Discrete particle simulations of bubble-to-emulsion phase mass transfer in single-bubble fluidized beds

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Discrete particle simulations of bubble-to-emulsion phase

mass transfer in single-bubble fluidized beds

Citation for published version (APA):

Tan, L., Roghair, I., & van Sint Annaland, M. (2017). Discrete particle simulations of bubble-to-emulsion phase

mass transfer in single-bubble fluidized beds. Particuology, 33, 80-90.

https://doi.org/10.1016/j.partic.2016.09.008

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10.1016/j.partic.2016.09.008

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ContentslistsavailableatScienceDirect

Particuology

jou rn al h om ep a g e :w w w . e l s e v i e r . c o m / l o c a t e / p a r t i c

Discrete

particle

simulations

of

bubble-to-emulsion

phase

mass

transfer

in

single-bubble

ﬂuidized

beds

Lianghui

Tan,

Ivo

Roghair,

Martin

van

Sint

Annaland

∗

ChemicalProcessIntensiﬁcation,MultiphaseReactorsGroup,DepartmentofChemicalEngineering&Chemistry,EindhovenUniversityofTechnology,P.O. Box513,5600MBEindhoven,TheNetherlands

a

r

t

i

c

l

e

i

n

f

o

Articlehistory: Received4July2016

Receivedinrevisedform22August2016 Accepted8September2016

Availableonline6February2017 Keywords:

Masstransfer Discreteparticlemodel Fluidizedbed Bubble-to-emulsion

a

b

s

t

r

a

c

t

AclassicalEuler–Lagrangianmodelforgas–solidflowswasextendedwithgascomponentmass conser-vationequationsandusedtoobtainfundamentalinsightsintobubble-to-emulsionphasemasstransferin bubblinggas–solidfluidizedbeds.Simulationsofinjectedsinglerisingbubblesunderincipient fluidiza-tionconditionswerecarriedout,usingGeldart-Aand-Bparticles.Phenomenaobservedinthesimulations andthoseofvarioustheoreticalmodelsusedtoderivephenomenologicalmodelswerecomparedto chal-lengetheassumptionsunderlyingthephenomenologicalmodels.Thebubble-to-emulsionphasemass transfercoefficientscalculatedforthesimulationsusingGeldart-Bparticleswereinagoodagreement withpredictionsmadeusingtheDavidsonandHarrison(1963)model.Thebubble-to-emulsionphase masstransfercoefficientsforGeldart-Aparticleswere,however,muchsmallerthanthepredictions obtainedfromtheoreticalmodels(e.g.ChibaandKobayashi(1970)).Thenewlydevelopedmodelallows adetailedanalysisofvarioushydrodynamicaspectsandtheireffectsonthemasstransfercharacteristics inandaroundrisingbubblesinfluidizedbeds.

Introduction

Gas–solid ﬂuidized bed reactors are often used in process

industriesowingtotheirexcellentmixingandheattransfer char-acteristics.Itiswellknownthatbubblesprevailinthesebedsand theirdynamicsareresponsiblefortheagitationofsolidsandthe accompanyingfavorableheatandmasstransfercharacteristicsof ﬂuidizedbeds.Animportantfoundationforarationaldesignof ﬂu-idizedbedreactorsisathoroughunderstandingofthemasstransfer

processes in ﬂuidized beds,speciﬁcallythe bubble-to-emulsion

phasemasstransfer.Thisphenomenonoccursviathecombined

effectsofgasdiffusion,coherentgasﬂowandsolidsmotion car-ryingadsorbedgasatoms (Davidson& Harrison,1963;Kunii&

Levenspiel,1991).

Single-bubbleﬂuidizedbedsandfreelybubblingﬂuidizedbeds

havebeenusedinpastdecadestostudythebubble-to-emulsion

phase masstransfer, both experimentally and numerically(a.o.

Dang,Kolkman,Gallucci,&vanSintAnnaland,2013;Deshmukh,

vanSintAnnaland,&Kuipers,2007;Hernández-Jiménez,

Gómez-García,Santana,&Acosta-Iborra,2013;Patil,vanSintAnnaland,&

∗ Correspondingauthor.Fax:+31402475833.

E-mailaddress:M.v.SintAnnaland@tue.nl(M.vanSintAnnaland).

Kuipers,2003;Pavlinetal.,2007).Phenomenologicalmodels,used

forthedesign ofindustrial-scalereactors,canonlyprovide reli-ablepredictionswhenaccuratemasstransfercoefﬁcientsareused. Untilnow,mostcorrelationsforthesecoefﬁcientshavebeenbased on(i)analytical considerationsand(ii)experimentsusing

inva-sivemeasurementtechniques.Severalproblemsarisewhenusing

phenomenologicalmodels.First,variousassumptionsaremadeto

reducethemathematicalanalysis,butthescopeoftheirvalidityhas notyetbeenanalyzedindetail.Second,theinvasiveexperimental

techniquesmaydisturbtheﬂow andare limitedtopoint

mea-surements.Noninvasiveopticaltechniques(e.g.,Dangetal.,2013;

Mülleretal.,2006;Pavlinetal.,2007Roels&Carmeliet,2006)have

beendevelopedinthemeantime,butdetailedunderstandingofthe

underlyingmechanismsremainsoutofreachparticularlyowing

todifﬁcultiesinmeasuringthegasconcentrationintheemulsion phase(Dangetal.,2013).

Numericalsimulations(i.e.,computationalﬂuiddynamics)can shedmorelightonthedetailedprocessofinterphasemasstransfer.

Patiletal.(2003)andHernández-Jiménezetal.(2013),forinstance,

employedatwo-ﬂuidmodel(TFM,employingtheEuler–Euler

tech-nique)forﬂuidizedbedscomprisingGeldart-Bparticles.Patiletal.

(2003)foundthattheDavidsonandHarrison(1963)model

pre-dictedthemasstransferforsingleinjectedbubblesreasonablywell, buttheirresultsgaveabubblesizeevolutionandtracergas

con-http://dx.doi.org/10.1016/j.partic.2016.09.008

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Nomenclature A Area(m2₎ c Numberofspecies d Particlediameter(m) Db Bubblediameter(m) D Diffusioncoefﬁcient(m2_/s) e Coefﬁcientofrestitution f Volumefraction

Fcontact,a Contactforceofparticlea(N)

g Gravitationalacceleration(m/s2₎

I Momentofinertia(kgm2₎

k Springstiffness(N/m)

K Masstransfercoefﬁcient(s−1)

ma Particlemass(kg)

M Molarmass(kg/mol)

Np Particlenumber

P Pressure(Pa)

R Gasconstant(J/molK)

Sp Particledragsourceterm(N/m3)

t Time(s)

T Temperature(K)

ug,va Gasandsolidvelocities(m/s)

U Velocity(m/s)

V Volume(m3₎

x Molefraction

y Massfraction

Greeksymbols

ˇ Inter-phase momentum exchange coefﬁcient

(kg/m3_s)

ε Volumefraction

Dampingcoefﬁcient

Gasphaseshearviscosity(Pas)

f Frictioncoefﬁcient

Density(kg/m3₎

Stresstensor(Pa)

Subscripts a,p Particle b Bubble bc Bubble-to-cloud be Bubble-to-emulsion ce Cloud-to-emulsion A,B Gascomponent g Gas i,j Component

mb Minimumbubblingﬂuidizationcondition

mf Minimumﬂuidizationcondition

n Normaldirection t Tangentialdirection w Wake inj Injection diff Diffusion Acronyms

CFD Computationalﬂuiddynamics

DPM Discreteparticlemodel

TFM Two-ﬂuidmodel

centrationthatwereinconsistentwiththeresultsofexperiments conductedbyDangetal.(2013).Hernández-Jiménezetal.(2013), meanwhile,obtainedresultsthatwereingoodagreementwiththe

DavidsonandHarrison(1963)modelforsingle-injectedbubbles

butfoundthatthemasstransfercoefﬁcientsweremorethantwice thosepredictedwhenusingfreelybubblingﬂuidizedbeds.

Itwillbepossibletoidentifythemostimportantaspectsofthe interphasemasstransferwithmodelsofferinggreaterdetail.While

theTFMmakesvariousassumptions todescribetherheologyof

theemulsion phase,theparticle–particle interactions aretaken intoaccountdeterministicallyinadiscreteparticlemodel(DPM,

employingaEuler–Lagrangetechnique).TheDPMmodelcan

there-foreprovidemoredetailedinsightintotheprevailingphenomena

thantheTFMmodelandallowsthesimulationofsmaller

parti-cles(e.g.,Geldart-Aparticles).Geldart-Aparticlesareoftenusedin industrialfluidizedbeds(typicallyfluidcatalyticcrackingcatalyst) andareofinterestinthedesignofmicrofluidizedbeds(e.g.,Tan,

Roghair,&vanSintAnnaland,2014,2016).

Thepresentworkusesastate-of-the-artDPMmodelextended

withgascomponentconservationequations tocharacterize the

interphasemasstransferprocessesingas–solidﬂuidizedbeds com-prisingGeldart-BandGeldart-Aparticles.Themodelwillbeused tosimulatesingleinjectedbubblesthatrisethroughanincipiently

ﬂuidizedbed,analogoustotheexperimentscarriedoutbyPatil

etal.(2003)andDangetal.(2013),butwithoutthespeciﬁc

lim-itationsinherenttotheirtechniques.Additionally,thetracergas concentrationintheemulsionphaseisnotneglectedbutanalyzed forthecomputationofthemasstransfercoefﬁcient.

Thissectioncontinueswithashortoverviewoftheavailable correlationsforthebubble-to-emulsionphasemasstransfer coefﬁ-cient,whichwillbeusedinthecomparisonwithsimulationresults.

Inthefollowingthreesections,theDPMmodelisthenoutlined

and a detailed analysisof mass transfer processesin Geldart-B

andsubsequentlyGeldart-Aparticlesisdescribed.Adiscussionand conclusionsareﬁnallypresented.

Phenomenologicalmodelsforbubble-to-emulsionmasstransfer Severalcorrelationshavebeenreportedintheliteratureforthe predictionofthemasstransfercoefﬁcients.Thederivationofthese correlationsusuallyassumesagascloudbetweenabubbleandthe emulsion(bulk)phase,originallydeemedasathinregion surround-ingthebubblewitharelativelyhighsolidsholdupcomparedwith

thebulkemulsion.Davidsonﬁrstsuggestedtheexistenceofthe

gascloudingas–solidbubblingﬂuidizedbeds(Rowe,Partridge,&

Lyall,1964).ThepioneeringmodelofDavidsonandHarrison(1963)

hasbeenwidelyusedinphenomenologicalmodelsforlarge-scale

ﬂuidizedbedreactors.Intheirmodel,thetotalmasstransfer con-sistsofaconvectiveﬂowfromthebubblestotheemulsionphase anddiffusionfromthebubblestothecloud.KuniiandLevenspiel

(1991)followedtheirapproachandproposedanextension

con-sideringtwoconsecutivetransfersteps,namelythetransferfrom thebubbletothecloudandthatfromthecloudtotheemulsion.

AccordingtothestreamfunctionderivedbyMurray(1965)and

ChibaandKobayashi(1970)assumedthatthegascompositionin

thecloudandbubbleisuniformandthatthemasstransfer limita-tionislargelygovernedbydiffusionthroughthesurfacebetween

thecloudandemulsionphases.Table1summarizestheequations

usedtoestimatethebubble-to-emulsionmasstransfercoefﬁcient

forthemostpopularphenomenologicalmodels,togetherwiththe

mainassumptionsuse

Numericalmethod

Extendeddiscreteparticlemodel

Thesoft-sphereDPMemployedinthisstudyisbasedonthe

pioneeringworkofTsuji,Kawaguchi,andTanaka(1993)andwas

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Table1

Phenomenologicalmodelsforthebubble-to-emulsionmasstransfercoefﬁcientKbe.

Reference Equations Dimension Mainassumptions

DavidsonandHarrison(1963) Kbe=

4 Db

0.6D1/2_i

g Db

1/4 +2Umf ␲

2D (1)masstransferfromthebubbletotheemulsionby moleculardiffusionandabulkﬂowtoandfromthe bubble;(2)perfectmixinginthebubbleandtheemulsion phase;(3)circularorsphericalbubblewithconstantsize; (4)bubblesinalargevolumebed;(5)diffusionfrom sphericalcapbubblesurfacewithanoseangleof100◦ Kbe=4.5

Umf Db

+5.85

D1/2_i g1/4 D5/4_b

3D

KuniiandLevenspiel(1991)

Kbc=4.5

Umf Db

+5.85

D1/2_i g1/4 D5/4 b

3D (1)Davidsonmodelforthebubbleandthegasflowatthe bubble;(2)theemulsionvoidageisobtainedatminimum fluidizingconditions;(3)Higbiepenetrationmodel (Hiegbe,1935)forthecloudtoemulsionmasstransfer coefficient;(4)twoconsecutivesteps:transferfromthe bubbletothecloudandfromthecloudtotheemulsion phase Kce=6.77

DiεmfUb D3 b

1/2 3D Kbe= 1 1 Kbc+ 1 Kce 3D

ChibaandKobayashi(1970)a Kbe=

4.52 1−fw

_D iε2_mfUb D3 b

1/2

2D (1)uniformgascompositioninthecloudandbubble;(2) constantbubblevolumeandbubblerisevelocity;(3) circularorsphericalcapbubbleandcircularorspherical clouds;(4)gasandparticleflowaroundthebubblefollow theanalysisbyMurray(1965);(5)thegasflowinthe emulsionphaseisinplugflow

Kbe= 6.78 1−fw

_D iε2mfUb D3 b

1/2 3D

a_In_the_literature,_the_Chiba_and_Kobayashi₍₁₉₇₀₎_model_is_often_cited_with_the_expression_for_the_mass_transfer_coefﬁcient_k

gintermsofthesurfaceareaofthecloud.In

thisstudy,forthesakeofconsistency,wedenotethemasstransfercoefﬁcientKbebasedontheunitarea(twodimensions)orvolume(threedimensions)ofthebubble.

(1996)andYe,VanderHoef,andKuipers(2004).Itisapopular

Euler–Lagrangemodelwithadiscretedescriptionoftheparticulate phaseandacontinuousdescriptionofthegasphase,andithasbeen

widelyusedinhydrodynamicstudiesofgas–solidﬂuidizedbeds.

Asetofthree-dimensionalvolume-averagedNavier–Stokes

equa-tionsforcompressibleﬂowaresolvedonthecellsofanEuleriangrid forthegas-phasehydrodynamics,assumingidealgasbehavior.For theparticlephase,Newton’ssecondlawofmotionisappliedtoeach individualparticletotraceitspositionandvelocitywhiletaking intoaccountparticle–particleandparticle-wallcollisions.Because theEuleriangridcellsarelargerthantheparticlediameter,the detailsoftheinteractionsbetweenthegasphaseandparticlesare unresolvedandconstitutivecorrelationsarerequiredtocompute

momentumexchangebetweenthetwophases.

Themainequationsofthemodelusedinthisstudyare summa-rizedinTable2.TheclassicdragforcecorrelationfromErgun(1952)

andWenandYu(1966)isusedforthegas–particleinteraction.The

contactforceresultingfromparticle–particleand/orparticle-wall interactionsiscalculatedusingthelinearspringanddashpotmodel proposedbyCundallandStrack(1979).Theparticlespring

stiff-nessconstantisanimportantinputparameteranditiscommon

practicetouseavaluemuchsmallerthanthetruevaluederived frommaterialproperties,becauseitallowsagreatertimestepto

beusedwithoutnoticeablyaffectingthehydrodynamicsandthus

reducestherequiredcomputationaltime(Tsujietal.,1993).Here,

wechoosevaluessuchthatthemaximumoverlapbetween

inter-actingparticlesandparticle-wallatanytimestepislessthan1% oftheparticlediameter.Foramoredetailedexplanationofthis model,wereferthereadertoourpreviouspapers(Tanetal.,2014, 2016)andreviewsbyDeen,vanSintAnnaland,VanderHoef,and

Kuipers(2007),andZhu,Zhou,Yang,andYu(2008).

The transport of chemical components is described using a

nonstationaryconvection–diffusionequationforcomponenti.For binarygassystemswithcomponentsAandB,Fick’slawcanbeused forthemoleculardiffusionmassﬂuxji=−DAB

∇

yA,usingyAto

denotethemassfractionofcomponentA,whichisemployedinthe gascomponentconservationequationusedinthisstudy(Table2). ThisnovelaspecthasbeenexplainedindetailbyTanetal.(2016).

TheDPMhasbeenextensivelyusedtoexaminethe

hydrody-namiccharacteristicsofgas–solidﬂuidizedbedsindetail(e.g.,Li

&Kuipers,2007;Tanetal.,2014;Wang,VanderHoef,&Kuipers,

2010;Xu&Yu,1997;Yeetal.,2004).Theextensionforthe

com-ponent conservation calculation has been carefully veriﬁed by

carryingoutsimulationsforsimpliﬁedsystemsandcomparingthe resultswiththeoreticalsolutions,aspresentedbyTanetal.(2016). Simulationconﬁguration

Bubble-to-emulsionphasemasstransferisinvestigatedby sim-ulatingsingle-bubbleinjectionsofCO2 tracergasintoaﬂuidized

Table2

Maingoverningequationsofthesoft-sphereDPMextendedwithgascomponent conservationequations.

Gasphasecontinuityequation: ∂(εgg)

∂t + (∇·εggug)=0 Gasphasemomentumequation:

∂(εggug)

∂t +∇· (εggugug)=−εg∇Pg−Sp−∇·(εgg)+εggg Gasphaseequationofstate:

g=MRTgPgwithMg=

yi Mi

−1 Gasphasestresstensor:

g=␮g

∇ug+∇uTg

−

␭g−23␮g

(∇·ug) I Gas–solidmomentumexchangerate:

ˇ=

⎧

⎨

⎩

3 4CD gεg(1−εg)ug−va dp ε−2.65g εg≥0.8 150(1−εg) 2 g εgd2p +1.75g(1−εg)ug−va dp εg<0.8 TheporosityinDPMsimulation:

εg,cell=1−Vcell1

∀a∈cell fa cellV a p

Equationsofmotionforeveryparticle: madv_dta =mad

2_r_a

dt2 =−Va∇Pg+Vεapˇ(ug−va)+mag+Fcontact,a Iadwdta=Ta

Gascomponentconservationequations: ∂

∂t(εggyi)+∇· (εggyiug)=∇· (εggDi∇yi) Closureequationforcomponenti=0:y0=1−

c

i=1 yi

Viscosityofagasmixture:g=

i xii

jxiij ij=

1+

i_j

1/2

_Mj Mi

1/4

2

8

1+Mi Mj

1/2

(5)

Fig.1. Snapshotsofthetracergasconcentrationinthecentralsliceoftheﬂuidizedbedatdifferentmomentsintimefromthebeginningoftheinjection(fromt=0to 0.16swithstepsof0.02s),wherethebubbleisdepictedwithdifferentthresholdvaluesoftheporosity(0.7−0.85)(bedwidth=6×10−2_m,_d_p₌_400–600_␮m,_U_inj₌₆_m/s, tinj=0.075s).

bedmaintainedatminimumﬂuidization(forbedsusingGeldart-B particles)orminimumbubblingconditions(forbedsusing Geldart-Aparticles).N2isusedasthebackgroundﬂuidizationgas.Afteran

initial1sofincipientﬂuidization,thebubbleisinjectedby set-tingthecentralcellsatthebottomboundary(i.e.,thenozzle)to theinﬂowofCO2 usingaprescribedinjectionvelocity.Thecells

areswitchedbacktoN2attheincipientﬂuidizationvelocityafter

theinjectionis ﬁnished.Theinjectiontime required for gener-atingspeciﬁcbubblesizesisdeterminedinseparatesimulations beforehand.

Aconstantmolecular(binary)diffusioncoefﬁcientofCO2inN2

isusedtodescribethetracergasdiffusivity.ForGeldart-Aparticle simulations,thediffusioncoefﬁcientofCO2isalsoincreasedbya

factorof3toinvestigatetheeffectofgasdiffusivity.Thenozzlefor gasinjectionisatthecenterofthebottomplate.

ThesimulationsusingGeldart-Bparticlesaresetupaccording totheconditionsofexperimentsconductedbyDangetal.(2013). Thebedwidthtakesvaluesof4cm(matchingthebedwidthinthe experiments),5and6cm.Similartothecaseintheexperiments, Geldart-Bparticleshaveadensityof2525kg/m3_and_average

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Fig.2.Snapshotsofthestreamlinesthroughthebubbleinthecentralsliceofaﬂuidizedbedfrom0.06to0.12safterthetracergasisinjected(bedwidth=6×10−2_m,

dp=400–600␮m,Uinj=6m/s, tinj=0.075s).

with=5.0×10−5m).Theminimumﬂuidizationvelocityofthese

particlesis determined bysimulations using thepressure drop

methodas0.22m/s(0.206m/s inthereferencedpaper).No-slip

boundaryconditionsareappliedtothesidewallsofthebedinthese simulations.

The Geldart-A particles employed in the present study are

monodispersedparticleswithadiameterof100␮manddensity

of1500kg/m3_._Van_der_Waals_forces_are_neglected_in_this_study.

Heretheminimumbubblingﬂuidizationvelocity(9.0×10−3m/s), determinedinsimulationsaccordingtothestandarddeviationof thepressuredropoverthebed,isusedasthebackground ﬂuidiza-tionvelocity.Becausemanyparticlesareusedtoallowthebubble

somerisingtime,pseudo-two-dimensional(2D)ﬂuidizedbedsare

simulatedtoreducethecomputationalcost.Thebeddepthisonly 6timestheparticlediameterandafree-slipboundaryconditionis thusappliedtothefrontandbackwalls.

Forallsimulations,thepressureatthetopoutletisspecifiedas theatmosphericpressure(101,325Pa).Particlesareinitiallyplaced regularlylayerbylayeratthebottomofthebedwithsmall ran-domfluctuating translationalandrotational velocitiestoensure thatthesystemisinanasymmetricfluidizedstatefromthestart, andthesystemisleftatminimumfluidizationconditionsfor1s.

Table3summarizesthemainparametersusedinthesimulations.

TheparametersintheDPMmodelforGeldart-Bparticles,suchas therestitutionandfrictioncoefﬁcients,wereﬁrsttestedwith sim-ulationsettingsclosesttotheexperimentalsettingsusedbyDang

etal.(2013).Goodagreementwasfoundwithregardtothemass

transferphenomena(seethesnapshotsshowninFig.1),andthe bubblesizeasafunctionoftimecompareswellwiththeresultsof theexperiments.Notethatdespitethegreatsimilarity,the simula-tionscannotbeusedforanexactone-to-onecomparisonwiththe experiments.Theanalysisandresultingmasstransfercoefﬁcient aresensitivetothegasinjectionandbubbleformationtime,and

Dangetal.(2013)usedahighinjectionvelocityandstoppedthe

injectionbeforethebubbleformationwascomplete.Additionally, theiranalysisforthebubble-to-emulsionmasstransfercoefﬁcients

startedbeforethebubblehadcompletelyformedanddetached.

Theseaspectscanbecontrolledmuchmoreaccuratelyinthe sim-ulationsandhencedifferfromthoseintheexperiments.

Calculationofthemasstransfercoefﬁcient

ThemasstransfercoefﬁcientKbecanbecalculatedfroman

inte-gralmassbalanceofthetracergasinthebubble(deﬁnedasan

enclosedregionwithporosityexceedingapredeﬁnedthreshold): d

CCO2,bVb

dt =−Kbe

CCO2,b−CCO2,e

Vb. (1)

ToanalyticallysolveEq.(1),theaverageconcentrationofthe tracergas(inthiscaseCO2)intheemulsionphaseisassumed

neg-ligible(andthus settozero). Wekeep thebubblevolume(and

thusdiameter)constantbyintegratingoverashorttimeduring

whichthebubblediameterisrelativelyconstant.Weusethe time-averageddiameterforfurtheranalysis.AspresentedbyDangetal.

(2013),forapseudo-2Dbedhavingasinglebubblewithaninitial

averagedtracergasconcentrationofCCO2,b(0),andassuminga neg-ligibleconcentrationintheemulsionphase(CCO2,e=0),integration yields CCO2,b(t) CCO2,b(0)

_D b(t) Db(0)

2 =exp (−Kbet) , (2)

whereDb(0) istheinitialbubblediameterandDb(t) isthebubble

diameterattimet.Byplottingln

CCO2,b(t) D

2

b(t) /CCO2,b(0) D

2 b(0)

asafunctionoftime,themasstransfercoefﬁcientKbecanbe

cal-culatedfromtheslopeofthelinearcorrelationaccordingtoEq.

(7)

Table3

SummaryofthemainsimulationparametersintheDPM.

Geldart-Btype Geldart-Atype

Particlediameter,dp(␮m) 400–600 100

Particlenumber,Np 72500 279000

Particledensity,g(kg/m3) 2525 1500

Normalrestitutioncoefﬁcients,en 0.97 0.95

Tangentialrestitutioncoefﬁcients,et 0.33 0.95

Frictioncoefﬁcient(particle–particle/wall),f 0.1 0.3

Normalspringstiffness,kn(N/m) 3500 28

Tangentialspringstiffness,kt(N/m) 1000 8

CFDtimestep,t(s) 1.0×10−5 _1.0_×₁₀−5

Particledynamicstimestep,tp(s) 1.0×10−6 1.0×10−6

Domainheight(m) 1.5×10−1 _3.6_×₁₀−2

Domainwidth(m) 4.0,5.0,6.0×10−2 2.32/3.0×10−2

Domaindepth(m) 5.0×10−3 6.0×10−4

Numberofgridcells 150×40/50/60×5 90×58/75×2

DiffusivityofCO2,Ddiff(m2/s) 1.65×10−5 1.65/4.95×10−5

Injectionvelocity,Uinj(m/s) 5.0,6.0 0.1,0.2

Injectiontime, tinj(s) 0.07,0.075 0.035,0.036

Nozzlearea(mm2₎ ₂₀₍₄_×₅_grid_cells) _0.96₍₄_×₂_grid_cells)

Minimumﬂuidizationvelocity,Umf(m/s) 0.22 5.9×10−3

Bubblingﬂuidizationvelocity,Umb(m/s) – 9.0×10−3

FluidizedbedscomprisingGeldart-Bparticles Characteristicsofbubble-to-emulsionphasemasstransfer

Thetracergasconcentrationinthecentralsliceofthefluidized bedis shown inFig. 1for thesimulation usinga bed widthof 6×10−2mandaninjectionvelocityof6m/s.Whiletheactualvalue issomewhatarbitrary,aporosityof0.85isusedtodefinethebubble contour.Inthefigure,arangeofporositycontoursforthebubble

isprovidedtoshowthepronouncedphenomenonofparticle

rain-ingthatleadstobubblecollapse.Thebubbleshapeandbubblesize continuetochangeovertimeasfoundbyPatiletal.(2003)and

Dangetal.(2013).Thesameholdsforthetracergasconcentration

insidethebubble.

ThreestagescanbediscernedfromthesnapshotsinFig.1.First, itisnotedthatthebubblestartstoformshortlyaftertheinjection andthetracergasappearsatthenozzleevenbeforethebubbleis distinguishable.Thegasﬂowtrajectories,showninFig.2,indicate thatboththeCO2tracergas(viathemiddle)andtheN2background

ﬂuidizationgas(viathebubblesides)contributetotheformationof thebubble,whiletheCO2tracergasimmediatelyﬂowsthroughthe

topofthebubbleintotheemulsionphase.Afterthebubbleforms, thesecondstagestarts,whereN2backgroundﬂuidizationgasﬂows

intothebubblealsoviathebottomandreplacestheCO2that

con-tinuestoleavethebubblethroughitsroof.Themasstransferisthus stronglydominatedbyconvectioninthiscaseofGeldart-B parti-cles,asreportedbyDangetal.(2013).AsmallpartoftheCO2that

leavestheemulsionphaseisrecycledthroughtheleftandright vortices,backintothebubble(seeFig.2).Duringthelaststage,the lasttracesofCO2 thataretrappedinthevorticescanonlyleave

throughdiffusion,which isaslowprocesssuchthatthebubble

concentrationremainsfairlyconstantwhilethebubblecollapses, asalsoshownbytheexperimentalresultsofDangetal.(2013).

Fig.3(a)shows theaverageCO2 concentrationinthebubble

andemulsionphases,wherethethreestagesdescribedabovecan

againbedistinguished.Theemulsionphaseconcentrationis non-zeroandattimesevenlargerthanthebubbleconcentration,but

becausethe majority of CO2 is founddownstream of the

bub-ble(andisincapableofre-enteringthebubble),theassumption ofanegligibleemulsionphaseCO2concentrationtodescribethe

bubble-to-emulsionphase masstransferseemsjustiﬁed.Fig.3b

showstheequivalentbubblediameterDb=

4Ab/␲andreveals

thatthebubblestartstocollapseshortlyafteraninitialgrowth,

whichisconsistentwithexperimentalresultsobtainedbyDang

Fig.3. (a)AverageCO2concentrationintheemulsionphaseandinthebubblewith

itsspatialstandarddeviation(shadedbar)asafunctionoftime;(b)equivalent bub-blediameterasafunctionoftimeaftertheinjectionstarts(bedwidth=6×10−2_m,

dp=400–600␮m,Uinj=6m/s, tinj=0.075s).

et al. (2013). However, Patil et al. (2003) failed to predict the

decreasingbubblesizeatthelater stagein 2DTFMsimulations

andpresentedaslowincreaseintheequivalentbubblediameter, explainingthediscrepancydescribedabove.

(8)

Fig.4.Linearcorrelationsofthebubble-to-emulsionmasstransfercoefﬁcientsfordifferentcasesusingparticlesof400–600␮mindiameter:(a)bedwidth=5×10−2_m,

Uinj=5m/s, tinj=0.07sandDb=0.0231m;(b)bedwidth=6×10−2m,Uinj=6m/s, tinj=0.075sandDb=0.0277m.

Fig.5.SnapshotsoftherisingbubbleinGeldart-Aparticlebedsatdifferenttimesteps.Atthetop,theCO2concentrationinthecentralsliceisshown.Thebubbleisplotted

byvaryingthethresholdvalueofporosityfrom0.8to0.85.Atthebottom,thestreamlinesoftherelativegasphaseinthebubbleandtheemulsionphasesareshown.From topandbottom:thestreamlinesinthelefthalfofthebubblearecoloredusingtherelativegasvelocityinthezdirectionwhilethoseintherighthalfofthebubblearecolored withtheCO2concentration.(Bedwidth=2.32×10−2m,dp=100␮m,Uinj=0.1m/s,Ddiff=1.65×10−5m2/s, tinj=0.035s).

Table4

Comparisonofthesimulatedbubble-to-emulsionmasstransfercoefﬁcientswithpredictionsmadeusingtheDavidsonandHarrison(1963)model,includingtheindividual contributionsofdiffusion-andconvection-basedmasstransfers.

Cases Averagedbubblediameter(m) MasstransfercoefﬁcientKbe(s−1)

DavidsonandHarrisonmodel(1963) Thisstudy Difference

Bedwidth=5cm,Uinj=5m/s 0.0231 27.08(Conv24.90+Diff2.18) 28.89 6.69%

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Fig.6. (a)Equivalentbubblediameterasafunctionoftimeafterthestartofinjection withthetracergaseshavingdiffusivitiesdifferingbyafactorof3and(b)theaverage CO2concentrationinthebubblesanditsspatialstandarddeviation(shadedbar)as

afunctionoftime,insingle-bubbleﬂuidizedbedscomprisingGeldart-Aparticles (dp=100␮m,p=1500kg/m3,Ddiff=1.65×10−5m2/s, tinj=0.035and0.036s).

Coefﬁcientofbubble-to-emulsionphasemasstransfer

Theprevioussectionhasjustiﬁedtheassumptionofa

negli-gibleemulsionCO2concentration,Eq.(2)canthusbeemployed

tocalculateKbe.Toexcludewalleffectsasmuchaspossible,the

evolutionofthegasstreamlinesisstudiedtoﬁndthetimethat

elapsesbeforewalleffectsbecomedominant.Subsequently,Kbe

isobtainedfromthelinearizedEq.(2),usingtwodifferentbubble sizes(Fig.4).Table4presentsthecomparisonofKbebetweenthe

simulationsandpredictionsmadewiththeDavidsonandHarrison

(1963)model.

In Table4, thedifferences between thevalues calculated in

thesimulationsandthepredictionsmadewiththeDavidsonand

Harrison(1963)modelarelessthan10%,showinggoodagreement.

TheshorttimeusedtocalculateKbeallowstherelevant

assump-tions,suchasaconstantbubblediameter,aconstantmasstransfer coefﬁcientandzerotracergasintheemulsionphase,tobemade.

Table4alsoshowsthattheconvectivecontributionisindeed

dom-Fig.7. AverageCO2concentrationinthebubbleandemulsionphasesfor

single-bubbleﬂuidizedbedscomprisingGeldart-Aparticlesandthetracergaseswith diffusivitiesdifferingbyafactorof3(dp=100␮m,p=1500kg/m3,Uinj=0.1m/s,

tinj=0.035s).

inantandthatalargervalueofKbeisobtainedforasmallerbubble,

whichisconsistentwiththeresultspresentedaboveandwiththe resultsofpreviousstudies(Patiletal.,2003;Dangetal.,2013).

FluidizedbedscomprisingGeldart-Aparticles

TheDPMtechniqueisalsosuitedtosimulatingGeldart-A parti-cles,whichareusedinmanyindustrialﬂuidizationprocesses.This sectionusesparticleswithadiameterof100␮mandadensityof 1500kg/m3_._Single_bubble_injections,_similar_to_those_in_the

pre-viouscases,areperformedusingaporosityof0.85todeﬁnethe bubblecontour.Thebubblesrisesteadilythroughthebeduntilthey breakupatthebedsurface.

Characteristicsofbubble-to-emulsionphasemasstransfer

Fig.5showstheconcentrationﬁeldandgasphasestreamlines forasimulationusinganinjectionvelocityof0.1m/s.TheCO2

con-centrationinthemiddleofthebubbleisrelativelyhigh.TheCO2

transferredtotheemulsionphaseisfoundmainlyalongthepath behindthebubble.Thisconcentrationproﬁleissimilartothe pro-ﬁleofgasbubblesrisinginaliquid(e.g.,Stöhr,Schanze,&Khalili, 2009)withtherebeingatailbehindthebubble.

Thesnapshotsindicateamasstransferprocessdifferentfrom

that whenusing Geldart-B particles.Thegas phase streamlines

recirculateinsidethebubbleandintheimmediatesurroundings

ofthebubble(i.e.,thecloud),keepingthetracergaslocaltothe bubble.Additionally,thereisnosignofbubblecollapseinthe snap-shots.ThesecharacteristicsarehighlightedinFig.6,showingthat thebubbleevencontinuestogrowwhilerisingthroughthebed.A simulationwith3timesthetracerdiffusivitybutotherwise iden-ticalsettingsshowsthatthediffusivityofthegashasanegligible effectonthebubblesize.TheaverageCO2concentrationinthe

bub-blecontinuestodecreasewithtimewhileitremainsmoreorless uniformlydistributed.Moreover,asshowninFig.7,overtime,the

averageCO2 concentrationintheemulsionphaseismuchlower

thantheaverageCO2 concentrationinthebubble,andthemass

transferrateincreaseswithincreaseddiffusivity.

Thebubblesaresurroundedbycloudsaccordingtothecriteria foracloudedbubblepresentedbyDavidsonandHarrison(1963).

Fig.5showsthatthegasﬂowingoutofthebubbleviaitsroofﬂows

downwardsalongthebubbleedgeandcirculatesbackthroughthe

cloudintothebubble.Themaindifferencesbetweenthecalculated

(10)

Fig.8.(a)GasandparticleﬂowpatternsnearabubblebasedonexperimentsandMurray’stheory(Roweetal.,1964)( 2.5).(ReprintedfromRoweetal.(1964)with permissionfromElsevier.)(b)ThesymbolizedmodelusedbyChibaandKobayashi(1970)(ReprintedfromChibaandKobayashi(1970)withpermissionfromElsevier).

cloudedbubbleinDavidsonandHarrison(1963)arethebubble

shapeandthegasflowprofilesinthewakeofthebubble.Thebubble isassumedtobeideallysphericalandnowakeisdefinedintheory whereastheoppositeisseeninsimulations.

Coefﬁcientofbubble-to-emulsionphasemasstransfer

ThestreamlinesshowninFig.5aresimilartopredictionsbased

onthetheoryofMurray(1965)andusedbyChibaandKobayashi

(1970)toprovidea correlationforthemasstransfercoefﬁcient

(Fig.8).

Eq.(1)isusedtocalculateKbe,basedontheaverageCO2

con-centrationthroughouttheemulsionphase,asafunctionoftime.

Fig.9showsthatwhilethebubblerises,Kbedecreaseswithtimeas

aresultofbubblegrowth(Fig.6(a)).Togetherwiththeaveraged

bubblediameter,thetime-averaged Kbe is presentedin Table5

(inthe“bulkemulsion”column)andcomparedwithpredictions

madeusingclassicphenomenologicalmodels:D&H(Davidson&

Harrison,1963),K&L(Kunii&Levenspiel,1991),andC&K(Chiba&

Kobayashi,1970).ThelargedifferencesinthepredictionsofKbeare

relatedtothedifferencesintheunderlyingassumptions.Notethat thebubblerisingvelocityUbcalculatedfromsimulationresultshas

beenusedinthecomputationofKbewiththesephenomenological

models.

Despitethesimilaritybetweenthegasstreamlines(Fig.5)and thetheoryofMurray(1965)(Fig.8)usedintheChibaandKobayashi

(1970)model,theC&K modeldoesnot givethebestprediction

ofKbe for thesesimulations.Thisis mostprobablybecausethe

phenomenaobservedinthesimulationsarenotconsistentwith

theassumptionsfortheprocessusedinthederivationoftheC&K model,suchasaconstantbubblevolume(andhencecloudvolume) andthesphericalshapeandsizeofthecloud.

TheeffectofbubblediameterontheaverageKbe isanalyzed

ﬁrst.Thechangeintheamountoftracergasinthebubbleconsists oftwoparts:acontributionduetothechangeinbubblesizeanda contributionduetothechangeinCO2concentrationinthebubble.

Fig.9.Bubble-to-emulsionphasemasstransfercoefﬁcientasafunctionoftime cal-culatedfromthesimulationsofasingleinjectedbubbleinaﬂuidizedbedcomprising Geldart-Aparticles(dp=100␮m,p=1500kg/m3,Uinj=0.1and0.2m/s, tinj=0.035

and0.036s,Ddiff=1.65×10−5m2/s).

Theanalysisisthuscarriedoutforthesetwotermsseparately.It turnsoutthattheﬁrsttermcontributesover60%ofthetotalchange intheamountofCO2inthebubble.

Eq.(1)computesthetotalmasstransferbasedonthemass trans-fercoefﬁcientandthedrivingforce,deﬁnedastheconcentration differencebetweenthebubbleandtheemulsionphaseintherestof thebed.AsshowninFig.5,however,amoreaccuratedrivingforce

canbetheconcentrationdifferencebetweenthebubbleandthe

emulsionphasearoundthebubble(orthecloud).Here,the emul-sionaroundthebubblewithaCO2concentrationhigherthan62.5%

and73%(correspondingtodifferentcloudsizes)oftheaverageCO2

(11)

Table5

ComparisonofmasstransfercoefﬁcientsKbecomputedfromthesimulationresultsandfromphenomenologicalmodels.

Uinj(m/s) Db(m) Kbe(s−1)(predictions) Kbe(s−1)(simulations)

D&H K&L C&K Bulkemulsion 62.5%Cb 73%Cb Diffusioncoefﬁcient=1.65×10−5 0.1 0.003 27.37 11.85 16.93 2.07 10.61 15.36 0.2 0.004 18.64 7.74 10.61 1.63 9.30 13.55 Diffusioncoefﬁcient=4.95×10−5 0.1 0.003 42.34 19.12 28.11 4.33 20.48 29.16 0.2 0.004 28.98 12.66 17.88 3.47 14.42 19.60

concentrationhasbeenusedtocalculateKbe.Thisgivesan

emul-sionphaseconcentrationthatismuchmorelocalthantheaverage overtheentirebed.InTable5,thevaluesofKbeincolumnslabelled

62.5%Cband73%CbcomputedusingtheCO2concentrationinthe

surroundingsofthebubbleforthedrivingforceareinmuchbetter

agreementwiththepredictionsmadeusingthemodelsofKunii

andLevenspiel(1991)andChibaandKobayashi(1970).

Discussionandconclusions

Discussion

Thesimulationsnapshotsandanalysisshowthatthemass trans-ferprocessisgreatlyaffectedbythegasﬂowpatternatandaround thebubble.AslearnedfromDavidsonandHarrison(1963),thegas ﬂowpatternatthebubbleisdeterminedbytheratioofthebubble risingvelocitytotheinterstitialgasvelocity(Ub/u0).Forslowly

ris-ingbubbles(Ub/u0<1)insingle-bubbleﬂuidizedbedscomprising

Geldart-Bparticles,themasstransferisdeterminedbythe

convec-tiveﬂowsfromtheemulsiontothebubbleandfromthebubble

backtotheemulsion.Thesimulationsinthepresentworkshow

thatthetracergasinthebubbletransfersintotheemulsionphase ﬁrstbythe(convective)depletionofthecenterofthebubbleand thenbydiffusionviathevorticesattheleftandrightsidesofthe

bubblebeforethebubblecollapses.Theemulsionphase

concen-trationcanbeassumedtobezerobecausetheinjectedtracergas escapesthroughthebubble.Animportantissueremains,however, intermsofhowtheupstreamconcentrationprofilecanbe incorpo-ratedinthecorrelations,especiallyiftheprofileisnotuniform.Any tracergasthatexistsupstream(i.e.,belowthebubble)willlikely traveltowardsthebubble,enterviathebottomandleavethrough theroof.Thisaspectofmasstransferisnotcapturedinexisting correlationsbutiscrucialforfreelybubblingfluidizedbeds.

Forquickly risingbubbles(Ub/u0∼5)inﬂuidized beds

com-prisingGeldart-Aparticles,thephenomenaobservedinthemass

transferprocessaredifferentfromthoseintheGeldart-Bparticle

simulations.Noobviousvolumetricﬂowratebetweenthebubble

andtheemulsionbulkisobserved,andratherastronggas circu-lationinthebubbleandthelocalemulsionphase (i.e.,cloud)is distinguished.Someofthetracergasisleftbehindviathewakeat thebottomofthebubble,butalsoalargepartofthetracergasis trappedinthebubbleandisreleasedonlywhenthebubblebreaks upatthebedsurface.Themasstransfercoefﬁcientscomputedwith thetheoreticalmodelsdonotagreewiththoseobtainedfromthe simulations,unlessalocalemulsionphaseconcentrationistaken intoaccount.Whilethelocalemulsionphaseisclearlythemost

inﬂuentialzoneandusingalocalemulsionphaseconcentration

canbejustiﬁed,itisimportanttoacknowledgethatthetheoretical modelsaretypicallyusedforfreelybubblingﬂuidizedbeds. Bub-blebreakupandcoalescencecangreatlyincreasethemasstransfer

betweenthebubbleandemulsionphases.Thismaygiveagood

additionalexplanationonwhyvaluesofK_becalculatedfromthe simulationresultsaremuchlowerthananyofthepredictionsmade usingthephenomenologicalmodelsderivedforfreelybubbling

ﬂu-idizedbeds.Thescopeofthisstudy,however,remainsassingle risingbubbles.

Theusedcorrelations incorporatethediffusioncoefﬁcientby includingD0.5

i inthediffusiveterms.Increasingthediffusion

coef-ﬁcientbyafactor3wouldthereforegiveapredictionofanincrease inthemasstransfercoefﬁcientbyafactorof30.5₌_1.73;_however,

thesimulationstypicallyyieldamasstransfercoefﬁcientthatis largerbyafactorof2,showingthattheeffectofthediffusiveﬂux ismoreimportantthananticipatedbythecorrelations.

Conclusions

ADPMextendedwithgascomponentconservationequations

wasusedtoinvestigatethebubble-to-emulsionphasemass

trans-ferby carrying out simulationsonsingle-bubble ﬂuidized beds

comprisingGeldart-Aand-Bparticles.Newinsightsonthespatial distributionandtemporalevolutionofthetracergasconcentration

inthebubbleandemulsionweredeveloped.Detailedinformation

onchangesinthebubblesize,tracergasconcentrationandgasﬂow

atthebubblewereusedtoaccuratelyanalyzethemasstransfer

characteristicsandtocalculatethemasstransfercoefﬁcient.

This work challenged the validity of the assumptions used

inpopularphenomenologicalmodelsforthebubble-to-emulsion

masstransfer coefﬁcient.For single-bubbleﬂuidized beds

com-prising Geldart-B particles,the bubble-to-emulsion phase mass

transfer process is dominated byconvective gasﬂow from the

emulsionphasetothebubblephaseandthenbacktothe

emul-sionphase.Nouniformgasconcentrationinthebubbleorconstant bubblesizeandshapecanbeassumedforthesecases.

TheDavidsonandHarrison(1963)modelassumesaconstant

volumetricﬂow-rateintothebubblefromtheemulsionbulkand

out of the bubble into the emulsion bulk. Because convective

transferis dominant(byfar)for Geldart-Bparticles,this model reasonablypredictsmasstransfercoefﬁcientsfortheseparticles, providedthataconstantbubblesizeandzerotracergas

concentra-tionintheemulsionphasecanbeassumed.Theﬁrstassumption

isassuredbytakingashorttimeinterval(whichkeepsthesize relativelyconstant),andtheconcentrationintheemulsionphase

canbeassumedtobezerobecausethetracergasconcentration

profileextendsonlydownstreamofthebubble.Infreelybubbling fluidizedbeds,however,concentrationprofilesmayexistupstream ofabubbleandshouldbetakenintoaccount.

Thebubblescannotbeassumedtobeconstantinsizeorhavea circular(orspherical)shapeinthecaseofGeldart-Aparticles,and itwasshownthatthebubble-to-emulsionmasstransfercoefﬁcient

changesovertime.Thesefindingsconfirmandextendthefindings

ofpreviousexperimentalandnumericalstudies.Notalltracergasis ultimatelyexchangedwiththeemulsioninsidethebeds,withsome beingreleasedatthebedsurfacewhenthebubblesbreakup.Hence, aproperdeﬁnitionofthecloudaroundthebubbleandtheuseof theaveragetracergasconcentrationinthecloudarenecessaryto calculatethemasstransfercoefﬁcient.Inthiscase,theKuniiand

Levenspiel(1991)andChibaandKobayashi(1970)modelsgivethe

(12)

hasadistinctiveeffectonthebubble-to-emulsionmasstransferin thebeds,whichisunderestimatedbythemodels.

Acknowledgment

ThisprojectisﬁnanciallysupportedbytheNetherlands

Orga-nizationforScientiﬁcResearch(NWO)underSTWVIDIGrantNo.

10244.

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