Experimental and numerical investigation on mixing and axial dispersion in Taylor-Couette flow patterns

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Experimental and numerical investigation on mixing and

axial dispersion in Taylor-Couette flow patterns

Marouan Nemri, Éric Climent, Sophie Charton, Jean-Yves Lanoe, Denis Ode

To cite this version:

Marouan Nemri, Éric Climent, Sophie Charton, Jean-Yves Lanoe, Denis Ode. Experimental and

numerical investigation on mixing and axial dispersion in Taylor-Couette flow patterns. Chemical

Engineering Research and Design, Elsevier, 2012, �10.1016/j.cherd.2012.11.010�. �hal-00918887�

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To link to this article : DOI:10.1016/j.cherd.2012.11.010

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http://dx.doi.org/10.1016/j.cherd.2012.11.010

To cite this version : Nemri, Marouan and Climent, Eric and Charton,

Sophie and Lanoe, Jean-Yves and Ode, Denis Experimental and

numerical investigation on mixing and axial dispersion in

Taylor-Couette flow patterns. (2012) Chemical Engineering Research and

Design . ISSN 0263-8762

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Experimental

and

numerical

investigation

on

mixing

and

axial

dispersion

in

Taylor–Couette

flow

patterns

Marouan

Nemri

_,

_Eric

_Climent

_,

_Sophie

_Charton

_,

_Jean-Yves

_Lanoë

_,

_Denis

_Ode

b_{InstitutdeMécaniquedesFluidesdeToulouse–CNRS–INP–UPS,AlléeduPr.CamilleSoula,31400Toulouse,France}

a

b

s

t

r

a

c

t

Taylor–Couetteflowsbetweentwoconcentriccylindershavegreatpotentialapplicationsinchemicalengineering. Theyareparticularlyconvenientfortwo-phasesmallscaledevicesenablingsolventextractionoperations.An exper-imentaldevicewasdesignedwiththisideainmind.Itconsistsoftwoconcentriccylinderswiththeinneronerotating andtheouteronefixed.Moreover,apressuredrivenaxialflowcanbesuperimposed.Taylor–Couetteflowisknown toevolvetowardsturbulencethroughasequenceofsuccessivehydrodynamicinstabilities.Mixingcharacterizedby anaxialdispersioncoefficientisextremelysensitivetotheseflowbifurcations,whichmayleadtoflawedmodelling ofthecouplingbetweenflowandmasstransfer.Thisparticularpointhasbeenstudiedusingexperimentaland numericalapproaches.Directnumericalsimulations(DNS)oftheflowhavebeencarriedout.Theeffectivediffusion coefficientwasestimatedusingparticlestrackinginthedifferentTaylor–Couetteregimes.Simulationresultshave beencomparedwithliteraturedataandalsowithourownexperimentalresults.Theexperimentalstudyfirst con-sistsinvisualizingthevorticeswithasmallamountofparticles(Kalliroscope)addedtothefluid.Tracerresidence timedistribution(RTD)isusedtodeterminedispersioncoefficients.Bothnumericalandexperimentalresultsshow asignificanteffectoftheflowstructureontheaxialdispersion.

Keywords: Taylor–Couetteflow;Axialdispersion;Directnumericalsimulation;Experiments

1.

Introduction

Annular centrifugal contactors based on Taylor–Couette flow geometry have a great potential in chemistry, met-allurgy and, since the prior works of Davies and Weber (1960), in the nuclear industry wherethey are particularly suitable for small-scale studies of solvent extraction pro-cesses.

Sincethework ofTaylor (1923),Taylor–Couetteflow has beenextensivelystudiedduetoitsnonlineardynamics.Itis awidelystudiedflow, knowntoevolve towardsturbulence througha sequence ofsuccessiveinstabilities asthe inner cylinderrotationrateincreases.

Beyond a critical speed of the rotating cylinder, pure azimuthalCouetteflowresultsintheformationoftoroidal vortices,calledTaylorvortexflow(TVF).Thiscritical condi-tionisexpressedbytheTaylor number(Ta) orequivalently

∗ _{Correspondingauthor.}

E-mailaddress:sophie.charton@cea.fr(S.Charton).

by the Reynolds number (Re) based on the gap width

e, the inner cylinder linear velocity ˝Ri and the fluid

viscosity . AthigherReynolds numbersasecondary insta-bility causes the flow to become time-dependent due to the appearance of an azimuthal wave (deformation of the vortices). This flow state, known as wavy vortex flow (WVF), is characterized by: an axial wave number

n (or alternatively a mean axial wavelength ) and an azimuthal wave number noted m. AsRe further increases, the wavy flow becomes increasingly modulated by addi-tional frequencies(MWVF) and eventually becomes turbu-lent.

These flow instabilities have been the topic of many studies (Coles,1965; Fenstermacheret al., 1979). Suchflow patterns provide high values of heat and mass transfer coefficients, whichexplainswhysomeimportantindustrial operations(emulsionpolymerization,heterogeneouscatalytic

Nomenclature

Notation

Dx dispersioncoefficient(cm2s−1)

D∗

x effectivedispersioncoefficient

e gapwidth

f temporaldominantfrequencyofthetravelling wave(s−1₎

f′ _{frequencyofmodulation(s}−1₎

G normalizedtorque

H annulusheight

m azimuthalwavenumber

n axialwavenumber

Re outercylinderradius

Re=(˝Rie)/ Reynoldsnumber

Reax=(uxe)/ axialReynoldsnumber

Rec criticalReynoldsnumber

Ri innercylinderradius

Sc=/Dx dispersionbasedSchmidtnumber

Tc durationofarotorcompleterotation(s)

Ta=Re

p

Vϕ angularspeedoftravellingwave(rads−1)

˝ rotationalratefortherotor  radiusratio(Ri/Re)

aspectratio(H/e)  axialwavelength f density(kgm−3)

 dynamicviscosity(Pas−1₎

 kinematicviscosity(m2_s−1₎

DNS directnumericalsimulation MWVF modulatedwavyvortexflow PIV particleimagevelocimetry RTD residencetimedistribution TVF Taylorvortexflow

WVF wavyvortexflow

reactions,andliquid–liquidextraction)cantakeadvantageof Taylor–Vortexequipment.

When a weak axial flow is superimposed upon Taylor–Couette flow (PTC, Poiseuille Taylor Couette), axial motion of the Taylor vortices transports the cellu-lar vortices wherein toroidal motion of fluid elements yields efficient mixing. Such a flow system can be con-sidered as a near-ideal plug-flow reactor (PFR). Many researchers have investigated the axial dispersion in Taylor–Couette flows (Tam and Swinney, 1987; Moore and Cooney,1995;Desmetetal., 1996;Campero andVigil,1997; Ohmura et al., 1997; Rudman, 1998). Experimental obser-vations referenced in the literature (Tam and Swinney, 1987; Ohmura et al., 1997) indicate that axial dispersion decreases to a minimum near the onset of Taylor vor-tices and increases further as the Taylor number (Ta) increases.

The present study was carried out in the general context of the application of Taylor–Couette flow for liquid–liquid extraction and particularly for the extrac-tion of nuclear waste (uranium, plutonium) from spent nuclear fuels. We focus on the mixing properties of the single phase flow. The flow bifurcations as well as the axial dispersion in the Taylor–Couette appara-tus were studied using both experimental and numerical approaches.

Table1–Apparatusgeometryandoperatingconditions.

Parameters Firstprototype Secondprototype

Rotorradius(Ri) 8.5mm 24mm Shaftradius(Re) 10mm 35mm Gapwidth 1.5mm 11mm Radiusratio() 0.85 0.68 Length(H) 720mm 640mm Aspectratio( ) 480 58 Rotationspeed 80–1600rpm 4–1000rpm Rerange(water) 107–2136 111–27,646

2.

Methodology

2.1.1. Descriptionoftheapparatus

Thedeviceconsistsoftwoconcentriccylinderswiththeinner cylinder rotating(the outer cylinderis fixed)and with the possibility ofsuperimposing a pressure driven-flow in the axialdirection.Twogeometricalconfigurationsareusedinthis study(Table1).

• _{Thefirstonewasdesignedtostudythemixing} perform-anceswithaminimumvolumeoffluid.Thisfirstdeviceis thereforecharacterizedbyasmallgapwidthe=1.5mm. • Thesecondprototype,withalargergapwidth,e=11mm,

wasdesignedforspecificopticalinvestigations(PIV,PLIF). Themainfeaturesoftheexperimentaldeviceareshownin Fig.1.

Astheflowstateisstronglydependentontheflowhistory, aspeedcontrolsystemwasused.Arampgeneratorcontrols therotoraccelerationduringtransientevolutiontothedesired

Table2–Fluidspropertiesfor“visualization”andRTD experiments.

Fluid Density(kgm−3_{) Viscosity(Pas)}

Kalliroscopesolution 1000 1.4×10−3

Nitricacid0.5N 1018 9.7×10−4

Solvent(TBP30%inTPH) 826 1.5×10−3

Reynoldsnumber.Indeed,forthesameReynoldsnumber,Re, andusingdifferentaccelerationrampsprovidesvariousflow states.Therefore,agivenwavestateisestablishedbyfollowing aprescribedstart-upprotocol,asdeterminedbytheflow visu-alizationtechnique(seeSection2.2.1).Thisprocedureensures reproducibilityofthemeasurements.

2.1.2. Measurementtechniques

Theflowpatternwasvisualized usinganaqueoussolution seededwithKalliroscopeAQ-1000flakes.Theseparticles con-sist of small,light reflecting slabs which align themselves alongstreamlines.Usingahigh-speedcamera,thedifferent transitionscanbevisualized.Inordertofullycharacterizethe differentwavy regimes,aspectralanalysisofthedatawas performedtodeterminethewavestate(axialandazimuthal wavenumbers).

Axialdispersioninsinglephaseflowwasinvestigatedwith twodifferentliquids:anorganicphase(TBP30%inTPH)and anaqueousphase(0.5N,HNO3).Thefluidspropertiesare

sum-marizedin Table2.Tracerexperiments wereperformed by placingopticalsensorsatthetopandbottomofthecolumn. Theannularcavitywasthenfilledwiththeappropriatefluid and thetracer (methyleneblueor Sudan red)wasinjected throughaspecificorifice.Thecolouredtracerdispersionwas monitoredonlinewithaspectrometermountedontoaCCD sensor.Allexperimentalconditionsweresettobewithinthe linearrangeoftheBeer–Lambertlaw(tracerconcentrationis proportionaltoitsabsorbance).Oncenormalized,thecurveof absorbancegivesdirectaccesstotheresidencetime distribu-tion(RTD).

Most models of axial mixing in single-phase Taylor–Couetteflowarebasedonadiffusionequation. Alter-natively, stirredtanks in series can modeltracer response curves.Bothmodellingapproachescanbeusedtodetermine theeffectiveaxialdispersioncoefficientsbymatchingmodel predictiontoexperimentalresults(TamandSwinney,1987). However,evenforsingle-phasesystems,simplemodelsare notalwayssuitable,especiallyatTaylornumbersbelowthe onset of turbulent Taylor vortex flow (Desmet et al., 1996; CamperoandVigil,1997).Inthispaper,mathematicalmodels basedonspeciesmassbalancesappliedtoanear-ideal plug-flowreactorareused.Numericalfittingbetweenexperimental tracerresponsecurves(RTD)andmathematicalmodelslead toadispersioncoefficientestimate(seeFig.2).

2.2. Numericalmethods

2.2.1. Directnumericalsimulationsofthecarryingflow Inthepresentstudy,directnumericalsimulations(DNS)was carriedoutwiththeJADIMfinitevolumecode developedat IMFT.

ThefluidisconsideredincompressibleandNewtonianwith constant physicalproperties. The governing Navier–Stokes equationsarethenwritten:

∇·u=_{0 (1)} 0,0E+00 2,0E-04 4,0E-04 6,0E-04 8,0E-04 1,0E-03 1,2E-03 1,4E-03 1,6E-03 1,8E-03 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Time (s) Abs o rba nc e ( -) input signal output signal plug-dispersion model tanks in series model

Fig.2–Dyetracerexperiments:input(green)andoutput (black)signalsandoutputsignalsmodelbyeitherplugflow withaxialdispersionmodel(red)ortanksinseriesmodel (blue).(Forinterpretationofthereferencestocolourinthis figurelegend,thereaderisreferredtothewebversionof thearticle.) f









The Navier–Stokes equations were solved using a velocity–pressure formulation (2) and were discretized on a staggered non-uniform grid, using a centred second-orderscheme.Thetimeintegrationwasachievedthrougha third-orderRunge–Kuttaschemeandasecond-orderimplicit Crank–Nicolsonschemefortheviscousterms.

The geometryis described in Fig.3. The radiusratio is =Ri/Re,theheightofthenumericaldomainisL=3(where

the axialwavelengthiscorrelatedwiththegapwidth from =2e to=2.29e dependingonthe Revalue) sothree axial wavelengths were simulated. Periodic boundary conditions were usedintheaxialdirection.Finally,thedomainlength

Fig.4–Meshgrid(planexy).

Lwasvariedinordertostudytheinfluenceoftheaxial wave-length.

Allsimulationswerecarriedoutusingdimensionlessunits. Allflow-relatedquantitieswerescaledusingtheoutercylinder radiusReandthereferencevelocity˝Riwhere˝istherotation

rateoftheinnercylinder.ThisyieldsRi=,Re=1and˝Ri=1.A

uniformmeshwasusedintheaxialandazimuthaldirections. Thegridwasstretchedintheradialdirectiontobetterdescribe theboundarylayersnearthewalls(Fig.4).

To quantify axial dispersion, the method proposed by Rudman(1998)wasapplied.An“effectiveparticle diffusion coefficient”,basedonthechaoticadvectionofalarge num-beroffluidparticles,wasderivedfromLagrangiantransient simulations.Anaxial dispersion coefficientwas definedby Eq.(3). Thoseparticles wereadvectedbythe carrying fluid flow.Transportwithinvorticessupplementedbyinter-vortex mixingresultedinenhancedaxialdispersion.

D∗ x=lim t→∞D ∗ x(t)=lim t→∞

(

X

)

InEq.(3)xi(t)istheaxialpositionoftheithparticleattime

stept,xi(0)itsinitialposition(bothxandtaredimensionless)

andNthetotalnumberofparticlesseededintheflow. A large number of particles (1000–10,000) are randomly seededthroughoutthecomputationaldomain,andtheir tra-jectoriesaretrackedovertime.TheinstantaneousvalueofD∗ x

increaseswithtimeuntilitreachesaplateauvalueindicative of long-time diffusive axial mixing (Fig. 5). The dimen-sional(or“effective”)dispersioncoefficient,Dxisobtainedby:

Dx=(ReD∗x)/(1−).Mixing property canbecompared with

momentumdiffusionthroughthedispersionbasedSchmidt numberdefinedasSc=/Dx.

Particles trajectories illustrate the different transport mechanisms: within vortex or between neighbouring vor-tices (see Fig.6). Intra-vortex mixing is dueto the chaotic

Fig.5–Effectivedispersioncoefficientevolutionwithtime.

Fig.6–Particlespositionsinthecomputationaldomain (top).Exampleofaparticlepath(bottom).

particle paths induced by the three-dimensional velocity fields.Inter-vortextransportoccurswhenthewavesinduces fluid exchange through neighbouring vortices. The combi-nation ofthesetwomechanismsresultsin enhancedaxial dispersion.

2.2.2. Noteonaxialflow-rateeffect

RTDmeasurementsarecarriedoutwithaweakaxialPoiseuille flow.InFig.7,itisclearthattheaxialdispersioncoefficient isonlyweakly influenced inthis rangeofaxial mean flow rate.Moreover,wehavecarriedoutseveralsimulationswith andwithoutaxialflowofthesamemagnitudeasin experi-ments.Sincenosignificantdeviationoftheflowpatternswas observedforaflowrateQaxlowerthan300mlh−1,theeffect

ofaxialflowwasassumedtobenegligible.Theaxialflowis nottakenintoaccountforthenumericalsimulationscoupled toLagrangianparticletracking.

3.

Results

and

discussion

3.1. Flowcharacteristicsfromexperiments

Thedifferentflowregimeshavebeenidentifiedbyan experi-mentalvisualizationtechniquedescribedinSection2.1.2.For thetwoconfigurationsstudied(e=1.5mmande=11mm)all flowtransitionswereidentified.Fig.8showsthevisualization

Fig.7–Evolutionofthedispersioncoefficientasafunction ofRefordifferentaxialflowrateQax(=0.85).

Fig.8–Illustrationofdifferentflowregimes(=0.85).(a)Visualizationofexperimentsand(b)numericalsimulations.

Table3–Summaryoftheflowcharacteristicsfordevicegeometry#1(=0.85).

Flowregimes Reoftransition Flowcharacteristics

Taylorvortexflow(TVF) Rec=125 =2.01e

Wavyvortexflow(WVF) 1.32Rec<Re<5.44Rec =2.19eto2.61e,m=2to8

Modulatedwavyvortexflow(WVF) 5.44Rec<Re<9.2Rec =2.63eto3.1e,m=4to7

Turbulentflow Re>9.2Rec >3.1e

ofthe flowstructureatfive differentRenumbers.Thefirst flowbifurcation(TVF)occurrednearRec=125forthe

geome-try=0.85.Thisvalueisinagreementwithtransitionpoint predictedbyRudman(1998)andColes(1965)ofRec=119.For

thesecondgeometry=0.68,thetransitionReynoldsnumber

Recwasestimatedat79whichisalsoinagreementwiththe

literature(Takedaetal.,1990).TheWVFregimewasobserved forRelargerthan1.32Recand4.9Recrespectivelyforthefirst

andsecondconfiguration.Allthecharacteristicoffurtherflow transitionsaresummarizedinTables3and4.Inordertofully characterizethedifferentwavyregimes,aspectralanalysis ofthe imagesequencesisachieved.Space–time plotswere generatedbyextractingsinglelinesofpixelintensityata par-ticularspatiallocation.Then2DfastFouriertransforms(FFT) oftheresultingspacetimeplotswereperformed. Character-isticfrequencyofthetravellingwave(denotedf)isextracted fromthetemporalpowerspectrum.Infact,theformationof wavyvortexflowyieldsawell-identifiedtemporalfrequency anditsharmonics.Fig.9(b)showsthepowerspectrum cor-respondingtotravellingwaveatRe=1192.Themostenergetic frequencypeakcorrespondstof.Forwavyregimesspace–time

diagramsgiveaccesstotheangularvelocity(phasespeed)of thetravelling wavenotedVϕ.Itcorrespondstotheslopeof

thecurvesseenonspace–timediagrams(seeFig.9(e)). Know-ing the temporalfrequencyf andthe angularspeed ofthe travelling waveVϕ theazimuthalwavenumber mbasedon

m=2f/Vϕwasdetermined.

AsReisincreasedfurther,amodulationofthewaveshape, initially perfectly periodic, appears and the flow becomes more complex. Theamplitude of the rotating wave varies periodicallywhileamodulation ofthewave amplitudecan be observed (Takeda et al., 1992). This corresponds to the modulatedwavyvortexflow(MWVF).Thedominant tempo-ralfrequencyismodulatedwithaloweradditionalfrequency

f′_.

Theturbulentflowregimebeginswiththeonsetofchaotic structuresthathavemultipledistinctfrequenciesaroundthe modulationfrequencyf′_{.Thisinitialphaseissometimes}

des-ignated as “weakly turbulent regime”. The regime “highly turbulent”ischaracterizedbyabroadbandspectrumrelated tothepresenceofturbulentstructures,withoutemergenceof adominantfrequency.

Table4–Summaryoftheflowcharacteristicsfordevicegeometry#2(=0.68).

Flowregimes Reoftransition Flowcharacteristics

Taylorvortexflow(TVF) Rec=79 =2.01e

Wavyvortexflow(WVF) 4.9Rec<Re<22.7Rec =2.21eto2.33e,m=2to7

Modulatedwavyvortexflow(WVF) 22.7Rec<Re<27Rec =2.33eto3.3e,m=3to9

Fig.9–Time–spacediagrams/spectrum.(a)TimehistoryofthetravellingwaveatRe=200(WVF).(b)Spectrumofthe travellingwaveatRe=200(WVF).(c)TimehistoryofthetravellingwaveatRe=800(MWVF).(d)Spectrumofthetravelling waveatRe=800(MWVF).(e)Time–spacediagramsRe=200.

3.2. Validationofnumericalsimulations

Thedifferentregimesidentifiedexperimentallywere repro-ducednumerically.Basedonthevelocityprofilesobtainedin DNS,thenormalizedviscoustorquewascalculated(Eq.(4)).

G= 2Riω

2 (4)

ω istheviscousshearstressonthe innercylinder. Fig.10

reportstheevolutionofGwiththeReynoldsnumberRe.The computedvaluesare ingoodagreementwiththeempirical correlationproposedbyWendt(1933).

Fig.11showstypicalsnapshotsofinstantaneousvelocity fieldsintheverticalplane,fromtoptobottomatReynolds numbersof140,200and800.Theoccurrenceofinter-vortex fluidtransferisstronglyrelatedtothetransitiontothewavy vortexflow,whereasinTVFregimevorticesareessentially cel-lular.Thevortexshapechangesgraduallyasitfillsupwith fluidfrom neighbouring vortices and it shrinks as it loses fluidtoitsadjacentvortices. AsReisincreasedfurther,the

vortexshapechangesmorerapidlyandoftendisplays dislo-cationswhenvortexpairsmerge.Thisisclearlyrelatedtothe modulationofthewaveamplitude.AstheReincreaseseven more,Taylorvorticesbecomeseverelydistortedandthe

large-Fig.10–Normalizedtorqueexperiencedbytheinner cylinderasafunctionofRe(=0.85).

Fig.11–Instantaneousvelocityfieldsonameridianplane(xy)atReynoldsnumbers(fromtoptobottom)Re=140,200and 800(=0.85).

scalevorticesarecontaminatedwithsmallscalestructuresin thewallboundarylayers.Thisgeneratesfluctuatingenergetic fluidmotions.

Thepropertyofnon-uniquenesshasbeendemonstratedby theexistenceofhysteresisphenomena(Coles,1965). Depend-ingontheflowhistory,multipleflowstatesarestablefora givenReynolds number.Theflow structurescan be distin-guishedbytheaxialwavelength,rangingfrom2eto2.36e andtheazimuthalwavenumberm,whichrangefrom2to8. Thewavenumbercharacterizesthenumberofwaveperiods overtheazimuthaldirection.Differentwavestates(,m)were observedwhenapproachingthetargetedReynoldsnumberRe

withdifferentaccelerationratesoftheinnercylinderrotation. Numerically,similarflowregimesareachievedbyseedingthe baseTVFflowwithweakperturbationsaccordingtothewave numbermobservedexperimentally.Theaxiallengthofthe domaincanalsobeadaptedto.Thenumericalresultsdid matchtheexperimentaldataveryprecisely.Thedifferentflow regimesareillustratedinFig.8.

3.3. Axialdispersion

Intheliterature,manystudiessuggestpowerlawsforthe evo-lutionofthedispersioncoefficientwiththeReorTanumbers (Tamand Swinney,1987;MooreandCooney,1995;Ohmura et al., 1997). In thesestudies, the authors didnot account for important flow parameters, such as the axial and the azimuthal wavelengths. Indeed,as illustrated by Fig. 12, a great variation ofaxial dispersion can be observed (either experimentallyornumerically)forthesameRenumber.This differenceisduetothenon-uniquenessoftheflowespecially forwavy regimes.Theflowstatedependency issignificant overtheWVFandMWVFregimeswhileitislessobviousfor turbulentflows.

While Rudman (1998) studied precisely this point only undertheWVFregime,inthepresentworktheinfluenceof differenthydrodynamicparameters (m, ) onthe axial dis-persion coefficient has been studied over a wide range of

Re.Usingwell-definedrampingprotocols,alongwith visual-izationtechniquesguaranteethereproducibilityoftheflow regime,andconsequentlyoftheresultingmixingbehaviour.

Fig.12–DispersioncoefficientDxasafunctionofRe

(=0.85).

DirectnumericalsimulationsoftheflowandLagrangian track-ing werecarried outover similarflowcharacteristics.Good agreementbetweenthenumericalresultsandthe experimen-taldatawasobtained,asillustratedbyFigs.12and13.The dispersionbasedSchmidtnumber(Sc)showsageneralRe−1

trendasproposedbyMooreandCooney(1995).Thetransition towavyregimesincreasesthedispersioncoefficient dramat-ically:44enhancementfactorfromTVFtoWVF.Actually,the wavyperturbationhastwosignificanteffectsontheparticle

Fig.13–EvolutionofthedispersionbasedSchmidtnumber asafunctionofRe(=0.85).

Fig.14–(Top)EvolutionofthedispersioncoefficientasafunctionofReandazimuthalwavenumbermfor=2.29e.(Bottom) EvolutionofthedispersioncoefficientasafunctionofReandaxialwavelengthform=4.Simulatedflowpatternsare shownontherightsideforthesakeofillustration.

motionleadingtoenhancedaxialdispersion.First,thelossof rotationalsymmetrymakesthe velocityfielddependenton allthreespatialcoordinatesandtime.Thisleadstochaotic particle motionand hence intravortex mixing. Second,the wavyperturbationbreakstheboundaryseparatingadjacent vortices.Thisprovidesflowstreamsforfluidelementstopass fromonevortextoanother,resultinginintervortexmixing. Rudman(1998)foundthat theScvaluereachedan asymp-toticvalueof0.155asReincreases,indicatingthatinwavy vortexflowbeyondacertainlimitoftherotationspeedthe axialdispersionvariesveryweakly.However,ourresultsshow thatthemodulationofthewavyflowwithadditional frequen-cies(MWVF)can influencethedispersion significantly.The dispersionbasedSchmidtnumbercangobelowthe asymp-toticvaluegivenbyRudman.Thiscouldbeexplainedbythe characteristicsofthisflowdiscussedearlierinSection3.1.The modulationincreasestheinteractionsbetweenvorticeswhich enhancesaxialdispersionandreducesSc.

Theeffectofthewavestateonaxialdispersionhasalso beenstudied. Differentwavestateswere examined experi-mentallyandnumericallyformanyReynoldsnumbers.The influenceofboththeaxialandtheazimuthalwavenumbers isshowninFig.14. Atfirst,wasfixedand mwasvaried, thenmwasfixedandwasvaried.Thedispersioncoefficient

Dxappearstobeadecreasingfunctionofm.Thisresultisin

agreementwithRudman(1998).Incontrast,Dxisan

increas-ingfunctionof(Fig.14b).Thisobservationwasconfirmedby RudmanforwavyregimesandbyTamandSwinney(1987)for turbulentregimes.TamandSwinneyhaveproposeda correla-tionbetweenDxand.Asaresultofthissignificantvariability,

findingarelationshipbetweenDxandtheReynoldsnumber

Recannotbeachievedwithouttakingintoaccountthewave state.

4.

Conclusion

Inthiswork,therelationshipbetweenthehydrodynamicsof Taylor–Couetteflowandaxialdispersionofpassivetracerwas investigated.Theinfluenceofthesuccessiveflowbifurcations ontheaxialdispersioncoefficientwasinvestigated numeri-callyaswellasexperimentally.

The sequence offlow instabilities hasbeen determined usingavisualization technique.Spectralanalysiswasused toidentifytransitionReynoldsnumbersandstructural char-acteristicsoftheflowstates.Directnumericalsimulationsof the flowwere carriedout aswell,andall the flow-regimes were accurately reproduced.The resultsshow how mixing is enhancedwithincreasing theRe. Thestructureofwavy

regimesincreasesthemixingwithinthevorticesandbetween adjacentvortices.

Theaxial dispersioncoefficientwasnumerically derived from Lagrangian tracking simulation and measured by RTDexperiments.Bothnumericalandexperimentalresults revealed a significant effect of the flow state on the mix-ing properties. Indeed, in the wavy flow regimes (WVF and MWVF) a multiplicity of stable wavy states can be achieved and Dx may vary by a factor of two for the

same Reynolds number (King and Rudman, 2001). The influence of azimuthal and axial wavelengths has been investigatedanda good agreementbetweenthe numerical results and experiments has also been achieved for these aspects.

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