincaseofcomplexpartswherethelocationofchannelsisnotintuitive.Thelocationoftheerosionlineinthepartcorrespondstotheminimumsolidifiedthicknessofpolymerattheendofcoolingstagesothatejectorscanremovethepartfromthemouldwithoutdamages.2.2OBJECTIVEFUNCTIONIncoolingsystemoptimization,thepartqualityshouldbeofprimarilyimportance.Becausetheminimumcoolingtimeoftheprocessisimposedbythethicknessandthematerialpropertiesofthepart,itisimportanttoreachtheoptimalqualityinthegiventime.Thefluidtemperatureimpactsdirectlythetemperatureofthemouldandthepart,andforturbulentfluidflowtheonlycontrolparameteristhecoolingfluidtemperature.Inthefollowing,theparametertobeoptimizedisthefluidtemperatureandthedeterminationoftheoptimaldistributionaroundthepartisformulatedastheminimizationofanobjectivefunctionScomposedoftwotermscomputedattheendofthecoolingperiodEquation1.ThegoalofthefirsttermS1istoreachatemperaturelevelalongtheerosionofthepart.ThesecondtermS2usedinmanyworks47aimstohomogenizethetemperaturedistributionatthesurfaceofthepartandthereforetoreducethecomponentsof14thermalgradientbothalongthesurface2Γandthroughthethicknessofthepart.ThesetwotermsareweightedbyintroducingthevariablerefTΔ.Itmustbenotedthatwhen∞→ΔrefTthecriterionisreducedtothefirstterm.Onthecontrarytheweightofthesecondtermisincreasedwhen0→ΔrefT.³³ΓΓΓ¸¸¹¨¨©§Δ−Γ¸¸¹¨¨©§−−222112...dTTTdTTTTTSréfejecinjejecfluid1ejecTEjectiontemperature,injTInjectiontemperature,refTΔReferencetemperature,infTFluidtemperature,TTemperaturefieldcomputedwiththeperiodicconditions,0,0XtTXTf21Ω∪Ω∈X,andft,0isthecoolingperiod,ΓΓ³ΓdTT22.1Averagesurfacetemperatureofthepartattheejectiontimeft.3NUMERICALRESULTSNumericalresultsarecomparedwiththoseofTangetal4whoconsidertheoptimalcoolingoftheTshapedpartbydeterminingtheoptimallocationof7coolingchannelsandtheoptimalfluidflowrateofthecoolant.ThefirststepwastoreproducetheirresultsleftpartofFigure2obtainedwiththefollowingconditionscasew0.75in4KTfluid303,fluidflowratescmQ/3643ineachcoolingchannels,s5.23ft.Figure2GeometryTangleftandcoolinglinerightCase1CoolinglineversusfinitenumberofchannelsforaconstantfluidtemperaturefluidT.Theaveragedistancecmd5.1betweenthe7channelsandthepartsurfaceinthecoolingsystemdeterminedbyTangisadoptedinoursystemforlocatingthecoolingline3Γ.Moreover,thefluidtemperatureandtheheattransfercoefficientvaluesissuedfromTangareimposedonthedilationofthepart3Γ.InFigure3thetemperatureprofilesalongthepartsurface2Γarecomparedattheejectiontimeft.Allthetemperatureprofilesalongthesurfaces3,2,1ΓiiareplottedcounterclockwiseonlyonthehalfpartfromiAtoiBFigure1andattheejectiontime.Weobservethatthemagnitudeofthetemperatureislessuniformwiththecoolinglinethanwiththe7channels15Kinsteadof5K.Hencetheoptimalcoolingconfigurationcomputedwithafinitenumberofchannelsisbetterthanthiswiththecoolinglineanditwillbethenconsideredasareference.Figure3Temperatureprofilesalongthepartsurface2ΓCase2CoolinglinewithavariablefluidtemperaturesTfluidandtheweightingfactor∞→ΔrefT.ThefluidtemperaturessTfluidarecomputedbyminimizingtheobjectivefunctionofEquation1wherethesecondtermisignored.TheresultsareplottedinFigures4and5.Figure4TemperatureprofilesalongtheerosionFigure5TemperatureprofilesalongthepartsurfaceInFigure4thetemperatureprofileontheerosionismuchuniformandclosetotheejectiontemperaturewithourmethod511.79.10SthanwithTangsmethod512.32.10S.Howeverinbothcasesapeakremainsbetween0.12mand0.14mwhichcorrespondstothetopoftheribB1inFigure1.Thishotspotisduetothegeometryofthepartandisverydifficulttocool.NeverthelessinFigure5wenoticethattheprofileoftemperatureatthepartsurfaceislessuniformthanin15case120Kinsteadof15K.Inconclusion,thefirsttermisnotsufficienttoimprovethehomogeneityatthepartsurfacebutitisadequateforachievingadesiredleveloftemperatureinthepart.Case3CoolinglinewithsTfluidandtheweightingfactorsKTref10ΔandKTref100Δ.ThefluidtemperaturessTfluidarenowcomputedbyminimizingtheobjectivefunctionofEquation1withKTref10ΔandKTref100Δ.ResultsareplottedinFigures6and7.Figure6TemperatureprofilesalongthepartsurfaceFigure7TemperatureprofilesalongtheerosionTheinfluenceofthetermS2isshowninFigure6.Thistermmakesthesurfacetemperatureofthepartuniform.IndeedincaseKTref10Δtemperatureisquasiconstantalloverthesurface2Γexceptforthehotspotasexplainedpreviously.HoweverforthisvalueofrefTΔ,thetemperatureontheerosionisnotacceptable,themeantemperaturebeingtoohigh340Kforadesiredlevelof336K.Thenthesecondtermimprovesthehomogeneityattheinterfacebuthedgesthesolution.Makinguniformthetemperatureattheinterfacemeanwhileextractingthetotalheatfluxneededtoobtainadesiredleveloftemperatureinthepart,becomeantagonisticproblemsifthislevelistoolow.Thebestsolutionwillbeacompromisebetweenqualityandefficiency.Forexample,bysettingKTref100ΔtheleveloftemperatureejecTinthepartisreachedwhereasthesolutionbecomeslessuniformthanwiththevalueofKTref10Δ.NonethelessthissolutionremainsmoreuniformthanthesolutiongivenbyTang.TheoptimalfluidtemperatureprofilealongthedilationofthehalfpartisplottedFigure8.Figure8Optimalfluidtemperatureprofile4CONCLUSIONSInthispaper,anoptimizationmethodwasdevelopedtodeterminethetemperaturedistributiononacoolinglinetoobtainauniformtemperaturefieldinthepartwhichleadstothesmallestgradientandtheminimalcoolingtime.Themethodologywascomparedwiththosefoundintheliteratureandshoweditsefficiencyandbenefits.Notablyitdoesnotrequirespecifyingapriorithenumberofcoolingchannels.FurtherworkwillconsistindecidingaposterioritheminimalnumberofchannelsneededtomatchthesolutiongivenbytheoptimalfluidtemperatureprofileREFERENCES1PichonJ.F.Injectiondesmatièresplastiques.Dunod,2001.2PlasticBusinessGroupBayer.Optimisedmouldtemperaturecontrol.ATI1104,1997.3S.Y.Hu,N.T.Cheng,S.C.Chen.EffectofcoolingsystemdesignandprocessparametersoncyclicvariationofmoldtemperaturessimulationbyDRBEM,Plastics,rubberandcompositesproc.andappl.,23221232,19954L.QTang,K.Pochiraju,C.Chassapis,S.Manoochehri.Acomputeraidedoptimizationapproachforthedesignofinjectionmoldcoolingsystems.J.ofMech.Design,120165174,1998.5J.Huang,G.M.Fadel.Biobjectiveoptimizationdesignofheterogeneousinjectionmoldcoolingsystems.ASME,123226239,2001.6J.C.Lin.Optimumcoolingsystemdesignofafreeforminjectionmoldusinganabductivenetwork.J.ofMat.Proc.Tech.,120226236,2002.7E.Mathey,L.Penazzi,F.M.Schmidt,F.RondéOustau.Automaticoptimizationofthecoolingofinjectionmoldbasedontheboundaryelementmethod.MaterialsProc.andDesign,NUMIFORM,pages222227,2004.8X.Xu,E.Sachs,S.Allen.Thedesignofconformalcoolingchannelsininjectionmoldingtooling.Polymerengineeringandscience,4112651279,2001.16
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