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JOURNALOFOPTOELECTRONICSANDADVANCEDMATERIALSVOL8,NO5,OCTOBER2006,P17361740THERESIDUALSTRESSESOFFEBSITYPEINANINGOTMOULDIAŞTEFĂNOAEI,DRADU,HCHIRIACAABA“ALICUZA”UNIVERSITY,CAROLIBLVD,11,700506,IASI,ROMANIABNATIONALINSTITUTEOFRESEARCHANDDEVELOPMENTFORTECHNICALPHYSICS,47MANGERONBLVD,700050IASI,ROMANIAINTHISPAPERWEANALYSEDTHEINTERNALRESIDUALSTRESSESTHATAPPEARDURINGTHERAPIDCOOLINGOFAFEBSITYPEALLOY,LAININTHEINTERIORCHANNELOFANINGOTMOULDWHOSEEXTERIORWALLSAREMAINTAINEDATTHEROOMTEMPERATURETHETHEORETICALMODELPRESENTEDINTHEPAPEREMPHASIZESINACOMPLETEANDSYNTHETICALMANNER,THESPATIOTEMPORALDISTRIBUTIONOFTHEINDUCEDSTRESSESINAFEBSITYPEALLOYDURINGITSCOOLINGTOTHEROOMTEMPERATURE,TAKINGINTOACCOUNTBOTHTHETHERMALBEHAVIOUROFTHEALLOYANDTHESUPPLEMENTARYSTRESSESINDUCEDBYTHEINGOTMOULD,ASARESULTOFTHEDIFFERENCEBETWEENTHETHERMALEXPANSIONCOEFFICIENTSOFTHETWOMATERIALSINCONTACTRECEIVEDSEPTEMBER6,2006;ACCEPTEDSEPTEMBER13,2006KEYWORDSRESIDUALSTRESSES,INGOTMOULD,THERMALGRADIENTS1INTRODUCTIONTHEAIMOFTHISPAPERISTOEVALUATEOFTHETHERMALSTRESSESDURINGTHECOOLINGOFANALLOYINTHEINNERVERTICALCHANNELOFANINGOTMOULD,WHOSEEXTERIORWALLSAREMAINTAINEDATTHEROOMTEMPERATURE,MOREPRECISELY,OURPURPOSEISTODETERMINETHESPATIOTEMPORALDISTRIBUTIONOFTHESTRESSESTHATAPPEARDURINGTHERAPIDCOOLINGOFTHEWHOLEMATERIALTOTHETEMPERATURETHISMODELCONTAINSTWOIMPORTANTTASKS1THEEVALUATIONOFTHESPATIOTEMPORALTEMPERATURE’SDISTRIBUTIONDURINGTHERAPIDCOOLING1,2,3OFTHEMATERIALTOTHEROOMTEMPERATUREFORDIFFERENTVALUESOFTHERADIUSOFTHEINNERCHANNEL,TAKINGINTOACCOUNTTHETHERMALCONDITIONSONTHEALLOYINGOTMOULDINTERFACE,WEANALYZEDTHESPATIALANDTEMPORALEVOLUTIONOFTHETEMPERATURE;2STARTINGFROMTHESPATIOTEMPORALDISTRIBUTIONOFTHETEMPERATURE,ONECANOBTAINTHESTRESSESDUETOBOTHRAPIDCOOLINGBIGTHERMALGRADIENTSANDCONSTRAINTSPRODUCEDTOTHEALLOYBYTHECOOLEDINGOTMOULDASARESULTOFADIFFERENCEBETWEENTHETHERMALEXPANSIONCOEFFICIENTSOFTHETWOMATERIALS300WTKWTTHETHEORETICALMODELPRESENTEDHEREISBASEDONTHEFOLLOWINGWORKINGHYPOTHESESITHEFORMTHEGEOMETRYOFTHEFLOWINGCHANNELOFTHEMELTEDMETAL/ALLOYWITHACYLINDRICALSYMMETRYDEMANDSACYLINDRICALCOORDINATESSYSTEM,,RΘZ,WITHTHEAXISPOINTEDVERTICALLYDOWNWARD,ALONGWITHTHEINGOTMOULDCHANNELANDTHEZRCOORDINATEALONGITSRADIALDIRECTION;IIWECONSIDERTHELENGTHLOFTHEINGOTMOULDFLOWINGCHANNELMUCHBIGGERTHANITSRADIUS,,2R2LR;IIITHEMATERIALMADEFROMAFEBSIALLOYHASATTHEINITIALMOMENTTHETEMPERATUREANDIVWEASSUMETHATTHEREARENOTEMPERATUREGRADIENTSALONGTHEINGOTMOULDCHANNELTHECHARACTERISTICSOFTHEALLOYARE2,3ISTHESPECIFICHEAT,ISTHETHERMALCONDUCTIBILITY,1200MTKKK3530/PCJKG130/KWM37210/MKGMΡ⋅ISTHEMASSDENSITY,112210/ALLOYENM⋅ISTHEYOUNG’SMODULUSANDISTHETHERMALEXPANSIONCOEFFICIENTTHECHARACTERISTICOFTHEINGOTMOULDARE2,368710ALLOYKΑ−−⋅1MK2383/KWISTHETHERMALCONDUCTIBILITY,15LCMISTHELENGTH,61710CU1KΑ−−⋅ISTHETHERMALEXPANSIONCOEFFICIENT,1021310/INGOTENM⋅ISTHEYOUNG’SMODULUS,11RMMISTHERADIUSOFTHEINNERVERTICALCHANNELAND25RMMISTHETOTALRADIUS2THERAPIDCOOLINGPROCESSOFTHESOLIDIFIEDALLOY21THETEMPERATUREDISTRIBUTIONINTHEALLOYINGOTMOULDSYSTEMTEMPERATUREDISTRIBUTIONINTHEALLOYINTHEFOLLOWINGITISANALYSEDTHERAPIDCOOLINGOFTHESOLIDIFIEDMETALFROMTHETEMPERATURETOTHEROOMTEMPERATUREASITWILLBESHOWN,FROMAMATHEMATICALPOINTOFVIEW,WEMAYCONSIDERTHISASAPROBLEMOFCONDUCTIONANDTHERMALTRANSFERWITHASOURCE,ANDITIMPLIESTHEDETERMINATIONOFTHESPATIOTEMPORALDISTRIBUTIONOFTHETEMPERATUREGT1TR,TOFTHESOLIDIFIEDMATERIAL,WHICHHASINTHECENTEROFTHEINGOTMOULD’SCHANNELTHETEMPERATUREGTGT800KISTHESOLIDIFICATIONTEMPERATURE;THESOURCEISDISTRIBUTEDONTHEINNERSURFACEOFTHEINGOTMOULD’SWALL,HAVINGTHETEMPERATURETHEHEATLOSSESOFTHEALLOYDUETOITSFORCEDCOOLINGMAYBECONSIDEREDASUNIFORMLYDISTRIBUTEDNEGATIVESOURCESTHEDETERMINATIONOFTHETEMPERATUREDEMANDSTOFINDTHESOLUTIONOFTHETHERMALBALANCEEQUATIONFORTHEMATERIALSUBMITTEDTOARAPIDCOOLINGPROCESSWT1TR,TTHERESIDUALSTRESSESOFFEBSITYPEINANINGOTMOULD17372111121WTTTABTTRRR⎛⎞∂∂∂−−⎜⎟∂⎝⎠1RR≤≤,FOR0,1WHERE1/ΡMPAKC,,/BAPLV21ARΠ,12PRΠANDLISTHEINGOTMOULD’SLENGTHTHEGENERALSOLUTIONOFEQ1ISOFTHEFORM2210,/−−MATWTRTTCIRBAME,WHEREISANINTEGRATIONCONSTANT,C20/−IRBAMAREMODIFIEDBESSELFUNCTIONSOFTHEZEROORDERANDTHECONSTANTWILLBEDETERMINEDFROMTHETHERMALCONDITIONSONTHEINTERFACEALLOY–INGOTMOULDBYIMPOSINGTHEPARTICULARCONDITIONSM10,0GTRTTAND100TRR∂∂ONECANOBTAINTHEEXPRESSIONFORTHECONSTANTCGWCTT−,WHICHLEADSUSTOTHEEXPRESSIONOFTHETEMPERATUREDISTRIBUTIONINTHEMATERIAL2210,/MATWGWTRTTTTIRBAME−−−2THETEMPERATUREDISTRIBUTIONINTHEINGOTMOULD’SWALLDURINGTHECOOLINGPROCESS,THEINGOTMOULD’SWALLRECEIVESTHEHEATAMOUNTFROMTHECOOLINGALLOYWEWILLCONSIDERTHETEMPERATUREDISTRIBUTIONINTOTHEINGOTMOULD’SWALLOFTHEFORM421,LNTRTARA2,3WHERE1AAND2AAREVARIABLESDEPENDINGONTHETIMET11AAT≡,22AAT≡BOUNDARYCONDITIONSFORTHEALLOY–INGOTMOULDINTERFACEINORDERTODETERMINETHEFINALEXPRESSIONSOFTHETEMPERATURESANDWEMUSTUSETHEFOLLOWINGBOUNDARYCONDITIONS1,TRT2,TRTITHEHEATFLUXFROMTHEALLOYISRECEIVEDBYTHEINGOTMOULDTHISHEATFLUXFROMTHEALLOYINGOTINTERFACEMUSTBECONTINOUSSO,FOR,WEMUSTHAVE1RR1211∂∂TTRRRRRRKK,4WHEREANDARETHECOEFFICIENTSOFTHERMALCONDUCTIVITYOFTHEALLOYANDINGOTMOLD,RESPECTIVELY;1K2KIIONTHEALLOYINGOTINTERFACE,THETEMPERATURESFROMTHEADJACENTREGIONSMUSTBEEQUAL1R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