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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
Simulation of spray coating in a spouted bed using recurrence CFD
Paul Kieckhefen
a,∗, Thomas Lichtenegger
b,c, Swantje Pietsch
a, Stefan Pirker
b, Stefan Heinrich
aaInstituteofSolidsProcessEngineeringandParticleTechnology,HamburgUniversityofTechnology,Hamburg,Germany
bDepartmentforParticulateFlowModelling,JohannesKeplerUniversity,Linz,Austria
cLinzInstituteofTechnology(LIT),JohannesKeplerUniversity,Linz,Austria
a r t i c l e i n f o
Articlehistory:
Received1November2017
Receivedinrevisedform5January2018 Accepted16January2018
Availableonline25June2018
Keywords:
CFD–DEM RecurrenceCFD Spraycoating Time-scaledecoupling Spoutedbed Draftplate
a b s t r a c t
Although numerical models such as the computational fluid dynamics–discrete element method (CFD–DEM)haveenabledtheaccuratesimulationoflaboratory-scaleapparatuses,theapplicationof thesemethodstolarge-scaleapparatuseswithmanyparticlesandtimescalesrangingfromminutesto hoursremainsachallenge.TherecentlydevelopedrecurrenceCFD(rCFD)methodseekstoovercome theseissuesinpseudo-periodicprocessesbyextrapolatinggloballyrecurringpatternsinaphysically meaningfulwayanddescribingthetransportandinteractionofpassivescalarsusingLagrangiantracers.
Spoutedbedsrepresentaninterestingtargetbecauseoftheassociatedvarietyofflowregimes.Theycan beeffectivelydescribedbyCFD–DEMonthetimescaleoftensofseconds,whereasindustriallyrelevant processestypicallytakehours.Inthiscontribution,weestablishedthevalidityofapplyingtheLagrangian rCFDmethodtospoutedbedsbydemonstratingtheaccuratereproductionoftheparticleresidencetime distributioninafictitioussprayzone.Thedepositionofspraydropletsontotracerparticleswassimulated for1h,andtheparticlesurfacecoveragedistributionwasestimatedusingastatisticalapproachforboth anunstabilizedprismaticspoutedbedandonestabilizedbydraftplates.
©2018ChineseSocietyofParticuologyandInstituteofProcessEngineering,ChineseAcademyof Sciences.PublishedbyElsevierB.V.Allrightsreserved.
Introduction
Spouted bed technology dates back to Mathur and Gishler (1955),whofirstappliedittotheproblemofdryingwheatgrains thatweretoocoarseandasphericaltobetreatedinregularfluidized beds.Today,thetechnologyhasexpandedtoregularuseindiverse areasincludingchemicalvapourdepositiononfinemetalpowders (Caussat,Juarez,&Vahlas,2006)andhigh-densitynuclearfuelpel- lets(Liuetal.,2017;Mollicketal.,2015;Marshall,2017), spray coatingofvariousmaterialssuchasfertilizers(daRosa&dosSantos Rocha,2010),coatingofaerogelparticles(Antonyuk,Heinrich,&
Smirnova,2012;Plawsky,Littman,&Paccione,2010),spraygranu- lationofcompositematerials(Wolff,Salikov,Antonyuk,Heinrich,
&Schneider,2014;Eichner,Salikov,Bassen,Heinrich,&Schneider, 2017),andcombustion/pyrolysis(Alvarez,Amutio,Lopez,Bilbao,
&Olazar,2015;Ochoaetal.,2017;Alvarezetal.,2017).
Althoughtheseprocessesarecommonlythedomainoffluidized beds,spoutedbedsofferhigherstabilitywhenusingdifficult-to-
∗Correspondingauthor.
E-mailaddress:paul.kieckhefen@tuhh.de(P.Kieckhefen).
fluidizematerialssuchasGeldartDparticlesorasphericalparticles suchasgrains.Thisrobustnessisattributedtotheuniquebeddistri- butionandcircularflowstructurewithintheapparatus.Thegasis introducedinthemiddleoftheapparatusandacceleratesparticles verticallyfromthesurroundingparticlebulk,creatingthetitular spout.Asthegasjetdiffusesabovethesurfaceoftheparticlebulk, entrainedparticlesdecelerateandfalldownontothebulk,whichis commonlycalledtheannulus.Theannulusrestsonslantedwalls, causingtheparticlestoflowtowardsthebulk.Thisexplanationis idealistic,astheflowpatternofrealapparatusesdeviatesfromthis patterndependingontheoperatingregime.Theapparatusescanbe constructedeitherasymmetrically,axisymmetrically/conically,or prismatically/slot-rectangularly(Piskova&Mörl,2008).Thefirst spoutedbedswereconical designsand weretypicallyoperated usinghighbedheights.Theprismatictypehasfoundcommercial adoption(Jacob,2009)becauseoftheadvantageofeasyscale-up byextrusionoftheprocesschamberandiscommonlyoperatedat shallowbedlevelstoensureintensemixingandphaseinteraction.
Fortheseprismaticspoutedbeds,researchistypicallyconducted usingpseudo-2Dreplicaofcommercialapparatusestoreducesys- temcomplexityandenablecomputationalmodelling.
https://doi.org/10.1016/j.partic.2018.01.008
1674-2001/©2018ChineseSocietyofParticuologyandInstituteofProcessEngineering,ChineseAcademyofSciences.PublishedbyElsevierB.V.Allrightsreserved.
Nomenclature Greeksymbols
ı˛ Excessphasevolumefraction
˛ Phasevolumefraction
Efficiency
ε Poisson’sratio Arbitraryfield Filtercoefficient
Kinematicviscosity,m2/s ϕ Surfacefraction
Density,kg/m3 Residencetime,s
Deviatoricstresstensor,N/m2 ω Angularvelocity,s−1
Latinsymbols
A Modelparameter(Kolakalurimodel) d Diameter,m
D0 Diffusioncoefficient,m2/s
f Fractionofparticlesurfacecoatedbyasingledroplet F Force,N
g Gravitationalacceleration,m/s2 J Momentofinertia,kg/m2 m Masspervolumekg/m3
M Mass,kg
N Number
p Pressure,Pa n Normalvector
t Time,s
t Timestep,s Symbol
T Torque,Nm u Phasevelocity,m/s V Volume,m3 x Width/length,m x Coordinate,m Y Young’smodulus,Pa Dimensionlessnumbers Co Courantnumber Re Reynoldsnumber St Stokesnumber Subscripts/superscripts coll Collected dep Deposition
D Droplet/sprayphase eff Effective
fluc Fluctuation fr Friction
G Gasphase
min Minimum
n Normal
P Particlephase
rec Recurrencefield/quantity rfr Rollingfriction
t Tangential W Apparatuswall Abbreviations
CFD Computationalfluiddynamics DEM Discreteelementmethod rCFD RecurrenceCFD
Gryczkaet al.(2008)characterizedthepneumatic behaviour of one suchprismatic pseudo-2D apparatus, and Salikov et al.
(2015)createdaregimemapforGeldartDparticles.Theyobserved thatwithincreasinggasvelocity,thepressuredropincreaseswith bubblingoccurringatacertainpoint.Attheminimumspouting velocity,thepressuredropdecreasesandspoutingisinitiated.With increasinggasflowrate,thebedexpandsfurtherintotheprocess chamberandinstabilitiessuchaslateralspoutdeflectionsoccur.
Thepressuredroposcillatesregularlyinthedensespoutingregime and becomes moreirregular withincreasinggas velocitywhile the fluctuation intensity decreases and the primary frequency increases.Theseoscillationsarecausedbythespout–annulusinter- actions,astheannulusflowingintothespoutregioncausesparticle accelerationaswellasaninitialincreaseinthepressuredropfol- lowedbyasubsequentdecreaseastheparticlesareclearedfrom theregion. Instabilitiesarise because ofthe higher bedexpan- sions, astheseequatetoa lowerannulusheightwithlessbulk solidloadstabilizingthespout,preventingself-amplifyingdeflec- tionandlateralbeddistributionasymmetry.Atveryhighflowrates, thepressuredropfluctuationsdisappearbecauseofthehomoge- neousdistributionofthebedintheprocesschamberandthelack ofanannulustocausefluctuations.Thisflowregimeiscalleddilute spouting.
Spoutedbedsarewellsuitedforspraycoatingbecauseofthe intenseheatandmasstransferbetweenthegasandparticlephase (Kucharski&Kmiec,1983).DeOliveira,Freire,andCoury(1997) usedacylindrical–conicalspoutedbedtocoataluminaparticles withasucrosesolutionwithhighhomogeneityandobservedthat thehydrodynamic operatingregime had asubstantial effecton particlegrowth.
Most previous works have applied the fully Eulerian two- fluidmodel(TFM)ortheEulerian–Lagrangiancomputationalfluid dynamics–discreteelementmethod(CFD–DEM)tospoutedbeds.
Bothmethodshaveadvantagesanddisadvantagesdependingon thecontext.TheTFMusesthekinetictheoryofgranularflow,as describedbyLun,Savage,Jeffrey,andChepurniy(1984),andmodels granularmotioninanEulerianframeofreferenceusingthecon- ceptofgranulartemperatureandclosuresforfrictionalandnormal stresses.Thisapproachenablestheefficienttreatmentofsystems encompassingbillionsofveryfineparticles,asthecomputational demandscaleswiththenumberofgridcellsinsteadofthenum- berofindividualparticles.Thereareconstraintsonthecellsizeas coarsegridsmaynotresolveallrelevantflowstructuresandthus introduceinaccuracies (Schneiderbauer&Pirker,2014).Gryczka etal.(2009)andJacob(2009)conductedvariousTFMstudieson apseudo-2DspoutedbedusingFluentandwereunsuccessfulin reproducingthepressuredropfluctuations.
UnresolvedCFD–DEM,aselaboratedintheworkofZhu,Zhou, Yang,andYu(2007),usesanEuleriandescriptionofthefluidflow andtracksthesolidphaseusingcomputationalparcels,whichrep- resentthemotionofindividualphysicalparticles,ormultitudesfor coarse-graining(Bierwisch,Kraft,Riedel,andMoseler,2009).Par- celcontactsarecommonlyresolvedusingasoft-sphereapproach, inwhichparcelsareallowedtooverlapandtimestepsontheorder of10−6sarerequiredbythecontactmodel.Volumefractionsare mappedtoanEuleriangrid,wheretheflowissolved.Dragclo- suresplaceaconstraintonthecellsize,astheymodeltheeffectof unresolvedflowfeaturessuchasswarmingtoacertainextent.A comprehensivetreatmentofthegridsensitivityofCFD–DEMcan befoundintheworkofRadlandSundaresan(2014).Themaincom- putationaldemandliesintheresolutionofinterparticlecontacts.
Althoughthisapproachallowsfortheaccuratedepictionofbulk solidmotion, industrialandevenpilot-scalesystemscommonly containtoomanyparcelsforsimulationwithouttheapplicationof coarsegraining(Nasato,Goniva,Pirker,&Kloss,2015).Salikovetal.
(2015) conducted CFD–DEM simulations of pseudo-2Dspouted
bedsinstableandunstableregimesandfoundexcellentagreement ofthepressuredropfluctuation.Pietschetal.(2017)extendedthis worktoGeldartBparticlesusingcoarse-grainingin3Dspouted beds.These worksonlysimulated tensofsecondsof operation, whereasspraycoatingrequiresontheorderoftensofminutesfor completionunderexperimentalconditions.
Inaseminalpaper(Lichtenegger&Pirker,2016),someofthe currentauthorsproposedthetimeextrapolationofrecurrentflows by first conductingfully resolved simulations, storing the flow fields,andsubsequentlyscoringthepairwisesimilarityofsystem statesintheformofa recurrenceplot(Eckmann,Kamphorst,&
Ruelle,1987).Fortheextrapolation,intervalsofcontiguousflow field states are alternated with random jumps to recreate the pseudo-periodicnatureofthesystemwhiledescribingthetrans- portofpassivescalars.Thistaskcanbeachievedbysolvingeither EuleriantransportequationsortransportingLagrangiantracers.
Theconceptwasfirsttestedonabubblecolumnsimulatedusing TFMand a steelmaking convertersimulated using thevolume- of-fluidapproach.Althoughthebubblecolumnshowedsuperior performancewhentheEulerianrCFDapproachwasapplied,the steelmakingconverterbenefitedfromLagrangiantreatment.Some ofthecurrentauthorsappliedtheLagrangianvarianttoafluidized bed(Lichtenegger&Pirker,2017;Lichtenegger,Peters,Kuipers,&
Pirker,2017)andextended themethodtopredictheattransfer;
theresults(extrapolatedfromafewsecondsofdata)showedexcel- lentagreementofthebeddistributionreproductionandglobalheat transfercharacteristicsonthescaleofminute,extrapolatedfrom fewsecondsofdata.
Inthiswork,weappliedtheLagrangianrCFDmethodtoathree- dimensionalspoutedbed simulatedusingCFD–DEM, paralleling theprevious works conducted by someof the current authors (Pietschetal.,2017).Wefirstvalidatedthemethodbyassessingthe reproductionofparticlevolumefractionsandthepredictionofpar- ticleholdupsandresidencetimesinafictitioussprayzone.Basedon thispreliminarywork,longer(1h≡3600s)spray-injectionsimula- tionswereusedtostudythecoatingqualityinathree-dimensional laboratory-scalespouted bed equipped withand without draft plates.
Simulations CFD–DEMsimulation
Aswe will only provide a short overview of theCFD–DEM method,thereaderisreferredtotheworkofZhou,Kuang,Chu,and Yu(2010)foranin-depththeoreticaltreatmentandtotheworkof Kloss,Goniva,Hager,Amberger,andPirker(2012)forimplemen- tationofthismethodintheCFDEMcouplingsoftware.
Gas-phasegoverningequations
Thegasphasewasassumedtobehaveasanincompressiblefluid atthelowvelocitiesinvolved.Itstimeevolutionwasmodelledin anEulerianframeofreferenceusingthecontinuumEq.(1) and momentumtransportEq.(2)
∂˛G
∂t +∇·(˛GuG)=0, (1)
∂(˛GuG)
∂t +∇·˛GuGuG=−˛G∇p
G +˛G∇·G
G +˛G
iFi,drag GV ,
(2) whereuG,˛G,andGarethevelocity,volumefraction,andden- sityofthegasphase,respectively;pisthepressure;andVisthe volumeofthecorrespondingmeshcell.Undertheassumptionof Newtonianfluidbehaviour,thedeviatoricstresstensorisgivenby
G=GvG
∇uG+(∇uG)†.Thenetinterphaseforceactingupona particleiisgivenbyFi,interphase=Fi,drag+Fi,p,whereFi,p=Vi∇pisthe pressuregradientforce.ThedragforceFi,dragwascalculatedusing thecorrelationdevelopedbyBeetstra,vanderHoef,andKuipers (2007).
Becauseofthefinemeshintheinletregion,thesolidphasevol- umefractionwascalculatedbydividingeachparticleinto29points andmappingthesepointstothemesh.Detailsofthisprocedurecan befoundinRadl,Gonzales,Goniva,andPirker(2014).Thevolume fractionfield and momentumexchange field weresmoothened usinganapproachdevelopedbyPirker,Kahrimanovic,andGoniva (2011)toensurestabilityandenabletheuseofhighertimesteps.
Fora field that istobesmoothened,a conservativediffusion equation
∂
∂t =L2smooth
tCFD∇2, (3)
issolved(Radletal.,2014),whereLsmoothisthesmoothinglength, whichiscommonlyselectedtobeontheorderofthreeparticle diameters.
Particleequationsofmotion
InDEM,asproposedbyCundallandStrack(1979)andimple- mentedinLIGGGHTS(Klossetal.,2012),particlemotionisresolved bysolvingtheNewtonianequationsofmotion
x¨i= 1 Mi
⎛
⎝
j
Fj→i+Fi,interphase
⎞
⎠
+g, (4)ω˙i= 1 Ji
j
Tj→i, (5)
wherexiandMiarethepositionandmassofaparticlei,respec- tively;gisthegravitationalaccelerationvector;ωistheangular velocity;Jithemomentofinertia;andTj→iisthetorqueacting oniduetoj.TheinterparticlecontactforceFj→i=Fnj→i+Ftj→iof particlejactinguponanotherparticleiiscomposedofanormal componentFnj→iandatangentialcomponentFtj→i.Thesoft-sphere model,asimplementedinLIGGGHTS,usesaglobaltimestepand integratesoverallforcesactingupontheparticles.Contactforces areresolvedbyallowingtheparticlestooverlapandbyapplying overlap-dependentforcemodelssuchastheHertz–Mindlin–Tsuji model(Tsuji,Tanaka,&Ishida,1992).Thesemodelsaccountfor thesingle-particlemechanicalpropertiessuchasthecoefficientof restitution,modulusofelasticity,andPoisson’sratio.Themaximum particleoverlapshouldbekeptunder0.3%oftheparticleradius (Lommen,Schott,&Lodewijks,2014)foraccuratereproductionof bulkbehaviour,whichputsrestraintsonthevalueoftheglobal timestep.Rollingfrictionwasmodelledusingtheconstantdirec- tionaltorquemodelasdescribedintheworkofAi,Chen,Rotter, andOoi(2011),inwhichatorqueproportionaltothedifferencein angularvelocityoftwointeractingparticleswasapplied.
RecurrenceCFDsimulations
As previously proposed by some of the current authors (Lichtenegger&Pirker,2016),therecurrentnatureofcertainflows canbeusedtotime-extrapolatetheseprocessesusingtoolsfrom recurrencestatistics, namelyrecurrence plotsand signalrecon- struction.Wewillonlyprovideabriefsummaryoftheaspectsof themethodthatpertaintothecurrentcase.
Recurrencestatistics
AsdescribedinLichteneggeretal.(2017),thecontinuousrecur- renceplotR˛Gisconstructedusingthegasphasevolumefraction
˛G:
R˛G(ti,tj)=1− 1 N˛G
V
(˛G(ti)−˛G(tj))
2
dV, (6)
N˛G=max
ti,tj
V
(˛G(ti)−˛G(tj))
2
dV, (7)
wheretiandtjaretwotimeswithcorrespondingflowstates,Vis theentiretyoftheflowdomain,andN˛Gisanormalizationfactor.
Becauseofthenormalization,themaindiagonalcontainsvaluesof 1andthemostdissimilarstateshavevaluesof0.
Thetemporalresolution neededforrecurrentpattern recon- structionislimitedbythetemporalrequirement
trec<
2˙2, (8)
foragivenfieldquantity,where·denotestimeaveraging.The field-samplingintervaltrecwasselectedtosatisfytheserequire- ments.
PreviousexperimentalworkonspoutedbedsbyWang,Zhong, and Jiaqiang(2012)appliedtheoriginal,binaryrecurrenceplot method(Eckmannetal.,1987)toattractorsreconstructedfrom thetimedelayembeddedfromthepressuredroptimeandsuccess- fullydetectedrecurrentpatternsspecifictocertainflowregimesin spoutedbeds.
Recurrentprocesses
Based onthe recurrenceplot and underlying field data,the recurrent flow patterns can be extrapolated by constructing a sequenceoftimeindicesthatcorrespondtofieldsinthesampled database.Practically,thistaskisrealizedbyseparatingthesampled fielddatabaseintotwohalves.Ifthecurrentsequenceofconsec- utivefieldsendsinthesecondhalfofthedatabase,asearchfor themaximuminsimilarityisperformedinthefirsthalf,orvice versa,andthismaximumisusedasastartingpointforthenext sequence.Intheremainderofthispaper,wewillrefertosuchfields as“recurrencefields”.
LagrangianrCFD
Foreachstepwithinasequence,thecorrespondinggasphase velocity urecG , particle phase velocity urecP , and particle volume fraction˛recP fieldsareloadedfromthedatabase.Basedonthese velocities,tracersareevolvedbyintegrating
˙
xi=urecP +ufluc, (9)
whereurecP istheparticlephasevelocityinterpolatedontherecur- rencefield and ufluc is the fluctuationvelocity. This additional velocitycomponentisintroducedtomodeltheeffectofinterpar- ticlecollisionsand toeffectivelypreventoverpackingof tracers relativetotherecurrencephasefractions.Anexpressionforthis componentwasderivedbyLichteneggerandPirker(2016)from firstprinciplesinspiredbyclassicalBrowniandiffusion:
ufluc=nrand
D0 ı˛P
6t˛P, (10)
wherenrand is arandomunit vectorwithnrand=1,D0 isthe diffusioncoefficient,ı˛P=max(0,˛P−˛recP )istheexcessvolume fraction,andtisthelocaltimestep.TheselectionofD0ishighly dependentontheflowsituationbecauseofitspurposeinmod- ellinggranulartemperature/collisions, andas such, D0 mustbe calibratedwithrespecttotheunderlyingCFD–DEMsimulation.For tracersinregionswithnorecurrenceinformation,asqualifiedby
˛recP <0.02,thetrajectorywascalculatedaccordingtotheeffectof single-spheredragforceandgravity.
Spraymodelling
TheinjectedspraywasmodelledasLagrangiandropletparcels withmassMDND withND dropletsin aparcel.Because oftheir smalldiameterandlowrelaxationtimetrelax,D=Dd2D/(18vGG)<
5ms, thedroplets wereassumed tomoveat thefluidvelocity, whichavoidstheneedforcalculationofthedragforce.Thedroplets wereinjectedatthenozzleinletpatch,whichwasangledtorepro- ducetheexperimentallymeasuredspraycone,asoutlinedinour previouswork(Pietschetal.,2018).Inaddition,weassumedno dropletevaporation,anddropletparcelswereremovedfromthe systemwhentheycontactedtheapparatuswallsoroutlet.
The droplet deposition was modelled using a filtercorrela- tionproposedbyKolakaluri(2013)toavoidnumericallyexpensive directcontactdetection.Thetargetquantitywasthedeposition efficiency
dep=1.5trCFDuG,rec−uP,rec˛P/dP, (11) withinasingletimestep,accordingtowhichmassisstrippedfrom thedropletparcels.AsimilarapproachwasusedbyAskarishahi, Salehi,andRadl(2017),albeitusinganEuleriandropletphase.The filtercoefficient,
= St3.2eff
St3.2eff+4.3, (12)
dependsontheeffectiveStokesnumber Steff= St
2(A+1.14Re1/5m ˛−3/2G ), (13) andthemodelparameter
A= 6−6˛5/3P
6−9˛1/3P +9˛5/3P −6˛2P, (14) where Rem=(1−˛P)uG,rec−uP,recdP/vG is the superficial Reynolds number andSt=uG−uPd2DP/(9dPvG)is theStokes number.
Ineverytimestep,themassstrippedfromeachdropletparcel iscalculated,mappedontoanEulerianfield,andthendistributed amongthetracersinthecells.Thisprocessstripsdropletsfromthe dropletparceluntiltheyareremovedwhenND<ND,min=1.
Droplet impacts coat the particlesurface. Kariuki, Freireich, Smith,Rhodes,and Hapgood(2013)proposedusingastatistical approachtocalculatethepercentageofaparticlesurfacecoated byimpactingdroplets.Thekeyparametersinthisapproacharethe areacoatedbyasingledropletAD,projandtheratioofthisareato thetotalparticlesurfacef=ADA,projP =(dD
2dP)2.Thedegreeofcoating canthusbeapproximatedas
ϕcoverage=1−(1−f)Ncoll, (15)
where Ncoll is the number of droplets collected by a particle.
Althoughthisapproachdoesnotconsiderspreadingduetosurface wetting,porosity,fluidflow,andparticlerotation,itiseffectivein representingthegeneralcharacterofcoatingqualityestimation, namelyasymptoticbehaviourregardinginjectionofmorecoating liquid.
Furthermore,weassumethatthecoatingprocesshasaneg- ligibleeffectonthegranulardynamicsbecauseofthelowmass injectedandgeneralresistanceofspoutedbedstochangingbed loads.Theevaporationofthecoatingliquidisassumedtobeinstan- taneous and to have noeffect onthefluid dynamics, which is reasonablegiventhelowinjectionrateandhighgasflowrates,
Table1
Processconditionsandgas(subscriptG)anddroplet(subscriptD)phasematerial properties.
Processconditions
Processairflowrate, ˙VG(m3/h) 230
Atomizationairflowrate, ˙VG,nozzle(m3/h) 5
Particlebed,MP(kg) 1.5
Sprayflowrate, ˙Mspray(kg/h) 0.3
Gasphaseproperties
Density,G(kg/m3) 1.225
Kinematicviscosity,vG(m2/s) 1.5×10−5
Dropletphaseproperties
Dropletdiameter,dD(m) 40
Dropletdensity,D(kg/m3) 1000
Dropletinjectionrate, ˙ND(s−1) 1×105
Table2
Meshandnumericalsetup.
CFDmesh
Numberofmeshcells,Ncells 73,647
Cellsizes
Inletregion,xcells,in 2.5mm≈1.4dP
Processchamber,xcells,pc 5mm≈2.8dP
Freeboard,xcells,fb 10mm≈5.6dP
CFD–DEMsimulation
CFDtimestep,tCFD(s) 2.5×10−5
MaximumCourantnumber,Comax,CFD 0.8
CFDwriteinterval,tW,CFD(s) 5×10−3
DEMtimestep,tDEM(s) 1×10−6
Numberofparticles,NP,DEM 472,850
RecurrenceCFDsimulation
Globaltimestep,trCFD(s) 5×10−3
ParticleCourantnumber,Comax,P,rCFD 1 Numberofparticletracers,NP,rCFD 472,850 Dropletparcelinjectionrate, ˙ND,rCFD(s−1) 1×105
equatingtoamaximumincreaseintheairhumidityofapproxi- mately1g/kg.
Simulationsetup
Theprocessconditionsandgasphasepropertiesarelistedin Table1.Theoverallsetupofthecasewasselectedtobeascale-up ofcase3discussedbySalikovetal.(2015).
Geometryandmeshgeneration
Thegeometryoftheapparatusisidenticaltothatofthecommer- cialProCell5(GlattGmbH,Germany)lab-scalespoutedbedwith aprismaticangleof60◦,widthof250mm,anddepthof200mm, similartothoseusedbySalikovetal.(2015),Gryczkaetal.(2009), andPietschetal.(2017).Theprecisedimensionsoftheinletgeom- etrycanbefoundinGryczkaetal.(2009).Thelowerregionofthe inletgeometrywassimplifiedbyremovingthecurvedregionand mergingthetwoinletslits.
Thesprayconehalf-anglewasexperimentallydeterminedtobe approximately17◦.Toreproducethesprayconeanditsflowveloc- ityprofile,theboundaryofthenozzlewascurvedtoreproducethe complementaryangleof73◦.Meshingwasperformedusingthe OpenFOAMhexahedralcut-cellmeshersnappyHexMesh.Thecell sidelengthsarelistedinTable2,andtheresultingmeshisshownin Fig.1.Theparticle(subscriptP)andapparatus(subscriptW)mate- rialandcontactpropertiesarelistedinTable3,whichareadapted fromSalikovetal.(2015).
Fig.1. Surfacemeshoftheapparatus.
Table3
Particle(subscriptP)andapparatus(subscriptW)materialandcontactproperties, adaptedfromSalikovetal.(2015).
Diameter,dP(mm) 1.8
Particledensity,P(kg/m3) 1040
Young’smodulus,YP=YW(Pa) 1×109
Poisson’sratio,εP=εW 0.25
Coefficientof restitution
eP–P 0.9
eP–W 0.75
Coefficientoffriction kfr,P–P 0.5
kfr,P–W 0.24
Coefficientofrolling friction
krfr,P–P 0.06
krfr,P–W 0.05
CFD–DEMsimulations
Thefullsimulationswereconductedusinganextendedversion oftheCFDEMcoupling(Klossetal.,2012)softwarepackage.
AnoverviewofthenumericalsetupisprovidedinTable2.The CFDtimesteptCFDwassettosatisfyCo=ut/x<1anddid not exceedmax(Co)≈0.8 duringtheentire simulation. Forthe DEMpart,thetimestepwassettobeapproximately20%ofthe Rayleightime.InterphasecouplingwasperformedonceperCFD timestep,orevery25DEMtimesteps.Thesmoothinglengthwas setto5×10−3mm.Theinletwasprescribedafixedvelocitybound- arycondition.Forturbulencemodelling,thek–εmodelwasused withaninletturbulenceintensityof5%.
First,1.5kgof1.8mm␥-Al2O3particleswereinsertedataheight of0.2m<y<0.3mwithinthefreeboard regionoftheapparatus.
Thesimulationwasstoppedafter3sandresumedusinga field samplingfrequencyof200Hzfor10s,resultinginafielddatabase comprising2000entriesandrequiring16GBofmemory.Thedata generatedinthisrunwasnotusedforresidencetimecomputation toavoidbiases.Thesimulationwascontinuedforanother10s,and theresidencetimewithinthecone,showninFig.2,wascalculated.
AnotherCFD–DEM simulationwasperformedusingidentical settingsandgeometrybutfeaturingdraftplates,asproposedby Pietschetal.(2017).Thesedraftplateswere60-mmhigh,located 10mmabovethemidprofile,anddistanced45-mmapart.
rCFDsimulations
ThetracerequationsofmotiondescribedinEq.(9)wereimple- mented based on the hard-sphere Lagrangian particle tracking (LPT)algorithm presentin OpenFOAM5.x(Weller,Tabor,Jasak,
Fig.2.ShapeanddimensionsofthesprayconeusedinbothfullCFD–DEMandrCFD simulations.
Fig.3. SchematicofinterprocesscommunicationinCFDEMcouplingandOpen- FOAMLagrangianparticletracking(LPT).
& Fureby, 1998), which is described in detail in Macpherson, Nordin, and Weller (2010). This implementation enabled effi- cient treatmentof thecomplex walls present in thegeometry.
For thewalls,simple elasticrestitutionwasassumed, unlikein ourfirstimplementationofthismethodinCFDEMcouplingusing CFDEM/LIGGGHTS.There,simplereflectivewallboundarycondi- tionswereappliedtoavoidcostlycontactdetection.
Anothernotabledifferenceis themodeofparallelization, as illustratedinFig.3.CFDEMcouplingusesdifferentdomainsubdivi- sionsfortheCFDandDEMsides,whichenablesefficientdynamic loadbalancingontheDEMsidebutintroducesabottleneck:during coupling,theDEMinformationisdistributedtoallCFDprocesses, whichiscommonlycalled“all-to-all”communicationandwhich canbeverycostlydependingonthenumberofpartitionsandpar- ticles.Additionally,datamustbecopiedbetweentheconstituent codesOpenFOAMandLIGGGHTS.Incontrast,OpenFOAMLPTuses thesamegeometricsubdivisionsforbothparticlesandthefluid domain, meaning that no furtherinter-process communication
Fig.4. OptimaldomaindecompositionsusingCFDEMcouplingemploying“all-to- all”communicationandOpenFOAMLagrangianparticletracking(LPT).
needstooccurwhileevolvingtheparticlecloud,exceptforinter- processboundarycrossingofparticles.Thisprocessrequirescareful selectionofthedomaindecompositiongeometrytoensureoptimal performanceforcasesinwhichbothCFDandDEMaresimilarly demanding;however,theLagrangianphaseisnothomogeneously distributed, asillustratedfor asimple case inFig.4. Thisprob- lemdoes notappear for mostLagrangian rCFDapplications,as thedecompositioncanbeselectedtorepresenta homogeneous distributionofparticlesamongdomains.Themaincomputational demandhereliesintheEuler–Lagrangemappingandsolvingthe underlyingphysicssuchastransportprocessesandintegrationof equationsofmotion.Thesetwostepsbenefitthemostfromaco- locationofCFDfielddataandparticleinformationwithinthesame process,makingOpenFOAMLPTthesuperiorchoice.
ShortrecurrenceCFDsimulationsof35swereperformedfor thediffusion coefficientcalibrationusing a samplingfrequency of200Hzfortheprobesand1Hzforthefields.Thefluctuation velocitieswerelimitedto1m/stopreventtracersfromdiffusing fartherthanapproximatelyonecelldiameterintheprocesscham- berwithinonetimestep.Thefirst5swerenotusedforresidence timecalculation.
Resultsanddiscussion CFD–DEMsimulation
Theverticalparticlevelocityasa functionoftime inpromi- nentprobinglocationsinthespoutandannulusisshowninFig.5.
Theparticlesinthespoutexperiencedmostlyupwardsmotion,as expected.Themaximumvelocitiesheredidnotexceed3m/swith arootmeansquare(RMS)velocityof1.2m/sandaRMStemporal derivativeof28.2m/s2,givinganupperboundofthefieldsampling intervaloftw,CFD<0.04s.Theparticlesintheannulusunderwent alternatingperiodsofupwardanddownwardmotion,indicating lateralspoutejectionsthatarethedefiningfeatureoftheinsta- bleoperatingregime.Notably,thelowresultingRMSvelocityand relativelyhightemporalvariabilityresultedinastrictercriterion forthefieldsamplingintervaloftw,CFD<0.02s.Therefore,afield samplingintervaloftw,CFD=0.005swasselectedtoensureappro- priatereproductionofthesystemdynamics.Thisintervalwasalso assumedtobesufficientforthecasecontainingdraftplates,asthese stabilizeflowpatternsinboththespoutandannulus.
TheresultingrecurrenceplotinFig.6hasmanylaminardiago- nals,especiallyintheregion4s≤trec≤8s,indicatingthepresence ofa highdegreeofrecurrentsystemstateswithinthesampled databaseaswellas aninterval withlowrecurrence withinthe sampledtimespanattrec≈3sandtrec≈9s.
Fig.5. VerticalparticlevelocitiesUP,y(straightline)atprobelocations1(spout) and8(annulus)aswellasthecorrespondingrootmeansquarevelocities
UP,y2 (dashedline)andtemporalderivatives
dUP,y/dt
2(dottedline).Thesignals werelow-passfilteredusingacut-offfrequencyof50Hztoremovenumericalnoise.
Fig.6. Recurrencematrixderivedfromthevolumefractionfield˛Psampledfor10s at200Hz.
ReproductionofbeddistributionanddynamicsinrCFD
ToensurethevalidityoftherCFDsimulationsandavoidover- packingof thetracer particles relative to thevolume fractions presentintherecurrencefields,theintensityofthevelocityfluc- tuationswascalibrated.Forthis process,35-ssimulationsusing diffusioncoefficientsin therangeof0≤D0≤1×10−2m2/swere conducted.Toeliminatetheeffectofrandomrecurrencepathcal- culation,apre-calculatedpathwasusedforthesesimulations.
Thesuccessof thesimulation canbejudgedbyitsabilityto reproducetheinstantaneousvolumefractionsatprobelocations relativetotherecurrencefieldsandthetime-averagedvolumefrac- tion,aswellastopredicttheresidencetimewithinthepreviously describedsprayzone.
Reproductionofdynamicvolumefraction
Theresultingtracervolumefractionsandcorrespondingrecur- renceparticlevolumefractionsatasamplelocationinthespout regionareplottedas afunctionof timeinFig.7. Agreementof therecurrenceparticlevolumefractionsandtracervolumefrac- tionsimpliesaccuratereproductionofboththebeddistribution andgranularfluxes,asforD0≥5×10−4m2/s,whereanincreaseof thediffusioncoefficientD0 doesnotimprovetheagreement.For D0<1×10−4m2/s,thetracervolumefractionfrequentlyexceeded therecurrencevolumefraction,whichindicatesinaccuraterepro- ductionofthebeddistribution.Here,thetracervolumefraction evenexceededtheclose-packedvolumefractionlimitof˛P>0.67.
Without the relaxation model (D0=0m2/s), the tracer volume fractionattheprobinglocationwasconsistentlylowerthanthe recurrencevolumefraction.Thismismatchindicatesthatthebed distributionwasseverelymisrepresentedduetoparticlesaccumu- latingelsewhereinthesystem.
Fig.7.RecurrenceCFDtracervolumefractionsfordifferentdiffusioncoefficientsD0andcorrespondingparticlevolumefractionsattheprobelocationinthespoutregionat x=0m,y=0.075m,z=0.05m,whichisapproximatelythesameheightandlateralpositionsasthefictitioussprayzone.
Fig.8.Penaltyscoresfordifferentdiffusioncoefficients.
Toremovesubjectivityfromtheselectionofthediffusioncoef- ficient,apenaltyfunction1/Ndatapoints˛P−˛P,rec2 wasapplied to all 10 sampling locations in the apparatus. The locations were in prominent regions within the annulus and spout to enable theaccuracy of thebed distributionreproduction tobe gauged.Thetime-averagedpenaltyscoresaveragedoverallloca- tionsfordifferentdiffusioncoefficients areshown inFig.8.The penaltyscoredecreasedwithincreasingdiffusioncoefficientupto D0=1×10−4m2/s,atwhichpointaminimumoccurred.Forhigher diffusioncoefficients,thescoreincreasedagain,possiblydueto artefactsinducedbythestrongdiffusion.Thisbehaviorisgener- allyinaccordancewiththequalitativeobservationsmadewhen comparingthetracerandrecurrencevolumefractionsatasingle samplingpoint.
Time-averagedvolumefractionreproduction
Reproducing the time-averaged recurrence volume fraction determinedbythetracersisarequirementforusingtheLagrangian rCFDapproach, as a failuretoreproduce this parameterwould
invalidatealltheresultsoffurthersimulations.Theresultingtime- averaged tracer and recurrence volume fractions in a planeat z=0.1marepresentedinFig.9.Applyingnovelocityfluctuations (D0=0)resultedinveryhigh,unphysicaltracervolumefractions closeto thewalls and, consequently,lowerbed concentrations andexpansionsinthespoutregions.Thisproblempersistedfor low diffusion coefficients, e.g., D0=1×10−5m2/s. Starting with D0=5×10−4m2/s, the significantly overpacked regions in the annulusdisappearedandtheappropriatemeanshapeofthebulk wasaccuratelyreproduced.Ashigherdiffusioncoefficientsintro- duceerrorssuchasunphysicalmixinginbulkregions,thelowest viablediffusioncoefficientD0=5×10−4m2/swasselectedforfur- therinvestigations.
Influenceonholdupandresidencetimedistributionwithina fictitioussprayzone
Togaugetheaccuracyofthesimulationsrelativetoareal-world targetvalue,asprayzonewasdefined.Thisstepwasvaluable,as unlikethe previousexamples,thefateof singleparticles could beassessed,enabling clarificationoftheeffectofrandom-walk- inducedunphysicalmixing.
Examplesoftheinstantaneoussprayzoneholdupfordifferent diffusioncoefficientsand thefullCFD–DEM simulationarepre- sentedin Fig.10.Thesignals haveverysimilarpseudo-periodic patterns.Exceptforthesimulationwithoutthediffusionmodel, wherepoordistributionofsolidsledtoanoverallholdupunderes- timation,thepeaksintherCFDsimulationswerehigherregardless ofthediffusioncoefficient.Themeanholdupwasgenerallyonly slightlyoverestimatedrelativetothatofthefullCFD–DEMsimu- lation.Theseminordeviationsmaybeexplainedbyoverpacking, indicatinganinsufficiencyofthediffusion-basedrelaxationmodel.
Thesprayzoneresidencetimedistributionsforvariousdiffu- sioncoefficientsandforthefullsimulationaregiveninFig.11.As inthepreviouscases,alackofdiffusiverelaxationleadstopoor agreementwiththeCFD–DEMsimulation.Theuseofhigherdif- fusioncoefficientsresultedingoodagreementwiththeCFD–DEM
Fig.9.Time-averagedparticlevolumefractionsatz=0.1m.Theplotstitled“D0”presenttheaveragedtracervolumefractionsforvariousdiffusioncoefficientsandthattitled
“Recurrence”presentsthetime-averagedrecurrencevolumefractionfield.
Fig.10.DependenceofinstantaneoussprayconeholdupondiffusioncoefficientforrCFDsimulationsandfullCFD–DEMsimulation.Thedottedlinesindicatethetime- averagedholdups.
Fig.11.Sprayzoneresidencetimedistributionsafter10-ssimulationtimeforvar- iousdiffusioncoefficients.NotethatthegraphsforD0=5×10−4m2/s(dotted)and D0=1×10−2m2/s(dash-dot)practicallyoverlap.
simulation,withsomeunderpredictionofthefractionofparticles withhighresidencetimes.Asevenexcessivediffusioncoefficients donotsubstantiallyaffecttheresidencetimedistribution,itcan beassumedthatunphysicalmixingdoesnotoccurtoadegreethat wouldadverselyaffecttheoutcome.Thelimitofthefluctuation velocityof1m/smight alsoplaya role inpreventingexcessive diffusion.
Spraycoating
Thesimulationpredictedoversprayof2.3%fortheunstabilised spoutedbedand0.8%forthestabilizedone,which,whilesomewhat optimistic,qualitativelyindicatesthattheprocessmightbenefit
Fig.12.Fractionalsurfacecoveragedistributionsafter1hofsprayinjection.
fromthedenserand lessvariableflow patterninthestabilized apparatus.
Thefractionalsurfacecoveragedistributionscalculatedusing Eq.(15)fortheunstabilisedspoutedbedandthatequippedwith draftplatesafter1hofsprayinjectionareshowninFig.12.Surpris- ingly,themedianfractionalcoveragewasidenticalat90%,withan overallnarrowerdistributionobservedfortheunstabilisedsystem.
Thisresultcanbeattributedtoinhibitedmixingalongtheappara- tusdepthaxis(z),astheprimarycomponentofparticlemotionis withinthex−yplane.Withtheejectionssuppressedbythedraft plates,thismodeofmixingisgreatlyreduced.
Basedonthesefindings,characterizationoftheregionsinwhich dropletdepositionoccurswasperformed,asshowninFig.13.Inthe unstabilisedapparatus,mostdepositionoccurreddirectlyabove
Fig.13.Time-averageddepositionratedensitydistributionsattheapparatusmidplanes(z=0.1mintheupperplotsandx=0minthelowerplots).
thenozzlewithinadiffusecloud,whichisinclearcontrasttothe apparatuswithdraftplates,forwhichthemaindepositionzone wasmoresharplydefinedandshiftedapproximately2-cmhigher totheupperendoftheplates,possiblyduetohigherbackground gasvelocities withinthechannelpresentedbytheplates.Com- parisonofthelateralprofilesindicatedthatthedepositionarea wasverycompactinitslongitudinalexpansioninthestabilized case,signifyinglesslongitudinalgasflow,whichcouldmoveeither dropletsorparticlesalongthisdirection.
Overall,these findings provideinsight intotheoptimization potentialofthespoutedbedapparatus.Althoughtheevaluated design(Pietschetal.,2017)didnotimprovethecoatingquality, it mightbe interestingfor otherapplications. Processesrequir- ingintensivecontactwithagranularcatalystshouldbenefitfrom theincreasedstability,resultinginanarrowgasphaseresidence timedistributionandlowbypass;however,furtherinvestigation isrequired.
ToreducethecalculationtimewithoutapplyingrCFD,onemay betempted touse time-averaged velocity and volume fraction fieldstomovetracersforbothparticlesanddropletswhilemod- ellingdropletdepositioninthesamewayperformedinthisstudy.
Althoughthisapproachmaybeviableforthestabilizedapparatus, itwouldintroducegreatinaccuraciesfortheunstabilizedvariant, asanygivenstateoftheflowfieldswoulddiffergreatlyfromthe averagebecauseofthelateraldeflectionsandirregularejections.
Performanceconsiderations
TheCFD–DEMsimulationsrequiredwalltimesof13daysfor 13-ssimulationson2×12coresofIntelE5-2680v3processors.The residencetimecalculationsusingrCFDwereperformedon12cores ofthesamehardwareandrequired36minofwalltimefor35s, with12minspentloadingtheapproximately16GBofrecurrence fieldsfromstorageintomemoryandcalculating therecurrence matrix.Theseparameterscorrespondtoa2100-foldnetspeedup whenexcludingtheloadingtimes,whichbecomenegligiblewhen consideringtheoveralldurationofthesimulationsforwhichthe applicationofrCFDisappropriateanddesirable.
Thespraysimulationsaddedtheburdenofanother(negligible) setofrecurrencetracers(sprayparcels)andthedropletdeposition calculations,whichreducedtheperformanceto≈1500s/daywith aspeedupof1500times.Asthedepositionalgorithmonlyinvolves
oneEulerianfieldcalculationandtwonon-nestedloopsoverall tracers,thisfindingprovidedfurtherevidenceoftheleannessand efficiencyoftheparticletrackingalgorithmpresentinOpenFOAM.
Discussionofmodellimitations
Aspromisingasthefindingsandperformanceappear,thepro- posedmethodinitscurrentstateisonlyapplicabletoaspecificclass ofprocesses.Recurrentflowpatternscanbefoundonanyscale inturbulentflowbutareinpracticemodelledusinga sub-grid- scalemodel.Morecomplexphysicalmodelssuchasheattransfer orchemistryareusuallysolvedusingasteady-statesolutionofthe systemathandortransientsimulationsforflowswithstrongly coupledphysicsforsubsequentextrapolationinpost-processing.
AlthoughrCFDcaneliminatetheneedtoextrapolatephysicsin post-processingbyinsteadextrapolatingthedynamicsanddirectly solvingthephysics,itisonlyappropriatewhennegligibleback- propagationfromthesolvedphysicstothefluiddynamicsoccurs.
Acentralweakpointofthemethodinitscurrentformisthe relaxationmechanism. Althoughdiffusion byrandom walkis a simple,inexpensive,andelegantsolutiontotheoverpackingprob- lem,italsointroducesartificialmixing.Here,amoresophisticated approachisneededthataccountsforadditionalinformationabout thestateofthesystemduringtherecurrencedatabasegeneration.
Forgranularsystems,appropriatequantitiesmaybethegranular temperatureorstatisticalmomentsofthegranularfluxacrosscell faces.
Becausethemethodtradestimecomplexityforspacecomplex- ity,memoryusagewillbeoneofthebottlenecksforapplicationof themethodtolargerapparatuses.Usingthemethodisonlyviable whenrecurrentpatternsoccurontimescaleswithinafeworders ofmagnitudeofthetemporalresolutioncriterion(Eq.(8)),asoth- erwise,thememoryrequirementswouldbeincreased.Thesizeof therecurrencedatabasemustbecarefullybalancedbetweenthe lowerboundofcontainingall prominentflow patternsandthe upperboundoftheavailablememorysize.
A case that would greatly benefit from therCFD method is oneexhibiting ideallystrongly periodicpatterns,which deviate strongly from the mean, as this would inhibit convergence of steady-statemodelsandthusrequirecomputationallyexpensive transientsimulations.
Conclusions
In this work,we successfully appliedtherCFD methodto a laboratory-scale3Dspoutedbedapparatus.Thiseffortpresented aparticularchallenge,astheentiresetofflowregimes,ranging frombulkflowintheannulustodenseanddiluteflowinthespout andfountain,wererepresented.Themethodwasshowntoaccu- ratelyreproduceflowpatterns,asevidentbyacomparisonofthe resultingaveragedvolumefractionsandresidencetimedistribu- tionwithinasprayconewiththoseobtainedfromafullCFD–DEM simulation,whilerequiringonly0.05%ofthecalculationtimeafter generatingtherecurrencedatabase.
Thisperformancegainallowedforthedirectinvestigationof sprayinjectiononthetimescaleof1hinlessthanthreedaysof walltime,whichwouldhaverequiredatotalof3600daysusing fullCFD–DEM.Theaddition ofdraftplatesledtoworsemixing performance,whichcanbeattributedtothestabilizedspouting pattern,andproducedawidersurfacecoveragedistributionamong theparticles.Thisworkisonlyafirststepinexploringthepoten- tialoftheapparatusanditsmodificationsforpracticalapplications.
Thereducedmixingalongtheapparatusdepthmayalsointroduce opportunitiesforcontinuousdryinginapparatusesscaledupby
increasingtheapparatusdepthorinchemicalreactorswherethe granularphaseactsasacatalyst.
Infutureworks, rCFDwillbeappliedtoadditionalproblems withdiversephysicalchallenges,and,inparticular,itslimitsmust besystematicallydetermined.Newrelaxationmodelsshouldbe developedusingsimplerdensegranularandbulkflowsituations becauseinmorecomplexsystems,inaccuraciesmaybemaskedby theinterplayofdifferentfactors.Furtherdevelopmentsregarding algorithmicreductionofsampleddatawillpavethewaytobroad applicationforindustrial-scaleproblems.Asforspoutedbeds,new approachesformeasuring,orideallymonitoring,coatingquality areneededandwouldprovideachancetovalidatetheassumptions madeandconclusionsdrawninthiswork.
Acknowledgements
T.LichteneggeracknowledgesfundingfromtheLinzInstitute ofTechnology(LIT),JohannesKeplerUniversity(projectLIT-2016- 1-YOU-007).S.PietschacknowledgesfinancialsupportfromBASF SE.
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