Modelling and simulations of a Monolith Reactor for three-phase hydrogenation reactions – Rules and recommendations for mass transfer analysis

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Duran-Martinez, Freddy and Julcour-Lebigue,

Carine and Billet, Anne-Marie and Larachi, Faïçal Modelling and

simulations of a Monolith Reactor for three-phase hydrogenation

reactions – Rules and recommendations for mass transfer analysis.

(2016) Catalysis Today, vol. 273. pp. 121-130. ISSN 0920-5861

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Modelling

and

simulations

of

a

monolith

reactor

for

three-phase

hydrogenation

reactions

—

Rules

and

recommendations

for

mass

transfer

analysis

Freddy

L.

Durán

Martínez

a

_,

_Carine

_Julcour

a

_,

_Anne-Marie

_Billet

a,∗

_,

_Faïc¸

_al

_Larachi

b

a_{LaboratoiredeGénieChimique,UniversitédeToulouse,CNRS,INPT,UPS,4AlléeEmileMonso,CS84234,F-31432Toulouse,France} b_{LavalUniversity,ChemicalEngineeringDepartment,2325Ruedel’Université,Québec,QCG1V0A6,Canada}

Keywords: Monoliths Catalysis Taylorflow Masstransfer Simulation

a

b

s

t

r

a

c

t

Astrategyforthescale-upofamonolithreactordedicatedtogas-liquidcatalyticreactionsisworked out;focusismadeonthecrucialstepofgas-liquidmasstransfermodellingviaasteady-statenumerical studybasedonasinglechannelandsingleunitcellrepresentation,usingaframemovingwiththe bubbleandsolvingtheliquidphaseonly.Therelevanceofthissimplifiedapproachisassessedthrough aspecificcase(givenbubbleshape,channeldiameterandfluidflowrates),andhydrodynamicsaswell asmasstransferresultsaresuccessfullycomparedtopreviouslypublishednumerical,semi-analytical andexperimentalworks.Influenceofunitcelllengthandofcatalyticsurfacereactionrateisthoroughly investigated.Inferredoverallmasstransfercoefficientsarefoundtoincreasewithbubblefrequency, duetohigherinterfacialareainunitcellandintensifiedrecirculationinslug.Filmcontributiontomass transferisusuallyfounddominantinthecaseofshortbubbleswithreactivewall,andhardlyvarieswith reactionrate.However,thiscontributionisstronglylinkedtobubblefrequency,andareliableevaluation oflocalmasstransferbycorrelationsdemandsaccurateknowledgeontheprecisedimensionsofbubble, slugandfilmentities.

1. Introduction

Chemicals and fuels are produced through catalytic gas-liquid-solid reactions in a wide range of industries, including petrochemicals,finechemicals,pharmaceuticalsandbiochemicals. Conventionaltechnologiestohostsuchgas-liquid-solidreactions are fixed-bed,slurrybubble columnand fluidized-bed reactors. Slurrybubblecolumnreactors,andfluidizedbedstoalesserextent, combinethreemajoradvantages:thepossibilityforcontinuous cat-alystreplacement,amuchreducedintra-particulardiffusionpath length (dueto thesmall size of catalyst particles),and a good heat transferefficiency.However,theysuffer fromsome draw-backssuchasliquidback-mixing,significantattritionofthecatalyst andneedforcatalystseparationandrecycling.Ontheotherhand, despite fixed-bed reactors can beoperated closer to plug flow withnegligible attrition of catalyst,restrictions quicklyemerge regardingsoaringpressuredropswhichultimatelyforceauseof large(millimetric)particleswhichunavoidablyimplysignificant

∗ CorrespondingAuthor.

E-mailaddress:annemarie.billet@ensiacet.fr(A.-M.Billet).

internaldiffusionallimitations.Inaddition,thetraditional cocur-rentdownwardconfiguration(trickle-bedreactors)cangiverise tomaldistributionoftheliquidresultingincatalystpartialwetting atlowflowrateswhichrendersuchreactorspronetohotspots inception,thus catalystdeactivationandeven thermalrunaway [1].Anotherdifficultyencounteredwithconventionalmultiphase reactorsistheirscale-uptoindustrialsizeunits.Althoughthese conventionalreactorsstillplayamajorroleinindustrialprocesses, researchersstrivelookingforadvantageousalternative technolo-gies.

Structuredreactors have beenclaimed over thepastseveral years to offer interesting possibilities; among them, Monolith Reactors(MRs),alsocalled‘honeycombreactors’,havebeen con-siderably studied for almost four decades as they represent a promisingcutting-edgetechnologytocircumventtheabove men-tionedproblemsenumeratedinthecaseofconventionalreactors. MRs were initially developed as catalytic converters for the automotiveindustry;theyhave beenextended toinclude other environmentalapplicationssuchasselectivereductions(DeNOx catalysts)usedinpowerplantsandincinerators[2].Morerecently, MRs have emerged as promising candidates competing with conventionalgas-liquid-solidreactors,astheyofferseveral

Nomenclature

a Bubbleinterfacialarea;(m2

Bm−3UC)

c Concentrationofdissolvedgas;(molm−3_L)

coverall Volumeaveragedconcentration,asdefinedinEq.

(6);(molm−3_L)

cs,mean Averageslugconcentration;(molm−3L)

cwall Wallconcentration;(molm−3L)

c* Dissolvedconcentrationatsaturation;(molm−3_L)

D Moleculardiffusionof dissolvedgasintheliquid phase;(m2_s−1₎

dB Bubblediameter;(m)

dc Channeldiameter;(m)

dS Elementarybubblesurface;(m2 B)

dV Elementaryvolume;(m3₎

g Gravityaccelerationvector;(ms−2₎

KC Rate constant of first order surface reaction;

(m3

Lm−2walls−1)

kLa Volumetric mass transfer coefficient;

(m3

Lm−3UCs−1)

LUC Unitcelllength;(m)

Lf Liquidfilmlength;(m)

MR Monolithreactor

N Gasmolarflux,asdefinedinEq.(5);(mols−1₎

n Normalvector;(−)

P Pressure;(Pa)

1P Pressuredrop;(Pa)

QL Volumetricflowrate;(m3s−1)

r Radialcoordinate;(m)

RB Bubbleradius;(m)

UB Velocityofbubblecenterofmass;(ms−1)

UTP Two-phasevelocity,uGs+uLs;(ms−1)

u Velocityvector;(ms−1₎

uGs Superficialgasvelocity;(ms−1)

uLs Superficialliquidvelocity;(ms−1)

uzG Axialcomponentofgasvelocity;(ms−1)

uzL Axialcomponentofliquidvelocity;(ms−1)

VL Liquidvolume;(m3L)

VUC Unitcellvolume;(m3UC)

z Axialcoordinate;(m)

Greeksymbols

«G Gashold-up;(−)

df Filmthickness;(m)

mG Gasdynamicviscosity;(Pas)

mL Liquiddynamicviscosity;(Pas)

rL Liquiddensity;(kgm−3) s Surfacetension;(Nm−1₎ DimensionlessGroups Ca Capillarynumber, mLUB  ;(−) Re Reynoldsnumber,LUBdc mL ;(−)

ReG SuperficialgasReynoldsnumber, Gu_mGsdc

G ;(−)

ReL SuperficialliquidReynoldsnumber,Lu_mLsdc

L ;(−)

ScL LiquidSchmidtnumber,_mL

LD;(−)

ShL LiquidSherwoodnumber,kLDdc;(−)

tages,e.g.,thecatalyticlayerdepositedonthewallofthenumerous MRchannelsisthinenough(ca.10mm)tominimizeinternal diffu-sionalresistances;thechannelshostspecificallytunablegas-liquid flowregimes(chiefamongthemtheso-calledTaylorortrain bub-bleorslugflow),whichcanproveparticularlyconvenientinterms

ofmasstransferinterfacialarea;pressuredropinMRislow; flu-idsflowfreelyavoidingreactorfoulingandcloggingandlimiting theoccurrenceofhotspots;MRsoffertheopportunitytoperform efficientreaction heat removalthrough themonolith backbone providedthatitisbuiltinhighlyheat-conductingmaterial.

ManyworkshavebeendedicatedtothestudyofMRoperation wheretheliteraturereportsexperimentalstudiesoffluid distri-butionintothechannelsofmonolithblocks[3–6],flowregimes inside thechannels[7–10],andmass transferbetweengasand liquidphasesovertheentireapparatus[11,12].Theoverall volu-metricgas-liquidmasstransfercoefficient,kLa,wasreportedtobe

muchlargerinMRoperatingintheTaylorflowregime(0.1–1s−1₎

[11,13,14] than in stirred tanks (0.03-0.4s−1_{), bubble columns}

(0.005–0.25s−1_{)orpackedbeds(0.004–1s}−1_{)[15].Thisenhanced}

masstransferwasattributedtotheexistenceofathinliquidfilm(a fewtensofmm)betweenthebubbleandthechannelwall,aswell astotheefficientconvectivemixingwithintheliquidslugs pro-videdtheyareshortenough[16].Ofpracticalinterest,itwasshown thatkLavaluesmeasuredinMRcorrelateratherwellwiththose

predictedfromsingle-channelmodels[11,14].Indeed,mostofthe experimentalandtheoretical worksongas-liquidmasstransfer havebeendevotedtosinglemillimetriccapillaries[12,16–23].The relativecontributionsofbubblecapsandlubricatingfilmtothe gas-liquidmasstransferwerediscussedmorespecifically,thoughthe conclusionsweremainlydrawnfornon-reactivesystemswhere likelihoodoffilmsaturationwiththetransferringspecies drasti-callyjeopardizessuchlevelofdiscrimination.Insuchasituation, transferthroughbubblecapsbecomestheonlyeffectivepathway turningkLa insensitivetobubblelengthorchannel diameteras

observedbyBerˇciˇcandPintar[18].Conversely,forshortunitcells (bubble+sluglengths<50mm)andbubblevelocities>0.15ms−1_,

simulationsfromvanBatenandKrishna[19]showedthatscalar transportthroughthefilmaccountsfor60–80%oftheoverallkLa

values.Experimental resultsofVanduetal. [12]alsoconfirmed a dominantfilm contribution for unit cells lower than 25mm. Thislatterscenariobecomesespeciallycrucialwhena heteroge-neousreactionoccursatthecatalystcoatedwallduethegenerated concentrationgradient, and in this case theproposedchemical engineeringmodelsoftenneglect(withvaryingdegreesofsuccess) thepossibleinteractionbetweenthedifferenttransferpathways [24–26].

Oneoftherareandcompleteexamplesofadevelopment strat-egyofaMRwasillustratedbyHaakanaetal.[27]whotooklactose oxidation as a study case. They used severaldifferent mockup experimentstostudyseparatelydifferentphenomena,e.g., hydro-dynamics,masstransferandintrinsickinetics,andultimately,the differentsub-modelswerecombinedforacompletemathematical description.Exceptthisrelativelydetailedstudy,amethodology for scale-up or design of a MR apparatus accounting for local inter-channeldisparitiesofthehydrodynamicsandconcentrations stemmingfromunequalflowdistributionintheparallelchannels israrelyproposed.

Inthepresentwork,astrategyformodellingaMRasawhole is described. The objective is to develop a pre-design tool for industrial-scalereactorsappliedtohighlyexothermalreactions. AscendingTaylorflow isassumedinthechannels,andamodel reactionrateisconsideredtooccuratchannelwalls.Thechosen strategyallowsfocusingongas-liquidmasstransferaspartand parceloftheentiremasstransportmechanismsintheunitcellsas akey-pointforMRperformance.Thusthesephenomenaare specif-icallymodelledandsimulatedbymeansofComputationalFluid Dynamics.Foragivensetofoperatingparameters(i.e.,fixedgas andliquidflowrates),theoverallandlocalmasstransferratesare quantifiedanddiscussedforvariousvaluesofunitcelllengthand reactionrate.

2. ProposedmethodforMRmodelling 2.1. Principles

In the present work, a pre-design tool for industrial-scale monolithreactorsisbuiltthankstosimplifyingassumptionsand numericalsimulations.Thestudiedtechnologyconsistsina mono-lithicmetalstructureofferingthincircularchannelsdedicatedto reactionandacoolingfluidcirculatinginthehollowstructure.The reactingcircularchannelsare0.2–1mlong–dependingon reac-tionyieldtobeachieved-,andhaveaninternaldiameterofafew millimetres.Thechannelscanbefedingasandliquidbymeansof standardfluiddistributionsystems(spraynozzleandshowerhead tonameafew)althoughtheirinherentimperfectionsareknown toimpactthedistributionoffluidsamongtheMRchannels.The channelsarecoatedwithafewmicronthincatalyticlayersin addi-tiontohostagas-liquidsegmentedflow(theso-calledTaylorflow) resultinginthefullproblemtobethree-dimensional,locally non-stationaryand stronglyintermingledwithcoupledmultiphysics phenomena(complexhydrodynamics,massandheattransfer, cat-alytic reaction)which necessitatedescriptionsaltogetherat the film/catalyticlayer,channelandreactorscales.Themodelling strat-egyconsistsinrepresentingeachphenomenonwiththerequired levelofcomplexitybyprogressingstep-wisefromthelocalscale tothereactorscale;forthatpurpose,theCFDsoftwareCOMSOL Multiphysics®_{ischosenasitallowstocouplethedifferentphysics}

aswellasdifferentscales.Ateachmodellingstage,the simula-tionresultswillbecomparedtotheoreticalresultsfromliterature, ortoexperimentalmeasurementsobtainedfromdedicated set-ups:jacketedsingle-channelreactor,coldtransparentmock-ups forhydrodynamicregimeandmal-distributionpurpose,and com-pleteMR.

Asafirstassumption,aspatiallyuniformtemperatureis con-sidered for themonolith framework owing toits highthermal conductivityandtothefastcoolantcirculation.Ontheotherhand, theeffectofunevenfluiddistributionwillbesimplyaccountedfor bycombiningoutflowsfromchannelsfedwithdifferentgasand liquidflowrates;thewaysinglechannelsarebeingfedisbasedon phase-retentionmappingandresidencetimedistributionstudies developedinthecoldmock-ups.

Thesetworulesaresufficienttouseasingle-channelapproach tomodelthereactor.

2.2. Implementationofcomputationalfluiddynamicsmodel 2.2.1. Generalapproach

Inthiswork,anapproachinspiredfromFukagataetal.[28]and Guptaetal.[29]ischosenfordescribingTaylorflowinmilli-and microchannelsusingtheso-calledunitcell(UC,Fig.1)inwhicha gasbubbleissurroundedbyaliquidfilmandseparatedbytwo liq-uidhalf-slugs.TheUCisrepresentedinareferenceframemoving withthebubble.Thisapproachisrelevantaslongastheconsidered UCisfarenoughfromtheinletandoutletofthechannel.Many com-putationalworks dedicatedtothemodelling offully-developed Taylorflowinmicro-andmilli-channelsusenumericaltrackingof thedeformationandthemotionofgas-liquidinterface[28,30,31]. Here,becauseofthelowcapillarynumber(O(10−3_)),thebubble

shapecanbeconsiderednon-deformable[32];itisdescribedby meansoftwohemisphericalcapsandacylindricalbodyofradius RB,asdepictedinFig.1.The channelcross-sectionbeing

circu-lar,a2D-axysimmetricrepresentationofthesystemisused.The filmthicknessdf beweenbubbleandwallisestimatedfromthe

semi-empiricalrelationdevelopedbyAussillousandQuéré[33]: ıf

dc =

0.66Ca2/3

1+3.33Ca2/3 (1)

Fig.1.Unit-cellTaylorflowrepresentation.

Ingas-liquidTaylorflow,thepressuregradientsintheliquid phasearemuchgreaterthanthoseinthegasphase,typicallylinked toviscositydifferences,i.e.,mL»mG.Duetothemuchsmaller

viscos-ityingasphase,viscouseffectsatthebubblesurfaceareneglected. Furthermore,thebubblebeingalsonon-deformable,theinfluence ofgasphasephenomenaonliquidphasebehaviorcanbeneglected inthisproblem.Hence,asassumedbyotherauthors[19–21,23], theliquidflowistheonlycomputedphase,andaslipboundary condition(orzero-shear-stresscondition)issetatbubblesurface. Therelevanceofthisapproachwillbevalidatedinthefollowingby comparisonofhydrodynamicresultstoliterature.

2.2.2. Mathematicalmodelling

Theliquidflowisupward,incompressible,andlaminar(liquid Re<840).Uponenablingmasstransfercalculations,the modifica-tionofliquidphasepropertiescansafelybeneglected(one-way coupling)toconsidertheliquidhydrodynamicsstationaryandthus fullyobtainable inadecoupledmanner.Todoso,theCFD soft-wareCOMSOLMultiphysics®_{5.1isusedfirsttosolvethefollowing}

equations:

Continuityequation:

∇

· (u) =0 (2)

Momentumequation:

Once hydrodynamics is solved, the velocity field is used to underlie the calculation of mass exchange between bubble interface and liquid phase. Unlike most of the previous works [19–21,23],asteady-stateconvection-diffusionequationissolved. Thetransportofthedissolvedgasisthusdescribedby:

∇

·(−D

∇

cL)+u·

∇

cL=0 (4)

Dis-solvedgascon-sump-tionisac-countedforattheUCwallto pre-ventthetriv-ialso-lu-tionofcom-pleteUCsat-u-ra-tion.The boundaryconditionsofEqs.(2),(3)and(4)aredetailednext. 2.2.3. Boundaryconditionsforhydrodynamicsmodelling

Forhydrodynamicscalculations,periodicboundaryconditions aresetonoppositefrontiersofthedomain:velocityprofilesare forcedtobeidenticaloninletandoutletboundaries,andpressure drop(1P)is imposed.Theotherboundaryconditions are:axial symmetry,movingwallonchannelwall,andperfectslipcondition onthegas-liquidinterface.Apressureconstraintpointischosen inthecomputationaldomaintoestablishareferenceforpressure fieldcalculation.Inthiswork,astheflowofgasphaseisnotsolved, thecomputedpressuredropcorrespondstosingle-phasepressure variationoftheliquidsurroundinganobjectwithwallperfectslip. Thepressuredropovertheunitcelldependsstronglyonviscous shearatthewallinthelubricationfilmandthusonfilmthickness andonviscosity,andcannotbeeasilydetermined;accordingto literature,themostcommonvalueusedinsimilarcomputational worksiszero.However,validityofsuchanassumptioncanbe chal-lengedbythefactthattheenergylossduetotheshearstresson thewallisdifficulttoignore.Toliftsuchaconstraint,numerical evaluation ofpressure dropis carriedout bymeansofanother simulationstrategy,theopenunitcellstrategy,whereavelocity profileisimposedattheinletboundary,whilerelativepressureis settozeroattheoutletboundarywithconditionofnormalflow. AHagen-Poiseuillevelocityprofileisimposedatinletboundary which,accordingtoliterature,insuresveryrelevantvelocityfield infilmandfullydevelopedslug[19,34].Hence,thisopenunitcell strategyallows evaluatingUCpressuredrop,while theperiodic strategycapturestheexactTaylorflowthatdevelopsinthechannel atanyaxialpositionfarfromthephysicalcapillaryinletandoutlet. 2.2.4. Masstransfermodellingandprocessing

At bubble interface, a Dirichlet boundary condition is used, wheredissolvedgasconcentrationissetequaltothe thermody-namicsaturation(c*).Thecatalyticreactionatthewallactingasa sinkinducesalocalconsumptionrateofthedissolvedgas.Asink fluxissetasboundaryconditionatthewall;thevalueofthisflux isequatedwithafirstordersurfacereactionrate,therateconstant (KC)ofwhichis6×10−5ms−1.Thefluxofdissolvedgastransferred

“interfacially”intheUCwascheckedtobe3ordersofmagnitude lowerthanitsadvectedfluxcounterpartsenteringandleavingthe cell.Thisfeatureprovidesprimafacieevidenceforsettingperiodic conditionsforinletandoutletboundariesoftheunitcell(identical radialprofilesindissolvedgasconcentration).Thus,the steady-stateconcentrationfieldisarrivedatbyviewingmasstransferflux atbubbleinterfaceandgasconsumptionatthewallasstrictlyequal. Itallowsevaluationofgas-liquidmasstransferfluxfortheunit cell:thegasmolarfluxleavingthebubble(N)iscalculatedby com-putingthefollowingsurfaceintegraloverthegas-liquidinterface (axisymmetricmode): N=

Z

Z

 −D



∂

c

∂

z.nz+

∂

c

∂

r.nr



dS (5)

wherecorrespondstothebubbleinterface.

A unit cell volumetric mass transfercoefficient can then be definedwithrespecttothevolumeaveragedconcentrationof dis-solvedgasintheunitcell:

coverall=

R R R

V_LcdV

R R R

VLdV (6) ThevolumetricmasstransfercoefficientkLaisthencalculated

from: kLa= N (c∗_−coverall)∗ 1 VUC (7) Thetransferredgasfluxcomponentsarelikewisedissectedfor specificzonesof thebubble interface:frontand backcaps,and cylindricalpart(lubricatingfilmzone),andtheircontributionto theoverallmasstransferevaluated.

2.2.5. Studiedgeometryandoperatingparameters

ThehydrodynamiccharacteristicsoftheTaylorflowin milli-andmicro-channels,astheratioofbubbletosluglength,the bub-bleshapeandthefilmthickness(df),mainlydependonchannel

diameter(dc),fluidpropertiesandsuperficialvelocities.The

rel-evantnon-dimensionalnumberstodescribetheproblemarethe Capillary,ReynoldsandEötvös/Bondnumbers:theycomparethe relativeimportancebetweenviscous,surfacetension,inertiaand gravitationaleffects[29,32,35].Thepresentworkfocusesonthe descriptionofTaylorflow inthereactingcircular channelsofa monolith,andthegeometrydetailsareinspiredfromastudycase developed by van Batenand Krishna [19] which characteristic parametersareshowninTable1(referencecaseofthiswork).

ThedevelopedTaylorflowdependsonthesetofgasand liq-uidflowrateswherebyvariousunitcelllengths,andthusvarious bubblefrequencies,aretooccur.Theunitcelllength(LUC)cannot

beaprioridetermined;inexperimentsthislengthiscontrolledby thefluidpropertiesandbythetechnologyofthefeedingsystem (TorY-junction,forinstance).Tochecktheinfluenceofunitcell length(orbubblefrequency)ontomasstransferefficiency,several unitcelllengthsaretested(Table1).

2.2.6. Meshfeatures,numericalparametersandsensitivitystudy CFDmodellingofTaylorflowneedsspecialcareregardingmesh resolutionespeciallynearbytheinterfaceswheresteepvelocityand concentrationgradientsemerge,inparticularintheverythin lubri-cationliquidfilm.Asdocumentedbyseveralauthors[30,36],poor meshresolutionpreventscaptureoftheexactdetailsoftheflow fieldaroundthebubble.Thus,Guptaetal.[30]recommendeda min-imumoffivemeshelementsacrosstheliquidfilm.Inthepresent work,auser-controlled meshwasused(Seeadditionalfigurein AppendixA):afreetriangularmeshisbuiltonthedomain,and optimizedbyadjustingelementsizeand growthratenear bub-blesurfaceandchannelwall.Aboundarylayermesh(quadrilateral elements)wasalsocreatedclosetotheseboundaries.

Tomakesurethattheimplementedmeshispreciseenough, ameshsensitivity studywasperformedbyvaryingthenumber andsizeofelementsandbycheckingthevariationsofthe calcu-latedpressuredropintheunitcell(UC).FortheUCofreference (samegeometry asvanBaten&Krishna’s case),thenumber of elementswasincreasedfrom33,347to226,789(Table2).Asa com-promisebetweencomputationaltimeandresultsaccuracy,agrid with77,318cellswherethesmallestelementsizeis0.8mm,was chosen.Forcomparison,vanBaten&Krishnaused72,890elements and1mmsmallestcellsize.ForeachUClength,thesamestrategy wasrepeatedtoobtainrelevantamesh.

Furthermore,toachieveresultswithhigheraccuracy,aP2–P1 mixed-order interpolation scheme has been then used with piecewisequadratic approximation ofvelocity componentsand

Table1

Summaryofinputdataforthenumericalstudy.

Inputparameters Computedparameters

Case dc(mm) LUC(mm) «G(−) Lf(mm) df(mm) UB(ms−1_{) KC(ms}−1_{) c*(molm}−3_{) D(m}2_s−1_{) UTP(ms}−1_{) QL (mls}−1₎

1:ref 3 40.0 0.17 5.320 48 0.3 6×10−5 _{1.3 1×10}−9 _{0.28 1.62}

2 3 20.0 0.17 1.692 48 0.3 6×10−5 _{1.3 1×10}−9 _{0.28 1.62}

3 3 13.3 0.17 0.483 48 0.3 6×10−5 _{1.3 1×10}−9 _{0.28 1.62}

Table2

MeshdetailsforthesensitivityanalysisforthecaseLUC=0.04m.

Totalmeshelements Smallestelementsize(mm) Biggestelementsize(mm) Elementsinliquidfilm 1POpenUC(Pa)

1 33,347 3.0 155 17 283

2 77,318 0.8 155 19 324

3 151,616 0.2 87 22 331

4 226,798 0.2 67 22 325

Fig.2. Streamlines(a)andliquidvelocityvectors(b)obtainedfromCFDsimulationforthereferencecase(Case1,seeTable1).Zoomedarea(c)correspondstotheentrance ofliquidfilm.

piecewiselinearapproximationofpressure.P3finiteelementsare appliedforconcentrationfield.Intheseconditions,lessthan2% differencehasbeenfoundbetweenMesh2and4forbothpressure gradientandvolumetricmasstransfercoefficient.

3. Resultsanddiscussion 3.1. Velocityfield

WerecallthatMesh2(seeTable2)isused.Boundaryconditions ofperfectslipandnosliparesetonbubbleinterfaceandchannel wall,respectively.TheUCpressuredropisevaluatedinadvanceby meansofanopenunitcellcalculation.

3.1.1. Velocitycontoursandvelocityprofilesinslugandinfilm Forthereferencecase(case1,Table1),Fig.2showsthe veloc-ityfieldobtainedinaframemovingwiththebubblewherethe recirculationstreamlinestakeindeedplaceintheslug(Fig.2a,b). In addition,Fig.2cshows that,inthelubricationfilm,theflow developsrapidlyshowcasingtranslationalinvarianceofthe veloc-ityprofilefromz=0.1mmfromfilmentrance;inthefilmtheliquid appearstomoveoppositetothedirectionofbubblemotion.

Slugvorticesareinducedbyviscouseffectsinitiatedinthe vicin-ityofwallandcanbeobservedinslugsinTaylorflowsforCa<0.5

[37].ThisbehaviorwasfirstreportedbyTaylorin1961[38]andhas beenconfirmedbynumerousexperimentalandnumerical stud-ies[39,40].Theslugrecirculatorymotionisattheoriginofthe notoriousintensemixing,heatandmasstransferobservedin Tay-lorflows.Transposingtheradialvelocityprofilesinthelaboratory frame(stationarywall,risingbubble)providespost-facto confirma-tionthataHagen-Poiseuilleflowisretrievedinthemajorpartof theslug(Fig.3a).Thisbehaviorisexpectedwhenconsideringthat theliquidslugissufficientlylongtoallowafullydevelopedflowto beattainedawayfrombubblenoseandtail[39].Inthestationary frame,themeanvelocityofliquidflow(i.e.,velocityaveragedover channelcrosssection)isequaltoUTP,whereUTPisdefinedasthe

sumofsuperficialvelocitiesofgasandliquid.However,as classi-callyobservedinTaylorflows[41],thebubblemovesslightlyfaster thanUTP and,tosatisfymassconservation,liquidmovesslower

thanthebubbleinthefilmregion,andinsomecasesitmovesin theoppositedirection.Inthepresentdescribedsituation, “down-ward”motionofliquidoccursinthefilm;asaconsequence,inthe frameofreferencemovingwiththebubble,liquidisobservedto movedownwardandslightlyfasterthanthewall.

3.1.2. Influenceofpressuredifferenceoverunitcell

Thevaluetobetakenforunitcellpressuredropinliquidphase 1Pishardlydiscussedinliteraturedescribingcomputational

stud-(a)

(b)

Fig.3.Comparisonofradialdistributionofliquidvelocityinslug(farfrombubble caps)inlaboratoryframe(a),andliquidfilmvelocityprofileinbubblereference frame(b).Simulationofreferencecase(case1,Table1).

iesbasedontheperiodicunitcellapproach.IntheworkofShao etal.[20]forinstance,gravitationalforcesareneglectedbecause thechannelishorizontal,andpressuredropduetoviscousshear istakento0overtheunitcell(“Pin=Pout”);inthatofvanBaten

andKrishna[19](upflowconfiguration)onlyhydrostaticpressure appearstohavebeenaccountedfor.Tochecktherelevanceofthis choice,the1Pvalueimposedaccrosscomputationaldomainis var-iedinthepresentwork;thevelocityprofileatoutletofthefilmzone isscrutinizedforbothscenarii.AsobservedonFig.3b,thevelocity profileobtainedwhenpressuredropisforcedto0andgravitational forceneglectedinEq.(3)isinexcellentagreementwiththeprofile foundbyvanBatenandKrishna;inparticular,avelocitymagnitude of−0.32ms−1_{isobservedatbubbleinterface.Whenpressuredrop}

overtheunitcellisforcedtothevaluepreviouslycomputedfrom theopenUnitCellapproach,liquidflowsslightlyfasterinthefilm andtheinterfacialvelocityreaches−0.33ms−1_.Thistinygap

rep-resentsalowrelativedifferenceof3%formaximumvelocityinfilm regionwhereas,inthemiddleoftheslug,wherePoiseuilleprofile isestablished,thetwoassumptionsleadtoverysimilarvelocity valuesonchannelaxistoo(0.84%ofrelativedifference).

Theseresultsexplainandapprovepracticesfoundinprevious CFDstudies:asfarashydrodynamicsareconcerned,pressuredrop overunitcellcanbeneglected.

3.1.3. Influenceofboundaryconditionusedatbubbleinterface Tocheckfurtherthereliabilityofthepresentcalculations,the velocityprofileinthedevelopedzoneofthelubricationfilmis com-paredtotheanalyticalmodelproposedbyAbiev[41].Thisauthor derived anexact lubricationsolutionofgasand liquid flowsin thechannelcross-sectionwherethefilmisfullydeveloped.

Con-Fig.4. Velocityprofilesinliquidfilmfordifferentboundaryconditionsatbubble surface(case1,seeTable1).

tinuityandmomentumequationsareanalyticallysolvedforboth phases.Atgas-liquidinterface,thecontinuityofthedistributionof thevelocityandshearstressissetasboundarycondition:

uzG|r=RB=uzL|r=RB (8) G

∂

uzG

∂

r |r=RB =L

∂

uzL

∂

r |r=RB (9) Inthelubricationsolution,thepressuredropisanoutputof the model. The velocity profile obtainedfor liquid phase with Abiev’smodelisplottedinFig.4alongwiththeoneobtainedwith COMSOL®_{usingperfectslipconditionatbubblesurfaceandavalue}

ofpressuredropissuedfromopenunitcellcalculation.Ascanbe seen,theperfect slip conditionleadstoresultsclose toAbiev’s model;inparticular,bothmaximumaxialvelocitiesinthefilm(at bubblesurface)differof6%only.Thereforetheperfectslip condi-tionisrelevanttomodelliquidflowintheunitcell.

Theinfluenceonmasstransportoftheslightdifferenceinliquid flowrateinthefilmhasstilltobefurtherinvestigatedtohighlight theimpactofthetwoboundaryconditionsdiscussedabove. 3.2. Concentrationfield

Contourplotsofsimulatedconcentration(Fig.5a)showthatthe liquidphasecontentintheslugisalmostuniformindissolvedgas, showinganaverageconcentrationof85%ofc*.Somethinzonesare closetosaturation(Fig.5b):adiffusionlayernearbubbleinterface, andabandalongchannelaxis,wheredissolvedgasisadvectedby liquidflowrecirculation.Zoomingnearthewall(Fig.5c)instructs onthefactthata concentrationgradienttakesplacecrosswisse throughtheentirefilmthicknessbyvirtueofthewallreaction,and thatthefinitereactioncharacteristictimeallowsatthewallalow (butnon-zero)dissolvedgasconcentration(20%ofc*).

3.3. Masstransfercharacteristics

Themasstransferbehaviorwillbedescribedindetailforthe referencecase(case1,Table1),andforadditional reparameter-izedcaseswherethereactionrateconstantandbubblefrequency arevaried.Thefirstobjectiveoutofthissensitivityexerciseisto understandwhichpartsofthebubbleinterfacearethemain con-tributors totheunitcelloverall masstransfer.Thesecondis to highlightany influenceof bubble frequency onUC mass trans-ferefficiency,inwhich instancethis parametershouldbetaken intoaccountinmonolithreactordesignandscale-up.Thispoint introducesapeculiarcomplexity intotheproblem,asitis diffi-cult,especiallyinindustrialoperations,toeitherpredictorcontrol howgasandliquidsplitandre-agregateintobubblesandslugsat channelinlet.Thedependencyofbubbleandsluglengthson oper-atingparametershasbeenextensivelystudied.However,despitea

Fig.5.ConcentrationfieldovertheentireUCdomain(a)withzoomedareanearthe bubble(b),andintheliquidfilm(c).Simulationforcase1(Table1).

numberofempiricalcorrelationsthusfarproposed,yetthe hetero-geneousoutcomesoutofthemstillsuggestthatthisaspectrequires morematureunderstandingtobeachievedinthefuture[42]. 3.3.1. Referencecase

Asexplainedearlier,thevolumetricmasstransfercoefficient, kLa, isderivedfromthefieldofdissolvedgasconcentration.An

overallvalueof0.08m3

Lm−3UCs−1(0.09s−1relatedtotheliquid

volume)is obtainedfor the unit cellfor a total bubble surface of7.50×10−5_m2_{tantamounttoaninterfacialareaof265.36m}2

perm3_ofunitcell.

Itisworthmentioningthatstationarysimulationsofasingle periodicunitcelldonottakeintoaccounttheentrancesectionof channel,whereliquid phaseisoftenalmostfreefromdissolved gasleadingtohighgas-liquidtransferrate.Asaconsequence,the presentapproachprobablyslightlyunder-estimatesthekLa

val-ues, in regard with the correspondingexperimental situations. However,thetubelengthallowingthedevelopmentofastable con-centrationfieldintheunitcellisprobablyveryshort:asdescribed byauthorsperformingdynamicsimulationofmasstranferinaunit cellwithhomogeneousreaction [20],this distancecorresponds roughlytothetimeneededfortheliquidintheslugtoenrichin dissolvedgasinthevicinityofbubblecaps,thatistodescribea completecirculationcyclewithintheslug.Forthepresentcaseof simulation,itrepresentslessthantwotimestheunitcelllength.

Withrespecttothevalidationofthechosenboundary condi-tionsandtotheslightdifferenceinliquidflowratetheyinducefor thefilmregion,itisimportanttonotethatthemeanslug concen-trationandkLavaluesdifferby0.6%and1.1%only,respectively,

whenpressuredropistakenintoaccountornot.Asaconsequence, itcanbestatedthatasmalldifferenceinthetransportofdissolved gasfromfilmtobackhalf-slughasnosignificantinfluenceofmass transfercharacteristics.Thisdefinitivelyvalidatestheconventional choicesfoundinliteratureforunitcellpressuredropandinterface boundaryconditions.

ThecalculatedkLavaluesarecomparedtothosederivedfrom

several literaturecorrelations, given in Table3. Theserelations wereobtainedfor differentconfigurations(regardingfilm satu-ration level)andeitherfromexperimentalornumericalresults. Berˇciˇc&Pintar’scorrelation(Eq.(A))wasbuiltbasedon experimen-talresultsofmethaneabsorptioninwaterobtainedincapillaries of1.5,2.5and3.1mmdiameter.Theauthorsreportedthatmajor partofmasstransferoccuredthroughthebubblecaps,probablyas liquidfilmwasquicklysaturatedintheirconditions[12]. There-fore,kLawasfoundtomostlydependuponliquidsluglengthand

velocity,whilegasbubblelengthandchanneldiameterhada neg-ligibleeffect.VanBaten&Krishna’scorrelation(Eqs.(B)and(C)) wasobtainedfromCFDsimulations.ItsplitskLaintotwoprincipal

contributions:onefromthebubblecapsandtheotherfromthe film.ThefirstoneisbasedonHigbie’spenetrationmodelandthe secondoneonfallingfilmmodel.Eq.(B)usestheexactdimensions of bubble,filmandslug, unlikeEq. (C)where thesedimensions areestimatedfromknowledgeoftheoperatingparameters.Eq.(D) proposedbyVanduetal.[12]considersthefilmcontributiononly, basedonvanBatenandKrishna’swork:theconstantfactorwas verifiedtobe4.5asbestfittingtheirexperimentaldataobtained fromexperimentsofairabsorptioninwaterin1–3mm capillar-ieswithcircularandsquarecross-sections.Thiscorrelationshould thenbevalidforTaylorflowsinwhichfilmcontributionis domi-nant.Eq.(E)fromYueetal.[15]wasderivedfornarrowchannels (<1mm)inadditiontohighgasandliquid superficialvelocities (1ms−1_<UTP<12ms−1_{).Shaoetal.[20]tunedthemultiplicative}

constantinEq.(D)tomatchCFDresultsforthecaseofCO2

absorp-tionintoanaqueoussolutionofNaOH.

ComparisonbetweenpresentkLavaluesandthosepredictedby

vanBatenandKrishna’scorrelation(Eq.(B)inTable3)showsa differenceof5.3%.Notethatthepresentsimulationscorrespondto theconditionsusedbyvanBatenandKrishna,i.e.,shortcontact timeoftheliquidfilm(tfilm<0.1df2D−1).Eq.(C)withestimated

parameters leadstolessaccurateresultsthanEq.(B),ascanbe seenonFig.6.

CorrelationsfromVanduetal.[12](Eq.(D))andShaoetal.[20] (Eq.(F))inwhichfilmcontributionispreponderantunderpredict thekLavalue.Incontrast,BerˇciˇcandPintarcorrelation[18](Eq.

(A))wasestablishedforlongbubbleswithalmostsaturatedfilms anddoesnotreflectthepresentsimulatedconditions;itleadsto anoverpredictionofmasstransfercoefficient.Finally,Eq.(E)by Yueetal.[15]underestimatesourcurrentresults,probablybecause itwasderivedforchannelsmuchnarrowerthaninthisstudyin additiontomuchhighergasandliquidsuperficialvelocities.

ThegoodagreementobtainedwithEq.(B)provestherelevance ofthesimplifiedapproachproposedinthepresentworktodescribe masstransferinthinchannels(stationnarymode,gasphasenot modelled).

Table4summarizesthecontributionsofdifferentpartsof bub-blesurfacetomasstransferasobtainedinoursimulation.Aslong asexperimentalstudiesareunabletoachievesuchcontributional dissections,ourapproachcanproveveryusefulingaininginsights in thiscomplextsubject.Table4shows thatfilmsurface repre-sents64%oftotalbubblesurfaceandcontributesto64%ofoverall transferredmolarflux.

Notwithstanding,oursimulation resultis inagreementwith generalobservationsthatlubricationfilmcontributionisthemajor oneinthecaseofwallreaction[12].Eachbubblecaprepresents 18%oftotalbubblesurfaceonly,butfrontcapcontributes4times morethanbackcaptomasstransfer,asthecorrespondingrelative masstransferfluxesare29%and7%,respectively.Thequitelimited contributionofbubblebackcapcanbeexplainedbythemoderate liquidvelocities(Fig.7)observedinthevicinityofbubblebackcap ascomparedtofrontcap,leadingtomoderatedrainageofdissolved gasinthisregion.

Table3

CorrelationsfromliteratureusedforcomparisonwithkLavalues.

Authors Correlation Equation

Berˇciˇc&Pintar[18] kLa=0.111 U 1.19 TP [(1−εG)LUC]0.57

(A)

vanBaten&Krishna[19] kLa=2 √ 2 

p

DUB dc 4d2 B LUCd2c+ 2 √_

q

DUB Lf 4dBLf d2 cLUC (B)

vanBaten&Krishna[19] kLa=2 √ 2 

p

DUB dc 4 LUC+ 2 √_

p

_εDUB GLUC 4εG dc (C)

Vanduetal.[12] kLa=4.5

p

DuGs LUC

1

dc (D)

Yueetal.[15] ShL·a·dc=0.084Re0.213 G Re

0.912 L Sc

0.5

L (E)

Shaoetal.[20] kLa=3

p

DuGs LUC

1

dc (F)

Fig.6.Comparisonofmasstransfercoefficientscomputedinthisworkagainstthosepredictedfromseveralliteraturecorrelations.

Table4

Comparisonofsurface,molarfluxanddensityfluxfordifferentzonesoftheUC(caseofreference).

Bubblenose Film Bubbletail UnitCell Units

Surface 1.32×10−5 _4.85×10−5 _1.32×10−5 _7.50×10−5 _[m2_]

18% 64% 18%

Molarflux 1.70×10−9 _3.71×10−9 _4.07×10−10 _5.82×10−9 _[mols−1_]

29% 64% 7%

kLa 0.02 0.01 0.01 0.08 [m3

Lm−3UCs−1]

Fig.7. Velocityfieldonbubblefrontandbackcapsforcase1(Table1).

Furthermore,oursteady-stateapproachoffersthepossibilityof calculationof“local”volumetricmasstransfercoefficients,related togas-liquidinterfacialareacalculatedfordifferentpartsofthe bubble,andtolocalaverageconcentrationinliquidphase.Ithasto beborneinmindthat,forthefilmarea,thedrivingforceusedin kLacalculationistakenas(c*-cwall).ThevaluesgatheredinTable4

showforthepresentcasewithwallreactionthatkLavalue,unlike

masstransferflux,ismore importantforbubblefront capwith regardtofilmandbackcapvalues.Thisobservationraisesthenthe followingquestion:isfilmcontributiontomasstransferfluxstill

dominantwhenfilmsurfaceissignificantlyreduced?Thispointis checkedinthenextsection.

3.3.2. Influenceofbubblefrequency

Withrespecttothereferencecase,additionalsituationsare sim-ulated,wheregasandliquidflowratesarekeptconstant,aswellas bubblevelocityandgasholdupinunitcell,andwherebubble fre-quencyistheonlyvariedparameter.LUC(referencevalueis40mm,

correspondingto25bubblespermeteronchannel)isdividedby factors2and3(hereinafterreferredtoas“case2”and“case3”, respectively),leadingtosmallerbubbles(Fig.8):bubblesurface reaches56%and41%oftotalreferencebubblesurface,respectively. Ontheotherhand,unitcelllengthdecreaseswithincreasing bub-blefrequencyinalessextentthanbubblesurface,leadingtohigher gas-liquidareapercubicmeterofchannel.Table5summarizesthe consideredcasesandthemainresults.

Asobserved,theoverallkLavalueisimprovedby35%fora

three-foldincreaseofbubblefrequency(fromcaseofreferencetocase 3).ThiskLaenhancementcannotbefullyattributedtotheincrease

ofinterfacialarea,whichisonlyof21%.Aprobableexplanationis thatshorterslugsleadtointensifiedliquidrecirculationandthus tomoreefficienttransportprocesses.

Backbubblecapremainspoorlycontributing,withonly19%of theoveralltransferredgasrateincase3.Bubblesurfaceinthefilm areadecreaseswithbubblefrequency(accountingforonly14%only

Fig.8.UnitCellcharacteristicsfordifferentstudiedcases.

Table5

ContributionstomasstransferofthedifferentzonesofthebubbleforthreeLUCtested.

Bubblenose Film Bubbletail UnitCell Units

Reference case

Surface 1.32×10−5 _4.85×10−5 _1.32×10−5 _7.50×10−5 _[m2_]

18% 64% 18%

Molarflux 1.70×10−9 _3.71×10−9 _4.07×10−10 _5.82×10−9 _[mols−1_]

29% 64% 7%

kLa 0.02 0.01 0.01 0.08 [m3

Lm−3UCs−1] Case2 Surface 1.32×10−5 _1.54×10−5 _1.32×10−5 _4.19×10−5 _[m2_]

32% 36% 32%

Molarflux 1.31×10−9 _1.35×10−9 _3.47×10−10 _3.01×10−9 _[mols−1_]

44% 45% 11%

kLa 0.04 0.01 0.01 0.10 [m3

Lm−3UCs−1] Case3 Surface 1.32×10−5 _4.40×10−6 _1.32×10−5 _3.09×10−5 _[m2_]

43% 14% 43%

Molarflux 1.58×10−9 _5.81×10−10 _5.14×10−10 _2.68×10−9 _[mols−1_]

59% 22% 19%

kLa 0.06 0.01 0.02 0.11 [m3

Lm−3UCs−1]

oftotalbubblesurfaceforthehighesttestedbubblefrequency), andsodoesitscontributiontooverallunitcellmasstransferrate, reaching22%onlyforcase3;filmcontributiondominancestops inbenefittobubblefrontcapcontribution(59%oftotaltransferred mass,incase3).Tooffsetthisphenomenon,filmcontributioncould beenhancedviaasteeperconcentrationgradientbetweenbubble andwall,i.e.,bythinningthefilmorinotherwordsbyslowingthe flow.However,fasterflowwouldenhancecirculationwithinslugs, asrecommendedintheliterature[42].In situationswherefilm contributiontomasstransferisdominant,lowsuperficialvelocities shouldstillbeprefered,keepinginmindthattotalsuperficialfluid velocitydirectlyimpactstheoverallresidencetimeandchannel reactionyield.

Considering thesteep concentrationgradientsnear thewall whenacatalyticreactionispresent,theinfluenceofreactionrate onfilmandcapcontributionsmaybeofprimaryimportance. 3.3.3. Influenceofreactionrateatthewall

Fromcaseofreference(LUC=0.040mmandKC=6×10−5ms−1),

threenewcasesaredescribed,wherereactionratecoefficientKCis

quenchedbyafactorof50(case(1′_{)),orinflatedbyafactorof5(case}

(2′_{)),orbroughttoinfinity,i.e.,cwall=0(case(3}′_{)).Forthefourcases}

examinedhere,bubblesurface,unitcelldimensions,andoperating parameters(fluidflowrates)areidentical.Theresultsinvolume averageconcentrationinslug, wallconcentration,masstransfer flux,andlocalorglobalkLavalues,aresummarizedinTable6.As

expected,theaverageslugconcentration(cs,mean)andwall

concen-trationdecreaseuponignitingfurtherthereactionrate.Itcanbe observedthat,inallcases,themeanconcentrationindissolvedgas isthesameforthefrontandthebackhalf-slugs.Forslow reac-tionrate(case(1′_{)),theslugaverageconcentrationequals98%of}

saturationconcentration,whereasitreaches74.8%forinfinite reac-tionrate(cwall=0).Similarly,theoverallmolartransferredfluxin

theunitcell(i.e.,molarfluxduetoreactionrateatwall)increases

sharplywithincreasingrateconstant.Interestingly,theseincreases arenotproportionalsincecwallalsosimultaneouslydecreasesfora

soaringKCvalue.

Irrespectiveofthestudiedcases,contributionsoffilmandcaps tomasstransferfluxarestrictlythesame.Thisobservationproves that,forshortnon-saturatedfilms,massfluxexchangedbetween bubbleandwallmaydependonoperatingparametersandon bub-bleandslugrelativedimensions,butnotonreactionrateatthe wall.

4. Conclusionsandperspectives

Thismasstransferstudywaspartofawidermodellingstrategy whichaimsatmodellingamonolithreactorasawholeby account-ingonlyfortherequiredlevelofcomplexityforthedescriptionof thephenomenaoccurringatthefilm,channelandreactorscales. Comparisonwithresultsofamultiphaseflowmodelprovedthat atlowcapillarynumbers,thehydrodynamicsofTaylorflowcan beadequatelyapproximatedbycalculationsonliquidphaseonly, neglectingunitcellpressuredropandusingslipconditionsatthe bubbleinterface.

Gas-liquidmasstransferratewasevaluatedforshortfilmsand reactiveconditions,closertothoseofinterest,whichalsoensure that bothfilmand slug regionsremaincontributivethroughout thecapillary;itwasshownthattransferredmassfluxandaverage concentrationvarywithsurfacereactionrate,butnottherelative contributionoffilmandcapstotheoverallmasstransfer.kLa

val-uescloseto0.1m3

Lm−3UCs−1wereobtainedfora3mmdiameter

channelandbubblevelocityof0.3ms−1_{,ingoodagreementwith}

thecorrelationofvanBatenandKrishnawhichwasalsoestablished forsteadystatevalues.Otherliteraturecorrelationseitherincluded theeffectoftheinletdissolvedgas-depletedzonesorwere devel-opedforratherdifferentbubblecontactingtime,leadingtolarger discrepancies.Despitesamehemisphericalshapeappliedforfront

Table6

Detailsofsimulationresultsforthekineticconstantdependence.

Bubblenose Film Bubbletail Total(UC) Units

Case(1′₎ MolarFlux 1.59×10−10 _3.48×10−10 _3.65×10−11 _5.44×10−10 _[mols−1_]

cs,mean 1.27 – 1.27 1.27 [molm−3_L]

cwall 1.20 1.20 1.20 1.20 [molm−3_L]

kLa 0.02 0.01 0.01 0.08 [m3

Lm−3UCs−1] Case(2′_{) MolarFlux 2.02×10}−9 _4.41×10−9 _4.84×10−10 _6.92×10−9 _[mols−1_]

cs,mean 0.99 – 0.99 0.99 [molm−3_L]

cwall 0.06 0.06 0.06 0.06 [molm−3_L]

kLa 0.02 0.01 0.01 0.08 [m3

Lm−3UCs−1] Case(3′₎ MolarFlux 2.12×10−9 _4.63×10−9 _5.08×10−10 _7.26×10−9 _[mols−1_]

cs,mean 0.97 – 0.97 0.97 [molm−3_L]

cwall 0 0 0 0 [molm−3_L]

kLa 0.02 0.01 0.01 0.08 [m3

Lm−3UCs−1]

andrearbubblecaps, thetwozoneswerenotfoundequivalent intermsofgastransferredrates.Therearbubblezoneexhibited, asa consequenceof a lowerlocal liquidvelocity, a moretepid activity,threetofourtimes lesserthanthebubble noseregion. Sucheffectshouldbeaccentuatedforamorerealisticbubbleshape (withelongatednoseandflattened back)duetosignificant dif-ferenceincorrespondingsurfaceareas.Fortherathershortunit cells investigated (withlength 4–14timesthe capillary diame-ter),filmcontributiontomasstransferfluxvariesinawiderrange thanusuallyreported,comingdownfrom64%(intheVanBaten andKrishna’sreferencecase)to20%whenunitcellsizeis signifi-cantlyreduced,andpointingoutthattheapproximationbasedon gasholdupandunitcelllengthforfilmlengthceasestobeavalid approximation.

Furtherworkwillexamineendeffectsbysimulatingseveralunit cellswithopenboundaryconditionsandwillextendthe paramet-ricstudytoshorterslugs,longerbubbles,largercapillarynumbers anddifferentbubblevelocitiestoassesstherelativecontribution ofbubble/filmzonesforawiderrangeofoperatingconditions. Acknowledgements

Authorsthankthe Frenchagency“Agence Nationale pourla Recherche” for financial support (grant number ANR-12-CDII-0011-01), and TOTAL S.A. company for financial and scientific support.

AppendixA. Supplementarydata

Supplementarydataassociatedwiththisarticlecanbefound, intheonlineversion,athttp://dx.doi.org/10.1016/j.cattod.2016.04. 009.

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