Numericalinvestigationofsolidparticlepenetrationintoliquiddroplet
NumerischeUntersuchungdesEindringverhaltensvonFeststoffpartikelninFlüssigkeitstropfen
X.-G.Li1,U.Fritsching1,2
Computationalfluiddynamics(CFD)modelisdevelopedtostudythepenetrationbehaviourofasolidparticleintoaliquiddropletwithrespecttotheproductionofmetal-matrix-composite(MMC)particles.Incontrasttoexistingtheoreticalmodels,thecomputationalfluiddynamicsmodelproperlydescribesthemultiphaseflowsi-tuation(gas-droplet-particle)bymeansofNavier-StokesequationscoupledwithVol-umeofFluidmethod,SixDegreesofFreedommethod,anddynamicmeshtechni-que,respectively.Thecomparisonwithexperimentaldataindicatesthatthecompu-tationalfluiddynamicsmodelcangiveaccuratedescriptionsoftheparticlepenetra-tionbehaviourbytakingintoaccountthepenetration-induceddropletdeformationandthecavityformationbehindthepenetratingparticle.Inthecaseofacubicparti-clepenetratingadroplet,twotypicaloutcomesobservedinsimulationsare:1)theparticlepartiallypenetratesintothedropletandthenisejectedbythedropletsur-face;2)theparticlecompletelypenetratesintothedroplet.Theoutcomesarecate-gorizedwithWebernumber(We)andReynoldsnumber(Re)inaregimemap.Thevariationoftheregimeboundaryisinvestigatedbyvaryingsolid-liquidcontactangle,particle/dropletsizeratio,particleorientation,andparticle-dropletcollisiondirection.Itisfoundthatthecriticalvelocityrequiredforcompletepenetrationincreasesra-pidlywithincreasingsolidfractioninsemi-soliddroplets.
Keywords:particle-dropletcollision/ComputationalFluidDynamicsCFD/multiphaseflow/VolumeofFluidVoF/dynamicmesh/metalmatrixcompositeMMC
BasierendaufeinemMehrphasen-computationalfluiddynamics(M-CFD)ModellwirddasEindringverhaltenvonFeststoffpartikelninFlüssigkeitstropfenuntersuchtaufdemHintergrundderHerstellungvonMetall-Matrix-Composite(MMC)Partikeln.ImGegensatzzueinemkonventionellenEnergie/Kräftebilanz-AnsatzbeschreibtdasM-CFDModelldieDreiphasenströmungsbedingungen(Gas-Tropfen-Partikel)währenddesPartikeleindringensunterBerücksichtigungderTropfendeformationundderKraterbildunghintereinemeindringendenPartikel.ImFalldesEindringenseineskubischenPartikelsineinenTropfenzeigtdieSimulationdasteilweiseoderdaskompletteEindringendesPartikelsindenTropfen.UnterschiedlichenRegimevonPartikel-Tropfen-KollisionenwerdenmittelsderWeber-Zahl(We)undderRey-nolds-Zahl(Re)klassifiziert.DieRegimegrenzenwerdeninAbhängigkeitdesFest-Flüssig-Kontaktwinkels,desPartikel/Tropfen-Größenverhältnisses,derPartikelori-entierungundderKollisionsrichtunguntersucht.AufgrunddesWe-Re-DiagrammsundeinesrheologischenModellswirddiekritischeGeschwindigkeitfüreinkomp-
12
StiftungInstitutfürWerkstofftechnik,BadgasteinerStraße3,28359Bremen,Germany
UniversitätBremen,BadgasteinerStraße3,28359Bremen,Germany
Correspondingauthor:U.Fritsching,StiftungInstitutfürWerkstofftechnik,BadgasteinerStraße3,28359Bre-men,Germany,E-mail:ufri@iwt.uni-bremen.de
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Mat.-wiss.u.Werkstofftech.2014,45,No.8Numericalinvestigationofsolidparticlepenetrationintoliquiddroplet
667lettesEindringeneinesPartikelsineinenhalbfestenerstarrendenTropfenabgelei-tet.
Schlüsselwörter:Partikel-TropfenKollision/numerischeStrömungsmechanikCFD/Mehrphasenströmung/VolumeofFluidVoF/dynamischeRechengitter/Metall-Matrix-Verbundwerkstoff
1Introduction
Metallicparticlescontainingceramicparticulatesaredefinedasparticulatereinforcedmetalmatrixcom-posite(MMC)particleswhichcanbeusedinpow-derprocesstogeneratecompositepartsorforther-malcoatingapplications.Inthepresentwork,theproposedrouteforthefabricationofmetalmatrixcompositeparticlesisbasedonthemixtureofcera-micparticulatereinforcementsandmetalmeltsdur-ingsprayatomizationandparticleco-injectionpro-cess,Fig.1(left).Heretheliquidmetalisdisinte-gratedintohundred-micrometer-sizeddropletsbygasjetsofhighvelocity.Ceramicparticulates(dp:5~20μm),conveyedbytheatomizationgasorviaaseparategas-assisteddeliverysystem,areinjectedintothemetallicdropletsprayandlikelytobeincor-poratedintothedropletsduringthefrequentparti-cle-dropletimpingements,formingmetalmatrixcompositedropletswhicharesubsequentlysolidifiedasmetalmatrixcompositeparticles.Thetermdro-pletisusedheretorefertothedispersedliquidphase,i.e.metallicdroplets;andparticleisusedtorefertothesoliddispersedphase,i.e.ceramicparti-
Figure1.(left)Schematicofgenerationofmetal-matrix-com-posite(MMC)particlesinsprayprocess;(right)MMCparti-cles:upper-right[1];lower-right[2]
Bild1.(links)SchemaderErzeugungvonMetall-Matrix-Kom-positpartikelnimSprayprozess;(rechts)Metall-Matrix-Kom-positpartikeln:obenrechts[1];untenrechts[2]
©2014WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim
clesandmetalmatrixcompositeparticles.Figure1(right)showsthemetalmatrixcompositeparticlesproducedinsprayprocesswherethematrixmateri-alsaremetaltinandaluminumalloy[1,2],andtheparticulatereinforcementsaresiliconcarbideparti-cleswhicharedistributedonthesurfaceoforinsidethemetalmatrixcompositeparticles.
Inthepresentstudy,computationalfluiddy-namics(CFD)modelsareemployedtodescribethepenetrationbehaviorofasmallsolidparticleintoalargeliquiddroplet.Aftertheparticle-dropletcolli-sion,thefollowingoutcomesmayexist:1)theparti-clepartiallypenetratesintothedropletandthenstaysonthedropletsurface;2)theparticlepartiallypenetratesintothedropletandthenisejectedbythedropletsurface;3)theparticlecompletelypenetratesintothedropletandremainsinsidethedroplet;and4)theparticlepenetratesthroughthedroplet.Thepossiblefactors,influencingthepenetrationpathofaparticle,includeanumberofintensivepropertiessuchasparticle/dropletdensity(ρp/ρd),liquidviscos-ity(μl),surfacetension(σ)andsolid-liquidcontactangle(θ),andextensivepropertieslikeimpactvelo-city,thesizeratioofparticletodroplet(δ=dp/dd)andimpactdirection.Basedontheparametersabove,twoimportantdimensionlessnumbers,i.e.dropletWebernumber(We)andReynoldsnumber(Re),aredefinedasfollows:ρdU2We¼
rel;0dp
andRe¼
ρdUrel;0dp
σμl
ð1Þ
whereUrel,0istheinitialparticle-dropletrelativeve-locitybeforecollision.Thedrivingforceforaparti-cletopenetrateintoadropletistheinitialkineticen-ergyofthecollisionpair.Thechangeinkineticen-ergyduringpenetrationisbalancedbythechangeindropletsurfaceenergyandtheworkdonebyviscousdragintheliquid.TheWebernumberreflectstherelativeimportanceoftheinertialforcetothesurfacetensionforcewhiletheReynoldsnumbertothevis-www.wiley-vch.de/home/muw
X.-G.Li,U.Fritschingcousforce.Therefore,itmakessensetoemployWeandRetocategorizetheparticle-dropletcollisionoutcomes.Otherimportantdimensionlessparam-etersassociatedwiththepenetrationprocesslikecapillarynumber(Ca)andOhnesorgenumber(Oh)whicharerespectivelydefinedasCa¼
μlUrel;0
σand
Oh¼pffiffiffiffiffiffiffiffiffiffiffiμl
ρffiddpσ
ð2Þ
cannotexplicitlydescribetherelativeimportanceofthefluid’sinertiatoitssurfacetensionandviscosity.
2Theoreticalmodeldescription
Theoreticalmodelsfordescriptionoftheparticle-dropletinteractionmechanismduringthepenetration
ofasolidparticleintoaliquid/semi-soliddropletarebasedonenergyand/orforcebalanceapproaches[3–5].InthispaperthemodelsinWuetal.andHoe-venaretestedforcomparisonwiththecomputa-tionalfluiddynamicsmodel[3,5].Boththeoreticalmodelsassumethatthedropletremainssphericalduringtheparticlepenetration.Theparticle-dropletisdefinedinaone-dimensionalcoordinatesystem.InWuetal.thevolumeofthedropletremainscon-stantduringthepenetrationwhileinHoeventhera-dialexpandingofthedropletvolumeduringthepe-netrationisconsidered[3,5].Figure2showsthegeometryofasphericalparticle(shadedgrey)pene-tratingadroplet(shadedyellow).Thedashedcirclerepresentstheinitialdropletsurfacebeforetheparti-clepenetration,rpistheparticleradiusandrdisthedropletradius.Thedistancebetweenthecollision
Figure2.Particlepenetrationmodelaccordingto[3–5]Bild2.PartikelPenetrationsmodellnach[3–5]
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Mat.-wiss.u.Werkstofftech.2014,45,No.8
pair(d)isgivenbytheabsolutedifferencebetweenthecoordinatesofthecentres(|xdure,xisthepenetrationaxisandφ–xp|).Inthisfig-isthesphericalcoordinate(0≤φ≤π).Thepenetrationdepthpandthepenetrationangleφarerelatedbyp¼rpð1ÀcosφÞ:
ð3Þ
Themainforcesactingonaparticleduringitspene-trationprocessarethesurfacetensionforceandtheviscousdragforce.InWuetal.thesurfacetensionforce(Fs)isobtainedbydifferentiatingthechangeinsurfaceenergywiththepenetrationdepth[3]:Fs¼σcosθ
Â
πrpð2r2dþ2rdrpÀ2rdpÀ2rppþp2
ÞðrdþrpÀpÞ2þσ
πrdð2r2pþ2rdrpÀ2rdpÀ2rppþp2
Þ
ðrdþrpÀpÞ
2;0 p 2rp;
ð4Þ
andthefluidviscousdragforce(Fd)ontheparticleisdescribedby
F¼À1
dρlU2rel8
2
ApCd;>0 d24=Re0:646>:; 1 areainthepenetratingdirectionandthedragcoeffi-cientCdisdependentontheReynoldsnumber.Itshouldbementionedthatthisdragformulationas-sumesanimmersedparticulateinthefluid.Theap-plicationofthisformulationtothecaseofapartiallyimmersedparticulatemayintroduceacertainamountofoverestimation,andthedragforcecalcu-latedthusmayserveasanupperlimitofthefluiddrag. InHoeventhefollowingcorrelationforthesur-facetensionforcewasemployed[5]:Fs¼2πrpσsinφsinðφþθÞ: ð6Þ Supposingthatthesolidparticulatemovesthroughacreepingviscousflow,theviscousdragforceisvar-www.wiley-vch.de/home/muw 668Mat.-wiss.u.Werkstofftech.2014,45,No.8iedbythepenetrationdepth(p),asfollows:Fd¼À3πμlUrelp: ð7ÞThetotalforceactingontheparticleisgivenbythesummationofdifferentforces.Thevelocityoftheparticleattimetmaythenbecalculatedfromanin-tegrationofthefollowingequation: ð tPUp¼Up;0þ Fi ð8Þ 0 ρdtpVp whereVpistheparticlevolumeandtistheelapsedtimefromthebeginningofthepenetration.Thecor-respondingparticlepenetrationdepth,p,attimetmaybecalculatedbyintegratingthevelocityfromt=0,correspondingtotheinitialvelocityUp,0,toatimet=t:p¼Ðt UpðtÞdt: ð9Þ 0 Itissupposedthattheparticleiscompletelysub-mergedbytheliquidifthepenetrationdepthsatis-fies p≥dpinWuetal.[3]orp≥rd,0inHoeven[5]. –rd+dp (10) Inthiscase,thesurfacetensionforceshouldturnzero,sincefurtherincreaseinthepenetrationdepthwillnolongerresultinanychangeinthesurfaceen-ergyandthepenetrationangleremainsatφ=180°. 3ComputationalfluiddynamicsmodeldescriptionThecomputationalfluiddynamicsmodelsimulatestheparticle-dropletinteractioninthesurroundinggasbysolvingincompressibleNavier-Stokes(N-S)equationscoupledwithVolumeofFluid(VoF)method,SixDegreesofFreedom(6-DoF)methodanddynamicmeshtechnique.Thesolidparticleisrepresentedasarigidbodywith6-DoFmotion,andthegas-liquidinterfaceisdescribedbytheVoFmethod.Abody-attachedmeshwhichfollowsthebodymotionisusedfortherigidbodymotionsimu-lation.Thecomputationalfluiddynamicsmodelhasbeendevelopedintheopensourcecomputational ©2014WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim Numericalinvestigationofsolidparticlepenetrationintoliquiddroplet 669fluiddynamicscodeOpenFOAMinwhichthegov-erningequationsaresolvedbyuseoffinitevolumemethod(FVM)[6]. 3.1Volumeoffluidmethod IntheVoFmethod,avolumefractionfisdefinedineachcomputationalcellasthefractionofthecellcontainingliquid.Ifthecelliscompletelyfilledwithliquidthenf=1andifitisfilledwithgasthenitsvalueshouldbe0.Atthegas-liquidinterfacetheval-ueoffisbetween0and1.Themotionoftheinter-faceiscapturedthroughtheevolutionofthevolumefractionthattakestheform[7]: @tfþrÁðf~U ÞþrÁðfð1ÀfÞU~relÞ¼0:ð11Þ ThethirdtermonthelefthandsideofEq.(11)isanartificialcompressiontermwhichisactiveonlyintheinterfaceregion.Ucompresstheinterface.relisavelocityfieldsuitabletoThegas-liquidflowisgovernedbytheN-Sequationsandconsideredlami-nar.Thevaluesofdensityρanddynamicviscosityμarecalculatedasweightedaveragesbythelinearin-terpolationofthevolumefraction.Thecontinuumsurfaceforce(CSF)method[8]isemployedtocal-culatethesurfacetensionforce.Therefore,theN-Sequationscanbeexpressedasfollows:Momentum:@tðρ~U ÞþrÁðρU~U~Þ¼rÁμr~Uþρ~gÀrpÀσκ~n;ð12Þ Continuity:rÁ~U ¼0:Thelastterminmomentumequationisfortheeva-luationofthesurfacetensionforce,whereσisthe surfacetensionandκistheinterfacecurvaturewhichrelates~totheunitinterfacenormalvectorbyκthe¼boundaryrÁn.Aconstantbetweencontactliquidphaseangleandisimposedrigidsolid.onThefluidsandsolidareassumedisothermal. Thegoverningequationsaresolvednumericallyusingafinitevolumemethodonafixed,uniformCartesianmesh.ThetimederivativeisdiscretizedusingtheEulerscheme.Thediscretizationofthedi-vergencetermsisbasedontheTVDschemes,e.g.thevanLeerschemeforthedivergencetermofthevolumefractioninEq.(11)andthevanLeerVschemeforthedivergencetermofthevelocityinEq.(12).Forthediscretizationoftheartificialcom-pressionterminEq.(11),aninterfacecompression www.wiley-vch.de/home/muw X.-G.Li,U.FritschingschemedevelopedinOpenFOAMisemployed.TheLaplaciantermisdiscretizedusingthecentraldiffer-encescheme.Thecorrectorstothemomentumequa-tionandthevolume-fractionconvectionequationarecalculatedthroughthePISOloop.Multi-dimen-sionaluniversallimiterwithexplicitsolution(MULES)isimplementedintoOpenFOAMsolvertomaintaintheboundednessofthephasefractionindependentofunderlyingnumericalschemes,meshstructure,andsoon.Bythisway,thesuper-fluousfluxescanbeclippedorcutoff,andtheso-lutionforthevolumefractioncanthereforebeboundedbetweenzeroandone.Thetimestepisad-justableautomaticallytokeeptheCourant(Co)numberbelowaspecifiedvalueduringcalculation(preferredCo<0.2). 3.26-DoFmethod Theterm‘DegreeofFreedom’(DoF)referstohowthemovementofabodyislimited.Havingallsixdegreesoffreedom(6-DoF)meansthatthebodycantranslateandrotatealongallthreeaxesinathreedi-mensionalsystem.SolvingN-Sequationsforfluidresultsinflowforcesandmomentswhicharecalcu-latedbyintegratingthepressurefieldandviscousstressesoverthebody.Foreachtimesteptherigidbodyvelocityanddisplacementwerecalculatedfromtheextractedforcesandmoments. 3.3Dynamicmeshsolver Thebodymotionisappliedonthebody-attachedmeshateachtimestep.Internalmeshpointmotionisnecessarytoaccommodatetheboundarymotion Figure3.Meshadaptationafterrigid-bodymotion(translationandrotation) Bild3.AnpassungdesGittersbeiStarrkörperbewegung(TranslationundRotation) ©2014WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim Mat.-wiss.u.Werkstofftech.2014,45,No.8 andthustomaintainmeshvalidityandquality.Themeshmotionschemeisbasedonthemeshdeforma-tionapproachinJasakandGosmanwheretheinter-nalmeshcellsaredeformed(stretchedorsqueezed)duetothemotionofbody-attachedmesh[9],Fig.3.MeshdeformationishandledbytheLaplaciansmoothingschemesdescribedinMoradnia[10].Thereferenceframeofthemovementcanbechangedtoreducethedistancetheparticlemoves,andtherebytomaintainthemeshquality.Theparticle-dropletcollisionresultsaredeterminedbythekineticenergychange(ΔEk)duringthepenetration.Supposingthecollisionpartnersmovewiththesamevelocityaftercoalescence,theΔEmomentumkcanbeobtainedbasedonlawofconservationofandtheoremofki-neticenergyasfollows:ΔE1mk¼Àmpd mmU2 2pþdrel;0 ð13Þ wherempandmdaretheparticlemassandthedro-pletmass,respectively.SincethekineticenergychangeonlyrelatestotheinitialrelativevelocityUrel,0,itisreasonabletogiveboththeparticleandthedropletappropriateinitialvelocitiesinordertoreducethedistancetheparticlemoves. 4Validation Thedevelopedmodelsmustbevalidatedbyexperi-ments.Assmallparticlesanddropletsaredifficulttotrackandcontrolinpractice,mostofexperimentalstudieswereconductedbyimpactingamillimetersizedballonaflat,undisturbedfluidsurface,andthepenetrationprocesswasrecordedbyahighspeedcamera.Therefore,thecomputationalfluiddy-namicsmodelandthetheoreticalmodelsarevali-datedbytheexperimentalresultsinHoeven[5]whereaglassbead(2mmindiameter)fallsduetogravityonaninfinitesiliconeoilsurface.Thephysi-calpropertiesofsiliconeoilandglassbeadarelistedinTable1. Figure4showsthe3Dcomputationaldomainforthecomputationalfluiddynamicssimulation.Thediameteroftheglassbead(dp)isresolvedby50cellsandthewholedomainconsistsofnearly4.5millioncells.Thetopofthedomainisopenandotherfacesareregardedasnon-slipwalls.Atthebeginningofthesimulation,theliquid(inred)insidethedomain www.wiley-vch.de/home/muw 670Mat.-wiss.u.Werkstofftech.2014,45,No.8Table1.Propertiesofsiliconeoilandglassbead[5] Tabelle1.StoffeigenschaftenvonSilikonölundGlasparti-kel[5] SiliconeoilGlassbeadDensityρ(kg/m3)9702600Contactangleθ(°) 10Surfacetensionσ(N/m)0.021–Viscosityμ(mPas) 100 – andthesurroundinggasareconsideredstatic.Theglassbeadisplacedatasmalldistanceof~0.2dpabovetheliquidplane,ensuringtheformationofthegaslayerbetweentheballandtheliquidplanebeforethecollisionoccurs. TheBondnumber(Bo)describestheratioofgravitationalbodyforcetosurfacetensionforce:Bo¼ρlgd2p σ: ð14Þ AhighBondnumber(e.g.Bo>1)indicatesthattheeffectofthegravitationalbodyforceisimportant.AstheBondnumberscaleswiththesquareoftheparticlesize,theinfluenceofthegravitationalbodyforceincreaseswiththeparticlesizeincreasing.SincetheBondnumberisabout2fortheglassbeadandsiliconeoilsystem,thegravitationalforce(Fg)andthebuoyancyforce(Fb)shouldbeaddedintoEq.(8),thelatterbeing:Fb¼ρlVpdg; ð15Þ Figure4.CasesetupforvalidationoftheCFDmodel:aglassbeadimpactsonaflat,undisturbedsiliconeoilsurface Bild4.Set-upfürdieValidierungdesCFD-Modells:einGlas-partikeltrifftaufeineflache,ungestörteSilikonöloberfläche ©2014WILEY-VCHVerlagGmbH&Co.KGaA,WeinheimNumericalinvestigationofsolidparticlepenetrationintoliquiddroplet 671whereVpdisthevolumeoftheparticlethatissub-merged.Inthecaseofaflatliquidsurface,thesub-mergedvolumecanbesimplifiedtoafunctionofthepenetrationdepth(p)forboththeoreticalmodels,asfollows:Vpd ¼πp23 ð3rpÀpÞ;0 p 2rp: ð16Þ Inthecomputationalfluiddynamicsmodel,thebuoy-ancyforceisimplicitlyconsideredinthetermofpres-suregradientinthemomentumequation(Eq.(12)).Figure5showstheframesfromtheactualhighspeedvideorecordingfromHoevenandthenumeri-calsimulationofaglassbeadimpactingwithavelo-cityof0.6m/sonasiliconeoilsurface[5].Thesi-mulationresultsin3Dformareshowninthesecondrow,whichexhibitsagreementwithexperimentalobservation.Thethirdrowshowsthecuttingplanethroughthecentreoftheballinthepenetratingdi-rection,wherethegasiscoloredwithblueandtheliquidwithred.Attheinitialstaget=1ms,theli-quidsurfaceisdeformedwiththeadvancingoftheballbutnotyetbrokenup.Agaslayerbetweentheballandtheliquidsurfaceisvisible,whichisgradu-allyexpandingaroundtheball.Usinghigh-speedimaging,Marstonetal.andThoroddsenetal.cap-turedtheformationofthisthinlayerduringsphere/dropletimpactontoliquidsurfaces[11,12].Thegaslayerdelaystheballfromgettingwettedbytheliq-uid.Atthemomentt=5ms,theballsinksbelowtheliquidsurface,buthasnotyetbeencompletelyimmersedintheliquid. Thelocationvariationoftheglassbeadduringpe-netrationisshowninFig.6wheretheverticalaxis Figure5.Impactofaglassbead(2mminsize)withaninitialvelocityof0.6m/sonasiliconeoilsurface(upperexperiment[5]) Bild5.AufpralleinerGlaskugel(Durchmesser2mm)mitei-nerAnfangsgeschwindigkeitvon0,6m/saufeineSilikonöl-oberfläche(obenExperiment[5]) www.wiley-vch.de/home/muw 672X.-G.Li,U.FritschingMat.-wiss.u.Werkstofftech.2014,45,No.8 Figure6.Positionchangesofaglassbead(2mminsize)duringthepenetrationprocessinsiliconeoilatdifferentinitialvelocities:(left)measurementsfromHoeven[5];(right)simulationresultsbasedontheCFDmodel,theWuetal.model[3]andtheHoevenmodel[5] Bild6.PositionsänderungeinesGlaspartikels(Durchmesser2mm)währenddesEindringpro-zessesinSilikonölbeiverschiedenenAnfangsgeschwindigkeiten:(links)MessungenausHoe-ven[5];(rechts)SimulationsergebnisseaufderBasisdesCFDModells,desWuetal.Modells[3]unddesHoevenModells[5] representsthedistancebetweenthetopoftheballandtheliquidsurface,Fig.5.Inexperiment,theballisreleasedatdifferentheightsandacceleratedbygravitytodifferentimpactvelocities[5].Thedecel-erationstartsatthemomenttheparticlehitsthesur-faceorwhenthelocationofthebottomoftheparti-cleequalsthelocationoftheliquidsurface.Forthehighinitialvelocities,e.g.atUp,0=0.6and0.9m/s,thecomputationalfluiddynamicsmodelandthemodelinWuetal.givemoreaccuratepredictionsonthepenetrationpaththanthemodelinHoeven[3,5].Thepathoftheglassbeadderivedfromthecomputationalfluiddynamicsmodelalmostcoin-cideswiththatfromthemodelinWuetal.[3].However,thetheoreticalmodelssupposethattheparticleistotallyimmersedintheliquidifthedis-tanceturnszerowhilethecomputationalfluiddy-namicsmodelindicatesthatonlypartialsolidsur-faceiswettedbytheliquid,Fig.5.Intheexperi-mentofHoeven,theinformationisnotobtainableiftheballfallsbelowtheliquidsurface[5].ForthecaseofUp,0=0.4m/s,thedeviationisapparentbe-tweenthecomputationalfluiddynamicsmodelandthetheoreticalmodels.Itseemsthatinthecomputa-tionalfluiddynamicssimulationthedeformedli-quidsurfaceisreleasedandpushestheballback-wards.Butthecontinuedcomputationalfluiddy-namicssimulationindicatesthattheballissub-mergedbytheliquidafteroscillationsofsmallamplitudeontheliquidsurface.Atlowvelocities,i.e.atUp,0=0.1m/s,thecomputationalfluiddy-namicsmodelgivesmorereasonableresults.As ©2014WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim measuredintheexperiment,thecomputationalfluiddynamicsmodelpredictsaquickinitialpenetration,followedbyaslowpenetration.Thismaybeascribedtotheliquidsurfacedeformationnearthepenetrationareawhichcanpartiallydissipatethekineticenergyforpenetration,thusslowingtheballdown. Thevariationofparticlevelocityduringpenetra-tionisshowninFig.7.Thecomputationalfluiddy-namicsmodelandthemodelinWuetal.showbetterpredictionsonthevelocityvariationbeforetheparti-clesinksbelowtheliquidsurface,whilethemodelinHoevenseemstounderestimatetheforceresistingtheparticlepenetration[3,5].Asmentionedabove,thedragexpressionofEq.(5)fromWuetal.maygiveanoverestimatedviscousdragforce,whichmaypartiallycompensatefortheresistancesduetoliquiddeformationthatarenotconsideredinthethe-oreticalmodel[3].Inthecomputationalfluiddy-namicsmodel,thedropletdeformationcanbewelldescribedbytheVoFapproach,andtheforcesre-sulted,likepressureforce,surfacetensionforceandviscousdragforce,canbeobtainedbysolvingtheN-Sequationsandimplicitlyimposedontherigidbody.ForthecaseofUp,0=0.1m/s,allthemodelspredictaninitialslowincreasefollowedbyasmoothdecreaseinpenetratingvelocity.However,thecom-putationalfluiddynamicsmodelgivesalowervelo-cityprofile.Thiscanexplainwhytheglassbeadatt=25msispartlybelowtheliquidsurfaceincompu-tationalfluiddynamicssimulationwhile,accordingtothetheoreticalmodels,itiscompletelysubmergedintheliquid,Fig.6. www.wiley-vch.de/home/muw Mat.-wiss.u.Werkstofftech.2014,45,No.8Numericalinvestigationofsolidparticlepenetrationintoliquiddroplet 673Figure7.Velocityvariationsofaglassbead(size2mm)duringthepenetrationprocessinsili-coneoilatdifferentinitialvelocities:(left)measurementsfromHoeven[5];(right)simulationre-sultsbasedontheCFDmodel,theWuetal.model[3]andtheHoevenmodel[5] Bild7.GeschwindigkeitsänderungeneinesGlaspartikels(Durchmesser2mm)währenddesEindringprozessesinSilikonölbeiverschiedenenAnfangsgeschwindigkeiten:(links)MessungenausHoeven[5];(rechts)SimulationsergebnisseaufderBasisdesCFDModells,desWuetal.Modells[3]unddesHoevenModells[5] Themainfeaturesdistinguishingthecomputa-tionalfluiddynamicsmodelfromthetheoreticalmodelscanbesummarizedasfollows: 1.The3Dexpansionanddeformationoftheliquidvolumeduringpenetration,aswellasthearousedcomplexflowwhichoccursinthefluidandaroundtheparticle,areintrinsicallyconsideredbythemultiphaseflowsolver.TheworkinLiandFritschingindicatesthattheVoFmodelcangiveexcellentperformanceindescribingdropletdeformationduringdroplet-dropletcollisionpro-cess[13].Figure8showstheinitialstagewhenacubicparticlecollideswithaliquiddroplet.Theinitialrelativevelocitybetweenthecollisionpart-ners(tindropletandtitaniumcarbideparticle)is15m/swiththeparticleof13m/sandthedropletof2m/smovinginoppositedirections,Fig.9.ThefirstrowinFig.8showsthe3Ddropletdefor-mationaswellasthevariationofthevelocitydis- Figure8.Dropletdeformationandvariationofvelocitydistributioncausedbytheparticleimpact:3Dform(top)andcuttingplane(bottom) Bild8.TropfendeformierungundVariationderGeschwindigkeitsverteilungwäh-renddesPartikelaufpralls:3D-Form(oben)undSchnittebene(unten) ©2014WILEY-VCHVerlagGmbH&Co.KGaA,Weinheimwww.wiley-vch.de/home/muw 674X.-G.Li,U.FritschingMat.-wiss.u.Werkstofftech.2014,45,No.8 Figure9.CasesetupforCFDsimulationofparticle-dropletcollision:(Left)3Dcomputationaldomain;(Right)Arrange-mentofthecollisionpartners(solidparticleandliquiddroplet)Bild9.Set-upfürdieCFD-SimulationderPartikel-Tropfen-Kollision:(links)3DRechengebiet;(rechts)AnordnungderKollisionspartner(FeststoffpartikelundFlüssigkeitstropfen) tributiononthedropletsurfaceafterparticle-dro-pletimpact.ThesecondrowinFig.8showsthecuttingplanethroughthecentresofthecubeandthedroplet,whichindicatesthevelocitydistribu-tioninthefluidandaroundthecube(thedropletcontourishighlighted).Theinitialimpactcausesanobviousvelocitygradientnearthepenetrationarea.Inthecomputationalfluiddynamicssimula-tion,itisfoundthatthekineticenergyforpenetra-tionislargelydissipatedbytheinitiallocalvol-umedeformationaroundtheparticle,especiallyin thecaseoflargeparticle/dropletsizeratios.TheresultsinFig.8weregeneratedbasedonthemeshresolutiona/Δ=10inFig.10whichindicatesthattheparticlehasnotyetrupturedthedropletsurfaceatthemomentt=2ms,buttheparticlevelocityhasbeendecreasedbyover60%. 2.Thecomputationalfluiddynamicsmodeldealswiththree-phaseflow(gas-liquid-solid)whilethetheoreticalmodelsdealswithquasi-two-phaseflow(liquid-solid).Thetheoreticalmodelscon-siderthattheparticleistotallyimmersedintheliquidiftheparticlesinksbelowtheliquidsur-face,Fig.11.Incontrast,thecavityformedbe-hindthepenetratingparticlecanbewellde-scribedbythecomputationalfluiddynamicsmodel,Fig.11(middle).Thecavitydynamicscaninfluencethefinalpenetrationoutcomes.Forexample,theexperimentalobservationinVere-zubetal.indicatesthatthecavitycandragtheparticlebacktothesurfaceatlowimpactveloci-ties,andfinallytheparticlestabilizesatthegas-liquidinterface,notabletopenetrateintothebulkliquid;AkersandBelmontepointedoutthatifthepenetratingparticlestopsbeforetheformedaircavityhasclosed,theenergystoredintheliq-uidcanbepartiallyreleased,whichcanresultintheparticlereboundingbeforecompletelysub- Figure10.Meshindependencetest:TiC-particleandSn-droplet;a/d=20:120;relativevelocityUrel=15m/s;Δrepresentingthecellsizeintheinitialuniformmesh,andtheverticalarrowindicatingthedirectionofparticlemotion Bild10.Netzunabhängigkeitsstudie:TiC-PartikelundSn-Tropfen;a/d=20:120;RelativgeschwindigkeitUrel=15m/s;Δreprä-sentiertdieursprünglicheZellgrößeimuniformenGitter;dervertikalePfeilrepräsentiertdieRichtungderPartikelbewegung ©2014WILEY-VCHVerlagGmbH&Co.KGaA,Weinheimwww.wiley-vch.de/home/muw Mat.-wiss.u.Werkstofftech.2014,45,No.8mergingintotheliquid[14,15].Thegaslayerformedbetweentheprojectileandthetargetcanberoughlydescribedbythecomputationalfluiddynamicsmodel.Thislayercancontractintoanearlyhemisphericbubbleatthebottomoftheprojectile,orstretchintoasubmicronhemi-sphericgasfilmwhicheithercausesreboundingoftheprojectileorrupturestoformamyriadofentrappedmicro-bubbles[11,12,16].Marstonetal.deducedanupperlimitfortheentrappedgaslayertocontractintoarelativelylarge-sizedbubbleatthebottomtipofasphereprojectileintermsofacapillarynumberCa=O(1)[11].Bub-bleentrapmentthroughgaslayercontractionmayoccurwhenaceramicparticlecollideswithaliq-uiddropletoflowviscosity,buthardlyhappenstoasemi-soliddropletwithintensifiedsolidifica-tionbecauseofarapidincreaseintheliquidvis-cositywithincreasingsolidfractioninthedropletandtherebyaveryhighCavalue.Thoroddsenetal.arguedthattheformationoftinysatellitebub-blesatthebottomoftheprojectileisrelatedtothegaslayerrupturewhichhappenswhenitsthicknessapproaches~100nm[12].Bubbleen-trapmentthroughgaslayerrupturecanbeex-pectedduetothehigherviscosityandmaybemainlyrelatedtothepenetrationprocessofaceramicparticleintoasemi-solidmetallicdro-plet,leadingtotheformationoftinyholesaroundtheceramicparticlesinthesolidifiedmetalma-trix[11].Thismayexplainwhytheconnectionbetweenparticulatereinforcementsandmetalma-trixisnotsotightandsmooth,Fig.1(lower-right).Itisnoteworthythatthesetinysatellitebubblesduetogaslayerrupturecannotbewellcapturedbasedonthepresentmeshresolution. 5Computationalfluiddynamicssimulationresultsanddiscussion 5.1Casesetup Inthecomputationalfluiddynamicssimulation,themodeledmaterialsincludemetaltinandthreekindsofceramicparticles,i.e.ferriferrousoxide,titaniumcarbideandsiliconcarbide.ThematerialpropertiesarelistedinTable2.Giventheirregularshapeofaceramicparticlewithsharpanglesinpractice,theparticleismodelledbyacubicrigidbody.Theratio ©2014WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim Numericalinvestigationofsolidparticlepenetrationintoliquiddroplet 675Table2.Propertiesofceramicparticlesandliquidtin(Sn)[17–20] Tabelle2.StoffeigenschaftenvonKeramikpartikelundZinn[17–20] Fe3O4TiCSiCSnDensityρ(kg/m3)5200493032106979Contactangleθ(°)5084152–Surfacetensionσ(N/m) –––0.544Viscosityμ(mPas) – – – 1.85 ofcubewidth(a,μm)todropletdiameter(d,μm)covers:20:120,20:240and20:+∞,where+∞meansaflattenedliquidsurface.Fig.9showsthe3Dcomputationaldomainandthearrangementofasolidparticleandaliquiddroplet.Threedifferentparticleorientations,i.e.I,IIandIII,areconsidered.Bothhead-oncollisionandoff-centrecollisionareinvestigatedinthepresentwork.Asindroplet-dro-pletcollisionstudies,animpactnumberBisdefinedtodescribetheoff-centredegreeduringparticle-dro-pletcollision,asfollows[13]:B¼ 2dddþdp ð17Þ wheredisthedistancebetweenthecentresofparti-cleanddropletnormaltotherelativevelocity,Fig.9.Forhead-oncollision,theimpactnumberBturnszero.Atthebeginningofthesimulation,thesur-roundinggasisstaticwhilethedropletandtheparti-clestartsuddenlywithauniformvelocity,andthecollisionpartnersareplacedatasmalldistancefromeachother( 5.2Collisionoutcomes ThevariablesofinterestareWeandRenumbers,solid-liquidcontactangle(θ),particlesizetodropletsizeratio(δ),particleorientation,andparticle-dro-www.wiley-vch.de/home/muw X.-G.Li,U.FritschingFigure11.CavityformationbehindapenetratingparticleBild11.KraterbildunghintereinemeindringendenPartikel pletcollisiondirection.Foracubicparticle,avol-ume-equivalent-spherediameterisusedtocalculateWeandRenumbersaccordingtoEq.(1).Intherangeof1 Figure12.Differentparticle-dropletcollisionresults,A-1:par-tialpenetrationfollowedbyejection,(We,Re,δ,θ)=(1.26,187.14,20:120,50°);A-2:partialpenetrationfollowedbyejection,(We,Re,δ,θ)=(53.2,1216.4,20:120,152°);B:fullpenetration,(We,Re,δ,θ)=(70.8,1403.5,20:120,50°);C:shootingthrough,(We,Re,δ,θ)=(283.3,2807.1,20:120,152°),cuttingplane Bild12.Partikel-TropfenKollisionenmitverschiedenenEr-gebnissen,A-1:teilweisesEindringenundAbprall,(We,Re,δund,θ)Abprall,=(1.26,(We,187.14,Re,20δ,:θ120,)=(53.2,50°);A-2:1216.4,teilweises20:120,Eindringen152°);B:komplettesEindringen,(We,Re,δ,θ)=(70.8,1403.5,20:120,50°);C:Durchstoßen,(We,Re,δ,θ)=(283.3,2807.1,20:120,152°),Schnittebene ©2014WILEY-VCHVerlagGmbH&Co.KGaA,WeinheimMat.-wiss.u.Werkstofftech.2014,45,No.8 ondifferentcontactangles,beingeitherwettedornon-wetted,twoscenariosareobtained:whenthecontactangleisbelow90°,theejectedparticlemaydragtheliquidoutwardstoformaneckorligamentwhichbecomesthinnerandthinnerwiththereversedparticlemotionandfinallybreaksuptoseparatetheparticleandthedroplet;whenthecontactangleislargerthan90°,thecubecompletelybouncesofftheliquidsurface,Fig.12A-1,A-2.Forbothcasesabove,thecubicparticlehaspartlysubmergedintheliquidbeforetheejectionoccursandthepenetrationdepthdependsontheinitialimpactvelocity.Astheimpactvelocityincreases,theparticlecompletelype-netratesthegas-liquidinterfaceandsinksintheliquid,Fig.12B.Withfurtherincreaseoftheimpactvelocity,theparticlecanevenpenetratethroughthedroplet,Fig.12C.Theshooting-thoughphenomen-onusuallytakesplaceinthecaseofalargeparticletodropletsizeratio(δ).Thepresentworkfocusesonthefirsttwomodes:partialpenetrationfollowedbyejection,andfullpenetration. Differentpenetrationmechanismsareobservedfromthecomputationalfluiddynamicssimulation:inthecaseofθ≤90°,theuppersurfaceofthecubecanbegraduallywettedbytheliquidbeforethecubeiscompletelyimmersedintheliquid;inthecaseofθthe>cavity90°,theclosure,cubeFig.isenclosed13(a,b)by. theliquiddueto5.3Re-WeRegimeMaps Thecollisionoutcomeswithregardtoarangeofin-putvariablescanbeplottedinaRe-Weregimemap.Figure14showssixtypicalregimemapsofhead-oncollisionswithdifferentsolid-liquidcontactangles(θ),particle/dropletsizeratios(δ)andparticleorien-tations.Giventhephysicalpropertieslikeliquidviscosity(μl)andsurfacetension(σ)areconstantfor Figure13.Differentpenetrationmechanismsatdifferentcon-tactangles,cuttingplane Bild13.VerschiedeneEindringmechanismenbeiverschiede-nenKontaktwinkeln,Schnittebene www.wiley-vch.de/home/muw 676Mat.-wiss.u.Werkstofftech.2014,45,No.8Figure14.Re-Weregimemapsfordifferentsolid-liquidcon-tactangles(θ),particle/dropletsizeratios(δ)andparticleor-ientations,head-oncollision Bild14.Re-WeRegimediagrammebeiverschiedenenFest-Flüssig-Kontaktwinkeln(θ),Partikel/Tropfen-Größenverhält-nissen(δ)undPartikelorientierungen,zentrischeKollision themodeledliquid,thedependencyofWe-number onRe-numberischangedonlythroughadjustingtheinitialcollisionvelocities.WithincreasinginitialcollisionvelocitiesandtherebyincreasingWeandRenumbers,atransitionoccursfromtheregimeofpartialpenetrationtotheregimeoffullpenetration.Intheseriesofsimulationsperformed,itisdifficulttocapturetheaccuratepoint(Recrit,Wecrit)wherethetransitionoccursexactly.Therefore,thecriticalWebernumberWecritisestimatedbythefollowingway: Wecrit¼ðWeeþWepÞ=2;ifðWepÀWeeÞ<5and ðRepÀReeÞ<5; ð18Þ wherethesubscriptseandpdenotethatthecollisionoutcomeslieintheregimesofejectionandpenetra-©2014WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim Numericalinvestigationofsolidparticlepenetrationintoliquiddroplet 677tion,respectively.AndthecriticalReynoldsnumberRecrit,correspondingtotheWecrit,canbederivedac-cordingtoEq.(1).Acriticalpointisthusdeter-mined.Ifliquidviscosity(μl)orsurfacetension(σ)ischanged,newcriticalpointscanbedetermined.Byconnectingthesecriticalpoints,acriticalbound-aryisbuiltwhichdividestheregimemapintotwoparts:belowistheregimeofpartialpenetrationfol-lowedbyejection,andaboveistheregimeoffullpenetration,Fig.14. AtlowReynoldsnumber,theviscousdragoftheliquidisdominant.ThecriticalimpactvelocityandtherebythecriticalWebernumberhastobehighfortheparticletocompletelypenetratethedroplet.AstheReynoldsnumberincreases,theWebernumberontheregimeboundarytendstodecreasetoamini-mumvalue(Wesat)asymptotically,Fig.14.Ontheotherhand,theReynoldsnumberontheregimeboundaryalsoseemstoapproachasaturationlevel(Resat)withincreasingWebernumber.Thisbehav-iourwasalsodiscoveredforthecaseofcontactangleθ≥90°inHoevenwhereregimemapswerebuiltbasedontheresultsfromaforcebalancemod-el[5].TheRe-Werelationontheregimeboundarycanbemodeledbythefollowingregressionequa-tion: We Wesat¼alnReblnResat ;b<0: ð19Þ Theconstantsintheaboveequation,Wesat,Resat,aandb,canbeestimatedbasedonthesimulationre-sults,Table3.CriticalpointsandfittingcurvesarepresentedinFig.15. Intheupper-leftofFig.15,comparisonismadebetweendifferentsolid-liquidcontactangles(θ),whichindicatesthattheregimeboundaryshiftstothehigherRe-Weregionwithincreasingcontactan-gle.Ingeneral,alargecontactanglecanleadtoalargedeformationfollowedbythecavityformationbehindthepenetratingparticle,andthusalargere-sistancetothepenetration,Fig.13.However,inthelowReregionwheretheviscousdraggovernsthepenetrationprocess,thedifferencecausedbydiffer-entcontactanglesisreduced,especiallywhenθ 678X.-G.Li,U.FritschingMat.-wiss.u.Werkstofftech.2014,45,No.8 Table3.ConstantsinregressionequationEq.(19),R2:coefficientofdeterminationTabelle3.KoeffizienteninRegressionsgleichungEq.(19),R2:Bestimmtheitsmaß (θ,δ,particleorientation,B)WesatResatabR2 (90°,20:120,I,0)3065(90°,20:240,I,0)2050(90°,20:+∞,I,0)1540(90°,20:240,III,0)530(50°,20:240,I,0)550(152°,20:240,I,0)6573(90°,20:240,I,0.2)2545pillaryforceexertedontheparticleontheparticle/dropletsizeratio.AccordingtoHoeven,thecapil-laryforceonaparticledecreaseswithdecreasingparticlesizeandincreasingdropletsize[5].Inthepresentcomputationalfluiddynamicssimulation,thesizeofthecubicparticleisconstant.Therefore,theshiftoftheregimeboundarywiththeparticle/dropletsizeratioshouldbeascribedtothechangeofthedropletsize. ThebottomofFig.15showsthecomparisonbe-tweendifferentparticleorientations,whichindicatesasignificantshiftoftheregimeboundarytowardsthelowerRe-WeregioniftheorientationvariesfromItoIIorIII,butthechangeisnotobviousbe-tweenorientationsIIandIII(Therefore,theresultsbasedonparticleorientationIIisnotpresentedhere).Itmakessensethatasharpedgeorpointdis-ruptsthegas-liquidinterfacemoreeasilythana Figure15.Shiftofregimeboundarywithvariationofsolid-li-quidcontactangle(θ),particle/dropletsizeratio(δ)andparti-cleorientation,head-oncollision Bild15.RegimegrenzeinAbhängigkeitdesFest-Flüssig-Kontaktwinkels(θ),desPartikel/Tropfen-Größenverhältnisses(δ)undderPartikelorientierung,zentrischeKollision ©2014WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim 3.73–2.590.99711.82–0.990.99982.30–1.000.99961.86–1.110.99973.15–0.560.99951.11–1.430.99881.54–1.390.9985 plane.Inmostcases,especiallyatlowormoderateimpactvelocities,agasbubbleformsbelowthepar-ticleandistrappedatthecubebottomiftheparticlehitstheliquidsurfacewithorientationI,whichmayincreasetheresistancetothepenetration.Thecom-putationalfluiddynamicssimulationfoundthatforthecasearoundaregimetransitionpoint,thetrappedbubblecanpushtheparticleoutofthedropleteventhoughtheparticlehassubmergedbelowthedropletsurface.Thecapillarynumber(Ca)correspondingtothecriticalWe-ReboundaryisinarangeofCa=0.001toCa=10andthebubbleformationisasso-ciatedwiththegaslayercontractionmechanism,whichisconsistentwiththeconclusioninMarstonetal.[11].However,nobubbleisobservedatthecubebottomforthelattertwocases,Fig.16.Itim-pliesthatthegaslayerbetweenparticleandliquidsurfaceiseasiertobesqueezedoutiftheparticlehitstheliquidsurfacewithorientationIIorIII. Figure17showstwocasesofoff-centrecollisionwithdifferentimpactnumbers(B).Inthefirstcase,theparticlecompletelypenetratesintothedroplet;inthesecondcase,theparticlepartiallypenetratesintothedroplet,andthenisejectedbythedropletsur-face.Itcanbefoundthatthepenetrationprocessforbothcasesischaracterizedbytherotationofthepar-ticle.Inthecaseofpartialpenetration,iftheparticlespinscounterclockwisetopenetratethedroplet,it Figure16.Bubbleformationatthebottomofacubicparticleduringthepenetrationinadroplet,Urel=9m/s,δ=20:240,θ=90° Bild16.BlasenbildungamBodeneineswürfelförmigenParti-kelswährenddesEindringensineinenTropfen,Urel=9m/s,δ=20:240,θ=90° www.wiley-vch.de/home/muw Mat.-wiss.u.Werkstofftech.2014,45,No.8Figure17.Off-centrecollisionprocessatdifferenttimemo-ments,(θ,δ,particleorientation)=(90°,20:240,I) Bild17.ExzentrischeKollisionenimzeitlichenVerlauf,(θ,δ,Partikelorientierung)=(90°,20:240,I) Figure18.Off-centrecollision:(left)Re-WeregimemapatB=0.2;(right)shiftofregimeboundarywithvariationofim-pactnumberfromB=0toB=0.2 Bild18.ExzentrischeKollisionen:(links)Re-WeRegimedia-grammbeiB=0,2;(rechts)RegimegrenzeinAbhängigkeitdesStoßparametersvonB=0bisB=0,2 willbeejectedfromtheliquidsurfacewithaclock-wiserotation.Unlikeinacollisionsystemoftwoequal-sizeddroplets[13],aparticleofsmallinertiaimposestrivialinfluenceonthemotionstateofitscollisionpartner.Figure18(left)showstheregimemapofoff-centrecollisionwithanimpactnumberB=0.2.TheregimeboundarycanalsobedescribedbyEq.(19).ThecorrespondingcoefficientsarelistedinTable3.ThefittingcurveandthecriticalpointsarepresentedinFig.18(right)forcompari-sonwiththecaseofhead-oncollision.Withincreas-ingimpactnumber,theregimeboundarymovesto-wardslowerWe-Reregionslightly. 5.4Criticalpenetrationvelocity Collisionsbetweensolidparticlesandsemi-soliddropletsinthestateofsolidificationarefrequentinsprayprocess,whichmayalsocontributetothein-corporationrateoftheparticulatereinforcementsinthematrixmaterials.Thesemi-solidormushystateofthedropletischaracterizedbyavariableviscosityrelatedtothesolidfractioninthedroplet.Thework ©2014WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim Numericalinvestigationofsolidparticlepenetrationintoliquiddroplet 679byWuetal.andZhangetal.indicatedthatthevis-cousdragexertedbyadropletonapenetratingparti-clemaybedramaticallyincreasedtobedominantoverthesurfacetensionresistancebyarapidin-creaseinviscosityduetothepresenceofsolidphasesofevenalowfractioninthesemi-soliddro-plet[3,4].Incalculatingtheeffectofthesolidfrac-tionfonthepenetrationbehaviour,theapparentviscosityisusedinplaceofviscosityinEq.(1).Asaresultoftheabsenceofexperimentaldataavailablefortheapparentviscosityofpuretin,therheologicalmodeldevelopedinChenandFanisemployedtoevaluatetheapparentviscosityinthepresentanaly-sis[21,22]:μ¼μ0ð1ÀfeffÞÀ5=2 ð20Þ whereμ0istheviscosityofthepureliquidatliqui-dustemperature(Table2)andfefftheeffectivesolidvolumefraction.Supposingazero-shear-rate,theef-fectivesolidfractioncanbeexpressedas 8> þ4:643fÀ0:16975; ð21Þ0:05 Bild19.ScheinbareViskositätdesZinns(Sn)alsFunktiondesFeststoffanteilsineinemhalbfestenTropfen www.wiley-vch.de/home/muw X.-G.Li,U.FritschingFigure20.Criticalpenetrationconditionvs.solidfractioninasemi-soliddropletatdifferentparticleorientations,solid-liquidcontactangles(θ)andparticle/dropletsizeratios(δ) Bild20.KritischeBedingungfüreinkomplettesEindringenalsFunktionvonFeststoffanteilineinemhalbfestenTropfenbeiverschiedenenPartikelorientierungen,Fest-Flüssig-Kon-taktwinkeln(θ)undPartikel/Tropfen-Größenverhältnissen(δ) Supposingthataparticleoftitaniumcarbide(6.1μminsize)hitsasemi-solidtindroplet,thepe-netrationbehaviourcanbedescribedbytheregimemaps(θ=90°,δ=20:240,I)and(θ=90°,δ=20:240,III),Fig.20(upper-left).ThedashedRe-Wecurve,plottedaccordingtothedefinitionsofWeandRenumbersinEq.(1),shiftstotheupper-leftcornerofthemapwithincreasingsolidfractionf.Thetransi-tionpointbetweenregimesejectionandpenetrationislocatedattheintersectionoftheRe-Wecurveandtheregimeboundarycurve.Asexpected,thecriticalWebernumberisdramaticallyincreasedwithincreas-ingsolidfraction.ThecriticalvelocityforaparticletopenetrateadropletcanbederivedbasedonEq.(1)andEq.(19).Thecalculatedcriticalpenetrationvelo-cityissummarizedasafunctionofsolidfraction,Fig.20(upper-right).Itcanbefoundthatthecriticalvelocityincreasesrapidlywithincreasingsolidfrac-tion.ThisisconsistentwiththeconclusionsfromWuetal.andZhangetal.[3,4].Forbothorienta-tions,thecriticalvelocityrequiredfora6.1μmsizedparticletopenetrateaSn-dropletisincreasedbyaboutfifteentimesasthesolidfractioninthedropletincreasesfrom0(pureliquid)to35%.ThecriticalvelocitybasedonorientationIcanbere-gardedasanupperlimitforparticlepenetration,whilethatbasedonorientationIIIasalowerlimit. ©2014WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim Mat.-wiss.u.Werkstofftech.2014,45,No.8 BasedontheregimeboundarycurvessummarizedinFig.15,(δ=20:240,I)and(θ=90°,I),thecriti-calpenetrationvelocitiesfora6.1μmsizedparticlewithdifferentsolid-liquidcontactanglesandparti-cle/dropletsizeratiosarederived,andpresentedasafunctionofsolidfractioninadropletinthelower-leftandlower-rightofFig.20,respectively.Thecri-ticalpenetrationvelocityincreasesexponentiallywithincreasingsolidfractionforallthecases.Anincreasingsolid-liquidcontactangleleadstoanin-creasingcriticalpenetrationvelocity,butthein-creaseisnotsignificantwhenθ≤90°.Thecriticalvelocityincreasesslightlywithincreasingparticle/dropletsizeratio.Ingeneral,whenthesolidfractioninaTin-dropletexceeds30%,thecriticalvelocityrequiredforacompletepenetrationisabove120m/sforallthecases. 6SummaryandOutlook Inthepresentinvestigation,acomputationalfluiddynamicsmodelhasbeendevelopedtosimulatethepenetrationbehaviourofasolidparticleintoaliquiddropletduringsprayprocessingofmetalmatrixcompositeparticles.Inthecomputationalfluiddy-namicsmodel,thegas-dropletsystemisdescribedbytheVoFapproachwhilethesolidparticleisre-presentedbyarigidbodywith6-DoFmotion.Thecouplingbetweenbodymotionandmeshmotionisrealizedbyuseofameshdeformationmethod.Comparedwiththetheoreticalmodels,thecomputa-tionalfluiddynamicsmodelcandealwithmoreis-sues,includingthedropletdeformationduringtheparticlepenetration,thecavityformationbehindthepenetratingparticle,theinducedcomplexflowinthefluidandaroundthepenetratingparticle,andtheruptureofthegaslayerbetweenparticleandliquidsurface.Thecomparisonwithexperimentaldataprovesthatthecomputationalfluiddynamicsmodelcangivemorereliablepredictionsontheparticlepe-netrationbehaviourthantheexistingtheoreticalmodels,suchastheparticlepenetrationpath,theparticlevelocityvariationduringthepenetrationpro-cess,andthesituationwhentheparticlesinksbelowtheliquidsurface. 3Dsimulationisconductedtoinvestigatethepe-netrationbehaviourofacubicceramicparticleintoametallictindroplet.Threedistincttypesofimpactbehaviourareobserved,i.e.partialpenetrationfol-www.wiley-vch.de/home/muw 680Mat.-wiss.u.Werkstofftech.2014,45,No.8lowedbyejection,fullpenetration,andshootingthroughthedroplet.ThecollisionoutcomesareplottedasafunctionoftheWeber(We)andRey-nolds(Re)numbersinaregimemap.Theboundarybetweenregimes‘partialpenetration’and‘fullpene-tration’canbesimulatedbythecorrelationln(We/Wesat)=a*(ln(Re/Resat))^b.TheconstantsWesat,Resat,aandbdependonsolid-liquidcontactangle,particle/dropletsizeratio,particleorientationandparticle-dropletcollisiondirection,andcanbedeter-minedfromthecomputationalfluiddynamicssimu-lationresults.Itisfoundthatthefullpenetrationhappensmoreeasilytothecollisionpartnerswithdecreasingsolid-liquidcontactanglesandparticle/dropletsizeratios.TheregimeboundaryshiftstothelowerRe-Weregiongreatlybyvaryingparticleor-ientationfromItoIIorIII.Thepenetrationprocessforoff-centrecollisionischaracterizedbytherota-tionofthecubicparticle,andwithincreasingimpactnumbertheregimeboundarymovestowardsthelowerWe-Reregionslightly.Thecriticalvelocityrequiredforcompletepenetrationofaparticleintoasemi-soliddropletincreasesdramaticallywithin-creasingsolidfractioninthedroplet. Infuture,thesimulationresultsbasedonthepre-sentpenetrationmodelwillbeincorporatedintotheparticle-dropletcollisionmodeldevelopedinLietal.toderivetheincorporationrateandthestick-ingrateofceramicparticulatesinmetalmatrixcom-positeparticles[1].TheRe-Weregimemapscreatedinthepresentworkcanalsobeusedtodescribetheparticle-dropletcollisionbehavioursinothersprayapplications,e.g.spraydryingprocessandwetscrubbingsystem. Itisnoteworthythatadirectsimulationofthein-teractionbetweenaceramicparticleandsolidsinasemi-soliddropletoranoxidelayeronadropletsur-faceishardlyrealizedbasedonthepresentcomputa-tionalfluiddynamicsmodel.Inthispaper,asemi-soliddropletisinessencemodeledasafull-liquiddropletwithanequivalentviscosity.Thecriticalpe-netrationvelocityisderivedbycombiningtheWe-ReregimemapswiththerheologicalmodelinChenandFan[21,22].Therheologicalmodelrelatestheapparentviscosityofpuretintothesolidfractioninadroplet.Thesolidformationmaybecausedbydro-pletsolidification,oxidelayerformation,etc.There-fore,thederivedcriticalpenetrationconditionisalsomeaningfulforthesituationwhenanoxidelayerformsonadropletsurface.However,someaspects ©2014WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim Numericalinvestigationofsolidparticlepenetrationintoliquiddroplet 681arenotabletobedealtwithbytherheologicalmod-elinChenandFan,e.g.theinfluenceofsolidmor-phology(dendriticstructure,oxidelayerorsolidpar-ticle),solidsizeandsoliddistributioninadropletontheviscosityofliquidmatrix[21,22].Amoreaccu-raterheologicalmodelisthusnecessaryinfuturetoobtainmorereliableresults. Acknowledgements Wegratefullyacknowledgethesupportofthispro-jectprovidedbytheGermanResearchFoundation(DFG)withinthescopeofSPP1423“Process-Spray”aswellasfromtheChinaScholarshipCoun-cil(CSC). 7References [1]X.G.Li,L.Heisterüber,L.Achelis,V.Uhlen-winkel,U.Fritsching,AtomizationSprays[2]2011Y.Wu,,21,E.J.933. Lavernia,Metall.Trans.A1992, 23,2923. [3]Y.Wu,J.Zhang,E.J.Lavernia,Metall.Mater. Trans.B1994,25,135. [4]J.Zhang,Y.Wu,E.J.Lavernia,Acta.Metall. Mater.1994,42,2955. [5]M.J.Hoeven,Ph.D.Thesis,Universityof Queensland,Australia,2008. [6]OpenFOAM,Version2.0.1,http://www.open-foam.com/. [7]H.Rusche,Ph.D.Thesis,ImperialCollegeof Science,TechnologyandMedicine,UK,2002.[8]J.U.Brackbill,D.B.Kothe,C.Zemach,J. Comp.Phys.1992,100,335. [9]H.Jasak,Z.Tukovic,Trans.FAMENA2006, 30,1. P.Moradnia,PhDcourseinCFDwithOpen-Sourcesoftware,2008. J.O.Marston,I.U.Vakarelski,S.T.Thorodd-sen,J.FluidMech.2011,680,660. S.T.Thoroddsen,M.J.Thoraval,K.Takehara, T.G.Etoh,J.FluidMech.2012,708,469. 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Nomenclature SymbolDescriptionSIunitacubewidthmaconstant –Aprojectionaream2bpconstant –Bimpactnumber–BoBondnumber–Cacapillarynumber–Cdragcoefficient–CodCourantnumber–ddiametermddistance mEkineticenergyJfkvolumefraction–Fb buoyancyforce N ©2014WILEY-VCHVerlagGmbH&Co.KGaA,WeinheimMat.-wiss.u.Werkstofftech.2014,45,No.8 SymbolDescription SIunitFviscousdragforceNFdgravitationalforceNFgsurfacetensionforceNgsgravitationalaccelerationm/s2mmass kgnunitnormalvector–ppressure Pappenetrationdepthmrradius mReReynoldsnumber–ttimesUvelocitym/sVvolume m3WeWebernumber–xx-coordinate mδΔparticle/dropletθcell–κsolid-liquidsize sizeratiomμinterfacecontactangle°ρdynamiccurvature1/mσdensity viscosityPaφsurfacekg/ms3penetrationtensionangleN/m° SubscriptSymbolDescription0initialconditioncritcriticalddropleteffeffectiveggaslliquidpparticlerelrelativesat saturation Receivedinfinalform:June17th2014T303 www.wiley-vch.de/home/muw 682 因篇幅问题不能全部显示，请点此查看更多更全内容