Large Eddy Simulation of the Gas-Liquid Flow in a

LargeEddySimulationoftheGas-LiquidFlowinaSquareCross-sectionedBubbleColumn1N.G.Deen2,T.SolbergandB.H.HjertagerChem.Eng.Lab.,AalborgUniv.Esbjerg;NielsBohrsVej8;DK-6700Esbjerg;DenmarkAbstractInthisworktheuseoflargeeddysimulations(LES)innumericalsimulationsofthegas-liquidflowinbubblecolumnsisstudied.TheEuler-Eulerapproachisusedtodescribetheequationsofmotionofthetwo-phaseflow.Itisfoundthat,whenthedrag,liftandvirtualmassforcesareused,thetransientbehaviourthatwasobservedinexperimentscanbecaptured.Goodquantitativeagreementwithexperimentaldataisobtainedbothforthemeanvelocitiesandthefluctuatingvelocities.TheLESshowsbetteragreementwiththeexperimentaldatathansimulationsusingthek-epsilonmodel.KeywordsLargeeddysimulation,Euler-Eulerapproach,bubblecolumn,gas-liquidflow,bubblyflow,computationalfluiddynamics1.IntroductionBubblecolumnsarewidelyusedinthechemicalandbiochemicalprocessindustry.Inordertodevelopdesigntoolsforengineeringpurposes,alargeamountofresearchhasbeencarriedoutintheareaofComputationalFluidDynamics(CFD)modellingofgas-liquidflows.Twoapproachesaremostlyusedtosimulatetheflowinbubblecolumns:theEuler-LagrangeandtheEuler-Eulerapproach.IntheEuler-Lagrangeapproach,thecontinuousliquidphaseisdescribedasacontinuuminanEulerianframework.Thedispersedgasphaseontheotherhandis1Tobepresentedat5thInt.Conf.onGas-LiquidandGas-Liquid-SolidReactorEnging,Melbourne,Australia,23-27Sept.20012Correspondingauthor;Fax[phone];Email:[e-mail]y;thatis,theindividualbubblesinthesystemaretrackedbysolvingNewton'ssecondlawfortheforcesactingonthebubbles.ThisapproachwasamongstothersusedbyDelnoij,Lammers,Kuipers&vanSwaaij(1997)andSokolichin,Eigenberger,Lapin,&Lübbert(1997).Anadvantageofthisapproachisthatbubble-bubbleinteractionsareeasilyaccountedfor.Adisadvantageisthatthelargerthesystemgets,themoreequationsneedtobesolved,i.e.oneforeverybubble.Anotherdisadvantageisthatthesizeofthenumericalgrid-cellsislimitedbythesizeofthebubbles,inordertodeterminetheinterfaceforcesactingonthebubblecorrectly.Thismeansthat,unlessthebubblesareverysmall,onecannotuselargeeddysimulations(LES)tomodeltheturbulenceoftheliquidphase.IntheEuler-Eulerapproachbubblesarenottrackedindividually,butthedynamicsofthedispersedphaseareensembleaveraged,toobtainasetofEulerianequations,similartotheequationsforthecontinuousphase.Sokolichinetal.(1997)showedthatprovidedthattheequationsforthegasphasearediscretizedinanadequateway,theEuler-EulerapproachgivessimilarresultsastheEuler-Lagrangeapproach.TheadvantageofthisapproachisthatthecomputationaldemandsaremuchlowercomparedtotheEuler-Lagrangeapproach,forsystemswithhigherdispersedvoidfractions.Forfurtherinformationofbothapproaches,thereaderisreferredtothedetailedreviewsofKuipers&vanSwaaij(1997),Jakobsen,Sannaes,Grevskott&Svendsen(1997)andHjertager(1998).Inthiswork,theEulerianapproachwillbeusedforthedescriptionofbothphases.Theturbulenceofthecontinuousphasecanbeincorporatedthroughdifferentmodels,likethek-εmodelorlargeeddysimulations(LES).Thelattermodelhasbeenidentifiedasagoodwaytomodeltheturbulenceby,amongstothers,Jakobsenetal.(1997).Thismodelhashoweversofarhardlybeenusedforthemodellingofgas-liquidflows.ToourknowledgeonlySmith&Milelli(1998)haveusedacombinationoftheEuler-EulerapproachandaLESmodelfortheturbulenceoftheliquidphase.Inthiswork,wewilltakeasimilarapproach.Euler-Eulersimulationsofthegas-liquidflowinabubblecolumnusingaLESturbulencemodelwillbecomparedwithsimulationincludingak-εmodelbyDeen,Solberg&Hjertager(2000b)andexperimentaldatafromDeen,Hjertager&Solberg(2000a).2.TheoryTheequationsofmotionforphasekinanEulerian-Euleriansimulationaregenerallygivenasfollows:0)()(=⋅∇+∂∂kkkkktuραρα(1)kIkkkkkkkkkkkkPt,)()()(Mgτuuu++∇−⋅−∇=⋅∇+∂∂ραααραρα(2)ThetermsontherighthandsideofEq.(2)arerespectivelyrepresentingthestress,thepressuregradient,gravityandtheensembleaveragedmomentumexchangebetweenthephases,duetointerfaceforces.Thestresstermandtheinterfacetermwillbediscussedlateron.ThevelocitiesinEqs.(1)and(2)aredefinedasfollows:kkk'~uuu−=(3)Here,ukisthepartofthevelocityforphasekthatwillberesolvedinthenumericalsimulations,ku~istheinstantaneousvelocityandu'kistheunresolvedpartofthenumericalsimulations.Theinterpretationofthetermsukandu'kdependsonthemethodofderivation.MostauthorsderiveEqs.(1)and(2)throughensembleaveraging.Thenukandu'krepresentthemeanvelocityandthefluctuatingvelocity.WhenEqs.(1)and(2)areobtainedthroughafilteringoperation,thesetermsarerespectivelythegridscale(GS)andthesub-gridscale(SGS)velocities.Fromnowon,whenspeakingaboutvelocities,werefertouk,unlessmentionedotherwise.Wheneitherensembleaveragingorfilteringisused,unclosedpartsoccurinthestresstermandtheinterfaceforces.Inthiswork,theunclosedpartoftheinterfaceforceswillbeneglected.Thestresstermofphasekisdescribedasfollows:())()(32,kTkkkeffkuIuuτ⋅∇−∇+∇−=µ(4)whereµeff,kistheeffectiveviscosity.Theeffectiveviscosityoftheliquidphaseiscomposedofthreecontributions;themolecularviscosity,theturbulentviscosityandanextratermduetobubbleinducedturbulence:lBIlTlLleff,,,,µµµµ++=(5)Thecalculati