A consistent dimensional analysis of gas-liquid mass transfer in an aerated stirred tank containing purely viscous fluids with shear-thinning properties

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A consistent dimensional analysis of gas-liquid mass

transfer in an aerated stirred tank containing purely

viscous fluids with shear-thinning properties

Raouf Hassan, Karine Loubiere, Jack Legrand, Guillaume Delaplace

To cite this version:

Raouf Hassan, Karine Loubiere, Jack Legrand, Guillaume Delaplace.

A consistent dimensional

analysis of gas-liquid mass transfer in an aerated stirred tank containing purely viscous fluids

with shear-thinning properties. Chemical Engineering Journal, Elsevier, 2012, vol. 184, pp.42-56.

�10.1016/j.cej.2011.12.066�. �hal-00744336�

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Hassan, Raouf and Loubiere, Karine and Legrand, Jack and

Delaplace, Guillaume A consistent dimensional analysis of gas–

liquid mass transfer in an aerated stirred tank containing purely

viscous fluids with shear-thinning properties. (2012) Chemical

Engineering Journal, vol. 184 (n°1). pp. 42-56. ISSN 1385-8947

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A

consistent

dimensional

analysis

of

gas–liquid

mass

transfer

in

an

aerated

stirred

tank

containing

purely

viscous

fluids

with

shear-thinning

properties

Raouf

Hassan

a,b

_,

_Karine

_Loubiere

c,d,∗

_,

_Jack

_Legrand

a,b

_,

_Guillaume

_Delaplace

e

a_{UniversitédeNantes,LaboratoireGEPEA,CRTT,37boulevarddel’université,BP406,F-44602Saint-Nazaire,France} b_{CNRS,LaboratoireGEPEA,F-44602Saint-Nazaire,France}

c_{UniversitédeToulouse,INPT,UPS,LaboratoiredeGénieChimique(LGC),4alléeEmileMonso,BP84234,F-31432Toulouse,France} d_{CNRS,LaboratoiredeGénieChimique(LGC),F-31432Toulouse,France}

e_{INRA,LaboratoirePIHMUR638,369RueJulesGuesde,BP39,F-59651Villeneuved’AscqCedex,France}

Keywords:

Dimensionalanalysis Variablematerialproperty Gas–liquidmasstransfer Shear-thinningfluids Aeratedstirredtank ModelofWilliamson–Cross

a

b

s

t

r

a

c

t

Thispaperdealswithgas–liquidmasstransferinanaeratedstirredtankcontainingNewtonianor

shear-thinningfluids.Theaimistodemonstratethat,foragivenmixingsystem,anuniquedimensionless

correlationgatheringallthemasstransferrates(150klameasurements)canbeobtainedifandonlyifthe

variabilityoftherheologicalmaterialparametersiscorrectlyconsideredwhenimplementingthetheory

ofsimilarity.Moreparticularly,itisclearlyillustratedthatatoogrosssimplificationintherelevantlist

oftheparameterscharacterizingthedependenceofapparentviscositywithshearratesleadstopitfalls

whenbuildingthe-spaceset.Thisisthenastrikingexampleshowingthatarobustpredictivecorrelation

canbeestablishedwhenthenon-constancyoffluidphysicalpropertiesceasestobeneglected.

1. Introduction

Thedispersionofagaseousphase inaliquidphaseformass transferpurposes isinvolved in manyprocesses, inthefield of chemicalreactionengineering(e.g.forchlorinations, hydrogena-tions,oxidations,alkylations,ammonolysisandsoforth)butalso inbiochemicalengineering(includingfermentation,wastewater treatment).Theuseofagitatedtanksisawidespreadpracticefor operatingsuchabsorptionprocessesasofferingtheadvantagesto generatehighinterfacialareasandintensemixingofliquidphase. Understandingandmodellingmasstransferbetweenphasesisof importance,becauseitmayoftenbecomethecriticalstep deter-miningtheachievementoftheapplication,andthusmaygivethe mainguidelinesonwhichthedesignandthescale-upofthe pro-cesswillbebased.Thetransferredmassquantitydependsonthe solutesolubilityintheliquidphase,butaboveallontheinterfacial area,a,andontheoverallliquid-phasemasstransfercoefficient,Kl. Theproductoftheselatterparametersiscommonlycalled absorp-tionratecoefficientoroverallvolumetricgas–liquidmasstransfer coefficient (Kla or kla for low soluble gases). The factors influ-encingof such unit operation (inparticularKla) arevery large,

∗ Correspondingauthor.Tel.:+330534323619;fax:+330534323697. E-mailaddress:Karine.Loubiere@ensiacet.fr(K.Loubiere).

includingtankgeometryanddimensions,impellertype, dimen-sionsandrotationalimpellerspeed,aerationsystemandgasflow rate,physicalandrheologicalpropertiesofgasandliquidphases, temperature,andpressure.Inanattempttoelucidatetheeffects ofthelatterparametersontheabsorptionratecoefficient,agreat amountofpublishedinvestigationsareencounteredinthe litera-ture;someinterestingoverviewsaregivenin[1–5].Theyprovide improvedknowledgeongas–liquidmasstransferthrough experi-mentaldata,empiricalcorrelations,mechanisticanalysisormore recentlynumericalsimulations.

Mostofthemdealwiththecaseswhentheliquidphaseisa Newtonianfluidwithlowviscosities.Fromvarioussetsof experi-mentscarriedoutatlab-scaleand/oratlargerscale,someempirical correlationsforklaareproposed,involvingeitherdimensionalor dimensionlessparameters.Atpresent, themostfrequentlyused dimensionalcorrelationremainstheoneofVan’tRiet[6]orsome variantsinwhichtheconstantandexponentshavebeenmodified. Theyareexpressedsuchas:

kla=C·(Ug)C1_·



P

Vl



C2

·()C3 ₍₁₎

wheretheconstantCisstronglyaffectedbythegeometrical param-etersoftheagitationsystem,and is theNewtonian viscosity. Garcia-Ochoa andGomez[3]recentlysummarized thedifferent exponentsassociatedwithEq.(1)thatareavailableintheliterature.

Nomenclature

a specificinterfacialarea(m−1₎

CO₂ concentrationindissolvedoxygen(kg/m3₎ C∗

O2 concentration in dissolved oxygen at saturation

(kg/m3₎

d airspargerdiameter(m) D impellerdiameter(m)

D oxygen diffusion coefficient in the liquid phase (m2_/s)

G/Vl massthroughoutperunitofliquidvolume(kg/s/m3) g gravityacceleration(m/s2₎

H Henry’sconstant(Pa) Ht tankheight(m)

k_l liquid-sidemasstransfercoefficient(m/s)

K consistencyindexfromthemodelof Ostwald–de-Waele(Eq.(14))(Pasnost₎

Kl overallmasstransfercoefficient(m/s)

kla overallvolumetricgas–liquidmasstransfer coeffi-cient(s−1₎

KMO Metzner–Ottoconstant(Eq.(5)) m Henry’sconstant

nost flow indexfromthemodelof Ostwald–de-Waele (Eq.(14))

nw flowindexfromthemodelofWilliamson–Cross(Eq.

(15))

N rotationalimpellerspeed(s−1₎

P/Vl powerdissipatedperunitofvolume(W/m3) Ps pressureinthesystem(Pa)

Qg gasflowrate(m3_/s) t time(s)

tw time parameter from the model of Williamson–Cross(Eq.(15))(s)

T_l temperatureoftheliquidphase(K) Tt vesseldiameter(m)

Ug superficialgasvelocity(m/s) V tankvolume(m3₎

Vl liquidtankvolume(m3) Greekletters

ε meanstandarddeviation(Eq.(50)) ˙ shearrate(s−1₎

˙ o referenceshearrate(s−1₎

˙ av averageshearratedefinedfromtheMetzner–Otto

concept(Eq.(4))(s−1₎  dynamicviscosity(Pas) a apparentviscosity(Pas)

o referenceapparentviscosityparameterdefinedat referenceshearrate(Pas)

w viscosity parameter from the model of Williamson–Cross(Eq.(15))(Pas)



cinematicviscosity(Pas)  density(kg/m3₎

 surfacetension(N/m) Dimensionlessnumbers

Fr Froudenumber(Eq.(27))

kla* dimensionlessvolumetricmasstransfercoefficient (Eq.(25))

* dimensionlessviscosity(Eq.(29))

i dimensionlessnumberdeducedfromthetheoryof similarity

* dimensionlessdensity(Eq.(28)) * dimensionlesssurfacetension(Eq.(30))

Sc Schmidt number defined according to gas phase properties(Eq.(31))

t∗

w dimensionless time number issued from the Williamson–Cross’smodel(Eq.(45))

U∗

g dimensionlesssuperficialgasvelocity(Eq.(26)) Subscripts

g gasphase l liquidphase

Another approach is to use correlations with dimensionless groups;contrarytodimensionalcorrelations,theyguaranteefirm basisforprocessscale-up,providedthattheymustbeestablished withrespecttothetheoreticalcontextofthetheoryofsimilarity. Thepioneerwork’sofZlokarnik[7]hasestablishedtherelevant listofinfluencingintensiveparametersandproposedthefollowing dependencebetweendimensionlessnumbers:

(kla)∗=f1

n

_P Qg



∗ ,



_Q g Vl



∗ ,∗_,Sc,Si∗

o

where











(kla)∗=kla·



 l g2



1/3 ,



_P Qg



∗ =



_P Qg



·[l·(l·g)2/3] −1 , ∗_=·[ l·(4_l·g) 1/3 ]−1,



_Q g Vl



∗ =



_Q g Vl



·



 l g2



1/3 (2)

Notethat,inEq.(2),ScistheSchmidtnumberandSi*isamaterial dimensionlessparameterwhichdescribescoalescencebehaviour ofsolutions(i.e.ionicstrength,electricalchargeofions,...).Thanks tothis approach, Zlokarnik[7] couldrigorously distinguish dif-ferent process relationships depending whether the system is coalescentornon-coalescent.

Fewyearslater,Judat[8]hascriticallyexaminedtheexisting publicationsongas–liquidmasstransfer(coalescingsystems)in stirredvessels.Descriptionofexperimentaldatawiththeaidof intensiveparametershasleadedthisauthorto(±30%deviation): (kla)∗=9.8×10−5· (P/Vl)

∗0.40

B−0.6_+0.81×10−0.65/B (3) whereB=(Qg/T2

t)·(l·g)−1/3,theothersnumbersbeingdefined asin[7].Judat[8]hasthenshownthatamonoparametric represen-tationof(kla)*versus(P/Vl)*isinadequate,andthatonlya-space representationcontainingbothdimensionlesspowerperunit vol-umeandsuperficialgasvelocitycansatisfactorilycorrelate(kla)*. Thisauthorhasalsoputforwardthatanother-space, contain-ingnonintensiveparameters (rotationalimpeller speedinstead ofpowerperunitliquid volume)couldbeusedtodescribethe measuresof(k_la)*,butthisi-spaceislargerthanthepreviousone. Fewauthors(forexample[9,10])haveconservedthe dimen-sionlessgroup(kla)∗

=kla·(l/g2₎1/3_{definedbythesetwopioneer} works for modelling absorption processes. Most of them have adopted,withorwithouttheoreticalbackgrounds,others defini-tionsformakingdimensionlesskla[11–17].Theyincludenotably a modified Sherwood number (kla·T2

t/D) or Stanton number (kla·Vl/Qg)[3].

Whenshear-thinningfluidsareinvolved,twoadditional ques-tions arise unfortunately in a point of view of the theory of similarity:

-Howshouldweproceedtoguaranteethattheresultsobtained withNewtonianfluidscanbeextendedtothesenon-Newtonian liquids?Thespatialdistributionofliquidviscosityinthetank,due toitsdependencywithshearrates,constitutesamajordifficulty withregardtothechoiceofarepresentativeviscosity.

Table1

Dimensionsoftheexperimentalsetup.

Tanksize Tt=0.212m Ht=0.316m V=10L Hl=0.212m Vl=7.4L Baffles wb=Tt/10(width,inm)

bb=Tt/50(distancefromwalls,inm)

Impeller(six-concave-blade turbine)

D=0.4·Tt(impellerdiameter,inm)

Ds=3·Tt/4(diskdiameter,inm)

C=Tt/4(clearancefromthebottom,inm)

b=D/5(bladeheight,inm) w=0.5D/5(bladewidth,inm) l=D/4(bladelength,inm) Sparger d=D(spargerdiameter,inm)

-Howshouldweproceedtoguaranteethattheresultsobtainedat lab-scalewillbealsoscalableatindustrialscale?

Untilnow,mostofworksencounteredintheliteraturedidnot takecareaboutthesequestions:theysimplyconsistinreplacing theNewtonianviscositybyanapparentviscositydefinedfromthe Ostwald–de-Waele’smodelinwhichanaverageshearrateis con-sideredaccordingtothewell-knownconceptofMetzner–Otto:

˙ av=KMO·N (4)

where theconstant KMO dependsontheagitationsystem.Such choiceofapparentviscosityisinmanycasesquestionable,in par-ticularwhen(i)theflowregimeisnotlaminar(theuseofEq.(4)

becomesthenquitehaphazard)and(ii)therheologicalbehaviour offluidscannotbedescribedinthewholerangeofshearratesby theOstwald–de-Waele’smodel.

Thepresentpaperaimsatrigorouslyansweringthesetwo lat-terquestions,startingfromthetheoreticalbackgroundunderlying thedimensionalanalysisandextendingittothecasesofvariable materialproperties.Moreaccurately,theobjectiveistoshowhow toproceed:(i)toconstructacompletelistofrelevantparameters abletoconsidervariablerheologicalparameters,andconsequently (ii)toelaborate,withoutpitfalls,asetofdimensionlessnumbers characterizingallthefactorsgoverningabsorptionratecoefficients (kla)inanaeratedstirredtankwherepurelyviscousfluidswith orwithoutshear-thinningpropertiesareinvolved.Tosupportthis theoreticalapproach,asetofexperimentswascarriedoutto mea-sureklainastirredtankaeratedinvolume.Differentoperating conditions(rotationalimpellerspeed,gasflowrate)werecovered aswellasvariousfluids(sevenpurelyviscousfluidsofwhichfour haveshear-thinningproperties).

2. Materialsandmethods 2.1. Experimentalset-up

AsshowninFig.1,theexperimentalset-upconsistedofa cylin-dricalPMMAvesselof10Lwithacurvedbottom.Itwasequipped withasquaredoublejacket(27.2cm×27.2cm)andfourbafflesin stainlesssteelmountedperpendiculartothevesselwall.Table1

presentsthegeometricaldetailsofthetank.Theagitationsystem wascomposedofahome-madesix-concave-blades diskturbine whichdimensions wererespectfulfor theonesimplementedin commercial CD6Chemineer®

impellers. Therotational impeller speed(N)wasregulatedbyusinganelectricalmotor(Ikavisc MR-D1 Messrührer, Janke &Kunkel, Ika®_{), and varied from200to} 1000rpm.

Gas(airornitrogen)wasfedintothetankusingaringsparger withadiameterequaltotheimpellerdiameter,asrecommendedby

[18].Thelatterwascomposedby20holesof0.5mmindiameter.

Thespargerwaslocated30mmfromthebottomofthetank,in theaxisoftheimpeller.Thegasflowrate(Qg)wasregulatedby using amanometer (Samson® _{4708-1155) and measuredwith} avolumetric flowmeter(Brooks® _{R2-25-C)withanaccuracyof} 0.05L/min.Rangedfrom0.33to3.33L/min,thesevaluesremained quitenarrowwhencomparedtotheavailableliterature,theywere initiallyimposedbytheapplicationunderlyingthiswork(namely the AutothermalThermophilic AerobicDigestion of sludge, see

[19]).Intermsofgas–liquidregime,itcanbenoticed(visual obser-vations) thattheoperating conditions undertest (N,Qg, fluids) leadedtoacompletedispersionregime,meaningthusthat bub-bleswerealmostuniformlydistributedthroughoutthetank.One exceptionwasforN=200rpmwheretheloadingregimetookplace (presenceofbubblesonlyintheupperpartofthetank).

2.2. Methodsofoverallvolumetricgas–liquidmasstransfer coefficientmeasurement

2.2.1. Standarddynamicmethod

For implementing the standard dynamic method,the liquid phase wasdeoxygenated by flushingwithnitrogen.Then, after replacingnitrogenbyair,thevariationindissolvedoxygen con-centrationswithtimewasmeasureduntilreachingthesaturation. Forthat,twoprobes(InPro-6050,Mettler-Toledo®_)andan acqui-sitioncardwereimplemented,aswellastheLabView®

software fordataacquisition.Thepositionsoftheprobesarerepresentedin

Fig.1:theyarelocatedverticallyat4cmand18cmabovethe bot-tomofthevessel,andhorizontallyat1.5cmfromwalls.Assuming awellmixedliquidphase,themassbalanceindissolvedoxygen concentrationisgivenby

dCO₂

dt =Kl·a·(C ∗

O2−CO2) (5)

whereKlistheoverallmasstransfercoefficientintheliquidside andaistheinterfacialareabetweengasandliquidphases.The two-filmtheoryofLewisandWhitman[20]assumesthatKlistheresult oftwolocalmasstransfercoefficients(klandkg):

1 Kl= 1 kl+ 1 m·kg (6)

where m is the Henry’s constant (m=H/Ps). The solubility of oxygen is low: H is equal to 4.05×109_{Pa (corresponding to} C*=9.09mgL−1_{)indeionisedwaterat293Kinequilibriumwith} airunderatmospheric pressure.So,alltheresistancetooxygen masstransferislocatedintheliquidfilm,leadingtoKl≈kl.

Asaconsequence,thevolumetricgas–liquidmasstransfer coef-ficient,kla,canbedirectlydeducedfromtheslope ofthecurve relatingln(C∗

O₂−CO2)totime,obtainedwhenintegratingEq.(5).

Thedynamicsoftheoxygenprobecanbedescribedusinga first-orderdifferentialequation[21]as:

dCp dt =

1

tp(CO2−Cp) (7)

whereCpisthedissolvedoxygenconcentrationinsidetheprobe. Thetimeconstantoftheoxygenprobe,tp,wasmeasuredusinga methodbasedonprobe responsetonegativeoxygensteps[22]

andfoundequal to16s.Thislattervalueremainedsmallwhen comparedtomasstransfercharacteristictimes,1/k_la.Asthe tem-peratureTlslightlyvaried(20±3◦C)withpowerdissipation,the usualtemperaturecorrectionwasapplied[23]:

Fig.1. Experimentalset-up.

Foragivenoperatingcondition(NandQg),themeanklainthetank wascalculatedbyaveragingthevaluesmeasuredbythetwoprobes andforthethreeruns(N′_=3),suchas:

kla=<kla>= 1 N′ ·

X

N′ |klatop+ klabottom| 2 (9)

Theaxialhomogeneity(h)ofvolumetricgas–liquidmasstransfer coefficientswasalsoevaluatedbyusingthecriterionhdefinedas follows[24]: h= 1 N′′ ·

X

N′′ |klatop−klabottom| <kla> (10)

whereN′′_{wasthenumberofexperiments(N}′′_{=60foreachliquid} phase).

2.2.2. Chemicalmethod

Whenapplyingthelatterdynamicmethodinviscousfluids,the impactoftheprobedynamicsandoftheliquidfilminfrontofthe membraneontheprobecannomorebeignored,aspossibly bias-ingthemeasurementsofkla[25].Inaddition,insuchfluids,the gashold-upstructureisknowntobeverydifferentfromtheone observedinwaterandotherlow-viscosityliquids:manytiny bub-blesappeartoaccumulateduringaerationandcirculatewiththe liquidwhilelargebubblesarealsoobserved(bimodalbubble popu-lation).Thesetinybubblescanactivelycontributetomasstransfer, dependingwhethertheyareinequilibriumwiththelevelof dis-solvedsoluteintheliquidphase(highresidencetimes)ornot[26]. Forthesereasons,ithasbeenchosentoimplementasecond methodforklameasurement.Itwillthenenabletotestthe valid-ity and accuracy of the dynamic method in the viscous fluids involved.Thisalternativemethodwasthechemicalmethod devel-opedby[27],basedonamassbalanceonsodiumsulphite(Na2SO3) concentrationsduringagivenaerationtime.Nitrogenwasfirstly injectedintotheliquidphaseinordertoeliminatethedissolved oxygenpresentinthetank.Whentheconcentrationofdissolved oxygenreachednearlyzero,anadequateamountofNa2SO3was introduced;air wasthenintroducedin thetankandwillreact,

duringanaerationtimetaeration,withthesmallquantityofNa2SO3 introduced:

Na2SO3+1

2O2→Na2SO4 (11)

Themass ofNa2SO3 toinitially introduce (mt)must bechosen carefully,asitshouldenabletokeepazerooxygenconcentration duringtheaerationtime(taeration)whileavoidinganexcessiveuse ofNa2SO3.Indeed,itisimportanttoguarantythatthecoalescing propertiesoftheliquidphasewerenotaffectedbythepresence ofNa2SO3.Agoodcompromisewastomaintainaninitial concen-trationbelow0.5g/L(i.e.mt<3.5gwithVl=7.4L)[25].Notethat tominimizemt,itwasalsopossibletoplayontheaerationtime (taeration),whichwasheretypicallyrangedfrom1.5minto9min. Theoptimizationofbothmt andtaerationwasmadeeasierbythe factthattheordersofmagnitudeofkLawereknownthankstothe measurementsissuedfromthedynamicmethod.

Whensuch conditionsarerespected,Painmanakuletal.[27]

haveshownthattheoverallvolumetricgas–liquidmasstransfer coefficientcanbededucedfrom:

kla=(1/2)(MO2/MNa2SO3)·(mt−mr) taeration·Vl·C∗

O₂

(12) wheremt isthetotalmassofNa2SO3 initiallyintroduced,mr is themassofNa2SO3remaininginthetankafteranaerationperiod taeration,andCO∗2istheconcentrationindissolvedoxygenat

satura-tion.Forglycerinesolutions,C∗

O2wasbydefaultconsideredequalto

8.8mgL−1_{asindeionisedwaterat20}◦_{C.Anidenticalassumption} wasmadeforCMCandxanthangumsolutionsinagreementwith thedatareportedby[28]whoshowedthat,intherangeof con-centrationshereinvolved,nomajorvariationofC∗

O₂ occurswhen comparedtowater.

Atlast,foreachcondition,threesamples(10mL)weretakenin thetank,immediatelymixedwith10mLofstandardiodinereagent at0.12equiv./L.Thetitrationofthesesampleswithasodium thio-sulphatesolution(0.05equiv./L)andastarchindicator(iodometry titration)gaveaccesstotheconcentrationofNa2SO3remainingin thetankafteranaerationperiodtaeration,andthustomr.

2.2.3. Surface-andvolume-aerations

Theoverallvolumetricgas–liquidmasstransfercoefficient mea-suredbythelattermethodsisinrealitytheglobalresultofboth contributions:

- thesurface-aeration:itcorrespondstothemasstransferoccurring atthefreesurfacewhichimportancedependsstronglyonthe surfacemotion.Italsoincludestheaerationassociatedwiththe bubblesentrainedfromthesurfaceintotheliquidbulk; -thevolume-aeration:itisinducedbythebubblesgeneratedatthe

spargerdirectlyinsidetheliquidbulk. Thiscanbeexpressedsuchas:

kla|t=kla|surf+kla|vol (13)

Somemeasurementsaremadewithmechanicalagitationand with-outbubblingatthesparger.Theyenabletogetanideaoftherelative importanceofeachcontribution.

2.3. Fluids

Theapplicationunderlyingthisstudydealtwithinvestigations onaerationperformancesinanAutothermalThermophilic Aero-bicDigestion(ATAD)processfortreatingsludgeissuedfromwaste watertreatmentplant[19].Assludgewasaverycomplex mate-rial,itwasdecidedinafirststeptoworkwithmodelfluidsinstead ofsludge.Theirformulationwaschosensoastoobtain rheologi-calbehavioursasclosetosludgeaspossible,inparticularinterms ofshear-thinningproperties.Forthesereasons,andwithregard toliterature,aqueoussolutionsofcarboxymethylcellulose(CMC) and xanthangumwereselected.Theuseoftwo typesoffluids offeredtheadvantagetocoverawiderrangeofcombinationsof flowandconsistencyindexes.Toensuretheirstabilityintime (dur-ingseveraldays),NaClwasaddedat0.1%(w/v)tothesolutions ofxanthangumsolutions[29]andNaHCO3(0.1mol/L)+Na2CO3, 10H2O(0.1mol/L)tothesolutionsofCMC[30].Various concentra-tionsofCMCandxanthangumweretested,andfinallyconverged towardsthefollowingones:4and6g/LforCMC,and1and2g/Lfor xanthangum.Inadditiontothesenon-Newtonianfluids,deionised waterandtwoaqueoussolutionsofglycerine(50%and70%,v/v) werealsochosenasNewtonianfluids.

The rheology of these fluids was measured, at 20◦_{C, by a} rotationalstress-controlledrheometrer(MCR500,PAARPhysica®₎ equippedacone-platedevice(50mmindiameter,3degreeincone angle).Theirdensityandsurfacetensionweredeterminedusing adensimeterERTCO®_{andatensiometerinvolvingtheWilhelmy} plate method (3S GBX®_{).The physical and rheological} proper-tiesofbothNewtonianandnon-Newtonianfluidsarecollectedin

Table2.

Therheologicalbehavioursofthenon-Newtonianfluidswere firstlycharacterisedbymeasuringthevariationofshearstress() orapparentviscosity(a)asafunctionofshearrates(˙ )which range variedfrom0.1 to3000s−1_{.Whencomparing thecurves} obtainedforincreasinganddecreasingshearrates,nodifference wasobservedwhateverthenon-Newtonianfluids:nohysteresis phenomenonthenexisted.ShowninFig.2,therheogramsobtained (issued fromseveral trials)clearly illustrate theshear-thinning propertiesofthesefluids.Foreachfluid,theyieldstresswasalso determined,byapplyingthemethodproposedby[31]which con-sisted in oscillating stress sweep testsat a constant frequency (1Hz);thedynamicyieldstresswasthendefinedattheendofthe linearviscoelasticregion,namelyfromtheabscissa correspond-ingtotheintersectionpointbetweenthetangenttotheplateau andthetangenttotheinflexionpointofthecurvelinkingcomplex modulusandshearstress.Dependingonthefluid,theyieldstress

0.001 0.01 0.1 1 10 100 0.1 1 10 100 1000 10000 shear rate (s-1) v is co si ty ( P a.s ) Exp. Ostwald Williamson

(a)

0.001 0.01 0.1 1 10 100 0.1 1 10 100 1000 10000 shear rate(s-1) v is co si ty ( P a. s) Exp. Ostwald Williamson

(b)

0.001 0.01 0.1 1 10 100 0.1 1 10 100 1000 10000 shear rate (s-1) v is co si ty ( P a. s) Exp. Ostwald Williamson

(c)

0.001 0.01 0.1 1 10 100 0.1 1 10 100 1000 10000 shear rate (s-1) v is co si ty ( P a. s) Exp. Ostwald Williamson

(d)

Fig.2.Rheograms:(a)xanthangumat1g/L,(b)xanthangumat2g/L,(c)CMCat 4g/L,and(d)CMCat6g/L.

wasfoundtovarybetween0.1and1Pa,andremainedthus neg-ligible.Tocompletetherheologicalcharacterisationofthefluids, theviscoelasticpropertieswereinvestigated,bymeansofcreeping tests,relaxationtestsand/ordynamicoscillatingmeasurements.In

Table2

Physicalandrheologicalpropertiesofthefluids.

l(kg/m3) l(Pas) (N/m) Ostwald–de-Waele’smodel: Williamson–Cross’smodel:

K(Pasn_{) n} ost w(Pas) tw(s) nw Air 1.18 1.85×10−5 _{– – – – – –} Newtonianfluids Deionisedwater 998 0.001 0.0728 – – – – – Glycerine50%[Gly50] 1145 0.0109 0.0456 – – – – – Glycerine70%[Gly70] 1195 0.0349 0.0503 – – – – – Non-Newtonianfluids CMC4g/L[CMC4] 997 – 0.0717 0.1914 0.642 0.091 0.029 0.546 CMC6g/L[CMC6] 1006 – 0.0771 0.9470 0.527 0.948 0.844 0.514 Xanthangum1g/L[XG1] 1013 – 0.0753 0.0890 0.543 1.885 57.85 0.411 Xanthangum2g/L[XG2] 1032 – 0.0767 0.5084 0.373 29.505 150.36 0.281

therangeofshearrateinvestigated,nomajorelasticpropertywas highlighted.

Basedonthesefindings,somerheologicalmodelswerechosen tomathematicallydescribethevariationofapparentviscosity(a) withshear ratesrangingfrom0.1 to3000s−1_{.Amongthelarge} variabilityavailableintheliterature,twomodelswereselected: - theOstwald–de-Waele’smodel

a=K· ˙ nost−1 ₍₁₄₎

whereKandnostarerespectivelytheconsistencyandflowindexes respectively.

-theWilliamson–Cross’smodel

a= w

1+ (tw· ˙ )1−nw (15)

where w is a parameter describing a pseudo-Newtonian behaviourforthesmallestshearrates,twisa timeparameter characterizingthetransitionbetweenthe“pseudo-Newtonian” andpurelyshear-thinningbehaviours,andnwistheconsistency index.

Thevalues of K,nost,w,tw,nw are reportedin Table2 for eachfluid;theyhavebeenobtainedbymulti-parameter optimiza-tionsusingthesoftwareAuto2fit®_{.Intermsofconsistencyindex,} themostshear-thinningfluidappearedtobethesolutionof xan-thangumat2g/L.Basedonthelatterparameters,theapparent viscositiespredictedbyEqs.(14)and(15)werecomparedtothe experimental ones in Fig. 2. For all the fluids, a better agree-mentwithexperimentswasobtainedwiththeWilliamson–Cross’s model,insofarasitenablestodescribethemostfaithfullypossible theshapeoftherheogramsoverthewholerangeofshearrates.This demonstratesthatthefluidsundertestwerenotshear-thinningon thewholerangeofshearratesinvestigated,threeparametersbeing requiredtowelldescribetheirbehaviour.

3. DimensionalanalysisforNewtonianandnon-Newtonian Fluids

3.1. Generationofi-setsgoverningaerationprocessfor Newtonianfluids

Inthepresentstirredtank, bubblesweredirectlygenerating insidetheliquidbulkbymeansofagassparger.Surface-aeration wasofcoursepresent,butitscontributionremainedminorwhen comparedtovolume-aeration(seeSection4).Consequently,the overallvolumetricgas–liquidmasstransfercoefficient,kla,canbe consideredasthetractablequantitywhich issignificantly influ-encedbyaerationconditions:itwillbethustakenastargetvariable. Remindthat such choiceisbased onthefollowing relationship

describingthephysicalabsorptionprocessaccordingtothe two-filmtheory:

G Vl

= kl· a· 1C or kl· a= G

Vl·1C (16)

whereG/Vlisthemassthroughputperunitvolumeofliquidand 1Cisacharacteristicconcentrationdifference.Eq.(16)implicitly assumesthat(i)theintensityofgas–liquidcontactingissohigh thataquasi-uniformsystemisproduced,(ii)thegas-phasemass coefficientkgisnegligiblewhencomparedtokl(lowsolublegases), (iii)theabsorptionrateattheinterfaceisextremelyfast,resultingin anequilibriumconcentrationofthedissolvedgasattheinterfaceC* (e.g.1C=C*− C).Hence,theestablishmentofthelistofparameters influencingthemainparameterk_lashouldrespectthefollowing rules[7]:(i)klamustbeindependentofallgeometricalparameters (i.e.diametersofstirrerandtank,etc.),(ii)klamustbeindependent ofthematerialparametersofgasphase,and(iii)k_laisanintensive quantitybecauseofitsvolume-relatedformulation.

Aspreviouslymentionedthenumberoftheparameters influ-encingkla,evenperformedinNewtonianliquids,islargeandcan bedecomposedaccordingto:

• Thegeometricparameters(seelegendinTable1),characterizing -thetank:Tt,Hl,curvatureradiusandangle(fortank’sbottom),

...

-theimpeller:D,Ds,C,w,b,l,...

-thesparger:d,number,diameterandshapeofholes,... • Thematerialparameters:

l,l,g,g,s,_D,C* • Theprocessparameters:

g,N,Ug= Qg ·T2

t/4

,Tl,Ps,...

Notethat,inthepresentstudy,alltheexperimentswere con-ductedatroomtemperatureandatmosphericpressure,inducing thusthattemperatureandabsolutepressurewillnotbelisted.The geometryofthetankwasalsounchanged,aswellasthetypeand positionofbothspargerandimpeller.Asaconsequence,thelist ofindividualphysicalquantitiescouldbereduced:Table3shows thedimensionalmatrixobtainedwiththereducedlistofrelevant parameters.

Itisvoluntarilychosentolistthenon-intensiveparametersN andDinsteadofpowerperunitofliquidvolume(P/Vl).Themain motivationisthat P/Vlisan intermediaryvariablewhichis not alwaysavailable(inparticularatindustrialscale),andthususing suchintensivevariablewouldrestrictthefieldofapplicationsof thefinaldimensionalcorrelationwhichwillbeestablished.

InTable3,thecolumnsareassigned totheindividual phys-icalquantities and therows to theexponents appearing when

Table3

Dimensionalmatrixoftheinfluencingparameters(Newtonianfluids).

Corematrix Remnantmatrix

g g g kla Ug N D l l  D

Mass,M(kg) 1 1 0 0 0 0 0 1 1 1 0

Length,L(m) −3 −1 1 0 1 0 1 −3 −1 0 2

Time,T(s) 0 −1 −2 −1 −1 −1 0 0 −1 −2 −1

each quantityis expressedasan appropriatepower productof thebasedimensions(mass,length,time).Thistableisstructured inacorematrixandaresidualmatrix.Thecorematrixregroups theindividualphysicalquantitiesputforwardbytheusertoform thedimensionlessratiosfromotherindividualphysicalquantities (namely kla,Ug, N, D,l,l,, D).Depending onthe individ-ualphysicalquantitiesassignedinthecorematrix,severalsetof dimensionlessnumbersicanbeobtained.Ithasbeenshown[32]

thatallthei-setsobtainedfromasingleandidenticalrelevance listareequivalenttoeachotherfromapointofviewof dimen-sionalanalysis,andcanbemutuallytransformedatleisure.The finalformforthei-setshouldbelaiddownbytheusersoastobe the“best”suitableforevaluatingandpresentingthe experimen-taldata.Contrarytowhatcommonlyfoundintheliterature,the gaspropertiesgandg(andnottheliquidproperties)werehere chosenasindividualphysicalquantities;theassociated motiva-tionwastogeneratedimensionlessnumbersdependentofasingle influencingparameter,thegasphasebeingkeptunchangedinthis study(air).

Generatingthesetofdimensionlessnumbers(andpossiblytheir futuretransformation)representsanextremelyeasyundertaking whencomparedtothedrawingupofareliableandasaccurateas possiblerelevancelist;thiscanbemadebymatrixtransformation. Thestartingpointconsistsincarryingouttheso-calledGaussian algorithminordertoobtainaunitcorematrixbylinear transfor-mations(zero-freemaindiagonal,beneathitzeros).Table4reports theunit corematrixassociated withthedimensionalmatrix of

Table3.Theanalysisoftheunitcorematrixleadstothe dimen-sionlessratios.Indeed,therowsofresidualmatrixareassignedto theexponentswithwhomtheelementsofthecorematrixappear whentheindividualphysicalquantitiesoftheresidualmatrixare expressedasanappropriatepowerproductofthephysical quanti-tiesofthecorematrix.Literature[33]offersdetailedexamplesof howtohandlematrixtransformationandrecombinationinorder toquicklyobtainthecompletesetofdimensionlessnumbers.This aspectwillbethenonlybrieflydescribedinthispaper.For New-tonianfluids,thematrixanalysisleadstotheeightdimensionless numbers,1to8: 1= kla g1/3·−1/3g ·g2/3 (17) 2= Ug g−1/3·1/3g ·g1/3 (18) 3= N g1/3·−1/3g ·g2/3 (19) 4= D g−2/3·2/3g ·g−1/3 (20) 5= l g (21) 6= l g (22) 7=  g−1/3· 4/3g · g1/3 (23) 8= D g−1·g (24) Notethat1isnothingotherthatthedimensionlessvolumetric masstransfercoefficientdefinedby[7],thecombinationof4with (3)2_{leadstotheFroudenumber,and8istheinverseofaSchmidt} numberdefinedaccordingtogasphaseproperties.Byintroducing thegaskinematicviscosity(g=g/g)andgivingexplicit nota-tions,thelatternumbersbecome:

1=kla∗_=kla·



g g2



1/3 (25) 2=U∗ g= Ug (g·g)1/3 (26) 3= Fr=N 2_·D g (27) 5=∗₌ l g (28) 6=∗₌ l g (29) 7=∗₌  (3 g·4g·g) 1/3 (30) 8=Sc= g D (31)

Thus,atagiventemperature,underagivenpressure,forthe geom-etryoftheaeratedstirredtankundertest(inparticularforHl/Tt=1, D/Tt=0.4, Ds/Tt=0.75and fortheothergeometricalratios char-acteristics of thesystem), the dimensionalanalysis states that, when Newtonian fluids are involved, dimensionless volumetric masstransfercoefficient(kla*)ispotentiallyaffectedbysix dimen-sionlessnumbers,respectivelydescribingtheeffectsofsuperficial gasvelocity,rotationalimpellerspeed,liquiddensity,Newtonian viscosity,liquidsurfacetensionandoxygendiffusivity:

kla∗_=f



U∗ g,Fr,∗,∗,∗,Sc

(32) Table4

Unitcorematrixobtainedbylineartransformationsofdimensionalmatrix(Newtonianfluids). Corematrix Residualmatrix

g g g kla Ug N D l l  D M+T+2A 1 0 0 1 3 − 1 3 1 3 − 2 3 1 0 − 1 3 −1 3M+L+T+A 0 1 0 −1₃ 1₃ −1₃ 2₃ 0 1 4 3 1 A=−1₃×(3M+L+2T) 0 0 1 2 3 13 23 −13 0 0 13 0

Suchformulationofdimensionlessnumbersofferstheadvantageto enabletheimpactofallinfluencingparameterstobestudied sepa-rately,orinotherswordseachdimensionlessnumberisdefinedfor asingleinfluencingvariable.Thisisnotthecaseintheliterature where,forexample,mostoftheauthorsusedtheaeration num-ber,Na=Flg=Qg/(N·D3_{),inwhichtheeffectofgasflowrateisnot} decoupledfromtheoneofrotationalimpellerspeed.

Atthisstate,thedependenceofEq.(32)isallthatcanbe con-tributedbythetheoryofsimilarity.Themathematicalexpression forthefunctionf,namelyfortheprocessrelationship,hastobe determinedexperimentally.

3.2. Extensionofthetheoryofsimilaritytothecasesofvariable materialproperties

Whenusingthedimensionalanalysistomodelsystemanswers, itisgenerallyassumedthatthematerialpropertiesremain unal-teredinthecourseoftheprocess.However,theinvariabilityof materialpropertiescannotbeassumedwhennon-Newtonian flu-idsareinvolved.Indeed,attheleastoneofthematerialproperties, theapparentviscosity,cannolongerbeconsideredasaconstant insidethewholevolumeoftheaeratedstirredtank,insofarasthe dependencyofthislatterwiththeshearrates(˙ )generatesa spa-tialdistributionoftheliquidviscosity.Theunderlyingquestion addressedtothedimensionalanalysisisnow:howmustthespace ofdimensionlessnumbers,i,bebuilt inpresenceofsuchvariable materialproperty?

Inthecaseofmaterialswithconstantproperties,nospecial pre-cautionshouldbemadetoguaranteethataprocessrelationship correlatingasetofdimensionlessratiosisalsoapplicabletoanother material.Thisisnottrueformaterialswithvariablesproperties,as demonstratedby[34].Inthiscase,weshouldfirstensureas prior-itythatacertainsimilarityexistsformaterialsinordertoextend therangeofvalidityoftheprocessrelationshiptoother materi-als.Despitethat,thetheoryofsimilarityhaslittlechangedsince itsbeginning,andthedimensionalmodellinginvolvingmaterials withvariablephysicalpropertiesremainstreated,inmostofthe papers,asthecasewithconstantmaterialproperties.Theauthors ignorethenthefactthatthespatio-temporalvariabilityof mate-rialpropertiesinfluencesthecourseoftheprocess!Oneexception isthemodellingofthetransformationprocesseswhere a mate-rialhavingatemperature-dependenceinviscosityissubmittedto heattransfercondition.Mostofattemptsmadetotakeintoaccount thevariabilityofproductpropertiesinthereactorhaveconsisted inaddinganewratioraisedtoa certainexponentto character-izethesystemresponse.Thisratioisdefinedbytheratiobetween theviscositiesatbulktemperatureandatwalltemperature. How-ever,thiskindofenlargementofthesetofdimensionlessratios istheoreticallyvalidonlyforfewlimitedcases,thatistosayonly ifthematerialfunction(hereviscosityversustemperature) sat-isfiesspecificcriteria[34].Inotherwords,suchmethodleadsto biasedpredictionsandthus,cannotbegeneralisedwhenhandling otherfluids. Despite this fact, this ratio remains systematically used,whatevertheproductsinvestigated,inmostofthestudies since[35].

The theoretically consistent way of proceeding has been introducedbyPawlowskiin1971[34]andrememberedby[32,36]. Themethodconsistsinintroducingsomeadditionalparametersin therelevancelistsoastotakeintoaccountthevariationinthe flowdomainofthephysicalproperty,noteds(forexample viscos-ity),asafunctionofaparameternotedp(forexampletemperature orshearrate);s(p)iscalledthematerialfunction(forexample(T) or(˙ )).Thisimpliesthatthei-spacegoverningtheprocesswill beextendedincomparison tofluidshavingconstantproperties. Theguidelinestointroducetherightnumberofadditional dimen-sionalparametersaredetailedin[34].Hence,whendealingwitha

material function s(p) which is apparent viscosity a(˙ ), the Pawlowski’swork[34]canbesummedup,asfollows:

• Firstly,allthedimensionalparametersdefinedintherelevantlist withfluidshavingconstantproperties(Newtoniancase)should beconserved,exceptforthevariablephysicalpropertiesin ques-tion(hereapparentviscositya(˙ )).

• Secondly,theNewtonianviscosityshouldbereplacedbya refer-enceapparentviscosity,o,calculatedatareferenceshearrate, ˙ o.Itisimportanttopointoutthatanyvalueforthereference shearratecanbechosen.

• Thirdly,thereferenceshearrateshouldbeaddedinthelistof relevantparameters.Atthisstage,wecanpointoutthatsome exceptionstothisruleexist.Indeed,ithasbeendemonstrated thataddingthereferencepointisnotnecessarywhenthe mate-rialfunctions(p),herea(˙ ),canbedescribedbythefollowing familyofcurves:

s(p)=(A+B·p)c or s(p)=exp(A+B·p) (33) whereA,BandCarethreeindependentconstants.

• Finally, a set of additional dimensionless numbers shouldbe addedintherelevant(dimensional)listtotakeintoaccountthe dependencyofmaterialfunction.Thesedimensionless parame-ters,calledrheol,correspondtoallthedimensionlessratiosi whichappearintheexpressionofthefunctionu,exceptforthe ratio ˙ /˙ 0: {rheol}=



{i}suchasu=(˙ − ˙ 0) 1 a(˙ 0)·

h

_da d˙

i

˙ =˙ o =g



˙ ˙ 0;{i}





(34) Whenintegrating the previousguidelines,the listof the influ-encingparametersestablishedforNewtonianfluidsbecomesfor non-Newtonianfluids:

{kla,g,g,g,Ug,N,D,l,,D,a(˙ o), ˙ o,rheol} (35) Thenextstepistofindthetypeofmaterialfunctiondescribingthe rheologicalbehaviourofviscousfluidshavingshear-thinning prop-erties.AspresentedinSection2,twomodelsarewelladaptedfor theinvestigatedfluids:theOstwald–de-Waele’smodel(Eq.(14)) andtheWilliamson–Cross’smodel(Eq.(15)).Asthesemodelsare abletodescribethevariationofviscositywithshearrateforallthe aqueoussolutionofCMCandxanthangum,eachofthemconstitute onepossiblematerialfunction.

3.2.1. CaseNo.1:Ostwald–de-Waele’smodel.

Whenthematerialfunction correspondstothe Ostwald–de-Waele’smodel,thefunctionucanbeexpressedas:

u=



_˙ ˙ 0−1



(nost−1) (36) Consequently, {rheol}={nost} (37)

TheOstwald–de-Waele’smodel, definingasa=K· ˙ nost−1 _(Eq.

(14)),verifiesEq.(33),implyingthusthatthereferenceshearrate, ˙ o,canberemovedfromtherelevancelist.Asaconsequence,for purelyviscousfluidshavingshear-thinningproperties,the addi-tionalparametersisrestrictedtonostandtheNewtonianviscosity isreplaced bya(˙ o).Eq.(32)establishedfor Newtonianfluids becomesthen: kla∗_=g



U∗_g,Fr,∗_,∗_,Sc,∗₌a( o) g ,nost



(38)

Table5

Dimensionalresultsforkla(expressedins−1):Newtonianandnon-Newtonianfluids.

N(rpm) Qg(L/min) Newtonianfluids Non-Newtonianfluids

Water Gly50 Gly70 CMC4 CMC6 XG1 XG2

200 0.33 9.05×10−4 _{– – – – – –} 0.9 1.91×10−3 _{– –– – – – –} 1.6 3.35×10−3 _7.75×10−4 _3.13×10−4 _1.64×10−3 _1.26×10−3 _1.55×10−3 _1.31×10−3 2.33 4.58×10−3 _1.05×10−3 _4.22×10−4 _2.08×10−3 _1.85×10−3 _2.06×10−3 _1.47×10−3 3 5.33×10−3 _1.45×10−3 _5.15×10−4 _2.58×10−3 _2.01×10−3 _2.68×10−3 _2.16×10−3 3.33 5.88×10−3 _1.77×10−3 _5.75×10−4 _2.82×10−3 _2.3×10−3 _2.85×10−3 _2.56×10−3 400 0.33 2.31×10−3 _{– – – – – –} 0.9 4.91×10−3 _{– – – – – –} 1.6 8.69×10−3 _2.90×10−3 _6.82×10−4 _3.68×10−3 _2.54×10−3 _4.83×10−3 _3.61×10−3 2.33 1.12×10−2 _3.50×10−3 _8.00×10−7 _4.43×10−3 _4.40×10−3 _5.93×10−3 _4.81×10−3 3 1.29×10−2 _4.28×10−3 _9.99×10−4 _4.98×10−3 _4.98×10−3 _6.59×10−3 _6.51×10−3 3.33 1.31×10−2 _4.63×10−3 _1.19×10−3 _5.19×10−3 _5.53×10−3 _7×10×10−3 _8.01×10−3 600 0.33 4.30×10−3 _{– – – – – –} 0.9 9.19×10−3 _{– – – – – –} 1.6 1.62×10−2 _4.72×10−3 _1.38×10−3 _7.51×10−3 _3.99×10−3 _9.52×10−3 _7.68×10−3 2.33 1.82×10−2 _5.73×10−3 _1.77×10−3 _8.60×10−3 _6.90×10−3 _1.13×10−2 _9.57×10−3 3 2.06×10−2 _6.39×10−3 _2.25×10−3 _9.23×10−3 _8.20×10−3 _1.30×10−2 _1.11×10−2 3.33 2.26×10−2 _6.78×10−3 _2.55×10−3 _9.44×10−3 _9.05×10−3 _1.39×10−2 _1.20×10−2 800 0.33 4.94×10−3 _{– – – – – –} 0.9 1.39×10−2 _{– – – – – –} 1.6 2.26×10−2 _6.87×10−3 _2.35×10−3 _1.14×10−2 _6.40×10−3 _1.29×10−2 _1.08×10−2 2.33 2.68×10−2 _7.61×10−3 _3.31×10−3 _1.32×10−2 _8.30×10−3 _1.51×10−2 _1.28×10−2 3 2.81×10−2 _8.14×10−3 _3.97×10−3 _1.45×10−2 _9.17×10−3 _1.79×10−2 _1.43×10−2 3.33 3.21×10−2 _8.45×10−3 _4.46×10−3 _1.48×10−2 _9.75×10−3 _1.87×10−2 _1.57×10−2 1000 0.33 1.05×10−2 _{– – – – – –} 0.9 1.43×10−2 _{– – – – – –} 1.6 2.25×10−2 _8.36×10−3 _4.41×10−3 _1.54×10−2 _8.90×10−3 _1.70×10−2 _1.38×10−2 2.33 2.70×10−2 _8.87×10−3 _5.21×10−3 _1.75×10−2 _1.05×10−2 _2.06×10−2 _1.68×10−2 3 2.84×10−2 _9.18×10−3 _5.74×10−3 _1.88×10−2 _1.17×10−2 _2.36×1010−2 _1.89×10−2 3.33 3.34×10−2 _9.48×10−3 _6.21×10−3 _1.90×10−2 _1.29×10−2 _2.61×10−2 _2.06×10−2

WhendegeneratedtotheNewtoniancase,thematerialfunction associatedwiththeOstwald–de-Waele’smodelleadstonost=1and *=a/gwherea=listheNewtonianviscosity.

3.2.2. CaseNo.2:Williamson–Cross’smodel

When the material function corresponds to the Williamson–Cross’s model, the function u can be expressed as: =



_{˙ − ˙ 0} ˙ 0



·(nw−1)·(tw· ˙ 0) (−nw) 1+(tw· ˙ 0)1−nw (39) Consequently, {rheol}={nw,tw· ˙ 0} (40)

The reference shear rate, ˙ o, should be listed as the Williamson–Cross’smodeldoesnotverifyEq.(33).Finally,thenew variablestoaddintherelevantdimensionallistare:

{˙ o,nw,tw· ˙ 0} (41)

Nevertheless,as ˙ ocantheoreticallytakeanyvalue,itispossible tochoose:

˙ o= 1

tw (42)

Bythisway,thesupplementaryvariablesarerestrictedto:

{tw,nw} (43)

As a consequence, for the fluids described by the Williamson–Cross’s model, the additional parameters are tw andnw,andtheNewtonianviscosityshouldbereplacedby a



1

tw



=0= w

2 (44)

After introducing these variables in the core matrix (Table 3) andapplyingthelineartransformationsfromTable4,thesetof

dimensionlessnumbersdefiningaeration processisenlargedby twodimensionlessnumbers:

9= nw and 10= t∗ w= 1 tw·



_g g2



1/3 (45) Eq.(32)establishedforNewtonianfluidsbecomesthen:

kla∗=h



U∗ g,Fr,∗,∗,Sc,∗= a(1/tw) g ,nw,t ∗ w



(46) Note that theWilliamson–Cross’s model leads toa Newtonian behaviourforparticularvaluesofnwandtw:namelynw=1and tw=1.Inthiscase,l=w/2.

4. Results

4.1. Validationofklameasurements

InTable5arecollectedthesetofexperimentswhichwillserve asdatabaseforthedimensionalanalysis.Itisconstitutedby150 measuresofkla(including70valuesfortheNewtoniancase), car-riedout at five rotational impellerspeeds (200≤N≤1000rpm) and fourflow rates (0.33≤Qg≤3.33L/min),and for several flu-ids(threeNewtonianfluids,fournon-Newtonianfluids).Notethat theseoverallvolumetricmasstransfercoefficientscorrespondto thevaluesobtainedwiththedynamicmethod,andaveragedfrom themeasuresforbothprobesandthreeruns(Eq.(9)).

Whateverthefluids,thecriterionforaxialhomogeneity(defined inEq.(10))hasbeenfoundvaryingfrom6to8%forN=200rpm, andfrom1to3%forN=1000rpm[19,37],thesmallestvaluebeing obtainedfortheprobelocatedatthetopofthetank.Then,no signif-icantspatialheterogeneitytakesplaceinthetank,andthesevalues ofklacanbeconsideredrepresentativeoftheaerationstateinthe wholetank.

kl a (s

-1

)

with physical method

kl a (s

-1

)

with chemical method Water 400 (1.6) Water 400 (3.33) Water 800 (1.6) Water 800 (3.33) Gly50 400 (3.33) Gly50 800 (3.33) XG2 400 (3.33) CMC4 400 (3.33) 0 10-2 2.10-2 3.10-2 0 10-2 2.10-2 3.10-2

Fig.3.Comparisonbetweenchemicalandphysicalmethodsformeasuringkla(the

dottedlinescorrespondtoadeviationof±15%;inthelegend,thefirstnumberisN inrpm,thenumberintobracketsQginL/min).

Fig.3 presents,for somerepresentativecases, a comparison betweenthephysicalandchemicalmethodsformeasuringk_la.A goodagreementbetweenbothmethodsisobserved:thedeviation never exceeds 15% which corresponds to the order of magni-tudeassociatedwiththeexperimentaluncertaintyinthechemical method[27].Asaconsequence,thevaluesofklareportedinTable5 canbeassumedrelevantasvalidatedbytwomethods.

Thecontributionofsurfaceaerationtotheoverallvolumetric gas–liquidmasstransfercoefficienthasbeenestimatedby mea-suringklawithoutbubblingattheringsparger(seeSection2).For 200≤N≤1000rpmand1.6≤Qg≤3.33L/min,thefollowingtrends havebeenobtained[37]:

- forwater,kla





_surf<0.14×kla





_t,

-forglycerineat50%, kla





_surf<0.15×kla





_t,andforglycerineat

70%,kla





_surf <0.23×kla





_t,

-for CMCat4g/L, kla





_surf <0.08×kla





_t,and for CMCat 6g/L,

kla





_surf≪kla





_t,

-forxanthangumat1g/L, kla





_surf<0.17kla





_t,andforxanthan

gumat2g/Lkla





_surf<0.14×kla





_t.

Thisdemonstratesthatthesurfaceaerationremainssmallwhen comparedtovolumeaeration,confirmingthusthatkla|tisthe ade-quatetargetparametertotractinthedimensionlessanalysisfor qualifyingtheaerationstateinthetank.

Whenanalysedindetail[19],Table5pointsthat,foragiven fluid, theoverall volumetric mass transfer coefficients logically increase for increasingrotational impeller speeds and gas flow rates.Therelativecontributionsofgassparging(Qg)and mechan-icalagitation(N)onthevariationsofklaarecomparable,evenif therotationalimpellerspeedplaysamorepronouncedrole.The presentinvestigationsarethenperformedundertheintermediary conditiondefinedby[13,14],namelytheconditionrangedbetween the bubbling-controlling condition (at relatively high gas flow rates)andtheagitation-controllingcondition(atrelativelyhigh rotationalimpellerspeeds).Moreimportantisthemajor reduc-tioninklaobservedinpresenceofviscousfluids(Newtonianand non-Newtonian)whencompared towater.Tobetterappreciate thisphenomenon,thefollowingratiocanbedefined:

R=kla|viscousfluid

kla|water (47)

Forinstance,atQg=3L/minandfor200≤N≤1000rpm,thelatter ratioRvaries:

-from27to32%inglycerineat50%,andfrom9.7to20%inglycerine at70%,

-from43to66%inCMCat4g/L,andfrom38to43%inCMCat6g/L, -from50to83%inxanthangumat1g/Landfrom40to66%in

xanthangumat2g/L.

Thecomparisonof suchvalues ofkla,or anyotherattempts for their modelling, is usually made (see Section 1) by calcu-latingtheapparentviscositybasingonthewell-knownconcept ofMetzner–Ottoand theOstwald–de-Waele’s model (Eq.(14)). Whenapplyingthismethod[19],itispossibleneithertoexplain nortounderstandsatisfactorilytheseresults,confirmingthusthe requirement to perform more consistent investigations on the influencingparameters.

4.2. DimensionlessresultsforNewtonianfluids

The oxygen diffusivity, D, was unknown for the viscous Newtonian and non-Newtonian fluids under test. Indeed, such informationremainsunavailableintheliterature,andtheusual correlations(forexampletheWilke–Changone)cannotbeapplied bylackofsomerequireddata(forexampletheassociation fac-tor of solvent). So, the contributionof Schmidt number in the process relationship (Eq. (32)) cannot be rigorously sought. By default,whatevertheliquidphases,theoxygendiffusivity,D,will beassumedequaltotheoneinwaterat20◦_{C(i.e.to2×10}−9_ms−2_). Inthiscase,theSchmidtnumberSc(definedwithrespecttogas cin-ematicviscosity,Eq.(31))isthenequalto7850.Asaconsequence, itseemsreasonabletoassertthattheprocessrelationship estab-lishedwillbeinsuredforvaluesofSchmidtnumbersclosetothis latter.Inaddition,thevariationofdensityforthefluidsinvestigated isnotimportant(Table2),implyingthusthatthechangein*is weak(847<*<1015).Asaconsequence,forsecuringtheprocess relationship,itischosentoignorethepossiblealterationsof*and ScinEq.(32),leadingto:

kla∗_{= f}′_{Ug,∗ _Fr,∗_,∗_{} (48)} Havingnomechanisticindicationontheformofthef′_{-relation,the} simplestmonomialformislookedfor:

kla∗_=˛·(Fr)a ·(U∗

g)b·(∗)c·(∗)d (49) where ˛, a, b, c and d are respectively the constant and the exponentstowhichthedimensionlessFroude,gasvelocity, vis-cosityandsurfacetensionnumbersareraised.Thedimensionless viscosity, *, is calculated using the Newtonian viscosity (l) reportedin Table2. ThesoftwareAuto2fit® _{is usedtoperform} themulti-parameteroptimizationrequiredtodetermine˛,a,b,c andd.Differentmathematicalalgorithmsaresystematicallytested (globalLevenberg–Marquardt,globalQuasi-Newton,standard dif-ferentialevaluation,geneticalgorithm)toverifythestabilityofthe resultsandtheirindependencywithinitialconditions.Themean standarddeviationiscalculatedfrom:

ε= 1 N

X

i=1,N









(kla)∗exp,i−(kla)∗mod,i (kla)∗ exp,i









(50)

Ina firsttime,it isinterestingtovisualizetheeffectof Newto-nianviscosity(*)on(kla)*separatelyfromtheothervariables. TheproblemisthatthegraphicrepresentationassociatedwithEq.

(49)requiresfivedimensionsas(kla)*dependsonFr,Ug∗,*and *.Asimplewaytosidestepthisproblemistocomedowntoa2D representation,inwhichtheimpactof*on(k_la)*isneglectedand anidenticalexponentisimposedfortheFroudeanddimensionless superficialgasvelocitynumbers.ThisisillustratedinFig.4,where (kla)*isplottedasafunctionof(Ug∗·Fr)2/3forNewtonianfluids. Thevalueof2/3ischosenasafirstapproximationfortheexponent

Fig.4. EffectofdimensionlessNewtonianviscosity,kla∗=kla·(g/g2)1/3,versus

(U∗ g·Fr)

2/3

(thedotted/continuouslinescorrespondtothevaluespredictedbyEq.

(51)).

onFrandU∗

g,asalreadyencounteredintheliterature(seereviewof

[3]).Thus,Fig.4offerstheadvantagetoeasilyappreciatethe neg-ativeeffectof*onaerationperformances:whatever(U∗

g· Fr)2/3, anincreaseof*from54(water)to1886(glycerine70%)leadsto adrasticreductionof(kla)*(morethan85%).

When imposing the exponent 2/3 to (U∗

g· Fr), the best fit-tingbetweenexperimentaldataandEq.(49)withouttakinginto account*leadsto:

kla∗_{=0.1420·(Fr·U}∗

g)2/3·(∗)−0.591 (51) Theassociatedmeanstandarddeviationisequalto17.1%.Totest therobustnessofsuchcorrelation(inparticularoftheexponents found),severalcasesarenowtestedwhenimplementingthe multi-parameteroptimization:

-CaseNo.1:alltheexponentsinEq.(49)arekeptfreeand*is neglected;

-CaseNo.2:theexponentsofFrandofU∗

gareimposedidentical withoutanyspecifiedvalue,and*isneglected;

-CaseNo.3:theexponentsofFrandU∗

gareimposedequalto2/3 (asinEq.(51)),butnowtheeffectof*istakenintoaccount.

InTable6arecollected,foreachcase,theconstantand expo-nentsofEq.(49)deducedfromthemulti-parameteroptimization. Inageneralpointofview,nomajordifferenceappearsbetweenthe differentcases:theexponentof*remainscloseto−0.59 (devia-tion<1.3%),andtheexponentsofFrandU∗

gdonotvarysignificantly whethertheyareimposedidenticalornot(deviation<6.5%).These findingsconfirmthattheordersofmagnitudeoftheexponents foundinEq. (51)are relevant,and thus,thatthis correlationis mathematicallyrobust.Notethat,evenifFrandU∗

ghavethesame exponent,thecontributionofNistwotimeshigherthantheone ofsuperficialgasvelocity,astheFroudenumberisexpressedin squaredrotationalimpellerspeed(Eq.(27));thisiscoherentwith theseobservationsmadeonthedimensionalresults(Section4.1). ThecasesNo.1andNo.2leadstoaslightimprovement(smaller than1%)ofthemeanstandarddeviation(ε);however,the expo-nentsofEq.(51)willbethereafterconserved,insofarastheyleadto themostsimpleformulation,andalsoasthevalueof2/3isalready

Table6

DimensionlessmodellingforNewtonianfluidsusingEq.(49)(inboldthecase retained). Case ˛ a b c d ε(%) No.1 0.02125 0.747 0.66 −0.605 – 16.1 No.2 0.01535 0.683 0.683 −0.592 – 16.8 No.3 0.2097 2/3 2/3 −0.591 −0.245 16.8 (kla )*exp (kla )*mod Water Gly50 Gly70 [µ * = 54, σ* = 73385] [µ * = 589, σ* = 45066] [µ * = 1886, σ* = 50704] 10-6 10-5 10-4 10-3 10-6 10-5 10-4 10-3

Fig.5.Experimentaldimensionlessoverallgas–liquidmass transfercoefficient versuskla*predictedfromEq.(52)(Newtonianfluids).Thedottedlinescorrespond

toadeviationof±20%.

encounteredintheliterature.Basedonthat,theeffectofsurface tension, ∗_=/(3

g·4g·g) 1/3

,hasbeenaddedin Eq.(51)(Case No.3):theconstantisinreturnincreasedandanegativeexponent appearsfor*(−0.245).Thislatterlogicallyconfirmsthepositive impactofdecreasingsurfacetensiononaeration:whensurface ten-sionisreduced(forexamplewhenaddingsurfactantoralcoholin theliquidphase),smallerbubblesizesaregenerated,leadingtoa riseininterfacialareaandthusinoverallvolumetricmass trans-fercoefficient.Intheliterature[7],thisusuallyresultsinpositive exponentsfortheWebernumber.

Atlast,atagiventemperature,underagivenpressure,forthe geometryoftheaeratedstirredtankundertest(inparticularfor Hl/Tt=1,D/Tt=0.4,Ds/Tt=0.75andfortheothergeometricalratios characteristicsofthesystem),thedimensionalanalysisstatesthat theprocessrelationshipforNewtonianfluidsisexpressedby:

kla∗=0.2097·(Fr·Ug∗) 2/3 ·(∗₎−0.591 ·(∗₎−0.245 validfor



0.096<Fr<2.4, 0.0029<U∗ g<0.029, 54<∗<1886 847<∗_{<1015, 50704<}∗_{<73385, Sc=7850} (52)

Fig.5comparestheexperimentaldatawiththedimensionless overallvolumetricgas–liquidmasstransfercoefficientspredicted byEq.(52).Agoodagreementisobserved,asthemeanstandard deviationremainssmallerthan17%.Itcanbeenobservedthatsome datacorrespondingtothesmallestk_la*(obtainedforlowspeedsin glycerine)tendstomoveawayfromthestraightlinesrepresenting (kla∗₎

exp= ±20%· (kla∗)pred.Thisshouldbelinkedtothefactthat, insuchconditions,thegas–liquidregimecorrespondstoaloading regime,andnottoacompletedispersionregimecharacterizingall theotheroperatingconditions.Themechanicalagitationisnotthen sufficienttodisperseuniformlythebubblesinthewholevolumeof tank,inparticularinthelowerpartofthetank(belowtheimpeller). Hence,therelativecontributionsofthemechanismscontrollingkla (mechanicalagitationagainstspargeraeration)deviatesfromthe onesactingwhenacompletedispersionregimetakesplace.When suchchangesinregimeoccurs,itbecomesthendifficulttodefine anuniqueandaccurateprocessrelationshipabletodescribethe entirerangeofoperatingconditions.Somedeviationscanbealso observedforthehighestvalesofkla;theycanbehereassociated withanincreasingcontributionofthesurfaceaeration(mass trans-feroccurringatthefreeliquidsurface)whenrotationalimpeller speedrises.Indeed,forthehighestN,asignificantvortexappears inthecentreofthetankandentrainsmanybubbles.Insuch condi-tions,themechanism(orregime)ofaerationisdeviatedtoapure volume-aeration,andthus,thechoiceofkla





_t(andnotkla





_surf)for

tractingvolume-aerationstateislessrepresentative.

Toconclude,itshouldbekeptinmindthatthevalidityofEq.

Table7

Dimensionlessnumberscharacteristicsforliquidproperties.

* * nost ∗ost nw t∗w ∗w Water 848 73385 1 54 1 1 108 Gly50 973 45966 1 589 1 1 1178 Gly70 1015 50704 1 1886 1 1 3773 CMC4 861 75905 0.642 1866 0.546 0.201 4944 CMC6 877 77316 0.527 5309 0.513 6.84×10−3 _5.12×104 XG1 847 72276 0.543 539 0.411 9.98×10−5 _1.02×105 XG2 855 77719 0.373 1366 0.281 3.84×10−5 _1.59×106

numbersabovementioned,atgiventemperatureandpressure,and inthegeometrydefinedinSection2.

4.3. Dimensionlessresultsfornon-Newtonianfluids 4.3.1. WhenconsideringthemodelofOstwald–de-Waele

Asany valuecan beconsidered(see Section 3.2), the refer-enceshearrate, ˙ o,hasbeenarbitrarychosenequalto120s−1_.The dimensionlessnumbersdescribingtherheologicalproperties asso-ciatedwiththeOstwald–de-Waele’smodel(Table2)andsuch ˙ o arecollectedinTable7(fourthandfifthcolumns).Itcanbe logi-callyobservedthattheflowindex(nost)decreaseswhenincreasing theconcentration,thesmallestvaluebeingobtainedforthemost concentrated solutionof xanthangum.Dimensionless apparent viscosity(∗

ost)arehigherforaqueoussolutionsofCMCthanthe onesofxanthangum.

BasingonthereasonsmentionedinthecaseofNewtonianfluids, thepossiblechangesin*andinScwillbeneglectedinEq.(38)

previouslyestablishedwhenthemodelofOstwald–de-Waeleis considered.Thus,Eq.(38)becomes:

kla∗_=g′



U∗_g,Fr,∗_,∗₌ a( o) g ,nost



(53) Thesimplestmonomialformisherealsolookedfortheg′_-relation (nomechanisticinformationavailable):

kla∗_=˛′_·(Fr)a′ ·(U∗ g)b ′ ·(∗₎c′ ·(∗₎d′ ·(nost)e (54) where ˛′_{, a}′_{, b}′_{, c}′_{, d}′ _{and e are respectively the constant and} theexponentstowhichthedimensionlessFroude,superficialgas velocity, viscosity,surface tensionand flow indexnumbers are raised(Auto2fit®_{).However,itisimportanttokeepinmindthat} themodellingforNewtonianfluidsisadegradedcaseoftheone fornon-Newtonianfluids.Asaconsequence,theexponentsofFr, U∗

g,*and*(namelya′,b′,c′,d′)arealreadydefined,andwillbe thustakenequalrespectivelyto2/3,2/3,−0.591and−0.245asin theNewtoniancase(Table6,CaseNo.3).

Fig. 6.Effect of dimensionless apparent viscosityand flow index when the Ostwald–de-Waele’smodelisconsidered:(kla)*versus(Ug∗·Fr)2/3.

InFig.6,(kla)*isplottedasafunctionof(Ug∗·Fr)2/3forallthe fluids.Suchfigureisparticularlyimportantastheeffectsof*and nostcanbevisualizedseparately.Indeed,atagiven*,andecrease in nost clearly induces an increase in (kla)*, or in otherwords, theshear-thinningcharacteroffluidsfavourstheaeration perfor-mances,probablyduetothespatialheterogeneityofviscosity.This isillustratedwhencomparingeither(i)theglycerineat70%and theaqueoussolutionsofCMCat4g/Lwhichhavealmostthesame dimensionlessapparentviscosity(1886against1866),but differ-entflowindexes(1against0.642),or(ii)theglycerineat50%and theaqueoussolutionsofxanthangumat1g/Lwhichhavealmost thesamedimensionlessapparentviscosity(589against538),but differentflowindexes(1against0.543).Theimpactofnostseems howeverlesspronouncedthantheoneof*.

Fig. 7 compares the dimensionless overall volumetric mass transfercoefficientsmeasuredinpresenceofnon-Newtonianfluids with the ones predicted using the process relationship estab-lishedforNewtonianfluids(Eq.(52)).Thepointsrelatedtoeach non-Newtonianfluidareregroupedalongindividualstraightlines which are parallelto each others and tothe curves previously obtainedforNewtonianfluids.Thisillustratesthatthe introduc-tionofnost intothedimensionlessmodelling(Eq.(54))couldbe a meanfor differentiatingthe non-Newtonian fluidsfromeach othersandfromtheNewtonianones.Nevertheless,thisgroupof fourstraightlines(oneforeachfluid)isnotclassifiedaccording tothevalues offlowindex,in particulartheassociatednost are notdecreasingwhendeviatingfromtheNewtoniancurve(nost=1). Indeed, thestraight linethe closestfrom theNewtonian curve correspondstonost=0.543(XG1g/L),followedbynost=0.373(XG 2g/L),thennost=0.642(CMC4g/L),untilreachingthemostdistant line,nost=0.527(CMC6g/L).Thisfindingtendstoshowthat gather-ingallthedatarelatedtonon-NewtonianfluidsovertheNewtonian oneswillbedifficult byadding onlythe flow indexnost inthe relevantlist.Inotherswords,thiswouldmeanthatthematerial functionassociatedwiththemodelofOstwald–de-Waelecouldbe insufficientand/orunsuitablefordescribingcompletelytheeffect oftheshear-thinningpropertieson(kla)*.

1.0E-04 (kla )*exp (kla )*mod Water Gly50 Gly70 CMC4 CMC6 XG1 XG2 nost=0.543 nost=0.373 nost=0.642 nost=0.527 10-6 10-5 10-4 10-3 10-6 10-5 10-4 10-3

Fig.7.ModellingusingtheOstwald–de-Waele’smodel:comparisonbetweenthe experimental(kla)*andthevaluespredictedbyEq.(52)forNewtonianfluids.The