_Correspondingauthor:AlbanAgazzi,UniversitdeNantes-LaboratoiredethermocintiquedeNantes,LaChantrerie,rueChristianPauc,BP50609,44306Nantescedex3-France,phone:+33240683171,fax:+33240683141email:alban.agazziuniv-nantes.frAMETHODOLOGYFORTHEDESIGNOFEFFECTIVECOOLINGSYSTEMININJECTIONMOULDINGA.Agazzi1*,V.Sobotka1,R.LeGoff2,D.Garcia2,Y.Jarny11UniversitdeNantes,NantesAtlantiqueUniversits,LaboratoiredeThermocintiquedeNantes,UMRCNRS6607,rueChristianPauc,BP50609,F-44306NANTEScedex3,France2PleEuropendePlasturgie,2ruePierreetMarieCurie,F-01100BELLIGNAT,FranceABSTRACT:Inthermoplasticinjectionmoulding,partqualityandcycletimedependstronglyonthecoolingstage.Numerousstrategieshavebeeninvestigatedinordertodeterminethecoolingconditionswhichminimizeundesireddefectssuchaswarpageanddifferentialshrinkage.Inthispaperweproposeamethodologyfortheoptimaldesignofthecoolingsystem.Basedongeometricalanalysis,thecoolinglineisdefinedbyusingconformalcoolingconcept.Itdefinesthelocationsofthecoolingchannels.Weonlyfocusonthedistributionandintensityofthefluidtemperaturealongthecoolinglinewhichisherefixed.Weformulatethedeterminationofthistemperaturedistribution,astheminimizationofanobjectivefunctioncomposedoftwoterms.Itisshownhowthistwoantagonisttermshavetobeweightedtomakethebestcompromise.Theexpectedresultisanimprovementofthepartqualityintermsofshrinkageandwarpage.KEYWORDS:Inverseproblem,heattransfer,injectionmoulding,coolingdesign1INTRODUCTIONInthefieldofplasticindustry,thermoplasticinjectionmouldingiswidelyused.Theprocessiscomposedoffouressentialstages:mouldcavityfilling,meltpacking,solidificationofthepartandejection.Aroundseventypercentofthetotaltimeoftheprocessisdedicatedtothecoolingofthepart.Moreoverthisphaseimpactsdirectlyonthequalityofthepart12.Asaconsequence,thepartmustbecooledasuniformlyaspossiblesothatundesireddefectssuchassinkmarks,warpage,shrinkage,thermalresidualstressesareminimized.Themostinfluentparameterstoachievetheseobjectivesarethecoolingtime,thenumber,thelocationandthesizeofthechannels,thetemperatureofthecoolantfluidandtheheattransfercoefficientbetweenthefluidandtheinnersurfaceofthechannels.Thecoolingsystemdesignwasprimarilybasedontheexperienceofthedesignerbutthedevelopmentofnewrapidprototypingprocessmakespossibletomanufactureverycomplexchannelshapeswhatmakesthisempiricalformermethodinadequate.Sothedesignofthecoolingsystemmustbeformulatedasanoptimizationproblem.1.1HEATTRANSFERANALYSISThestudyofheattransferconductionininjectiontoolsisanonlinearproblemduetothedependenceofparameterstothetemperature.Howeverthermophysicalparametersofthemouldsuchasthermalconductivityandheatcapacityremainconstantintheconsideredtemperaturerange.Inadditiontheeffectofpolymercrystallisationisoftenneglectedandthermalcontactresistancebetweenthemouldandthepartisconsideredmoreoftenasconstant.TheevolutionofthetemperaturefieldisobtainedbysolvingtheFouriersequationwithperiodicboundaryconditions.Thisevolutioncanbesplitintwoparts:acyclicpartandanaveragetransitorypart.Thecyclicpartisoftenignoredbecausethedepthofthermalpenetrationdoesnotaffectsignificantlythetemperaturefield3.Manyauthorsusedanaveragecyclicanalysiswhichsimplifiesthecalculus,butthefluctuationsaroundtheaveragecanbecomprisedbetween15%and40%3.Thecloserofthepartthechannelsare,thehigherthefluctuationsaroundtheaverageare.Henceinthatconfigurationitbecomesveryimportanttomodelthetransientheattransfereveninstationaryperiodicstate.Inthisstudy,theperiodictransientanalysisoftemperaturewillbepreferredtotheaveragecycletimeanalysis.Itshouldbenoticedthatinpracticethedesignofthecoolingsystemisthelaststepforthetooldesign.Neverthelesscoolingbeingofprimaryimportanceforthequalityofthepart,thethermaldesignshouldbeoneofthefirststagesofthedesignofthetools.DOI10.1007/s12289-010-0695-2Springer-VerlagFrance2010IntJMaterForm(2010)Vol.3Suppl1:16131.2OPTIMIZATIONTECHNIQUESINMOULDINGIntheliterature,variousoptimizationprocedureshavebeenusedbutallfocusedonthesameobjectives.Tangetal.4usedanoptimizationprocesstoobtainauniformtemperaturedistributioninthepartwhichgivesthesmallestgradientandtheminimalcoolingtime.Huang5triedtoobtainuniformtemperaturedistributioninthepartandhighproductionefficiencyi.eaminimalcoolingtime.Lin6summarizedtheobjectivesofthemoulddesignerin3facts.Coolthepartthemostuniformly,achieveadesiredmouldtemperaturesothatthenextpartcanbeinjectedandminimizethecycletime.Theoptimalcoolingsystemconfigurationisacompromisebetweenuniformityandcycletime.Indeedthelongerthedistancebetweenthemouldsurfacecavityandthecoolingchannelsis,thehighertheuniformityofthetemperaturedistributionwillbe6.Inversely,theshorterthedistanceis,thefastertheheatisremovedfromthepolymer.Howevernonuniformtemperaturesatthemouldsurfacecanleadtodefectsinthepart.Thecontrolparameterstogettheseobjectivesarethenthelocationandthesizeofthechannels,thecoolantfluidflowrateandthefluidtemperature.Twokindsofmethodologyareemployed.Thefirstoneconsistsinfindingtheoptimallocationofthechannelsinordertominimizeanobjectivefunction47.Thesecondapproachisbasedonaconformalcoolingline.Lin6definesacoolinglinerepresentingtheenvelopofthepartwherethecoolingchannelsarelocated.Optimalconditions(locationonthecoolingandsizeofthechannels)aresearchedonthiscoolingline.Xuetal.8gofurtherandcutthepartincoolingcellsandperformtheoptimizationoneachcoolingcell.1.3COMPUTATIONALALGORITHMSTocomputethesolution,numericalmethodsareneeded.Theheattransferanalysisisperformedeitherbyboundaryelements7orfiniteelementsmethod4.Themainadvantageofthefirstoneisthatthenumberofunknownstobecomputedislowerthanwithfiniteelements.Onlytheboundariesoftheproblemaremeshedhencethetimespenttocomputethesolutionisshorterthanwithfiniteelements.Howeverthismethodonlyprovidesresultsontheboundariesoftheproblem.Inthisstudyafiniteelementmethodispreferredbecausetemperatureshistoryinsidethepartisneededtoformulatetheoptimalproblem.TocomputeoptimalparameterswhichminimizetheobjectivefunctionTangetal.4usethePowellsconjugatedirectionsearchmethod.Matheyetal.7usetheSequentialQuadraticProgrammingwhichisamethodbasedongradients.Itcanbefoundnotonlydeterministicmethodsbutalsoevolutionarymethods.Huangetal.5useageneticalgorithmtoreachthesolution.Thislastkindofalgorithmisverytimeconsumingbecauseittriesalotofrangeofsolution.Inpracticetimespentformoulddesignmustbeminimizedhenceadeterministicmethod(conjugategradient)whichreachesanacceptablelocalsolutionmorerapidlyispreferred.2METHODOLOGY2.1GOALSThemethodologydescribedinthispaperisappliedtooptimizethecoolingsystemdesignofaT-shapedpart(Figure1).ThisshapeisencounteredinmanypaperssocomparisoncaneasilybedoneinparticularlywithTangetal.4.Figure1:HalfT-shapedgeometryBasedonamorphologicalanalysisofthepart,twosurfaces1and3areintroducedrespectivelyastheerosionandthedilation(coolingline)ofthepart(Figure1).Theboundaryconditionoftheheatconductionproblemalongthecoolingline3isathirdkindconditionwithinfinitetemperaturesfixedasfluidtemperatures.Theoptimizationconsistsinfindingthesefluidtemperatures.Usingacoolinglinepreventstochoosethenumberandsizeofcoolingchannelsbeforeoptimizationiscarriedout.Thisrepresentsanimportantadvantageincaseofcomplexpartswherethelocationofchannelsisnotintuitive.Thelocationoftheerosionlineinthepartcorrespondstotheminimumsolidifiedthicknessofpolymerattheendofcoolingstagesothatejectorscanremovethepartfromthemouldwithoutdamages.2.2OBJECTIVEFUNCTIONIncoolingsystemoptimization,thepartqualityshouldbeofprimarilyimportance.Becausetheminimumcoolingtimeoftheprocessisimposedbythethicknessandthematerialpropertiesofthepart,itisimportanttoreachtheoptimalqualityinthegiventime.Thefluidtemperatureimpactsdirectlythetemperatureofthemouldandthepart,andforturbulentfluidflowtheonlycontrolparameteristhecoolingfluidtemperature.Inthefollowing,theparametertobeoptimizedisthefluidtemperatureandthedeterminationoftheoptimaldistributionaroundthepartisformulatedastheminimizationofanobjectivefunctionScomposedoftwotermscomputedattheendofthecoolingperiod(Equation(1).ThegoalofthefirsttermS1istoreachatemperaturelevelalongtheerosionofthepart.ThesecondtermS2usedinmanyworks47aimstohomogenizethetemperaturedistributionatthesurfaceofthepartandthereforetoreducethecomponentsof14thermalgradientbothalongthesurface2andthroughthethicknessofthepart.ThesetwotermsareweightedbyintroducingthevariablerefT.ItmustbenotedthatwhenrefTthecriterionisreducedtothefirstterm.Onthecontrarytheweightofthesecondtermisincreasedwhen0refT.()+=222112.dTTTdTTTTTSrfejecinjejecfluid(1)ejecT:Ejectiontemperature,injT:Injectiontemperature,refT:Referencetemperature,infT:Fluidtemperature,T:Temperaturefieldcomputedwiththeperiodicconditions(),0(,0XtTXTf+=21X,andft,0isthecoolingperiod,=dTT22.1:Averagesurfacetemperatureofthepartattheejectiontimeft.3NUMERICALRESULTSNumericalresultsarecomparedwiththoseofTangetal4whoconsidertheoptimalcoolingoftheT-shapedpartbydeterminingtheoptimallocationof7coolingchannelsandtheoptimalfluidflowrateofthecoolant.Thefirststepwastoreproducetheirresults(leftpartofFigure2)obtainedwiththefollowingconditions(casew=0.75in4):KTfluid303=,fluidflowratescmQ/3643=ineachcoolingchannels,s5.23=ft.Figure2:GeometryTang(left)andcoolingline(right)Case1:Coolinglineversusfinitenumberofchannelsforaconstantfluidtemperature(fluidT).Theaveragedistance(cmd5.1=)betweenthe7channelsandthepartsurfaceinthecoolingsystemdeterminedbyTangisadoptedinoursystemforlocatingthecoolingline3.Moreover,thefluidtemperatureandtheheattransfercoefficientvaluesissuedfromTangareimposedonthedilationofthepart3.InFigure3thetemperatureprofilesalongthepartsurface2arecomparedattheejectiontimeft.Allthetemperatureprofilesalongthesurfaces3,2,1=iiareplottedcounter-clockwiseonlyonthehalfpartfromiAtoiB(Figure1)andattheejectiontime.Weobservethatthemagnitudeofthetemperatureislessuniformwiththecoolinglinethanwiththe7channels(15Kinsteadof5K).Hencetheoptimalcoolingconfigurationcomputedwithafinitenumberofchannelsisbetterthanthiswiththecoolinglineanditwillbethenconsideredasareference.Figure3:Temperatureprofilesalongthepartsurface2Case2:Coolinglinewithavariablefluidtemperature()(sTfluid)andtheweightingfactorrefT.Thefluidtemperatures)(sTfluidarecomputedbyminimizingtheobjectivefunctionofEquation1wherethesecondtermisignored.TheresultsareplottedinFigures4and5.Figure4:TemperatureprofilesalongtheerosionFigure5:TemperatureprofilesalongthepartsurfaceInFigure4thetemperatureprofileontheerosionismuchuniformandclosetotheejectiontemperaturewithourmethod(-511.79.10S=)thanwithTangsmethod(-512.32.10=S).Howeverinbothcasesapeakremainsbetween0.12mand0.14mwhichcorrespondstothetopoftherib(B1inFigure1).Thishotspotisduetothegeometryofthepartandisverydifficulttocool.NeverthelessinFigure5wenoticethattheprofileoftemperatureatthepartsurfaceislessuniformthanin15case1(20Kinsteadof15K).Inconclusion,thefirsttermisnotsufficienttoimprovethehomogeneityatthepartsurfacebutitisadequateforachievingadesiredleveloftemperatureinthepart.Case3:Coolinglinewith()(sTfluid)andtheweightingfactorsKTref10=andKTref100=.Thefluidtemperatures)(sTfluidarenowcomputedbyminimizingtheobjectivefunctionofEquation1withKTref10=andKTref100=.ResultsareplottedinFigures6and7.Figure6:TemperatureprofilesalongthepartsurfaceFigure7:TemperatureprofilesalongtheerosionTheinfluenceofthetermS2isshowninFigure6.Thistermmakesthesurfacetemperatureofthepartuniform.IndeedincaseKTref10=temperatureisquasi-constantalloverthesurface2exceptforthehotspotasexplainedpreviously.HoweverforthisvalueofrefT,thetemperatureontheerosionisnotacceptable,themeantemperaturebeingtoohigh(340Kforadesiredlevelof336K).Thenthesecondtermimprovesthehomogeneityattheinterfacebuthedgesthesolution.Makinguniformthetemperatureattheinterfacemeanwhileextractingthetotalheatfluxneededtoobtainadesiredleveloftemperatureinthepart,becomeantagonisticproblemsifthislevelistoolow.Thebestsolutionwillbeacompromisebetweenqualityandefficiency.Forexample,bysettingKTref100=theleveloftemperature(ejecT)inthepartisreachedwhereasthesolutionbecomeslessuniformthanwiththevalueofKTref10=.NonethelessthissolutionremainsmoreuniformthanthesolutiongivenbyTang.Theoptimalfluidtemperatureprofilealongthedilationofthehalfpartisplotted(Figure8).Figure8:Optimalfluidtemperatureprofile4CONCLUSIONSInthispaper,anoptimizationmethodwasdevelopedtodeterminethetemperaturedistributiononacoolinglinetoobtainauniformtemperaturefieldinthepartwhichlea