Granular Segregation in Tapered Rotating Drums

1.6 MB·English

by  Sebastian Gonzalez

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 re-
search [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 veloc-
ity 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
wheretomeasurethePe´cletnumber. 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 Pe´clet 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- thePe´cletnumberexperimentally. 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.WechosetofitthePe´cletnumberPe(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]
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