Transitory powder flow dynamics during emptying of a continuous mixer

(1)

This is an author-deposited version published in :

http://oatao.univ-toulouse.fr/

Eprints ID : 9910

To link to this article : DOI:10.1016/j.cep.2012.12.004

URL :

http://dx.doi.org/10.1016/j.cep.2012.12.004

To cite this version :

Ammarcha, Chawki and Gatumel, Cendrine and Dirion,

Jean-Louis and Cabassud, Michel and Mizonov, Vadim and

Berthiaux,

Henri

Transitory powder flow dynamics during emptying of a

continuous mixer.

(2013) Chemical Engineering and Processing :

Process Intensification, vol. 65 . pp. 68-75. ISSN 0255-2701

Any correspondence concerning this service should be sent to the repository

(2)

Transitory

powder

flow

dynamics

during

emptying

of

a

continuous

mixer

C. Ammarcha

a

_{, C.}

_Gatumel

a

_{, J.L.}

_Dirion

a

_,

_M.

_Cabassud

b

_,

_V.

_Mizonov

c

_,

_H.

_Berthiaux

a,_∗

a_Centre_RAPSODEE,_FRE_CNRS,_Ecole_des_Mines_{d’Albi-Carmaux,}_Campus_Jarlard,_route_de_Teillet,₈₁₀₀₀_Albi,_France b_Laboratoire_de_Génie_Chimique,_UMR₅₅₀₃_{CNRS-INPT-UPS,}_BP_1301,₅_rue_Paulin_Talabot,₃₁₁₀₆_Toulouse,_France c_Ivanovo_State_Power_Engineering_University,_Department_of_Applied_Mathematics,_Rabfakoskaya_34,₁₅₃₀₀₃_Ivanovo,_Russia

Keywords: Semi-batchmixing Powder Markovchain Transitoryregime

a

b

s

t

r

a

c

t

Thisarticleinvestigatestheemptyingprocessofacontinuouspowdermixer,frombothexperimental andmodellingpointsofview.TheapparatususedinthisworkisapilotscalecommercialmixerGericke GCM500,forwhichaspecificexperimentalprotocolhasbeendevelopedtodeterminetheholdupin themixerandtherealoutflow.Wedemonstratethatthedynamicsoftheprocessisgovernedbythe rotationalspeedofthestirrer,asitfixescharacteristicvaluesofthehold-upweight,suchasathreshold hold-upweight.ThisisintegratedintoaMarkovchainmatrixrepresentationthatcanpredictthe evo-lutionofthehold-upweight,aswellasthatoftheoutflowrateduringemptyingthemixer.Depending ontheadvancementoftheprocess,theMarkovchainmustbeconsideredasnon-homogeneous.The comparisonofmodelresultswithexperimentaldatanotusedintheestimationprocedureofthe param-eterscontributestovalidatingtheviabilityofthismodel.Inparticular,wereportresultsobtainedwhen emptyingthemixeratvariablerotationalspeed,throughstepchanges.

1. Aquickstate-of-the-art

Themixingofsolidsisacommonoperationinseveral indus-trialapplications.Itfrequentlyrepresentsacrucialunitoperationin manyindustries(foodstuffs,cosmetics,ceramics,detergents, pow-deredmetals,plastics,drugs,etc.).Whilemixersareoperatedeither inbatchorincontinuousmodeforindustrialapplications,itis

cur-renttoseesuchmachinesoperatinginsemi-batchmodeduring

limitedperiodsoftime,eitherbecausethefeedhasbeenstopped

while keeping a continuous withdrawal, or conversely keeping

acontinuousfeedwitha blockedoutlet.Thishasbeenrecently pointed out by Berthiauxet al. [4], who studiedthe impactof feedingperturbationsonmixturequalityforanOTCdrug.Because loss-in-weightfeedersloosetheirregularitywhenthemassof pow-derinthehopperbecomeslessthanapproximately15–20%ofits apparentvolume,the fillingperiod of thefeedersresultsin an importantsourceofmixtureheterogeneity. Itwassuggestedto stopthefeederswhenfillingandoperatethecontinuousmixerin semi-batchmode,atleastnottodisturbthefollowingoperations liketabletting.Anotherexampleoftheimportanceofsemi-batch modeintheindustryisthedischargeofbatchblendersattheend ofoperation.Duringemptying,thebatchmixeristherefore trans-formedintoasemi-batchsystem.Inthecaseofbatchconvective mixers,itisworthnotingthatoperatorswillruntheengineathigh

∗ Correspondingauthor.Tel.:+33563493144.

E-mailaddresses:henri.berthiaux@mines-albi.fr,henri.berthiaux@mines-albi.fr

(H.Berthiaux).

speedtoacceleratetheemptyingprocesswithoutpaying atten-tiontotheeffectthisprocedurecanhaveonmixturequality.Inthe pharmaceuticalindustry,thisisallthemoreimportantbecauseof the“principleofreconciliation”statingthateachportionofeach materialshouldbetrackedfromitsenteringintothesystemtoits leaving.

IfweexcludethepaperbySudahetal.[13],veryfewstudieshave beenpublishedconcerningtheemptyingprocessofamixerandits effectonblendhomogeneityorpowderflow.Moregenerally, con-tinuousprocessingofpowdershasneverbeenextensivelystudied fromtheviewpointoftransitoryregimephasesthatarecurrently takingplaceduringemptying,startingordosagemodechanging. Thisisallthemoreastonishingthatthediscretenatureofgranular medialogicallyinducestransitorymotionandtransitoryflow.

Behaviour of granular materials in mixers or in other

pro-cess equipment,is still poorly understood from a fundamental

standpoint, as it is hardly possible to portray perfectly all the detailsoftheprocessinamathematicalformandinareasonable time.However,somedynamicmodelsarepresentinthechemical engineeringliterature,tryingtoexplainanddescribegloballythe processratherthanintofulldetails.Markovchainspertaintothis system’sapproachtoolbox.

Twocasesareunderlinedbehindthetermmixingandmustbe distinguishedwhendealingwithmodelsofanytype:bulkparticle flowandtransportinanytypeofvessel,includinginmixers; mix-ing/blendingofvariousflowsrelatedtodifferenttypesofparticle

[3].SeveralresearchersusedtheMarkovchaintheorytodescribe “pure”powderflowpatternsinmixers.Some40yearsago,Inoue andYamaguchi[8]describedthegeneralflowpatternofglassbeads

(3)

insideabatchtumblermixer.Asinglecolouredbeadwasadded anditspositionwasrecordedateverytransition(revolutionofthe mixer).Theexperimentaldatawereusedtodeterminethe transi-tionmatrix,andservedtoderiveatwodimensionalhomogeneous Markovchainmodel.Morerecently,Aounetal.[1]appliedthistype ofmodelforalaboratoryhoopmixer,whichwasdividedinto33 elementarycells(11axialcompartmentsthatwereinturn sepa-ratedinto3radialcells).Theexperimentaldataallowedtoderive thetransitionmatrixandservedagaintodiagnosetheflow pat-ternofcouscousparticlesinthevessel.OtherexamplesofMarkov chainsforthedescriptionofpowderflowinmixerscanbefoundin Chenetal.[5]andLaiandFan[9].

Attheverysametime,otherresearchersweremodellingthe mixingofdifferentparticulateflows,principallybinarymixtures.At theendoftheseventies,FanandShin[7]studiedthebinarymixing processofsphericalluciteparticleswithtwodifferentparticlesize inatumblermixerdividedinto10sectionsofequalvolume.They linkedthetransitionprobabilitiesofaMarkovchaintothediffusion coefficientandthedriftvelocity.Thetransitionmatrixwas deter-minedexperimentallyandallowedtodescribetheconcentration profilesforeachsystem.Morerecently,atwo-dimensionalmodel oftheflowandmixingofparticulatesolidshasbeendevelopedon thebasisoftheMarkovchaintheoryforanalternatelyrevolving staticmixer[12].Thismodelrepresentedthedistributionof com-ponentduringmixingoperation,in bothverticalandhorizontal directions.Severalsimulationswereperformedtoinvestigatethe effectoftheinitialloadingofthecomponents,andtheeffectofthe valuesofthetransitionprobabilities.Otherexamplesarepublished inOyamaandAgaki[11],WangandFan[14]orWangandFan[15]. Inthepresentarticle,wewillconcentrateonstudyingthe dis-chargeprocessofacontinuouspilot-scaleconvectivemixer.We willfocusontheeffectoftheinitialpowdermass,aswellasonthe rotationalspeedofthestirrerwhileemptyingthemixeronboththe hold-upweightofa“pure”bulkmaterialanditsoutletflowrate. Thisisapreliminarytoabetterunderstandingofthedynamicsof blenddischarginganditsimpactonmixturehomogeneity.Wewill alsoinvestigatetheeffectofstepvariationsintherotational stir-rer’sspeedduringemptyingofthemixer.Allthiswillbeintegrated intoaMarkovchainasatoolforpredictionofhold-upweightand outflowratefromoperatingvariablesvaluesandvariability.

2. Experimentalset-upsandmethods

TheexperimentswereperformedinaGerickeGCM500 continu-ouspilotscalepowdermixer,whichcanberuneitherincontinuous, batchorsemi-batchmode.Thismixerisequippedwiththree loss-in-weightfeederstoensureaverypreciseandregulardosage.In

thiswork,allexperimentswereperformedinsemi-batchmode,

whichmeansthatparticleshavebeeninitiallyplacedintheblender andfeederswerenotused.Attheoutletofthemixeran analyti-calbalanceisplacedtomeasuretheoutflowmass inrealtime. Thisbalancecommunicateswithacomputerthroughaserialport RS232wehaveconfiguredintheMatlabControlToolbox. There-forealldataaremeasuredandsavedinrealtime.Experimentshave beenperformedwithfoodproducts,namelycouscousparticles.Full descriptionofequipmentandpowderusedcanbeseeninMarikh etal.[10]andinFig.1a.Themixeritselfisahemi-cylindricaltankof 50cmlong,16.5cmheightand20cmdiameter.Particlemotionis duetothestirringactionofthemobile,sothismixercanbe classi-fiedintothecategoryofconvectivemixers.Thestirrerisconstituted of14rectangularpaddlesinclinedata45◦_angle_and_installed_on_a

framesupportinganinternalscrew.Thedimensionsofthisframe are45cmlengthand18cmwidth.Thescrewisplacedatthecentre oftheframeandallowsaconvectiveaxialmotionoftheparticles insidethevessel.

Theoutletofthemixerconsistsinagatevalvewhoseposition playsanimportantroleasitdeterminesthesectionthroughwhich particleshavetoflowoutoftheapparatus.Ifthissectionisreduced, theholdupinthemixerwillbehigherforfixedconditions,asless particlesareallowedtoexitfromthemixer.Inthepresentwork,all experimentshavebeenperformedwiththeoutletgatevalveatthe maximumofopening.Insemi-batchmode,theholdupweightM1

isafunctioninfluencedaprioribythestirrer’sspeedofrotationNm

andtheinitialloadingmass(seeFig.1b).Inthisstudytheeffectof thesevariablesontheholdupM1willbeexaminedforthefollowing

ranges:

–fivevaluesoftherotationalspeedofthemobileNm,whichcan

beexpressedbythefrequencyoftheengine,N:10,20,30,40and 50Hz;

– threeinitialmassesintroduced4,6and8kg.

TherelationshipbetweentherotationalspeedofthemobileNm

andthefrequencyoftheengineisgivenby:Nm(rpm)=2.6N(Hz).

3. Markovchainmodellingofthesemi-batchprocess

3.1. Generalarchitectureofthemodel

The essenceof a Markov chain model is that ifthe present

statesofa systemareknown—forexamplethepowdermassin

eachstate—andiftheprobabilities—forexampleflowrateratios—to transittoanyotherstatesarealsogiven,thenthefuturestateof thesystemcanbepredicted.Whenprobabilitiesareunchanged,the chainissaidtobehomogeneousandasimplematrixrelationcan bederivedtolinkthefuturestatestotheinitialstate.Conversely, time-dependentorstate-dependentprobabilitieswillresultina step-by-stepcalculationofthestatesofthesystem.

Whenappliedtomixing,thegeneralprinciplelaysindividinga vesselintoncells(Fig.2),orstatesofthesystem,andtoexaminethe possibletransitionsofaparticleproperty(massorconcentrationin akeycomponent)betweeneachcellsduringafixedtimeinterval 1t.

LetthepropertyunderobservationbethemassMn(t)of

parti-clesincellnattimet,andletM(t)bethevectorofelementsMn(t).

Aftereachtransition,aparticlecantransittoanothercellandthe observedpropertycanchangeineachcell.Thisissummarisedbya (n×n)matrixP(t)oftransitionprobabilities,thepj,k(t)

correspond-ingtothetransitionfromstatektostatejattimet.Ifweassume thattransitionscanonlyoccurfromonecelltoaneighbouringcell, thefuturestateM(t+1t)canbepredictedbythefollowingmatrix equation: M(t₊1t)₌P(t)M(t) (1) P(t)₌







P1,1(t) P1,2(t) .... .... 0 0 0 P2,1(t) P2,2(t) ··· ··· 0 0 0 0 P3,2(t) ··· ··· 0 0 0 ··· ··· ··· 0 ··· ··· ··· ··· ··· P_i−1,i(t) Pi,i(t) P_i+1,i(t) ··· ··· ··· ··· ··· ··· ··· ··· ··· 0 0 _·_·_· _·_·_· _·_·_· P_n−2,n−1(t) 0 0 0 ··· ··· ··· P_n−1,n−1(t) 0 0 0 _·_·_· _·_·_· 0 P_n,n−1(t) 1







Therepresentationofthisprocedureundermatrixform,givesa practicaldescriptionandaneasyhandlingoftheprocessdynamics

(4)

Fig.1. GerickeGCM500equipmentshowingthestirringmobile(a).Generalprocessvariables(b).

Fig.2.GeneralMarkovchainrepresentationofthesemi-batchoperationofthemixer,includingaflowmodelinsidethemixer.

whatevertheregimeconsidered:batch,continuous,semi-batch, multiplestagedfeed,transitoryfeed,etc.Fulldescriptionof

blend-ing modelling using Markov chains is detailed in thework by

Berthiauxetal.[2].

Asaforementioned,themixeris usuallydividedinto several cells.Inthepresentrepresentation,wewillonlyconcentrateona generalframeformodellingabletogiveinformationonthe“black box”behaviourofthemixerunderunsteadyconditions,andusea simplifiedMarkovchainmodelwithonlytwocells.Thisfirstmodel willbeimplementedinthefuturetoenterintomoredetailsofthe flowstructure,uptotheleveloftheparticlesthroughdistinct ele-mentmodelling(DEM),givingthenrisetoahybridmodelsuchas theonerecentlydevelopedbyDoucetetal.[6].

Thefirstcellrepresentsthewholemixer,thelevelof discreti-sationbeingthewholeapparatus.The secondcellis theoutlet of the mixer, acting as an absorbingstate. This cell is shown, for example,in Fig.3 as thelast cellon theright.If a particle reachestheabsorbingstate,itstaysinitforever,i.e.,pii=1exists

forthisstate.Inourcase,theabsorbingstateisjoinedtothechain artificially.

LetM1bethemassofparticlesinmixer(orincell1),andM2the

massofparticlesintheabsorbingstate(orincell2).Thetransition ofaparticlefromstate1tostate2willobeytothegeneralMarkov chainrelation: M1(t+1t) M2(t+1t)

!

= p11(t) p12(t) p21(t) p22(t)

!

× M1(t) M2(t)

!

= 1−p21(t) 0 p21(t) 1

!

× M1(t) M2(t)

!

(2)

InordertorepresenttheemptyingprocessthroughEq.(2),itis necessarytoknowtheinitialloadingmassM1(0).Astheabsorbing

statedoesnotcontainparticlesinitially,thenM2(0)isequaltozero.

(5)

Fig.3.SimplifiedMakovchainrepresentationoftheemptyingprocessofthemixer.

canbedonebythefollowingstep-by-stepprocedure: P21(t+1t)= M2(t+_M1t)−M2(t)

1(t)

= [M1(0)−M1(t+1t)]_M −[M1(0)−M1(t)]

1(t)

(3)

3.2. Implementingthemodelfromexperimentalresults

AsstatedbyEq.(3),p21(t)canbedeterminedfromthe

experi-mentalresultsofM1(t).Fig.4givesanexampleoftheexperimental

resultsobtainedatN=20Hzfortheevolutionofthehold-upweight incell1(mixer)fordifferentinitialweight.P21isalsopresentedin

thesamegraphforatransitiontime1t=1.6s.Thistransitiontime hasbeenarbitrarilyfixedafterseveraltries,toensureasufficient

resolutionofthemodel.Twoothersimilarexamplesareshown

inFig.5forhigherrotationalspeeds.Itcanbeseenthatthe ini-tialmassM1(0)donotaffecttheresultofprobabilityp21,which

meansthatthedynamicsofemptyingofthemixeronlydependson therotationalspeed.Whenexaminingtheseresults,wecandefine threeregimesofprobabilitycorrespondingtothecurrentholdup weightinthemixer.Thereexistsaminimumholdupbelowwhich

nopowdercanflowoutofthemixer,andathresholdvalueabove whichp21isconstant(andequalsp21max).LetMminandMthreshold

bethesecharacteristicholdups.Intermsoftransitionprobability,

wehave

for M1>Mthreshold, p21=p21max (4)

and for M1<Mmin, p21=0 (5)

WhenM1 ishigherthanMthreshold,thevalueofp21isstablewith

time,sothattheMarkovchainisa timehomogeneouschainor

aMarkovchainswithtime-homogeneoustransitionprobabilities. ThetransitionmatrixPalwaysremainsthesameateachstep,so thek-steptransitionprobabilitycanbecomputedasthek’thpower ofthetransitionmatrix,Pk_.

WhenthemassM1islowerthanMthresholdandhigherthanMmin,

theprobabilityP21increasesalmostlinearlywithhold-upweight

M1:

for Mmin<M1<Mthreshold, p21(t)=aM1(t)+b0 (6) Inthiscase,theprobabilityoftransitioninasinglestepfromstate 1tostate2dependsontheparticlemassinstate1,whichmakes thetransitionstime-dependentthroughitsstate-dependency.The

Fig.4.ExperimentalevolutionofM1andp21duringemptyingofthemixer(N=20Hz).

(6)

Fig.6.EvidenceofalinearrelationbetweenparameteraandN(a);relationbetweenMminandN(b).

Markovchainisthereforenothomogeneous,andstatesmustbe

calculatedstep-by-step.However,this isfarfrombeinga prob-lemfromacomputationalpointofview.AsfarasM1isinferiorto

Mthreshold,thenewprobabilityp21(t)dependsonlyontheprevious

calculationofthestatesthroughM1(t)andbothparameters(aand

b)oflinearEq.(6).

ParameteraonlydependsontherotationalspeedNasitcan beseeninFig.6a,andthesecondparameterbcanbecalculated forM1=Mminasb=−a×Mmin.Inturn,MminhasbeenrelatedtoN

throughapowerlaw,asindicatedinFig.6b.Thepowerhasbeen identifiedtobeclosedto0.5,sothatweconsideredthisvalueas beenfixedandrecalculatetheotherparametervaluesk1andk2:

a₌k1×N (7)

b=−a×Mmin=−k1×k2N0.5 (8)

Inthisnon-linearregime,theemptyingprocessofthemixer there-forevarieswiththehold-upweightandtherotational speedof themobilethroughthefollowinggeneralrelationshipthatincludes twoconstantsdeterminedformthewholesetofexperimentaldata: p21(t)=k1N

M1(t)−√k2 N

(9) Itmustbeemphasizedthatthislatterequationmakesthe over-allMarkovchainnon-homogeneous,asthetransitionprobability dependsonthestateofthesystem.Ifthereisnoapriori obvi-ousphysicalexplanationforthis, itis clearthat fromaprocess standpoint,thisbehaviourmustbeknowntomonitortheprocess. IfM1ishighertoMthreshold,theprobabilityp21becomessteady

andcanbelinearlyrelatedtoN,asshowninFig.7a:

p21max=k3×N (10)

Finally,thethresholdmassMthresholdcanbedeterminedfromthe

aboveprocedureandlinkedtotherotationalspeed:

a_×Mthreshold+b=p21max (11) k1×N×Mthreshold+(−k1×N×k2×N0.5)=k3×N (12) Mthreshold= (k3+k1×k2×N 0.5₎ k1 (13)

4. Resultsanddiscussion

Despitethefactthattheabovemodelhasbeendevelopedusing

ageneralframework,itcontainsunknownparametersthathave

beenfoundfromtheexperimentsdescribedintheprevious sec-tion.Inthissection,wewilltestthevalidityofthisprocedureby eitherconsideringtheabilityofthemodeltodescribesecondary variables,suchasoutflowrate,oroperatethemixerunder condi-tionsfarfromthoseunderwhichparametershavebeenobtained, likeduringstep-likeperturbations.

4.1. Predictingoutletflowrateduringemptying

Fortheproductselected,e.g.,afree-flowingmaterial,the rota-tionalspeedofthestirrercompletelydefinesthedynamicsofthe emptyingprocessofthemixerbyfixingthevalueofthe

thresh-oldhold-upweight.Belowthisvalue,anon-homogeneouschain

canbeemployed.Aboveit,ahomogeneouschaingivestheruleof semi-batchprocessing.Thisprocedurealsogivesfullaccesstothe evolutionoftheoutletflowrateQout,throughthefollowing:

Qout(t)= M2(t+1t)_1t−M2(t) (14)

As M2(t+1t)=M2(t)+p21(t)×M1(t) (15)

Qoutcanthenbelinkedtop21:

Qout(t)= p21_1t(t)×M1(t) (16)

Anexampleofcomparisonbetweenexperimentalresultsofthe

flow rate,obtainedby Eq.(14),and modelsimulation formthe MarkovchainrepresentationisshowninFig.8fordifferent rota-tionalspeedsofthestirrer.Thedifferentflowregimes,depending

onthe value ofthe actualhold-up weightas compared to the

thresholdvalue,areagainpointedout: –ForM1>Mthreshold

Qout(t)= p21_1tmaxM1(t) (17)

Qout(t)= k3_1t×NM1(t) (18)

(7)

0 20 40 60 80 100 120 140 0 50 100 150 200 250 300 350 400 M1(g) Qout kg/h) c 0 50 100 150 200 250 300 350 400 0 200 400 600 800 1000 1200 1400 M1(g) Qout model: 10Hz Qout exp: 10Hz Qout model: 20Hz Qout exp: 20Hz Qout model: 30Hz Qout exp:30Hz Qout model:35Hz Qout exp: 35Hz Qout moedel: 40Hz Qout exp: 40Hz Qout (kg/h) c 0 20 40 60 80 100 120 140 0 50 100 150 200 250 300 350 400 M1(g) Qout kg/h) c 0 50 100 150 200 250 300 350 400 0 200 400 600 800 1000 1200 1400 M1(g) Qout model: 10Hz Qout exp: 10Hz Qout model: 20Hz Qout exp: 20Hz Qout model: 30Hz Qout exp:30Hz Qout model:35Hz Qout exp: 35Hz Qout moedel: 40Hz Qout exp: 40Hz Qout (kg/h) c

Fig.8.ComparisonofsimulatedandmeasuredoutflowrateaccordingtoretainedmassM1.

–ForMmin<M1<Mthreshold

Qout(t)=p21_1t(t)M1(t) (19)

Qout(t)=k1_1t×N

#

M21(t)−Mmin×M1(t)

(20) Inthislattercase,asthetransitionprobabilitychangeslinearlywith M1,theoutflowrateincreaseswiththehold-upweightaccording

toasecondorderpolynomiallaw.

4.2. Predictingtheeffectofstep-likeperturbations

Thepredictabilityofthemodelcanbebetterdemonstratedif itsresultsareplottedagainstdatathathavenotbeenusedinthe determinationofitsparameters.Inthislastpart,wewillcompare modelresultsandexperimentaldataobtainedwhenemptyingthe mixerthroughnegativeorpositivestepsontherotationalspeed, asitmaybethecaseintheindustry.

Fig.9 reports theresults obtainedwhile changing instanta-neouslytherotationalspeedfrom20to40Hzduringanemptying process.Asitcanbeobserved,thepredictionofbothoutflowrate

Qout and holdupweightM1 isexcellentwhencomparedtothe

experimentaldata.Thestepinrotationalspeedinducesajumpin flowrate,butalightdecreaseinhold-up.Thisisvalidated experi-mentallyforawiderangeofrotationalspeedofthestirrerdevice (10–50Hz),notreportedhereforclarity.Anotherexampleof com-parisonisshowninFig.10,forwhichanegativestepchangeinthe rotationalspeedfrom35to15Hzhasbeenperformed.Thesame conclusionscanbedrawnonthemodel’sperformance.

The evolution of the outflow rate with the hold up weigh

M1 during thesetwotypes ofperturbationis shownin Fig.11.

The lines predicted by the model with or without

perturba-tion are also figured to better illustrate the effect of the step changes.

Thetransitionprobabilities,usedinthesesimulations,are deter-minedfromtheempiricalequations(4)–(6),andarerepresentedin

Fig.12forthepositivestep.Inthisgraph,wecandifferentiatethree phasesdependingonNandM1(seeFig.13forarepresentationin

termsofoutflowrate):

• Initially,Nisfixedat20Hz.Inthiscase,asthehold-upweightin themixerM1ishigherthanMthreshold,thevalueofp21isstable

(a)

(b)

0 100 200 300 400 500 600 700 800 900 1000 0 20 40 60 80 100 Temps(s) 0 10 20 30 40 50 Qout model: 20_40Hz Qout exp: 20_40Hz N(Hz) Qout (kg/h) N 20Hz 40Hz 0 1000 2000 3000 4000 5000 6000 0 20 40 60 80 100 Temps (s) 0 10 20 30 M1 model:20_40Hz M1exp: 20_40Hz N (Hz) M1 ( N (H 20Hz 40Hz 0 100 200 300 400 500 600 700 800 900 1000 0 20 40 60 80 100 Temps(s) 0 10 20 30 40 50 Qout model: 20_40Hz Qout exp: 20_40Hz N(Hz) Qout (kg/h) N 20Hz 40Hz 0 1000 2000 3000 4000 5000 6000 0 20 40 60 80 100 Temps (s) 0 10 20 30 40 50 M1 model:20_40Hz M1exp: 20_40Hz N (Hz) M1 (g) N (Hz) 20Hz 40Hz

Fig.9.Comparisonofsimulatedandmeasuredresultsduringemptyingwithapositivestepchangeinrotationalspeed(N=20–40Hz).Effectontheoutflowrate(a)andon theholdupweight(b).

(8)

0 100 200 300 400 500 600 700 800 900 1000 0 20 40 60 80 100 120 140 0 10 20 30 40 50 Qout model: 35_15Hz Qout exp: 35_15Hz N(Hz) Qout (kg/h) (kg/h) N (Hz) 35Hz 15Hz 1000 2000 3000 4000 5000 6000 0 20 40 60 80 100 0 10 20 30 40 50 M1 model:35_15Hz M1exp: 35_15Hz N (Hz) M1 (g) N (Hz) 35Hz 15Hz 0 100 200 300 400 500 600 700 800 900 1000

(a)

(b)

0 20 40 60 80 100 120 140 Temps(s) 0 10 20 30 40 50 Qout model: 35_15Hz Qout exp: 35_15Hz N(Hz) Qout (kg/h) (kg/h) N (Hz) 35Hz 15Hz 0 1000 2000 3000 4000 5000 6000 0 20 40 60 80 100 Temps (s) 0 10 20 30 40 50 M1 model:35_15Hz M1exp: 35_15Hz N (Hz) M1 (g) N (Hz) 35Hz 15Hz

Fig.10. Comparisonofsimulatedandmeasuredresultsduringemptyingwithanegativestepchangeinrotationalspeed(N=35–15Hz).Effectontheoutflowrate(a)and ontheholdupweight(b).

(a)

(b)

0 200 400 600 800 1000 1200 1400 0 500 1000 1500 2000 2500 3000 3500 4000 M1 (g)₀ 10 20 30 40 Qout model: 35Hz Qout model: 15Hz Qout model: 35_15Hz Qout exp: 35_15Hz N(Hz) Qout (kg/h) N(Hz) 35Hz 15Hz 0 200 400 600 800 1000 1200 1400 0 1000 2000 3000 4000 5000 M1 (g) 0 10 20 30 40 Qout model: 40Hz Qout model: 20Hz Qout model: 20_40Hz Qout exp: 20_40Hz N(Hz) Qout (kg/h) N(Hz) 40Hz 20Hz 0 200 400 600 800 1000 1200 1400 0 500 1000 1500 2000 2500 3000 3500 4000 M1 (g)₀ 10 20 30 40 Qout model: 35Hz Qout model: 15Hz Qout model: 35_15Hz Qout exp: 35_15Hz N(Hz) Qout (kg/h) N(Hz) 35Hz 15Hz 0 200 400 600 800 1000 1200 1400 0 1000 2000 3000 4000 5000 M1 (g) 0 10 20 30 40 Qout model: 40Hz Qout model: 20Hz Qout model: 20_40Hz Qout exp: 20_40Hz N(Hz) Qout (kg/h) N(Hz) 40Hz 20Hz

Fig.11.Evolutionoftheoutflowrateaccordingtohold-upweightafterapositive(a)andoranegative(b)stepchangeinrotationalspeedN.

(a)

(b)

0 200 400 600 800 1000 1200 1400 0 500 1000 1500 2000 2500 3000 3500 4000 M1 (g) 0 10 20 30 40 Qout model: 35Hz Qout model: 15Hz Qout model: 35_15Hz Qout exp: 35_15Hz N(Hz) Qout (kg/h) N(Hz) 35Hz 15Hz 0 200 400 600 800 1000 1200 1400 0 1000 2000 3000 4000 5000 M1 (g) 0 10 20 30 40 Qout model: 40Hz Qout model: 20Hz Qout model: 20_40Hz Qout exp: 20_40Hz N(Hz) Qout (kg/h) N(Hz) 40Hz 20Hz 0 200 400 600 800 1000 1200 1400 0 500 1000 1500 2000 2500 3000 3500 4000 M1 (g) 0 10 20 30 40 Qout model: 35Hz Qout model: 15Hz Qout model: 35_15Hz Qout exp: 35_15Hz N(Hz) Qout (kg/h) N(Hz) 35Hz 15Hz 0 200 400 600 800 1000 1200 1400 0 1000 2000 3000 4000 5000 M1 (g) 0 10 20 30 40 Qout model: 40Hz Qout model: 20Hz Qout model: 20_40Hz Qout exp: 20_40Hz N(Hz) Qout (kg/h) N(Hz) 40Hz 20Hz

Fig.12.Comparisonofsimulatedandmeasuredvaluesofp21duringtheemptyingprocessafterapositivestepchangeinrotationalspeedN,onthebasisoftimechange(a)

orhold-upchange(b).

(p21max=0.07),andthecorrespondingoutflowratechanges

lin-earlywithM1accordingtoEq.(18).

• ThenextphasebeginswhenNchangesfrom20to40Hz,att=40s. Immediatelyafterthisstepchange,thevalueofp21increasesto

remainstable(p21=0.14)aslongasM1ishigherthanMthreshold

forthisnewvalueofN.

• Finally,whenM1becomeslowerthanthethresholdmass,the

probabilityincreasesalmostlinearlywiththehold-upweight, andtheoutflowratechangeswithM1accordingtoasecondorder

polynomialfunction(Eq.(20)).

The resultsobtainedfor thenegative step changein rotational speed,whilenotpresentedherforclarityreasons,alsosupports theparameteridentificationresults,andconfirmsthatthe proba-bilitytoleavethemixerdependsonlyontherotationalspeedand onthecurrentparticlemassinthemixer.Theempiricalcorrelations thathavebeendevelopedinthisworkseemtoberobustenough todescribethedynamicsoftheemptyingprocess,evenwhenit includesastrongvariationontheprocessvariables.Inotherwords, the“history”oftheprocess,doesnotinfluenceitsdynamics,which correspondstotheMarkovproperty.

(9)

0 50 100 150 200 250 300 0 200 400 600 800 1000 1200 1400 1600 M1 (g) 10 20 30 40 50 60 70 Qout model: 20_40Hz Qout exp: 20_40Hz N Qout (kg/h) N(Hz)

Fig.13.Simulatedandmeasuredevolutionoftheoutflowratewiththehold-upweightM1duringapositivestepchangeinrotationalspeedN,focussingonthesmallmasses.

5. Concludingremarks

In this study, we have developed a Markovchain approach

torepresentthedynamicsoftheemptyingprocessofa contin-uouspowdermixer.Thechainisonlyheretoprovideageneral frameworkoftheproblem,andincludescorrelationsobtainedfrom experimentaldatafitting.Thetransitionmatrixonlydependson therotationalspeedofthestirrerandthehold-upweight,asthe

chaincaneitherbehomogeneousornon-homogeneousduringthe

emptyingprocess.Allobtainedfunctionalrelationshipsarebased

oninformationgatheredexperimentallyfromcontinuoussystem

operatinginsemi-batchmode,andmayservetothebetter under-standingandmodellingofcontinuousregime.

Animportantindustrialissueisconcernedwiththeeventuality ofstoppingthefeedingsystemduringitsfillingtoavoid pertur-bationduetolow-qualitydosage,lettingthemixeroperateinthe meantime.Fromthepresentstudy,itcanbearguedthat,assoonas theholdupweightisabovethethresholdvalue,thechainis homo-geneousandthemixermaystilloperateadequately,butwitha changedoutflowrate.Butiftheholdupweightpassesthislimit,the chainbecomesnon-homogeneous,andthenon-linearcharacterof theemptyingprocessmaygiverisetosomefluctuationsand per-hapshomogeneitylossofthefinalproduct,astransitionmatrices arechangingateverystep.

Thisworkand themethodologypresented alsosuggest that

simple emptying experimentscan berun to determinetheset

ofgoverningparameters, atleastfor similarmixinggeometries andfree-flowingpowders.Thiswouldsufficetoelaboratea sim-plemodelofglobalbehaviourofacontinuousmixerthatmaybe usefulforaprocessengineer,asafirststep.

Finally, it may be thought that a number of principles lay-ingbehindpowderblendinginsemi-batchmode,orintransitory regimesingeneral,wouldplayanimportantroleinunderstanding andmodellingthesteady-stateoperationofacontinuoussystem anditsprocesscontrol.Futureworkfromourteamwillbe concen-tratedonthispoint.

References

[1]M. Aoun Habbache, M. Aoun, H. Berthiaux, V. Mizonov, An exper-imental method and a Markov chain model to describe axial and radial mixing in a hoop mixer, Powder Technology 128 (2002) 159–167.

[2]H.Berthiaux,K.Marikh,V.Mizonov,D.Ponomarev,E.Barantzeva,Modelling continuouspowdermixingbymeansofthetheoryofMarkovchains, Particu-lateScienceandTechnology22(2004)379–389.

[3]H.Berthiaux,V.Mizonov,ApplicationofMarkovchainsinparticulateprocess engineering:areview,TheCanadianJournalofChemicalEngineering82(6) (2004)1143–1168.

[4] H.Berthiaux,K.Marikh, C.Gatumel,Continuous mixingof powder mix-tures with pharmaceuticalprocessconstraints,Chemical Engineeringand Processing:ProcessIntensification47(12)(2008)2315–2322.

[5]S.J. Chen, L.T. Fan, C.A. Watson, The mixing of solid particles in a motionless mixer—a stochastic approach, AIChE Journal 18 (5) (1972) 984–989.

[6]J.Doucet,N.Hudon,F.Bertrand,J.Chaouki,Modelingofthemixingof monodis-perseparticlesusingastationaryDEM-basedMarkovprocess,Computersand ChemicalEngineering32(2008)1334–1341.

[7] L.T.Fan,S.H.Shin,Stochasticdiffusionmodelofnon-idealmixinginahorizontal drummixer,ChemicalEngineeringScience34(1979)811–820.

[8]I.Inoue,K.Yamaguchi,Particlemotioninamixer.Mixinginatwo dimen-sional V-type mixer, International Chemical Engineering 10 (3) (1970) 490–497.

[9]F.S.Lai,L.T.Fan,Applicationofdiscretemixingmodeltothestudyofmixingof multicomponentsolidparticles,Industrial&EngineeringChemistry:Process DesignandDevelopment14(4)(1975)403–411.

[10] K.Marikh,H.Berthiaux,V.Mizonov,E.Barantseva,Experimentalstudyofthe stirringconditionstakingplaceinapilotplantcontinuousmixerofparticulate solids,PowderTechnology157(2005)138–143.

[11] Y.Oyama,K.Agaki,Studiesonthemixingofparticulatesolids,KagakuKogaku (Japan)20(1956)148–154.

[12]D.Ponomarev,V.Mizonov,H.Berthiaux,C.Gatumel,J.Gyenis,E.Barantseva,A 2DMarkovchainformodellingpowdermixinginalternatelyrevolvingstatic mixersofSysmix®

type,ChemicalEngineeringandProcessing:Process Inten-sification48(2009)1495–1505.

[13]O.Sudah,D.Coffin-Beach,F.Muzzio,Effectsofblenderrotationalspeedand dischargeonthehomogeneityofcohesiveandfree-flowingmixtures, Interna-tionalJournalofPharmaceutics27(2002)57–68.

[14]R.Wang,L.T.Fan,AxialmixingofgrainsinamotionlessSulzer(Koch)mixer, Industrial&EngineeringChemistry:ProcessDesignandDevelopment15(3) (1976)381–388.

[15]R.Wang,L.T.Fan,Stochasticmodellingofsegregationinamotionlessmixer, ChemicalEngineeringScience32(1977)695–701.