Inter-compartment interaction in multi-impeller mixing

Inter-compartment interaction in multi-impeller mixing

Part II. Experiments, sliding mesh and large Eddy simulations

Haringa, Cees; Vandewijer, Ruben; Mudde, Robert F.

DOI

10.1016/j.cherd.2018.06.007

Publication date

2018

Document Version

Final published version

Published in

Chemical Engineering Research and Design

Citation (APA)

Haringa, C., Vandewijer, R., & Mudde, R. F. (2018). Inter-compartment interaction in multi-impeller mixing:

Part II. Experiments, sliding mesh and large Eddy simulations. Chemical Engineering Research and Design,

136, 886-899. https://doi.org/10.1016/j.cherd.2018.06.007

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ContentslistsavailableatScienceDirect

Chemical

Engineering

Research

and

Design

j o u r n a l ho me p a g e : w w w . e l s e v i e r . c o m / l o c a t e / c h e r d

Inter-compartment

interaction

in

multi-impeller

mixing.

Part

II.

Experiments,

sliding

mesh

and

large

Eddy

simulations

Cees

Haringa

_,

_Ruben

_Vandewijer,

_Robert

_F.

_Mudde

TransportPhenomena,DepartmentofChemicalEngineering,DelftUniversityofTechnology,vanderMaasweg9,

2629HZDelft,TheNetherlands

a

r

t

i

c

l

e

i

n

f

o

Articlehistory:

Received19November2017 Receivedinrevisedform24April 2018

Accepted1June2018 Availableonline13June2018

MSC: 00-01 99-00Keywords: Mixing Schmidtnumber CFD Multipleimpellers Rushton

a

b

s

t

r

a

c

t

Steady state multiplereference frame-RANS (MRF-RANS)simulations frequently show strongover-predictionsofthemixingtimeinsingle-phase,multi-impellermixingtanks, whichissometimespatchedbyadhoctuningoftheturbulentSchmidt-number.InPartI ofthiswork,weexperimentallyrevealedthepresenceofmacro-instabilitiesintheregion betweentheimpellers,aswellasapeakintheturbulentkineticenergyintheregionwhere theflowfromtheindividualimpellersconverges.TheMRF-RANSmethodwasfoundunable tocaptureboth.Inthissecondpaper,weshowthatthesliding-meshRANS(SM-RANS) approachdoescapturetheeffectofmacro-instabilities,whilestillunderestimatingthe tur-bulentkineticenergy.Consequently,theSM-RANSmethodmildlyover-estimatesthemixing time,whilebeinglesssensitivetotheexactmeshgeometry.Largeeddysimulationswith thedynamicSmagorinskymodelreasonablycapturethekineticenergycontainedin macro-instabilities,andproperlyassesstheturbulentkineticenergyintheregionbetweenthe impellers,evenforcrudemeshes.Consequently,themixingtimeisreasonablyassessed, andevenunder-predictedatthecrudestmeshes.However,theturbulentkineticenergyand energydissipationintheimpellerdischargestreamarepoorlyassessedbythedynamic Smagorinskymodel.

©2018InstitutionofChemicalEngineers.PublishedbyElsevierB.V.Allrightsreserved.

1.

Introduction

Computational fluid dynamics (CFD) has frequently been appliedtosimulatemixingprocessesinstirredtanks. Espe-ciallyReynoldsaveragedNavierStokes(RANS)modelsprovide thepossibility toevaluatethe mixingperformanceof vari-ousimpeller configurations without requiringan extensive experimentalcampaign.Anassessmentofmixingliterature focusing on Rushton turbines, conducted in Part I of this work,reveals thatRANSsimulationsarecapableof

reason-Abbreviations: MI, macro-instability;LDA,laserDoppleranemometry;PIV,particleimagevelocimetry; RANS,Reynoldsaveraged

NavierStokes;MRF,multiplereferenceframes;SM,slidingmesh;(S/R)KE,standard/realizablek−;RSM,Reynoldsstressmodel;(D/L)ES, detached/largeEddysimulation.

∗ _{Correspondingauthor.}

E-mailaddress:cees.haringa@DSM.com(C.Haringa).

1 _{Currentaddress:DSMBiotechnologyCenter,AlexanderFleminglaan1,2613AXDelft,TheNetherlands.}

ablypredictingthemixingbehaviorintanksstirredbyasingle impeller,albeitwiththerequirementofhighmeshdensities. Incontrast,thedimensionlessmixingtime95is

systemati-callyover-predictedinmulti-impellertankswithhighmutual impellerspacingwhenusingRANSmodels.Inparticular,for multiplereferenceframe(MRF)simulationsincreasingmesh densitiesresultsinanincreasingover-predictionof95

com-paredtoexperimentalresults.

Itiswellknownthatcompartmentsformaroundthe indi-vidualimpellersofamulti-Rushtontank.InPartIofthiswork,

https://doi.org/10.1016/j.cherd.2018.06.007

Nomenclature

Roman

Ct tracerconcentration,kg/m3

C off-bottomclearanceimpeller,m

CS Smagorniskyconstant,–

C inter-impellerclearance,m

D impellerdiameter,m D diffusioncoefficient,m2_/s

Dt diffusioncoefficient,turbulent,m2/s

Ep/Et fractionofperiodicenergyinspectrumofu,–

f frequency,s−1 f1.1 basefrequency1,s−1 f2.1 basefrequency2,s−1 f1.2 harmonicfrequency1,s−1 f2.2 harmonicfrequency2,s−1 FQ pumpingnumber,– h axialcoordinate,m H tankheight,m

k slotnumber(autocorrelation),–

kt turbulentkineticenergy,m2/s2

kMI macro-instabilitykineticenergy,m2/s2

kt,* kt+kMI,m2/s2

kt,fit ktcomputedviaauto-correlationfit,m2/s2

kt,SI ktcomputedviaspectralintegration,m2/s2

kMI,fit kMIcomputedviaauto-correlationfit,m2/s2

kMI,SI kMIcomputedviaspectralintegration,m2/s2

N impellerrevolutions,1/s

P power,W

Po powernumber,W

Qax axialflowrate,L/s

r radialposition,m

R tankradius,m

Sij shearrate,s−1

Sct turbulentSchmidtnumber,–

T tankdiameter,m t timestepsize,s

u fluctuatingvelocity,m/s

U meanvelocitym/s

Utip impellertipspeed,m/s

V tankvolume,m3

Vi gridcellvolume,m3

w Tukey–Hanningwindow,–

Greek

 laserwavelength,nm  viscosity,dynamic,Pas

t viscosity,dynamic,turbulent,Pas

 viscosity,kinematic,m2_/s

t viscosity,kinematic,turbulent,m2/s

 turbulentdissipationrate,m2_/s3

ˆ auto-correlationcoefficient,– l density,kg/m3

noisecomponent,– lag lagtime,s

95 mixingtime,s

ij stresstensor,Pa

95 mixingnumberN·95,–

we posed the hypothesis that the over-predicted 95

origi-natesfrom anunder-predictioninmassexchangebetween thesecompartments,whilemixinginsidethecompartments is likelyproperly captured (Coroneo et al., 2011). Although stirredtankshavebeenstudiednumeroustimes,theregion betweenthe impellershadpreviouslyattractedlittle atten-tion.Wehencefocusedourstudyonthisregion,toprovide insightin theinter-compartment hydrodynamicsand their role in mixing. Using laser-Doppler anemometry (LDA),we observed thatthis transport isgoverned bytwo processes: macro-instabilities(MIs)atfrequenciesf/N=0.02–0.06(Nthe agitationrateins−1)inthehorizontalplaneseparatingthe compartments,andapeakinturbulentkineticenergy(kt)in

theplanesegregatingthecompartments,generatedbythe col-lisionoftheflow-loopsintheconvergingflowsectionnearthe wall.Boththeseeffectsareexpectedtoenhanceaxialmass exchange.

Besides experimental results, we reported steady state MRF simulationsinPartIofthis study.Due totheirsteady statenature,theyare inherentlyincapableofcapturing the influence of macro-instabilities. Combined with an under-estimated turbulent kinetic energy (kt), attributed to poor

capturingofturbulencegenerationinthecollidingflowregion byvirtueofReynoldsaveraging,thisleadstoahighly over-estimated95; 95wasfurthermore stronglyaffectedbythe

axial flowrate between the compartments, which in turn was highly mesh sensitive. This explained the increasing 95 with increasing mesh density. In this second part, we

assesswhethertransientsimulations,bothslidingmesh(SM) RANSandlargeEddysimulations(LES),docaptureallrelevant hydrodynamicsbetweenthe impellers,and properly assess themixingtimesinmulti-impellerstirredtanksystems.

2.

Literature

review:

Rushton

turbines,

LES

and

DES

Inthepreviouspartofthisworkwediscussedtheliterature regarding experimental assessment of flows in Rushton-stirredtanks,andtheuseofRANSmodelstomodelsuchflows. MixingresultsforLESwerealsoincludedinPartI,andarenot listedhere.Here,wediscussLESanddetachededdy simula-tions(DES)forflowmodeling.Toourknowledge,DEShasnot beenusedformixingstudiesinbaffledRushton-stirredtanks. 2.1. LES

Revstedt et al.(1998) usedfinitevolumeLES(FV-LES), with 2.12×105 _{gridcells,animplicitclosure modeland}

momen-tumsourcetermsforimpellermodeling.DecentresultsforU

andktwerereportedinthebulk,withpooreragreementnear

theimpeller.Yeohetal.(2004,2005)appliedtheSmagorinsky (SGS)subgridmodelwithconstantCS=0.1(FV-LES,4.9×105

cellswithsliding-deforminggrid),reportinggoodresultsfor

U and kt, but providing no data on . Zhang et al. (2006)

reported similarresults. Alattice-Boltzmannapproach (LB-LES)isfrequentlyappliedinLESstudies(Eggels,1996).Eggels (LB-LES,SGS,CS=0.1)showedsomelocalunder-predictionof

Uax,butoverallagoodagreementwithdatabyBakker(1996)

wasreported.DerksenandVandenAkker(1999)(SGS-model,

CS=0.12,6×106gridnodes)reportedaccurateresultsforthe

dischargestreamvelocities,kt,andtrailingvortexbehavior.

Theyreportedaphase-averagedmaximumenergydissipation intheimpellerdischargeofmax¯ /N3D2≈4.6,over50%lower thanmeasuredexperimentally(DucciandYianneskis,2005).

Hartmannetal.(2004)reportedasimilarunder-estimationof withLB-LES,resultsthatarefurthersupportedbyMicheletti etal.(2004)(SGS,CS=0.1)withFV-LES.

Delafosse etal. (2008,2009)(sliding mesh,106 _cells,

FV-LES,SGSmodel)explicitlynotedthatsettingCS=0.1leadsto

asignificant under-predictionof inthedischarge stream.

CS=0.1 was selected based on testing for awide range of

flows,withhighervaluesofCSleadingtoexcessiveturbulence

dampening.SettingCS=0.2stronglyimprovespredictionsfor

,withoutsignificantlyaffectingthepredictionsforvelocity andkt.Soosetal.(2013)(slidingmesh,1.6×106cells)alsoused

CS=0.2.ComparedtothedataofbothEscudiéetal.(2004),

and Wu and Patterson(1989), theyreported amild under-predictionofthevelocities andthe periodickinetic energy, whilektwaswellpredicted.Thevaluesforareinaccordance

withDelafosseetal.

TheworkofDelafosseetal.andSoosetal.indicatesa case-by-casetuningofCSmay berequired,whichisundesirable

fromtheperspectiveofpredictivecapabilities.Thedynamic SGS (DYN)model aims atsolving this issue by computing ratherthanprescribingCS.MurthyandJoshi(2008)(FV-LES,

DYN,1.3×106_{cells,slidingmesh)reportgoodresultsforthe}

dissipationbasedpowernumberPo,butshownoprofilesof

orvaluesofCS.Jahodaetal.(2007)applythedynamicmodel

formixingin1and2impellergeometriesbutdidnotreport orPo.TheydidshowlocalvaluesofCS,whichwereinthe

rangeof0.05–0.1,belowthedefaultCS=0.1.

AdirectcomputationofCSusingdirectnumerical

simu-lation(DNS) byGillissen and Vanden Akker(2012) yielded

CS≈0.1,inagreementwiththeirowndynamicLES.This

indi-catesthattheCStuningconductedbyDelafosseetal.andSoos

etal.isnotinaccordancewithDNSobservations.Gillisenand vandenAkkernotedthattheunder-predictioninmaybethe resultofanunder-predictedkt-productionduetoinsufficient

meshresolutioninthevicinityofwalls.

2.1.1. DES

Detachededdysimulations(DES)blendaLESapproachinthe free-streamwithRANSinunder-resolved(wall)regions,and maytherebyreduceanydependenceofthebulkflowonwall effects,possiblyimprovingthepredictionsforifthe hypoth-esisbyGillisseniscorrect.Ofcourse,theaccuracyofthewall flowitselfwillstillbelimited,duetotheinherentassumptions oftheRANSmethodology.

Gimbunetal.(2012)presentedSpalart-Allmaras-DES sim-ulationsofaRushton-stirredtank,extensivelycomparingthe resultswith bothSKE-RANS and FV-LES (SGS, CS=0.1). The

bulkvelocitypredictionwasverysimilarbetweenthemodels. DESgenerallyyieldsthebestagreementwithexperimentalkt

data(Derksenetal.,1999),predictingslightlyhighervalues thanSKEandLES.LESandDESperformedsimilarinassessing thequalitativetrailingvortexbehavior,withtheSKEmodel predictingsignificantlylower radialspreadingofthevortex core,similartothestudyofSinghetal.(2011).DEScompared favorabletotheothermodelsinpredictingvelocitiesandkt

inthevortexcore.Overall,DESoutperformedRANS,and out-performedLESinregionswherewalleffectsaresignificant.

Charaetal.(2016)cametosimilarconclusions,observinggood agreementindischargevelocitiesandtrailingvortexbehavior. Theynotedthatthetangentialspreadofthetrailingvortexis slightlynarrowerthanexperimental(PIV)resultsshow.

Lane (2015) reported the energy dissipation behavior of variousturbulencemodelswithanA-310impeller(13.1×106

gridcells).Apowerrecoveryof69%wasobserved,i.e.PDES=

Table1–Meshesusedinthiswork.2IFrepresentsa 360◦domain.Thelastletter(s)representthemesh quality(C=crude,M=medium,F=fine,SF=super-fine).

Name Cells Domain Methods

2IF-C 648k 360◦ LES

2IF-M 1997k 360◦ SM-RKE,SM-RSM,LES

2IF-F 5884k 360◦ SM-RKE

2IF-SF 10584k 360◦ LES



(+t)S2_ijdVis69%ofthepowerinputbasedontorque.For

various SST and KE formulations,theenergy recoverywas 68–91%andstronglymeshdependent,supportingthe obser-vationsbyCoroneoetal.(2011).Thelowenergyrecoveryfor DES,atthefinestmeshused,doeshintthatthewall treat-mentofDESdoesnotprovideasignificantimprovementover LESintermsofresolving.Acomprehensivecomparisonof LESandDESwithvariouslevelsofwallresolution,possibly supportedbyDNS,isrequiredtoprovidefurtherinsightin thereportedunder-predictionsof.Thatisoutofthecurrent scope,however.

3.

Materials

and

methods

3.1. CFDsetup

3.1.1. Geometry

A2-impellerstirredtankwithH=2T,D=T/3,C=T/3andC=T

wasmodeled,withT=0.29m,andanagitationrateN=5s−1. Note that our own experiments were conducted after the CFD work, and because ofthe equipment availablein our labwereperformedwithT=0.26andN=5.78s−1.Thisgives

Re=4.34×104_{,comparedto4.67×10}4_{byJahodaetal.(2007)}

and inour CFD work. We donot expect the results tobe affected;allresultsarepresentedindimensionlessform,and allworkwasconductedinthefullyturbulentregime.Detailed informationontheexperimentsispresentedinPartIofthis work.Allinternalsweremodeledassheetbodies(Gunyoland Mudde,2009;Coroneoetal.,2011).LESandSM-RANS Simula-tionswereconductedwitha360◦domain,asusingaperiodic meshwillconstrainthemotionofmacro-instabilities(seePart I ofthis study).In accordance withthe experimentalwork ofJahodaetal.,thetracerconcentrationistrackedwithtwo probes,thebottomprobeatheightT/4andtopprobeatheight 1.25T,placedbetweenthebaffles,atT/20fromthewall.

3.1.2. Numericalsetup

Thenumericalsettingswerelargelysimilartothoseusedin PartI.Spatialdiscretizationwassetto2ndupwind(Gunyoland Mudde,2009;Coroneoetal.,2011)withRANS,andbounded centraldifferenceswithLESsimulations.Standardwall func-tionswereemployedforallsimulations.Thefreesurfaceat thetopwasmimickedbyusingano-shearsurface,allwalls weresettono-slip.Convergencewasdeclaredwhenthe resid-ualswerebelow10−5withinatimestep.Themeshesusedin thiswork areequaltothoseinPartIofthe work,withcell countslistedinTable1.Duetothecomputationtimerequired

forSM-RANSandSM-LES,itwasnotpossibletotestall

mod-elswithallmeshes.Additionalinformationonthemeshesis providedinSupplementarymaterialA.

Tracerwasinjectedinasphericalvolumewitharadiusof 0.0125m;thevolumecenterwasaty=0.551m(fromthe bot-tom),atr=0.0725m,inthebaffleplane.Wesett=0.00333s withslidingmeshandthe2IF-C LESsimulation.The2IF-SF

LESsimulationwasconductedwitht=0.001667s.Mixingin mesh2IF-M was studiedin triplicate withLES; twice with t=0.00333s,andoncewitht=0.001667s,totesthow repro-duciblethe resultsare.Temporaldiscretizationwassecond orderimplicit.The tracerand bulkfluidhad equal proper-ties,=1000and=0.001,suchthatthetracerwillnotdisturb theflowfield.Subgridspeciesdiffusioniscoupledtothe sub-gridturbulentviscositytviatheturbulentSchmidtnumber,

Sct=t/(Dt)=0.7,givingaspeciesfluxJi=−(lD+t/Sct)∇Ci

withD=10−9themoleculardiffusioncoefficientandCi the

scalarconcentration. 3.2. Turbulencemodels

Therealizable k− (RKE)and Reynolds stressmodel (RSM) used in SM-RANS modeling are well established, we refer to,e.g.GunyolandMudde(2009)fortheirdescription.Based onpreviousliterature,wefocus onthedynamic Smagorin-sky(LES-DYN)modelforLES.Thismodelassumesisotropic subgrid turbulence, using an eddy viscosity formulation with t being the subgrid turbulent viscosity. The filtered

Navier–Stokesequationsforanincompressiblefluidread: ∂ ¯ui ∂t + ¯ui ∂ ¯ui ∂xj =− 1  ∂ ¯p ∂xi + ∂ ∂xj





Here, t is the kinematic turbulent viscosity, t/. The

dynamicturbulentviscositytiscalculatedfromEq.(2)

ij−

1

3kkıij=−2tS¯ij (2) With Sij is the resolved-scalerate ofstrain tensor. The

subgrid-scalestresstensorijiscalculatedbytheapplied

sub-gridmodel.Thebasisofthedynamicsubgridmodel(LES-DYN) isformedbytheturbulentviscosityformulationofthe stan-dardSmagorinskymodel,Eq.(3):

t=L2s·



2 ¯SijS¯ij (3)

where Ls is the mixing length calculated as Ls=

min(d,CSV1/3c ) with  the Von Karman constant, d the

nearest wall distance and CS the Smagorinsky constant.

Whereas the standard Smagorinsky model typically pre-scribesCS=0.1,thedynamicmodelcomputesCSbasedonthe

localresolvedmotionsusingatest-filterapproachbasedon theworkofGermanoetal.(1990)andLilly(1992).Wereferto

Kimetal.(2004)fordetailsontheimplementationinFLUENT. Forstability,CSiscappedbetween0and0.23.

3.3. Analysismethods

WerecordedvelocitytimeseriesinthebaffleplaneinourLES simulations.Intheimpelleroutflow,theimpellerfrequency andthefirstandsecondharmonicthereofwereremovedvia Eq.(6),similartotheexperimentalprocedurereportedinPart I.Dataintheregionbetweentheimpellers,referredtoasthe inter-compartmentregion(withthehorizontalplane segre-gatingthecompartmentsreferredtoasinter-compartment plane),wasanalyzedforthepresenceofMIs.Thevelocitytime seriesatthepointscoincidingwiththeexperimentallyused gridwereanalyzedbyfittingtheauto-correlationsignal(Eq.

(4))andbyanalysisofthespectraldensityfunction(Eq.(5)), equaltotheexperimentalassessment:

ˆfit()=b+c0e−˛0+˙ni=1cne−˛ncos(2·nf ) (4)

S(f)= t



2ˆ(klag)w(klag)cos(kflag)



(5) Forthesemethods,theinfluenceofperiodiccomponents canberemovedfromthefluctuatingvelocityviaEq.(6)incase oftheauto-correlationsignal,andEq.(7)incaseofthespectral densityfunction: u_t=u·



HereEp/Etisthefractionofenergycontainedinthe

macro-instability componentsofthe u-spectrum;weassumed all frequencies below f/N=0.1 contribute to Ep. The influence

ofmacro-instabilities inthe inter-compartmentregion was addressedwithbothmethodsforLESsimulations.Inthiscase, twobasefrequenciesandtheirfirstharmonicswereincluded inEq.(6).Intheinter-compartmentregion,ktistheturbulent

kineticenergybasedonu_t,kt*thetotalfluctuatingenergy,and

the macro-instability kineticenergykMI=kt*−kt. Subscripts

fitandSIareusedtodistinguishbetweenresultscalculated using Eqs. (6) and (7), respectively. For sliding mesh simu-lations, thereisnoturbulent componentinthe fluctuating velocity,hencekMIwasdirectlycomputedfromthefluctuating

Reynolds-averagedvelocity,whilektisprovidedbythe

turbu-lencemodel,andkt*followsfromtheirsum.Thelocationy=0

isusedtoindicatethehorizontalplaneexactlybetweenthe twoimpellers,ataheight5T/6.Adetaileddiscussionofthe experimentalanalysismethodsisprovidedinPart Iofthis paper.

4.

Results

and

discussion

4.1. Validation:theimpelleroutflow

InFig.1,theimpellerdischargestreamprofilesareshown,in comparisontoexperimentalresultsfromliteratureaswellas PartIofthiswork.Experimentaldataonwasnotcollected inthecurrentstudyduetolimitationsintheequipment.As fortheMRFsimulationsinPartI,theSMsimulationsshow goodagreementwithexperimentaldataintheimpeller out-flow.Again,theRSMsimulationpredictsadecreaseinktand

nearthebladetip,wheretheRKEmodeldoesnot.Inthebulk oftheoutflow,RKEandRSMareinexcellentagreement.LESis wellcapableofcapturingUrad,butperformspoorerforktand

especiallyfor.Powernumbersandadditionaldiscussionon meshdependencyareavailableinSupplementarymaterialA. For the LES simulations, kt is computed based on the

resolved scales, removingperiodiccomponents (blade pas-sages)byEq.(6).Subgridkineticenergyisnotincludedinthe figures,whichmeans2IF-SFisexpectedtoyieldhigherktthan

2IF-Cand2IF-M,asisindeedobserved.However,2IF-SF,which isstillinsufficientlyfinetocaptureallenergycarryingeddies, doesover-estimatetheexperimentalkt.Thedissipationrate

ontheotherhandisstronglyunderestimated.Overall,the powernumberonenergydissipationfluctuatesaroundPo≈8,

Fig.1–ProfilesofUrad,ktandinthetopimpelleroutflowcomparedwithLDAdata.Bottomimpellerresultsareomitteddue tosimilarity.Toprow:SMsimulations.Solidline:2IF-F,SM-RKE.Dashedline:2IF-M,SM-RKE.Dottedline:2IF-M,SM-RSM.

Bottomrow:LESsimulations.Solidline:2IF-SF.Dashedline:2IF-M.Dottedline:2IF-C.Symbolsrepresentexperimental data.Abbreviations:W.P.=WuandPatterson(1989),M.J.=MurthyandJoshi(2008),D.Y.=DucciandYianneskis(2005).The bluecrossesintheUradplotrepresenttheupper-andlowerboundofthestudiesreviewedbyRanadeandJoshi(1990).

(2004) and Derksenand Van den Akker (1999) reported an under-estimationofwiththestandardSmagorninskymodel usingconstantCS=0.1.Delafosseetal.andSoosetal.found

thatsettingCS=0.2increasedagreement;ishighlysensitive

toCS.InagreementwithJahodaetal.(2007),weobservethe

dynamicmodelyieldsCS inthe range0.01–0.06.Theselow

CSvaluesareconsistentwiththestrongunder-predictionof

(Delafosse etal.,2008).Unfortunately,Jahodaetal.donot reportktandforverification.Clearly,accuratepredictionof

instirredtanksusingLESrequiresfurtherattention. 4.2. Mixingtimes

Themixingtimeisreportedindimensionlessform,95=N95,

with95beingthetimeinsecondsbeyondwhichCt/ ¯Ctisbound

between0.95and1.05,withCtthetracerconcentration.The

CoMquantifiesmixingintheentiredomain,andisdefinedas Eq.(8): CoM=







where95,CoMisachievedwhenCoM<0.0283(Hartmannetal.,

2006). The results for all SM-RANS and LES simulations are giveninTable2.SM simulationspredict aprobe-based 95≈110forRKEbased onthe bottomprobe, inagreement

with the SM simulation of Jahoda et al. (2007), and 15% abovetheexperimentalvalue,bothforthebottomandtop probe.Overall,95predictedwithSM-RKEisconsistentlylower

thanwithMRF-RKE(PartI)atthesamemesh,whichimplies

thateitherturbulentmassexchangebetweenthe compart-ments (t) ishigher inSM simulations,or MIs, which are

inherentlysuppressedinMRFsimulations,resultinahigher

Table2–Comparisonofdimensionlessmixingtimes95

fortheSM-RANSandSM-LESmethods,forthemeshes studiedinthiswork.botandtoprepresentthe

probe-basedresults,CoMthecoefficientofmixing.The workofHartmannisfollowedtosettheCoM-boundary.

Mesh bot/top CoM

2IF-M-SMRKE 112.7/89.5 126.5 2IF-F-SMRKE 110.1/84.9 122.0 2IF-M-SMRSM 133.3/107.9 147.0 2IF-C-LES 80.0/71.1 86.0 2IF-M-LES(t=1.67ms) 80.7/69.6 88.0 2IF-M-LES(t=3.33ms,run1) 81.6/69.6 n.m. 2IF-M-LES,(t=3.33ms,run2) 78.5/71.4 n.m. 2IF-SF-LES 94.0/79.1 99.0 Jahoda(LES) 81.5/n.r. n.m. Jahoda(EXP) 92.0/≈75 n.m.

n.m.=notmeasured,n.r.=notreported.

inter-compartmentmassexchange.Probeprofiles,reportedin

Fig.2AforMRFandSM,hintatthelatter:whereastheMRF

resultsshowaconstantincreaseindimensionless concentra-tionCt/ ¯Ct,wigglesintheprofilesforSMhintatthepresence

ofoscillatorymotionsintheinter-compartmentplane.

TheSM-RKEresultsaresimilarforbothmeshes,forprobe

dynamicsaswellas95.Reasonablesimilarityisobservedin

inter-compartmenthydrodynamicsbetweenthemeshes(see Section4.3),incontrasttoMRF-RKEinPartI.Thisgivessome confidencethatthenear-equal95isasignofmesh

indepen-denceforSM-RKE.TheresultsfromPartI,however,showthat

dataforSM-RKEonmesh2IF-SFisrequiredinordertomake

suchastatementwithconfidence;thecomputational require-mentsdidnotallowustoconductsuchasimulationwithin thisproject.Moregenerally,theMRFresultswerehighly

sen-Fig.2–Mixingprofilesfordifferentsimulations(bottomprobesignal).Black:typicalexperimentaldata,Jahodaetal.(2007). Top:RANSsimulations,including2MRFsimulations(reportedinPaperI).Bottom:LESsimulations.

sitivetodifferencesinQaxbetweenthemeshes,duetothe

flowfieldbeingfrozenduringmixing.Forslidingmesh sim-ulations,thedynamicratherthanfrozeninter-compartment flowfieldisexpectedtoleadtoalowermeshsensitivitythan wasobserved withMRF. OnlyoneSM-RSM simulation was conducted, yielding a higher 95, as well as shifted

inter-compartmentplane(seeSection4.3).Ingeneral,theinclusion ofturbulenceanisotropydoesnotseemtoimprovethe agree-ment withexperimentaldata, in the SM simulations,MRF simulationsofPartI,andsingle-impellerworkofGunyoland Mudde(2009).Hence,thereappearstobelittlereasontoopt forthemorecomputationallyexpensiveRSMmodel Rushton-stirredtankapplications,andwedidnotexploreSM-RSMat finermeshes.

TheLESsimulations with 2IF-C and 2IF-Myield95≈80

based on the bottom probe, in agreement with 95=81.5

observedbyJahodaetal.,usingasimilarmeshdensity.No effectofthetimestepsizeisobservedfor2IF-M.Simulation 2IF-SFyields95=94,inverygoodagreementwiththe

exper-imentalvalue 95=92 reported byJahoda et al. (2007). The

bottom-proberesponseprofilesforLESareshowninFig.2B. Itmustbekeptinmindthatthefiguresshowsingle realiza-tions,exceptforLESrunswithmesh2IF-M.Theresultsforthis caseshowlimitedvariationin95,whichgivesconfidencethe

differencein95betweenthismeshand2IF-SFisnotdueto

regularvariabilityintheLESmethod.Forthetopprobes,95

isinlinewiththeroughestimation madefromthemixing profilesofJahodaetal.Additionalproberesults,studyingthe sensitivityoftheresponsetoprobelocation,arereportedin SupplementarymaterialB.

4.3. Inter-compartmentdynamics:slidingmesh

SMsimulationspredictalower95 thanMRF,althoughthey

stillover-estimatetheexperimentalvalue.ThewigglesinFig.2

hintthedifferencebetweenMRFandSMliesintheinclusionof MIs.Inter-compartmentdata(Fig.3)showskt*ishigherforSM

simulationsthanforMRFduetotheinclusionofkMI,whichis

0inMRFsimulationsbyconstruction.Comparedtothe exper-iments,kt*isstillunder-estimatedeverywhereexceptnearest

to theshaft; hence 95 inSM simulations stillexceeds the

experimentalobservation. For SM−RKE, kt* isquite similar

betweenthetwomeshes,butthecontributionbykMIdiffers:

approx.10–25%formesh2IF-F,and5–60%for2IF-M.Closeto thewall,kMIisindecentagreementwithexperimentaldata

for2IF-M,butduetoanunder-estimationofkt,kt*islower

overall.

UradiswellcapturedbySM-RKE,althoughtheMRF

simu-lationsperformedslightlybetterinquantitativelycapturing

Uax;still,thereversalfromconvergingtodivergingflowis

rea-sonably captured withSM-RKE.SM-RSM performspoorly:a shiftininter-compartmentplanepositionisclearlyobserved in theprofiles for Urad, Uax and kt*. In contrastto theMRF

simulations,reportedinPartI,thisdoesnotreduce95for

SM-RSM.IntheMRFsimulations,ashiftintheinter-compartment plane positionwas associatedwith ahigher axial flowrate

Qax between the compartments.Thefrozen planeposition

meantthishigherQaxledtoconsistentlyfastermixing.In

con-trast,inSMsimulationstheinter-compartmentplaneposition isdynamic,whichimpliesthateveniftheplanepositionis shiftedsomewhat,thisdoesnotresultinaconsistentincrease ofmassexchangebetweenthecompartments.

Probing Uax at several points inthe inter-compartment

region(SM-RKE)revealsstrong oscillationsinthe Reynolds-averagedvelocity,asshowninFig.4AandB.Sincethereare no turbulent oscillations inthe Reynolds-averagedvelocity profilebyconstruction,thereisnoneedforauto-correlation fittingtodeterminekMI.Therefore, theMIfrequencieswere

Fig.3–Axialprofiles(baffleplane)of(A):Urad,(B):Uax,(C):kt,comparingLDAresults(symbols)withSM-CFDdata(lines).In (C(,theblackrectanglesrepresentthetotalkineticenergykt*,thebluediamondstheturbulentkineticenergykt,andthered diamondstheMIenergykMI.Theblacklinesrepresentkt*,theredlineskMI.Lines:CFDresultsatdifferentmeshdensities (dotted:2IF-M,RSM,dashed:2IF-M,RKE,solid:2IF-F,RKE).(Forinterpretationofthereferencestocolorinthisfigurelegend, thereaderisreferredtothewebversionofthisarticle.)

theFourierspectrum.Thespectrumshowsvelocity oscilla-tionsarehighlyperiodicfor2IF-M(Fig.4C),withadominant frequency f/N=0.058 and its harmonics. This is in excel-lent agreement with the jet instability frequency reported byPagliantietal.(2008),andthevalueobserved experimen-tallyinPartIofthiswork.Experimentally,wealsoobserved aweakcontributionoff/N=0.02,andstrongcontributionof

f/N≈0.04inthe parallelflowregion. Averyweak f/N=0.02 canbeobservedwithSM,whilef/N=0.04isabsent.Case

2IF-Fshowsmorescatter(Fig.4D);thedominantpeaknowisat

f/N=0.045withastrongshoulderatf/N=0.06forr/R=0.512 andr/R=0.694.Thisisqualitativelymoreinlinewith exper-imentalobservations,althoughthecontributionoff/N=0.06 istoo low.Atr/R=0.694, anadditional peakis observedat

f/N≈0.01,whichrepresentstheveryslowoscillationvisible inFig.4BthatisabsentinFig.4A.

Thequalitativeexperimentalobservationthataxial oscil-lationsincrease withincreasing radialpositioniscaptured bySMsimulations.ThenextquestionishowmuchMIs con-tribute to mixing. Fig. 5 shows the velocity vectors in an axialcrosssection(baffle-plane)atseveralmomentsthough amacro-oscillation.Duringtheoscillatorymotion,cross-over flowfromthetoptobottomcompartmentconnectsthe down-wardnear-wall flowinthetopcompartment,toflowalong theshafttowardstheimpellerinthebottomcompartment (Fig.5Band D).Tracerconcentrationsnapshots(Fig.6) con-firmthattracertransportbetweenthecompartmentsoccurs

dominantlyalongtheshaft.Nearthewall,theparallel-flow planesegregatingthecompartmentsisdisplacedasawhole withnovisibleinter-compartmenttransport.Videosof inter-compartment mixing are availableonline, for anMRF-RKE,

SM-RKEandLESsimulation.

4.4. Inter-compartmentdynamics:largeEddy simulations

ThevelocitycomponentsresolvedinLESspanamuchwider rangeoffrequencies,wellintotheturbulentdomain.Hence, weanalyzethemwiththesamemethodsastheLDAdatain PartI:fittingofthevelocitysignalauto-correlationwith damp-ened cosines(Eq. (4)),and spectralintegration(Eq. (5)). For spectralintegration,thecut-off frequencybetweenMIsand regularturbulencewassetatf/N=0.1;theenergycontainedin thespectralrangef/N<0.1isfullyattributedtokMI.Thismay

beanover-estimationofthetrueMIenergy,butbecausethe samecut-offwasusedinLDA,thismethodprovidesthemost directcomparison.Duetothenegligibledifferencebetween simulations witht=0.0033andt=0.001667, thereported resultsfor2IF-Mareobtainedwitht=0.0033,forwhichmore datewasavailable.

Similar to the experimental procedure, the auto-correlation signal was fitted with2frequency components (f1.1 and f2.1) and their first harmonics (f1.2 and f2.2).Fig. 7

Fig.4–(A)Axialvelocityaty=0versustime,2IF-MSM-RKE.(B)Axialvelocityaty=0versustime,2IF-FSM-RKE.(C)Fourier transformofA;insetshowsthesamegraphonalogscale.(D)FouriertransformofB;insetshowsthesamegraphonalog scale.Blackline:r/R=0.2.Blueline:r/R=0.512.Lightblueline:r/R=0.694.(Forinterpretationofthereferencestocolorinthis figurelegend,thereaderisreferredtothewebversionofthisarticle.)

Table3–AxialRMS-velocity,oscillationfrequenciesandtheircontributionsfor2IF-M-LESand2IF-SF-LESintheplane

y=0atdifferentradialpositions,usingtheperiodic-fittingapproach.Thecoefficientcrepresentthecontributiontothe Reynoldsstress ¯uu,withci.jthejthharmonicoffrequencycomponenti.Forbrevity,onlyuaxisreported,tablesforurad andu_tanareinsupplementarymaterialC.Toprows:2IF-M.Bottomrows:2IF-SF.

r/R 0.200 0.358 0.435 0.512 0.566 0.694 0.765 u_ax,M(m/s) 0.089 0.111 0.126 0.147 0.164 0.181 0.148 c1.1,ax,M 0 0.026 0 0.001 0.022 0 0 c1.2,ax,M 0.033 0.011 0.033 0.08 0.117 0.110 0.049 c2.1,ax,M 0.002 0.005 0 0.007 0 0.015 0.023 c2.2,ax,M 0.003 0.071 0.017 0.141 0.065 0.045 0.039 f1.1,ax,M/N 0.014 0.010 0.012 0.012 0.012 0.012 0.013 f2.1,ax,M/N 0.019 0.022 0.017 0.021 0.018 0.022 0.023 103_·k t,fit,M/U2_tip 9.35 8.98 9.59 9.36 9.84 9.91 7.00 103_·k MI,fit,M/U2tip 1.21 2.32 1.42 1.92 1.89 1.41 0.63 u_ax,SF (m/s) 0.088 0.106 0.117 0.132 0.148 0.208 0.174 c1.1,ax,SF 0 0 0.042 0.068 0.086 0.135 0.088 c1.2,ax,SF 0.011 0.007 0 0.014 0 0 0.039 c2.1,ax,SF 0 0 0.005 0 0 0.013 0 c2.2,ax,SF 0.022 0.09 0.006 0.030 0.006 0.210 0.108 f1.1,ax,SF/N 0.034 0.026 0.010 0.024 0.020 0.022 0.023 f2.1,ax,SF/N 0.043 0.042 0.042 0.036 0.040 0.036 0.035 103_·k t,fit,SF/U2_tip 6.48 8.22 8.90 7.37 8.28 10.2 8.06 103_·k MI,fit,SF/U2tip 1.80 1.01 0.60 1.89 1.62 4.00 2.10

positionsat axial positiony=0. Thedominant frequencies foraxialoscillationsinLESsimulationsareless sharpthan the experimentalfrequenciesreported inPart I, leading to a poorer fit quality. On average, 2IF-SF gives f1.1/N≈0.023

and f2.1/N≈0.039, albeit with significant variation between

radial locations. For f2, the harmonic f2.2/N≈0.078 has a

highermagnitude thatf2.1/N. Thesefrequencies liearound

the experimental value of f2.1/N≈0.06, which itself is not

observed in 2IF-SF. TheLES data furthermore hints at the presence of higher frequency oscillations, which are not captured in the current auto-correlation fit. For the axial oscillationsatmesh2IF-M,f1.2/N≈0.024hasarelativelyhigh

magnitude,andisnearlyequalinfrequencytof2.1.Harmonic f2.2/N≈0.04alsohasahighmagnitude.Whilethisfrequency

was relatively prominent experimentally, the strongest experimental component f/N≈0.06 is again not observed

Fig.5–Velocityvectorfields(2IM-RKE)duringfourstagesofamacro-oscillation.Thedashedredlinesindicatetheregions whereUax≈0,showingthatthroughouttheoscillationstheaxialseparationbetweenthecompartmentsislocallybroken, resultinginenhancedinter-compartmentmixing.

Fig.6–Snapshotsofmixingat6timepoints,2IF-M-SM,showingtracerexchangealongtheshaftisinfluencedbyMIs(video availableonline).

Fig.7–Fittedauto-correlationfunctionsfor2IF-M-LES(top)and2IF-SF-LES(bottom)aty=0,at3radialpositions.Blueline: rawdata.Redline:fitteddampedcosinefunction(seePartIfordetails).(Forinterpretationofthereferencestocolorinthis figurelegend,thereaderisreferredtothewebversionofthisarticle.)

in the simulation. As with the SM simulations, the radial oscillationshaveahighmagnitudeneartheshaft,dampening withincreasingradialposition,whiletheoppositeisobserved foraxialoscillations(Table3).

Thespectraldensity functions(Fig.8)show anf−1 scal-ingfor0.1<f/N<1,andanf−5/3scalingforf/N>1,inlinewith experimentalobservations.Themoderate spatialresolution ofthemeshesresultsinaquickdeviationfrom−5/3scaling atthehigh-frequencyendofthespectra.Forthe2IF-SF sim-ulations,thefilterlengthratiowasaround/=10−25with theKolmogorovlengthscale,thelowervalueneartheshaft andthehighernearthebaffle.Afinermeshmaybedesired, butforroutineusewithoutsuper-computingfacilities,2IF-SF isalreadymuchtoodemanding.

Inthelowfrequencyrange,thespectraldensityfunctions confirmthe observationsmadein theauto-correlation fits. Thespectrumfor2IF-Mdoesshowthebi-modalpeakobserved

experimentally,butatthelowerfrequenciesf1.2/N≈0.024and

f2.2/N≈0.04,whereasexperimentallyf/N=0.045and0.06were

observed. In2IF-SFthebimodal peakisnotobserved.Here

f1.1/N≈0.026andf2.2/N≈0.078contributemost,but,asinthe

auto-correlationdata,f/N≈0.045isabsent.

Boththeauto-correlationfitandspectraldensitymethod provideanestimateofthepercentageoffluctuatingkinetic energythatiscontainedinMIs.Firstwereportresultsforthe auto-correlationmethod.Usingthismethod,kMI,fit10–20%of

kt*atthemeasuredlocationsintheplaney=0for2IF-M,with

themaximumatr/R=0.358.2IF-SFyielskMI,fitis5–30%ofkt*,

withthemaximumatr/R=0.694.Forcomparison, experimen-taldatayieldedacontributionof8–20%withthemaximum atr/R=0.694whenusingtheauto-correlationmethod. Spec-tral integration gave anMI contribution iskMI,SI is25–30%

ofkt* for2IF-M,and 25–40%ofkt* for2IF-SF,comparedtoa

inte-Fig.8–Spectraldensityfunctionsofuax(darkblue)andurad (lightblue)at3radialpositions,intheplaney=0.(A)2IF-MLES.

(B)2IF-SFLES.(C)Experimentalresults.Dashedline:S(f/N)∝(f/N)−5/3.Dash-dotline:S(f/N)∝(f/N)−1.Dottedline:cut-off frequencybetweenMIandturbulence.(Forinterpretationofthereferencestocolorinthisfigurelegend,thereaderis referredtothewebversionofthisarticle.)

gration.Thedifferencebetweenauto-correlationfittingand spectralintegrationarisingduetotheinclusionoflimited fre-quencies intheformer, and alllow-frequencycomponents inthelatter. Next,wecomparethe profiles ofvelocityand kineticenergyinthe inter-compartmentregion.Becauseof themorestraightforwardcomparisonwithexperimentaldata, thespectralintegrationmethodisusedtodeterminetheMI contributioninthekineticenergydatapresentedinFig.9.

Aside from an offset at the baffle position, which is attributed tothe 2D baffle geometry,Urad is well captured

(Fig.9).ThechangefromconvergingtodivergingflowinUaxis

excellentlyassessedbyallbutthecrudestmesh.Overallgood agreementinktisobserved(despitethepoorerperformance

intheimpellerdischargestream),with2IF-Mover-estimative neartheshaft,whereas2IF-SFissonearthebaffle.Both simu-lationsreasonablycapturekMI,SIneartheshaft,butonlywith

mesh2IF-SFthepeakatr/R=0.694isproperlycaptured.Even thoughthefrequencydistributiondoesnotcompletelyagree withexperiments,theoverallkMI,SIandkt,SIarewellcaptured

by2IF-SF,inlinewiththegoodassessmentof95.Theslight

under-estimationof95bythecrudeandmediummeshmay

bebyvirtueoftheirhigherktneartheshaft:axialtransportof

tracerisstrongestatthislocation,aswasshowninthemixing snapshotsfortheSM-RANSsimulations.

5.

Concluding

remarks

Inthiswork,wereportslidingmesh(SM)andlargeeddy sim-ulations(LES)ofastirredtankwith2Rushtonimpellersat largemutualclearance.Earlierstudiesreported(1) experimen-talevidenceformacro-instabilities(MIs)ina2-Rushtonstirred tank,(2)anover-estimationofmixingtime95withthe

multi-plereferenceframe(MRF)simulationmodel,increasingwith meshdensity,(3)amilderover-estimationof95withSM,and

slightunder-predictionof95withLES.

Compartmentformationaroundtheimpellersleadstoa parallel radial flow in the inter-compartment plane, caus-ingpoormassexchangebetweenthecompartments,thereby formingaratelimiting stepinmixing. Ourhypothesiswas thatRANSsimulationsover-predict95,asReynoldsaveraging

leadstoanear-shearfreeflowintheinter-compartmentplane, whichfailstoproperlycapturethegenerationofturbulence by the colliding flow. This in turn leads to a local under-predicting turbulent viscosity t. Both SM simulations and

MRFsimulations(PartI)indeedunder-predictedktinthe

inter-compartmentregion,comparedtoLDAmeasurements,which supports this hypothesis. Additionally, macro-instabilities (MIs)were observedtocontainaround30% ofthe fluctuat-ingkineticenergykt*intheinter-compartmentplane;these

pre-Fig.9–Axialprofiles(baffleplane)of(A):Urad,(B):Uax,(C):kt,comparingLDAresults(symbols)withLES-CFDdata(lines).In (C),theblackrectanglesrepresentthetotalkineticenergykt*,thebluediamondstheturbulentkineticenergykt,andthered diamondstheMIenergykMI.Theblacklinesrepresentkt*,thebluelineskt,theredlineskMI.Lines:CFDresultsatdifferent meshdensities(dotted:2IF−C,dashed:2IF−M,solid:2IF−SF.(Forinterpretationofthereferencestocolorinthisfigure legend,thereaderisreferredtothewebversionofthisarticle.)

dictthepresenceofsuchMIs,albeitslightlyunder-estimating theirkineticenergy,whereasthe frozen-flowfieldMRF sim-ulations byconstruction donot predict them. Overall, the under-predictedkt causes 95 to beover-predicted by both

SM-RANSandMRF-RANS,buttheinclusionofMIsmeansthe

over-predictionbySM-RANSislesssevere;approximately20% bySM-RKEwithrealizablek−(whichoutperformedSM-RSM),

whileanover-estimationofnear60%wasobservedwithMRF atsamemesh.Inthiswork,wehaveusedstructured hexa-hedralmeshes.Theresultsmaybedifferentatunstructured meshes of similar resolution, but the over-predictions are expectedtoholdoncemeshindependenceisachieved.

TurbulentSchmidtnumber(Sct)tuningwassuggestedto

improveagreementin95onsomeoccasions(Montanteetal.,

2005;GunyolandMudde,2009).Thecurrentresultssuggests suchtuningisnotbasedonphysicalconsiderations,but as apatchwork solutiontorepairaninadequateestimationof

kt (hence t and the effect ofMIs on multi-impeller

mix-ing behavior. Whilesuch a patch bean adequate solution if approximatemixing dynamicssuffice, the limitations of multi-impellerRANSsimulationsmustbekeptinmind,and theRANSmodelsstudiedinthecurrentwork(realizablek− and RSM)seeminadequate forquantitativemixing assess-mentinmulti-impellersystems.

Themixingtimewasunder-estimatedapprox.13% with LESatcrudemeshes(below2milliongridcells),inaccordance

withJahodaetal.(2007).Withameshofapprox.10million cells,95wasinexcellentagreementwithexperiments.The

MIfrequenciesintheinter-compartmentregionaspredicted inLESsimulationsarenotassharplydistinguishableasinthe LDAdata, and differencesinthe dominantfrequenciesare observed.Atthecrude-andmediummesh,kMIwasstill

under-predictedingeneral,buthigherneartheshaft,highervalues wereobservedthanexperimentally.Theregionneartheshaft iswheremostmassexchangebetweenthecompartmentsis predictedtotakeplace,whichmayexplainthelower95with

acrudermesh.Despitethedifferenceinfrequencies,kMIisin

goodagreementbetweenthefine-meshLESandexperiments, asiskt.

TheappliedLESapproachdidover-predictkt and

under-predictintheimpellerdischargestream.Thisisattributed totheemployeddynamicSmagorinskymodel,whichpredicts lowvaluesofconstantCS.WhiletakingCSconstantand

fine-tuningitmay increaseagreement(Delafosse etal.,2014),a moreuniversalapproachisdesired.Assuch,thereisroom forfurtherexplorationofLESandothertransientsimulation approaches ((ID)DES, hybrid LES-RANS,etc.) instirredtank applications. Strictlyspeaking,the finestcurrent LESstudy wasunder-resolved,especiallyintheimpellerregion.Higher resolutionstudiesmaybedesirableforfurtherfundamental studies,butalreadythecurrentfinemeshisconsideredtoo demandingforroutinemixingstudies.Despitethein

predict-ing,LESperformedwellintermsofmixingassessment,even onthecrudestmesheswhichfarfromresolveallenergy car-ryingturbulencescales.AsbothSMand LESmispredict95

toasimilardegreeatthecrudemeshes,andbothhave simi-larcomputationaldemands,botharecurrentlyconsideredto bereasonablyoptionsformixingassessmentprovidedtheir degreeoferroriskeptinaccount;bothareclearlypreferred aboveMRF.However,ifcomputationalfacilitiesallowforfiner meshes,LESisthepreferredchoice.

Acknowledgements

Wewish tothank our colleagues at ECUST, Shanghai and DSM Sinochempharmaceuticals forourongoing collabora-tion.WearegratefultoProf.H.vandenAkker,Dr.S.Kenjeres, Dr. L. Portela and Prof. H.J. Noorman for their advice and insights.CHacknowledgesallstudentsfollowingM.Sc.course CH3421, who contributed to the simulation results aspart oftheircoursereports.EvertWagner,JosThieme,Christiaan Schinkel,StefantenHagen,YoupvanGoozenandRuudvan Tolaregratefullyacknowledgedfortheirtechnicalassistance. Thisworkhasbeenconductedwithinamulti-partyresearch project,betweenDSM-SinochemPharmaceuticals,TUDelft, East ChinaUniversityofScienceand Technologyand Guo-jia,subsidizedbyNWOandMoST(NWO-MoSTJointprogram 2013DFG32630).Allsponsorsaregratefullyacknowledged.

Appendix

A.

Supplementary

data

Supplementary data associated with this arti-cle can be found, in the online version, at

https://doi.org/10.1016/j.cherd.2018.06.007.

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