LES of turb ulent channelflow of a binaryelectrolyte

F. Gurniki 1, K. Fukagata1&3, S.Zahrai2 andF. H. Bark1

1 FaxenLaboratoriet,Royal Instituteof Technology,

Stockholm,S-10044Sweden

2 ABB, CorporateResearch,S-72178Vasteras,Sweden

3 Departmentof QuantumEngineeringandSystemsScience,

Universityof Tokyo, 7-3-1Hongo,Bunkyo-ku,Tokyo 113-8656,Japan

Abstract

Theturbulentdiffusionboundarylayer in a binaryelectrolyteis consideredat Schmidtnum-bersof 1, 10 and100 andexchangecurrentdensitiesbetween10

� 4A m� 2 and10

� 2A m� 2. A

numericalschemeis developedfor efficient investigationof thedynamicsby meansof largeeddysimulations.Themethodologyis examinedby detailedcomparisonswith documenteddatafromearlierlargeeddyanddirectnumericalsimulationsandgoodagreementwasfound. Applicationof themethodologyto electrochemicalmasstransferindicatedthat theexchangecurrentdensityseemsto havenegligible effecton themeanconcentrationprofilebut it influencesthestructureofthefluctuatingfield in avisiblemanner.

Key words: Butler-Volmer, large eddysimulation,near-wall region, high Schmidt number,forcedconvection.

Publishedin J. Appl. Electrochem.30 (2000),1335-1343.c�

Kluwer

1

1 Intr oduction

Numericalsimulationsof turbulentflowsfor industrialuseareusuallymadeby consider-ing averagefields.In suchsimulations,influenceof turbulentfluctuationsonthetransportof differentquantitiesmustbemodeled.Modelsareusuallyconstructedbasedon infor-mationon thestatisticalstructureof theflow. Making useof experimentaldatais ausualapproachfor developmentandverificationof models.

Levich [1], basedon theanalogywith laminarboundarylayerproposedthat in a tur-bulentboundarylayer, the thicknessof thediffusion layer, δc, would beof theorderofmagnitudeof δh � µ� ρD ��� 1� 3 whereδh is thethicknessof theviscousboundarylayer, ρ isthedensityof theelectrolyteandD andµ denotethemassdiffusivity andanddynamicviscosityof theelectrolyte,respectively. Levich’s proposalhasbeenverifiedexperimen-tally in differentstudies,for exampleby Lin etal. [2]. Thehighvalueof Schmidtnumber,µ� � ρD � , appearingin mostof thepracticalapplicationsis oneof importantfactorsresult-ing in thin diffusionboundarylayerandmakingexperimentalinvestigationof flowswithmasstransferproblematic.

Unfortunately, althoughelectrochemicalsystemsseemto besuitablefor experimentalinvestigations,seee.g.[3], themeasureddataareusuallyin form of integratedquantitiesrather than detailedinformation on the structureof the flow. As examples,Fouadetal. [4] andNewman[5] studiedmasstransferin electrochemicalsystemsandreportedglobal informationon theprocesssuchasthe total cell potentialdrop,electricalcurrentandglobal densityvariationof ionic species.As a result,other tools mustbe usedinorderto providedetaileddataon thestructureof theflow.

With improvementof digital computersandcomputationalmethodsduring the pastyears,simulationscouldbeusedasanalternative to physicalexperiments.Studyingofturbulent channelflow, for example,hasplayedan importantrole for modelingturbu-lence,in particularin vicinity of arigid wall. A purenumericalsolutionof themathemat-ical equationswithout any model is usuallyreferredto asdirect numericalsimulation.Clearly, the advantageis that no modelsare involved with the costof being limited tolow Reynolds numbers. An alternative route is to uselarge eddy simulations,whereonly smalleddieswhich have a moreuniversalbehavior aremodeled.Using largeeddysimulations,higherReynoldsnumbercanbestudied.

Lyons et al. [6] useddirect numericalsimulationsfor studyingthe turbulent heattransferin a channelat Prandtlnumbersequalto 0 � 7 and1. Theresultswerecomparedwith experimentaldataandgoodagreementwasfoundfor first andsecondordermomentsbut deviationswereobservedfor higherordermoments.

Papavassiliouet al. [7] performeddirectnumericalsimulationof a turbulentchannelflow with masstransferanda differentialvalueof thepassive scalarat thewalls. In thatwork, a Lagrangianapproachwasusedfor simulationof the flow at Schmidtnumbersvarying from 0 � 1 to 2400. Goodagreementwasfound betweenthosesimulationsandothersimulationsusingEulerianformulation.

Calmetet al. [8] reportedaboutstatisticalstructureof a turbulentchannelwith masstransferusinglargeeddysimulations.In thatstudy, thevalueof thepassivescalarwasset

2

to a givenvalueat thewall andsimulationswereperformedat Schmidtnumbers1, 100and200. It wasfoundthatthelocationof themaximumfluctuationsof theconcentrationfield wasdependenton bothSchmidtnumberandReynoldsnumber.

In theearlierstudies,boundaryconditionsfor thepassivescalarweresuchthatdirectuseof presenteddatafor modelingof masstransferat theelectrodein anelectrochemicalsystemcould not be possible. The goal of this study is to provide detailedinforma-tion aboutthestatisticalstructureof a turbulentboundarylayerwith masstransferusingboundaryconditionsof interestfor electrochemicalsystems.Statisticaldatapresentingthe structureof the boundarylayer are to be obtainedby performinglarge eddysimu-lations. The hydrodynamicpart of the computationperformedby Zahraiet al. [9] iscompletedin thepresentstudyto treatthemassconservationequation.

Thepaperis organizedin thefollowing way: themathematicalproblemis formulatedin section2 andthenumericalprocedureis detailedin section3. Resultsfrom numericalstudiesarepresentedin section4 andconclusionspresentedin section5.

2 Formulation of the problem

In an electrochemicalsystem,the passageof electricalcurrentthroughthe electrolyteis ensuredby transportof ions from oneelectrodeto theother. The chemicalreactionstake placein a thin sheath,sometimescalleddoublelayer, at thevicinity of thesurfaceof electrodes.In this thin region, no equilibriumcanbeassumed.In this study, no freechargesarepresentandno chemicalreactionstake placein the bulk of the electrolyte.The electrolytein a region far from the electrodescan be assumedto be in chemicalequilibrium andelectricallyneutral. For an electrochemicalcell, the maximumcurrentthat canpassthroughthe cell dependsdirectly on the efficiency of masstransportneartheelectrodes.

In general,therearethreecontributionsto thetransportof mass:diffusion,convectionandmigration[10]. In many applications,migrationis of importanceonly in thedoublelayerwherestrongelectricfieldsarepresent.Verycloseto thewalls,wherethevelocityofthefluid is negligibly small,thecontribution from convectionis negligible in comparisonwith theothermechanisms,while in thecoreof theflow, convectioncanbeconsideredasthemaintransportmechanism.Diffusionis usuallythemechanismwhich is strongneartheelectrodesandis responsiblefor transportof massfrom thewall regionout to thebulkof theflow.

In this study, turbulent flow of a binary electrolyteoutsidethe doublelayersin aninfinitely large cell is considered.The electrodesareassumedto be at a distanceof 2δandtheflow is drivenbyaconstantpressuregradient,strongenoughto makegravitationaleffectsnegligible. Thecoordinatesystemis chosensothattheelectrodeswill belocatedat y � δ.

Theelectricalneutralityof theelectrolytecanbeexpressedasz1c1 � z2c2 0, whereci andzi denotetheconcentrationandthechargenumberof speciesi, respectively. The

3

transportequationsfor the massin a binary electrolytecan be reducedto one singlediffusion-convectionequationfor a concentrationfield definedby c z1c1 � z2c2. Itcaneasilybeshown thatthechargeneutralitywill thenbeidenticallysatisfied.

Undertheaboveconditions,theflow canbedescribedby theNavier-Stokesequationsfor anincompressiblefluid, thelaw of conservationof massfor afluid atconstantdensityandatransportequationfor thementionedreducedconcentrationfield. Usinguτ, thefric-tion velocity, asthecharacteristicvelocityscale,l �� ν � uτ, thetypical lengthscalein wallunits,whereν is thekinematicviscosityof theelectrolyteand l � � uτ asthe typical timescale,the continuity andNavier-Stokesequationscanbe written for the instantaneousvelocityvectoras

∇ ��� u �� 0 (1)

and∂u �∂t � ��� u � � ∇ ��� u � � ∇ � p� � ∆ � u � , (2)

whereu denotestheinstantaneousvelocity field andp is theinstantaneouspressurefieldandis nondimensionalizedby ρu2

τ. Thetransportequationfor thereducedinstantaneousconcentrationfield, usingtheabovescales,canbewrittenas

∂c�∂t � � � u ��� ∇ ��� c�� 1

Sc∆ � c� , (3)

whereSc ν � D is theSchmidtnumberandD is thediffusivity constantcorrespondingto thereducedconcentration,seee.g. [11]. Theconcentrationfield canbemadedimen-

sionlessby usingthemassflux at thewall, i.e. C ��� Duτ

�∂c∂y wall

, wherec corresponds

to theaveragedvalueof theinstantaneousconcentration.All quantitiesarethusscaledinwall units thereforethesuperscript’+’ canbedroppedthroughouttheremainderof thispaper. Nevertheless,for moreclarity, thesuperscriptcanbemaintainedin thecaptionsofthefigures.

For thehydrodynamicalequations,no-slipconditionis usedat theelectrodes,i.e.

u � x !" Reτ ! z! t �# 0. (4)

Notethatthehalf-channelwidth, δ, will beequalto Reτ in wall units.For theconcentra-tion field differentboundaryconditionsareconsidered.As a first step,in orderto makecomparisonswith earlier numericaland experimentalinvestigations,the caseof masstransferwhentheconcentrationis setto agivenvalueat thewall is considered,i.e.

c � x !" Reτ ! z! t ��� co. (5)

For electrochemicalsystems,otherboundaryconditionsarealsoof interest. Here,twosimplifiedconditionsareusedfor modelingthemassflux from theelectrodes:

∂c∂y � x !$ Reτ ! z! t ��%'& o, (6)

4

i.e. the flux is given at the electrodes,andfinally a boundaryconditionwhich allowsfluctuationsof theflux with thevalueof theconcentrationat theelectrodes,

∂c∂y � x !" Reτ ! z! t ��%)( o * c � x !" Reτ ! z! t �+ c1 , . (7)

In the above relations,c1, ( o and & o are given constants.The form of the boundaryconditionin (7) comesdirectly from the first term in a Taylor expansionof the Butler-Volmermasstransferlaw, seee.g.[11]. ( o isproportionalto theexchangecurrentdensityof thechemicalreactionat thewall, andis equalto( o io

2FD1c1Reτ, (8)

whereio is theexchangecurrentdensity, F is theFaraday’s constant,D1 is thesaltdiffu-sivity constantfor themetallic ionic speciesof thebinaryelectrolyte,andc1 a referencevaluefor thereducedconcentrationatequilibrium,which is thesameasin thecoreof thechannel.If anelectrolytewith low electricalpotentialgradientsis considered,thebound-arycondition(7) is agoodapproximationof theelectrochemicalmasstransferlaw at theelectrodes.Experimentally, sucha configurationcanbeobtainedwith anadditionalionthatdoesnot take partof thechemicalreactionsandreducesconsiderablytheelectricalresistanceof theelectrolyte[5].

Equation(7) canalsobewritten as

∂c∂y � x !$ Reτ ! z! t ��% * ( o � c � x !" Reτ ! z! t �- �& o, , (9)

where & o io2FD1Reτ

. In this form, it is clear that for low valuesof ( o, this boundaryconditionapproachesequation(6).

It is worth noting that theexposedboundaryconditionsabove arelinearandremainthesamewhenconsideringthetime-averagedvalues.

Thesetof equations(1), (2) and(3) togetherwith (4) andoneof theconditions(5),(6) or (7) make a well-posedmathematicalproblemwhich is to be solved numerically.Thetimedependentsolutionof theequationsystematsufficiently largeReynoldsnumbergive thedesireddetailedinformationon thestructureof thediffusionlayerin anelectro-chemicalsystemdrivenby forcedconvection.

3 Numerical procedure

In this sectionanoverview of the largeeddysimulationmethodologiesfor thefluid ve-locity field andtheconcentrationfield is presented.Themethodologyusedfor thefluidvelocity field is takenfrom theearlierwork on thesimulationof turbulentchannelflowsby Zahraiet al. [9].

In this large eddysimulation,averagingover the volumeof the computationalcellsis usedas the filtering function. The subgridmodel is an anisotropicversionof the

5

Smagorinsky eddyviscositymodel.A rectangularcomputationalcell, numberedM, hasthewidth in i-direction,∆xM

i , theareaof surfacenormalto i-direction,∆sMi ∆xM

j ∆xMk ,

anda volumesize,∆vM ∆xM1 ∆xM

2 ∆xM3 . ThemodeledNavier-Stokesequation(2) aver-

agedoverthevolumeusingthecell M, whichgivesthedynamicsof largeeddiesresolvedon thegivenmesh,canbeexpressedas

∆vM ∂uMi

∂t . 3

∑j / 1

∆sMj 02143 u576 j 8 M

i u 596 j 8 Mj 1 u 5 � j 8 M

i u 5 � j 8 Mj : 1;3 p 596 j 8 M 1 p5 � j 8 M : δi j <

= 3

∑j / 1

∆sMj 0 = 3 τ 596 j 8 M

i j 1 τ 5 � j 8 Mi j :

= 1>2 ? 2@ 2

j A�BBB sMi j BBB 596 j 8 M C

s596 j 8 Mi j 1 BBB sM

i j BBB 5 � j 8 M Cs5 � j 8 Mi j D�E ,

(10)

whereτi j is the resolved stresstensor, and si j the instantaneousrate of strain tensorF∂ui

∂xj� ∂u j

∂xi G , HHH sMi j HHH F�I

∑3i J 1∑3

j J 1sMi j

2 G and K 2j � ∆xM

1 ∆xM2 ∆xM

3 � 2� 9 � ∆xMj � 4� 3. Con-

sideringaninstantaneousfunction f � x� , fM

standsfor thefilteredversionof f at thecen-ter of cell M, whosevolumeis ∆vM. Thesuperscript� j � M denotesthattheconsideredfilteredquantityis evaluatedon thecell surfacewhosecenteris thevectorxM 1

2∆xMj ej

andwhosenormaldirectionis ej .It is worth mentioningthat theabove equationis not differencedbut only filteredby

integratingtheNavier-Stokesequationson a cell volume.Thetransportthroughthecellsurfacesdueto fluctuationsof smallersizesthanthe sizeof the cell aremodeledby aSmagorinsky-like subgridviscosity. ThemodifiedSmagorinsky modelhasthepropertyof vanishingin theregionswheretheresolutionis goodenoughin ananisotropicmanner.As a result, in the wall-region wherethe meshsize is chosenso that the detailsof theflow could be studied,the influenceof the model becomesweak without any explicitdamping. The fine resolutionof the flow nearthe wall eliminatesthe needof any wallfunctions.ThemodelconstantL wassetto 0 � 08,assuggestedin [9]. For moredetailedinformation aboutlarge eddy simulationand accuracy of the predictionthe interestedreaderis referredto Zahraiet al [9]. Theflow is drivenby a bodyforce,or by a pressuregradientconstantin spaceandtime. In otherwords,thepressuretermis decomposedintoa meanpressuregradient,i.e. a givenconstantequalto 1 in wall units,anda fluctuatingpartwhich is to besolvedtogetherwith thevelocity field.

Similarly, themasstransportequationcanbefilteredat eachcell M, which candifferfrom onesusedfor thevelocity field, andwill bereducedto

∆vM ∂cM

∂t . 3

∑k/ 1

∆sMk 0 1 3 c 596 k8 Mu 596 k 8 M

k 1 c 5 � k8 Mu 5 � k8 Mk : <

= 3

∑k / 1

∆sMk MON 1

Sc= 1>

2 ? 2@ 2k

Sct P 3 BB sMi j BB 596 k8 M C

j 596 k 8 Mk 1 BB sM

i j BB 5 � k 8 M Cj 5 � k8 Mk :RQ ,

(11)

6

Table1: Meshspecification.

Nx Ny Nz ∆x ∆z ∆ymin ∆ymax

Mesh1 S 1 S 1 32 42 96 70.6858 11.781 1.4062 18.92Mesh1 S 2 S 1 32 84 96 70.6858 11.781 0.7031 9.46Mesh1 S 6 S 1 32 252 96 70.6858 11.781 0.2344 3.16

where jk is theinstantaneousmassgradient,jk ∂c∂xk

, and j TVU k W Mk standsfor thefiltered

massgradientevaluatedon thecell surfacewhosecenteris thevectorxM 12∆xM

k ek andwhosenormaldirectionis ek. TheturbulentSchmidtnumber, Sct , wassetto 0 � 25,whichmaybea reasonablevalue,sincetheturbulentPrandtlnumberPrt , in thecasesof largeeddysimulationwith heattransfer, is usuallyproposedto bebetween1� 2 and1� 3, seee.g.[12].

Thecomputationaldomainis periodicin thestreamwiseandthespanwisedirectionswith correspondingperiodicity lengths,λx and λz. The distancebetweenthe channelwalls is 2Reτ. λx andλz arerespectively 4 S π S Reτ and2 S π S Reτ, seefigure1. Peri-odic boundaryconditionsareappliedfor thepressurefluctuationsandtheinstantaneousvelocityandconcentrationin thestreamwiseandspanwisedirection,i.e.,

u � x � λx ! y! z � λz ! t �X u � x ! y! z! t � , (12)

p � x � λx ! y! z � λz ! t �X p � x ! y! z! t � , (13)

c � x � λx ! y! z � λz ! t �X c � x ! y! z! t � . (14)

As a uniquefeatureof this study, theuseof differentmeshsystemsfor thefluid velocityfield andtheconcentrationfield shouldbepointedout. Table1 shows thecharacteristicsof threemeshes.TheonedenotedasMesh1 S 1 S 1 is usedfor thefluid velocity field.The meshesusedfor the concentrationfield areMesh1 S 2 S 1, andMesh1 S 6 S 1.Thefinestmesh,Mesh1 S 6 S 1, wasusedfor thethecalculationsperformedatSc 100only. Thegridsareuniformin themeanflow direction.They arestretchedin thedirectionnormalto thewall, with thefinestspacingat thewalls. Accordingto Calmetet al. [8],at leastthreemeshpointsin the wall-normaldirectionarerequiredwithin the diffusive

sublayer, y Y 5

Sc1� 3 , to resolve the unlinearbehavior nearthe wall. The meshsystems

usedin thestudyfulfill this requirement.Theturbulentvelocityfield andtheconcentrationfield werecomputedsimultaneously

in the following way: First, the volumeaverageNavier-Stokesequation(10) is solved,and thenthe fluid velocity and the eddyviscosityare interpolatedto the meshsystemusedfor theconcentrationfield. With theknown velocityfield, thevolumeaveragemasstransportequation(11) is solved.Thisprocedureis repeatedby returningto thefirst stepandcomputationfor thenext time step. It is worth mentioningthat theeddydiffusivity

7

Z Z Z Z Z Z Z ZZ Z Z Z Z Z Z Z

Z Z Z Z Z Z Z ZZ Z Z Z Z Z Z Z

[\λx

Z Z Z Z Z Z Z Z^]ZZZZZZZZ _ λz [`

aaa b x,u

y,v

z,w

`c 2Reτ[Reτ 180

Meanflow

Figure1: Thegeometryof thecell andthechosencoordinatesystem.

is computedonly on the grid wherethe velocity field is treated. The filter scalesareprescribedby this grid, andtheeddydiffusivity is theninterpolatedto themeshsystemusedfor theconcentrationfield.

A fourth orderLagrangianinterpolationschemeis usedto ensurea sufficient accu-racy in interpolationof turbulent velocity field, seee.g. Wang & Squires[13]. Thefilteredmasstransportequationis integratedusinga threestep,third ordertensorviscos-ity scheme,[14] [15]. Theadvectionterm in theright handsideof thefilteredequationwasapproximatedwith QUICK scheme[16] to keepasecondorderspatialaccuracy anda numericalstability at thesametime. Thediffusiontermwasdiscretizedusingcentraldifferentiation.

Theinitial profile in acrosssectionfor thevelocitywassetparabolic.Themasstrans-fer calculationwasnotaddeduntil thehydrodynamicsimulationhadreachedstatisticallyequilibriumstate.Theinitial profile in a crosssectionfor thereducedconcentrationwaslinearfor thefirst calculation,at Sc 1 andwith a fixedvalueastheboundaryconditionat thewalls. Thecalculationswith otherboundaryconditionsor higherSc wereinitial-izedwith thecalculatedconcentrationfield obtainedwith thehighestavailableSchmidtnumber. Thecalculatedvariableswereconsideredto beat equilibriumwhenvariationofthestatisticalpropertiesin time weresmall.

4 Results

In thissectionresultsfrom largeeddysimulationsof amodelproblemfor electrochemicalsystemwith a binary electrodearepresented.Specialattentionis paid to the influenceof theexchangecurrentdensityon thestructureof turbulentboundarylayeranddetailedinformationis provided. Suchinformation,which is traditionallyfoundby experiments,is neededfor constructionof turbulencemodels. For masstransferin electrochemicalsystems,experimentaldataarevery limited andtherefore”numericalexperiments”mustbecarriedout.

The flow is assumedto take placeat Reynolds numberequalto 180 basedon the

8

cwall � ccwall

a)

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

y� δ

b)

0.1 0.2 0.3

0.3

0.4

0.5

0.6

0.7

y� δFigure2: a) The meanconcentrationat Sc 1 in the normaldirectionto the walls. .

: Presentstudy, LES, Reτ 180,cwall � 1� C � . : Presentstudy,�

∂c∂y wall

�'& o. dded : Presentstudy, ( o 1� Reτ. : Presentstudy, ( o 10� Reτ. fgfgf : Presentstudy,( o 100� Reτ. hihjhjh : Lyons[6], DNS,Reτ 150. ������� : Papavassiliou[7], DNS,Reτ 150.b) Sameasin a)zoomedon thenear-wall region.

wall friction velocity andthechannelhalf width. Theconditionsof computationof thevelocity field are exactly the sameas thosereportedin [9], and correspondto case2presentedin table 1 of that paper. The choiceof this Reynolds numberis motivatedby the fact that firstly, the hydrodynamicalfields are well studiedand documentedinliterature,andsecondly, thatdespitethelow valueof Reynoldsnumber, theflow is fullyturbulent.

Thepresentstudyfocuseson themeanconcentrationfield, RMS intensities,theeddydiffusivity, andthestructureof the instantaneousconcentrationfield. Of specialinterestin this studyaretheRMS intensityof theresolvableturbulentfluctuationfor concentra-tion, andtheaveragecorrelationsbetweenconcentrationandvelocityfluctuations.Thesequantitiesarenotedascrms, uc andvc, respectively. Transportequationsaresolved fortheconcentrationfield atSchmidtnumber1,10and100.Threedifferentboundarycondi-tions,asdefinedby equations(5), (6) and(7), areusedfor theconcentrationfield atsolidwalls.

Thehydrodynamicalpartof thenumericalcodeusedin this study, wasvalidatedbyZahraiet al. [9]. In thatwork it wasshown that theproposedmethodologyandmodelwereableto capturethe detailsof the flow in an accurateway. Simulationwith a res-olution astheonein thepresentwork predictedthemomentsof thevelocity field up tothird orderin goodagreementswith experimentaldata.As will befoundbelow, themasstransferpart is successfullytestedby comparisonswith previouswork [6, 7], for two ofthreeboundaryconditionsmentionedabove.

In figure2, themeanconcentrationprofile in thedirectionnormalto thewalls is pre-sentedfor flow atSchmidtnumberequalto one.Theconcentrationis madedimensionless

9

cwall c

100

1020

5

10

15

20

25

y

Figure3: ThemeanconcentrationatSc 1 in thenormaldirectionto thewallsandin the

near-wall region. . : Presentstudy, cwall � 1� C � . : Presentstudy,�

∂c∂y wall

�'& o. dded : Presentstudy, ( o 1� Reτ. : Presentstudy, ( o 10� Reτ. fgfgf : Presentstudy,( o 100� Reτ. klkik : c� 3 � 6 � ln � y� � � 1 � 7. hihlhjh : Lyons[6], DNS,Reτ 150. ����m� : Papavassiliou[7], DNS,Reτ 150.

by usingthereferenceconcentrationc� � Duτ

�∂c∂y w

. Goodagreementis foundwith re-

sultsfrom directnumericalsimulationsby Lyonset al. [6] andPapavassiliouet al. [7].The methodologyandsubgrid-scalemodelusedin the presentstudyareconsequentlyfoundto beaccurateenoughto modeltheturbulencemasstransporton smallscales.

In arealelectrochemicalcell, theexchangecurrentdensity, io, variesovermany ordersof magnitudefor differentmetallic ions. This canbe illustratedby consideringtypicalvaluesof io for hydrogenevolutiononvariouselectrodematerialsin acommonelectrolytewhich vary between10� 9 and10A m� 2, see[10]. For a referenceconcentrationin thecenterof thechannel,co in equation(8), equalto oneandasaltdiffusivity for themetallicion of 5 � 0 S 10� 10m2 s� 1, an exchangecurrentdensityof 10� 4A m� 2 correspondsto avaluefor thenon-dimensionalparameter( o in theButler-Volmer-like law, equation(7),of ( o 1� Reτ, and10� 2A m� 2 to a valueof ( o 100� Reτ.

Here,threedifferentvaluesof ( o, 1� Reτ, 10� Reτ and100� Reτ, in theButler-Volmer-likelaw, equation(7) havebeenexamined.Accordingto theabovediscussion,thechosenrangefor ( o in this studyis consequentlyappropriateto concernindustrialapplications.

Figure 3 indicatessimilar variation of the meanconcentrationprofile in the loga-rithmic diagram. With increasingy� , the concentrationprofile exhibits a buffer layercharacter, followedby a logarithmicregion. As shown onfigure3, theconcentrationpro-file in theouterregion fits well with the logarithmiclaw of c� 3 � 6 � ln � y� � � 1 � 7 aftery� 30. Theconstantsof thelogarithmiclaw foundin thepresentstudyagreewell withthemeanprofilespredictedby Lyonset al. [6] andPapavassiliouet al. [7]. At thesame

10

crms

100

1020

1

2

3

4

5

y

Figure4: RMS levelsof theconcentrationfield at Sc 1 in thenormaldirectionof the

walls. hihihih : Presentstudy, cwall m 1� C � . SnSoSoS : Presentstudy,�

∂c∂y wall

�'& o.dod�d : Presentstudy, ( o 1� Reτ. : Presentstudy, ( o 10� Reτ. flflf : Presentstudy,( o 100� Reτ. � � m :Lyons[6].

Schmidtnumber, Calmetet al. [8] andKaderet al. [17] founda logarithmiclaw fittingwith 2 � 12 � ln � y� � � 7 � 2. This differenceobtainedin the predictionof the constantsforthe logarithmiclaw canbe explainedin the following way. In a recentstudy, Xu et al.[18] foundthat theconstantof proportionalityin the logarithmiclaw for heattransferisproportionalto Prt � κ, whereκ is Karmanconstant,andequalto 0 � 41. This observationwasconfirmedby Meignenet al. [19]. Thespace-averagedturbulentSchmidtnumberinthelogarithmiclayerwascomputedin thisstudy, andwasfoundto beequalto 1 � 35. Theresultingvaluefor theratio Sct � κ is 3 � 29,a valuecloseto 3 � 6. Kaderet al. [17] foundaspace-averagedturbulentSchmidtnumberof 0 � 9, anda ratio Sct � κ of 2 � 19.

Figures2 and 3 presentalso the averageconcentrationcalculatedwith the Butler-Volmer-likeboundarycondition(7) andthethreedifferentvaluesof ( o. No influenceof( o, or theexchangecurrentdensity, canbeobservedon themeanconcentrationprofiles.

TheRMSlevelsfor theconcentrationfield arecalculatedatSchmidtnumbersand ( o

mentionedabove, andpresentedin figure 4. Very goodagreementis obtainedwith theresultspresentedby Lyonset al. [6], performedwith a fixedvalueof concentrationsetat thewalls. Moreover, RMS levelscomputedwith theboundarycondition(7) and ( o 10� Reτ, have intermediaryvaluesbetweenthetwo resultsobtainedwith ( o 1� Reτ and( o 100� Reτ. Therefore,atSchmidtnumberequalto one,RMS levelscloseto thewallarelikely to beamonotonefunctionof ( o, or io, andseemto decreasewhentheexchangecurrentdensityincreases.

An issueof interestcouldbetheasymptoticbehavior of theButler-Volmer-like,equa-tion (7), with respectto variationsof ( o, definedin (8). At high valuesof ( o, or at highexchangecurrentdensities,theright handsideof (7) becomesdominant.Thesystembe-

11

crms

100

1020

0.5

1

1.5

y

Figure5: RMS levelsof theconcentrationfield at Sc 10 in thenormaldirectionof the

walls. : Presentstudy, cwall � 1� C � . fpflf : Presentstudy,�

∂c∂y wall

%'& o.

crms

100

1020

0.2

0.4

0.6

0.8

1

y

Figure6: RMS levelsof theconcentrationfield atSc 100in thenormaldirectionof the

walls. : Presentstudy, cwall � 1� C � . fpflf : Presentstudy,�

∂c∂y wall

%'& o.

12

Ec

101

102

10−2

100

102

y

Figure7: The meaneddydiffusivity in the near-wall region at Sc 1. - h - h - h -: Ec 0 � 000775y3 [7]. : Presentstudy, cwall � 1� C � . flfjf : Papavassiliou[7]. dndod : Lyons[6].

comessensitive to smallvariationsof theconcentrationat thewall andtendsto balancethemfast.In thelimiting casewhere( o approachesinfinity, equation(7) becomesequiv-alentto equation(5), i.e. thesystemactsasonewith a fixedvaluesetfor concentrationat thewall. At low valuesof ( o, thefluctuationsof theconcentrationgradientat thewallaredumpedandthe behavior of equation(7) becomessimilar to a constantflux at thewall.

As a resultof theabove discussion,thediffusionboundarylayersimulatedwith theButler-Volmer-likeboundarycondition(7) ata low exchangecurrentdensity, is expectedto have similar behavior asthatwith a constantflux for the boundarycondition,andata high exchangecurrentdensity, similar to the casewith a fixed valueat the walls. Acomparisonbetweenprofilesfoundwith differentvaluesof ( o in figure4, confirmstheproposedbehavior.

Figures5 and6 show RMSintensitiescalculatedwith equation(5) and(6), atSchmidtnumberequalto 10 and100, respectively. As expected,valuespredictedwith a fixedvalue set at the walls, are lower than RMS intensitiespredictedwith a constantflux.Similarbehavior waspredictedat Schmidtnumberequalto one.

Sincetheabove discussiondoesnot accountfor valuesof Schmidtnumber, it is pos-sible,at any Schmidtnumber, to draw conclusionsabouttheeffect of very low andveryhigh exchangecurrentdensitieson RMS intensities,on the basisof calculationsmadeonly with afixedvalueandaconstantflux setat thewalls.

It is worthmentioningthatlargeeddysimulationswith afixedvalueaccordingto (5),or a constantflux (6), aremuchlesstimedemandingthansimulationswith theboundarycondition(7). Therefore,the above discussionhasan importantrelevancefor efficientstudiesof turbulentflow in electrochemicalcells.

13

Ec

101

102

10−2

100

102

y

Figure 8: The mean eddy diffusivity in the near-wall region at Sc 1. - h - h - h -:Ec 0 � 000775y3 [7]. : cwall n 1� C � . fqfqf : � ∂c

∂y wallo'& o. drdrd :

�∂c∂y wall

� cwallReτ

�& o .

Ec

101

102

10−2

100

y

Figure9: Themeaneddydiffusivity in thenear-wall regionatSc 1. : ( o 100� Reτ. fflf : ( o 10� Reτ. d�dod : ( o 1� Reτ.

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ucSc

y

Figure 10: At Sc 1, the streamwiseturbulent heatflux. Resultscomputedwith theButler-Volmer-like boundarycondition. Solid line, ( o 10� Reτ. Dashedlines, ( o 100� Reτ. The starsaccountfor the numericalresultsof Kawamuraet al. [20], withReτ 180andSc 1 � 5.

Themeaneddydiffusivity, definedasEc o vcsdcdy t , is presentedfor differentcasesin

figures7, 8 and9. Figure7 shows goodagreementbetweenthepredictionof thepresentstudy, the predictionof Papavassiliouet al. [7], andthe empiricalprofile proposedbyPapavassiliouetal. [7]. Figures8 and9 indicatethattheeddydiffusivity is not influencedby thevaluesof ( o, or theexchangecurrentdensityio. Thefactthattheeddy-diffusivityis found to be independentof ( o canbe relatedto the sameobservation for the meanconcentrationprofile,seefigures2 and3.

Streamwiseandwall-normalturbulent passive massflux arepresentedin figures10and11. Goodagreementis foundwith thedirectnumericalsimulationof Kawamuraetal. [20]. No datafrom previousresultswereavailablefor comparisonwith thepredictionsof thepresentstudyatSchmidtnumberhigherthan5. Thepresentstudyfinds,asthelastmentionedstudy, thatthescalarfluctuationscorrelatemorestronglywith thestreamwisevelocity thanwith its normal-wall component.Figure10showsthattheexchangecurrentseemsto influencethe rangeof strongcorrelationbetweenthe streamwisevelocity andthescalarfluctuations.Inversely, no influenceof theexchangecurrentis noticedon thecorrelationbetweenthewall-normalvelocityandthescalarfluctuations,seefigure11.

Iso-linesof theinstantaneousconcentrationfluctuationsfor Schmidtnumberequaltoone in a (x-z) sectionat y� 6 � 76 arepresentedin figure 12. The presenceof well-known streaky structureselongatedin thestreamwisedirectionis clearlyobserved.Suchstructuresare typical for turbulent flows in the viscoussublayer[21]. For the viscoussublayer, Kline et al. [21] consideredthosestructures,and their interactionswith theouterregion,ascentralin theenergy andmomentumtransferbetweentheinnerandouter

15

vcSc

y

Figure11: At Sc 1, the wall-normal turbulent heatflux. Resultscomputedwith theButler-Volmer-like boundarycondition. Solid line, ( o 10� Reτ. Dashedlines, ( o 100� Reτ. The starsaccountfor the numericalresultsof Kawamuraet al. [20], withReτ 180andSc 1 � 5.

x

a)

z

x

b)

z

Figure12: At Sc 1, snapshotcontourplotsof theconcentrationfluctuationsin a (x-z)planeat y� 6 � 76. Resultscomputedwith the Butler-Volmer-like boundarycondition.The incrementin scalaris 0 � 02. Solid lines represent0 � 0 u cv v�u cv vmax. Dashedlinescv vmin u cv v�u 0 � 0. a) ( o 10� Reτ. cv vmin w 0 � 323, cv vmax � 0 � 21. b) ( o 100� Reτ.cv vmin � 0 � 457,cv vmax � 0 � 186.

16

Rcv v cv v

rz

Figure 13: At Sc 1, spanwisetwo-point correlationcoefficients at y� 6 � 76. Re-sultscomputedwith theButler-Volmer-likeboundarycondition.Solid line, ( o 10� Reτ.Dashedlines, ( o 100� Reτ. Thestarsaccountfor thenumericalresultsof Calmetet al.[8] at y� 1 � 7, Reτ 640,andaDirichlet boundaryconditionat thewalls.

layers.Formationanddynamicsof thesestreaksarealsoof greatimportancein thestudyof turbulent masstransfer. They canbe consideredasvorticesin meanflow directionwhichcontributeto turbulenttransferin thedirectionnormalto thewall.

To computethe resultsin figure 12, the Butler-Volmer-like law hasbeenused. Theinfluenceof the exchangecurrenton the streaksis studied,by comparing12.a,whichpresentsafield computedwith ( o 10,and12.bwhichcorrespondsto ( o 100.Evenif it is notpossiblehereto giveaclearquantitativeapproximationof thespacing,it seemsthatthespanwisespacingdecreaseswith theincreasingexchangecurrent.

This result is confirmedby the computationof the two-point spanwisecorrelationcoefficientsof theconcentrationfluctuations,seefigure13. After reachinga minimum,thecoefficientsslightly oscillate.It is consequentlydifficult to determinethemeanstreakhalf-spacingr �z . For this reason,the presentstudychosethe methodof determinationusedby Calmetetal. [8]. Consideringthenormaldistancefrom thewall to thefirst localminimumof thespanwisecorrelationcoefficientasthemeanstreakhalf-spacing,Calmetetal. foundr �z 100.With thesamemethod,thepresentstudyfoundatSchmidtnumberequalto one,r �z around65 with anintermediaryexchangecurrent,andaround55 with ahigh exchangecurrent,seefigure13. Thesetwo last resultsarein goodagreementwiththecomputedspacingfoundby Kline et al. [21]. Calmetet al. [8] claimedthatthey haduseda spanwisespacingof themeshgreaterthanthetypical diameterof thestreamwisevorticesresponsiblefor thestreaks.They concludedthatthey couldnot captureproperlytherealsizeof thesevortices,andthattheir predictionof r �z wasoverestimated.

Figure15 presentstheinstantaneousconcentrationfluctuationsin thelogarithmicre-gion at y� 54� 1. The streaky structuresarenot visible any more. Here, ( o is taken

17

x

z

Figure14: At Sc 100,snapshotcontourplotsof theconcentrationfluctuationsin a(x-z)planeat y� 1 � 17. Resultscomputedwith a constantflux for concentrationat thewalls.The incrementin scalaris 0 � 02. Solid lines represent0 � 04 u cv vxu cv vmax. Dashedlinescv vmin u cv v u 0 � 04. cv vmin m 0 � 687,cv vmax 0 � 379.

x

a)

z

x

b)

z

Figure15: At Sc 1, snapshotcontourplotsof theconcentrationfluctuationsin a (x-z)planeat y� 54� 1. Resultscomputedwith the Butler-Volmer-like boundarycondition.The incrementin scalaris 0 � 02. Solid lines represent0 � 0 u cv v�u cv vmax. Dashedlinescv vmin u cv v u 0 � 0. a) ( o 10� Reτ. cv vmin w 0 � 43, cv vmax � 0 � 262. b) ( o 100� Reτ.cv vmin � 0 � 575,cv vmax � 0 � 325.

18

equalto 10 and100 only. At thesevalues,the exchangecurrentdensityhasnot influ-encedthe wall-normal locationof the streaky structures.This wasexpected,sincethestreaky structuresare traditionally observed in the diffusive laminarsublayer[8]. Butsupplementarycomputationswith differentvaluesof exchangecurrentwould allow toconfirmthis result,andextendit to abiggerrangeof exchangecurrents.

At Schmidtnumberequalto 100,theconcentrationfluctuationsareplottedin thevis-coussublayerin a section(x-z) at y� 6 � 76� Sc1� 3 1 � 17, seefigure 14. Again, thestreaky structureof fluctuationsareclearlyobserved.

5 Conclusions

A model problemfor studyingthe mechanismsof turbulent masstransferin an elec-trochemicalsystemby meansof large eddysimulationsis proposed.In order to makenumericalintegrationof the governingequationsat high Schmidtnumberseconomicanumericalschemeis developedin whichtwo differentmeshesareusedfor hydrodynamicvariablesandfor theconcentrationfield.

Firstly, it wasshown that themethodologywasableto successfullyreproducedocu-mentedliteratureresults.Both meanprofilesandfluctuationswerepredictedaccurately.

The influenceof the exchangecurrentdensityon the turbulent diffusion layer wasinvestigatedat Schmidtnumbersof 1, 10 and100. The studiedrangeof the exchangecurrentdensitywasbetweenio 10� 4A m� 2 andio 10� 2A m� 2.

Threedifferentboundaryconditionsfor theconcentrationfield wereusedat thewalls,i.e. thesurfaceof electrodes,a linearizedButler-Volmer-likecondition,agivenvalueanda givenflux. At Schmidtnumberequalto one,theexchangecurrentwasshown to haveno influenceon thestructureof themeanconcentration,neitheron theeddydiffusivity,noronthewall-normalturbulentheatflux. However, theRMSintensities,thestreamwiseturbulent heatflux, andthe lateraldimensionsof turbulencestructuresin the near-wallregionwereinfluenced.TheButler-Volmer-likeboundaryconditionpredictsresultscloseto thoseobtainedwith a givenvalueat thewall if theexchangecurrentdensityis high,andcloseto thoseobtainedwith agivenflux if theexchangecurrentdensityis low. Thisbehavior canbejustifiedmathematicallyandis likely to bevalid independentof thevalueSchmidtnumber.

In futurework themethodologywill beusedto provide anaccuratemodelingof thediffusion layer in anelectrochemicalsystem.Thepresentresultssuggestthat it is suffi-cientto considerthelimiting casesof agivenflux andagivenvalueat thewallsasafirststep.

19

Nomenclaturec Thepassivescalarcc Thepassivescalarat thecenterof thechannelcw Thepassivescalarat awallD Thepassivescalarmoleculardiffusivity � m2 s� 1 �

For thereducedconcentration,D is a functionof themoleculardiffusivities for cationandanionspecies

Ec Theeddydiffusivity � m2 s� 1 �F TheFaradayconstant(96485C mol � 1)g Thegravity � 9 � 81ms� 2 �io Theexchangecurrentdensitydueto masstransport� A m� 2 �k Thetemperatureconductivity � Wm� 1 K � 1 �( non-dimensionalizedboundaryconditionconstantl � Theviscouslengthscaleν � uτ � m�Nu TheNusseltnumberp Thepressure� Nm� 2 �Pr ThePrandtlnumberν

αPrt TheturbulentPrandtlnumberνt

αt

R Thegasconstant(8.31 J mol � 1 K � 1)Reτ TheturbulentReynoldsnumberuτδ

νsi j Therateof strain � ∂ui

∂x j� ∂u j

∂xi�

Sc TheSchmidtnumberνD

Sct TheturbulentSchmidtnumberνtDt

∆xMi Thesidelengthin thei-directionof thecell centeredin xM

T Theabsolutetemperatureat theelectrode(K)t Thetime � s�uτ Thefriction velocity

Iτwρ � ms� 1 �

u Thex-componentof thevelocity vector � ms� 1 �v They-componentof thevelocity vector � ms� 1 �w Thez-componentof thevelocityvector � ms� 1 �z Thenumberof electronsinvolvedin electrodereaction

20

Greeksymbolsα Theheatmoleculardiffusivity � m2 s� 1 �δ Thehalf-widthof thechannelin they-directionδi j TheKronecker deltaλ Theperiodicitylength

Exceptwhenspecifieddifferentlyin thetext.µ Thedynamicviscosityof solution � kg m� 1s� 1 �ν Thekinematicviscosity µ

ρ � m2 s� 1 �π Thetrigonometricpiρ Thefluid density � kg m� 3 �τw Theshearstressat thewall

�µ∂u

∂y wall� kg ms� 2 �

τi j Thestresstensorµsi j � kg ms� 2 �Subscripts� �y�y�9� Theinstantaneousvalueof � �y�y�9�� �y�y�9� Theaveragevalueof � �y�z�7� on time,

exceptwhenspecifieddifferentlyin thetextg A vector� �y�y�9� v = � �y�y�{� - � �y�y�9� Thefluctuationwith timeof � �y�y�{�� �y�y�9� M Thevalue( �y�z� ) at thecenterof thenumericalcell xM� �y�y�9� x Thex-direction� �y�y�9� y They-direction� �y�y�9� z TheThez-direction� �y�y�9�y� Thenon-dimensionalvalueof � �y�y�9�� �y�y�9� � Thevaluemadedimensionlesswith thewall parameters� �y�y�9� t Thecontribution from turbulence� �y�y�9� c Thevalueat thecenterof thechannel� �y�y�9� wall Thevalueat thewall� �y�y�9� o Thevalueof reference

Exceptwhenspecifieddifferentlyin thetext� �y�y�9� 1 A valuereferredto themetallicionic speciesExceptwhenspecifieddifferentlyin thetext� �y�y�9� 2 A valuereferredto thenonmetallicionic speciesExceptwhenspecifieddifferentlyin thetext

21

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