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GranularSegregation inTapered Rotating Drums S. Gonza´lez1,∗ and L. Orefice1 1Research Center Pharmaceutical Engineering GmbH, Inffeldg. 13, Graz, Austria Westudythegranularsegregationinataperedrotatingdrumbymeansofsimulations.Inthisgeometry,both radialandaxialsegregationappear,thestrengthofwhichdependsonthefillingfraction. Westudytheeffect ofthedrum’staperingangleonthesegregationspeedandshowthat,coherentlywithseveralrecentstudies,the axialsegregationisduetothecombinedeffectoftheradialsegregationandtheshapeofthedrum.Weshowthat theaxialsegregationbehavesinawayanalogoustotheonefoundinchuteflows,andcorrespondstoprevious theoriesforshallowgravitydrivensurfaceflowswithdiffusion.Bymeansofthisanalogy,weshowthattapered 5 drumscouldbethesimplestexperimentalset-uptoobtainthefreeparametersofthetheory. 1 0 PACSnumbers: 45.70.Mg 2 n a It has been hinted that axial segregationin rotating drums butwithafundamentaldifference: theshapeofthedrumin- J is mainly the result of the shape of the free-surface, the thin creases the curvature of the flow lines on the flowing layer 4 layerofflowingparticlesinthedrum[1–4]. Thefree-surface thustransportingmoresmallparticlesthaninaparalleldrum, 2 is notflat, but slightly curvedalong the axialdirection, with andincontrasttothelatter,inadefinitedirection. peaks near the parallel walls of the drum. Due to the fric- ] Thestructureofthepaperisasfollow. First, thedrumge- t tionwiththem,theparticlesonthesidesareliftedmorethan f ometryandandsimulationdetailsareexplained. Second,we o the particles in the centre, thus creatinga convectiveflow in compareourresultsforhalf-filleddrumswithexperimentsto s the drum that inducesthe axial segregation. How to harvest . validateourmodel.Third,tominimizetheeffectofradialseg- at thisrelationbetweenflowandthesegregationpatternremains regation,alowerfillingfractionisusedandtotalsegregation m an open question: a question whose answer could have far- is observed. The effect of the drum’s tapering angle on the reachingpracticalimplications. - speedofthesegregationisalsostudied. Finally,comparisons d Motivatedbytheresultsof[5,6]–whereaxialsegregation willbedrawnwithsimulationsofmixturesin2and3Dchutes n ofbinarymixturesinataperedrotatingdrumisreportedalbeit o andwithatheoryforshallowgravitydrivensurfaceflowswith c not explained – and how they relate to our own results [7] diffusion. [ – where the shape of the rotating drum is used to speed up and control the axial segregation– we study the segregation 1 patternintaperedrotatingdrumsbymeansofsimulations,and v 0 seetowhatextentitispossibletocontrolitbymodifyingthe 4 drum’sshape. 0 Ithasrecentlybeenshownthat,contrarytothecommonbe- 6 liefthat3Drotatingdrumsarethesumofindependentquasi- 0 2Dslicesinteractingonlyviadiffusion,theshapeofthedrum . 1 doesinfluencethesurfaceflowcreatingaslowaxialdrift[3]. 0 This drift is caused by the axial slope of the drum’s walls, 5 1 whichdeterminesthe3Dstructureoftheflow.Ifabi-disperse : mixtureisused,theparticleswillfollowtheflowstructureand v willsegregateaccordingly. i X Intheparticularcaseofatapereddrum,whenmixedparti- r clesentertheflowinglayerintheupperpartoftheavalanche, a gravitydrivesthemtowardsthesmallsideofthedrumbefore returningthemtothelargeone. Smallparticlescanpercolate trough the voids created by the big ones leaving the flowing layerbeforeitchangesitsdirection. Thisleadstoanettrans- port of small particles to the small side of the drum and of largeparticlestothebigside: axialsegregationoccurs. Since FIG.1: (Color online) Snapshots for half-filledsystem at different the shape of the flowing layer depends on the shape of the positions along the rotation axis (from left to right) and different drum, one would also expect it to affect the segregation. In times (from top to bottom, the number of revolutions indicated in thispaperwe showthatthis is indeedthe case andwe study theleft). Smallparticlesdark,largeparticleslight. Onecanseethat firsttheradialsegregationappears,andthisslowlyandcontinuously howithappens. transforms into axial segregation, that here is visible in the domi- The coupling of axial segregation with the velocity pro- nanceofsmallparticleinthesmallsideofthedrum. file has been alreadyreportedfor the classical rotatingdrum [4]. Inthetaperedrotatingdrumthesamecouplingisinplay 2 SIMULATIONS towardsthe smallside. Thisisinvertedwhenparticlesreach the lowerpartof the avalanche: particlesin the largeside of Our simulations are in a slightly different geometry than thedrumarelowerthanthoseinthesmallside.Increasingthe thatof[5, 6](shorterdrum)tospeedupthesimulationtime. angleofthedrum,increasesthecurvatureandthemagnitude We use an equalvolumemixtureof 5 and10mmradii parti- of this flow. This can be seen in Fig. 3 for three different clesinatapereddrum150mmlength,withcapsofradiir=50 angles. The surfaceis definedas the planewhoseprojection mm(atz=0mm)andvariableR(atz=150mm),intheleft canbeseeninFig.2(solidline). andrightside,respectively. Theirrelationdefinetheangleof Whenamixtureofparticlesenterstheavalancheintheup- thedrum,suchthatR=r(1+tan(a )). Inthesimulations,par- perpart, they willsegregateas theytraveldownslope. If the ticles’ contactforcesare representedusing a standardspring drumislargeenough,somesmallparticleswillreachthecore dash-potmodel[8]asimplementedin[9]. Theangularveloc- beforetheflowchangesitsdirection,creatinganettransport ityissettop s−1,i.e.30rpm,whichimpliesanaverageFroude of small particles. (Large enough here means in relation to numberofFr=w 2(r+R)/2g∼0.1. Thesystemiseitherat the segregation speed of the mixture. For size-driven segre- 50 or 25% filling fraction. As initial condition the particles gation, the smaller the size ratio of the particles, the slower are arranged in a square lattice, alternating one big particle segregation is, and thus the drums needs to be larger.) The andeightsmalltoobtainaninitialmixedstate. Snapshotsof largeparticles,sincetheytendtostayonthesurface,canflow thesystemfordifferenttimesandaxialpositionsareshownin backtothelargesideofthedrum,followingtheflowlines. In Fig.1(cf.Fig.5inRef.[6]foranexperimentalcomparison). thisway,axialsegregationoccurs. The simulations were performed using Mercury-DPM [9] with the same material properties used in our previous research [7], since they proved to reproduce the experimental results. ShapeoftheFreeSurface In thissection we show the three-dimensionalstructureof theflow. Wefocusonthreedifferentcharacteristics: thelevel of the surface along the axis of rotation (Fig. 2), the velocity field on the free surface (Fig. 3) and the density profile, as seen by a vertical cut in the centre of the drum (Fig. 4). Together,theyprovidearathercompletewaytovisualisethe structure.Dataareaveragedfor30consecutivesnapshots(one every0.8s)andcoarse-grainedtoobtainacontinuousfunction followingtheprocedureofRef. [10]. Theweightfunctionis aLucypolynomial[11]withawidthof2d. Unlessotherwise noted, every coarse-grainingpresented here is done with the same parameters. We measuredatdifferenttimes duringthe FIG.2: Contour linesfor adensity r ∗=r particles/10 at threedif- evolution of the system, and the results do not significantly ferent axial positions for the same systems as in Fig. 1. The level differencecanbeseenmostlyintheupperpartoftheavalanche.One change, so we conclude that what we observe depends only shouldthinkofthecurvesasbeingdisplacedinthedirectionnormal onthegeometryandnotinthedegreeofsegregation. tothe page. Compare the qualitativeagreement withthe model of The free surface along the axial directionshows leveldif- Ref. [12]. Note that since the flow velocity increases along z, the ferences. This phenomenon, to the best of our knowledge, bluecurveshowsaslightlymoreS-shapedprofile. Solidlineshows was first describedby Zik et al. for a drum modulatedalong theplanewherethevelocityismeasuredforFig.3anddashedverti- the axial direction with a sinusoidal function [12]. This can callinetheplanewherethedensityismeasureforFig.4. beseeninoursimulationsfromFig.1, orlookingattheiso- linesofthedensityatdifferentaxialpositions,inFig.2. This Finally, the packingfraction of the system givesus an in- leveldifferencealongtheaxismakesparticlestomovealong sightontheinnerstructureoftheflow,andshowsthedepen- the gradient of the surface, that is as particles go down the denceofthefreesurfaceonthedrum’sangle,Fig.4. Despite avalanche,theymoveforwardsandbackwardsintheaxisper- thepresenceofsegregation,thedensityisuniformduringthe pendiculartothepage. evolutionofthesystem,withthedensestregionatthebottom Particles enter the flowing layer with a small velocityand ofthedrum.Thedensityismostlyhomogeneousalongtheax- they are mainly directedtowardsthe small side of the drum. ialdirectionwithincreasedspatialfluctuationsasa increases. Thisisduetothetaperedgeometry:whenenteringtheflowing The density remainsconstantsduringthe segregationpro- layer, the particles in the large side of the drum are higher cess,andismaximuminthelowerregionofthedrumforall thanthoseinthesmallsideofthedrum,thuscreatingaflow the angles studied. We confirmed that the density does not 3 FIG.3:Averagevelocityfieldoftheparticles’freesurfaceforthetapereddrum(fromlefttoright,taperinganglea =10◦,15◦,20◦),thecolour representingtheparticles’meanvelocityinm/s. Thelengthofthecomponentofthevectorfieldparalleltotheaxisofrotationisincreased tostresstheeffectofthetaperedgeometryontheflow. Asexpected,theconvectiveflowsincreasesitsamplitudeandreachwiththeangleof tapering. varybymeasuringitatdifferenttimesduringthesegregation saturatessoontoaroughlyconstantvalue.Ontheotherhand, process. the axial segregationis slower but steadily increases beyond the pointof saturation for the radialsegregation. Eventually (aroundthe 10threvolution)the axial segregationcauses the RadialandAxialSegregation valueofq todecreaseslightly,sinceinthesmallsideof radial thedrumthenumberoflargeparticlesdecreasesgreatly. Tocorroboratethattheproposedmechanismisresponsible If a is increased the segregation speed should increase oftheaxialsegregation,wemeasuretheradialandaxialseg- since the level difference on the free surface is also increas- regation in the drum as a function of time. In order to have ing[12]. Thisisindeedthecaseas[5]showsexperimentally orderparametersforboth,theradialsegregationisdefinedas (see inparticularFig. 8). Thesame principlecanbe seen in follows: other configurations such as the spherical drum [14], or the doubleconeconfiguration[15]. qradial=Ns<z∗/NTs+NL>z∗/NTL−1, (1) Tobetterunderstandtheroleofgeometryonthesegregation process, several simulations were run for different a . It is whereN isthenumberofsmall(large)particleswhose s<(L>)r∗ expected that the radial segregation will not depend on the radial coordinateis smaller (larger)than the half volume ra- geometryofsystem. However,fortheaxialsegregation,since diusr∗ (notethatthisradiusdependontheaxialcoordinate), itdependsonthevelocityprofileandonthegeometryofthe N beingthetotalnumberofsmall(large)particles. That Ts(L) flowinglayer,oneexpectstofindastrongdependenceonthe is, we countthe fractionof particlesthat are in the expected angleofthedrumsincethisdeterminestheleveldifferencein sideofthedrum.Whenq =1allthelargeparticlesarein radial theflow. IndeedthisisthecaseasFig.5shows. theoutsideandthesmallinthecoreofthedrum. Itmustbe notedthat,duetotheflowinglayer,theradialsegregationcan neverreachthemaximumvalue. Tomeasuretheaxialsegre- LessthanHalfFilledDrum gationweuseaquantitythatalsoworkswithbanding(which appearedinoneofthesystemsstudied)[13]: Our previous result shed light on the relation between a andtheaxialsegregation,butthisisstronginfluencedbythe 1 l |c (z,t)−c (z)| radial segregation in half filled drum. So large in fact, that s s q (t)= dz, (2) axial at this filling fraction some systems segregate the other way l Z0 cs(z) around, that is, large particles in the small side of the drum, wherec (z,t)isthenumberofsmallparticlesalongtheaxis, as reportedin [5] and confirmedby some of oursimulations s andc (z)thenumberofsmallbeadsthatcorrespondtoaslice (datanotshown). Weransimulationsat25%fillingfraction, s of the drumat position z for the totally mixed state. It must fordifferenta ,andweobserveindeedthattheaxialsegrega- benotedthatsincewearesimulatingataperedrotatingdrum, tionisbolder.Whatismore,theradialsegregationdisappears the mean numberof small particleschangeswith z since the almost completely since the mixing at this filling fraction is crosssectionofthedrumincreasesaccordingly. maximum [16]. In this configuration, a pure phase of small Figure5showsthetemporalevolutionforeachkindofseg- particles appears in the small side of the drum and a steady regation for three different angles. As expected, the radial stateforthesegregationisreached,whileincreasingtheangle segregationsetsinquitefast,essentiallyafterfourturns,butit ofthedrumincreasesthesegregationspeed(seeFig.6). 4 0.60 0.48 0.36 0.24 0.12 FIG.4:(Coloronline)Packingfractiononthemiddleplaneofthedrumforincreasingdrumanglesa =10◦,15◦,20◦,fromlefttoright.The increasingslopeofthefreesurfacecanbeseenfromlefttoright,asa increases. 0.4 wayasthesizeratiobetweentheparticles[18,19]orthevol- umefractionbetweenthetwospecies[17]. 0.3 al di a al,r 0.2 xi a q 0.1 0.0 0 10 20 30 40 50 60 turns FIG.5: Axial (solid-lines) and radial (dashed-lines) segregation as afunctionoftimeforthetaperedrotatingdrum. Fromlightgrayto blackinincreasingangleoftapering. Asitcanbeseen,thereisno correlationbetweentheangleandtheradialsegregation. However, axialsegregationshowsaclearcorrelationwitha . FIG. 6: Displacement of the centre of mass of large particles for systemswithdifferenta ,normalizedbytheasymptoticvalueD zin a log-linear scale. Straight lines correspond to the exponential fit. This behaviour is strikingly similar to that of bi-disperse Compare with Ref. [17]. If plotted in a linear scale, one obtains chuteflows,eitherin2D[17]or3D[18]. Numerically,those resultsasinRef.[18]. are simulated with a box with periodic boundary conditions in the flowing plane, and a rough inclined bottom. Depend- This is valid for a wide range of drum angles. We define ing on the inclination of the bottom with respect to gravity, the segregation speed as the inverse of the exponent of the a steadyflowcanbeobtained. Thisisastraightforwardway tostudysegregationforafixedvelocityprofile,howevertheir exponential fit D z=exp(−t/t ). In general, the segregation experimentalrealisationisrathercomplex. speedisamonotonicfunctionoftheangle,asonecanseein Figure6showsthedisplacementofthecentreofmassofthe Fig.7,thelinebeingalineareye-guide. Asimpleestimation large particles normalized by its asymptotic value, for three tellsusthatthedistanceaparticletravelsonthefreesurfaceis differentdrumangles Ls(cid:181) tan(a ). Iftheparticlestraveltwiceasmuchduringone avalanche they will segregate twice as fast, and this would D z= zcm(¥ )−zcm(t) , (3) explainthelinearbehaviourofthesegregationspeedwiththe zcm(¥ )−zcm(0) angle. wherez (t)isthecentreofmasspositionontheaxialdirec- Sincewehavethisqualitativeagreementbetweentwocom- cm tionforthelargeparticles. Asa increases,thesystemsegre- pletelydifferentsystems, theobviousnextstepistoseehow gatesfastersincetheshearrateincreaseswithit. Onecansee well the theoretical predictionsdevelopedfor the chute flow that the drum’s angle influences the segregation in the same workonthetapereddrum. 5 comparedtodiffusiveremixing)toreproducetheevolutionof the system rather than the steady state. Does this mean that thePeinthetapereddrumchangeswithtime?Answeringthis questiongoesbeyondthescopeofthispaper. Itmustbenotedthatwearecomparingtheresultsofonly onerealisation,andourdatapointsarenotaveragedontime, which would decrease the noise and give a better fit, but we prefer to show the raw results to highlightthe quality of the datathatispossibletoobtainwiththisset-up. Weconfirmed thatthesamequalitativebehaviourholdsfordifferentangles anddrumlengths(datanotshown). Finally, it must be noted that Gray & Chugonov’s theory canbeusedtoexplainthecentreofmassmovementreported FIG.7:Normalisedsegregationspeedmeasuredastheinverseofthe in[17,18],togetherwithourresults.Considerthesolutionfor Ftyipgi.c6alfotirmdeif,fte0re/nttotabptaeirninegdafrnogmlesth(et0ceisntthreeotyfpmicaaslstdimiseplfaocream=ent2◦o)n. the concentration of small particles in the chute, f (x,t). To Thesolidlineisalineareyeguide. obtainthecentreofmassofthesmallparticlesoneintegrates theconcentrationtimestheposition,z (t)= 1zf (z,t)dz, cm 0 which can be solved numerically in a very sRtraightforward way. Itiseasyto verifythatthe predictedsolutionshowsan Comparisonwiththeory exponential decay for small values of Pe∼4, but seems to showtwodifferentslopesforlargervaluesofit. Inthisway, Letuslookatthephenomenonfromanotherpointofview. Pecanbemeasurednotonlyforthesteadystate solution,as Since the freesurfaceis at anangle with respectto the hori- donein[18]butfromtheevolutionofthesystem. Theresults zontal(goinguptowardsthelargeside ofthe drum,see Fig. presentedhereindicatethatbothmaybedifferent. 2) the projection of gravity on the free surface is non-zero. AdetailedstudyofPefordifferentsystemparametersisbe- Thistranslatesintoaneffectivegravitytowardsthesmallside yondthescopeofthispaper.Ourobjectivebeingtoshowhow ofthedrum.Fromthetheoryforshallowgranularavalanches wellthetheorydescribetheevolutionofthesystemandhow [20,21]weknowthetimeevolutionfortheconcentrationof the tapereddrum couldbecome a trivial experimentalset-up eachspecies,andweshouldbeabletoapplythisresulttothe wheretomeasurethePećletnumber. Adetailedstudyofthis tapereddrum. shouldbehopefullyasubjecttobeaddressedbothexperimen- Whereasthe chuteflow developsverticalsegregationas it tallyandtheoreticallyinthefuture. movesdownslope, the rotating drumdevelopsaxialsegrega- tion as time passes. This is due to the structure of the flow as seen in Fig. 3. The tapered drum turns the kinetic siev- CONCLUSIONS ingalongtheflowlinesintoaxialsegregationatthesystem’s level.Accordingtothissimpleanalogy,smallparticlesshould We havestudiedthe size-drivensegregationof bi-disperse goto the smallsideof thedrum, whilelargeparticlewillgo mixturesintaperedrotatingdrums. Therelationbetweenve- towardsthelargeside. Furthermore,sincethefree-surfaceis locityfield,freesurface,andthesegregationpatternwasstud- ageometriccharacteristicoftheflowanddoesnotdependon ied. Theresultsconfirmthatleveldifferencesonthefreesur- thedegreeofsegregationofthemixture,thetapereddrumwill facecreateaxialflows,whichinturncauseaxialsegregation. segregatethe particles into stable configurationsin the same Forthefirsttimewereporttotalaxialsegregationinthispar- mannerthatachuteflowdoes. ticular geometry and we studied its dependenceon different The temporal evolution of the concentration is something parameters; in particular, we showed that the axial segrega- particularly difficult to measure experimentally and, to the tionspeedsupwiththeangleofthedrum. bestofourknowledge,thereisonlyonerealisation[22]. Our Wehaveshownthattapereddrumscanbeusedtodetermine resultsshowastrikinglygoodfitwiththetheory,Fig.8. For the Pećlet number used in previoustheories from a straight- thesakeofspace,wereferthereadertotheoriginalpaperfor forward experimentalset-up. This has been done before but thederivationofthesolution[23]andtotheappendixforthe the experiments are very difficult to set up “with both well explicitformulaweuse. Notethatforatapereddrumthehalf controlled initial conditions and a steady uniform bulk flow volume height is at z50 =L/ (2), the spatial coordinate of field” [22]. And indeed, to the best of our knowledge, the thetheoreticalsolutionhasbepenscaledtoaccountforthis. results from Wiederseiner et al. are the only ones that give Theresultsareremarkablysimilarconcerningboththeevo- thePećletnumberexperimentally. Onthecontrary,toobtain lutionoftheconcentrationandthespatialdistribution. How- well controlled results on a tapered drum is extremely easy, ever,wecannotfitboththetemporalevolutionandthesteady for example, using the same set-up that motivatedthis study statesolution.WechosetofitthePećletnumberPe(adimen- [5]. sionless number that quantifies how large the segregation is Besidesthesetheoreticalimplications,theresultspresented 6 1.0 1.0 0.8 0.8 0.6 0.6 L (cid:144) Z z 0.4 0.4 0.2 0.2 0.0 0.0 5 10 15 20 25 30 5 10 15 20 25 30 turns t FIG.8:Leftpanel,concentrationofsmallparticlesalongthenormalizedrotationaxisasafunctionoftimeforournumericalexperimentswith a =10◦andL=200mm.Rightpanel,theoreticalsolutionusingtheresultsofRef.[23]withPe=14. herehavedirectapplicationsinthe separationandsortingof forthesteadysolution,whilethetemporalpartisgivenby granulatesinareasrangingfrompowderhandlinginthephar- mshaecdeulitgichatloinndtuhsetriymtpooorrtaenmtirlolilnegoifngtehoemmeitnryingininsdegursetrgyatainodn w t = (cid:229)¥ Anexp(cid:18)−(nx2p22+14)t (cid:19)sin(nxpx ). n=1 0 0 processes, not only for rotating drums but everywhere tridi- mensionalflowsappear. wherethetransformedtimeandspacevariablesaregivenby t =(S2/D )t and x =(S /D )z and x =Pe. For an equal r r r r 0 volumemixture,asinourcase,c (x )=1. UsingEq.(5.46) 0 0 from[23] forourmeanconcentrationf =0.5, we find that ACKNOWLEDGMENTS m thecoefficientsintheFouriersineseriesaresimplifiedinto This work was supported by the IPROCOM Marie Curie initial training network, funded through the People Pro- 2Pe2 gramme (Marie Curie Actions) of the European Union’s An= np (Pe2+4n2p 2)(1−(−1)n). SeventhFrameworkProgrammeFP7/2007-2013/underREA grantagreementNo. 316555. WewouldliketothankJ.Kih- In our solution, we take only the first hundred terms of the nast,N.Rivas,W.denOtter,S.LudingandA.R.Thorntonfor series. fruitfuldiscussions,andthekindhospitalityofA.Michrafyat theEcoledeMines,Albi,France,wherepartofworkwasre- alised. Thesimulationsperformedforthispaperwereunder- takeninMercury-DPM.ItisprimarilydevelopedbyT.Wein- hart, A.R. Thornton and D. Krijgsman at the University of ∗ [email protected] Twente. 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