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外文翻译--数值模拟与影响铝电解槽磁热耦合问题 英文版.pdf

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外文翻译--数值模拟与影响铝电解槽磁热耦合问题 英文版.pdf

AbstractqSponsorsAlcanPechineyCompanyandSwissNationalScienceFoundationGrantNo.200020101391.Correspondingauthor.Tel.41223792366fax41223792205.Emailaddressesyasser.safaepfl.ch,yasser.safaobs.unige.chY.Safa.Availableonlineatwww.sciencedirect.comAppliedMathematicalModelling3320091479–1492www.elsevier.com/locate/apm0307904X/seefrontmatterC2112008ElsevierInc.Allrightsreserved.Aphasechangingproblemmotivatedbythemodellingofthermalproblemcoupledwithmagnetohydrodynamiceffectsinareductioncellisstudied.InasmeltingcelloperatingwithHall–He´roultprocess,themetalpartisproducedbytheelectrolysisofaluminiumoxidedissolvedinabathbasedonmoltencryolite1.VariousphenomenatakeplaceinsuchacellforwhichatransversesectionisschematicallypicturedinFig.1.Runningfromtheanodesthroughliquidaluminiumandcollectorbars,thesteadyelectriccurrentspreadsintheelectrolyticbath.Theimportantmagneticfieldgeneratedbythecurrentscarriedtothealignmentofcells,coupledwiththecurrentsrunningthroughthecellsthemselvesgivesrisetoafieldofLaplaceforceswhichmaintainsamotionwithinthesetwoconductingliquids.Amagnetohydrodynamicinteractiontakesplaceinthecell.IntheotherhandaheatingsourceisproducedbytheJouleeffectduetotheelectricresistivityofthebath.Asystemofpartialdifferentialequationsdescribingthethermalbehaviorofaluminiumcellcoupledwithmagnetohydrodynamiceffectsisnumericallysolved.ThethermalmodelisconsideredasatwophasesStefanproblemwhichconsistsofanonlinearconvection–diffusionheatequationwithJouleeffectasasource.ThemagnetohydrodynamicfieldsaregovernedbyNavier–StokesandbystaticMaxwellequations.ApseudoevolutionaryschemeChernoffisusedtoobtainthestationarysolutiongivingthetemperatureandthefrozenlayerprofileforthesimulationoftheledgesinthecell.Anumericalapproximationusingafiniteelementmethodisformulatedtoobtainthefluidvelocity,electricalpotential,magneticinductionandtemperature.Aniterativealgorithmand3Dnumericalresultsarepresented.C2112008ElsevierInc.Allrightsreserved.KeywordsAluminiumelectrolysisChernoffschemeHeatequationMagnetohydrodynamicsLedgeSolidification1.IntroductionNumericalsimulationofthermalproblemscoupledwithmagnetohydrodynamiceffectsinaluminiumcellqY.Safa,M.Flueck,J.RappazInstituteofAnalysisandScientificComputing,E´colePolytechniqueFe´de´raledeLausanne,Station8,1015Lausanne,SwitzerlandReceived27December2006receivedinrevisedform4February2008accepted8February2008Availableonline29February2008doi10.1016/j.apm.2008.02.011ElectrolyteAnodeBlocksFig.1.Transversecrosssectionofaluminiumreductioncell.1480Y.Safaetal./AppliedMathematicalModelling3320091479–1492Onthewallofthecell,asolidifiedbathlayer,thesocalledledgeiscreated.Theseledgesprotectthecellsidewallfromcorrosiveelectrolyticbathandreducetheheatlossfromthecellsee2page23.Moreover,itsprofilestronglyinfluencesthemagnetohydrodynamicstabilitycausingoscillationsofthealuminium–bathinterfacewhichcoulddecreasethecurrentefficiency.Consequentlyanoptimalledgeprofileisoneoftheobjectivesofcellsidewalldesign.Thethermalsolidificationprobleminsmeltingcellhasbeentreatedbyseveralauthors3–5.Asfarasweareaware,thisproblemhasneverbeenconsideredwhencoupledwiththemagnetohydrodynamicfields.Theaimofthispaperistodealwithsuchfieldsinteraction.LetusmentionthatthedetailsonthisproblemcanbefoundinSafasthesis6.Mathematically,theproblemistosolveacoupledsystemofpartialdifferentialequationsconsistingoftheheatequationwithJouleeffectasasource,MaxwelllawequationswithelectricalconductivityasafunctionoftemperatureandNavier–Stokesequations.Theinterfacebetweenaluminiumandbathisanunknown.Theledgeisconsideredaselectricalinsulator,thethermalmodelisastationarytwophasesStefanproblem.TheoutlineofthispaperisasfollowinSection2weintroducethephysicalmodel,thealgorithmispresentedAluminiumCathodeLiningFrozenledgeFrozenledgeinSection3andwegivethenumericalresultsinSection4.2.ThemodelInordertointroducethemodelwefirstdescribesomegeometricalandphysicalquantities.2.1.GeneraldescriptionsThegeometryisschematicallydefinedbyFig.1.WeintroducethefollowingnotationsC15X¼X1X2fluidsandsolidledge,C15N¼N1N2electrodes,C15K¼XNdomainrepresentingthecellandwedefinetheinterfacesC15C¼oX1\oX2freeinterfacebetweenaluminiumandbath,whichisanunknown,C15Ri¼oK\oNii¼12,C15R¼R1R2outerboundaryoftheelectrodes.Y.Safaetal./AppliedMathematicalModelling3320091479–14921481C15Cpspecificheat,C15latentheat.2.2.PhysicalassumptionsThemodelleansonthefollowingbasichypotheses1.Thefluidsareimmiscible,incompressibleandNewtonian.2.IneachdomainXi,i1,2,thefluidsaregovernedbythestationaryNavier–Stokesequations.3.TheelectromagneticfieldssatisfythestationaryMaxwellsequations,OhmslawismoreoversupposedtobevalidinallthecellK.4.Theelectricalcurrentdensityoutsidethecellisgivencurrentinthecollectorbars.5.Theelectricalconductivityrisfunctionoftemperaturehinthefluidsandelectrodesparts.6.Theviscosityg,thedensityqandthespecificheatCparetemperatureindependent.7.ThevolumesofthedomainsX1andX2havegivenvaluesmassconservation.8.TheonlyheatsourceisproducedbytheJouleeffectduetothecurrentcrossingthecell.9.Effectsofchemicalreactions7,Marangonieffect8,9,surfacetensionaswellasthepresenceofgasflowareneglected.2.3.ThehydrodynamicproblemInthispartweconsiderthetemperaturefieldhandtheelectromagneticfieldsjandbasknown.WechoosetorepresenttheunknowninterfacebetweenaluminiumandbathbyaparametrizationoftheformCðC22hÞ¼½ðxyzÞz¼C22hðxyÞðxyÞ2DC138,whereDisusuallyarectanglecorrespondingtotheparametrizationofaluminium–cathodeinterface.WedenotethedependenceofX1X2andCwithrespecttoC22hbyusingTheunknownphysicalfieldswithwhichweshalldealarelistedasfollowsHydrodynamicfieldsC15uvelocityfieldinXii¼12u¼0insolidledges,C15ppressure.ElectromagneticfieldsC15bmagneticinductionfield,C15eelectricfield,C15jelectriccurrentdensity.ThermalfieldsC15Henthalpy,C15htemperature.ThematerialpropertiesaredefinedasC15qmassdensity,C15rbandrelectricalconductivityinand,respectively,outsidethebath,C15gviscosityofthefluids,C15l0magneticpermeabilityofthevoid,C15kthermalconductivity,Xi¼XiðC22hÞi¼12C¼CðC22hÞhðxyÞdxdy¼V1whereV1isthevolumeofaluminium1C22C22Here3thosethefluids.fieldsThefluidC22C22Inorderinvolvingapenalizationtool.Thevelocityandthepressurewillthenbedefinedinbothliquidsandsolids.WefunctionKisgivenbyCarmanKozenylawtheDarcy1482Y.Safaetal./AppliedMathematicalModelling3320091479–1492Whenfs1,wegetKðfsÞ1andthenu¼0inthesolidzone.lawrðpþqgzÞ¼C0KuþjbIfonlyliquidphaseispresentwehaveK¼0andtheaboveequationreducestotheusualNavier–Stokesequation.InsidethemushyzoneKmaybeverylarge,comparedtotheotherterms,andtheaboveequationmimicsqðurÞuC0divð2lDðuÞC0ðpþqgzÞIÞþKu¼jbinX1ðC22hÞX2ðC22hÞð7ÞwherePisthemeanporesizeandCisaconstantobtainedexperimentallysee10.Eq.1maythenbemodifiedtoKðfsÞ¼lCf2sP2ð1C0fsÞ3addtoNavier–StokesequationthetermKðfsÞufsisthesolidfractionwhichisafunctionoftemperature.ThepartofXiðhÞi1,2isonlyasubdomainofthedomainXiðhÞdelimitedbythefrontofsolidification.tosolvethehydrodynamicprobleminafixeddomainXi,weusethemethodoffictitiousdomainu¼0onoXð4Þ½uC138CðC22hÞ¼0ð5Þ½ðC0pIþ2lDðuÞÞnC138CðC22hÞ¼0ð6Þ.,.istheusualscalarproductonR.Eqs.1–3correspondto1stand2ndassumptions.WecompleteequationsbyintroducingtheconditionsontheboundariesofthedomainsX1ðC22hÞandX2ðC22hÞcontainingForanyfieldw,½wC138CðC22hÞdenotesthejumpofwacrossCðC22hÞ,i.e.½wC138CðC22hÞ¼wbathC0waluminium.FortheuandpwehavewithDðuÞ¼12ðruþðruÞTÞI¼ðdijÞij¼123qðurÞuC0divð2lDðuÞC0ðpþqgzÞIÞ¼jbinX1ðhÞX2ðhÞð1Þdivu¼0inX1ðC22hÞX2ðC22hÞð2ÞðurðzC0C22hÞÞ¼0onCðC22hÞð3ÞWeconsiderthefollowingstandardsetofequationsforhydrodynamicfieldsn¼krðzC0C22hÞkrðzC0C22hÞDTheunitnormaltoCðC22hÞpointingintoX2ðC22hÞisgivenbyZC22Fromassumptionviiwegetthefollowingrelation

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