Development of a commercial code-based two-fluid model for bubble plumes - 图文

EnvironmentalModelling&Software22(2007)536e547

www.elsevier.com/locate/envsoft

Developmentofacommercialcode-basedtwo-?uidmodelfor

bubbleplumes

HectorR.Bravoa,*,JohnS.Gulliverb,MikiHondzobaDepartmentofCivilEngineeringandMechanics,UniversityofWisconsin-Milwaukee,Milwaukee,WI53201-0784,USA

bDepartmentofCivilEngineering,UniversityofMinnesota,St.AnthonyFallsLaboratory,MN55414,USA

Received18October2004;receivedinrevisedform6September2005;accepted12February2006

Availableonline4May2006

Abstract

Bubbleplumemodelsareappliedtostudythede-strati?cationoflakewater,aerationofreservoirs,wastewatertreatment,andgasinjectionintoliquidmetals.Severalexistingmodelsexemplifynumericalmodelingasproblem-speci?cart,solveamixturemomentumequation,andhavelimitationsintheavailabilityofdocumentation,de?nitionofboundaryconditions,andpost-andpre-processingcapabilities.Thetransferofproblem-speci?cmodelstoaclientortoamultidisciplinaryresearchanddevelopmentteamisadif?cultprocess.Thequestionsaddressedinthisstudywereasfollows:(a)canoneuseacommercialcodeasabasistodevelopauser-friendly,ef?cientmodelthatsimulatestwo-phase?owinbubbleplumes?(b)whatarethecapabilitiesandlimitationsofsuchamodel?

Thetwo-?uidmodeldevelopedhas?exibilityinthede?nitionofthemultiphaseandviscousmodels,easilyunderstoodde?nitionofboundaryconditions,simplede?nitionofspatialdimensionalityandtimedependency,ef?cientnumericalsolution,cleardocumentationanduser-friendlypre-andpost-processingcapabilities.Waterandairphasevelocities,waterturbulentkineticenergy,andairvolumefractionarepredictedwithaccuracysimilartothatofexistingproblem-speci?cmodels.Astrategytoovercomesomeunder-dispersionandincludeairewatermasstransfereffectsthroughuser-de?nedfunctionsisdiscussed.ó2006ElsevierLtd.Allrightsreserved.

Keywords:Two-phase?ow;Bubbleplumes;Two-?uidmodel

1.Introduction

Bubbleplumemodelsareappliedtostudythede-strati?cationoflakewater,aerationofreservoirs,wastewatertreatment,andgasinjectionintoliquidmetals.Thedevelopmentofabubbleplumemodelfortheaerationofdeepreservoirsinatypicalcasecanbestudiedinitself.Thedesigneroftheaerationsystemwantstousethemodeltoreproducethegeometryofbubblediffusersandtesttheeffectivenessofdifferentaerationdevices.Interpretationofmodelresultsrequiresknowledgeofthemodelhypotheses,limitationsandstructureonpartoftheuser.Thespecialistinairewatermasstransfermaywant

*Correspondingauthor.

E-mailaddress:hrbravo@uwm.edu(H.R.Bravo).

1364-8152/$-seefrontmatteró2006ElsevierLtd.Allrightsreserved.doi:10.1016/j.envsoft.2006.02.009

amodelthatallowshim/hertoeasilyexploredifferentformu-lationsforgastransfercoef?cientsandtheloadimposedbyoxygendemand,andtheirrelationwith?owturbulence.Themodelbuyerwantsauser-friendlymodelthatcanef?cientlymodelawholedeepreservoir.Everyinvolvedpartywantsamodelthatproducesquantitative,reliableresultsonwhichdecisionscanbebased.

Severalexistingmodelsaredevelopedspeci?callyforaproblem,aredif?culttousebythedesignerorthespecialistinairewatermasstransfer,andhavelimitationsintheavail-abilityofdocumentationandpost-andpre-processingcapabil-ities.Thede?nitionofboundaryconditionsinproblem-speci?ccodescanleadtoinconsistenciesordif?cultiesintheexplora-tionofairewatermasstransfer.Thistypicaldisparitybetweenexpectationsandexistingsoftwareoccurredintheresearchanddevelopmenteffortsaimedtodesigntheaerationsystem

H.R.Bravoetal./EnvironmentalModelling&Software22(2007)536e547537

fortheMcCookReservoir.Thisscenarioleadstothemainques-tionsaddressedinthisstudyasfollows:(a)canoneuseacom-mercialcodeasabasistodevelopanef?cientuser-friendlymodelthatsimulatestwo-phase?owinbubbleplumes?(b)whatarethecapabilitiesandlimitationsofsuchamodel?

Letusbrie?ydiscussbubbly?owmodelsingeneral,aera-tionofadeepreservoirinparticular,andacoupleofmodelsdevelopedspeci?callyforthisapplication.

Modelsforstudyingbubbly?owshavebeengenerallydividedintoEulereLagrange,orEeL(ShengandIrons,1995),andEulereEulerorEeEmodels(Bernardetal.,2000;Buscagliaetal.,2002;MuddeandSimonin,1999;SokolichinandEigenberger,1995;Smith,1998).

IntheEeLformulation,theliquidphaseistreatedasacon-tinuum,whilethedispersedgasphaseissolvedbytrackingbubbles.Theliquid?ow?eldiscalculatedwithaninitiales-timateforthevoidfraction.Alargenumberofbubbletrajec-toriesarethencalculated,sothatanewvoidfractioncanbecalculated.Thisnewvoidfractionisthenusedtorecalculatetheliquid?ow?eld,anditerationcontinuesuntilthevoidfrac-tionandliquid?ow?eldconverge(ShengandIrons,1995).IntheEeErepresentation,thetwophasesaretreatedasin-terpenetrating?uids.Theconceptofphasicvolumefractionisintroduced,sincethevolumeofaphasecannotbeoccupiedbyanotherphase.TheEeEmodelfamilyincludesthemixturemodel(Wuestetal.,1992;Manninenetal.,1996)andthetwo-?uidmodel.Themixturemodelsolvesthemixturemo-mentumequationandprescribesrelativevelocitiestodescribethedispersedphase.Inthetwo-?uidmodel,thetwophasesarerepresentedbytwosetsofmassandmomentumbalances,whichhavesimilarstructuresforallphases(MuddeandSimo-nin,1999).Theseequationsaresolvedbyprovidingcertainconstitutiverelationsthatareobtainedfromempiricalinformation.

Thedesignofdeepreservoirsforcombined(stormwaterandwastewater)?owsleadstovariouswaterqualitychal-lenges,andaerationisonepossiblesolution.Inmostcasesthedepthofthereservoirsisbeyondtheprovenrangeofval-idityofexistingdesignmethodsforaerationsystems.Oneno-tablecaseisthealmost80-mdeepMcCookReservoirbeingplannedbytheUSArmyCorpsofEngineers(USACE)forChicago,Illinois.Thedesignofaerationsystemsfordeepres-ervoirsrequiresapracticalcapabilityfornumericallysimulat-ingtheeffectofsubmergeddiffusersonreservoir?owandaeration(Bernard,1998).

ThedesignoftheMcCookReservoirissupportedbyacombinationofexperimentsandcomputationalmodeling.Large-scaleexperimentsonaeration-induced?ows(SogaandRehmann,2004),waterquality,andairewatermasstransferareunderway.ComputationaleffortsincludethosebyBernard(1995,1998,2002),Bernardetal.(2000),andBuscagliaetal.(2002).ThePAR3DmixturemodeldevelopedbyBernard(1995,1998)andBernardetal.(2000)iscapableofreproduc-ingobservedvelocitieswithinafactoroftwo.Thecurrentver-sionofthePAR3Dcodeincludesthesimulationofairewatermasstransfer(Bernard,2002).Theauthors’reviewofthePAR3Dmodel(Bernard,2002)revealedthelimitationsin

termsofdocumentationavailability,de?nitionofboundaryconditions,post-andpre-processingcapabilities,andmodelef?ciency.Buscagliaetal.(2002)developedamixturemodelcoupledtoamasstransfermodel,andappliedittotypicalcon-ditionsofisolatedaerationplumesindeepwastewaterreser-voirs.Buscagliaetal.’s(2002)modelpredictionscomparedwellwiththoseofthe1DmodelproposedbyWuestetal.(1992),butcombinesamixturemodelwithboundarycondi-tionsde?nedfortheindividualphases.

Thepurposeofthepresentstudywastobuildauser-friendlymodelthatoffers?exibilityinthede?nitionofthemultiphaseandviscousmodels,easilyunderstoodde?nitionofboundaryconditions,andisbasedonacommercial,well-documented,ef?cientcodetofacilitatecommunicationamongstamultidisciplinaryresearchanddevelopmentteam.TheFluent(2001)codewasadoptedandasix-equation,two-?uid,dispersedturbulencemodelwasconstructed.Themodelfacilitatesuserinteractionregardingmodelgeometryandgridconstruction,de?nitionofspatialdimensionality,def-initionofimplicitorexplicitandsteadyorunsteadyformula-tion,typeofmultiphaseandviscousmodels,interactionbetweenphases,andnumericalsolveroptions.Thereis,how-ever,nosubstituteforthedepthofphysicalinsightrequiredtomodelbubbleplumes.

Themodeldevelopedinthisstudypredictsmean?owandturbulentquantitieswithaccuracysimilartothatofexistingcodes,andshowssomeunder-dispersionoftheairphase.Theimprovementinmodelaccuracyandtheairewatermasstransferacrossthesurfaceareaofthebubblesorthefreesur-facewillbethesubjectofafuturestudy.

Section2ofthispaperdiscussesthemodelselectionandpresentsthetwo-?uidmodel.Section3includesacomparisonofmodelpredictionswithexistingdatarelevanttothedesignofdeepwastewaterreservoirs,suchasdetailedmeasurementstakeninsmalltanks(ShengandIrons,1992,1993,1995;AnagboandBrimacombe,1990),measurementstakenina1.5-mdeepbubblecolumnthatexhibitedoscillatorybehavior(Beckeretal.,1995),andmeasurementstakeninalargetank(JohnsonandDuncker,2001).Section4containsthemainconclusionsofthisstudy.2.Overviewofthemodel2.1.Modelselection

Therequirementtosimulatefreesurface?owsandthecon-siderationofthetypeofboundaryconditionsavailableinFlu-entmodelledtotheselectionofthetwo-?uidmultiphasemodeloverthemixturemultiphasemodel.Differentauthorshaveuseddifferentcombinationsofmathematicalboundaryconditionstorepresenttheliquidfreesurface.Bernardetal.(2000)representeditasafrictionlesslid(slipsurface),wheretheyimposednullvaluesoftheverticalcomponentofthevelocityandthevertical(z)derivativesofothervariables(uz?vz?Pz?kz?3z?0).P?egerandBecker(2001)simu-latedthedynamicbehaviorinabubblecolumn,modelingthefreesurfacebyasymmetryconditionwithadegassingsink.

538H.R.Bravoetal./EnvironmentalModelling&Software22(2007)536e547

Theygavenodetailsaboutthedistributionofsinkstrengthacrossthefreesurface.Anunspeci?edsinkseemsanimprecisede?nitionfortheboundaryconditionforgasvolumefraction.Buscagliaetal.(2002)usedasymmetryconditionatthetopboundaryfortheliquidphase,whileforthegasphasethetopsurfaceisanout?owboundary.Thatcombinationofgoverningequationsandboundaryconditionsseemscomplicated.

Fluentde?nesauniquesetofmathematicalboundarycon-ditionsateachphysical(orarti?cial)boundary.InaFluentmodel,asymmetryboundaryconditionentailszeronormalve-locityandzeronormalgradientofallothervariablesatthesymmetryplane.Thissymmetryboundaryconditiondoesnotappropriatelyrepresentthefreesurface,becauseitassumeszero(convectiveanddiffusive)?uxofallquantities,includingairvolumefraction,acrossasymmetryboundary.Aconsistentmodelwasthereforeconstructedbymodelingthe?ow?eldintheaeratedtankandanairlayerlocatedabovetheliquidsurface.Thetopoftheairlayerisde?nedasa(con-stant)pressureoutletboundary.Smith(1998)usedasimilarap-proachtomodelthefreesurface.Theairlayerisdeepenoughtosimulatethedeformationofthefreesurfaceandminimizetheeffectsofthepressureoutletboundaryconditiononthe?ownearandbelowthefreesurface.Theairlayerdepth/watercolumndepthratiosusedinthemodelswere0.3,0.5,0.33,and0.17,respectively,inthefourtestcases.Theactualdepthoftheairlayerwas1.52minthefourthcase.Smith(1998)modeledthethirdtestcasepresentedhereinandusedanairlayerdepth/watercolumndepthratioof0.33.

Intheadoptedmodel,airisthedispersedphaseinthewatertankandbecomesthecontinuousphaseinthetopairlayer.Amixturemultiphasemodelthatincludedatopairlayerprovedtobeunfeasiblebecauseofthesharpdiscontinuityinvolumefractionatthedeformingfreesurface.Themixturemodelpre-dictedanunphysicalriseinthefreesurfacewhenthebubbleplumereachedtheinterface.2.2.Two-?uidEulerianmodel

Thederivationoftheconservationequationscanbedonebyensembleaveragingthelocalinstantaneousbalanceforeachofthephasesorbyusingthemixturetheoryapproach.Masstransferbetweenairandwaterisnotconsideredandtheconservationofmassequationforphaseq(water:q?1,air:q?2)is

v?

vt

aqrqátV$?aqrquqá?0e1T

whereaq,rqanduqarethevolumefraction,densityandveloc-ityofphaseq,respectively.

Themomentum-conservationequationforphaseqisv?á?ávt

aqrquqtV$aqrququq??aqVptV$tqtaqrqg?R12e2Twherepispressuresharedbybothphases,tqistheqthphase(laminarandturbulent)stressestraintensor,gisthe

accelerationduetogravity,R12istheinteractionforcebe-tweenphases,whichcomprisedrag,lift,andvirtualmass.Thevirtualmasseffectoccurswhenasecondaryphaseaccel-eratesrelativetotheprimaryphase.Theinertiaoftheprimaryphasemassencounteredbytheacceleratingparticles(orbub-bles)exertsavirtualmassforceontheparticles(DrewandLahey,1993).Fluent(2001)hasthecapabilitytoincludeadriftvelocitycorrectiontotheinterphaseturbulentmomentumtransfer.Thatcorrectionisnotdiscussedfurtherhereinbecausepreliminaryexplorationofthatcapabilityresultedinnoimprovementinthedispersionofair.Thesignoftheinter-facialforceR12ispositivefortheairphaseandnegativeforthewaterphase.Thedragforcedependsontheairvolumefraction,bubblediameter,andrelativeReynoldsnumber.Thelatteriscomputedwiththewaterdensity,relativeorslipvelocity,u2?u1,bubblediameter,anddynamicviscosityofwater(Fluent,2001).Informationonthebubblediameterwasavailableforthefourtestcases,asdiscussedbelow.HabermanandMorton(1954)foundthattheslipvelocityincreaseswithbubblediameter,butisfairlyconstantforbub-blediameterbetween1.4mmand10.2mm.Weexpectthattherelationbetweenbubblediameterandthecomputedslipvelocitywillfollowthattrend.Onlydragforcewasconsideredin2Dmodels,whiledragandvirtualmasswereconsideredinthe3Dmodelinthethirdtestcase,inordertoobtainstablemodelsolutions.Thedefaultoption(SchillerandNaumannmodel)wasusedfordragforce,andinthe3Dmodelaliftco-ef?cientof0.1wasusedalongwiththedefaultvalueforthecoef?cientofvirtualmass(0.5).

Thismodelsimulatesthetransportoftheairphasebyad-vection,itsdispersion,andtheinteractionforcebetweenphases,butdoesnotyetincludeairewatermasstransferacrossinterfaces.Wuestetal.(1992)showedthatbubbledis-solutioncanbeignoredforradiilargerthanthatabout2cmbycalculatingthedepthofmaximumplumeriseasafunctionofinitialbubblesizewithandwithoutgasexchange.Theyshowed‘‘theimportanceofgasexchangeforbubbleplumemodels,iftheinitialbubbleradiusislessthanabout5mm,indeepwater.’’MasstransfereffectscanbeignoredwhenmodelingShengandIrons’(1992,1993)tests(?rsttestcaseinTable1),andwouldbedif?culttoestimateinJohnsonandDuncker’s(2001)test(fourthtestcaseinTable1)becauseofuncertaintyinthemeasurements.Thewatercolumndepthsinthefourtestcasesusedforcomparisonare0.42m,0.40m,1.5m,and9.14m,respectively(seeTable1).Thewaterdepthwasnotdeepinthesecondandthirdtestcases(experimentsperformedbyAnagboandBrimacombe,1990andBeckeretal.,1995,respectively).

Theinclusionofmasstransferinthemodelmaychangethe?owpredictionsandtheconsequentcomparisoninthesecondandthirdtestcasespresentedherein.Thenumerousliteratureprecedentsofsuccessfulmodelsthatdonotincludemasstransfer,andthecharacteristicsofthetestcasespresentedhereleadustobelievethatthosechangeswouldbesecondaryimprovementstothe?owscalculatedherethroughthesimula-tionofthetransportoftheairphasebyadvection,itsdis-persion,andtheinteractionforcebetweenphases.The

H.R.Bravoetal./EnvironmentalModelling&Software22(2007)536e547

539

Table1

Summaryofmaingeometricand?owparametersinthefourtestcasesTestcase

WatercolumnAir?owrate

Airinjectordimensionsdimensions

ShengandIrons

0.42mdeep,1.5?10?4m3/s0.024m

(1992,1993,1995)0.50mdiameterdiameterori?ce

Anagboand

0.40mdeep,2?10?4m3/s0.06mdiameter

Brimacombe(1990)0.50mdiameterporousplugBeckeretal.(1995)

1.5mdeep,

2.667?10?50.04msquare0.50mby0.08mm3/sori?cecross-sectionJohnsonand

9.14mdeep,1.41?10?20.61mdiameterDuncker(2001)7.62mdiameter

standardm3/s

diffuser

exclusionofmasstransfer,however,preventsquantifyingthesigni?canceofsuchpossiblechanges.

Thisstudyacknowledgesthatthemasstransferisimpor-tant,andisoneofthemainreasonsthatbubbleplumesarein-stalledinlakesandreservoirs.However,untilthemeanandturbulent?owparametersareknowninabubbleplume,thereisnoreasontoevenattemptadetailedmodelofmasstransferinabubbleplume.Wemightaswellstickwiththeempiricalmodelswithcurve?tcoef?cientsthatareparticulartoagivenapplication,becausethereisnochanceofdoingabetterjobofpredictingmasstransfer.Theturbulent?owbubbleplumemodelpresentedhereinisaprecursortoinstallingadetailedmasstransfermodel,andutilizingfundamentalmasstransfercharacterizationstopredictconcentrationsthatresultfromabubbleplume.

Fluent(2001)providesthreemethodsformodelingturbu-lenceinmultiphase?owswithinthecontextoftheke3models:mixtureturbulencemodel,dispersedturbulencemodel(usedinthisstudy),andturbulencemodelforeachphase.LiuandDucoste(2005)combinedaCFDmodelwiththeRNGke3modelturbulencemodel,withamulti-?uidmi-croscalemodeltoinvestigatetheimpactofmicromixingonthefreechlorineresidual.Themixtureturbulencemodelisap-plicablewhenphasesseparate,forstrati?ed(ornearlystrati-?ed)multiphase?ows,andwhenthedensityratiobetweenphasesiscloseto1.Inthesecases,usingmixturepropertiesandmixturevelocitiesissuf?cienttocaptureimportantfea-turesoftheturbulent?ow.Themixturemodelisnotappropri-atetomodelbubbleplumes.Themostgeneralmultiphaseturbulencemodelsolvesasetofkand3transportequationsforeachphase.Thisturbulencemodelistheappropriatechoicewhentheturbulencetransferamongthephasesplaysadominantrole.Theturbulencemodelforeachphaseisnotpresentedherebecausethesimulationsaremorecomputation-allyintensiveandtheresultsarenotsigni?cantlydifferentfromthoseusingthedispersedphaseturbulencemodel.

Thedispersedturbulencemodelistheappropriatemodelwhenthesecondaryphasesaredilute.Inthiscase,inter-parti-clecollisionsarenegligibleandthedominantin?uenceintherandommotionofthesecondaryphasesistheprimary-phaseturbulence.Fluctuatingquantitiesofthesecondaryphasescanthereforebegivenintermsofthemeancharacteristicsoftheprimaryphaseandtheratiooftheparticlerelaxation

timeandeddy-particleinteractiontime.Themodelisapplica-blewhenthereisclearlyoneprimarycontinuousphaseandtherestaredisperseddilutesecondaryphases.Thedispersedtur-bulencemodelissummarizedbelow.

Theeddyviscositymodelisusedtocalculateaverage?uc-tuatingquantities.TheReynoldsstresstensorforcontinuousphaseqtakesthefollowingform:

t00q??2?3rq

kqtaqmt;qV$uqáItaqmt;q?VuqtVuT

q

áe3T

ThesuperscriptTinthelasttermdenotesthetransposeofthevectoruq.Theturbulentviscositymt,qiswrittenintermsofthekineticenergyofphaseq:

2mkq

t;q?rqCm

3e4T

q

where3qisthedissipationrateandCmisaconstant.

Theturbulentkineticenergyanddissipationrateforthecontinuousphaseqareobtainedfromthemodi?edke3modelv?vtaqrqkqátV$?aqrqUqkqá?V$??

amt;q??qsVkqtaqGk;qk

?aqrq3qtaqrqPkq

e5T

and

v?vta3qátV$?aqrqUq3qá?V$??amqt;q??

3qqrq?

sV3qtaqC13Gk;q3k?C23rq3qá

q

taqrqP3qe6THerePkonqandP3thecontinuousqrepresentthein?uenceofthedispersed

phasesphaseq,andGk,qistheproductionrateforturbulentkineticenergy.ThetermPkfromtheinstantaneousequationofthecontinuousqcanbederivedphase.Pkdependsonvolumefractionsanddensities,relativeReynoldsqnumber,bubblediameter,dynamicviscosityofwater,theco-varianceofthevelocitiesofthecontinuousphaseqanddis-persedphasel,therelativeorslipvelocity,andthedrift(ordispersion)velocity(Fluent,2001;MuddeandSimonin,1999).Thedriftvelocityaccountsforthedispersionofthebubblesduetotransportbyturbulent?uidmotionandiscal-culatedbasedonthegradientsofthevolumefractions,diffu-sivitiesandtheturbulentSchmidtnumberspq.ThetermP3ismodeledintermsofP,3qkq,kqq,andtheconstantC33(ElghobashiandAbouArab,1983).Timeandlengthscalesthatcharacterizethemotionareusedtoevaluatedispersioncoef?cients,correlationfunctions,andtheturbulentkineticenergyofthedispersedphase(Fluent,2001).

Theconstantsintheturbulencemodelareasfollows:Cm?0.09,C13?1.44,C23?1.92,C33?1.3,sk?1.0,s3?1.3,andspq?0.75.