Numerical investigation of collision dynamics of wet particles via force balance

via force balance

Citation for published version (APA):

Buck, B., Lunewski, J., Tang, Y., Deen, N. G., Kuipers, J. A. M., & Heinrich, S. (2018). Numerical investigation of

collision dynamics of wet particles via force balance. Chemical Engineering Research and Design, 132,

1143-1159. https://doi.org/10.1016/j.cherd.2018.02.026

DOI:

10.1016/j.cherd.2018.02.026

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Published: 01/04/2018

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Pleasecitethisarticleinpressas:Buck,B.,etal., Numericalinvestigationofcollisiondynamicsofwetparticlesviaforcebalance.Chem.Eng. ContentslistsavailableatScienceDirect

Chemical

Engineering

Research

and

Design

jo u r n al ho m e 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

Numerical

investigation

of

collision

dynamics

of

wet

particles

via

force

balance

Britta

Buck

a,∗

_,

_Johannes

_Lunewski

a

_,

_Yali

_Tang

b,c

_,

_Niels

_G.

_Deen

c

_,

J.A.M.

Kuipers

b

_,

_Stefan

_Heinrich

a

a_{InstituteofSolidsProcessEngineeringandParticleTechnology,HamburgUniversityofTechnology,Denickestrasse} 15,21073Hamburg,Germany

b_{MultiphaseReactorsGroup,DepartmentofChemicalEngineeringandChemistry,EindhovenUniversityof} Technology,P.O.Box513,5600MBEindhoven,TheNetherlands

c_{Multiphase&ReactiveFlowsGroup,DepartmentofMechanicalEngineering,EindhovenUniversityofTechnology,} P.O.Box513,5600MBEindhoven,TheNetherlands

a

r

t

i

c

l

e

i

n

f

o

Articlehistory:

Received1September2017 Receivedinrevisedform15January 2018 Accepted16February2018 Availableonlinexxx Keywords: Coefficientofrestitution Forcebalance Collision Liquidlayer Numericalmodel

a

b

s

t

r

a

c

t

Knowledgeofcollisiondynamicsofsolidmaterialsisfundamentaltounderstandand pre-dictthe behaviorofparticulatemacroprocessessuchas influidizedbeds,mixersand granulators. Especially,particlecollisionswiththepresenceofliquidsarestillnotfully understood.Manyexperimentalinvestigationsaddressenergydissipationduetothe colli-sionandtheliquidinvolved.Forthistheso-calledcoefficientofrestitutionisoftenused, whichisdefinedasratioofreboundtoimpactvelocity,assuchdescribingdissipationof kineticenergy.Inthisworkanumericalmodelbasedonforcebalancesisproposed,which predictsthecoefficientofrestitutionfornormalandobliquecollisionsofaparticleandawet plate.Themodelisvalidatedbyextensiveexperimentsregardingtheinfluenceofcollision parameterssuchascollisionvelocityandangle,liquidpropertiesaswellasinitial parti-clerotation.Goodagreementbetweenmodelandexperimentsisfoundforallinvestigated parameters.

©2018InstitutionofChemicalEngineers.PublishedbyElsevierB.V.Allrightsreserved.

1.

Introduction

Collision dynamics of particles and between particles and wallsarefundamentalknowledgeforunderstandingthe over-all dynamic behavior of particulate processes, such as in fluidizedbedsormixers.Ifliquidsareadditionallyinvolvedin theprocessasliquidlayersordropletsontheparticle,e.g.in granulationoragglomeration,orasmoistureindrying appli-cations,thecollisiondynamicsbecomeevenmorecomplex. Theparticles may stick together aftercollision forming an agglomerate,ortheymightreboundresultinginadistribution oftheliquidbetweentheparticles.Consequently,thisleadsto muchhighercomplexityalsoforunderstandingormodeling ofwetparticulateprocesses.

∗ _{Correspondingauthor.}

E-mailaddress:britta.buck@tuhh.de(B.Buck).

To characterize collision dynamics many authors, e.g.

Luding (1998), Davis et al. (2002), Antonyuk et al. (2009),

Hogekampetal.(1994)andGollwitzeretal.(2012),usethe so-calledcoefficientofrestitution(CoR),whichisdefinedasratio oftherelativereboundvelocityoftheparticletotherelative impactvelocity: e=|vR v|=



Ekin,R Ekin =



1−Ediss Ekin (1)

Thus, the coefficient ofrestitution describesthe energy dissipated during a collision.In industrial processes parti-cle collisions can occur normally (perpendicular) but also obliquely,inwhichcasetheCoRcanbedividedintonormal, tangentialandrotationalcomponents:

en=|vR,n_v n |

(2)

https://doi.org/10.1016/j.cherd.2018.02.026

Pleasecitethisarticleinpressas:Buck,B.,etal., Numericalinvestigationofcollisiondynamicsofwetparticlesviaforcebalance.Chem.Eng. Nomenclature ˛ collisionangle[◦] ˛d empiricalconstant[–] ıl layerthickness[m] ın normaldisplacement[m] ε surfaceroughness[m]  viscosity[Pas] ϕ half-fillingangle[◦]

sl slidingfrictioncoefficient[–] i Poissonratio[–]

ω rotationalvelocity[rads−1]  density[kgm−3]

surfacetension[Nm−1] liquidcontactangle[◦]

A,B,C non-dimensionalcoefficients[–] a* _{non-dimensionaldistance[–]} Ca capillarynumber[–]

E* _{Young’selasticmodulus[Nm}−2_] Ekin kineticenergy[J]

Ediss dissipativeenergy[J] e coefficientofrestitution[–] Fc contactforce[N]

Fc,d dampingcontactforce[N] Fc,el elasticcontactforce[N] Fcap capillaryforce[N] Fcap, surfacetensionforce[N] Fcap,p capillarypressureforce[N] Fg gravitationalforce[N] Fvis viscousforce[N] F_vis,∞ viscousdragforce[N] Fvis,wall viscouswallforce[N] G* _{shearmodulus[Pa]} IP momentofinertia[kgm2] kd dampingconstant[Nsm−1] kel elasticspringconstant[Nm−3/2]

m mass[kg]

Mvis viscousmoment[Nm] pc capillarypressure[Pa] Re Reynoldsnumber[–] RP particleradius[m] ra azimuthalradius[m] rm meridionalradius[m]

rc contactradiusbetweenliquidandparticle[m] Vb bridgevolume[m3]

v velocity[ms−1]

vc tangentialvelocityinthecontactpoint[ms−1] y distancefromy-axis[m]

Indices n normal R rebound t tangential ω rotational et= vR,t vt (3) eω= ωR ω (4)

Additionally,initialandpost-collisionrotation, ωandωR respectively,canbeanalyzedseparately.

Dryparticle–particleaswellasparticle–wallcollisionswere already extensively investigatedbyseveralauthors. Experi-mentalinvestigationswereforexampleconductedbyKharaz et al. (2001), Dong and Moys(2006), Antonyuket al. (2010)

andFoersteretal.(1994).Fordifferentmaterialsthenormal coefficientofrestitutionwaseitherfoundtobeindependent ofimpact velocityforelasticmaterialsortodecrease with increasingimpactvelocityfor(partly)plasticdeforming mate-rials.Forobliquecollisionstangentialmovementdependson thematerialpropertiesofthecollidingsurfaces.Theparticles might roll/stickor slideonthe walldepending onthe fric-tioncoefficientofthesurfacesandoncollisionangleFoerster etal.(1994).Furthermore,DongandMoys(2006)foundastrong dependenceofthetangentialcoefficientofrestitutionon ini-tialrotationoftheparticles.

Various contact models were developed to predict the reboundbehavior ofdifferentmaterials.Alinearmodelfor normalcontactforcesduringelasticcontactswasdeveloped by Hertz (1882), which was extended to oblique collisions byMindlinandDeresiewicz(1953).Tsujietal.(1992)further extendedthemodelsofHertzandMindlin&Deresiewiczto non-linearvisco-elasticcollisionsbyapplyingadampingforce totheelasticcontactforces.ThemodelsofHertzaswellas Tsujietal.bothfeatureaconstantnormalCoRindependence ofcollisionvelocity.Brilliantovetal.(1996), asanexample, developedamodelalsobasedonHertz,whichincludes dis-sipativeviscoelasticeffectsresultinginavelocitydependent normalCoR.Thornton(2009)developedalternativemodelsfor elasticmaterialshavingacloserlookontheinfluenceof ini-tialrotationaswellasforparticleswithinelasticdeformation behavior(Thorntonetal.,2013).Theinfluenceofadhesionon micro-particleswasincludedintomodelsforexamplebyLiu etal.(2011).Kruggel-Emdenetal.(2007)summarizedvarious dry normalforcemodels(linear,non-linear,hysteretic)ina reviewandfurtherextendedthemodels.DiRenzoandDiMaio (2004) comparednormalandtangentialcontactforce mod-els fortheirapplicabilityindiscreteelementmethod(DEM) simulations.

Deformation behavior of dry particles during collisions werefurtherinvestigatedviafiniteelement(FEM)simulations byWuetal.(2003b,2003a,2009)andZhengetal.(2012).Feng etal.(2009)modeledtheadhesionofmicro-particles.

Knowledge about dry contact forces is the basis for understanding wet particle collisions. However, the liquid introduces additional forces such as viscous and capillary forces,whichleadtothecomplexbehaviorofwetparticulate processes.Therefore,extensiveresearchregardingthe influ-enceoftheliquidoncollisionsdynamicsisalsonecessary.

Several authors considered particle–wall or particle–particle collisions immersed in liquid in normal direction (Josephet al., 2001; Davis et al., 1986)as well as foroblique impacts(e.g. Josephand Hunt(2004)aswell as

YangandHunt(2006)).Theyfoundthenormalcoefficientof restitutiontobestronglyreducedduetoviscousforcesinthe liquidandadependenceontheStokesnumberSt= mPvn

6R2 P

was described. For oblique collisions the tangential movement was found todiffer if the surfaces are eithervery smooth and someliquidispresentbetweenthe surfaceduringthe complete collision,or if asolid–solid contact happens due tosurfaceroughness.Forroughsurfacestheinteractionsof immersedparticlesintangentialdirectionnearlyequalthose ofdrycollisions,whileforsmoothsurfacefrictionisstrongly

Pleasecitethisarticleinpressas:Buck,B.,etal., Numericalinvestigationofcollisiondynamicsofwetparticlesviaforcebalance.Chem.Eng. reducedbyuptooneorderofmagnitude(JosephandHunt,

2004).

Considerableworkwasalsodoneinexperimental investi-gationsofparticlescollidingwithwallscoveredbythinliquid layers,e.g.innormaldirectionbyBarnockyandDavis(1988),

Hogekampetal.(1994),Davisetal.(2002),Kantaketal.(2005),

Antonyuketal.(2009), Sutkaretal.(2015), Gollwitzeretal. (2012),Fuetal.(2004)aswellasinCrügeretal.(2016a).Wet oblique collision experiments with thin liquid layers were exemplaryconductedbyKantakandDavis(2004),Maetal. (2013,2015,2016),Crügeretal.(2016b),Bucketal.(2017).In comparisontocollisionsimmersedinliquid,inthecaseof thinlayersthelayerthicknesshasadditionallytobetaken intoaccount.Furthermore,capillaryforcesmaynotbe neg-ligible,sincealiquidbridgeformsbetweenliquidlayerand reboundingparticles,rupturingatacriticallength.

Besidesexperiments,volumeoffluid(VOF)combinedwith immersedboundary(IB)methodsareusedforsimulatingwet particle collisionstofurtherinvestigate collisiondynamics, especially for configurations, which are difficult to realize experimentally(verythinlayers,smallvelocities).LinandLin (2013) forexample performed immersed boundary simula-tionsinvestigatingtheflowfieldofaparticlemovingnormally inthedirectionofawall.Jainetal.(2012)combinedVOFandIB methodstomodelaparticleimpactingnormallywithaplate, whichiscoveredbyaliquidlayer. TheresultsofCoRagree wellwithexperiments,howevertheformationoftheliquid bridgedidnot.Hence,Tangetal.(2017)extendedthisVOF/IB modelbyfurtherimplementingatensileforcemodelanda contactmodel.Thisledtoamuchbetteragreementbetween modelandexperimentsregardingtheoverallcollision dynam-icsincludingliquidbridgeshapeandlifetimeaswellasthe dependenceoftheCoRonseveralparameters(collision veloc-ity,layerthickness,liquidviscosity,surfacetension).Kanetal. (2015)usedacomputationalfluiddynamicsapproachto inves-tigatetheeffectofcollisionvelocityonthecapillarybridge forceaswellastheinfluenceofparticlewettabilityon colli-siondynamicsduringnormalparticle–particlecollisions(Kan etal.,2016).Wuetal.(2016)useddirectnumericalsimulation (DNS)topredictliquidbridgeformationbetweenwetparticles andliquidtransportduringcollisionofparticles (Wuetal., 2017).

Severalotherauthorsworkedonadescriptionofcollision dynamicsforwetparticlecollisionsbasedonforceorenergy balances.Enniset al.(1991)proposedalimiting criteriafor stickingoftwoparticlesduetoviscouseffectscharacterizedby acriticalstokesnumber.Capillaryforcesareneglectedthough, whilethesolid–solidcontactisaccountedforbyagiven“dry CoR”.Darabiet al.(2009) usedasimilar approachasEnnis etal.predictingnormal,limitingcasesforcollisions,where capillaryforcesorviscousforcesaredominating.Antonyuk etal.(2009)proposedaforce balancemodelincluding con-tactforces,drag,capillaryandviscousforcestopredictthe reboundbehaviorofaparticlenormallycollidingwithawet targetplate.AlsoMüllerandHuang(2016)developedanenergy balanceincludingviscouseffectsfornormalwetparticle–wall collisions.Mikami etal.(1998)implementedcontactforces, dragaswellascapillaryforcesintoaDEMsimulationto simu-lateawetfluidizationprocess.AlsoNaseetal.(2001)proposed aDEMmodelforwetgranularsystemsincludingviscousand capillaryforcesandvalidatedthemodelbycomparing simu-lationandexperimentofthestaticangleofreposeaswellas tumblerexperiments.

However,noforcebalancemodelfornormalandespecially obliquewetparticle-wallorparticle-particlecollisionsexistin literatureyet,whichisfullyvalidatedbyextensivesingle par-ticlecollision experiments.Therefore,this workproposesa forcebalancemodelthatpredictsnormalandoblique colli-siondynamicsofparticlesandwallscoveredbyaliquidlayer. Contactforcesandgravitationalforceareconsideredaswell asviscousandcapillaryforcestorepresentthedissipationof energyintheliquidlayer.Previousexperimentalresultsofour groupCrügeretal.(2016a),Crügeretal.(2016b)andBucketal. (2017)areusedtovalidatetheforcebalancemodelforalarge rangeofparameters.

2.

Methodology

2.1. Forcebalancemodel

Aforcebalanceisusedforsimulatingcollisiondynamicsfora particleimpactingawetplateinthiswork.Forthisthenormal partofacollisionisdividedintofoursubsections(Fig.1)ascan alsobefoundinAntonyuketal.(2009)orSutkaretal.(2015). Firsttheparticlepenetratesintotheliquidlayerresultingin viscousFvis,n andcapillaryforcesFcap,n actingonthe parti-cle.Whenreachingthesolidsurfaceorthewallelasticand dissipativecontactforcesFc,nleadtoareboundoftheparticle changingthedirectionofmotion.Thewallisassumedtohave aninfinitemasscomparedtotheparticleandthusgainsno movement.Duringthethirdphasetheparticleemergesback tothesurfaceoftheliquidagainaccompaniedbyviscousand capillaryforces.Whenemergingfromthesurfaceofthe liq-uidaliquidbridgeisformed,whichstretchesuntilacritical rupture lengthofthe bridgeisreachedand thebridge rup-tures.Duringthisphasestillviscousandcapillaryforcesact. Afterruptureapartoftheliquidstaysonthelowerpartofthe particle,whiletheremainingfractionfallsbackintotheliquid layer.Furthermore,throughoutallphasesagravitationalforce Fg,nactsontheparticle.

Formodelinganobliquecollisionasuperpositionofnormal andtangentialmovementisassumed,exceptforthe tangen-tialcontactforce(phaseII),whichdependsonthedeformation innormaldirection.Thus,themotioninnormaldirectionis modeledasdescribedabove,whileintangentialdirection dur-ingphaseIIanadditionalcontactforceintangentialdirection isadded(Fc,t),whichincludesCoulomb’sfrictionlawas lim-itingfactor,andinphaseIandIIIatangentialviscousforce (Fvis,t)andviscousmomentum(Mvis)areincluded.Fig.2shows schematicallytheactingforcesduringphaseIIIandIVofan obliquecollision.

Theparticle’smovementwithtimeiscalculatedbysolving Newton’slawofmotioninMatlabforallphasesofthecollision viausingthefunctionode45:

mPdvn dt =



Fn (5) mPdvt dt =



Ft (6) IP dω dt =Fc,tRP+Mvis (7)

Contactforcesactingduringacontactbetweentwosolid surfacescanbecalculatedbythemodelofTsujietal.(1992), whichisbasedonthemodelsofHertz(1882)andMindlinand Deresiewicz(1953).Thecontactforceconsistofoneelasticpart

Pleasecitethisarticleinpressas:Buck,B.,etal., Numericalinvestigationofcollisiondynamicsofwetparticlesviaforcebalance.Chem.Eng.

Fig.1–Schematicofcollisionsubsections:(phaseI)penetrationofparticleintoliquid;(phaseII)contactofsolidsurfaces; (phaseIII)particleemergingfromliquid;(phaseIV)formationandruptureofliquidbridge.

Fig.2–SchematicrepresentationofthemodelassumptionsinphaseIIIandIV.

Fc,elandonedissipativepartFc,d.Innormaldirectionitcanbe expressedas:

Fc,n=Fc,el,n+Fc,d,n (8)

=kel,n·ı3/2n +kd,n·vn (9) withınrepresentingthedisplacementinnormaldirection. Withtheassumptionoftheplatenotgettingdeformedand beingstatic,thenormaldisplacementcanbecalculatedfrom particleradiusRPandthepositionoftheparticleiny-direction:

ın=RP−y (10)

kel,nandkd,naretheelasticspringrateanddamping coef-ficient,respectively: kel,n= 4 3 E∗ √ R∗ (11) kd,n=˛d



m∗_·kel.n ı1/4n (12)

Thestar*indicatesaveraged propertiesofbothcollision partnersregardingmassm*_{,radiusinthecontactareaR}*_and

elasticmodulusE*_.˛

disadampingconstantdependingonthe drycoefficientofrestitutioninnormaldirectionen,dry:

m∗=



1 mi + 1 mj



−1 (13) R∗=



1 Ri + 1 Rj



−1 (14) E∗=



1−2_i Ei + 1−2 j Ej



−1 (15) ˛d=−ln(en,dry)·



5 ln2(en,dry)+2 (16)

Thecontactforceintangentialdirectionissimilardefined asthatinnormaldirection:

Fc,t=Fc,el,t+Fc,d,t (17) =kel,t·ıt+kd,t·vt (18)

withelasticstiffness kel,t=8·G∗

√

Pleasecitethisarticleinpressas:Buck,B.,etal., Numericalinvestigationofcollisiondynamicsofwetparticlesviaforcebalance.Chem.Eng.

Tsujietal.(1992)proposethatthetangentialdamping coef-ficientcanbeassumedtoequalthatofthenormaldirection: kd,t=kd,n=˛d·



m∗_·kel,n·ı1/4n (20) whereinG*_{istheaverageshearmoduluscomposedofthe} shearmodulusofbothcollidingmaterialsGi:

Gi=

_E i 2· (1+i)

(21) G∗=



2−2 i Gi + 2−2 j Gj



−1 (22)

Thetangentialdisplacementıtaccountsforthetangential movementoftheparticleinthecontactpointincluding rota-tionoftheparticleandiscalculatedaccordingtoDiRenzoand DiMaio(2004):

ıt= (x−x0)+rot·RP (23) withx0asthex-coordinateoftheparticlecenter,when par-ticleandwallaretouchingthefirsttime,androttheangleof rotation.

The tangential contact force however is limited by Coulomb’s friction law. When the tangential contact force reachestheproductofslidingfrictioncoefficientsland nor-malcontactforce,slidingoccurs:

Fc,t=



Fc,el,t+Fc,d,t ifFc,t<sl·Fc,n sl·Fc,n ifFc,t≥sl·Fc,n

(24)

Viscous(ordrag)forcesaccountfortheresistancedueto the liquid shear flow, which results from the moving par-ticle displacing the liquid. They are investigated by many researchers,e.g.ShiandMcCarthy(2008),LinandLin(2013)

andPitoisetal.(2000).White(2006)proposeamodelfora par-ticlemovinginaninfiniteliquid,whichisvalidinawiderange ofReynoldsnumberRe<2×105_{.ApplyingaparticleReynolds} number(Eq.(26)),wherethecharacteristiclengthisdefined asdiameteroftheparticlefraction,whichissubmergedina liquidlayeronaplate,thismodelwasapproximatedforthe caseofaparticleimpactingawetplate:

Fvis,n,∞=6lRPsin(ϕ)vn·



1+ ReP 4·



1+√ReP

+ReP 60



(25) ReP=2RPsin(ϕ)vnl_ l (26) withϕastheso-calledhalf-fillingangle,characterizingthe fractionoftheparticle,whichiscoveredbyliquid.

Foraparticleapproachingclosetoawallseveralauthors (ChanandHorn,1985;Matthewson,1988;Lianetal.,2001) pro-poseadifferentequationdependingonthedistancebetween bothsolidsurfaces:

Fvis,n,wall=

6 lR2Pvn ıl−y

(27)

Since this equation tends to infinity for the gap size approacheszero,acriticalminimumgapsizeıl−y=0.5␮m

wasdefinedaccordingtoanaverageroughnessofthe parti-cles.Afterreachingthiscriticalgapsizetheviscousforceis keptconstantuntilcontactofparticleandwall.

Inthismodelacombinationofbothmodelsforthenormal viscousforceisused,wherealwaysthelargerofbothisacting:

Fvis,n=



F_vis,n,∞ ifFvis,n,wall≤Fvis,n,∞ Fvis,n,wall ifFvis,n,wall>Fvis,n,∞

(28)

In tangential direction a viscous lubrication force and a viscous momentum are implemented, which were first introducedbyGoldmanetal.(1967).Thesemodelequations considertheforceandmomentuminthegapbetweena par-ticlemovingparalleltoawallinsidealiquid.

Fvis,t =6RP ·

8 15ln

_RP ıl−y

+0.9588

vt +

2 15ln

_RP ıl−y

+0.2526

ωRP

(29) Mvis =8R2P ·

₁₀1 ln

_RP ıl−y

+0.1895

vt +

2₅ln

_RP ıl−y

+0.3817

ωRP

(30)

Capillaryforcesaremodeledonlyinnormaldirectionand inaccordancetothetoroidapproximationbyKralchevskyand Nagayama(2001).Theyarecomposedoftwoparts,onepart resultingfromthesurfacetensionFcap, andtheotherfrom capillarypressureFcap,p:

Fcap,n=Fcap, +Fcap,p (31)

=−2 rcsin(ϕ+)+r2

cpc (32)

withtheliquidcontactangle.Herercrepresentstheradius ofthatpartoftheparticle,whichisincontactwiththe liq-uidasindicatedinFig.2.Thecapillarypressurepcdepends onsurfacetensionaswellasthemeridionalradiusrm and theazimuthalradiusra,whichrepresentsthesmallestcross sectionalradiusoftheliquidbridge:

pc= ·

1 rm− 1 ra

(33) ra=



rc+rm· [sin(ϕ+)−1] ifϕ+<90◦ rc ifϕ+≥90◦ (34) rm= y−RP·cosϕ−ıl 1+cos(ϕ+) (35)

Theliquidbridgebreaksafterreachingacertainlengthlrupt endingphaseIVaswellasthecalculationoftheforcebalance. Themaximumrupturelengthiscalculatedviatheequation ofPitoisetal.(2001)dependingoncapillarynumberCa,liquid contactangleandthevolumeofthebridgeVb:

lrupt=

1+ 2

·



1+Ca1/2

·V1/3_b (36) Ca=vn (37)

Pleasecitethisarticleinpressas:Buck,B.,etal., Numericalinvestigationofcollisiondynamicsofwetparticlesviaforcebalance.Chem.Eng.

Table1–Propertiesoftheparticles.

Material Diameter[mm] Density[kgm−3] E-modulus[GPa] Poissonratio[–] DryCoR[–]

Glass(SWARCO-Vestglas,2018a,b) 0.91 2500 63.0 0.23 0.96

1.74

␥-Al2O3(Antonyuketal.,2009) 1.74 1040 14.5 0.25 0.76

Table2–Liquidpropertiesat20◦C.

Liquid Density[kgm−3] Viscosity[mPas] Surfacetension[mNm−1]

Water 998 1.0 72.80

60gL−1Tween20 998 1.0 37.30

35wt%glycerol(GlycerineProducers’Association,1963) 1086 3.0 69.61

51wt%glycerol(GlycerineProducers’Association,1963) 1130 6.7 68.20

Adetaileddescriptionofthecalculationofthebridge vol-umecanbefoundinAppendixA.

For gravitational force the mass ofthe particle and the addedmassbytheliquidbridgeisconsidered:

Fg,n=

4 3R 3 Ps+Vbl

·g (38)

Thebuoyancyforcewasfoundtobeneglectableandwas thereforenotintegratedintothemodel.

2.2. Experimentalvalidation

For experimentalvalidation mainly resultsof ourprevious works(Crügeretal.,2016a;Crügeretal.,2016b;Bucketal., 2017)were used.Additionally, somenewexperimentswere conducted.

2.2.1. Experimentalsetup

The setupused forconducting experiments forvalidation mainlyconsistsofaparticleimpactingonawettargetplate. Theparticleeitherfallsdownontotheplatebygravity(Crüger etal.,2016a),shotobliquelyontotheplatebypressurizedair (Crügeretal.,2016b)orrollsdownarampcollidingwiththe platehavinginitialparticle rotation(Bucket al.,2017).The plateiseitherdryor coveredwithaliquidlayerofdefined thickness.Thelayerthicknessiscontrolledbeforeeach col-lision bya confocal sensor. Thecollision is captured from twodirectionsbysynchronizedhigh-speedcamerasallowing athree-dimensionalanalysisofthereboundbehavior(Fig.3). Impactvelocitiesaredefinedasthevelocityoftheparticle centerdirectlybeforetouchingtheliquidlayer,whilerebound velocity isdefined right after rupture of the liquid bridge. Tomeasurerotationalvelocities,theparticlesaremarkedby several dots before the experiments. Rotational velocity is calculated,bytrackingthesedotsalongwiththeparticle cen-ter.Foreachconfigurationofcollisionparametersatleast15 experimentswereconductedtogetreliableresults.Standard deviationsareshownforeachmeanvalue.

Duringobliquecollisionsthenormalimpactvelocitywas keptconstantat1ms−1toachievecomparability.Thus,with increasing collision angle from the vertical direction the tangentialimpactvelocityand accordinglythetotalimpact velocityincreaseaswell.

2.2.2. Materials

Glass beads (Swarco Type S) of two different sizes and ␥-aluminumoxidespheres(␥-Al2O3,companySasol)wereused asparticles.Table 1summarizesthe propertiesofthe par-ticles.Bothparticles featureahighsphericity ofabove0.91

Fig.3–Schematicrepresentationoftheexperimentalsetup formeasuringthecoefficientofrestitutionofaparticle impactingobliquelyonatargetplatecoveredwithaliquid layer.

and an average surface roughness in the range of a few micrometer.Thetargetplatewasalsomadeofglassandwas 80mm×80mm×10mm(W×L×H)insize.Thethicknessof theplatewasproventobelargeenoughtoneglecteffectsof elasticwavesinsidethematerialbyequation18ofAntonyuk etal.(2010).

VariousNewtonianliquidswereusedforthelayeronthe wall:Water,glycerol-watersolutionswithdifferentviscosities andasolutionofasurfactant,inthiscase60gL−1Tween20– watersolution,withhalfthesurfacetensionofwater(Table2). TheconcentrationoftheTween20 solutionwaschosenas criticalmicelleconcentrationasgiveninHeleniusetal.(1979). ViscosityofeachliquidusedwasmeasuredinaRHEOTESTRN 4.1(companyRHEOTESTMedingenGmbH).Surfacetensionof theTween20solutionwasdeterminedbyDu-Noüy-method inaK100ForceTensiometer(companyKrüssGmbH).

3.

Results

and

discussion

3.1. Assumptionsforcapillarybridgegeometry

Forcalculationoftheforcebalanceofwetcollisionsthe geom-etryofthecapillarybridgeisimportant.Asdiscussedabove, theshapeofthebridgeisassumedastoroid.InFig.4the for-mationofacapillarybridgeduringreboundofaparticlefroma wetwallisshownexemplaryforaglassparticle.The assump-tionofacircularcontourofthebridgeisreasonablefornearly thecompletelifetimeofthebridge.Tocalculatethe geome-trytwoinputparametersarerequired:thehalf-fillingangleϕ

Pleasecitethisarticleinpressas:Buck,B.,etal., Numericalinvestigationofcollisiondynamicsofwetparticlesviaforcebalance.Chem.Eng.

Fig.4–Evolutionofcapillarybrideshapeduringlifetime.

Fig.5–Half-fillingangleandcontactangleoverbridge lifetimeforglassparticlesimpactingonaglasswall.

andthecontactangle,forwhichapproximatevalueswere determinedfromexemplaryimagesofcollisionexperiments.

Fig.5displaysmeasuredhalf-fillingandcontactanglesover bridgelifetime.Thecontactangleisnearlyconstantover a largepartofbridgelifetime,exceptforaboutthefirst millisec-ond,somescatteroftheresultsisquitehigh.Furthermore,it doesnotchangewithvariationoflayerthicknessorimpact velocity.Hence,thecontactanglewasassumedconstantin the force balance model. Measurements for glass particles aswell asfor␥-Al2O3particlesand variousliquidsresulted ina(dynamic)contactangle ofapproximately 15◦ forboth materialsandallliquids,whichwasthenassumedinthe sim-ulations.

Thehalf-fillinganglefeaturesamorecomplextrendwith bridgelifetime.Inthebeginning,whentheparticleisstill par-tiallysubmergedintheliquidlayeronthewall,thehalf-filling angledecreases.Assoonasaliquidbridgeisfullybuilt,the half-fillingangleisapproximatelyconstantoveralongperiod ofbridgelifetime.Only,whentheshapeoftheliquidbridge constricts(rightimageinFig.4)thehalf-fillingangleslightly increasesagain,whiletheupperpartofthecapillarybridge

Fig.6–Studyofconvergence.

gains adroplet-like form.Adecrease ofthelayerthickness resultsinandecreaseofthehalf-fillingangle,becausethe par-ticlecannotpenetrateasfarintothethinnerlayer.Sincethe exacttrendofhalf-fillinganglewithbridgelifetimevarieswith layerthicknessandliquidviscosityandnomodelsarepresent in literature to predict these dependencies, the half-filling angleisapproximated byaconstantaveragevaluefor sim-plification. Withincreasingviscositythe averagehalf-filling angleslightlydecreasedintheexperimentsbyafewdegrees. Thevaluesapproximated forthemodelaresummarizedin

AppendixA(Table5).

3.2. Convergencestudy

Beforeusingtheforcebalancemodelthecalculationhasto betestedregardingnumericalstability.Asmentionedabove, theforcebalanceissolvedinMatlabviafunctionode45,which setsthetimestepautomatically,whilekeepingarelativeand absolute tolerance for the results. Both relative and abso-lutetoleranceshavetobespecifiedforthesimulation.Fig.6

shows the numerical resultsfor aglass particle impacting with1ms−1onaglassplatecoveredbya400␮mthickwater layerforvaryingrelativetolerance.Theabsolutetolerancewas alwaysthreedecimalpowerssmallerthantherelative toler-ance.Sincetheresultingcoefficientofrestitutionisconstant forrelativetolerancessmallerthan10−13,forallsimulations thefollowingtolerancesareused:relativetolerance10−13and absolutetolerance10−16.

3.3. Normalcollisions

Usingtheparticleandliquidpropertiesaswellashalf-filling and contact anglesas inputparameters, rebound behavior can now be predicted via the force balance model. Fig. 7

showsacomparisonbetweenexperimentsofglassparticles impacting ona glassplate and theproposed force balance

Pleasecitethisarticleinpressas:Buck,B.,etal., Numericalinvestigationofcollisiondynamicsofwetparticlesviaforcebalance.Chem.Eng.

Fig.7–ValidationofthenormalCoRindependenceofcollisionvelocityfor1.74mmglassparticlesanddifferentlayer thicknessofwater(a),0.91mmglassparticlesanddifferentlayerthicknessofwater(b),1.74mmglassparticles,200␮m

layerthicknessandvaryingviscosity(c)and1.74mmglassparticle,100␮mlayerthicknessandvaryingsurfacetension(d). Solidline:forcebalance;circles:ownexperiments.

model.Theconstantbehaviorofthedrycoefficientof restitu-tion(Antonyuketal.,2010)regardingcollisionvelocity(black signsin(a))arenicelyrepresentedbythemodelascouldbe assumed,sincethecoefficientofrestitutionisaninput param-eterofthedrycontactmodelbyTsujietal.(1992).Applying aliquidlayerontheplateresultsexperimentallyinastrong dependenceofthenormalcoefficientofrestitutiononthe col-lisionvelocityaswellasthelayerthickness(Fig.7(a)):Belowa critical“sticking”velocitythenormalcoefficientofrestitution iszero,becausethereboundenergyoftheparticleistoosmall tomaketheliquidbridgerupture.Abovethissticking veloc-ity,thenormalcoefficientofrestitutionisfoundtoincrease steeplyfirst.Whenincreasingtheimpactvelocityfurther,the slopedecreasesuntilaconstantvalueisreached asymptot-ically,whichalwaysstayssmallerthanthedrycoefficientof restitution.Thistypicaltrendwasalsoobservedforexample byKantaketal.(2005), Gollwitzeretal.(2012),Mülleret al. (2013)andMüllerandHuang(2016).Inaccordancetoprevious experimentalworks(Antonyuketal.,2009;Gollwitzeretal., 2012;Crügeretal.,2016a;Maetal.,2016),anincreaseinlayer thickness,leadstoadecreaseofthenormalcoefficientof resti-tution,becausetheparticlepenetratesdeeperintothehigher liquidlayer,resultinginalongertime,thedissipativeforces havetimetoact.Theforcebalance modelproposed inthis workisabletorepresentthesedependencies.Indetail,the forcebalancenicelypredictstheoveralltrendswithincreasing collisionvelocityaswellaslayerthickness.Ithoweverslightly overpredictsthecoefficientofrestitutionforhighvelocities. Thesesmalldifferencesmightbefurtherreducedbyapplying otherviscousandcapillaryforcemodels.Theviscousforce modelsinliteratureweredevelopedforparticlesinaninfinite fluid,howevernomodelsexist,whichcoveraparticlepartly submergedinaliquidlayer.Thus,theexistingmodelshadto beadjustedtofindanapproximationfortherealphysics.

Sim-ilarly,existingcapillarymodelsarefor(quasi)staticbridges. Thebridgeformationandelongationintheinvestigatedcase ofanimpacting and reboundingparticle howeverishighly dynamic.Therefore,thesmalldifferencesbetweenmodeland experimentinthisworkmightstillbereducibleby develop-ingmoreappropriatemodelsforcapillaryandviscousforces, whichhoweverisoutofthescopeofthiswork.Furthermore, theoverallconformityisalreadyassumedsufficientlyhigh.

Fig.7(b)displaysthecoefficientofrestitutionforsmaller glassparticlesof0.91mmindiameterattwodifferentlayer thicknesses.Theexperimentswiththesesmallerparticlesare evenbetter predicted bythe forcebalance model than the largerparticles.Stickingvelocitiesaswellastheoveralltrends are well captured. Overallthe normalCoRshows asimilar trendforboth particle sizes(comparing(a) and(b)),where thesmallerparticlesizealsofeaturessmallernormalCoRat thesamelayerthickness,aswasalsopredictedbyGollwitzer et al.(2012).Thisresultsmainly,becauseatthesamelayer thicknessasmallerparticlesubmergesdeeperintotheliquid thanalargerone.Comparinginsteadexperimentswiththe same ratio oflayerthicknessto particle size(ıl/dP=const., e.g.400␮m/1.74mmto200␮m/0.91mm)the normalCoRis similar,butstillslightlylowerforthesmallerparticles,both experimentallyaswellasbyforcebalancemodel.

Theeffectofhigherviscosityonthecoefficientof restitu-tion (Fig.7(c))isalsowell representedbytheforce balance model. Increasing viscosity leadstoa decrease ofthe nor-malCoRexperimentally(comparee.g.Antonyuketal.(2009),

Ma et al. (2013)), which is qualitatively and quantitatively accountedforinthemodelbyincreasingviscousforces.

Fig.7(d)featuresthecoefficientofrestitutionfortwo dif-ferent surface tensions. Theexperimental resultsshow no considerabledifferenceduetosurfacetension.Theforce bal-ancemodelpredictsslightlyhighercoefficientsofrestitution

Pleasecitethisarticleinpressas:Buck,B.,etal., Numericalinvestigationofcollisiondynamicsofwetparticlesviaforcebalance.Chem.Eng. forsmallersurfacetensionatcollisionvelocitiesclosetothe

stickingvelocity.Theinfluencehoweverissmallandisinside thestandarddeviationsoftheexperiments.Therefore,good agreementcanbeconcluded.

Fig.8summarizesresultsifinstead ofaglassparticle a ␥-Al2O3particleisused.Alsofor␥-Al2O3 particlesnormally impactingonaglassplatethemodelgivesagoodpredictionof thenormalcoefficientofrestitution.Mostpredictedsticking velocitiesaswellasdependenciesregardingcollision veloc-ityandlayerthicknessfitthe experimentalresults.Liquids withhigherviscositythanwater(Fig.8(b))canbepredicted similarlywell.

Fig.9displays thedistribution ofthemass-relatedtotal dissipatedenergyduringacollisionintofractionsdueto con-tactforces,gravitational,viscousandcapillaryforcesfortwo particle sizes (ıl/dP=const.) ofglass and different collision velocities.Totaldissipatedenergyincreaseswithincreasing collisionvelocityaccording toits relationtothecoefficient ofrestitution(Eq.(1)).Overall,energydissipationdueto vis-cousandcapillaryforcesisdominant,whilecontactforcesand gravityfeatureaminoreffectinenergydissipation. Dissipa-tionduetogravitationalandcontactforceshowevercannot beneglected,especiallyathighcollisionvelocities.Atacloser lookthefractionofdissipationviacontactforcesaswellas vis-cousforcesincreasewithincreasingcollisionvelocity,while energydissipationowingtogravitationalandcapillaryforces are nearlyconstant. Comparingdifferent particle sizes, for smallerparticles(Fig.9(b))thefractionofdissipatedenergy, whichresultsfromcapillaryforces,isconsiderablylargerthan forbiggerparticles(Fig.9(a)).Thus,capillaryforcesincrease withdecreasingparticlesize,whichinfactleadstotheslightly smallernormalCoRforsmallerparticlediametersmentioned above.

Overall,the proposedforcebalancemodelappearstobe suitableforpredictingthenormalcoefficientofrestitutionfor wetnormalcollisionsofdifferentmaterialsandparticlesizes independenceoflayerthickness,viscosityaswellassurface tension.

Fig.8–ValidationofthenormalCoRof␥-Al2O3particlesin

dependenceofcollisionvelocityfordifferentlayerthickness ofwater(a)andviscosityatalayerthicknessof200␮m(b). Solidline:forcebalancemodel;circles:experiments.

Fig.9–Dissipativeenergiesduetocontactforces,gravitational,viscousandcapillaryforcesnormalizedbyparticlemassat differentcollisionvelocitiesfor1.74mmglassparticlesanda400␮mwaterlayer(a)comparedtocollisionsof0.91mmglass particlesanda200␮mwaterlayer(b).

Pleasecitethisarticleinpressas:Buck,B.,etal., Numericalinvestigationofcollisiondynamicsofwetparticlesviaforcebalance.Chem.Eng.

Fig.10–ComparisonofexperimentalandmodelresultsofnormalCoR(a),tangentialCoR(b)androtationvelocityafter rebound(c)ofdry␥-Al2O3particlesobliquelyimpactingonaglassplate,whichiseitherdry(black)orcoveredbya100␮m

thicklayerofTween20(orange),independenceofcollisionangle.(Forinterpretationofthereferencestocolorinthisfigure legend,thereaderisreferredtothewebversionofthearticle.)

Table3–Slidingfrictioncoefficientslusedfortheforce

balanceofa␥-Al2O3particleobliquelyimpactingaglass

plate. Layer sl[–] Dry 0.195 Tween20 0.160 35%glycerol 0.140 3.4. Obliquecollision

Forobliquecollisions theslidingfrictioncoefficientsl acts

asanotherinputparameter.Itwasmeasuredfordry␥-Al2O3

particles ina Schulzering sheartester (ASTM, 2018)to be approximately 0.195. For example Joseph and Hunt (2004)

aswell as McFarlaneand Tabor (1950) showed experimen-tallythatfrictionisreducedifthecollidingsurfacesarewet, becausethe liquid actsaslubricant. Åhrströmet al.(2003)

foundthefrictioncoefficientstodecreasefurther,whenthe liquidviscosityisincreased.However,inliteraturenomodel exist,whichpredictstheinfluenceofliquidlayersonthe fric-tioncoefficientbetweenaparticleandawallquantitatively. Hence,thefrictioncoefficientsforwetcollisionsarefittedto theexperimentalresultsbyslightlyreducing themeasured dryfrictioncoefficient.Thevaluesusedinthisforcebalance aresummarizedinTable3.Asalreadymentioned,forall inves-tigatedconfigurationsofobliquecollisionsaconstantnormal velocityof1ms−1wasused.Thus,withincreasingcollision anglethetangentialvelocityaswellastotalimpactvelocity increase.

Fig.10compares resultsoftheforce balancemodeland experimentsofa␥-Al2O3particle collidingobliquelywitha dryorwettarget.Sinceinitialrotationoftheparticlevaried between180rads−1 and −252rads−1 simulation resultsare

shownforthreecasesofinitialrotation:200rads−1,0rads−1 and−300rads−1.Thenormalcoefficientofrestitutionshown indiagram(a) appearstobeindependentofcollisionangle andthusoftangentialimpactvelocitybothfromthe experi-mentsaswellasviatheforcebalancemodel.Theinfluence oftheliquidlayeronthenormalCoRisalsowellpredictedby theforce balancemodel.Furthermore,the modelindicates, thatthenormalCoRisindependentofinitialrotation,since allthreesimulationscoincideinthesameline.Amoredetailed lookattheinfluenceofinitialparticlerotationistakenlaterin

Fig.11.Theindependenceofthenormalcoefficientof restitu-tion regardingtangentialimpactvelocitywasalsofoundby

Antonyuk etal.(2010) fordry collisions ande.g.byKantak andDavis(2004),JosephandHunt(2004)aswellasYangand Hunt(2006)forwetcollisionswithliquidlayersorimmersed inliquid,respectively.

The tangential coefficient of restitution is displayed in diagram (b)of Fig. 10. The force balance model predicts a strong dependenceofthe tangentialCoRoncollisionangle aswellasoninitialrotation.Inthemodelcasewithoutany initialrotationoftheparticlethetangentialCoRis approxi-matelyconstantatabout0.82forsmallcollisionangles.With increasingcollisionangleitfirstdecreasesandafterreaching aminimumincreasesuntilitreachesthevalue1ata colli-sionangleof90◦.Acollisionatacollision angleof90◦ isa limiting case,wheretheparticle onlyslideson thesurface andnorealcontacthappens.Thepositionoftheminimum isstronglydependentontheslidingfrictioncoefficient(not directlyshowninthediagram).Comparableminimaarealso foundexperimentallybyGorhamandKharaz(2000),Kharaz etal.(1999),Foersteretal.(1994)andDongandMoys(2006)

duringdrycollisions,aswellasbyseveralauthorsforwet col-lisionsofroughsurfaces(e.g.JosephandHunt(2004)andYang andHunt(2006)).Thisminimumisaresultofchangingcontact

Pleasecitethisarticleinpressas:Buck,B.,etal., Numericalinvestigationofcollisiondynamicsofwetparticlesviaforcebalance.Chem.Eng.

Fig.11–InfluenceofinitialrotationonnormalCoR(a),tangentialCoR(b),rotationalvelocityafterrebound(c)androtational CoR(d)of␥-Al2O3particlesobliquelyimpactingonaglassplatecoveredbya100␮mthicklayerofTween20independence

ofcollisionangle.

motionbetweenrolling/stickingandslidingcontact(Foerster etal.,1994).

If a particle impacts with positive initialrotation (anti-clockwiseforaparticlemovingfromlefttoright)thepredicted tangentialCoRisbelowzeroforverysmallcollisionangles, whichimpliesachangeofdirectionforthetangential move-ment.ThetangentialCoRthenincreaseswithcollisionangle untilthelinecoincideswiththatofthecasewithoutinitial rotationataspecificcollision angle.Atthis collisionangle theparticle fullyslideson thewallsurface.Independence ofthevalueofpositiveinitialrotationthetangential coeffi-cientofrestitutionmightalsorunthroughamaximumand aminimumindependenceofcollisionangle.Inthiscasea specificchangebetweenrollingandslidingoftheparticleon thewalltakes placecomparabletothe minimumfor colli-sionswithoutinitialrotation.Similar resultswerereported fromexperimentsofDongandMoys(2006),wherethe tangen-tialcoefficientalsoincreasedfromnegativevaluestonearly oneforasteelballimpacting withnegativeinitialrotation onasteelwallatcollisionanglesrangingfrom0◦to60◦.For aparticleinitiallyrotatinginnegativedirectionthe tangen-tialCoRdecreasesfromvaluesaboveoneatsmallcollision anglesuntilitreachesaminimum.Alsoforpositiverotation

DongandMoys(2006)foundasimilartrendforthetangential coefficientofrestitutionwithcollisionangleaspredictedby theforcebalancemodel.AtangentialCoRlargerthan1means thataftercollisiontheparticlehasmorekineticenergyin tan-gentialdirectionthanbeforethecollision,becauserotation isconvertedtotangentialmovementduringthecollision.At thecollisionanglewiththeminimumintangentialCoRthe particleslidesontheplateandthetangentialcoefficientof restitutionisequaltothatofthecasewithoutinitialrotation. DuringthedecreaseofthetangentialCoRwithcollisionangle theslopechangesstronglyatcertaincollisionangles.These turningpointsresultwhentheparticlepartlyrolls(sticks)and

partlyslidesonthewallsurface.Dependingonthedurationof rollingcontactandslidingcontacttheslopechanges. Appli-cationofaliquidlayerleadstoaslightdecreaseofthesliding frictioncoefficient(inputparameterintomodel).Additionally, theliquidlayercausesaviscousdissipationofenergydirected againstmomentarytangentialmovement.Thus,insumthe valuesoftangentialcoefficientofrestitutionslightlyshiftto theleft.AthighcollisionanglesthetangentialCoRofawet collisiongetssmallerthanthedryone.Thisdecreaseresults from anincreaseofviscousdissipationintangential direc-tion.Withcollisionanglestendingtowards90◦thetangential impactvelocitytendstoinfinity(vn=const.)thusresulting instronglyincreasingviscousforcesintangentialdirection. Theexperimentalresultsfitquiteniceintothemodelresults. Atlowcollisionanglestheexperimentsshowlargestandard deviations, whichhowevercanbeexplainedbythe experi-mentallyvaryinginitialrotation:Atsmallcollisionanglesthe forcebalancemodelrevealsahighsensitivityofthetangential CoRregardinginitialrotation,becauserotationalenergyisin thesameorderofmagnitudeaskineticenergyintangential direction.Thissensitivityisrepresentedintheexperiments asstandarddeviation.Therotationalvelocityafterthe colli-sion(Fig.10(c))isalsostronglydependentoncollisionangles accordingtotheforcebalancemodel.Exceptforpositiveinitial collisionvelocitiesatsmallcollisionangles(<10◦)ωR is pre-dictednegative.Withincreasingcollisionangletheabsolute valueofthepost-collisionalrotationincreasesuntilcomplete slidingoccurs,whichthenresultsinaconstantωR.Theslope oftheincreasechangesindependenceoftherolling-sliding regimesaccordingtothetangentialcoefficientofrestitution. Thisisreasonable,sinceintheforcebalancemodel tangen-tial,translationalmovementandrotationarecoupledinthe velocity atthecontact point.Differentinitialparticle rota-tionshaveonlysmalleffectonpost-rotationatsmallcollision angles,e.g.onlyslightlyshiftingtheotherwisesimilartrends.

Pleasecitethisarticleinpressas:Buck,B.,etal., Numericalinvestigationofcollisiondynamicsofwetparticlesviaforcebalance.Chem.Eng. Thisshiftultimately leadstosmallerabsolutepost-rotation

forpositiveinitialrotationandhigherpost-rotationfor nega-tiveinitialrotationcomparedtothecasewithoutanyinitial rotation. Aliquidlayerleadstosmaller absoluterotational velocityduetoasmallerslidingfrictioncoefficient. Further-more,theliquidlayerresultsinanincreaseoftheabsolute rotationafterreboundathighcollisionangles(>80◦).As men-tionedbefore,the tangentialvelocityapproachesinfinityat veryhigh collision anglesleading tohigh tangentialforces insidethe liquid layerand thus amomentum on the par-ticle. This momentum increases strongly atlarge collision angles resulting in the detected increase of rotation after rebound. The tangentialforce duringsolid-solid contact is inthese cases limited byCoulomb’s friction and therefore constant. Comparing the model with experiments reason-ableagreementisfound.Smalldeviations(e.g.drycollision at60◦)betweenmodelandexperimentmightalsoresultfrom inhomogeneitiesofthe␥-Al2O3particles(e.g.different rough-ness and therefore varying friction) and the experimental variationsof the initialrotation. In literaturesometimes a minimumormaximum,dependingondefinitionofrotation direction,isreportedforthepost-rotationfordrycollisions (Kharazetal.,2001;Antonyuketal.,2010),whichisnotfound hereinthesimulationorintheexperiments. However,the experiments from literature featuring an extrema in post-collisionalrotationarealwaysconductedwithconstanttotal impactvelocityinsteadofconstantnormalvelocityasinthis work.Therefore,inthoseexperimentsthenormalvelocityand thusnormalcontactforceandcontactdurationarereduced withincreasingcollisionangle.Sinceduringslidingtangential contactforcesaredefinedbythenormalcontactforce, post-rotationdecreasesatlargecontactangles,wherethecontact ispurelysliding.Inthisworkthenormalimpactvelocityis constantleadingtoaconstantpost-collisionalrotation veloc-ity.Arotationalcoefficientofrestitutionwasnotanalyzedfor theseresults,sinceforaninitialrotationvelocityof0rads−1 itisnotdefined.

Theinfluenceofinitialrotationoncollisiondynamicsis furtherinvestigatedin Fig. 11. The normal,tangential and rotationalcoefficientsofrestitutionaswellastherotational velocityafterthecollisionaredisplayedforfourdifferent ini-tialrotationvelocitiesrangingfromabout−600rads−1upto −1800rads−1.Allotherparameterswerekeptthesame.For eachinitialrotationvelocityonevalidationexperimentwas conducted.Withthedescribed setupit wasnotpossibleto changethecollisionanglewhilekeepingtherotational veloc-ity constant,so noadditional validationexperiments were carriedout.

AlsoforhighinitialrotationoftheparticlethenormalCoR staysconstantbothinthemodelandintheexperiments.This resultconfirmstheassumptionofpossiblesuperpositionof normalandtangentialmovementinthecontactpoint,which wasalsoreportedbysomeotherresearcher(DongandMoys, 2006;JosephandHunt,2004;KantakandDavis,2004).

ThetangentialCoRshowssimilartrendsforallfourinitial rotationvelocities.Withincreasing(negative)rotationvelocity priortotheimpactthetrendmostlyshiftstohighercollision angles.The collisionangle, wherecomplete sliding occurs, alsoshiftstohighervalues,althoughateachcriticalsliding pointthelinescoincideintoone.Hence,thecriticalcollision angle,whenslidingstarts,dependsoninitialrotation veloc-ity.Howeverinitialrotationhasnoeffectonthe tangential coefficientofrestitutionduringcompletesliding.The experi-mentsagreewellwiththesimulations.Simulationswiththe

Fig.12–Rollingandslidingphasesduringobliquecollision withinitialrotation.(a)TangentialCoR;(b)rotationvelocity afterrebound;(c)rotationalCoR.

exactinputvelocities,anglesandrotationalvelocitiesofthe experiments, shown as stars, are mostly close to the real experimentalvaluesandliealwaysatleastinsidethe stan-darddeviations.Especiallyforthetwolower(absolute)initial rotation velocities the experimentally measured tangential coefficient ofrestitutionisnearlyexact thesame asinthe simulation.

Fig.11(c)and(d)respectivelyshowtherotationalvelocity aftercollisionandtherotationalcoefficientofrestitution(eω=

ωR

ω).Fivedifferentregimescanbedetectedfortherotational velocity: Atlow collision anglesthe rotation afterrebound andtherotationalCoRareconstant.Withincreasingcollision angletheabsoluterotationvelocityincreasessteeply,asdoes therotationalCoR(initialrotationisconstant).Ataspecific collisionangle,theslopegetssmaller.Exactlyinthemiddle ofthisthirdregion(smallslopeofωR)therotational coeffi-cient ofrestitutionreaches thevalue 1andthen increases above 1. Thus, rotationalenergy isgained. Ateven higher collision anglesthe slopeincreasesagain untilatacritical

Pleasecitethisarticleinpressas:Buck,B.,etal., Numericalinvestigationofcollisiondynamicsofwetparticlesviaforcebalance.Chem.Eng.

Fig.13–InfluenceofliquidpropertiesonnormalCoR(a),tangentialCoR(b),rotationalvelocityafterrebound(c)and rotationalCoR(d)predictedbytheforcebalancemodelfora␥-Al2O3particleobliquelyimpactingonaglassplatewithan

initialparticlerotationof−500rads−1.(Forinterpretationofthereferencestocolorinthecitationofthisfigure,thereaderis referredtothewebversionofthearticle.)

collisionangletherotationvelocityaftercollisionaswellas rotationalCoRgetconstantagain.Withfurtherincreaseof col-lisionanglecloseto90◦post-rotationisstronglyenhanceddue toincreasedviscouseffectsasmentionedbefore.

Thesefiveregimes,whicharepresentforrotational veloc-ity, tangential and rotational CoR (exemplary marked for aninitialrotationvelocityof−960rads−1 inFig. 12)canbe explainedbyswitchingbetweenrollingandslidingcontacts. Inthefirstregime(constantωR)initialrotationstrongly dom-inatesthe velocityinthe contact point(vc=vt+ω·RP). As aresult the tangentialcontactforce is largerthan sl·Fc,n andtheparticleslides.Sincethenormalcontactforceis con-stant(vn=const.)thetangentialcontactforceisconstantas wellisthiscase,leadingtoconstantdissipationofrotational energy.Partofthisenergyisalsoconvertedtotranslational movementintangentialdirection(et>1).Withincreasing col-lisionanglethetranslationaltangentialvelocityvtincreases. Intheinvestigatedcasesrotationisnegativewhile transla-tionalmovementintangentialdirectionispositive.Therefore, thevelocityinthecontactdecreaseswithincreasingcollision angle.Thus,thetangentialcontactforcedecreasesandata specificcollisionangletheparticlestartstorollonthewall surface.Herethesecondregionstarts,whichisdescribedbya transition,fromthecase,wheretheparticleonalargepartof thecontacttimeslidesandonlyashorttimerolls,tothecase, whentheparticlemostlyrolls.Thethirdregimestarts,when rolling duringthe complete contact time isreached. Pass-ingthroughthisregime,thetranslationaltangentialvelocity onacriticalcollisionanglegetsequaltoω·RPresultingina tangentialcontactforceofzero.Therefore,noenergyis con-vertedbetweentranslationaland rotationalmovementand bothtangentialCoRandrotationalCoRareequaltothevalue1. Furtherincreasingthecollisionanglevtgetslargerthanω·RP andtranslationalmovementisconvertedtorotationduring

the contact,leadingtoarotationCoRlargerthan one(and et<1).Whenthecollisionangleislargeenoughtheparticle againstartstopartiallyslideonthewallresultingina sec-ondtransitionregimeuntilagaincompleteslidingisreached with constant rotationalvelocityafter reboundinthe fifth regime.Allvalidationexperimentsagreewellwiththemodel, althoughtheyonlyrepresentthefirsttransitionregime. Fur-thervalidationmightbebeneficialforcompletevalidationof themodel.

To further discuss the influence ofliquid properties on reboundbehaviorFig.13showsresultsofthe forcebalance modelfora␥-Al2O3particleinitiallyrotatingwith−500rads−1 impactingonaglassplate.Innormaldirectiontheliquid prop-erties have astrong influencein the force balance model. Application ofa liquid layeron the wall leadsto a strong decrease ofthenormalCoR.Anincreaseofthelayer thick-ness (green) as well as an increase of the liquid viscosity (orange)reducesthenormalCoRevenfurther.Theseresults werealreadydiscussedindetailfornormalcollisions.The tan-gentialCoRaswellasrotationalvelocityafterreboundand rotational CoRhowever are onlyslightly influenced by the liquid.Itwasassumed,thataliquidlayerwouldreducethe slidingfrictioncoefficient.Thus,thetransitionbetweenthe differentregimesofrollingandsliding isslightlyshiftedto smallercollisionangles.Anincreaseofviscositywasassumed to further reducethe sliding friction coefficient leading to anothersmallshiftofthecollisionangles,wheretransition between rollingand sliding happens.Ultimately, the influ-enceofliquids,especiallyofthelayerthickness,ontangential movementandrotationhoweverispredictedtobeverysmall.

Fig. 14 shows experimentalresults ofthe same system atcomparableparameterconfigurations.Theinitialrotation usedintheseexperimentsissummarizedinTable4.Aswas predictedbytheforcebalancemodelthenormalcoefficientof

Pleasecitethisarticleinpressas:Buck,B.,etal., Numericalinvestigationofcollisiondynamicsofwetparticlesviaforcebalance.Chem.Eng.

Fig.14–InfluenceofliquidpropertiesonnormalCoR(a),tangentialCoR(b),rotationalvelocityafterrebound(c)and rotationalCoR(d)experimentallyfoundfora␥-Al2O3particleobliquelyimpactingonaglassplatewithcomparableinitial

particlerotation(summarizedinTable4).

Table4–InitialrotationduringexperimentsinFigs.11 and14.

Dry Tween20 50%glycerol

– l=1mPas l=3mPas – ıl=100␮m ıl=150␮m ıl=100␮m 20◦ −698±172 −623±188 −741±102 −704±96 30◦ −1056±179 −961±199 −1125±178 −1330±109 40◦ −1777±411 −1366±196 −1803±172 −1564±135 50◦ −2171±569 −1777±344 −2035±152 −2181±151

restitutionisstronglyinfluencedbythepropertiesofthe liq-uidlayer,whilethevaluesoftangentialCoR,rotationvelocity afterreboundandrotationalcoefficientofrestitutionshowno considerableinfluence.Variationsbetweentheexperimental resultsallliewithineachstandarddeviation.Similarresults were alsoreportedbyKantak andDavis (2004), who found thenormalcoefficientofrestitutiontobestronglydependent onliquid properties,whiletangential coefficientof restitu-tionandrotationareonlymarginallyinfluencedbythinliquid layers.Consequently, the force balance model can be con-cludedtopredictthe(mostlysmall)influenceofliquidlayers onobliquecollisiondynamicsreasonablywell.

4.

Conclusion

Inthisworkweproposeaforcebalancemodel,which pre-dicts therebound characteristics ofparticles normally and obliquelycollidingwithadryorwetwall.Normal,tangential and rotational velocities are taken into account and ana-lyzedviacoefficientsofrestitution.Themodeliscompared toextensiveexperimentalresearchandthemodelwasfound topredicttheexperimentalresultswell.Themajorfindings andlimitationsaresummarizedinthefollowing:

1. Thepropertiesoftheliquidlayerfeatureastronginfluence onthenormalcoefficientofrestitutionbutarenearly neg-ligible regardingtangential coefficientofrestitution and rotationforthinliquidlayersintheinvestigatedrangeof parameters.

2. For oblique collisions normal and tangential forces and velocitiescanbesuperimposed.Thenormalcoefficientof restitutionisindependentofcollisionangleandinitial rota-tion ofthe particle.Tangential coefficient ofrestitution aswellasrotationalcoefficientofrestitutionarestrongly dependentonbothcollisionangleandinitialrotation. 3. Acleartransitionbetweenrollingandslidingregimescan

bedistinguishedinthetangentialandrotationalCoR.To validate the complete dependence somefurther experi-mentsmight bebeneficial.Neverthelessallexperiments sofararewellpredictedbythemodel.

4. Theproposedforcebalancemodelisabletofullypredict thereboundbehaviorofwetparticle–wallcollisions.It how-everstronglydependsonpreciseknowledgeofthematerial parametersandgeometryoftheliquidbridge,especially half-fillingangleandslidingfrictioncoefficient.

Next,themodelshouldtobeextendedandvalidatedfor particle–particle collision, where slight modification to the modelmightbenecessary(e.g.assumptionsforthecapillary force).Furthermore,thethirddimensionshouldbeincluded. Ultimately, this model, especially the influence of liquids, couldbeincludedintoaDEMcodetobeabletosimulatenot onlytwo-bodycontacts,butalsolargerprocesses.

Acknowledgements

We gratefully acknowledge for the financial support: Ger-manResearchFoundation(DFG),Germany,andNWOdomain

Pleasecitethisarticleinpressas:Buck,B.,etal., Numericalinvestigationofcollisiondynamicsofwetparticlesviaforcebalance.Chem.Eng. Applied and Engineering Sciences TTW, The Netherlands.

ProjectnumberHE4526/9-2.

Appendix

A.

Liquid

volume

Forcalculatingtheliquidbridgevolumethegeometricsofa toroidapproximationareusedandcanbeexpressedinfour fractionsshowninFig.15:acylindricalapproximationofthe bridgeVb,1,minustherotationalvolumeofthefractionofthe toroidsectionVb,2,rotationalvolumeofthetriangleVb,3and thevolumesectionoftheparticleimmersedinliquidVb,4: Vb=Vb,1−Vb,2−Vb,3−Vb,4 (39)

Approximatingthecontactanglebetweenliquidlayerand liquidbridgetobezeroVb,1iscalculatedbythemaximalradius ofthebridge(ra+rm)andthetotalheightofthebridge: Vb,1=(ra+rm)2· (y−RP·cosϕ−ıl) (40)

Thevolumeofabodyofrotationiscalculatedbyaproduct ofthecross-sectionalarea,whichisrotated,andtheperimeter oftherotation,whichresultsinthefollowingexpressionsfor Vb,2andVb,3: Vb,2= 1 2r 2 m(− (+))·2



ra+rm−4rmsin −(+ϕ) 2 3 (−(+ϕ)) ·sin

_{− (+ϕ)} 2



(41) Vb,3= 1 2r 2 msin (+ϕ)·cos (+ϕ)·2



rc+ 2 3rmsin (+ϕ)



(42)

Thevolumeofthesphericalcapoftheparticleiscalculated via Vb,4= 1 3(RP(1−cosϕ)) 2 · (3RP−RP(1−cosϕ)) (43)

Appendix

B.

Half-filling

angle

Table5–Half-fillingangleusedintheforcebalance modelfordifferentexperimentalconfigurations.

Particle Liquid ıl[␮m] ϕ[◦] 1.8mmglass Water 100 35 200 45 400 55 35%glycerol 400 53 50%glycerol 400 51 Tween20 100 35 0.91mmglass Water 100 45 200 55 Al2O3 Water 100 28 200 45 400 53 35%glycerol 100 34 200 44 50%glycerol 200 43 Tween20 100 35 150 42

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