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Hydrodynamics and scale-up of bubble columns in theheterogeneous regime: Comparison of bubble size, gas

holdup and liquid velocity measured in 4 bubblecolumns from 0.15 m to 3 m in diameter

Pedro Maximiano Raimundo, Ann Cloupet, Alain H. Cartellier, DavideBeneventi, Frédéric Augier

To cite this version:Pedro Maximiano Raimundo, Ann Cloupet, Alain H. Cartellier, Davide Beneventi, Frédéric Augier.Hydrodynamics and scale-up of bubble columns in the heterogeneous regime: Comparison of bub-ble size, gas holdup and liquid velocity measured in 4 bubble columns from 0.15 m to 3 m in di-ameter. Chemical Engineering Science, Elsevier, 2019, 198, pp.52-61. �10.1016/j.ces.2018.12.043�.�hal-01999459�

1

Hydrodynamicsandscale-upofbubblecolumnsintheheterogeneousregime:2

comparisonofbubblesize,gasholdupandliquidvelocitymeasured3

in4bubblecolumnsfrom0.15mto3mindiameter4

5

P.MaximianoRaimundo1,2,A.Cloupet1,A.Cartellier2,D.Beneventi3,F.Augier1,*6

7

1-IFPEnergiesnouvelles,rond-pointdel’échangeurdeSolaize,69360Solaize8

2-UniversitéGrenobleAlpes,CNRS,GrenobleINP**LEGI,F-38000Grenoble,France,9

3-EcoleFrançaisedePapeterieetdesIndustriesGraphiques,INPG,BP65,F-38402St.Martind'Hères,10

France.11

*Correspondingauthor,frederic.augier@ifpen.fr12

**InstituteofEngineeringUniv.GrenobleAlpes13

14

Keywords15

Bubble Column, Sauter Diameter, Scale-up, Velocimetry, Heterogeneous Regime, Experimental,16

Cross-correlation,Multiphasereactor,Clustering17

18

Highlights19

• BubbleSauterdiameter,gasholdupandaxialliquidvelocitymeasurementsareperformed20

• 0.15m,0.4m,1mand3mdiameterbubblecolumnsareinvestigated21

• Awideexperimentaldatabaseisfurnishedtoassistfurthermodeldevelopments22

• Void fraction and liquid velocities profiles happen to be self-similar in the heterogeneous23

regime24

• Theentrainedliquidflowrate,proportionaltoD3/2,isonlysetbythecolumndiameter25

• Thequantity(gD)1/2appearsasanaturalscaleformeanandfluctuationsvelocities.26

• Stronggasholdupgradientsarepresentintheflowduetoconvectiveinstabilities27

28

Abstract29

The development of CFD models coupled with Population Balance is a very promising topic30

concerningmultiphase reactors. In the caseofbubbly flowsandbubble columns, a serious lackof31

localhydrodynamiccharacterizationsstillharmsdevelopmentandvalidationofrelevantmodels.To32

fillpartiallythisgap,anewbubblesizemeasurementtechnique,previouslyintroducedbyMaximiano33

Raimundoetal. (2016),hasbeenappliedonaverywide rangeofbubblecolumndiameters (from34

0.15mto3m)andsuperficialgasvelocities(from0.06m/sto0.35m/s).Sizemeasurementshavebeen35

coupled with others concerning gas holdup and axial liquid velocity, in order to provide an36

experimental database allowing to clarify the scale-up rules and to assist futuremodellingworks.37

Averagebubblesizeshavebeenmeasuredasglobally similarateveryscale.Measuredholdupand38

average liquid velocity confirm already reported behaviours at lower column diameters. Liquid39

velocityfluctuationsalsofollowself-similarradialprofilesandareproportionaltotheaverageliquid40

velocity at the centre of the column leading to a strong turbulence intensity. The fact that the41

quantity(gD)1/2appearsasanaturalvelocityscaleandthepresenceofstronggas-holdupgradients42

underlinethesimilaritybetweenbubblecolumnsoperatedheterogeneousregimeandfreethermal43

convectioninpipes.44

1. Introduction4546

Although being a major research topic for decades, the design and scale-up of bubble column47

reactors is still a difficult task for chemical engineers, as associated hydrodynamic and transfer48

phenomenaareverysensitivetophysicalpropertiesoffluids,operatingconditionsandgeometrical49

settings.Ahugenumberofcorrelationshavebeendevelopedinthepasttoassistscale-upbymeans50

of 0D or 1D approaches (Deckwer, 1992; Kantarci et al., 2005, Besagni et al., 2018). Available51

correlationsaregenerallyvalidinrathernarrowrangesofdesignparameters,physicalpropertiesand52

operating conditions.Nowadays it is generally admitted that CFD canbe used to secure industrial53

designs when outside of validation ranges of empirical correlations. But preliminary parameter54

adjustmentsaregenerallynecessarytofit initialsimulationsonagivenwell investigatedscale.The55

fittingstepkeepsCFD far frombeinga fullpredictive tool.Parameters tobeadjustedcanconcern56

turbulencemodelling, includingBubbleInducedTurbulence(McClureetal.,2014;Joshi,2001),but57

aboveall,interfacialforces(Jakobsenetal.,2005).Thereisnorealconsensusconcerningtheforces58

to consider to achieve realistic simulations, except for the drag law that all authors point out as59

impacting mostly the calculated gas holdup. The use of lift, added mass, wall lubrication and60

turbulent dispersion are sometimes suggested to improve the agreement between experimental61

data and CFD simulations (Krishna et al., 2001;McClure et al., 2015). However, during a previous62

work (Gemelloet al., 2018b), itwas found that time-averagedhydrodynamics (i.e. gasholdupand63

liquid velocity profiles) of several bubble columnswere satisfactorily predicted using an apparent64

drag formulation as the only interfacial force.Nevertheless, themean bubble diameter had to be65

beforehandknowntoperformsuchsimulations,asitconditionsdirectlythedragforce,andthusgas66

holdup(Guedonetal.,2017).Toovercomethis limitationinpresenceofbreakageandcoalescence67

phenomena,onepowerfulpossibilityconsistsinassociatingCFDwithapopulationbalanceonBubble68

SizeDistribution(BSD)(Buffoetal.,2013,Lehretal.,2004,Sanyaetal.,2005).Ifefficient,thiswould69

makeCFDamuchmorepredictivetool. 70

Whether to develop interfacial forcemodels for CFD or Population Balance kernels, experimental71

data are essential, including information concerning bubble sizes in flow regimes of industrial72

interest. Only a few experimental data are usable for this purpose. Major of them are based on73

multitips optical probes (Chaumat et al., 2005, Xue, 2004, Xue et al., 2008,McClure et al., 2017).74

Multitips technique ideally allows to measure both bubble chord and velocity distributions. The75

techniqueisbasedontheprocessingofthephase-signaldelaybetweenprobeslocatedatdifferent76

heights. As often pointed out by authors, its accuracy is acceptable at the centre of the column,77

wherebubblehavemostlyverticaltrajectories,butitrapidlydecreaseswiththedistancefromit,as78

bubble trajectories become more chaotic and include downward motion and the delay between79

signals is not linkable with the axial bubble velocity anymore. In addition, the performances of80

multitips probes in terms of minimum detectable bubble size are not clearly determined. An81

alternative approach consists in the use of ameasurement technique independent of the bubble82

trajectory. This is the case of the cross correlation (CC) technique developed by Maximiano83

Raimundoetal. (2016)andvalidated intheheterogeneousregimebycomparisonwithendoscopic84

measurements. The CC technique does not measure the global BSD, but only the mean Sauter85

diameter, i.e. the ratiobetween3rdand2ndmomentsof theBSD.But themeasurementshave the86

sameaccuracyregardlessoftheradialposition.Ithasbeenappliedrecentlytothestudyoftheeffect87

ofwaterqualityandthegasspargerdesign,ina400mmcolumndiameter(Gemelloetal.,2018a).88

Inthepresentwork,thesamecrosscorrelationtechniquehasbeenusedtomeasurebubblesizesin89

fourbubblescolumnsfrom0.15mto3mininternaldiameter,withthepurposeofinvestigatingthe90

effectofscale-uponbubblesizes,foranidenticalgas/liquidsystem.Distilledordemineralizedwater91

aregenerallyused inacademicstudies,butas tapwaterandairwere theonly fluidsusable in the92

biggestcolumn,asimilarsystemhasbeenusedatallscales.Thetapwaterusedinthepresentwork93

can be qualified as partially contaminated, as it exhibits a less coalescent behaviour than94

demineralizedwater.Onelimitationofusingtapwateristhatitsqualitydependsonthelocationand95

canleadtodifficultiestorepeatexperimentselsewhere.Toovercomethisissue,Ithasbeenshown96

in (Gemello et al., 2018a) that the tap water used in the present study behaves as a solution of97

demineralizedwatercontaining0.01%inweightofethanol(Gemelloetal.,2018a).Inadditiontothe98

bubble size, the global (<αG>) and local (αG) gas fractions have been measured via the liquid99

elevationmeasurementandtheopticalprobesusedforthecrosscorrelationtechnique,respectively.100

Axial liquidvelocityprofiles (averageand fluctuationRMSvalues)havebeenalsomeasuredwitha101

modifiedPitottube,so-calledPavlovtubetechnique(Forretetal.,2003).Thistechniqueistheonly102

oneusableathighgasholdup inall columnsizes.MeasurementshavebeendoneatdifferentVsg103

from 0.03m/s to 0.35m/s depending on bubble columns. In the following, the four experimental104

setupsareintroduced,aswellastheinvolvedmeasurementtechniques.Thenresultsconcerninggas105

fraction,liquidvelocityandbubblesizesaresummarizedanddiscussed.Adiscussionconcerningthe106

presenceofbubbleclustersandofvoids(regionswithfewbubbles)andtheroleofthesemeso-scale107

structuresonthehydrodynamicsoftheheterogeneousregimeisfinallyproposed.108

2. Experimentalsetups109

Inthepresentwork,fourdifferentcolumnswith0.15m,0.4mand1mand3minnerdiametershave110

been tested.All theexperimentswere conductedwith compressedair,whichhadbeendried and111

cooled, as the gas phase andwithwater,with no net liquid flow rate (batchmode), as the liquid112

phase. In thedifferent columns, apartially contaminated tapwater is used. Table1 reportswater113

analysis.114

Table1:tapwateranalysis@20°C115

Surfacetension(mN/m) 67Conductivity@25°C(µS/cm) 559Carbonate(mg/L) ≈0Hydrogenocarbonate(mg/L) 251pH 7.9116

Inthefourcolumns,thegaswasintroducedintothecolumnfromthebase.Thevolumetricflowrate117

wasdeterminedbya rowofseveralcalibrated flow-meters,withanuncertaintygivenbysuppliers118

equalto1.6%.Thegasdistributorsareperforatedplates,thenumberofholesandtheirdiametersin119

thedifferentcolumnsaregiven in the following table.The injectorshavebeendesignedtoensure120

comparableporosities(atavaluesmallenoughtouncouplethegasinjectionandthebubblecolumn121

dynamics),andalsoto formgas jetswithcomparableejectionvelocities (as theejectionvelocity is122

equal to the superficial gas velocity divided by the porosity of the injection plates) which are123

expected to experience similar break-up process and thus to deliver comparable bubble size124

distributionsatinjectioninallcolumns.Columnsof0.15and0.4minnerdiameterexhibitexactlythe125

sameconditionsofgasspargingwithorifices1mmindiameterand10mmlong.Theinjectorforthe126

1minnerdiametercolumnhasthesameporosity,i.e.thesamegasvelocityattheoutletofinjections127

holes(from13to140m/s),butthediameterofholesisdoubled.Indeed,theeffectoftheholesize128

canbeconsideredassmallasholediametersstandmuchsmallerthanbubblesizesinthecolumns.129

Thebiggestcolumnisequippedwithadifferentsparger,withlargerandshorterinjectionholesand130

with a lower porosity, but leading to comparable ejection gas velocities (from 37 to 100m/s).131

However,Gemelloet al. (2018a)have studied the impactof the spargeron thebubble size in the132

0.4minnerdiametercolumnusedinthepresentstudy.Ithasbeenfoundthatthespargerdesignhas133

an important effect at the bottom of the column. This effect is due both to the spatial gas134

distribution, conditioned by the number of holes, and the bubble sizes generated by different135

spargers, which can be very disparate. But the effect of the sparger becomes negligible above a136

distance of 0.4m from the bottom of the column because of the dominant breakup phenomena137

whatever thewaterquality.Thisbehaviour is inagreementwith theusualobservation that, in the138

heterogeneous regime,quantities suchas local void fraction and velocities areweakly sensitive to139

injection conditions when 𝐻!/D > 2 and when the data are gathered within the (almost) fully140

developedregion(Forretetal.,2006).Inaddition,itwillbeshowninsection4thatthemeanbubble141

sizes are similar for all columns and all flow conditions, probably thanks to an efficient break-up142

processofthegasjetsformedatinjection.143

Table2:gasdistributorscharacteristicsinthedifferentcolumns144

Column Numberofholes

Holediameter(mm)

Length/diameterratioofinjectors

Porosity(%) Pitch(mm)

Ø0.15m 55 1 10 0.24 15(triangular)Ø0.4m 391 1 10 0.24 15(triangular)Ø1m 613 2 10 0.25 37(triangular)Ø3m 164 9 0.5 0.15 200(squared)145

Thecolumnsused in thisstudypresentofpairsofdiametricallyopposedholesatdifferentheights146

(0.5m,1m,1.5m,2.6m,3.4mand4.6mabovethegasdistributorinthecaseofthe1mindiameter147

column),enablingthepositionofdifferenttechnicalmeans(Pavlovtube,opticalprobe…)orcanbe148

used as pressure taps. The scheme of the 1 m in diameter column is given in the Figure 1. The149

schemeoftheothercolumnsisverysimilartothisone.150

151

Figure1:schematicviewofthe1mindiametercolumnanditsperforatedplate152

153

In addition, the 0.15 m, 0.4 m and 1 m diameter columns were operated with a liquid static154

height/columndiameter“aspectratio”(H0/D)of4.The3mdiametercolumnhasoperatedwith2.2155

liquid static height/diameter ratio because the column is only 12 m high. Besagni et al. (2017a)156

studied theeffectof theaspect ratioon thegasholdupand suggesteda critical aspect ratioof 5.157

Nevertheless,exhaustiveexperimental results reportedby theseauthors show thatabove3-4, the158

effect of the aspect ratio on gas holdup becomes very weak. This is also confirmed by the159

measurementsofSasakietal.(2016).Measurementshavebeenperformedatanelevationheightof160

2.5timesthecolumndiameter(H/D=2.5),exceptinthe3mcolumn,insidewhichmeasurementsare161

performed atH/D=2.According to Forret et al. (2006), such elevations are indeedwithin the fully162

developed flowregion.However, therecentworkofGuanetal. (2016)may leadtomoderate this163

conclusion. They found that gas holdup and liquid velocity radial profiles were never totally164

independentoftheaxialpositioninacolumnof0.8mindiameter.Yet,usingauniformgassparger,165

the flowhappens to be almost fully developed betweenH/D from2 to 4. Guan et al. (2016) also166

suggested that theheightof flowdevelopmentdependson thecolumndiameter.Figure2 reports167

gasholdupprofilesmeasuredintheØ0.4mcolumnatVsg=0.16m/sandatdifferentheights.Profiles168

measuredatH/D=2.5and3.75aresimilar,whilesomediscrepanciesareobservedbelow,especially169

inthemiddleofthecolumn.Insummary,theaspectratioofthestudiedcolumnsisconsideredtobe170

sufficienttoneglectitseffectonhydrodynamics,andtheheightofmeasurementiscomprisedwithin171

therangeoffullydevelopedflow.172

173

174

Figure2:ComparisonofgasholdupprofilesmeasuredintheØ0.4mcolumnatdifferentH/D.Vsg=0.15m/s(H0/D=4)175176

177

3. Measurementtechniques178

3.1. Gasholdup179

Theglobalgasholdupcanbedirectlycalculatedbythevisualobservationoftheexpansionofliquid180

heightby:181

182

𝛼! = 1 − !!!!

(1)183

184

Where 𝛼! representstheglobalvoidfraction,𝐻!representsthenon-aeratedliquidheightand𝐻!185

theaerated liquidheight. In thepresentstudy, the liquidheight ismeasuredbyvisualobservation186

along a graduated rule, and fluctuations of liquid level induce a relatively high uncertainty of the187

measurement,estimatedat10%.Sasakietal.(2016)stronglydecreasedtheuncertaintywhileusing188

image processing to detect liquid level. The local gas holdup was measured by a light reflective189

opticalprobe.Thebeamisgeneratedbyalaserandissentthroughtheopticalfibertothepointof190

theprobe. If theprobe is incontactwith the liquidphase, the lightbeam is refracted through the191

liquidmedia.Otherwise,iftheprobetipisincontactwiththegasphase,thelightbeamisreflected192

bythebubbleintotheprobetipandthendetectedbyaphotodiodewherethelightbeamintensityis193

convertedintovoltage.Addingthetimethattheprobedetectsthegasphase,thelocalgasholdupis194

determinedbythefollowingequation.195

196

𝛼! =!"#"$%&'( !"# !"#$!"#$%&'$()*+ !"#$

(2)197

198

The local gas holdup detected with the optical probe was compared with global gas hold-up for199

heterogeneousconditionsinthe0.4mbubblecolumn.Theagreementwaswithin±15%(Maximiano200

Raimundoetal.,2016),whichissatisfactoryowingtotheuncertaintyonthedeterminationof 𝛼! .201

202

3.2. Cross-Correlation(Bubblesize)203

Inordertocharacterizebubblesindense,heterogeneousbubblyflowssuchasthoseencounteredin204

industrial bubble columns, a recently developed measuring technique based on the spatial205

correlation of phase indicator functions is proposed (Maximiano Raimundo et al., 2016). The206

normalized cross-correlation is a function that quantifies the similarity of two binary signals by207

analysingbothsignalssimultaneously:208

𝐶𝐶 =!"#$%& ! ! ×!"#$%& ! ! !"!!"#

!!!

!"#$%& ! ! !"!!"#!!!

(3)209

210

Here𝑆𝑖𝑔𝑛𝑎𝑙 !and𝑆𝑖𝑔𝑛𝑎𝑙 !representtherawsignalscomingfromtwodifferentprobesatthesame211

elevation in the column at a radial distance (d) and t!"# represents the time of registration. The212

cross-correlationvalue ismaximal ifbothprobesareatthesamepoint inspace(distancebetween213

probesd of 0mm), since the signalswill be identical. In a single bubble-probe interaction, as the214

distancebetweentheprobesincreases,thecross-correlationoftheprobesignalswilldecrease.The215

cross-correlationwill be zerowhen the distance between probes becomes larger than the bubble216

horizontaldiameterduetothefactthatthesamebubblecannotbedetectedbybothprobesatthe217

sametime.Nevertheless,inabubblecolumn,thecross-correlationisneverzeroevenifthedistance218

betweenprobes reachesavaluemuch larger thanthe largestbubblehorizontaldiameter.At large219

distances, the cross-correlation tends to the local gas hold-up αG. It has been shown that some220

informationrelatedtothehorizontalbubblesizecanbeextractedfromtheinitiallinearslopeofthe221

cross-correlation curve as a function of probes distances (Maximiano Raimundo et al., 2016). The222

relationship between the Sauter mean value of horizontal bubble diameters, noted d32,h, the223

correlationcoefficientCCandtheinter-probesdistancedalwaysexpressesasfollows:224

225

𝑑!",! = a !!!(!)!! (4)226

227

wheretheprefactora is functionof thebubbleshape,asdetailed in the followingtable (details in228

MaximianoRaimundoetal.,2016).229

230

Table3:Prefactorsaforvariousbubbleshapes231

Sphere Oblate ProlatewithEcc=0,7Prefactora -1.472 -1.5978 -1.7

232

Tovalidatethismethod, ithasbeencomparedtoendoscopic imaging.TheSautermeanhorizontal233

diameters detected with the correlation technique happen to be in good agreement with those234

providedbyendoscopic imaging.This latestmethodenablesalsothedeterminationoftheaverage235

eccentricity(ecc) :the latterhasbeenmeasuredequalto0.7forall flowconditionsandallcolumn236

dimensionsexcept forthe3mI.D.columnforwhichtheeccentricitywasnotmeasured.Therefore,237

theSautermeandiameterd32canbelinkedtothed32,hmeasuredbythecross-correlationasfollows.238

239

𝑑!" = 𝑑!",! ∗ 𝑒𝑐𝑐!/! (5)240

241

withecc=0.7.242

243

3.3. Pavlovtube(axialliquidvelocity)244

ThePavlovtubeusedinthisworkiscomposedbyfour5mmdiametertubes,arrangedasitisshown245

intheright-handsideoftheFigure3.Thefourtubesareplacedinsidea30mmdiametertube,which246

crosseshorizontallythecolumn,inordertoobtainrigidityandtoallowhorizontaldisplacement.The247

5mmdiametertubesarecompletelyclosed,exceptfora0.5mmdiameterlateralholethatassures248

thatthepressureinsidethetubeisequaltotheoneinthecolumn.249

250

Figure3:SchematicrepresentationofthePavlovtube(Axialmeasurementsareperformedalongaverticalaxis,radial251measurementsareperformedalongahorizontalaxis)252

253

The holes of the tubes on the vertical plan are placed in the same axis but in opposite254

directions,asdepicted inFigure3.Additionally,eachtubeontheverticaldirection isconnectedto255

one of the chambers of a differential pressure sensor. These tubes serve to measure the axial256

pressure difference, since the holes on the tubes are aligned with the column vertical axis. The257

Pavlov tube allows also themeasurement of the radial liquid velocity, quite low in this study, not258

discussed in this paper. Themeasurements of the axial pressure differences (ΔP) can be used to259

calculatetheinstantaneousaxialliquidvelocityrecurringtothefollowingequation.260

𝑈(𝑥,𝑡)=

!Δ!(!,!)!!

if ΔP(x, t) ≥ 0

− !!Δ! !,!!!

if ΔP(x, t) < 0 (6)261

TheaxialΔPmeasurementwasmadewithadifferentialpressuretransmitterRosemount3051Twith262

arangeof±60mbar,afrequencyof8Hzandresolutionof0.01mbarLetusmentionthatcorrections263

ofeq.(6)accountingforlocalvoidfractionhavebeensometimesproposed.Ononehand,thelargest264

ofthesecorrectionsistheoneproposedbyBosioandMalnesasdiscussedbyRiemannetal.(1984).265

According to that correction, the difference in velocity prediction remains smaller than 3% since266

thevoid fractionarealwaysbelow35% inallourexperiments.Onanotherhand,these corrections267

consider that the dynamic pressure collected from pitot or pavlov tubes in two-phase flows is268

associatedwiththedensityandthevelocityofthemixture.However,inbubblyflows,theorificesof269

pitot or pavlov tubes are always wetted, meaning that the gas phase is never detected as such.270

Therefore, the local stoppingpressuredetectedwith such sensorsprovidesameasureof the local271

liquidvelocityevenwhenthesensoris inthevicinityofabubble,andnocorrectionisrequired.All272

Pavlovtubemeasurementspresentedhereafteraregroundedoneq.(6).273

The radial profiles are easily obtainedbymoving themeasuring cell along the column radius. The274

axial velocity fluctuations can be calculated through the instantaneous and the averaged velocity,275

usingReynoldsdecomposition,aspresentedinthefollowingequation.276

277

),(')(),( txuxUtxU += (7)278

279

where )(xU representsthemeanliquidvelocity, ),(' txu representstheliquidfluctuationvelocity280

attheinstanttand ),( txU representstheliquidvelocityattheinstantt.281

282

4. Results283

Thereafter,thediscussionfocusesontheheterogeneousregime.Onallcolumns,onecanobservein284

Figure 4, that the almost linear behaviour of 𝛼! with Vsg is observed only below 4-5cm/s,285

approximately.Thisvaluecorrespondstothetransitiontotheheterogeneousregime.286

4.1. Gasholdup287

GlobalgasvolumefractionshavebeenmeasuredforallcolumnsandVsg.Radialgasholdupprofiles288

havebeenmeasuredwith theopticalprobesatH/D=2.5 (Figure4). Schweitzeret al. (2001) found289

similar profiles of𝛼!(x)when normalized by the average gas holdup in the column 𝛼! . Authors290

suggest the correlation reported in eq.(8) to represent holdup profiles. The correlation has been291

initiallydevelopedandvalidatedina50mminnerdiametercolumnforVsgbetween0.03to0.25m/s.292

ItsrangeofvalidityhasbeenextendedbyForretetal.(2006)tocolumnsupto1mofinnerdiameter.293

294

𝛼! 𝑥 = 𝛼! −1.638 𝑥! − 1 + 1.228(𝑥! − 1) − 0.939(𝑥! − 1) (8)295

296

OnFigure4(left),unsurprisingly it isverifiedthatgasholdupfollowsthesameprofileatanyscale,297

withanaverageerrorbelow5%.Concerningtheaveragegasholdup,keepinginmindthattheliquid298

elevation measurement method eq.(1) is relatively imprecise (+/- 10%), different correlations299

validated in air/water systemsmay be found in acceptable agreement with present experimental300

data.Asthegasholdupshavebeenmeasuredwiththesametechniqueandusingthesamewater,301

experimentaldatahavebeenusedtofitanewcorrelationintheaimtoextractthespecificeffectof302

scale up in the heterogeneous regime. Remember that existing models have been generally303

developedbasedonasinglecolumndiameter,generallylowerthan0.4m.Thefollowingcorrelation304

hasbefoundtopredict 𝛼! withanerrorbelow6%whenVsg>0.1m/s:305

306

𝛼! = 0.49.𝑉𝑠𝑔!.!".𝐷!!.!"# (9)307

308

Measured and predicted average gas holdups are compared in Figure 4 (right). Only experiments309

correspondingtoVsg>0.05m/sareconsideredtofiteq.(9),giventhatgasholdupisproportionalto310

Vsg in the homogeneous regime. It is observed a small, but not negligible effect of the column311

diameter on the gas holdup, especially in intermediate range of Vsg, between 0.1 and 0.2 m/s.312

Although most of existing correlations do not include the effect of D (Kantarci et al., 2005), the313

questionof the impactof thecolumndiameteron thegasholdup isstillnotcompletely settledas314

notably discussed by Rollbush et al. (2015). Additional experimentsmay although be necessary at315

higherDtoconfirmthistrend,asathighVsgmeasurementsin0.4and1mdiametersarerelatively316

close.Alternativelytheeq.(9)canbereplacedbyanequationinvolvingonlyVsgwithanexponentof317

0.41andaprefactorof0.5insteadof0.49,withameanerrorof8%.318

319

Figure4:GasholdupprofilesatH/D=2.5,normalizedbytheaveragegasholdup(left),andaveragegasholdup320measurements(right).Comparisonwithcorrelations(eq.8&9).321

322

0

0,2

0,4

0,6

0,8

1

1,2

1,4

-1 -0,5 0 0,5 1

αG/<αG>

r/R

Ø0.15mVsg=0.15m/sØ0.4mVsg=0.16m/sØ1mVsg=0.16m/sØ3mVsg=0.2m/sSchweitzeretal.(2001)

0

0,05

0,1

0,15

0,2

0,25

0,3

0,35

0 0,1 0,2 0,3 0,4

<αG>

Vsg(m/s)

Ø0.15mØ0.4mØ1mØ3mØ0.15mmodelØ0.4mmodelØ1mmodelØ3mmodel

4.2. Liquidvelocity323

Inasimilarmanner,Forretetal.(2006)suggestedtonormalizeaxialliquidaveragevelocityprofiles324

bythe liquidvelocityat thecenterU0,andfoundagoodagreementwiththe followingpolynomial325

model:326

327

𝑈 𝑥 = !!!!!

𝑎. exp −𝑏. 𝑥! − 𝑐 (10)328

329

With a=2.976, b=0.943, c=1.848. The eq. (10) has been compared to the present experimental330

measurements of U(x), and a very good agreement has been found, except in the largest column331

wherethemagnitudeoftheliquidvelocityclosetothewallsisslightlyunderestimatedbythemodel.332

The comparison between U(x)/U0 and eq.(10) are reported in Figure 5. The liquid velocity at the333

centercanalsobecorrelatedwithVsgandD.Theequation(11)presentstheresultoftheregression,334

whichpredictsU0withanaverageerrorbelow8%:335

336

𝑈! = 1.35.𝑉𝑠𝑔!.!".𝐷!.!" (11)337

338

Figure 6 (left) presents the parity diagram between eq.(11) and experimental measurements in339

differentcolumns.InFigure6(right),experimentalresultsarealsocomparedtoashortselectionof340

existingcorrelationsofMiyauchi&Shyu (1970),Riquarts (1981),Nottenkämper (1983)andZehner341

(1986).Inthisfigure,eachexperimentalcentrelinevelocity,reportedinabscissa,iscomparedtothe342

velocitygivenbythedifferentlistedcorrelations,includingeq.11,atthesameoperatingVsgandD,343

whicharereportedinordinate.Thiscomparisonshowsthatourexperimentsareingoodagreement344

with previous works of Miyauchi & Shyu (1970) and Nottenkämper (1983), but not with others,345

pointingoutthedisparityofexperimentalresultsobtainedindifferentlaboratories.Thisdisparitycan346

havedifferentorigins, as thedifferentmeasurementmethods,differentqualityofwater,different347

gas spargingconditionsordifferent columnsizes. Thepresent studyallows toeliminate the3 first348

possibilities.349

WhiletheexponentonDineq.(11)isquiteclassical–intherangeof0.3-0.7,theexponentonVsg350

canbeconsideredlowerthangenerallyobserved(0.3-0.5).Thislaterresultmaybeaconsequenceof351

thequalityofwater,whichpartiallyprevents coalescencewhencomparedwithnon-contaminated352

water, resulting in a less pronounced heterogeneity of bubble sizes and possibly a lower liquid353

recirculationassuggestedbyRollbuschetal. (2015).Thus, fromtheabove results, ithappens that354

the flows in bubble column operated in the heterogeneous regime self-organise, leading to self-355

similarradialprofilesforthevoidfractionandforthemeanliquidvelocities.356

357

Figure5:Normalizedliquidvelocityprofiles(left)andcomparisonbetweentheaxialaverageliquidvelocityatthecenter358U0andeq.(11).359

360

361

Figure6:Centre-linevelocityparitydiagramofeq.11(left);paritydiagramofdifferentcorrelations(right)362

363

364

Besides,awell-knownbehaviourconcerningtheliquidvelocityprofileisobserved:thesectionofthe365

columnscanbedividedintotwosurfacesofequalarea:thecoreregionwhenr< !!Dorx<0.71,where366

Uispositive,andtheexternalregionwhereUisnegative.Inthecoreregion,theliquidupwardflux367

canbecalculatedas:368

369

𝑄!,!" = 2𝜋. 𝑟. 1 − 𝛼! .𝑈 𝑟 .𝑑𝑟!! !! (12)370

371

ThecalculatedliquidupwardfluxforthreeVsgandforalltheinvestigatedcolumnsarereportedin372

Figure.Theliquidupwardfluxsurprisinglyhappenstobeindependentofthesuperficialgasvelocity373

in the heterogeneous regime. Results correlates very well with∝ D2,5, implying that the average374

liquidvelocityinthecoreregionfollowsatrend∝ 𝐷.Onepossibleexplanationofthatcouldbethe375

predominant role of the natural velocity scale 𝑔𝐷. If we define a Froude number based on the376

mean liquid velocity in the core regionof section Score (= 𝜋𝐷! 8), a constant valueof the Froude377

numberisobserved:378

379

𝐹𝑟 = !!,!"!!"#$ !"

≈ 0.024 (13)380

381

Thisresultalsoillustratesthecloselinkbetweenliquidvolumefractionandvelocityprofiles,asboth382

change with the gas flow rate but not their mutual product. The above velocity scaling and the383

observedfloworganisationarereminiscentoffreethermalconvectioninpipesforwhichthenatural384

scaleforvelocityhasbeenshowntobethefreefallvelocityunderbuoyancyaccelerationevaluated385

foralengthscaleequaltothepipediameter(Tisserandetal.,2010,Rusaouenetal.,2014).386

387

388

Figure7:Liquidupwardfluxinthecoreregionasafunctionofthebubblecolumndiameterfortheheterogeneous389regime.390

391

Relativevelocity392

In thecore region, themean flow isdirectedupwardand it is thus similar toa co-currentupward393

two-phase flow. According to classical kinematic approaches (e.g. Zuber and Findlay 1965), the394

differencebetweenthegasflowratefractiondefinedasthevolumetricgasflowdividedbythesum395

of liquid and gas volumetric flow rates and the void fraction is controlled by the relative velocity396

between phases. In the same spirit, let us introduce the apparent (in the sense that it is a global397

quantityatthescaleofthecoreregionofthecolumnandnotalocalone)relativevelocitybetween398

gasandliquid<Ur>inthecoreregionasthedifferenceofmeanvelocitiesbetweengasandliquidin399

thecoresection.Eachmeanvelocity iscalculatedasthephasicvolumetricflowratedividedbythe400

associated cross section, i.e. the core sectionmultiplied by the volume fraction of the concerned401

phaseinthissection:402

403

𝑈𝑟 = !!,!"#$!! !"#$∙!!"#$

− !!,!"!! !! !"#$ ∙!!"#$

(14)404

405

Where𝑄!,!"#$ isthegasflowrateinthecoreregion,𝑄!,!"#$ theliquidflowrateinthecoreregionis406

givenby (eq.(13) and 𝛼! !"#$ is computed fromeq. (8) and is found≈ 1.33 𝛼! .𝑄!,!"#$ canbe407

roughlyestimatedbythetotalgasflowrate𝑄! .Notethatthisapproximationtendstominimizethe408

estimation of the relative velocity as the recirculation of a part of the bubbles is not taken into409

account.TheestimationofrelativevelocitiesarereportedinFigure8forVsg>0.09m/sandforthree410

columns.TherelativevelocityincreaseswithVsgandachievesvaluesthataremuchhigherthanthe411

terminal bubble velocity (≈0.22m/s). Such an increase in the bubble relative velocity in the412

heterogeneous regime has been evoked by Ruzicka (2013). Possible origin of this phenomenon is413

discussedbelow.414

415

Figure8:Estimationoftherelativevelocityinthecoreregion,consideringthatthegasflowrateinthecoreregionequals416thetotality(Urmax)orthehalf(Urmin)ofthetotalgasflowrate.417

418

Fluctuatingvelocity419

RMS of axial liquid velocity fluctuations (u’) have beenmeasured in various operating conditions.420

ResultspresentedinFigure9(left)pointoutthatu’radialprofilesalsofollowself-similarprofiles.The421

followingpolynomialfunctionhasbeenregressedineq.(15)andisalsoreportedinFigure9(left)to422

illustratetheself-similarityofprofiles.423

424

𝑢! 𝑥 = 𝑢! 0.71 . 𝑎 + 𝑏𝑥! + 𝑐𝑥! + 𝑑𝑥! (15)425

426

Witha=0.48,b=2.25,c=-2.6,d=0.40.Thenormalizationoftheu’profilescanbedonewhateverthe427

positionofreference(x)butithasbeenfoundthatusingthemaximalu’valueminimizestheerrorof428

residues of the regression. The maximal value of u’ is always located at the boundary between429

positiveandnegativeliquidvelocities(x= 2 2 ≈0.71).Whencomparingu’maxatdifferentscales,it430

isfoundproportionaltoU0.Asaconsequence,themaximalu’valuecanbewrittenasfollows:431

432

𝑢! 𝑥 = 2 2 = 𝑢′!"# = 0.695 𝑈! (16)433

434

u’maxisreportedinFigure9(right)forthedifferentcolumns.TheexperimentaldataofMenzeletal.435

(1990) using a hot wire probe is also reported on the same figure. The consistency between the436

values measured with different techniques confirms the reliability of Pavlov tube measurements437

concerningaxial liquidvelocityfluctuations.Notethatinthecoreregion,theliquidfluctuationsare438

abouthalftheirmaximum,correspondingtoaturbulentintensityu’/Uoabout25-30%.Hence,large439

velocityfluctuationsarepresentovertheentirecross-sectionatallflowconditions:thatfeatureisa440

clearcharacteristicoftheheterogeneousregime.441

442

443

Figure9:u’radialprofiles(left)andcorrelationbetweenu’(x=0,71)andU0(right).444

445

4.3. Bubblediameters446

Sautermeandiametersmeasuredwiththecross-correlationtechniquearereportedinFigure10.On447

the left, bubble Sauter diameter (d32) profiles are reported at a given Vsg. On the right, volume448

averageSauterdiameters<d32>forthe4columnsaregivenasafunctionofVsg.<d32>iscomputed449

asthegas-volumeaveragedSauterdiameterovercolumnsections.Letusrecallthat,intheup-flow450

region, the typical relative uncertainty on the Sauter diameter is less than 10% for the large451

superficial velocitiesconsideredhere (MaximianoRaimundoetal.,2016). Important results canbe452

summarizedasfollows:453

• Themeanbubblediameterincreasesslowlyfrom≈4.5-5mmatlowVsgto5.5-6mmathigh454

Vsg455

• Themeanbubblediameterisalmostinsensitivetothecolumndiameter,456

• Bubblesizeprofilesfollowroughlyaparabolicshape,assuggestedbytheparabolicfunction457

(y=5.8-2x²)reportedinFigure10.458

• Biggerbubblesaremeasuredatthecenterofthecolumn,andthebubblesizedecreaseswith459

xtypicallyby1to2mmindicatingthatasmallamountofspatialsegregationbetweenupflow460

anddownflowregionsoccursintheseflows.461

Inallourexperiments,theterminalvelocityoftheaveragebubblesvariesoveralimitedrange(0.21-462

0.23m/s in cleanwater), and thebubbles alwayspertain to the same regime i.e. ellipsoids athigh463

particulateReynoldsnumber(thelatterrangesbetween950and1300)experiencingwobbling(Clift464

etal.,1978).Hence, thepresentdatabasehasbeenbuilt-upfornearly thesamemeanbubblesize465

irrespectiveofthedimensionofthebubblecolumnandoftheamountofgasinjected.466

467

Figure10:Sauterdiameterprofilesinthe4columns(right)andvolumeaverageSauterdiameterforthe4columnsversus468Vsg.(H/D=2.5exceptinthe3mcolumnwhereH/D=2).469

470

AsthemeanbubblesizedoesnotexhibitstrongvariationsforVsgintherangeof0.03to0.35m/s,471

this quantity does not explain by itself the transition between the homogeneous and the472

heterogeneousregime.Further investigationbasedonbubblesizedistributionsmayberequiredto473

identifyapresumedlinkbetweencoalescencephenomenaandflowregimetransitionassuggested474

andreviewedbyBesagniandInzoli(2017b).475

476

4.4.Discussion477

Bubblesizemeasurementssuggest informationconcerningturbulence.Classicalpopulationbalance478

kernels(Buffoetal.,2013)usethedissipationrate(ε)asamajorparametertopredictbreakageand479

coalescence.ε canclassicallybewrittenas∼𝑢′! Λ,Λbeinga turbulencemacroscale.Besides, the480

globaldissipationrateinsidebubblecolumnscanbeestimatedviadifferentways,butitisgenerally481

considered as not depending on the column diameter (Deckwer, 1992; Roels and Heijnen, 1980).482

Indeed in bubble columns ε is usually calculated as 𝑔. 𝜌! − 𝜌! 𝜌! .𝑉𝑠𝑔. The observed almost483

constantbubble sizewith scale-up is consistentwith this theory. In thepresent studyu’hasbeen484

foundtofollow∼D0,4,thisimpliesthatΛincreasesduringscaleupas∼D1.2,letsay∼D,whichmakes485

sense from a pure geometrical consideration. Further investigations concerning the size of486

turbulence macroscales in bubble columns may be useful to understand the link between flow487

structures and turbulence production, as well as with breakage and coalescence phenomena.488

Concerningvelocityscales,theabovediscussionontheapparentrelativevelocityindicatesthatthe489

terminal velocityofbubbles isnot themost relevant scale in thesebuoyancydrivenbubbly flows.490

Indeed,anaturalvelocityscale(gD)0.5arisesfromtheanalysisoftheentrainedliquidflowrateinthe491

centralportionofthecolumn.Suchascaleisreminiscentofturbulentflowsdrivenbyconvection.In492

buoyancydrivenflowssuchasinthermalconvection,theequilibriumbetweenflowconvectionand493

Archimedian forcing leads to a velocity scaling as𝑉² ≈ 𝑔𝐿∆𝜌 𝜌,where the density gradient ∆ρ is494

evaluatedat the lengthscaleL. L is tentatively takenhereas thecolumndiameter.Thatchoice is495

supported here by the fact that we are considering the fully developed region, and, as the flow496

happenstobeself-similarinthatzone,theheightofthecolumnisnolongerarelevantparameter.497

Inadditiontotheselarge-scalegradients,localisedvariationsinthedensityofthegas-liquidmixture,498

equivalent then to variations in the local gas concentration, canalsoarise. Indeed, dispersed two-499

phaseflowscanbepronetotheformationofclusters,andVoronoïtessellationsthatareanefficient500

waytodetectandcharacterisesuchclusters(Monchauxetal.,2012)havebeenexploitedhere.501

In the present experiments,we exploited the signals delivered by optical probes that provide the502

arrivaltimeofbubbles.Fromsuchatimehistory, it isstraightforwardtodefinesuccessiveVoronoï503

time intervals ∆Tbubble containing a single bubble centred into it. Figure 11 provides the statistics504

gatheredontheaxisofthe1mI.D.columnatVsg=25cm/s:thedatacorrespondtoarecordof600505

seconds during which 85000 bubbles have been detected. The abscissa in Figure 11 is the time506

interval∆Tbubblenormalisedbythemeanvalue<∆Tbubble>.Thequantity∆Tbubble/<∆Tbubble>represents507

theinverseoftheratioofthelocalandinstantaneousgasconcentrationtotheaveragegashold-up.508

Clearly, themeasured probability density function differs from that of a random Poisson process509

referredtoasRPPinthesequel(FerencandNéda,2007).Inparticular,theprobabilitytofoundsmall510

timeintervals∆Tbubble,below0.6<∆Tbubble>,ishigherthaninRPPindicatingthatclusters(i.e.regions511

wherebubblesaccumulate)aremoreprobablethaninarandomprocess.Similarly,theprobabilityto512

foundlargetimeintervals∆Tbubble,above4<∆Tbubble>,ishigherthaninRPPindicatingthatvoids(i.e.513

regionswithfewbubbles)arealsomoreprobablethaninRPP.Thisdemonstratesthatclustersand514

voidsareindeedpresentintheheterogeneousregime(thisbehaviourholdsforotherscolumnssize515

and superficial velocities). Moreover, that plot indicates that local, instantaneous concentrations516

evolve between 0.1 and 10 times the average gas holdup. Local, instantaneous concentrations517

varying over a so wide range imply quite strong fluctuations in density and thus in buoyancy.518

Therefore,oneexpectsstronglocalvelocitydifferencesbetweendenseanddiluteregions.Thelatter519

cancontributetoenhancetheapparentrelativevelocitybetweenphasesasthebubblesaremostly520

located in clusterswhile the liquid inmostlypresent in voids. Thismayexplain the largeapparent521

relativevelocitydetectedsincethebubbleterminalvelocity inastill fluid isno longertherelevant522

scale.Suchstructuresarealsopronetocontributetoturbulenceproductionastheyproducestrong523

localand instantaneousshearrates.ThevelocityfluctuationsshowninFigure9combinedwiththe524

correlation(13)supportthatstatement.Finally,ifonestickstothemeanvaluesofbubblesizes,the525

transitionbetweenhomogeneousandheterogeneousregimesobservedinthepresentexperiments526

isnotrelatedwithcoalescence;instead,convectiveinstabilitiesthatleadtotheformationofdense527

anddiluteregionsarebelievedtoberesponsibleforthattransition.Topursuealongthese lines, it528

would be worthwhile to characterise these clusters and voids in terms of gas holdup and size529

distributionsand toexaminehow theyevolvewith flowconditionsandcolumndiameter.Another530

key issueishowtoquantitativelyrelatetheseclustersandvoidswiththeapparentrelativemotion531

between phases andwith the turbulence production in the liquid phase. In particular itwould be532

relevant to examine the connections between clusters and void dimensions and the correlation533

lengthscaleΛdiscussedabove.534

535

Figure11:Voronoïdiagramdeducedfromanopticalprobesignalcollectedinthecentreofthe1mdiametercolumn536operatedatVsg=25cm/s(dots).ThedashedlinerepresentsthestatisticsofaRandomPoissonProcess.537

538

5. Conclusions539

The objective of this work was to study specifically the scale-up of bubble columns in the540

heterogeneousregimeusingsimilar fluidsandthesamemeasurementtechniquesatverydifferent541

scales,whichhasneverbeenreportedbeforeinliterature.Thewiderangeofcolumndiameter,from542

0.15to3mandofsuperficialgasvelocities, from0.05m/sto0.35m/s, involved inthisstudymakes543

thereporteddataanimportantmaterialforvalidationofPopulationBalanceandCFDmodelsandfor544

up-scalingissues.Correlationsconcerninggasholdup,meanliquidvelocityandturbulencehavebeen545

validatedoverawiderangeofgeometryandoperatingconditions.Inparticular,theself-similarityof546

the flowstructure in termsofvoid fraction, liquidmeanaswellas fluctuatingvelocitieshavebeen547

demonstrated in the heterogeneous regime. This is believed to holdwhile the aspect ratio of the548

bubblecolumnremainslargeenoughsothatendeffectsnolongeraffectthefloworganisationinthe549

centralportionofthecolumn.Thatself-similarityleadstoanentrainedliquidflowrateproportional550

toD2(gD)1/2,meaningthattheentrainmentcapabilityofabubblecolumnisonlysetby itssizeand551

does not depend on the injected gas superficial velocity. This result also demonstrates that the552

velocityscale(gD)1/2 isakeydescriptorforbubblecolumnscale-up.Also,boththeaverageandthe553

1,E-03

1,E-02

1,E-01

1,E+001,E-01 1,E+00 1,E+01

PDF(-)

ΔTbubble/<ΔTbubble>

EXPVsg=0.25m/sRandomPoissonProcess

fluctuations of the liquid velocity are very sensitive to the scale-up. Small dependencies to the554

column diameter have been found on local and global gas holdup. The above results hold for all555

columns and flow conditions pertaining to the heterogeneous regime, and for almost the same556

average Sauter bubble diameters. The presence of strong concentration gradients has also been557

demonstrated, and the impact of these clusters and voids on the flow characteristics and on558

turbulence production deserves to be analysed further. As the turbulent dissipation rate plays an559

importantroleontheBSD,theturbulenceinducedintheheterogeneousregimeneedstobebetter560

characterisedinparticularbydeterminingitsintegrallengthscale.561

562

Abbreviations563

d distancebetweenprobes(mormm)564

d32 LocalmeanSauterdiameter(mormm)565

<d32> volumeaveragedmeanSauterdiameter(mormm)566

d32,h LocalmeanSauterhorizontaldiameter(mormm)567

CC cross-correlation568

D Columndiameter(m)569

ecc bubbleeccentricity570

H heightfromgasdistributor(m)571

H0 static(nonaerated)liquidheight(m)572

HD aeratedliquidheight(m)573

QG gasflowrate(m3/s)574

QG,core gasflowrateinthecoreregion(m3/s)575

QL,up Upwardliquidflowrateinthecoreregion(m3/s)576

r Radialposition(m)577

R radiusofthecolumn(m)578

Sc Sectionofthecoreregion(m²)579

),( txU Liquidaxialinstantaneousvelocity(m/s)580

),( txU Liquidaxialaveragevelocity(m/s)581

U0 Liquidaxialvelocityatthecenterofthecolumn(m/s)582

Ur Relativevelocitybetweengasandliquid(m/s)583

u' RMSofliquidaxialvelocityfluctuation(m/s)584

Vsg Superficialgasvelocity(m/s)585

x normalizedradialposition(=2r/D)586

Symbols587

αG Localgasholdup588

<αG> Averagegasholdup589

ε Turbulencedissipationrate(w/kg)590

Λ Turbulencemacroscale(m)591

ρ Liquiddensity(kg/m3)592

593

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