The effect of column tilt on flow homogeneity and particle agitation in a liquid fluidized bed



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http://doi.org/10.1016/j.ijmultiphaseflow.2017.02.008

Aguilar-Corona, Alicia and Masbernat, Olivier and Figueroa, Bernardo and Zenit, Roberto The effect of column tilt on

flow homogeneity and particle agitation in a liquid fluidized bed. (2017) International Journal of Multiphase Flow,

92. 50-60. ISSN 0301-9322

The

effect

of

column

tilt

on

flow

homogeneity

and

particle

agitation

in

a

liquid

fluidized

bed

A. Aguilar-Corona

a

_,

_O.

_Masbernat

b

_,

_B.

_Figueroa

c

_,

_R.

_Zenit

d,∗

a Facultad de Ingeniería Mecánica, Universidad Michoacana de San Nicolás de Hidalgo, Francisco J. Mujica s/n C.P. 58030, Morelia-Michoacán, México b Laboratoire de Génie Chimique, Université de Toulouse, CNRS/INPT-UPS, 4, allée Emile Monso BP 44362, 31030 Toulouse Cedex 4, France

c Laboratorio de Ingeniería y Procesos Costeros, Instituto de Ingeniería, Universidad Nacional Autónoma de México, Puerto de Abrigo S/N, Sisal, Yucatán

97355, México

d _{Instituto de Investigaciones en Materiales, Universidad Nacional Autónoma de México, Apdo, Postal 70-360, México D.F., 04510, México}

Keywords:

Agitation Liquid fluidized bed Homogenization Solid fraction Column tilt

a

b

s

t

r

a

c

t

Themotionofparticles inasolid-liquidfluidizedbedwasexperimentallystudiedbyvideotrackingof markedparticlesinamatchedrefractiveindexmedium.Inthisstudy,twofluidizedstatesarecompared, onecarefullyalignedintheverticaldirectionensuringahomogeneousfluidisationandanotheronewith anon-homogeneousfluidisationregimethatresultsfromaslighttiltofthefluidisationcolumnof0.3° withrespecttothevertical.Asaresultofthemisalignment,largerecirculationloopsdevelopwithinthe bedinawell-definedspatialregion.Itisfoundthatinthatrangeofsolidfraction(between0.3and0.4), theinhomogeneousmotionoftheparticlesleadstosignificantdifferencesinvelocityfluctuationsaswell as inself-diffusioncoefficientoftheparticles intheverticaldirection, whereasthe fluidisationheight remainsunaffected.Atlower(lessthan0.2)orhigher(higherthan0.5)concentration,particleagitation characteristicsarealmostunchangedintheverticaldirection.

1. Introduction

Thestudyofliquidfluidisationissignificantfromboth theoret-ical andpractical viewpoints. It allows forthe experimental ver-ification of two-fluid modelling concepts in gravity driven solid-liquidflows, wheretheslipvelocity isofthesameorderof mag-nitudeasthatofthecontinuousphaseandwheresolidphase agi-tationisinducedbycollisionalandhydrodynamicinteractions(the contribution dueto thecontinuous phaseturbulencebeing negli-giblecompared totheaforementionedinteractions).Evenifa liq-uidfluidizedbeddoesnotexhibit chaoticmixingorsharpregime transitions(asinthecaseofbubblyflows),astateofhomogeneous fluidisationisseldomobserved.

The concept ofa homogeneous bed is usually referred to the appearance of solid fraction fluctuationswhich are observable at macroscopic scale, characterized by particle-free regions or voids ofdifferentshapes:dependingonthefluidisationconditions,a flu-idizedbedcan destabilizeinthesense oftheappearanceof one-dimensionaltravelingwaves(ODTW)(AndersonandJackson,1969) whichmaylaterdevelopintotwo-dimensionalstructuresand tran-sitionstootherflowregimessuchastheformationofbubbles,and

∗ _{Corresponding author.}

E-mail address: zenit@unam.mx (R. Zenit).

structuresthatremindobliquewaves(DidwaniaandHomsy,1981;

Duruand Guazzelli,2002; El-Kaissy andHomsy, 1976; Homsyet al.,1980).Therearemanyinvestigationsthattrytomapsuch tran-sitionsthroughexperimental,theoreticalandnumericalsimulation intheliterature,thereadermayreferto(DiFelice, 1995; Homsy, 1998;Sundaresan,2003)foramorecomprehensivereview.

Inthisworkwestudyhomogeneityfromadifferentviewpoint; the column is (or is not) homogeneous in terms ofthe absence oflargescalerecirculationandthebreakofsymmetryoftheflow velocityfield interms ofsome statisticalparametersthat charac-terizesitinspaceorintime(Gordon,1963;Handleyetal., 1966). Thehomogenizationconceptisnoteasilydefinedbecauselow fre-quency motion is always present due to confinement (Buyevich, 1994). Thiseffect is evident when the particlemotion is studied forrelativelylongtime,so theparticleshaveenough timeto tra-verse many times the column length and diameter. Two dimen-sionalanalysesoftenturnouttobeinsufficientbecausethesystem isnot guaranteedto be symmetricandtheformation oflow fre-quencystructures maynot be observedfroma givenobservation direction. This iswhy a full three dimensional analysisis funda-mentallyimportantto characterizea liquidfluidisation system. It alsoallowsfordirectcomparisonswithnumericalresults.

The effect of inclination on a fluidized bed is of particular relevance in the context of this study. It has been investigated from several viewpoints in the literature: for gas-solid

fluidisa-tion one canfindmany previous relevantworks:in particular,in

Yamazaki et al.(1989) the minimum fluidisation velocity for in-clined columns was studied. The authors observed three distinct flow regimes (fixed bed, partially fluidized and completely flu-idized), andnoted the effectof columninclination on the corre-spondingregimetransitions.Yakubovetal.(2005) studiedthe ef-fectofinclinationofaliquid-solidfluidizedbedonseveralworking parameterssuchascriticalflowrate,bedheightanddynamic pres-suredrop.Theyobservedapatternofconcentrationwaves(forthe effectofinclinationinthecaseofcohesivepowders,see(Valverde etal.,2008)).Numericalinvestigations havealsobeencarriedout onthesubject;inChaikittisilpetal.(2006)DiscreteElement Sim-ulations (DEM) were used to study gas-solid two-phase flow, in orderto investigatethe mixingbehaviorofthesolid phasein in-clined fluidized beds. A large scale recirculationpattern was ob-served. Low concentration bubbles tend to move upwards along the uppermost wall, contrary to the particles that moved down-ward,closertothewallbelowit,enhancingbackmixing.This be-havior has also been observed experimentally in gas-liquid bub-blyflows, dueto buoyancy,forvery smalltiltangles(Zenit etal., 2004).

Forliquid-solidfluidizedbeds,littleattentionhasbeenpaidto the effect of inclination on the columnhomogeneity. Hudsonet al.(1996)usedsalttracermeasurementstoconcludethatfluidized bedinclinationstronglyaffectsthecolumnhydrodynamics. More-over, inDel Pozo etal.(1992) itwas shownthat a smalltilt an-gleof1.5°onathree-phasefluidizedbedaffectstheparticle-liquid mass and heat transfer coefficientssignificantly. Other important aspectinthecomplexinteractionbetweensolidandliquidphases istheeffectoftheinclinationangleonmixinganddiffusion,asa functionoftherelevantparameterssuchastheReynoldsnumber, Stokes number, andparticle-column width (orheight) ratio. Sev-eralstudieshavebeendevotedtothediffusioninaliquid-solid flu-idizedbed(Al-DibouniandGarside,1979;CarlosC.andRichardson J.,1968;Dorgeloetal.,1985;JumaandRichardson,1983;Kennedy S. andBretton R.,1966;Van DerMeeretal.,1984; Willus, 1970). Twotrendscanbeidentifiedintheliteratureconcerningdiffusion: firstly, withrespecttosolid concentration,andsecondly,with re-spectto theparticle-to-columndiameterratio.Insome investiga-tions (CarlosC. andRichardson J., 1968; Willus, 1970; Dorgeloet al.,1985)itisfoundthatthediffusioncoefficientdecreasesassolid fractionincreases.Ontheotherhand,otherstudiesfromthe liter-ature(Kangetal.,1990;Yutanietal.,1982)foundasmallpeakon theauto-diffusioncoefficientasthesolidfractionconcentration in-creases.Concerningtheparticlesizeratio,theexperimentsshowed that diffusiondecreases astheparticlesize ratioincreases.Those experimentswere carriedout fordifferentflowregimes compris-ing superficial Reynolds numbers of O(10–1000) and two-phase StokesnumbersofO(1–10).Althoughtherearemanyinvestigations devoted tothe effectofinclination onliquid-solid fluidizedbeds, none ofthe aforementioned worksremarked the highsensitivity ofthefluidized bedcharacteristics toasmallinclination; mostof those studies comprisedranges ofinclination betweenhorizontal tovertical,butincrementedthetiltinlargesteps,ignoringthe ef-fectsofverysmallinclinationangles.

Thisworkisdevotedtostudytheeffectofasmalltiltofthe flu-idisationcolumn(0.3°withthevertical),comparedto avertically aligned column. Low frequencystructures are detected andtheir effectonthedispersedphasevelocityisassessedthroughthe anal-ysisof:a)Theparticletrajectories,b)Thespatialdistributionofthe verticalspeed,c) Theparticlevelocityvariancesandd)The diffu-sioncoefficient.Thetechniqueusedtocalculatethemeanvelocity andagitation(velocityvariances)alongthethreedirectionsis sim-ilartothatusedinHandleyetal.(1966)andCarlosandRichardson (1968) and later revisited in Buyevich (1994), Willus (1970) and

Latif andRichardson(1972), who useda Lagrangiantracking ofa

colored particleinthebulkofa transparentbed.Morerecentlya similarparticletrackingtechniquewasusedbutinacarefully con-trolledopticallymatchedsystem(Aguilar,2008;AguilarCoronaet al., 2011;HassanandDominguez-Ontiveros,2008).Acamerawith highresolutionwas used(bothintimeandspace),whichallowed for the determination of detailed information about the particle phasemotionwithinthefluidizedbed.

2. Experimentalset-up

The experimental device isshownschematically inFig. 1.The fluidisation section iscomposed ofa 60cmhighcylindricalglass columnof8cminnerdiameter.Aflowhomogenizer,consistingof a fixed bedof packedbeads covered by syntheticfoam layers,is mounted atthe bottom ofthe columnto ensurea homogeneous flowentry.Theflowtemperatureismaintainedat20°Cbya con-trolled heat exchanger. Twoparticular cases were studied during thiswork: 1) Averticallyalignedcolumnand2)Atilted column, forminganangleinthe(y,z)planeof0.3°withrespecttothe ver-ticalaxisz.ThereferenceframeisshowninFig.2.

2.1. Particlesandfluid

Calibrated 3mmpyrexbeadswere fluidizedby aconcentrated aqueous solution of Potassium Thiocyanate (KSCN, 64%w/w). At 20°C,thefluidandtheparticlesandfluidhavethesamerefractive index(∼1.474),sothatatagged(colored)particlecouldbetracked individually inanearly transparentsuspension (Aguilar-Corona et al.,2011).ParticleandfluidpropertiesarereportedinTable1.The particleStokesandReynoldsnumbers,basedontheterminal (sed-imentation)velocity,areSt=4.8andRe=160,respectively.

2.2. Particletrackingtechnique

The analysisof particle motion in the fluidized bedwas per-formed by means of highspeed 3-D trajectography. The fluidisa-tioncolumnisequippedwithanexternalglassboxfilledwiththe aqueousphaseinordertoreduce opticaldistortion(seeFig.2).A mirror orientedat45°totheside ofthebox allowed forthe ob-servationoftheparticlepathinthreedimensions,providingan ad-ditionalside view.APhotronAPXcameraequippedwithaCMOS sensorwasusedtorecordthefront((x,z)plane)andthesideview fromthemirror((y,z)plane)inasingleframe(512pix×1024pix). Images were recorded over periods of 204 seconds, starting af-terthestationaryregimehadbeenreached.Takingacharacteristic particlevelocityof3cm/s(fromthestandarddeviationofthe par-ticlevelocityofatypicalexperiment),thistotaltimewould repre-sent morethan 70timesthetimea particlewouldtake totravel one columndiameter.The averageresidencetime(the averageof the time it takes to a particle to travel one column height) for thealignedcasewas 6seconds,whilethecorrespondingvaluefor the tilted case was 4.5s, so one canexpect the average absolute speedtoincreasewithinclination.Ablackcoloredparticlewas in-troducedinthe bedandits trajectory wasrecorded at60frames persecond(fps).

Fig. 3showsboth front (x-z plane)andside (y-zplane) views as captured by the camera,for solid fractions of

h

α

p

i

=0.50 and

h

α

p

i

=0.14 (sub-figures (a) and (b), respectively). All the experi-mentswerecarriedoutwithaparticlesizeofdp=3mm.Theblack linebetweentheimagesis justthe spacebetweenthe frontwall and themirror, whichwas maskedin orderto avoida confusing view oftheadjacentwallofthecolumn.Theimage fromthe mir-ror hadaslightlydifferentscaleduetotheoptical pathsbetween the (direct)front view andthat comingfromthe mirror,so each planehaditscorresponding(horizontalandvertical)scalefactorin

Fig. 1. Scheme of the experimental set-up and the entry section.

Table 1.

Fluid and particle properties at 20 °C.

Pyrex beads dp = 3mm ρp = 2230 kgm −3 nD = 1.474 KSCN solution 64% w/w µf = 3.8 × 10 −3 Pa s ρf = 1400kgm −3 nD = 1.474

Fig. 2. Top view scheme of the trajectography 3D system.

ordertoobtaintheactualpositionincentimeters afterimage dig-italprocessing.Notethatthecoloredparticleisclearly discernible evenwhen itislocateddeepinsidethebulk ofthecolumn(even forlargesolidfraction).ItcanbeseenfromtheFigure(d.1andd.2) thattheparticlediameteroccupiesapproximately13to15pixels.

3. Results

3.1. Globalsolidfraction

The initialvolume ofparticlesinthefixed bedcorrespondsto an initial height h0 of9.5cm,slightlylarger thanthe column

di-ameter. The maximumcompactness concentration

h

α

c

i

was esti-matedtobe0.56,correspondingtoarandompacking.Eventhough thesystemisopticallyhomogeneous,beadsinterfacesarestill de-tectable;acarefulobservationoftheimagesallowedforthe deter-mination ofthemaximum height reachedby theparticles h_b for eachsolidfraction.Themeansolidfractionwasobtainedas:

h

α

p

i

=

h

α

c

i

h0

hb

. (1)

The bracket symbol represents the time and space averaging over the bed volume of the local instantaneous solid fraction. It wasobservedthat thehbfluctuationsdecreasedasthesolid frac-tion increased, with a relative error of lessthan 5% in all cases. Inthiswork, fivefluidisation velocitiesweretested (0.095,0.078, 0.053,0.038 and0.02m/s) correspondingto globalsolid fractions

h

α

p

i

of0.14,0.2,0.3,0.4and0.5respectively.Forthehomogeneous case,the fluidisation velocityUF (fluidvelocity inempty column) wasfoundtobeadecreasingpowerlawofvoidfraction:

UF=0.145

(

1−

h

α

p

i

)

2.78 (2) where the prefactor is close to the particle terminal velocity (0.135m/s) andtheexponentvalue, n=2.78, iscloseto that pre-dictedbyRichardson-Zaki’scorrelation(n=4.4Ret−0.1=2.67).

A first visual observationindicates that the slight tilt didnot haveanymeasurableeffectonthebedexpansion,soforeach flu-idisation velocity studied, the global solid fractionremained un-changedinbothhomogeneous (verticallyaligned) and inhomoge-neous(tiltedcolumn)cases.Thisobservationisconsistentwiththe averaged momentum balance in the bed volume. At first order, theeffectoffluidandparticlefrictionatthewallbeingneglected, thisbalancereducestoequilibriumbetweenbuoyancyforce term anddragforce termbaseduponmeanslipvelocity, i.e. themean liquidvelocity. Theaveragesolid fraction, orequivalently thebed height,resultsfromthisbalance.However,whenspatially

averag-Fig. 3. Raw images as captured by the camera: a) Large solid fraction: < αp > = 0.5; to the left of the black division: front view ( x-z plane). To the right of the division is the lateral view ( y-z plane). b) Moderate solid fraction: < αp > = 0.14 (same views as in (a)); c) Particle close-up; c.1) front view and c.2) side view; d) Binarized particle image. (d.1) front view and (d.2) side view; e) Centroid detection: (e.1) and (e.2) correspond to front view and mirror image, respectively.

ing thelocal two-phase momentumtransport equation, two con-tributions arisingfromthefluctuatingmotionoftheparticlesand the fluid need alsoto be considered:one is the non-linear drag forcetermthroughthevelocityfluctuationsandthesecondisthe cross-correlationbetweenthespatial fluctuationsofsolidfraction andpressuregradientinthebed.Itcanbeshownthatinallrange offluidisationvelocities,thefirstcontributionisalwayslargerthan thesecond one,whichroughlyscales asfew percentofthemean drag term. As a consequence, in a homogeneous liquid fluidized bed,thebedheightisweaklydependentuponphaseagitation.In areference framewheretheaxialdirection istheaxisofthe col-umn,thebuoyancyforcecomponentis

1

ρ

gcos

θ

whereisthe an-gle with the vertical (0.3°), so therelative variation of thisterm is of order of 10−5 _{and can be neglected. Therefore, tilting the}

columna smallangle(0.3°)will notmodify thebedheight,even though thisperturbationinduces importantflow inhomogeneities and significant variations of particle fluctuations, as discussed in thenextsections.

3.2. Particletrajectories

Fig. 4 shows particle trajectories recorded at three different concentrations. The left panel of the figure shows the trajecto-ries projection in thehorizontal(x,y) plane ofthe column, while therightpanel displaysthe projectionsinthe vertical(x,z) plane. For moderatesolid fractions (<

α

p>=0.14and<

α

p>=0.20)the particle path was observedto span the wholebed volume with-outexhibitingclearcoherentstructures.Atlargersolidfractions,a toroidalstructurewasobservedinthelowerpartofthebed,along with a corresponding increase ofthe low frequency fluctuations. The originofthissteadystructurehasnotbeenclarifiedyet.One possibleexplanationcould bethat duetoawall effect:a slip ve-locitydifferencebetweenthemiddleandthenear-wallregion de-velops intheentrysection,resultinginasolid fractionhorizontal gradient.Thissolidfractiongradientwouldtheninducea horizon-talpressuregradientthatwouldgeneratethisrecirculationpattern. But such a mechanismneeds a more indepth analysis, which is beyondthescopeofthispaper.

Fig.5 showsa comparisonbetweentheparticletrajectoriesof theverticalcolumnandthetiltedone, correspondingtothe(x,z), (y,z) and(x,y) planes fora solid fractionof <

α

p>=0.30. In the tilted columncase(forthatconcentration)awell-defined recircu-lationloop inthe(y,z) planewas observed,wherethetracer par-ticle trajectory forms an annulus. For thesame case, inthe (x,z) plane the particle path spans across the whole column volume

without any preferential motion of the dispersed phase. Inclina-tioninducesabuoyancyforcecomponentnormaltothewall; how-ever,thecounterbalanceofthisforcecannotbereadilyidentifiedif therearenosignificantchangesinconcentrationorvelocity. There-fore, thissmall imbalancemay generatea radial drift velocity at thescaleofeachparticle.Nowasthisdriftvelocityislikelyto in-duce a radial concentration gradient, collectiveeffects (such asa radial apparentdensitygradient) areprobablydrivingthe recircu-latingmotionatthebedscale thatis observedontrajectory pat-terns(similarinthatsensetotheso-calledBoycotteffect).

3.3. Testofhomogeneity

In order to characterize fluidisation homogeneity, for each mean fluidisation velocity studied,the spatial distribution of up-ward and downward particle motion was analyzed in four dis-tinct cross sections Si, (i=1 to 4) regularly distributed along the bed height (0≤zS1≤0.25hb; 0.25hb<zS2≤0.50hb; 0.50hb<

zS3≤0.75hb;0.75hb<zS4≤hb),asschematizedinFig.6.

Fig. 7shows thevelocity signdistributions following particles trajectoriesineachtestsectionSi,foraglobalsolidfractionof0.3. Bothaligned(leftcolumn)andtilted(rightcolumn)casesare dis-playedinthisfigure.Thedirectionofthemotionisindicatedwith a crosssymbol if the particlemoved downwards ora circle ifit movedupwardsasitcrossedtheplaneSi.Fortheverticallyaligned casethesignatureofan axisymmetrictoroidalmotionatthe low-ermostpartofthebedcanbe identified,withapreferential con-centration ofascending velocities atthe centerof thebed cross-section,anddescendingvelocityinthenear-wallregion.Inthe up-permost section, the distribution appears homogeneous over the cross-section. Forthetilted casethere isapreferential motionin all test sections,whichconsistsofa large-scalerecirculationover thewholebedvolume,wheretheparticletendstoriseinone half-sectioninFig.7-(ii),andtodescendintheotherone.Notethatthis motionisquiteparalleltothe(y,z)plane asexpected. Theseplots clearly demonstratetheeffectofthetiltontheparticlemotionin thefluidizedbed.

In order to quantify homogeneityin the cylindricalgeometry, the (circular) cross-section Si was divided into 12 sectors of 30° eachintheangulardirection.Theprobabilityofparticlecrossingin aparticularsectorjwithascendingverticalmotionwascalculated as:

ϕ

up,j=

nup,j

nj

Fig. 4. Particle trajectories projection for the vertically aligned case at a) < αp > = 0.14, b) < αp > = 0.20 and c) <αp > = 0.40: left and right columns correspond to the horizontal ( x,y ) and vertical ( x,z ) planes, respectively.

withnjthetotalnumberofparticlecrossinginsectorj.This prob-abilityismutuallyexclusivewithrespecttoitscounterpart (down-wards crossing)

ϕ

down,j. A perfectly homogeneous columnwould attainavalueof

ϕ

up, j=0.5forallsectors(j=1to12inthiscase), which ina polarrepresentations wouldgive a circleof radius ½.

Figs. 8and9showthistypeofrepresentation,forthecaseofthe

alignedandtiltedcolumns,respectively. Thedotted circlehas ra-diusr=0.5,forcomparison.

Fig.8showsanangulardistributionclosetohomogeneous for

S4,whilethereseemstobemoredownwardsmovingparticlesfor

S1. Thereare smalldeviationsfrom½forS2 andS3.Fig.9shows thedistribution

ϕ

up,jforthetiltedcolumn.Itcanbeobservedthat

Fig. 5. Particle trajectories projection at < αp > = 0.30 in a) ( x,z ) plane; b) ( y,z ) plane; c) ( x,y ) plane. Left and right columns correspond to the aligned and tilted cases, respectively.

thedistributionisclosetothecenterforthefirstthreesectors,and morehomogeneous inS4. Howeverthereisstill atrend,showing

ϕ

up,j> 0.5inthesecond quadrant,withvaluesbelow0.5forthe third andfourthquadrants(values closertoone meanthat more particlesmove upwards,consistentwithFig.7(ii)).Iftheparticles were less dense than the liquid,the particles would descend (in average)onthefirstandthirdquadrants.

Another way to represent Figs. 8 and 9 is to plot the angu-larstandard deviationof

ϕ

up,j ineach sector forthe verticaland tiltedcases.ThisquantityisplottedalongbedheightinFig.10for

the verticalandtiltedcases.Anincrease bya factorcloseto 3or 4 of thereference values inthe vertical casecan be observed in the tilted case. Note there isa correspondence betweenthe four points inFigs. 10and9forS1,S2,S3 andS4.Themore heteroge-neousdistributionoftheverticalvelocitycomponentwasobserved at S2 and S3, where the value of

ϕ

up,j is very small in the 3rd

and4thquadrants,indicatingdownwardsverticalmotioninthese quadrants.ThemosthomogeneoussectionwasS4,consistentwith

Fig. 9, where

ϕ

up,j is close to 0.5 in the 3rd and 4th quadrants.

s

1

s

2

s

3

1h

_b

s

4

0.75

h

_b

0.50

h

_b

0.25

h

_b

Fig. 6. Bed test cross sections S 1 , S 2 , S 3 and S 4. .

Consequences of column tilt-induced flow inhomogeneities upon particleagitationareexamined inwhatfollows,by comput-ingtheparticlesvelocityvarianceandself-diffusioncoefficientand comparingtheirintensitieswiththoseofthealignedcase.

3.4. Effectofcolumntiltingonparticlevelocityvariance

The varianceoftheithparticle velocitycomponentinthebed iscomputedas:

D

u′2 pi

E

=

D

_¡

upi

(

t,x

(

t

)

)

−

­

upi

®¢

2

E

₍₆₎

whereupi(t,x(t))istheinstantaneousvelocityi-component follow-ing particle trajectory x(t), and the bracketsymbol denotes here theaverageofparticlevelocityith-componentoveralltrajectories (equivalent to an ensemble average operator). Note that

h

upi

i

is closetozero,theaverageparticlevelocityinthebedbeingzerofor a steadyfluidizedbed(sou′

piisvery closetoupi). InFig.11,the variance ofeach velocity componentis reportedas a function of globalsolidfraction,inbothhomogeneous(verticallyaligned)and inhomogeneous (tilted) cases. In both cases, particle agitation is stronglyanisotropicasexpectedingravity-driventwo-phaseflows. In thehomogeneous case, theaxial componentofparticle ve-locityvariance(z-component)beingabout2timeslargerthanthe components in the horizontal plane (x,y) for all concentrations. Particle velocity variance isa continuouslydecreasing functionin therangeofconcentrationinvestigated[0.14–0.5],withaprobable maximumlyingintherange[0–0.14].

In the non-homogeneous case, the evolution of the axial ve-locity variance is quite different compared to the homogeneous case, mainly in the range ofsolid fraction[0.3–0.4]. At the low-est concentration (<

α

p>=0.14), the axial velocity variance is smallerthaninthehomogeneouscase,thenabruptlyincreasesup to a maximum at <

α

p>=0.3, then strongly decreases between <

α

p>=0.3 and 0.5. This evolution results fromthe progressive development of the large-scale loop induced by the column tilt as the particle concentration increases. In the horizontal plane, particle velocity variance is a continuous decreasing function of solid fraction, slightlybelowthehomogeneous casevaluesinthe range [0.14–0.3] with a similar behavior at higher concentration (<

α

p>=0.5).

Table2reportstherelativedifference betweentheparticle ve-locity component variances forthe homogeneous (noted

h

u

′

2_pi

i

H) andnon-homogenous (noted

h

u

′

2_pi

i

H) fluidisation cases,measured

Table 2.

Relative difference δupi of h u ′2piiH between homogeneous and inho- mogeneous cases. <αp > δux δuy δuz 0 .14 0 .12 0 .22 0 .15 0 .2 0 .10 0 .16 −0 .015 0 .3 0 .20 0 .08 −0 .59 0 .4 0 .18 −0 .04 −0 .6 0 .5 −0 .08 −0 .1 0 .09

atfivedifferentglobalsolidfractions

h

α

p

i

,anddefinedas:

δ

upi=1−

D

u′2 pi

E

nH

­

u′2 pi

®

H (7)

Positivevaluesof

δ

u_piindicatethattheith-componentvelocity varianceinthenon-homogeneous caseissmallerthanthatofthe homogeneouscasewhereasnegative valuesof

δ

upireveal the op-positetrend.Notethatinthehomogeneouscase,thevelocity vari-ance inxandy-direction shouldbe equal.The relativedifference betweenthesevaluesisinaverageoftheorderof0.05forall con-centrations,so the relative difference betweennon-homogeneous andhomogeneouscaseisconsidered assignificantwhenitsvalue exceeds0.1.Largenegativevaluesareobservedfortheaxial com-ponents of the variance, reaching

δ

upz=0.6 at<

α

p>=0.3 and 0.4, which confirms the predominance of large-scale motions in thatrange ofconcentration.In thehorizontal(x,y) plane, the rel-ative difference is smaller than in the homogeneous case when <

α

p>≤0.3. At the highestconcentration (<

α

p>=0.5), the dif-ferencebecomesslightlynegative.

If, in the non-homogeneous case, particles are globally accel-erated in the vertical direction by a large scale motion induced bycollective effects,thenit canbe understoodthat velocity fluc-tuations in the transverse directions will diminish in the ascen-dantand descendantparts ofthe loop, andincrease in the hori-zontalpart.Inaverage,thetransversecomponentvariancewill de-crease,probablybecausetheweightoftheascendingand descend-ingpartsisstrongerthanthatinthehorizontalplane.Athigh con-centration(0.5), thesignofthecriterionisreversed,likelydueto anaspectratioeffect(inthiscase,theheightofthebedisindeed closetothecolumndiameter).Intherange[0.14,0.2],the concen-tration seem tobe too smallto induce a large recirculationloop inthebed,butanon-zerotransversecomponentofbuoyancystill existsandisabletodamp inthehorizontalplane thefluctuating motionofparticlesproducedbythemeandragforce.

3.4.Effectofcolumntiltingonparticlediffusioncoefficient

Particle diffusioncoefficient is determined fromthe computa-tionof Lagrangianvelocity autocorrelationcoefficient, definedfor eachvelocitycomponentupias:

Rii

(

t

)

=

­

upi

(

τ

)

upi

(

τ

+t

)

®

­

u2_pi

(

τ

)

®

(8)

In the range of globalsolid fraction investigated, particle La-grangianvelocity decorrelateswithin atime intervalsmallerthan 4seconds,asillustrated in Fig.12. Thecurves insuch figure can be fitted to a decaying exponential of the form Rii(t)=exp(-bt); Thefitted curve hasan exponentb=5.541s−1 (withR square of 0.990)forthez component, whileforthe xandycomponents it givesb=11.15 s−1 (withR square of 0.987).The time integration oftheautocorrelationcoefficient overthistime intervalgives the Lagrangianintegraltimescaleforeachcomponent:

TL ii =

Tmax

∫

Fig. 7. Projection of trajectories in test sections S i for < αp > = 0.30 a) S 1 ; b) S 2 ; c) S 3 ; d) S 4 ; in the case of i) aligned case and ii) tilted case. Symbols (o) and ( + ) indicate

the directions (ascending and descending, respectively). < αp > = 0.3.

Fig. 9. Homogeneity analysis in terms of particle crossing moving upwards ϕup, j , for different cross sections S i , for the tilted column.

Fig. 10. Evolution of the standard deviation of ϕup, j as a function of the normalized bed height, z/hb .

Thediffusioncoefficientineachdirectionisthengivenby:

Dii=

­

u2_pi

(

t

)

®

TL

ii (10)

It isthenclearfromplotsofFigs. 11and12thatthe diffusion in the vertical direction z is stronger than that ofthe transverse plane (x,y), resultingfromboth alarger decorrelationtime anda largervelocityvarianceinz-directionthaninxandy-directions.

Diffusion coefficients in transverse (Dxx and Dyy) and vertical (Dzz) directions asa function of global solid fraction are plotted inFig. 13,forboth homogeneousandnon-homogeneouscases.In

Fig. 11. Variance of particle velocity component as a function of <αp > . Comparison between homogeneous (vertically aligned) and non-homogeneous (tilted column).

both cases as expected, particle diffusion is strongly anisotropic, the diffusion in z-direction being an order of magnitude larger thanthatinxandy-directionsatallsolidfractions.Inthe homo-geneous case, thediffusion coefficientis a decreasingfunction of solid fraction butexhibitsa slightmaximum around <

α

p>=0.2 (open symbols in Fig. 13) for the three components. As for the evolutionofparticleaxialvelocityvariancewithsolidfraction(Fig. 11),thismaximumisaround<

α

p>=0.3inthenon-homogeneous case, and maximum differenceswith the homogeneous case are

Fig. 12. Particle Lagrangian velocity autocorrelation coefficient versus time for < αp > = 0.3. Homogeneous case. ( ο) z, ( 1) y and (—) x components.

Fig. 13. Particle diffusion coefficient D ii in 3 directions ( D zz , D yy and D xx ). Compari- son between homogeneous (open symbols) and non-homogeneous (filled symbols) cases.

Table 3.

Relative difference δDii between homo- geneous and inhomogeneous cases.

<αp > δDxx δDyy δDzz 0 .14 0 .17 0 .10 −0 .05 0 .2 0 .02 0 .06 −0 .07 0 .3 −0 .07 −0 .23 −2 .62 0 .4 −0 .07 −0 .75 −2 .14 0 .5 −0 .02 0 .14 0 .07

observed in thesame rangeof

h

α

p

i

, between0.3and0.4. Maxi-mum relativedifferencesare reachedforthez-componentinthat rangeofconcentration,duetothedevelopmentofalarge recircu-lationpatternevidencedby thetrajectoriesenvelopedisplayedin

Fig.5.

Relativedifferences

δ

Dii=1-DiinH/DiiHarereportedinTable3for allsolidfractioninvestigated.Atlowerconcentration(<

α

p>=0.1 and0.2),differencesbetweenbothcasesarenotsignificantinthe vertical direction, suggesting that the large-scale coherent struc-tureis notfullydeveloped,probablydueto atoo smallapparent density-induced collective effect.However, at the lowest concen-tration,theeffectofthetiltistodecreasethediffusivityof parti-cles inthe horizontalplane. Athighconcentration (<

α

p>=0.5),

this coherentmotion ofparticles is damped probablydueto the bedaspect ratio(heightofthebedcompares withcolumn diam-eter inthat case) andthe differencesbetweenthe two casesare also negligible. The maximum difference is reached in therange ofconcentration0.3–0.4,thediffusioncoefficientinz-direction be-ing morethan2timeslargerinthenon-homogeneous (tilted col-umn) casethan in the homogeneous (vertically aligned column) case. Note alsothat in that range ofconcentration, the diffusion inthe(x,y) planeisnotisotropicduetothefact thattheplaneof inclination isthe (y,z) plane, andthe relative difference of diffu-sion coefficient in the y-direction is larger than that observedin the x-direction.Inthe tiltedcase, thetransverse component vari-anceisveryclosetothevalueobtainedintheverticalcase.Itwas alsoshownpreviouslythatthevelocityfluctuationinthe horizon-tal plane was the correct scaling velocity for collisions ( Aguilar-Corona et al., 2011); or in other words, the horizontal fluctua-tionsdeterminedtheuncorrelatedmotionoftheparticles(notonly Gaussian but also Maxwellian, hence isotropic). As a result, tilt-ing the columndoes not significantly affectthe velocity variance (hencethepdf)oftheuncorrelatedpartofparticlesmotion.In re-turn,asshownbyourmeasurements, thediffusivemotioninthe

y-direction is slightly affected by the column tilt. Therefore, the decorrelation time isincreasedby the tiltdue tothe small grav-itycomponentnormaltothewall.

Itcanbe concludedthatwhenarecirculationloopdevelopsin thewholebedvolume,itmainlycontributestotheincreaseof par-ticlevelocity fluctuationsinthe verticaldirection andalsointhe decorrelationtime,leadingtoasignificantincreaseofparticle dif-fusivity in that direction. In this range of highparticle Reynolds and finiteStokes numbers, thevertical alignment ofthe fluidisa-tion column isan important criterion regardingthe validationof numerical methodsinconcentrated two-phase flows.The present experimentshavebeencarriedoutinaliquidfluidizedbedwhere theagitationoffluid,andconsequentlythatofparticles,ismainly induced by wake effects (also referred to aspseudo-turbulence). Note that thissituation is quite differentfrom gas-solid fluidisa-tion whereasgeneralcase, particleagitationisdrivenby the tur-bulenceofthecontinuousphase,modulatedbyparticleinertiaand finitesizeeffects.Inthelattercase,theeffectofasmalltiltofthe columnwouldprobablynotbethesame,becauseoftheturbulent large-scale induced intensemixingthat wouldprevent the devel-opmentofcoherentstructures atthebedscale.Hence,this situa-tion isparticulartogravitydrivendispersed flowathigh concen-tration forwhichproperturbulenceofthecarryingphaseremains smallcomparedtothatinducedbywakeeffects.

4. Conclusions

In this work, we carried out an experimental investigation of the 3-D particle fluctuating motion in a liquid fluidized bed andcomparetwodifferentsituations:ahomogeneous fluidisation regime (homogeneous feeding in the entry section and carefully verticallyalignedbed)andanon-homogeneousfluidisationregime resultingfromasmalltilt(0.3°)ofthefluidisationcolumnwiththe vertical.

The bedexpansion is not modifiedsignificantly by thetilt, as a resultof momentumconservation averagedin thebedvolume, whichatfirstorderbalancesthebuoyancyforceandthedragforce basedonaveragedslipvelocity.Inturn,weshowthattheparticle trajectoriesinthebedarestronglymodified,shiftingfroman over-all uniformlydistributedrandommotioninthebedwith axisym-metrictoroidalstructureinthebottompart,tolarge-scale recircu-lation patternsina givenrangeofbedexpansion(solidfraction). As aconsequence,theparticlevelocityvariance andself-diffusion coefficientaresignificantlyaffectedbythetiltinverticaldirection. Inparticular,itisshownthatthechangeobservedonthese

quan-titiesdependsonthe fluidisationvelocity andthatthemaximum variationoccursforsolidfractionsrangingbetween0.3and0.4.It is also possible, although not investigated in thisstudy, that the particle velocity fluctuations depend also on the particle inertia (Stokesnumber).Theseresultsarerelevantwhencomparisons be-tweenexperimentsandnumericalsimulationsareconductedatan industrial scale. Ifthe experiment is not accurately aligned with thevertical,mixingofpassivescalarand/or thetransport ofmass or heat could be significantly affected by the non-homogeneous state offluidisation. It isalso clearfromthe resultspresentedin this work that thethree dimensionalcharacter of thefluctuating motionhastobetakenintoaccountwhencomparingwiththe nu-mericalsimulationsandmodels.

This investigationshowedevidence oflargesensitivity tovery smallmisalignmentswithrespecttotheverticalinfluidizedbeds. The implications of such an effect are very important when de-signingamodelexperimentsforthevalidationofnumerical simu-lations.

Acknowledgements

TheauthorswishtoexpresstheirgratitudetotheNational Sci-ence and Technology Council of Mexico (CONACYT) and the re-searchfederationFERMaT(FRCNRS3089)forfundingthisproject.

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