temperaturePujos,Cedex,greatmoldingnumercoolingistoeffectandqualityfastestlarindustrieincreasewellknowneconomicallymermeltsufficientlysothatthepartcanbeejectedwithoutanysignificantdeformation2.Anefficientcoolingsystemdesignofthecoolingchannelsaimingatreducingcycletimemustminimizesuchundesireddefectsassinkmarks,differentialshrinkage,ther-malresidualstressbuilt-upandpartwarpage.Duringthepost-fill-ingandcoolingstagesofinjectionmolding,hotmoltenpolymertouchesthecoldmoldwall,andasolidlayerformsonthewall.tiontothecoolantmovingthroughthecoolingchannelsandbynaturalconvectiontotheairaroundtheexteriormoldsurface.Thecoolantisflowingthroughthechannelsatagivenflowrateandagiventemperaturewhichisconsideredconstantthroughoutthelengthofthechannel.Inthiswork,time-dependenttwo-dimensionalmodelisconsideredwhichconsistsofanentirecomputationaldomainofthecavity,moldandcoolingchannelsurfaces.ThecyclictransienttemperaturedistributionofthemoldandpolymerT-shapecanbeobtainedbysolvingthetransientenergyequation.*Correspondingauthor.Tel.:+330540006348；fax:+330540002731.AppliedThermalEngineering29(2009)17861791ContentslistsavailableE-mailaddress:hassanenscpb.fr(H.Hassan).cesswherepolymerisinjectedintoamouldcavity,andsolidifiestoformaplasticpart.Therearethreesignificantstagesineachcy-cle.Thefirststageisfillingthecavitywithmelthotpolymerataninjectiontemperature(fillingandpost-fillingstage).Itisfollowedbytakingawaytheheatofthepolymertothecoolingchannels(coolingstage),finallythesolidifiedpartisejected(ejectionstage).Thecoolingstageisofthegreatestimportancebecauseitsignifi-cantlyaffectstheproductivityandthequalityofthefinalproduct.Itiswellknownthatmorethanseventypercentofthecycletimeintheinjectionmoldingprocessisspentincoolingthehotpoly-distributionofthemoldandpolymer,therefore,theireffectonthesolidificationdegreeofthatpolymer.AfullytransientmoldcoolinganalysisisperformedusingthefinitevolumemethodforaT-shapeplasticmoldwithsimilardimensionsto5,asshowninFig.1.Differentcoolingchannelspositionsandformsarestudied.2.MathematicalmodelTheheatofthemoltenpolymeristakenawaybyforcedconvec-1.IntroductionPlasticindustryisoneoftheworldsrankedasoneofthefewbillion-dolinjectionmoldedpartscontinuestoplasticinjectionmoldingprocessiscientmanufacturingtechniquesforprecisionplasticpartswithvariousshapesatlowcost1.Theplasticinjectionmolding1359-4311/$-seefrontmatterC2112008ElsevierLtd.Alldoi:10.1016/j.applthermaleng.2008.08.011growingindustries,s.Demandforeveryyearbecauseasthemosteffi-producingofandcomplexgeometryprocessisacyclicpro-Asthematerialcoolsdown,thesolidskinbeginstogrowwithincreasingtimeasthecoolingcontinuesuntiltheentirematerialsolidifies.Overtheyears,manystudiesontheproblemoftheopti-mizationofthecoolingsystemlayoutininjectionmoldingandphasechangeofmoldingprocesshavebeenmadebyvariousresearchersandoneswhichfocusedintensityonthesetopicsandwillusedinoursystemdesignandvalidationsare36.ThemainpurposeofthispaperistostudytheeffectofthecoolingchannelspositionanditscrosssectionshapeonthetemperatureCoolingsystemleadstominimumcoolingtimeisnotachievinguniformcoolingthroughoutthemould.C2112008ElsevierLtd.Allrightsreserved.EffectofcoolingsystemonthepolymerduringinjectionmoldingHamdyHassan*,NicolasRegnier,CedricLebot,CyrilLaboratoireTREFLE-Bordeaux1-UMR8508,SiteENSCPB,16Av.PeyBerland,33607PessacarticleinfoArticlehistory:Received15November2007Accepted19August2008Availableonline30August2008Keywords:PolymerSolidificationInjectionmoldingabstractCoolingsystemdesignisofiscrucialnotonlytoreduceityofthefinalproduct.Aperformed.Acyclictransientofthemoldcoolingstudycoolingsystemdesign.Theturedistributionofthemoldtivityoftheprocess,thecoolingshouldbenecessaryfortheAppliedThermaljournalhomepage:www.elsevirightsreserved.GuyDefayeFranceimportanceforplasticproductsindustrybyinjectionmoldingbecauseitcycletimebutalsoitsignificantlyaffectstheproductivityandqual-icalmodelingforaT-moldplasticparthavingfourcoolingchannelsisanalysisusingafinitevolumeapproachiscarriedout.Theobjectivedeterminethetemperatureprofilealongthecavitywalltoimprovetheofcoolingchannelsformandtheeffecttheirlocationonthetempera-thesolidificationdegreeofpolymerarestudied.Toimprovetheproduc-timeshouldbeminimizedandatthesametimeahomogeneouscoolingoftheproduct.TheresultsindicatethatthecoolingsystemwhichandsolidificationatScienceDirectE/locate/apthermengdissipationoftheheatthroughphasechangeprocess.Thistech-plicit/implicittechniquealreadyvalidatedinpreviousstudiesbyVincent8,andLeBot9thatisbasedonthetechniqueNewSource”ofVoller10.Thismethodproposestomaintainthenodeswherephasechangeoccurstothemeltingtemperature.Thissolu-tionisrepeateduntiltheconvergenceofthetemperaturewiththesourcetermequalstothelatentheat.Thesourcetermisdiscret-izedby:ScqLfofsotqLffn1sC0fnsDt5Thesolidfractionwhichisfunctionofthetemperatureisline-arizedas:NomenclatureCP(J/kgK)specificheatatconstantpressurefssolidfractionh(W/m2K)heattransfercoefficientKnumberoftheinternaliterationsLlatentheatoffusion,J/kgnnumberoftheexternaliterationsNnormaldirectionScsourcetermT(K)temperaturet(s)timeH.Hassanetal./AppliedThermalEngineeringniqueisappliedonfixednodesandtheenergyequationinthiscaseisrepresentedasfollow:qCPoTotr:krTSc2AndthesourcetermScisrepresentedby:ScqLfofsot3wherefs(T)=0.0atTC31Tf,(fullliquidregion)0C30fsC301,atT=Tf(iso-thermalphasechangeregion)and,fs(T)=1atTC30Tf(fullsolidregion).Onthewholedomain,thefollowingboundaryconditionsareappliedC0koToNhcTC0Tc2C1；andC0koToNhaTC0Ta2C2:43.NumericalsolutionThenumericalsolutionofthemathematicalmodelgoverningthebehaviorofthephysicalsystemiscomputedbyfinitevolumemethod.TheequationsaresolvedbyanimplicittreatmentforqCPoTotr:krT1Inordertotakeintoaccountthesolidification,asourcetermisaddedtotheenergyequationcorrespondingtoheatabsorptionorheatrelease7,whichtakesinconsiderationtheabsorptionorthethedifferenttermsoftheequationssystem.Whenwetakeincon-siderationthesolidificationeffect,theenergyequationissolvedwithafixedpointalgorithmforthesolidfraction.Foreach,itera-tionofthatfixedpoint,weusediscretizationwithtimehybridex-0.0040.030.004P2P3P4P1P6P7P5Exteriorair,freeconvection,haCoolingchannels,forcedconvection,hfFig.1.MoldstructurewithaT-shapeproductandfourcoolingchannels(Dim.Inm).Greeksymbolsk(W/mK)thermalconductivityq(kg/m3)densityC1interiorsurfaceofthecoolingchannelsC2exteriorsurfaceofthemoldSubscriptsaambientairccoolingfluidfphasechange0.010.010.010.010.010.02A1A2A3A4A5A7B1B2B3B4B5B7C1C2C3C4C5D1D2D3D4D50.040.020.010.015PolymerFig.2.Differentcoolingchannelspositions(Dim.Inm).29(2009)178617911787fnk1KsfnkKsdFsdTC18C19nkKTnk1KC0TnkK6Then,weforcethetemperaturetotendtothemeltingtemper-aturewherethesourcetermisnotnullbyupdatingthesourceterm:Sk1cSkcqCpTC0TfDt7Theenergyequationisdiscretizedasfollow:qCPDtC0qLfDtdFdTC18C19nkK!Tnk1KC0r:krTnk1KqLfDtfnk1KsC0fnsC0qLfDtdFdTC18C19nkKTfqCPDtTn8With:dFdT!C01if0C30fnkKsC301anddFdT0iffnkKs0or19Thisprocessallowsdifferentiatingthetemperaturefieldandso-lidfractioncalculatedatthesameinstantandthelinearsystemissolvedbycentraldiscretizationmethod11.Foreachinternaliter-ation,theresolutionofthatequationprovidesfnk1KsandTnk1K.Theconvergenceisachievedwhenthecriteriaofthesolidfractionandtemperatureareverifiedby:fnk1KsC0fnkKsC13C13C13C13C13C13C302fand；Tnk1KC0TnkKC13C13C13C13C13C13C302T10Furtherdetailsonthenumericalmodelanditsvalidationarepresentedin9.thehorizontaldirection(betweenpositionsB2andB5orpositionsA2andA5whichhavethemaximumsolidificationpercent).WhenwecomparethesolidificationpercentfordifferentlocationsoftheupperpositionsCandD,wefindthatasthechannelapproachestotheproductinthehorizontaldirectionthesolidificationpercentincreases,andthecoolingrateincreaserapidlycomparedwiththeeffectoflowerposition.Wenoticethat,theeffectofthecoolingchannelpositiononthetemperaturedistributionandsolidificationdecreasesasthecoolingtimeaugmentstohighervalueanditsef-1788H.Hassanetal./AppliedThermalEngineering4.ResultsanddiscussionAfulltwo-dimensionaltime-dependentmoldcoolinganalysisininjectionmoldingiscarriedoutforaplatemouldmodelwithT-shapeplasticmoldandfourcoolingchannelsasindicatedinFig.1.Duetothesymmetry,halfofthemoldismodeledandana-lyzed.Allthecoolingchannelshavethesamesizeandtheyhavediameterof10-mmeachincaseofcircularchannels.ThecoolingoperatingparametersandthematerialpropertiesarelistedinTa-bles1and2,respectively,andtheyareconsideredconstantduringallnumericalresults5,7.Eachnumericalcycleconsistsoftwostages,coolingstagewherethecavityisfilledwithhotpolymerinitiallyatpolymerinjectedtemperature,theejectionstagewherethecavityisfilledwithairinitiallyatambienttemperature.Figs.3and4showthecyclictransientvariationsofthemouldtempera-turewithtimefor16smoldcoolingtimeatlocations；(P1,P2,P3,P4)besidethemouldwallsandP5toP7insidethemouldwalls,respectively(Fig.1)andthatincaseofappliedthesolidifica-tionandwithoutappliedsolidification.Theyaresimulatedforthefirst30cyclesincaseofcircularcoolingchannelsposition(A5,D3)asshowninFig.2.Wefindthat,thesimulatedresultsareingoodagreementwiththetransientcharacteristicofthecyclicmoldtem-peraturevariationsdescribedin5.Itisfoundthatthereisaslightlydifferenceintemperaturesvaluesbetweenthetworesults,thusduetothedifferenceinnumericalmethodusedandtheaccu-racyinthenumericalcalculations.Thefiguresshowthat,therela-tivelytemperaturefluctuationislargestnearthecavitysurfaceanddiminishesawayfromthecavitysurface.Wefindthatthemaxi-mumamplitudeoftemperaturefluctuationduringthesteadycyclecanreach10C176Cwithoutapplyingsolidificationand15C176Cincaseofapplyingthesolidification.4.1.EffectofcoolingchannelsformAnefficientcoolingsystemdesignprovidinguniformtempera-turedistributionthroughouttheentirepartduringthecoolingpro-cessshouldensureproductqualitybypreventingdifferentialshrinkage,internalstresses,andmouldreleaseproblems.Italsoshouldreducetimeofcoolingandacceleratethesolidificationpro-cessoftheproducttoaugmenttheproductivityofthemoldingTable1CoolingoperatingparametersCoolingoperatingparameterCoolingoperatingparameterCoolantfluidtemperature30C176CAmbientairtemperature30C176CPolymerinjectedtemperature220C176CHeattransfercoefficientofambientair77W/m2KTemperatureoffusionofpolymer110C176CHeattransfercoefficientinsidecoolingchannel3650W/m2KLatentheat115kJ/Moldopeningtime4skgprocess.Todemonstratetheinfluenceofthecoolingchannelsformonthetemperaturedistributionthroughoutthemouldandsolidi-ficationprocessoftheproduct,weproposedthreedifferentcrosssectionalformsofthecoolingchannels,circular,square,rectangu-lar1withlongtowidthratioof0.5andrectangular2withwidthtolongratioof0.25.Twocasesarestudied；firstcase,allthecoolingchannelshavethesamecrosssectionalarea,andthesecondcase,theyhavethesameperimeter.Thecomparisoniscarriedoutforthesamecoolingchannelsposition(A5,D3).Fig.5showsthesolidificationpercent(calculatednumericallyasthesummationofthesolidfractionofeachelementmultipliedbytheareaofthatelementtototalareaoftheproduct)fordiffer-entformswithdifferentcoolingtime.Thefigureindicatesthattheeffectofcoolingchannelsformonthecoolingratedecreaseswithincreasingthecoolingtime.Italsoshowsthatthecoolingchannelformrectangle2hasthemaximumsolidificationpercentforcase1,andincase2thechangingofthecoolingchannelsformhasnotasensibleeffectonthesolidificationpercent.Thesameresultscanbeobtainedwhenwecomparedthesolidificationintheprod-uctandthetemperaturedistributionthoughthemouldfordiffer-entformswiththesamecrosssectionalareaattheendofthecoolingstageforcoolingtime24sforcoolingcycle25,asshowninFigs.6and7,respectively.Theresultsindicatethatthecoolingprocessisimprovedasthecoolingchannelstendtotaketheformoftheproduct.4.2.EffectofcoolingchannelspositionToinvestigatetheeffectofthecoolingchannelsposition,wedi-videdtheproposedpositionsintofourgroups,groupsAandBfordifferentpositionsofthebottomcoolingchannel,withafixedpo-sitionofthetopcoolingchannel,andwithviceversaforgroupsCandDforthesamecoolingchannelform(circular)asillustratedinFig.2.Fig.8representstheeffectofdifferentcoolingchannelpositionsontheofsolidificationpercentattheendof25thcoolingcycleforgroupsAandB(lowercoolingchanneleffect),CandD(uppercool-ingchanneleffect)withcoolingtime.Itindicatesthatforlowercoolingchannelpositioneffect,thecoolingrateincreasesandhencethesolidificationpercentofthepolymerincreasesasthecoolingchannelapproachesthepolymerintheverticaldirection(positionBhassolidificationpercentgreaterthanpositionA,andwiththesamepositionsCandD).Thefigureshowsalsothemostefficientcoolingrateisobtainedasthecoolingchanneltakesthepositionbetween20%and50%throughtheproductlengthforTable2MaterialpropertiesMaterialDensity(kg/m3)Specificheat(J/kgK)Conductivity(W/mK)Mould767042636.5Polymer93818000.25Air1.1710060.026329(2009)17861791fectonthecoolingrateoftheproductisnotthesamefordifferentpositions.Engineering6065abH.Hassanetal./AppliedThermalThesolidificationdegreedistributionthroughtheproductatt