Mixing assessment by chemical probe

Mixing assessment by chemical probe

Charbel Habchi

^a

, Thierry Lemenand

^b

, Dominique Della Valle

^b,c

, Mahmoud Khaled

^a

, Ahmed Elmarakbi

^d

, Hassan Peerhossaini

^e,

*

aEnergyandThermo-FluidsGroupETF,SchoolofEngineering,LebaneseInternationalUniversityLIU,146404Mazraa,Beirut,Lebanon

bLUNAMUniversite´,ThermofluidComplexFlowsandEnergyResearchGroup,LaboratoiredeThermocine´tiquedeNantes,CNRSUMR6607,44306Nantes, France

cONIRIS,44322Nantes,France

dDepartmentofComputing,EngineeringandTechnology,FacultyofAppliedSciences,UniversityofSunderland,SunderlandSR60DD,UnitedKingdom

eUnivParisDiderot,SorbonneParisCite´,InstitutdesEnergiesdeDemain(IED),CNRSFRE3597,75013Paris,France

1. Introduction

Characterizing micro-mixing is an important issue in the

‘‘Green Process’’ scheme, since it governs, in a broad class of industrialprocesses,byproducteffluentsandconsequentlyprocess efficiency.The selectivityof fastchemicalreactions dependson reagent mixing at the molecular scale. In turbulent flows, the speciesaggregatesarereducedinsizebytheturbulentcascade.In thisprocess,thelimitingmechanismoccursatsmallerturbulence scales[1].Thus,thesequenceofmicro-mixingis(i)engulfmentin the energetic vortices at Kolmogorov scale, (ii) stirring in the viscous-convective subrange, wherethe fluid particlesare sub- jectedtolaminarstretching[2],and (iii)moleculardiffusionat sub-Batchelor scales that rapidly dissipates the variance in concentration.Understandingandquantifyingthismechanismis essentialindesigningindustrialprocessesinvolvingfastreactions that canpresent characteristic reactiontimes smaller than the characteristicmicro-mixingtime.

Thetwofinalstepsinthemicro-mixingmechanismdescribed aboveare‘‘faster’’[1–4]thanengulfmentattheKolmogorovscale:

asaconsequence,micro-mixingdependsontheturbulenceenergy dissipationrate,whichgovernsthetimeandlengthscaleofthe smallereddies.Thisfundamentalpropertyoftheturbulentfield canbedeterminedbyclassicalvelocimetrymethodssuchaslaser Doppleranemometry(LDA),particleimagevelocimetry(PIV),or hot-wire anemometry, all of which give access, in three- dimensional space, to the nine contributions of the turbulent energydissipationrate[5].

Alternative methods to characterize micro-mixingbased on observationsofachemicalsystemhavebeendevelopedoverthe lastfewyears[6–10],mostlyforthecaseswherethereisnooptical accesstotheflowtocarryoutreferencemethodslikeLDAorPIV, butalsofortheirabilitytogiveaccesstotheresultofachemical reaction, and thus to the straightforward result of the mass transfer.Thesemethods wereinvestigatedespecially byBourne [11](couplingofnaphtol-1and-2withdiazotsulfanilicacid),and Fournieretal.[12](Villermaux–Dushmanreactionsortheiodide/

iodatemethod).Thesetechniques,called‘‘chemicalprobemeth- ods’’, are based onthecompetition between micro-mixingand well-knownchemicalkineticsbythestraightforwardobservation ofreactionselectivity,i.e.thesecondaryproductconcentrations.

Suchexperimentsmustbeperformedundercontrolledconditions, firstbyensuringthatthemainreactionisnotfullyachieved:the selectivitymustbe‘‘far’’from0and1tomakethereactionproduct ARTICLE INFO

Articlehistory:

Received3February2013

Receivedinrevisedform24June2013 Accepted17July2013

Availableonline9August2013

Keywords:

Micro-mixing

Turbulenceenergydissipationrate Chemicalprobemethod Mixingmeasurement Iodide–iodatechemicalsystem

ABSTRACT

Quantificationofmicro-mixingisafundamentalissueinindustrialchemicalprocesses.Localmixingthat isnot‘‘fastenough’’comparedwiththereactionkineticsreducestheselectivityofthereaction.Micro- mixingcanbecharacterizedbychemicalprobemethods basedonobservationofalocal chemical reactionthatresultsfromthecompetitionbetweenturbulentmixingatmicro-scalesandthereaction kinetics.However,real-worldexperimentalconditionsrarelycomplywiththegroundingassumptions of this method. Starting from physical considerations, the present study aims to establish some guidelines forobtaining quantitativeinformation fromthe chemical probeand forimprovingthe accuracyofthemethodbyanadaptiveprotocol.Forthefirstaspect,ananalyticalapproachisproposedto definethevaliditydomainbasedonanalysisoftheturbulenttimescales.Forthesecondpurpose,anovel experimentalprocedureissuggestedthatentailstargetingtheconcentrationsofthechemicalspecies thatcanprovidetheoptimalconditionsforarelevantuseofthechemicalprobe.

ß2013TheKoreanSocietyofIndustrialandEngineeringChemistry.PublishedbyElsevierB.V.Allrights reserved.

* Correspondingauthor.Tel.:+33607533161.

E-mailaddress:hassan.peerhossaini@univ-paris-diderot.fr(H.Peerhossaini).

ContentslistsavailableatScienceDirect

Journal of Industrial and Engineering Chemistry

j o urna l hom e pa ge :ww w. e l s e v i e r. c om/ l o ca t e / j i e c

1226-086X/$–seefrontmatterß2013TheKoreanSocietyofIndustrialandEngineeringChemistry.PublishedbyElsevierB.V.Allrightsreserved.

http://dx.doi.org/10.1016/j.jiec.2013.07.026

sensitive to the micro-mixing. Under optimal conditions the slowest reactiontime is equal to themicro-mixingtime. From knowledge of the chemical reaction (mechanism, kinetics and stoichiometry), the local turbulent energy dissipation rate can readilybederivedfromthemeasuredselectivityviaphenomeno- logicalmicro-mixingmodels[3].

Theappropriatechoiceofoperatingconditions(initialreagent concentrations,injectionflowrate,stoichiometry...)isnottrivial andthereisasyetnoclearmethodologyforusingchemicalprobes, especiallyforopen-loopflows.Thechoiceofinitialconcentrations is generally made by ‘‘trial and error’’ and sometimes is not convenientwithrespecttothereactionkinetics[13–15].Accurate quantitativeresultscanbeobtainedifcertainconditionsonthe flow and the chemical systemare fulfilled.The purpose ofthe presentworkistoestablishguidelinesforobtainingquantitative information from the chemical probe when possible and for improvingtheaccuracyofthemethodbyanadaptiveprotocol.For the first aim, an analytical approach based on analysis of the competition between the different turbulent time scales is proposed in order to assess the feasibility of chemical probe method,i.e.toanswerthequestionifthechemicalprobeisableor nottogiverelevantinformationonthetransfersinthesepeculiar conditions.Forthesecondpurpose,anovelexperimentaladaptive procedure is suggested, that entails targeting the optimal concentrationsof thechemical speciesprovidingmore accurate and more ‘‘localized’’ observation of the mixing time, and subsequently oftheenergy dissipationrate.Theserules canbe generalizedtoanychemical-probesystemandcanbeappliedin anyreactorgeometry.

In the present work, the mixing scales analysis and the adaptive procedure are applied to an inline heat exchanger–

reactor equipped with aligned vortex generators. The iodide/

iodate reaction system [12,16,17] and the micro-mixing E- model [18] employed here, are widely used in batch and continuous-flow reactors[19–22]. Thechemical probemethod andthemicro-mixingmodelaresuccinctlyreprisedinSection2.

In Section 3, a scaling analysis of turbulence and of the interactions amongthe differentscalesleads to thedefinition ofavaliditydomainforthechemicalprobemethod.InSection4, anovelexperimentalprocedureisproposedtoadaptthereagent

concentrations to the turbulence level and to check the measurementvolumeattheinjectionpoint.Section5isdevoted tosomesampleimprovementsthatmayresultfromthepresent analysis, specifically comparing a static mixer equipped with aligned vortex generators to previous experiments [23].

Concludingremarksabouttheapplicationopportunities ofthe methodare givenin Section6.

2. Chemicalprobe:chemicalsystemandmicro-mixingmodel 2.1. Theiodide/iodatemethod

Theiodide/iodate system[12,16,17]is based oncompetitive parallelreactions:thequasi-instantaneousborateneutralization [Eq.(1)]andtheDushmanreaction[24][Eq.(2)],whichismuch slower.Thebalancedreactionscanbemodeledasfollows:

H2BO₃þH^þ$H3BO3 (1)

5IþIO₃þ6H^þ$3I2þ3H2O (2)

TheiodineI2furtherreactswithiodideionsI,yieldingI₃ ions followingthequasi-instantaneousequilibriumreaction:

I2þI$I₃ (3)

The kinetics of the three reactions was established by Guichardonetal.[17].Onlythecharacteristictimeoftheslowest reactioninEq. (2)is describedhere, sinceit isusedfor further calculations:

tr2¼Min³₅½I;3IO₃

;¹₂H^þ

r2

(4) wherethebracketsdenotethereagentconcentrationandr2isthe rateofthesecondreaction:

r2¼K2½I²H^þ2

IO₃

(5) wheretheconstantK2isafunctionoftheionicstrengthasgivenby Palmeretal.[25]andGuichardonetal.[17].

Whenthemicro-mixingislimiting,alocaloverconcentrationof H⁺canreactafterreaction(2)andproduceiodineI₂,whichitself reactswithiodideIandyieldsI₃ions[Eq.(3)].ThepresenceofI2

andI₃ishencethemanifestationofamixingtimesmallerthanthe secondreactiontimeandcanbequantifiedbyasegregationindex.

In continuoussystems, it isdefined byFournieret al. [26]and Ferrouillatetal.[27]as:

Xs¼2½I2þI₃ H^þ

0

1þ^Q_Q^P

Hþ

1þH2BO₃

0

6IO₃

0

!

(6) whereQpandQ_Hþarerespectivelytheprincipalflowrateofthe initialmixtureflow andtheinjection flowrateof sulfuricacid.

XS=0forperfectmicro-mixingandXS=1fortotalsegregationon themolecularscale.

InordertodeterminetheI2andI₃concentrationsintheEq.(6), theoutputflowisdriventoaspectrophotometercellwherethe lightabsorption,whichisproportionaltotheconcentrationofI₃ ions,isrecorded.Theiodine[I2]isderivedfromthemassbalance oniodine[26].

2.2. Micro-mixingmodel

ThesegregationindexXSprovidesonlyqualitativeinformation onmicro-mixingsinceitdependsontheinitialconcentrationsand ontheratiobetween themainand injectionflows.The related quantitative parameter is the intrinsic micro-mixing time tm, Nomenclature

ci concentrationofspeciesi(mol) d needlediameter(m)

D reactordiameter(m) Dt turbulentdiffusivity(m²/s) TKE,k turbulencekineticenergy(m²/s²)

t time(s)

t_m micro-mixingtime(s) W localflowvelocity(m/s) Wi injectionvelocity(m/s) W¯ meanflowvelocity(m/s) Q flowrate(m³/s)

Xs segregationindex [] concentration(mol) Greeksymbols

e

turbulenceenergydissipationrate(m²/s³)

L

integrallengthscale(m)

y

kinematicviscosity(m²/s)

t

timeconstant(s)

independentofthechemicalsystem,whichcanbeidentifiedbya micro-mixingmodel.

Several micro-mixing models exist for determining micro- mixingtime[26,28,29].Inthepresentwork,theengulfmentmodel (E-model)developedbyBaldygaandBourne[18]forfluidswith highSchmidtnumberisused,sinceitgivesmorereliableresults thanothermodelsintheliterature[30].

Themassbalance,involvingthegrowthoftheuniformmixing zone of concentration ci, is given by the first-order temporal differentialequation:

dci

dt ¼ 1 tm

¯cici

ð ÞþRi (7)

where ¯ci is the mean concentration value in the environment before mixing is completed and R_i is therate of formation of substanceibyachemicalreaction[18].Thesystemiscomposedof ninenonlinearequations(foreachspecies);Eq.(7)isintegrated using the Newton–Raphson iterative method. Actually, the solutiontmistherateofthereactantformation yieldingtothe observedexperimentalvalues.Inpractice,oneintegratesontime the setofEq. (7) for a rangeof tm values:for each value,a XS

‘‘computed value’’ can be derived from the final species concentrations (tillthe whole consumption of H⁺). The ‘‘right’’

tmvalueinthisrangeisdeterminedbyprovidingacomputedXS

equaltotheexperimentalXS.Thisisdonethankstothe‘‘calibration curves’’ specificofthechemicalsystem(Fig.1)which globalize XS¼FðtmÞ,whereFisconceptuallythetemporalintegrationstep.

Examples of calibrationcurves XS¼FðtmÞfor three different initialconcentrationsofsulfuricacidareshowninFig.1.Itcanbe observed that the curveis flat for both ‘‘small’’ and ‘‘large’’ tm

values,denotingalackofsensitivityofXStothemicro-mixingtime.

This can be explained from phenomenological considerations.

For small micro-mixing time, tm<<tr2, micro-mixing is very efficientandhasnosignificantinfluenceonXS,becauselocally thereagentsare‘‘rapidly’’homogenized.Eventually,ifthelarge- scaletransfersarelimiting,thesecondreactionmaydevelopona larger volume and its selectivity is hence determined by the global mixing. Atthe other limit, for verylarge micro-mixing time, tm>>tr2, reaction selectivity is completely governed by thekineticsofthesecondreaction,andthehydrodynamichave onlyaweakeffectonXS.Thus,themethodwillbelessaccurate whentr2isverydifferentoftm.

Whentmisdetermined,theBaldygaandBourne’sanalysis[18]

givesaccesstotheTKEdissipationrate

e

:

e

¼297:22

y

tm

(8) where

y

isthekinematicviscosity.

3. Analysisofthemeso-mixingscales

Twomeso-mixingmechanismshavebeenidentifiedbyBaldyga etal.[31]:first,theturbulentdispersionofthefeedstreamintothe main flow, in both the radial and streamwise directions, and second thebreak-down of injectedaggregates in the turbulent cascade,fromthelargeintegralscaletotheKolmogorovscales.The correspondingtimescalesarecomputedinthissection.

3.1. Injectiontimescales

First, following concepts of Batchelor [2] and Corrsin [32], Baldygaetal.[30,33]defined,byscalinganalysis,twocharacteris- tic times for the turbulent dispersion mechanism, in the streamwise and radial directions relative to the injection, tD1

andtD2respectively:

tD1¼ Q_i

WDt (9)

tD2¼ d² 4Dt

(10) wheredisthefeedpipeinternaldiameter,Qiistheinjectionflow rate,WisthelocalflowvelocityandDtistheturbulentdiffusivity ofthemainflow,whichisclassicallymodeledwiththeturbulent kineticenergykandthedissipationrate

e

[24]:

Dt¼0:1k²

e

⁽¹¹⁾

TheinjectionflowrateQiisexpressedbytheinjectionvelocity Wi:

Qi¼

p

d²

4 Wi (12)

Defining

f

bytheratiobetweenthemainflowvelocityandthe injection velocity,

f

=W/Wiand theturbulentdiffusivity coeffi- cientbyitsexpressioninEq.(11),tD1andtD2canbeexpressedby:

tD1¼

p

0:4 d²

f e

k² (13)

tD2¼ d² 0:4

e

k² (14)

Second,whenthescaleofinjectedfeedstreamislargerthanthe turbulenteddies,theconcentrationfluctuationsbreakdownfrom the integral scale

L

(convective-inertial large scale) toward Kolmogorov’smicro-scalebyamechanismcalledinertial-convec- tivemeso-mixing.Forfullydevelopedturbulence,thecharacteristic time(cascadetime)isgivenbyBatchelor[2]andCorrsin[32]:

tC¼2

L

²

e

!1=3

(15) If the injection momentum is moderate, the injected fluid particlesareinstantaneouslysubmittedtothemain-flowvelocity field.Inthiscase,withL0theinitialradiusofthereactivejetthatis convectedwithlocalvelocityW,itfollows[31]bycontinuitythat

0.01 0.1 1 10

0.0 0.2 0.4 0.6 0.8 1.0

XS

[H⁺]= 0.02 mol L^-1 [H⁺]= 0.10 mol L^-1 [H⁺]= 1.00 mol L^-1

t_m ^(s)

Fig.1.CalibrationcurvesfromtheE-modelfordifferentinitialacidconcentrations andforthefollowingconcentrationsofinitialreagents:[KI]0=0.01165molL¹, [KIO3]0=0.00233molL¹,[NaOH]0= [H3BO3]0=0.001512molL¹,forf=15,the ratiobetweenthemainflowvelocityandtheinjectionvelocity.

Qi

p

L²₀W,byassuming[35]that

L

=L0,leadingto:

L

¼ ffiffiffiffiffiffiffi d² 4

f

s

(16) By combining Eq. (15) and Eq. (16), the inertial-convective meso-mixingcharacteristictimecanbeestimatedby:

tC¼1:26 d²

fe

!1=3

(17) Inordertocomparethesethreedifferenttimescaleswiththe micro-mixingtime,threedimensionlessparametersaredefinedby theratioofthethreecharacteristictimes(tD1,tD2,tC)bythemicro- mixingtimetm,calledrespectively

t

1,

t

2and

t

3,writtenasfollows:

t

1¼tD1

tm ¼0:145

p

^d

2

fy

¹⁼²

e

³⁼²

k² (18)

t

2¼tD2

tm ¼0:145 d²

y

¹⁼²

e

³⁼²

k² (19)

t

3¼tC

tm

¼0:073 d²⁼³

f

¹⁼³

y

¹⁼²

e

¹⁼⁶ (20)

Theseparametersprovideacriterionforthedomaininwhich thechemicalprobemethodcanreachthemicro-mixingtime,that iswhenthethreecharacteristictimesaresmallerthanthemicro- mixingtime,i.e.theconditionis:

t

1,

t

2and

t

3<1.Inthissystem, theindependentparametersthatmayaffectturbulentmixingnear theinjectionarethefeedneedle(jet)diameterdandthevelocity ratio

f

fora given mainflow rate.Eqs.(21)–(22)highlightthe dependence on theseparameters by expressing the turbulence kineticenergyanditsdissipationratewithscalinganalysis,with thetypicalscales:W¯ theaverageflowvelocityandDthereactor diameter.Inthesameway,thelocalvelocityWisalsonormalized bytheaveragevelocity:

k¼CkW¯² (21)

e

¼Ce W¯³

D (22)

W¼CWW¯ (23)

AssumingdynamicalsimilarityforanarrowrangeofReynolds number,wheretheflowpatterndoesnotchangesignificantly,the coefficientsCk,C_eandCWdependonlyonlocationandnotonthe Reynoldsnumber.By substitutingEqs.(21)–(22)intoEqs.(18)–

(20),thecharacteristictimeratiosread:

t

1¼0:145

p

^C

3=2 e C_k²D³⁼²

y

¹⁼²

d²

f

^W¯

1=2 (24)

t

2¼0:145 C_e³⁼² C_k²D³⁼²

y

^1=2d

2W¯¹⁼² (25)

t

3¼0:073 Ce¹⁼⁶ D¹⁼⁶

y

¹⁼²

d²⁼³

f

¹⁼³

W¯¹⁼² (26)

An analytical study is carried out with Eqs. (24)–(26) to investigatethevaliditydomaininanarbitrarycase.Theexample used here is the flow in a straight pipe of hydraulic diameter D=20mm,withassumedlocalconstantsCk=0.455,C_e=0.971and CW=0.125(correspondingresultsareshowninFig.2).Thevalues oftheseconstantsareadaptedfromHabchietal.[22].InFig.2,the characteristictimeratiosareplottedsuccessivelyasafunctionof

the three independent parameters

f

, d, and W,¯ two of them assumedtobeinthenominalstateforeachcase.Theregionwhere allthetimeratiosarelessthan1ishatched.InFig.2(a)acritical valuefor thevelocity ratio

f

o=7 is obtainedfromthe limiting criterion

t

3<1,sinceitisthehighestrelativeto

t

1and

t

2.Fig.2(b) showsthatifthefeedpipediameterdisincreased,meso-mixing overcomesmicro-mixingas

t

2becomesgreaterthan1.Thecritical feedpipediameterisfoundtobearound2.20mminthiscase.The highermainflowratesaswellleadto

t

2>1inFig.2(c),sincethe

0 5 10 15 20 25 30

0.0 0.5 1.0 1.5 2.0

d= 2 mm, D= 20 mm, W= 0.75 ms^-1

1 2 3

1, 2, 3

Velocity ratio

(a)

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.0 0.5 1.0 1.5 2.0

o= 15, D= 20 mm, W= 0.75 ms^-1

1 2 3

1, 2, 3

d (mm)

(b)

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.0 0.5 1.0 1.5 2.0

W (ms^-1) o= 15, d= 2 mm, D= 20 mm

1 2 3

1, 2, 3

(c)

Fig.2. Validity domainofthe chemicalprobemethodin a straightpipefor characterizingmicro-mixingtimedependingon(a)velocityratio,(b)feedpipe diameterand(c)meanflowvelocity.

timeratiosareincreasingfunctionsofW,¯ acriticalaverageflow velocityW¯ isaround1.20ms¹inthiscase.Thescalinganalysis proposed in this work shows that it is possible to evaluate theoreticallytheoptimalvalueofthevelocityratio

f

andofthe feed pipediameterdforgivenconditions,andit thendefinesa validitydomainoftheuseofthechemicalprobe.

3.2. Measurementvolumerequirements

Micro-mixingisalocalandsmall-scalemechanism,andthus shouldbeachievedata shortdistancefromtheinjectionpoint, meaningthatconvectivetransferbythemainstreamvelocityis small.SincetheresidencetimescaleinthereactoristResL=W,¯ it is self-evidentthat theconditiontr2<<tResmustbeverifiedin ordertogetalocalmeasurement.

Reciprocally,atvelocityW,thesecondreactionlengthscalecan beevaluatedbyLr2Wtr2.Itispossibleto‘‘tune’’tr2sothatLr2isat the reactor scale, and in that case, the selectivity will be the signatureoftheglobalmixingproperties.Thiscouldbeadeliberate choiceinexperimentaldesigninordertocharacterizetheglobal mixingefficiency.

3.3. Secondreactiontime-scalerequirement

The sensitivity of the chemical probe is linked to partial achievement of thesecond reaction,requiring that thissecond reactiontimebeclosetotm.Asdiscussedbelow,arecommenda- tionoftheadaptivemethodistoreachconditionsinwhichtr2tm. Inthiswork,thespeciesconcentrationsarefittedtothereaction kinetics,whileotherworkintheliteratureusesconstantspecies concentrations.

4. Anadaptivemethod

Thissectionproposes apractical methodfor experimentsby exploring the different degrees of freedom influencing the selectivityof thechemicalprobe system,as summarizedin the flowchartofFig.3.

4.1. Measurablemixingtimeinagiveninjectionflow

Thefirststepconsiststochecktheinjectionhydrodynamics:

for a given feed needle (jet) diameter d, to ensure that the flow is not perturbed by the injection, the velocity ratio

f

must be higher than the critical value, and this condition must be checked for all operating conditions. For a full rig design, thesamelogicallowsevaluationofthemaximal value ofthefeedpipediameterdforagiven

f

.Thiscaseisrelatively rare andthereasoninghereisconfined toagivenneedlesize.

The limitingmixingtimesiscomputedfromEqs.(24) through (26).

4.2. Acidconcentrationbasedonsecondreactiontime-scale Thesecondstepconsiststooptimizespeciesconcentrations:

the reagent concentrations are varied to target the greatest sensitivityofthesegregationindexXStotheflowdynamics,and finallytothemeasurementoftm,whentm=tr2.Hence,theinitial reagent concentrations must be adapted for each location, especially if the hydrodynamic and turbulencelevels are not homogeneousinthereactor.Sincealltheinitialconcentrations arelinked(exceptfor thesulfuricacid,whichiscontainedina separatereservoir),itismoretractabletomodifytheconcentra- tion of the H^þ

ions, although the procedure might also be applied by modifying the other reactive concentrations (that wouldbe necessary in anyevent ifthe convenient H^þ

would not be in stoichiometric defect). Then the measured micro-mixing time tm, obtained by the engulfment model, is plottedwiththecharacteristictimeforthesecondreactiontr2, computed from Eq. (4), versus the H⁺ concentrations. The intersectionbetweenthecurvestmandtr2indicatestheoptimal H⁺ concentration, andthe corresponding segregation indexXS

canberetainedasshowninFig.1.

The experimental range of H^þ

can be estimated from the turbulenceenergydissipationrateintheflowfromthepressure drop.Forexample,ifitisknownthatinagivenflowtheturbulence energydissipationrateisoftheorderof10m²s³,thenthemicro- mixingtimeisabout5.5ms.Thecriteriontm=tr2givestheorderof magnitudeoftr2andthusprovidestheconcentrationofH⁺,about 0.5molL¹inthisexample.

Whenthetmvalueresultingfromthechemicalprobemethod yieldsaturbulenceenergydissipationrate

e

farfromtheestimated one

e

ˆ,thenanewestimateoftheconcentrationmustbemadeto iteratetheprocedureforamoreaccuratevalueof

e

.

4.3. Measurementvolume:localorglobal

Thethirdstepconsiststoexaminethemeasurementvolumein order tocheckthelocalcharacterofthechemical probe.If the measurement lengthscale L_r2Wt_r2 is smallerthanthe meso- scalelengthscalecomputedfromEq.(16),thenthemeasurement canbeconsideredlocalandthemicro-mixingprocessmightby targetedbythechemicalprobe.Conversely,ifthemeasurement lengthscaleLr2ismuchlargerthanthemeso-scalelengthscale, Fig.3.Flowchartoftheadaptiveprocedureforthechemicalprobemethodfor micro-mixingcharacterization.

then the measurement cannot be considered local and the chemical probe characterizes in this case, not only the micro- mixingprocessbutalsothelarge-scalemixing.

5. Experiments

Themeso-mixingscalesanalysisandtheadaptiveprocedure areappliedtoaninlinemixerofHEVtype(Chemineers^TM)thatis widelyusedin severalpracticalapplicationsand asa reference geometry in many laboratory research works [39,41,44]. This mixerpresentsverydifferentlocalhydrodynamicconditionsinthe volume,allowingtestingawiderangeofconditionsfollowingthe injectionlocation.

5.1. Testingtheprocedureinacomplexflow

TheHEVgeometryconsistsofa circularpipeequippedwith sevenalignedrowsoflongitudinalvortexgeneratorsplacedatthe wall, asshowninFig.4.Theso-calledvortexgeneratorsinduce coherentstructurestopologicallysimilartothosefoundinconcave boundary layers and in wall turbulence [34–42]. Two main featuresareidentified(Fig.5)[45]:

astreamwisecounter-rotatingvortexpair(primaryCVP)witha commonup-flowinthesymmetryplanofthevortexgenerator, andapairofsecondaryCVPofverysmallsizeclosetothewall, withanassociatedcommoninflow,

aperiodicsequenceofhairpinvorticesconvectingdownstream and riding on top of the CVP [39,41], corresponding to a maximumturbulencekineticenergydissipationratelocatedata radialpositionofaroundy/R=0.4[43].

ThemeasurementsaretakeninthetypicallocationsB,SandW, inthesymmetryplaneofthevortexgenerator,atthecross-section 3mmdownstreamfromthefirsttabarray,asshowninFigs.4and 5.Thesethreemeasurementlocationshavebeenchoseninorderto getcompletelydifferentflowconditions,giveninTable1,soasto checkthevalidityoftheadaptiveproceduretoprovidereliableand accuratemeasurements[22].

5.2. Experimentalsetupandmethods

Aschematicdiagramoftheexperimentalsetupisshownin Fig.6.ThechemicalsolutionofthereactionsysteminEqs.(1) and(2)ismixedbyanimmersedpumpanditstemperatureis kept constant at 298K by a helical heat exchanger whose temperature is controlled by a thermostat (Crythermostat71 Huber). As shownin Fig. 6,the mixtureis drivenby a rotary pumpintothehydraulicloop.Themainflowrateismeasuredby

a precision balance and data acquisition is realized by Labview^TMsoftware.

Thesulfuricacidinjectionsystemconsistsofaregulatedstep motorconnectedtoamultipush-syringesystem.Thesulfuricacid is injected into thetest section through an injection needle of 0.6mminnerdiameterconnectedtothesyringesbyflexibletubes.

The location of the needle in the reactor cross-section is determined bya displacementmechanism, as shown in Fig. 4, withanaccuracyof10

m

m.Theflowrateoftheacidinjectionis controlled by a speed regulatoron theangular velocity of the stepper motorvia a PC. The test sectionis incorporatedin the hydraulicloopelementsbyflexibletubestoavoidfluctuationsdue to pump vibration. The reactor is preceded by a straight-pipe preconditioner of 1.50m length to ensure fully developed turbulent pipe flow at the reactor inlet, and followed by a postconditionerof0.30mlength.Thefinalproductsofthereaction areanalyzedincontinuousflowthroughachannelplaced0.3m downstreamfromthereactoroutletand branchedtoaspectro- photometer(Jenway6505^TM)ofresolution0.1nmandbandwidth

Fig.4.Schematicviewsof(a)thestaticmixer(referencemodel)cross-sectionshowingthreevortexgeneratorrowsandinjectionneedlelocationand(b)dimensionsofthe trapezoidalvortexgenerator.

Fig.5.Schematicviewoftheflowstructuredownstreamfromthevortexgenerator andlocationsofthemeasurementpoints.

Table1

Measurementlocations.

B Bulkflowregion y/R=1.0 Highvelocity Lowturbulence Lowgradients S Shearlayerregion y/R=0.2 Moderatevelocity Highturbulence

Highgradients W Wakeofthetab y/R=0.4 Lowvelocity Moderateturbulence

Lowgradients

1.8nmwhosewavelengthrangesbetween190nmand1100nm.

Themeasurableabsorbancerangeofthespectrophotometerlies between0and3,with0.1%accuracy.ThemeasurementsoftheI₃ ionabsorbanceareperformedintheUVdomain.

Allrunsarecarefullyperformedwiththeconstantconcentra- tionsgiveninTable2[23].Themeasurementsaremadeusingthe adaptive procedure described in Section 4 where the H⁺ concentrationissweptbetween0.2and1.0molL¹.

Theadaptiveprocedureisthenappliedintheseconditions,for Reynoldsnumber12,500andforthethreelocationsB,S,andW representedinFig.5.Table3summarizestheconstantsCk,C_eand CWobtainedfrompreviousnumericalsimulations[41].Infact,for anyoftheexperimentalstudiestheseparameterscanbeassumed asaninitialguessanddonotneedtobeveryaccurateeventoget accurate results. However, to improve the convergence of the measurements,theseavailabledataareusedinthisspecificcase.

Theeffectivevelocityratio

f

0usedforthemeasurementsandthe localvelocityWforeachlocationarealsopresentedinTable3.In thepresentstudy,theinjectioninnerdiameterisd=0.6mmand themeanflowvelocityisW¯ ¼0:625 ms¹.

6. Resultsanddiscussion

Theadaptivemethodisappliedinordertoassesstheresults whenthechemicalprobeiscarriedout:verificationofthemeso- mixing scales at the injection location, the appropriate acid concentration,andfinallythedeterminationofthemeasurement volume.

6.1. Injectionhydrodynamics

Thevalidity domainofthechemicalprobe isdeterminedby sweepingtheonly‘‘free’’parameter,thevelocityratio

f

,asshown in Fig. 7. Following the approach described in Section 4, the hatched regionsrepresentthevalidity domainin which micro- mixingisunaffectedbytheinjection.

Formeasurementsinthebulkregion(locationB)inFig.7(a),the velocity ratio

f

othat definesthevalidity domaintendstozero underthenominalconditions,meaningthatthemicro-mixingis alwaysthelimitingprocessatthislocationandthatanyvelocity ratio

f

valuecanbeusedformeasurements;here

f

measurement=7.6.

InFig.7(b),formeasurementsintheshearregion(locationS),the velocityratio

f

0=0.65isgivenbythetimescale

t

1,sothatthelocal operatingvalue

f

measurement=1.31isconsistentwiththedomain ofvalidity.InFig.7(c),inthewakeregion(locationW),itcanbe seen thatthetime scale

t

2 isalwaysgreaterthan 1.Therefore, whatever the velocity ratio in this region, the chemical probe methodcannotbeappliedbecausethecondition

t

2<1cannever Fig.6.Schematicdiagramofthehydraulicloopandinjectionsystem.

Table2

Speciesconcentrations.

Reagents H3BO3 NaOH KIO3 KI H⁺

Concentrations (10³molL¹)

1.512 1.512 2.33 11.65 200–1000

Table3

LocalhydrodynamicparametersindependentofReynoldsnumberandlocalvaluesforRe=12,500.

Measurementlocation Ck Ce CW k(m²s²) e(m²s³) W(ms¹) f0

Bulk(B) 4.4210³ 4.2110⁴ 0.983 1.7310³ 5.1410³ 0.614 7.61

Shear(S) 0.24 0.70 0.169 9.5010² 8.60 0.106 1.31

Wake(W) 0.13 0.51 0.312 5.0710² 6.20 0.195 2.42

Source:AdaptedfromHabchietal.[41]).

be reached. The solution suggested to satisfy the time scales condition would beto usea finer needle diameter in orderto decreasethetimescale

t

2below1.

6.2. Optimal[H⁺]forthemicro-mixingmeasurement

By applyingtheadaptivemethoddescribedin Section4,the measurements are performed in the three flow conditions by sweepingtheH⁺concentrationbetween0.2and1.0molL¹.The secondreactiontimeiscomputedbyEq.(4)asafunctionof[H⁺], andbothareplottedinFig.8.Thesulfuricacidconcentrationis takenwherethesecondreactiontimecurvetr2crossesthemicro- mixingtimecurvetm.Hence,aconvenient[H⁺]isdeterminedfor

thethreelocationsB,S,andW,andthecorrespondingmixingtimes anddissipationratescanbederived.Fromthesevaluesof[H⁺]itis thenpossibletoobtaintheabsolutevalueofthesegregationindex X_S,asshowninFig.9.ThesevaluesaresummarizedinTable4for thethreelocations.

6.3. Measurementvolume

Thissectiondiscussesthemeasurementvolumebycalculating Lr2=Wtr2andcomparingittothemeso-scaleofEq.(16).

Inthebulkregion(locationB),theeffectiveturbulenceenergy dissipationrateobtainedfromnumericalsimulations[41]israther small,about0.70m²s³,yieldingamixingtimeofabout0.020s.

Sincethelocalflowvelocityishigh(0.614ms¹),thelengthofthe measurementvolumeintheaxialdirectionisLr2=Wtr2=18.9mm,

1 2 3 4 5 6 7 8 9 10

0.00 0.25 0.50 0.75 1.00 1.25 1.50

1, 2, 3

Bulk region (y/R=1.0)

1 2 3

measurement

(a)

1 2 3 4 5 6 7 8 9 10

0.00 0.25 0.50 0.75 1.00 1.25 1.50

measurement

Shear zone (y/R=0.4)

1 2 3

1, 2, 3

(b)

1 2 3 4 5 6 7 8 9 10

0.00 0.25 0.50 0.75 1.00 1.25 1.50

measurement

Wake zone (y/R=0.2)

1 2 3

1, 2, 3

(c)

Fig.7.Timeratiosinthe(a)bulkregion,(b)shearregionand(c)wakeregionas functionofthevelocityratiof,forRe=12,500.

0.0 0.2 0.4 0.6 0.8 1.0

0 2 4 6 8 10 12 14

[H⁺] (mol L^-1) t_r2

Micro-mixing time t_m : B:y/R= 1.0 S: y/R= 0.4 W: y/R= 0.2 tm , tr2 (ms)

t_m = t_r2

Fig.8.DeterminationofsuitableH⁺concentrationfortm=tr2inthebulk(B),shear (S)andwake(W)locations,forRe=12,500.

Table4

Nominal[H⁺]andresultsobtainedfromthechemicalprobemeasurementsinthe threelocationsB,SandW,forRe=12,500.

Measurementlocation [H⁺](molL¹) XS tm=tr2(ms) e(m²s³)

Bulk(B) 0.336 0.130 9.111 3.581

Shear(S) 0.434 0.106 5.741 9.019

Wake(W) 0.462 0.110 6.338 7.400

0.0 0.2 0.4 0.6 0.8 1.0

0.09 0.10 0.11 0.12 0.13 0.14 0.15 0.16

B: y/R= 1.0 S: y/R= ^0.4 W:y/R= 0.2

[H⁺] (mol L^-1) XS

Fig.9.Determinationofthesegregationindexfor[H⁺]inthebulk(B),shear(S)and wake(W)locations,forRe=12,500.

much larger than the 3.25mm meso-scale computed at this location.Thus,themeasurementisnotlocalandthedetermination of the turbulence energy dissipation rate in the bulk region, locationB,isnotpossiblewiththisinjectionneedle(jet)diameter.

IndeedthemeasurementbychemicalprobeobtainedinBisnot local, but influenced by the turbulence downstream from the injectionpoint.

Intheshearlayer(locationS),theturbulenceenergydissipation rateishigher8:65 m² s³

andthecorrespondingmixingtimeis about 5.88ms; the measurement length is close to Lr2=Wtt2=0.77mm.Meanwhilethemeso-scaleis6.89mm,the chemicalreactiontakesplacelocallyandthedeterminationofthe micro-mixingtimeandthelocal

e

ismeaningful.

In the wake (location W), the chemical probe is not valid because the turbulence in the main flow is dominated by the injection flow. Nevertheless, the analysis of the measurement volumecanbecarriedout:

e

=6.06m²s³providesamixingtime of7.0ms.Themeasurementlengthisaround0.88mm,lowerthan the meso-scale which is found to be 5.61mm. It would be necessarytodecreasetheneedleinnerdiametertominimizethe injectionturbulenceforobtainingsatisfactoryresults,inspiteof theperfectlylocalaspectofthemeasurement.

6.4. Discussionontheadaptiveprocedure

In Fig. 10, it is possible to compare the turbulent energy dissipation rates obtained by the adaptive method with those obtainedfrompreviouschemicalprobemeasurementsusingthe concentrationvaluessuggestedintheliterature[15,23];thelatter were carefully carried out by Mohand Kaci [23] but without assessmentofthevaliditycriteriaproposedhere(byMohandKaci [23];Habchi etal.[15]).Independentarbitrationvalues,quoted from CFD and LDV (laser Doppler velocimetry) measurements, allowanobjectivediscussionofthevaluesaccuracy.

ThelocationsBandWaredisqualifiedfortwodifferentreasons:

-inthebulkzone(locationB),wehaveshownthatthelimitationis due to the non-local character of the measurement. The turbulenceisweakandthevelocityishigh,sothattheconvective effectsdominatethebehaviorofthechemicalspecies.Thenew measurementis,asexpected,notbetterthantheformerwiththe conventionalconcentrations,andcannotbeimprovedwiththe presentneedlediameter.

-inthewakezone(locationW),theadaptivemethodcannotbring betterinformation,as mentionedinthelocalanalysis oftime

scales(Section4),but itcanbenotedthat allthevelocimetry methodslosetheirreliabilityinthecaseofzeroaveragevelocity fields. It couldprovide reliable measurementsonly if a finer needlediameterdpermitstoobtainacharacteristictime

t

2<1.

Finally,onlythelocationSconditionsisqualifiedtoprovidea localmeasurementof

e

.Indeed,asignificantaccuracyimprove- ment is obtained by following the adaptive procedure. The conventional chemical probe method gives more than 50%

deviationfromLDV results, while theadaptiveprocedure gives valueswithin15%ofCFDandLDVreferences.

Additional measurements are performed at the location S (shearlayer)fordifferentReynoldsnumbersrangingfrom7500to 15,000toassesstheaccuracygain.Theresults,plottedinFig.11, arealsocomparedtothoseobtainedbytheclassicalmethod[23]

andbyLDVmeasurements[41].Thenewresultsaremuchcloserto LDVonesandtherelativeerrorisabout50%atRe=7500and8%at Re=15,000,while,fortheclassicalmethod,theerroris163%at Re=7500and35%atRe=15,000.

7. Concludingremarks

Thepresentanalyticalandexperimentalinvestigationofmicro- mixing quantificationusing chemicalprobemethods provides a betterunderstandingofthedifferentlimitationsthatariseinthe finaldeterminationoflocalturbulenceenergydissipationrate.The firstpartofthisstudydetailsthechemicalprobeprinciple,focusing on the iodide/iodate method and its two parallel-competitive chemicalreactions.Thesegregationindexleadstoameasureofthe mixing time by the engulfment micro-mixing model, and the correspondingturbulenceenergydissipationrateisdeterminedbya phenomenologicalmodeldevelopedbyBaldygaandBourne[18].

Thisstudyproposesagenericexperimentalproceduretoensure thevalidityandenhancetheaccuracyofsuchmethods.Inthefirst step,scalinganalysisbasedonthehydrodynamicsoftheinjection suggestscheckingifthreeindependentkeyparameters(ratiosof timescales),determiningwhichmechanismcontrolstheturbulence mixing,aresimplylessthanunity,criteriathatdefinethevalidity domainofthechemicalprobe.Inthesecondstep,optimalvaluesfor the operating conditions are set by sweeping the injected acid concentrationinanappropriaterange,estimatedbytheorderof magnitude of the mass energy dissipation rate. The retained concentrations correspond to equality of the micro-mixing timeandthereactiontime.Thethirdstepisaimedatcheckingif the measurement volume size ensures local measurement of

W S B

0 1 2 3 4 5 6 7 8 9 10 11 12 13

(m2 s-3)

Wake Shear Bulk

B: out of domain conditions for quantitative measurement S: significant

accuracy improvement W: all methods

are questionable

Present results with adaptive procedure

Chemical probe with classical procedure (Mohand Kaci 2007) LDV measurements (Habchi et al. 2010b)

Numerical simulations (Habchi et al. 2010b)

Fig.10.ImprovementinedeterminationbytheadaptivemethodforRe=12,500.

6000 8000 10000 12000 14000 16000

0 2 4 6 8 10 12 14 16 18 20

(m2 s-3)

Re Present results with adaptive procedure

Chemical probe with classical procedure (Mohand Kaci 2007) LDV measurements (Habchi et al. 2010b)

Fig.11.TKEdissipationrateeobtainedbythreedifferentmethodsattheshearlayer region(locationS)fordifferentReynoldsnumbers.