AbstractqSponsors:Alcan-PechineyCompanyandSwissNationalScienceFoundation；GrantNo.200020-101391.*Correspondingauthor.Tel.:+41223792366；fax:+41223792205.E-mailaddresses:yasser.safaepfl.ch,yasser.safaobs.unige.ch(Y.Safa).AAppliedMathematicalModelling33(2009)14791492/locate/apm0307-904X/$-seefrontmatterC2112008ElsevierInc.Allrightsreserved.Aphasechangingproblemmotivatedbythemodellingofthermalproblemcoupledwithmagnetohydro-dynamiceectsinareductioncellisstudied.InasmeltingcelloperatingwithHallHeroultprocess,themetalpartisproducedbytheelectrolysisofaluminiumoxidedissolvedinabathbasedonmoltencryolite1.Var-iousphenomenatakeplaceinsuchacellforwhichatransversesectionisschematicallypicturedinFig.1.Runningfromtheanodesthroughliquidaluminiumandcollectorbars,thesteadyelectriccurrentspreadsintheelectrolyticbath.Theimportantmagneticfieldgeneratedbythecurrentscarriedtothealignmentofcells,coupledwiththecurrentsrunningthroughthecellsthemselvesgivesrisetoafieldofLaplaceforceswhichmaintainsamotionwithinthesetwoconductingliquids.Amagnetohydrodynamicinteractiontakesplaceinthecell.IntheotherhandaheatingsourceisproducedbytheJouleeectduetotheelectricresistivityofthebath.Asystemofpartialdierentialequationsdescribingthethermalbehaviorofaluminiumcellcoupledwithmagnetohy-drodynamiceectsisnumericallysolved.Thethermalmodelisconsideredasatwo-phasesStefanproblemwhichconsistsofanon-linearconvectiondiusionheatequationwithJouleeectasasource.Themagnetohydrodynamicfieldsaregov-ernedbyNavierStokesandbystaticMaxwellequations.Apseudo-evolutionaryscheme(Cherno)isusedtoobtainthestationarysolutiongivingthetemperatureandthefrozenlayerprofileforthesimulationoftheledgesinthecell.Anumer-icalapproximationusingafiniteelementmethodisformulatedtoobtainthefluidvelocity,electricalpotential,magneticinductionandtemperature.Aniterativealgorithmand3-Dnumericalresultsarepresented.C2112008ElsevierInc.Allrightsreserved.Keywords:Aluminiumelectrolysis；Chernoscheme；Heatequation；Magnetohydrodynamics；Ledge；Solidification1.IntroductionNumericalsimulationofthermalproblemscoupledwithmagnetohydrodynamiceectsinaluminiumcellqY.Safa*,M.Flueck,J.RappazInstituteofAnalysisandScientificComputing,EcolePolytechniqueFederaledeLausanne,Station8,1015Lausanne,SwitzerlandReceived27December2006；receivedinrevisedform4February2008；accepted8February2008Availableonline29February2008doi:10.1016/j.apm.2008.02.011ElectrolyteAnodeBlocksFig.1.Transversecrosssectionofaluminiumreductioncell.1480Y.Safaetal./AppliedMathematicalModelling33(2009)14791492Onthewallofthecell,asolidifiedbathlayer,theso-calledledgeiscreated.Theseledgesprotectthecellsidewallfromcorrosiveelectrolyticbathandreducetheheatlossfromthecell(see2page23).Moreover,itsprofilestronglyinfluencesthemagnetohydrodynamicstabilitycausingoscillationsofthealuminiumbathinterfacewhichcoulddecreasethecurrenteciency.Consequentlyanoptimalledgeprofileisoneoftheobjec-tivesofcellsidewalldesign.Thethermalsolidificationprobleminsmeltingcellhasbeentreatedbyseveralauthors35.Asfarasweareaware,thisproblemhasneverbeenconsideredwhencoupledwiththemagnetohydrodynamicfields.Theaimofthispaperistodealwithsuchfieldsinteraction.LetusmentionthatthedetailsonthisproblemcanbefoundinSafasthesis6.Mathematically,theproblemistosolveacoupledsystemofpartialdierentialequationsconsistingoftheheatequationwithJouleeectasasource,MaxwelllawequationswithelectricalconductivityasafunctionoftemperatureandNavierStokesequations.Theinterfacebetweenaluminiumandbathisanunknown.Theledgeisconsideredaselectricalinsulator,thethermalmodelisastationarytwo-phasesStefanproblem.Theoutlineofthispaperisasfollow:inSection2weintroducethephysicalmodel,thealgorithmispresentedAluminiumCathodeLiningFrozenledgeFrozenledgeinSection3andwegivethenumericalresultsinSection4.2.ThemodelInordertointroducethemodelwefirstdescribesomegeometricalandphysicalquantities.2.1.GeneraldescriptionsThegeometryisschematicallydefinedbyFig.1.Weintroducethefollowingnotations:C15XX1X2:fluidsandsolidledge,C15NN1N2:electrodes,C15KXN:domainrepresentingthecellandwedefinetheinterfaces:C15CoX1oX2:freeinterfacebetweenaluminiumandbath,whichisanunknown,C15RioKoNi；i1；2,C15RR1R2:outerboundaryoftheelectrodes.Y.Safaetal./AppliedMathematicalModelling33(2009)147914921481C15Cp:specificheat,C15:latentheat.2.2.PhysicalassumptionsThemodelleansonthefollowingbasichypotheses:1.Thefluidsareimmiscible,incompressibleandNewtonian.2.IneachdomainXi,i=1,2,thefluidsaregovernedbythestationaryNavierStokesequations.3.TheelectromagneticfieldssatisfythestationaryMaxwellsequations,OhmslawismoreoversupposedtobevalidinallthecellK.4.Theelectricalcurrentdensityoutsidethecellisgiven(currentinthecollectorbars).5.Theelectricalconductivityrisfunctionoftemperaturehinthefluidsandelectrodesparts.6.Theviscosityg,thedensityqandthespecificheatCparetemperatureindependent.7.ThevolumesofthedomainsX1andX2havegivenvalues(massconservation).8.TheonlyheatsourceisproducedbytheJouleeectduetothecurrentcrossingthecell.9.Eectsofchemicalreactions7,Marangonieect8,9,surfacetensionaswellasthepresenceofgasflowareneglected.2.3.ThehydrodynamicproblemInthispartweconsiderthetemperaturefieldhandtheelectromagneticfieldsjandbasknown.WechoosetorepresenttheunknowninterfacebetweenaluminiumandbathbyaparametrizationoftheformCC22hx；y；z:zC22hx；y；x；y2DC138,whereDisusuallyarectanglecorrespondingtotheparametrizationofaluminiumcathodeinterface.WedenotethedependenceofX1；X2andCwithrespecttoC22hbyusingTheunknownphysicalfieldswithwhichweshalldealarelistedasfollows:Hydrodynamicfields:C15u:velocityfieldinXi；i1；2；(u0insolidledges),C15p:pressure.Electromagneticfields:C15b:magneticinductionfield,C15e:electricfield,C15j:electriccurrentdensity.Thermalfields:C15H:enthalpy,C15h:temperature.ThematerialpropertiesaredefinedasC15q:massdensity,C15rbandr:electricalconductivityinand,respectively,outsidethebath,C15g:viscosityofthefluids,C15l0:magneticpermeabilityofthevoid,C15k:thermalconductivity,XiXiC22h；i1；2；CCC22h:hx；ydxdyV1；whereV1isthevolumeofaluminium:1C22C22Here3thosethefluids.fieldsThefluidC22C22Inorderinvolvingapenalizationtool.Thevelocityandthepressurewillthenbedefinedinbothliquidsandsolids.WefunctionKisgivenbyCarmanKozeny”law:theDarcy1482Y.Safaetal./AppliedMathematicalModelling33(2009)14791492Whenfs!1,wegetKfs!1andthenu0inthesolidzone.law:rpqgzC0Kujb:IfonlyliquidphaseispresentwehaveK0andtheaboveequationreducestotheusualNavierStokesequa-tion.InsidethemushyzoneKmaybeverylarge,comparedtotheotherterms,andtheaboveequationmimicsqu；ruC0div2lDuC0pqgzIKujbinX1C22hX2C22h:7wherePisthemeanporesizeandCisaconstantobtainedexperimentally(see10).Eq.(1)maythenbemod-ifiedtoKfslCf2sP21C0fs3；addtoNavierStokesequationthetermKfsu；fsisthesolidfractionwhichisafunctionoftemperature.ThepartofXihi=