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
AbstractqSponsors:Alcan-PechineyCompanyandSwissNationalScienceFoundation;GrantNo.200020-101391.*Correspondingauthor.Tel.:+41223792366;fax:+41223792205.E-mailaddresses:yasser.safaepfl.ch,yasser.safaobs.unige.ch(Y.Safa).Availableonlineatwww.sciencedirect.comAppliedMathematicalModelling33(2009)14791492www.elsevier.com/locate/apm0307-904X/$-seefrontmatterC2112008ElsevierInc.Allrightsreserved.Aphasechangingproblemmotivatedbythemodellingofthermalproblemcoupledwithmagnetohydro-dynamiceectsinareductioncellisstudied.InasmeltingcelloperatingwithHallHe´roultprocess,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,E´colePolytechniqueFe´de´raledeLausanne,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:C15X¼X1X2:fluidsandsolidledge,C15N¼N1N2:electrodes,C15K¼XN:domainrepresentingthecellandwedefinetheinterfaces:C15C¼oX1oX2:freeinterfacebetweenaluminiumandbath,whichisanunknown,C15Ri¼oKoNi;i¼1;2,C15R¼R1R2: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.WechoosetorepresenttheunknowninterfacebetweenaluminiumandbathbyaparametrizationoftheformCðC22hÞ¼½ðx;y;zÞ:z¼C22hðx;yÞ;ðx;yÞ2DC138,whereDisusuallyarectanglecorrespondingtotheparametrizationofaluminiumcathodeinterface.WedenotethedependenceofX1;X2andCwithrespecttoC22hbyusingTheunknownphysicalfieldswithwhichweshalldealarelistedasfollows:Hydrodynamicfields:C15u:velocityfieldinXi;i¼1;2;(u¼0insolidledges),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,Xi¼XiðC22hÞ;i¼1;2;C¼CðC22hÞ:hðx;yÞdxdy¼V1;whereV1isthevolumeofaluminium:1C22C22Here3thosethefluids.fieldsThefluidC22C22Inorderinvolvingapenalizationtool.Thevelocityandthepressurewillthenbedefinedinbothliquidsandsolids.WefunctionKisgivenbyCarmanKozeny”law:theDarcy1482Y.Safaetal./AppliedMathematicalModelling33(2009)14791492Whenfs!1,wegetKðfsÞ!1andthenu¼0inthesolidzone.law:rðpþqgzÞ¼C0Kuþjb:IfonlyliquidphaseispresentwehaveK¼0andtheaboveequationreducestotheusualNavierStokesequa-tion.InsidethemushyzoneKmaybeverylarge,comparedtotheotherterms,andtheaboveequationmimicsqðu;rÞuC0divð2lDðuÞC0ðpþqgzÞIÞþKu¼jbinX1ðC22hÞX2ðC22hÞ:ð7ÞwherePisthemeanporesizeandCisaconstantobtainedexperimentally(see10).Eq.(1)maythenbemod-ifiedtoKðfsÞ¼lCf2sP2ð1C0fsÞ3;addtoNavierStokesequationthetermKðfsÞu;fsisthesolidfractionwhichisafunctionoftemperature.ThepartofXiðhÞi=1,2isonlyasubdomainofthedomainXiðhÞdelimitedbythefrontofsolidification.tosolvethehydrodynamicprobleminafixeddomainXi,weusethemethodoffictitiousdomain”u¼0onoX;ð4Þ½uC138CðC22hÞ¼0;ð5Þ½ðC0pIþ2lDðuÞÞnC138CðC22hÞ¼0:ð6Þ(.,.)istheusualscalarproductonR.Eqs.(1)(3)correspondto1stand2ndassumptions.WecompleteequationsbyintroducingtheconditionsontheboundariesofthedomainsX1ðC22hÞandX2ðC22hÞcontainingForanyfieldw,½wC138CðC22hÞdenotesthejumpofwacrossCðC22hÞ,i.e.½wC138CðC22hÞ¼wbathC0waluminium.FortheuandpwehavewithDðuÞ¼12ðruþðruÞTÞ;I¼ðdijÞi;j¼1;2;3:qðu;rÞuC0divð2lDðuÞC0ðpþqgzÞIÞ¼jbinX1ðhÞX2ðhÞ;ð1Þdivu¼0inX1ðC22hÞX2ðC22hÞ;ð2Þðu;rðzC0C22hÞÞ¼0onCðC22hÞ;ð3ÞWeconsiderthefollowingstandardsetofequationsforhydrodynamicfields:n¼krðzC0C22hÞkrðzC0C22hÞ:DTheunitnormaltoCðC22hÞpointingintoX2ðC22hÞisgivenbyZC22Fromassumption(vii)wegetthefollowingrelation: