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

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The effect of column tilt on flow homogeneity and

particle agitation in a liquid fluidized bed

Alicia Aguilar-Corona, Olivier Masbernat, Bernardo Figueroa-Espinoza,

Roberto Zenit

To cite this version:

Alicia Aguilar-Corona, Olivier Masbernat, Bernardo Figueroa-Espinoza, Roberto Zenit. The effect

of column tilt on flow homogeneity and particle agitation in a liquid fluidized bed. International

Journal of Multiphase Flow, Elsevier, 2017, 92, pp.50-60. �10.1016/j.ijmultiphaseflow.2017.02.008�.

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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

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

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

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

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

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.

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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