modelling and calculation of the current density distribution evolution at vertical gas-evolving...

Electrochimica Acta 51 (2005) 1140–1156

Modelling and calculation of the current density distributionevolution at vertical gas-evolving electrodes

Mandin Philippe∗, Hamburger Jerome, Bessou Sebastien, Picard GerardENSCP-LECA U.M.R. C.N.R.S. 7575 Laboratoire d’Electrochimie et de Chimie Analytique 11,

Rue Pierre et Marie Curie, 75005 Paris, France

Received 13 April 2005; received in revised form 1 June 2005; accepted 1 June 2005Available online 15 July 2005

Abstract

During industrial electrolysis, for hydrogen, dichloride or aluminium production, there is bubbles creation at one or two electrodes whichimply a great hydrodynamic acceleration but also a quite important electrical field disturbance. This disturbance can lead to the modificationof the local current density and to anode effects for example. There is few works concerning the local modelling of coupled electro activespecies transport and electrochemical processes in a biphasic electrolyte. There are also few local experimental measurements in term of

ts like thepective roles

not known.lectrolysisd the currentith speciescell with a

r

ect issts likeden-tions alu-tese gasorderith

eredt hasensity

chemical composition, temperature or current density which would allow the numerical calculations validation. Nevertheless, effecanode effect, particularly expensive on the point of the process efficiency, should need a better understanding. Nowadays, the resof the local temperature increases, the electro active specie composition or the transport properties modification due to bubbles are

The goal of the present work is the modelling and the numerical simulation of the vertical electrode configuration for a biphasic eprocess. Bubbles presence is supposed to modify the electrical properties, and then the electro active species diffusive transport andensity. Bubbles are also motion sources for the electrolysis cell flow, and then hydrodynamic properties are strongly coupled wtransport and electrical field. The present work shows hydrodynamic and electrical properties in a laboratory scale electrolysisvertical electrode. The numerical algorithm used was the finite volume used in the computational fluid dynamic software Fluent®.© 2005 Elsevier Ltd. All rights reserved.

Keywords: Electrochemical; Modelling; Vertical; Biphasic; Electrolysis

1. Introduction

Gas release and induced fluid flow over electrodes existin many electrochemical processes such as chlorine produc-tion, water electrolysis, alumina reduction, and many otherchemical processes. The hydrodynamical properties and thegas-flow motion in electrochemical cells is of great practi-cal interest in electrochemical engineering science since thedispersed phase modifies the electrical properties of the elec-trolyte (as well as mass and heat transfers), and thereforemodifies the macroscopic cell performances. In most cases,this phenomenon has to be avoided, but, in some other pro-cesses, the gas flow rate has only to be controled; this is the

∗ Corresponding author. Tel.: +33 155426380; fax: +33 144276780.E-mail address: [email protected] (M. Philippe).

case namely for gas production (H2, O2, Cl2, . . .) [1] and othespecial processes such as, e.g. chemical engraving[2,3]. In allthese different electrochemical processes, a coupling effparticularity strong (as shown by Hine[4], e.g. for simple gaevolving electrodes) because bubble-dispersed phase acan electrical shield, the shielding effect depending on thesity of the bubbles, which is namely the gas volume fracof the dispersion. Except in some special cases such aminium extraction (Heroult-Hall process), where high raof coalescence occur under the horizontal anode face, thphase is clearly composed of small dispersed bubbles (of magnitude 100�m) comparable to little rigid spheres wa constant radius ([5–7]). Though many authors[8–10]men-tion that the diameter of small bubbles can be considconstant but dependent with the local current density, ibeen here considered a constant diameter with current d

0013-4686/$ – see front matter © 2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.electacta.2005.06.007

M. Philippe et al. / Electrochimica Acta 51 (2005) 1140–1156 1141

Fig. 1. Vertical gas-evolving electrode: left is the geometric configuration and right is a current distribution example (y: electrode height;jx(y): local currentdensity).

value. As mentioned by many authors, the absolute velocityof bubbles (in the cell reference frame) is a key parameterof the process ([4] and[11,12]) as well as bubble diameter:the higher the local bubble velocity, the lower the local gasfraction. Though the hydrodynamical two phase flow prob-lem is not easy to solve a priori, some coarse assumptionsare usually made to calculate the hydrodynamic flow in elec-trochemical cells. But some precious information are lost,especially in region where there are strong velocity gradientsor some void fraction non uniformity. Moreover, experimentsare often very different from one paper to another, and it istherefore often difficult to extract some general information,especially for flow structure in cells.

1.1. Problem standing

Fig. 1is a scheme of a classical electrochemical cell, witha single vertical gas-evolving electrode configuration.

The purpose of this work is to compute a numerical solu-tion of the coupled electrical and hydrodynamical propertiesin such an electrochemical cell. An accurate tool for coupledtwo-phase flow and electrochemical properties seems to bean important task to develop for industrial applications andtherefore the aim of the present study is also to validate anoriginal coupled numerical simulation strategy, using compu-tational fluid dynamic (CFD) software to solve the fluid flowr ase.T tima-t

( nsityden-gianlsorow-

( ndt ofbe

necessary and helpful for a better understanding of thebubble release effect and help for further modelling. Atlast, cell performances could be optimised and enhanced.According with small current density assumption (meancurrent densityI/A lower than 4000 A m−2, accordingwith Tobias experiments[13]), the electrical resistanceincrease due to bubble release is possible to estimate.The anode effect modelling need to work at larger cur-rent density values and need to take into account thebubble shadowing and coalescence (see Vogt et al. andWuethrich et al.[10] and[15,16]).

2. Hydrodynamic and electrical coupling

The primary current density distributionj (A m−2) in anhomogeneous electrolyte cell is, at the macroscopic scale,uniform along electrode, and is horizontally flowing throughthe cell from anode to cathode. But, as a well known effect, thegas dispersion along electrode is not uniform, since bubblesrise vertically due to Archimede acceleration. Thus, the voidfraction increases with increasing electrode height (seeFig. 1)and the conductivity decreases with altitude. Then currentdistribution at anode is modified, and also local gas flow ratedue to Faraday’s law of electrolysis, and so on (Fig. 2).

gasp ith al sfl nalt -t use o bep longe andt fort voidf t thee by

esulting from an electrochemically driven bubble-relehe main purpose of this study is namely to have an es

ion of:

1) the coupling between bubble release and current deat electrode surface, at the beginning for low currentsity values. This assumption allows the use of Lagranmodelling (see biphasic flow modelling part) and aallows neglecting the shadowing resistance due to ging bubbles.

2) Orders of magnitude of induced fluid flow in the cell aits effect on current distribution. This developpemennumerical simulation using CFD software seems to

In the case of an electrochemical reaction withroduction, gas is produced at the electrode surface w

ocal gas mass flow rateq [kg m−2 s−1]. The local gas masow rate q due to electrochemical reaction is proportioo the local current densityj using the first law of elecrolysis. The distributionq(y) is also, for a homogeneolectrolyte, uniform, if the electrodes are considered terfectly conducting. Since gas bubbles rise upwards alectrode, non-uniformity is generated along electrode

he void fraction of gas, which is a hindering effecthe electrical current, increases along electrode. Theraction distribution depends on local gas production alectrode surface, and on the fluid flow motion induced

1142 M. Philippe et al. / Electrochimica Acta 51 (2005) 1140–1156

Fig. 2. Calculation flow-sheet for the coupling effect in the electrochemicalcell due to the presence of bubble release.

the bubbles. Therefore there is a two way coupling of thephysical problem (seeFig. 2), and the two coupling relationsare:1-The Faraday 1 aw of electrolysis:

q = Mjn

(nF )(1)

which determinates the local rate of produced gas by theelectrochemical reaction, in the case of no back reaction(efficiency = 100%).q is the gas phase local mass flowrate, jn the current density normal to boundary,n is thecharge number,F (96485 C mol−1) is the Faraday constant,and M is the molar weight (2 10−3 kg mol−1 for hydrogenbubbles).2-Bruggeman relation:

κ = κ0(1 − ε)3/2 (2)

This is the phenomenological law chosen in the presentwork for the local electrical conductivity sensitivity with bub-bles presence.κ0 stands for the electrical conductivity of thepure electrolyte, andε stands for the local void fraction.

The calculations, as shown with algorithm presentedFig. 2, focused the effort in the coupling programmingbetween bubble electrode mass flow rate and the local currentdensity j. Surface shadowing leads to the adherence resis-tance contribution, well described with Wuethrich discretemodelling works[10], whereas volumetric shadowing leadst intoa cov-e orsl ni lowc mod-e tionl ctualm du

The adherence resistance has been neglected accordingwith the low current density assumption.

To take into account the adherence contribution (the sur-face one), it should be necessary to measure or suppose thelaw betweenθ andε value at surface relation. This will bedone in future work. TheFig. 2 algorithm will be modifiedwith a stage between void fraction and potential field cal-culation with the surface shadowing functionθ calculation.For simplicity, assumptions are made in both hydrodynamicand electrical problem solution procedures in order to focusmainly on the coupling aspect we are interested in, particu-larity gas evolution on electrodes.

1. Electrical conductivity of the electrodes is supposed verylarge (infinite). Therefore, the imposed electrical potentialfor each electrode is constant.

2. Liquid electrolyte is supposed to be homogenous in termof dissolved species chemical composition and tempera-ture.

3. Heat exchanges and therefore energy equation are not con-sidered here, assuming that the temperature is uniform,though bubble layer modify heat transfer.

In this two-way coupling (hydrodynamic-electrical), thefluid flow itself also needs a two-way coupling calcula-tion between continuous and discrete phase. From this, andreminding that the local gas void fraction calculation is asineq int hasefl le tod vol-ui

thorsi recenty pro-c icals.T ringb elec-t ts inac h oft icalc mati-c in thep vans[

stlyw vent beeni tedn rrentd tedg rticalg tind[ ic-e rode

o the diffusion resistance effect which will be here takenccount alone. In the present work, the classical surfacerage functionθ (surface void fraction), often used by auth

ike Vogt [11], Tobias[13] or Wuethrich[10], has not beentroduced. The area shadowing effect is negligible forurrent density values studied here. The discrete phaselling can only be used under this assumption (void frac

ower than 10%). Due to the reduced active area, the aaximum current density is larger (j/θ) than this calculatender the present work assumptions.

ua none condition to determine the current distributionhe electrolytic cell, it is necessary to calculate the two-puid dynamic (gas–liquid) properties. It should be possibevelop a numerical tool; in the present work, the finiteme numerical method programmed in the Fluent® software

s used.This coupled problem has been treated by many au

n the past years, and has had an increasing interest inears for the purpose of optimizing electrochemicaless through better design of cells, materials and chemobias [13] developed a mathematical model, consideubble release in electrochemical cell with a stagnantrolyte medium (1959). This theoretical approach resul

differential equation for the void fractionε of gas in theell, which was considered to be uniform over the widthe cell. It was a one dimensional modelling with the vertoordinate alone to be taken into account. Other matheal approaches and experimental works have been doneast and are summerized in the paper of Ziegler and E

17].The most recent works about this subject deal mo

ith numerical simulation but are not numerous, ehough the expansion of computer performance hasmportant. Byrne and al[18,19]have nevertheless calculaumerically primary, secondary and pseudo tertiary cuistribution in chlorate membrane cell with a complicaeometry, and provided some experimental data on veas-evolving electrodes under forced convection. We

20], and Dahlkild[5], solved the coupled hydrodynamlectrical problem in the cases of a single vertical elect


and in the case of Byrne’s experiments ([18,19]), with aparticular attention to diffusion forces acting perpendicularto the wall (normal dispersion forces). The commercial codeCFX was used. Bech[21] solved the coupled problem in thecase of simplified aluminium reduction cell geometry andusing a discrete phase model in the code FLUENT 4.4.

3. Electrical model

3.1. Voltage components

The electrochemical cell is defined by the applied voltage�U/(V) for given overall electrical current flowing throughthe cell. The applied voltage decomposes as follows:

�U = E0 + ηr,a + ηc,a + RI + ηr,c + ηc,c + [�Ua+�Uc]

(3)

where E0 is the reversible thermodynamical potential ofchemical reaction involved in the cell,RI is the ohmic voltagedrop in the biphasic electrolyte,ηr is the reaction overpoten-tial, respectively for anode and cathode (respective subscripts(a) and (c)),ηc the concentration over potential, respec-tively, for anode and cathode (subscripts (a) and (c)), and�Ua +�Uc ohmic voltage drop in the anode and cathodem eo tfl ndb sual,m otent daryt

j

w sur-f ity,α ripts( -s

thed culed facei thee con-d s tot rationo tra-t eededT ds tot

withc g thek s, isd s, the

first stage consists to solve the electrical problem with sim-plified boundary conditions which allow the calculation ofthe primary current density distribution. Under this assump-tion, reaction and concentration over potentials at electrodesare neglected. With this electrical modelling, only the ohmiccomponent of resistance is taken into account. Thus, Eq.(3)reduces to:

�U = E0 + RI (5)

which is the simplest cell electrolysis modelling. The ohmicvoltage is then simply the applied voltage minus the thermo-dynamical potential.

3.2. Current distribution

The current distribution is determined by solving the cur-rent conservation Eq.(6) associated with phenomenologicalOhm’s law (7):

div j = 0 (6)

j = κE = −κ gradΦ (7)

wherej (A m−2) is the current density vector,Φ (V) is theelectrical potential defined in the whole domain of the cell,κ the electrical conductivity (S m−1) defined in the wholedomain of the cell. Combining Eqs.(6) and (7) leads tot ntial(

d

f er-i Ther andB tionε formo elec-t hasicl

then l area mod-e foren

4

blesr omep duc-t xistsa ere-f hasefl rsedp rface).

aterial, which are neglected here.R is the overall resistancf the electrolyte, andI = ∫ ∫

Aj ds is the integral curren

owing through the cell. In this work, current density aubble mass flow rate are locally calculated and not, as uean values over the electrode area. The reaction over p

ial ηs depends on the current density normal to the bounhrough Butler–Volmer Eq.(4):

n = j0

(exp

(αηsF

RT

)− exp

(− (1 − α)ηsF

RT

))(4)

here,jn is the local current density normal to electrodeace component (A m−2), j0 is the exchange current densis the transfer coefficient for anode and cathode (subsc

a) and (c)). Further isR = 8.314 J mol−1 K−1 the gas contant,F = 96487 C mol−1 the Faraday constant.

Over voltage due to surface reaction is confined inouble layer, which size is of order magnitude of a moleimension, which is, macroscopically speaking, the sur

tself. Therefore, relations (3) and (4) should be used inlectrical modelling to define the electrode boundaryition. The numerical resolution of this equation lead

he secondary current distribution. Because the concentver potentialηc implies the electro active species concenion at electrode surface, the species mass balance is nhe resolution of this coupled mathematical system lea

he ternary current distribution.The strategy to solve such non linear problems

oupled equations and boundary conditions, expressininetics of the electrochemical reaction at the electrodeifficult and then need patience. To ensure robustnes

-

.

he classical transport equation for the electrical pote8):

iv(κ gradΦ) = 0 (8)

In the particular frame of this study, conductivityκ is aunction of the local void fraction, and therefore only a numcal method is likely to be used to compute a solution.elationκ = (�(x, y, z)) has been defined by many authorsruggeman’s one is the most commonly used. Void fracdepends on many factors, and is often strongly non univer cell section since gas evolution is often confined atrode surface (see next section). There then exists a bipayer in the electrode vicinity.

In this procedure, using a finite volume algorithm forumerical solving, the boundary values of the potentiassumed to be constant at electrodes. Then, under thislling hypothesis, there is no potential gradient and thereo tangential current along electrode.

. Hydrodynamics

Fluid flow in the cell depends on the gaseous bubelease, which depends strongly on the cell design. In sarticular industrial configurations, as e.g. aluminium re

ion cells, the anode surface is not purely vertical (there enon negligible horizontal electro active surface) and th

ore bubble rise is hindered and lead to special two-pow patterns (coupling between bubbles forming a dispehase and bubbles packet formed under the anode su


However, the simple geometry of a vertical electrode con-sidered here has already its own complications, and will bedeveloped here in a first step.

4.1. Fluid flow

The fluid dynamics problem consists in solving the twophase flow in the electrochemical cell due to the electro-chemically generated bubble release at electrode surface.Gas–liquid two phase flow modelling is usually encounteredin bubble columns or bubbly pipe flows, in the nuclear fieldparticularly or other industrial applications. The two-phaseflow can be treated in two different ways, either consider-ing the gas phase as discrete particules (lagrangian model)or as a continuum (eulerian models). When there is a cleardispersed phase, the flow is called bubble flow. A Lagrangemethod is then suited, since bubbles can be considered asparticles. If the gas phase void fraction is too high (the limitvoid fraction is said to be 10%), as in slug or churn flow, thegas phase should better be treated as a continuous phase andthen, a two-continuous phase (Euler–Euler) model is thenmore suitable.

In the classical vertical gas-evolving electrodes configura-tion considered here (Fig. 1), if the inter electrode gap is nottoo small, with a moderate current distribution, the gas evolu-t thors(to withc ctrodei anda iame-t roded ne

4

resi-d dv ene indi-v igids em less,t lec-t ptedlt manyf

----

Therefore, the above condition will have to be checkedafter computation, and will be determinating the validity ofthe two-phase calculation.

4.3. Equations

Navier-Stokes equations are solved for the liquidphase (continuum)(9) together with continuity Eq.(10):Momentum:

∂(ρui)

∂t+ uj ∂(ρui)

∂xj

= − ∂p

∂xi

+ ∂

∂xi

(µ∂u

∂xi

)= ρgi + Km

(9)

Continuity:

∂ρ

∂t+ ∂(ρui)

∂xi

= 0 (10)

where in Eq.(10), the right hand terms are the forces exercedon the fluid volume:

- pressure force;- continuous viscous force;- gravity;- and finally the coupling termKm which is in fact the

momentum exchange due to bubbles passing through a unit

qua-te ent[ da kedE g thefl areg

4

ivenm en-s bero ctionp thec llings ands tes toc n. Int ver-a argert teadys amice rticlev

ion is clearly a dispersed flow as observed by many aue.g. Hine[4], Boissonneau[7], Schneider[6]). The flow inhese works is composed of tiny bubbles of released H2, O2,r Cl2. These bubbles are assimilated to rigid spheresonstant radius. This hypothesis assumes that the eles not too high to neglect the effect of pressure variationsssumes also a uniform temperature field. The bubble d

er will be supposed small compared with the inter electistance. Then diameterdp = 10−4 and 10−3 m have beexplored in the present study.

.2. Two phase flow-bubble flow

For vertical electrodes configuration, the bubblesence time is small due to the Archimede acceleration anoid fraction is often sufficiently small. Bubbles are thverywhere not numerous and can be considered asidual particles (not as eulerian flow) comparable to rphere with small radius (dp = 10−4 m for example). Thodel employed here is the lagrangian model. Neverthe

his model remains valid if the void fraction near the erode is smaller than 10%, which is the commonly acceimit for lagrangian models. The condition ofε < 10% shouldherefore be respected, since void fraction depends onactors, especially:

gas flow rate (or indirectlyjn);bubble diameter;liquid density, liquid flow;electrode height.

volume.

This set of equations constitutes the Navier-Stockes eions with the coupling exchange termKm due to bubblentrainment force. These equations are solved in the Flu®

22] software using the Patankar[23] finite volume methond the SIMPLEC (Semi Implicit Method for pressure Linquations Corrected) algorithm. More details concerninow modelling and the numerical parameters for solvingiven in[24].

.4. Lagrangian model

Particles are injected from a given surface with a gass flow rateq which depend on the local current d

ity value jn. Therefore, the program computes a numf bubbles per second (stream) issuing from one injeoint, which is, if the injection surface is a boundary,ell center of the face boundary. A more realistic modehould use as input data the nucleation sites densityhould deduce the average distance between two sihoose the electrode surface partitioning cell dimensiohe present work, the cell dimension is larger than this age distance and then the bubble injection frequency is l

han the actual one. But calculations are done under the state assumption. For each particle, The Newton dynquation (force balance) is solved to calculate each paelocity:

d(up)i

dt= FD(u − up)

i+ gi(ρp − ρ)

ρp + Fx,i

(11)


Fig. 3. Drag coefficientCD vs.Rep number (after[25]).

FD(ul − up)i

: drag

gi(ρp − ρ)/ρp : buoyancy

}

are the two main forces when gravity is concerned

Fx,i: other forces;i is the coordinate index (i = l is x andi = 2is y)

The two main forces acting on bubbles in the case ofvertical gas-evolving electrodes are the buoyancy force (orAchimede force) and the drag force. The former one dependsonly on the liquid–gas density ratio, and the drag forceFDis defined by:FD = 18�CD Rep/24ρp dp

2 which dependson bubble diameter and bubble Reynolds number (based onrelative velocity) andCD (drag coefficient). The coefficientvariationsCD versusRep are well known for rigid spheres(Fig. 3) and can be defined mathematically by some semi-empirical relations. In this study, the Morsi and Alexander[25] (seeAppendix A) modelling and values has been used.CD is an important coefficient to determine and its valuedepends onRep and on other factors as bubble shape, liq-uid properties (pure water or contaminated water). A typicalCD versusRep evolution scheme is shown inFig. 3. It can beseen that drag coefficientCD decreases withRep and switchesto a constant value for high Reynolds numbers (which cor-r thisst thee fterc bblert alle Tct ttomf

twop tualm andi suref highp kei tionf thew

artificial force perpendicular to the wall (close to the wall)has been implemented by Bech[21] to allow bubbles to moveaway from the wall at a distance of approximately one par-ticle radius. This force was used to avoid mesh dependence,bubbles being confined in the first cell column. Under theLagrange modelling assumption, each particle is supposedalone whereas each particle is actually in a cloud of particles,near the electrode wall and there are non negligible interac-tion forces. Then, there is an important need of an accuratedescription of bubble-wall and bubble-bubbles interactions.

However, since bubbles are growing and departing fromelectrode surface all along the surface, there is, in this case, noneed to ensure zero void fractions at electrode wall. Moreover,presence of bubbles is suited since bubble cover the electrodesurface with a given covering factor. For all the reasons above,it was chosen, in this study, to neglect these forces, and tointroduce a dispersion force perpendicular to gravity as aparameter of the numerical study.

4.5. Coupling term Km

The coupling term for motion exchange between contin-uous and discrete phase in Eq.(9) is therefore:

Km =∑

(FD(ul − up)i+ Fx,i)q �t (12)

W thec nf

5

5

edo ghty -tso rma-t angede rre-s n forh tion,i dt ue tot areao alle

ectlyc rodehc pperb us

esponds to a transition to turbulence). However, fortudy, bubbles are considered to remain spherical[5] andhe flow over the bubble to remain laminar everywhere inlectrochemical cell. This hypothesis will be verified aalculation, using numerical results. The steady-state buise velocity has been calculated using Eq.(11), neglectinghe Fx,i forces which are related to special problems (wffects, lift forces, vitual mass). A validation of the FLUENode is done in Appendix B for the (up, CD , Rep) triplet inhe case of different bubble radius injections at the boace of the cell.

Additional forces are used in special problems ofhase flow modelling. For example, the classical “virass force” is particularily important near gas injectors

nfluences the bubble motion in bubble columns. Presorce (due to pressure gradients) is taken in account forressure variations. . . Close to walls, one should also ta

nto account the lift force towards the wall and a lubricaorce [26] acting to make the bubbles move away fromall. These forces act very close to the wall[27,28]. Another

hereq is the mass flow rate of the bubble stream inonsidered computational cell;Fx,i is an arbitrary dispersioorce, perpendicular to gravity.

. Results

.1. Nominal Case

Computations of the coupled problem were performn a nominal two-dimensional geometry with heimax= 0.45 m and thicknesshg = 0.03 m (Fig. 4). The elecrolyte conductivity was set toκ = 60 S m−1 (which corre-ponds, for example, to a 10 wt.% KOH solution at 25◦C). Inur model, we have supposed that the electrochemical fo

ion of a gaseous mole necessitates a number of exchlectrons equal to two (this in order to fix idea, could copond for example to the overall heterogeneous reactioydrogen evolution at cathode surface in alkaline solu

s: 2H2O + 2e− → H2(g) + 2OH−. Moreover it is supposehat at the anode surface there is no bubbles creation dhe oxidation reaction of the electrolyte. At last, bubblesssumed to be spherical, with a constant diameterdp = 10−4

r 10−3 m. This unknown bubble diameter should be smnough to agree with modelling assumptions.

The electrode materials are supposed to be perfonducting so that the electrical potential over the electeight is constant (see electrical modelling)[29]. Thisondition can be reached experimentally by using cous bars, e.g. as used by Hine[4]. Thus, in a homogeneo


Fig. 4. Geometry for computed case (see text for details).

liquid electrolyte (with a constant electrical conductivity,current is also uniformly distributed over anode height.Therefore, changes in current distribution are only possibleby a changing of the electrical conductivity over electrodeheight in the cell, which is actually the case in verticallygas-evolving electrodes considered here.

The calculation domain mesh is formed with 10× 45 uni-form cells and has been obtained with the software Gambit®.The fluid mechanics and electrical modelling have been setin the Fluent® software. Particularly, the electrical poten-tial Φa and Φc have been fixed to an initial value of 2and 0 V, respectively, which corresponds, without any bub-bles exhaust, to a uniform current density :I/A = jav =κ

(Φa− Φc)/hg = 4.10+3 A m−2.Eq. (8) was solved with software FLUENT® as a user

defined scalar transport equation for the electrical potentialwith a diffusivity coefficient being equal to the local electri-cal conductivityκ. The local current density and the local gasflow rate creations along the vertical electrode were coupledaccording with the first law of electrolysis and were com-puted and set through user defined functions written in Clanguage, compiled with MS Visual Studio Pro® and hookedto be taken into account in the FLUENT® calculation proce-dure. Physical properties for the liquid electrolyte in nominalcase are summarized inTable 1.

Terminal rise velocities for different bubble sizes (com-p iso edesfi bles< a

Fig. 5. Terminal Rise velocity of single bubbles vs. bubble diameterdp.

bubble in a stagnant medium can be considered as constantand equal to this value.

Bubbles are created at the electrode boundary surface, inthe middle of each face-element of the mesh. Bubbles actuallymove up vertically immediately after detachment becauseof the Archimedes force and therefore, without any relevantdispersion force perpendicular to the flow, the bubble trajec-tory is confined in the first mesh cells column near the wall(Fig. 41).

5.2. Dispersed phase

The dispersed phase is first considered alone (stagnantliquid assumption) to be a sum of bubbles rising vertically attheir intrinsic terminal rising velocity. This becomes a fictiouscase when bubbles are in a sufficient number, and fluid flow isentrained, and therefore bubble velocity is changed (absolutevelocity). However, computations were performed in the caseof 10−3 m size bubbles, with the nominal properties of gas andliquid electrolyte defined above, and mesh already defined(cell size 1 cm).

Fig. 6a depicts the local current density distribution alongthe electrode length (ymax= 45 cm) in the case of coupled andnon coupled computations. Corresponding values of mass-flow rate injection of bubbles are shown onFig. 6b. As acomparison, Tobias analytical model (coupled) based on ther up-p idtho tionm byt ticalT lesa tu long

TP

N

uted) are plotted inFig. 5. The terminal rise velocitybtained and is a result of a balance between Archim

orce and drag force on a bubble (Eq.(11)). This velocitys reached very rapidly (almost instantaneously for bub10−3 m) as shown inAppendix B, so that the velocity of

able 1hysical properties for the two phase flow

ominal case M(g) (kg mol−1) ρp (kg m−3)

2.10−3 0.07

ising velocity of the bubbles, is added. This model soses that the void fraction is distributed over the wf the cell (see Appendix C). A simple manual calculaade from void fraction profile along electrode divided

he number of cells shows that the profile fit the analyobias profile (values for the calculation of Tobias profire summarized inTable 2). Fig. 6shows modelling impacpon current density and local gas flow rate distribution a

dp (m) ρ1 (kg m−3) µ1 (kg m−1 s−1)

1.10−3 1.10+3 1.10−3


Fig. 6. Left: comparison of current density distribution along electrode for the case of confined bubbles, distributed bubbles, and with Tobias theoretical model(Appendix C). Bubble diameter: 10−3 m. Gas flow rate: uniform (no coupling) or non uniform (coupled). Right: gas mass flow rate profile along electrode.

Table 2Numerical values computed for comparison with Tobias’s model[13]

us (m s−1) C (m A−1) K- jav (A m−2) �y (m) �q(g) (kg s−1)

1135.10−1 4.35× 10−5 8× 10−2 4× 10+3 1× 10−2 4.14× 10−7

the vertical electrode, under the stagnant liquid (continuousphase) assumption. The first step for the rigorous coupledcalculation is called “uniform”: the gas flow rate is supposeduniform and independent of the vertical coordinate y, withthe constant value deduced from the average current densityI/A = jav = 4000 A m−2. The resulting current density distri-bution due to this constant gas injection, which could be dueto a non-electrochemical origin injection, is shown in theFig. 6left. In the Tobias work case, the bubble creation is dueto the electrochemical reaction and then to the local currentdensity. Because this last is not constant, the bubble creationis not constant and has to be calculated. It has to be doneiteratively, till the convergence and the calculation stabilisa-tion is reached. This is the so-called “coupled” calculation.In this case, because current density decreases with y, thebubble creation also decreases.

The knowledge of the gas flow rate distribution allows thevoid fraction distribution calculation along electrode: this is

plotted inFig. 7: at left, vertical profile; at right 3 horizontalprofiles fory = 0, 0.225 and 0.45 m. For the uniform injectioncase, the gas flow rate of bubbles is constant at each injectioncell (since mesh is regular) and therefore void fraction profileincreases linearly. Each injection cell adds its contribution tothe local void fraction (void volume/cell volume) accordingwith:

�ε = 1

ρp �q(g) �y �z

(�y

up

)(�x �y �z)−1

= 1

ρp �q(g)

(�y

up

)(�x)−1 = 1.74× 10−2

where�q(g) is the gas flow rate in one cell adjacent to theelectrode, defined by:

�q(g) = jav

nFM(g)�y

F ed calc eh

ig. 7. Left: void fraction along electrode for the no coupled and coupleights for the coupled calculation. Bubble diameter: 10−3 m.

ulation. Right: void fraction distribution in the width of the cell at three electrod


The final value ofε (the mass fraction of bubbles) at thetop of the electrode is:

ε =∑

�ε = 45�ε = 0.78

which corresponds to the value computed by the Fluent®

software (Fig. 7).For the coupled calculation and the comparison with

Tobias experimental results, there is a problem: the currentdistribution is predicted lower than the Tobias measurementsand correlation. This should be due to the unrealistic bubbleconcentration near the electrode which is quite different thanthe uniformly distribution at positiony assumption used inTobias modelling. In the present modelling work, the bub-bles are confined in the first column of cell (adjacent to theelectrode), as seen on cross-sectional void fraction profiles(Fig. 7right), and therefore the void fraction depends actuallyon mesh size of the first cell column. This figure shows wellthe mesh dependence of the horizontal void fraction profiles:only one value is calculated at the surface and then, a linearlaw is used in the first cell to access the zero void fractionbulk value.

Hence, an additional artificial dispersion force shouldactually exist to ensure the bubble dispersion in the lat-eral x direction. Numerically, this horizontal force has tob enceA ver-s aren o notk PM)a and ab elledf s inc haseA alisticd e iss suret n.

5.3. Dispersion force

The simplest modelling of the lateral forces due to wallinteraction or bubble group effect which act upon individualparticles is a constant horizontal force which accelerationvalue is called bforce, and then:Fx 1 = bforce (=constant) (xdirection)Fx 2 = 0 (y direction)

We can find in the literature there exists some interestingtentative of simplified modelling of the dispersion force witharbitrary acceleration or with interaction potential law with anarbitrary form and identified parameters values. In fact, onlybubble scale works such these of the Lausanne polytechnicschool[2,3], should allow the determination of realistic inter-action and forces between bubbles and wall. In this first work,only an order of range of the horizontal force value is waited.For the dispersion study with the applied force, a three timerefined mesh has been used to have a better description of theprofile.

Fig. 8 shows at the left the bubble boundary layer in thecell under the coupled calculation modelling assumption, forthe three explored bforce values (1500, 3000 and 5000) m s−2

for the bubble diameterdp = l0−3 m. At the right is the relatedvoid fraction profile at the exit height cross section (Fig. 8b).Notice that the void fraction horizontal profile is almost con-stant. The boundary layer thickness increases with altitudeand also with the transversal force intensity. This is well con-fil cur-r cellr ly, forb taina bblesw hati ngeof lts:v odeo n is

F l force( exit of

e modelled to avoid the encountered meshing dependctually there are many forces acting on bubbles transally, particularly bubbles group force. But these forcesot yet mathematically formalised, because they are alsnown precisely. Under the discrete phase modelling (Dssumptions, each particle is supposed independentubble group effect should be programmed as a modorce. The Eulerian modelling for bubble group consistonsidering the discrete group as a second continuous ps it can be understood here, it is not easy to choose a reispersion force. In this work, an arbitrary unknown forcet in horizontal direction, perpendicular to gravity, to enhe bubble layer development in the transversal directio

ig. 8. Bubble boundary layer development evolution with transversaa) boundary layer frontier (ε = 10−2); (b) void fraction distribution at the

.

.

rmed by visual observations of Fascio et al.[9,10,14]. Fig. 9eft shows that there is an effect of dispersion force onent density profiles. If bubbles are confined in the firstaw, the electrical resistance is increased, and, inverseforce = 5000 m s−2, bubbles are distributed enough to obresistance close to the Tobias results, in which case buere homogeneously distributed over the whole cell. W

s very important to notice here is that the order of raf the lateral force seems to be about half the Archimede

orce. It is one of the more important quantitative resuertical Archimede force is famous but the bubble-electrr bubble-bubble or bubble-group of bubbles interactio

value bforce applied to bubbles: (1) 1500 m s−2; (2) 3000 m s−2; (3) 5000 m s−2:the cell. Bubble diameter: 10−3 m.


Fig. 9. (a) Current density profiles along electrode for three different values of bforce (bforce = 0 calculated with initial mesh, other cases are calculated withthe three time refined mesh); (b) void fraction evolution along the electrode. Bubble diameter: 10−3 m.

Fig. 10. Typical flow pattern for a Poiseuille forced flow with bubble diameterdv = 10−3 m (top) anddp = 10−4 m (bottom) A Poiseuille profile is set at thebottom entrance withuy,max= uy(x = 0.015) = 0.3 m s−1.


Fig. 11. Void fraction for (a)dp = 10−3 m; (b) dp = 10−4 m, in the cross section of the cell at 7 different heights.

much difficult to quantify. It appears that the equivalent effectof these interactions lead to a normal to the electrode force,which acceleration value is about half the Archimede value.At right,Fig. 9shows that the existence of a constant transver-sal acceleration leads to an asymptotic void fraction valueafter an establishment length. The asymptotic final void frac-tion at electrode decreases with the acceleration value. TheTobias profile, also given, is very smaller because the bubblerepartition in the x direction is not considered and is averagedupon the 3 cm width.

Remark:Here, only bubbles of size 10−3 m are consid-ered, since, for bubbles of size 10−4 m, the velocity of bub-bles under the stagnant liquid assumption,up = 0.0052 m s−1,lead to unphysical void fraction values, even if bubbles arehomogenously distributed in the width of the cell.

These previous results have been obtained with the cou-pled calculation of bubble trajectories and electrical field.But the continuous liquid phase motion has not been cal-culated: it was the stagnant liquid assumption. In fact, cou-pling between continuous and discrete phase is very strongand very important in confined electrochemical cells. Actu-ally, the fluid flow entrained, or is entrained, due to bub-bles motions which are like motion sources or turbulenceproviders. The actual modelling and calculation of the twophases is crucial to obtain a realistic void fraction calculation,knowing that hydrogen bubbles are effectively of typical size1

In the more rigorous calculation of coupled flow motion,the drag coupling term is a momentum source for the con-tinuous phase (electrolyte) close to the wall.Fig. 10shows atypical flow pattern for an fixed Poiseuille flow (uy,max= uy(x = 0.015) = 0.3 m s−1) at the bottom entrance (y = 0). In thisfigure, results have been obtained with the bubble diametersdp 10−3 and 10−4 m. It is remarkable that, even if lateralacceleration bforce remains at value 5000 m s−2, the velocityof bubbles is enhanced, but the thickness of bubble bound-ary layer is reduced. The vertical motion source modifies thePoiseuille profile. The continuous phase mass conservationis obeyed. The maximum velocity is no longer obtained atcentre but near the gas evolving electrode. It is 0.39 m s−1

for dp = l0−3 m and 0.48 m s−1 for dp = l0−4 m. These val-ues should be compared with terminal velocities valuesobtained under the stagnant liquid assumption: 11.3 cm s−1

for dp = l0−3 m and 5.2 mm s−1 for dp = 10−4 m. The lowerthe bubble size is, the higher is the drag force interacting withthe continuous phase. Finally, liquid velocity reaches almostthe same values for these two different bubble sizes. What isimportant is the large decrease of the boundary layer thick-ness which leads to a large increase of mass or heat transfers.

The void fraction profiles are shown inFig. 11. The voidfraction calculated with rigorous coupling with continuousphase coupling remains to the value found under the “stag-nant” liquid assumption (15%) for diameterd = 10−3 m,w ls

files alo

0−4 m.

Fig. 12. (a) Current density pro

phich is slightly over acceptable limit for DPM mode

ng electrode; (b) reduced resistance.


(which is said to be 10%). For diameterdp = l0−4 m, the voidfraction values are larger but becomes realistic which was notthe case under the stagnant liquid assumption.

The major effect of this increased rigor modelling of thecoupled phenomena is the void fraction lowering due to thevelocity enhancing. Then, as shown inFig. 12left, the currentdensity profile is increased when the two primary distributionfor dp = l0−3 m are compared. Because the casedp = 10−4 mleads, under the stagnant liquid hypothesis, to unrealistic voidfractions, it was not possible to compare. Then,Fig. 12rightshows the macroscopic result in term of reduced resistivitywhich is the ratio between the integral ohmic resistance withbubble and the same resistance for electrolyte without bub-bles, for a uniform distributionj = 4000 A m−2. It has to benoticed that the smaller the bubble is, the larger the resis-tance increase is: almost 20% fordp = l0−4 m and 10% fordp = 10−3 m, which is an important value for rigorous elec-trochemical process modelling.

6. Conclusion

A computational study was performed on selected elec-trochemical cell geometry with a gas-evolving electrode. Atypical electrochemical reaction with an exchange of twoe cterist .T thec pu-t ingt e ofc gasfl day’sl d atfi se,b dragc pir-i lids small( cencei verys eri-c

1 bleslesns

theo-o bewasod-llyof

ode

length is strongly influenced by the bubble layer thicknesswhen bubbles are slumped towards the electrode surface(seeAppendix D). Transversal diffusion and dispersionmodelling is needed because of the sensibility of Laplaceequation to strong variations in space (Appendix E).

2. With an interaction with electrolyte flow and the computa-tion of the fluid flow. Bubble drag is a momentum sourcein the continuous phase equation of motion when bubblescross an electrochemical cell. The coupled computationswith fluid flow entrainement by bubbles show a very dif-ferent trend. The fluid motion reaches values which areseveral times higher than bubble velocity in a stagnantfluid (up to 100 times for 10−4 m sized bubbles). There-fore, the liquid velocity becomes the “driving” velocity,the void fraction is drastically reduced and the electri-cal problem is no longer strongly dependent on bubblesize. The DPM model is usable under the small void frac-tion value assumption (about 10%), according with thebubble-bubbles neglected interaction assumption. For theexplored average current density value, this criterion hasbeen obeyed. However, the lack of dispersion force modeldue to packing of bubbles remains, and should be imple-mented in lagrangian models. As a comparison, eulerianmixture models used by Dahlkild[5] on the based modelof Ishii include the maximum packing of bubbles in theupward superficial velocity formulation but the problem of

herpti-

bleand

this

A

andc

A

EIjΦ

κ

η

PqMµ

ρ

lectrons for generating one gaseous mole and a charaic electrical conductivity ofκ = 10.6 S m−1 were consideredhe purpose was to compute two-phase flow solution inell by using computational fluid dynamic software. Comations were performed with FLUENT 6.0 version allowo customize gas flow injection properties through the usustom function compiled with a C compilator. Hence,ow rate depends on the current density satisfying Faraaw (first law of electrolysis). A lagrangian model is userst for a dicrete particle modelling of bubbles. In this caubbles are assumed to be spherical and rigid, thereforeoefficient is calculated from Morsi and Alexander’s emcal functions (Appendix A). This assumption is fairly vaince electrochemically generated bubbles are usually<10−3 m) and the gas phase is clearly dispersed (coaless rarely observed except in some particular cases ofmall interelectrode gap or high current densities). Numal simulations were performed in two steps:

. With the factious case of a stagnant liquid, where bubdon’t interact with the electrolyte. In this case, bubbmove up with their terminal rise velocity. Calculatiowere performed with bubbles of size 10−3 m, and theresults were compared to theoretical values of Tobias[13].A parametric dispersion force was used to approachretical values of Tobias which consider the bubbles tdistributed homogeneously over the cell width. Thisdone since there is an absence, in this kind of DPM melling, of any dispersion and diffusion force (speciarelated to “packing dispersion”). The results in termcurrent density profile show that profile over the electr

-maximum packing formulation remains. Though furtmodelling should be developed for the purpose of omising chemical cells. Fluid flow put in motion by bubseems to be an important factor for cell performancesthe development of computational fluid dynamic infield should also be strongly promoted.

cknowledgement

We want to acknowledge reviewers for their helpfulonstructive work.

ppendix A. List of symbols

Variables

lectrical problem/A mean current density (A m−2)

current density (A m−2)electric potential (V)electrical conductivity (S m−1)over potential (V)

hysical propertiesevolving gas mass flow rate (kg m−2 s−1)molecular weight (kg mol−1)viscosity (kg m−1 s−1)density (kg m−3)


GeometryA area (m2)hg interelectrode gap (m)ymax electrode height (m)�x element size (m)�y element size (m)ds differential area vector (m2)

Two phase flowu velocity (m s−1 )CD drag coefficient (-)Re Reynolds number (-)ε gas-phase volume fraction (-)d diameter (m)V volume (m3)

ConstantsF Faraday (96 485 C mol−1)C Tobias’ s constantn electron number

Subscripts:p bubble particleg0(((((((

Au

Caaaaaaa

Appendix C. Terminal velocity and relaxation timeof bubbles obtained with Discrete Phase Modelling.

For three different particle diametersdp = 10−4, 10−3 and10−2 m, terminal rise velocity and relaxation time have beencalculated and are presented inFig. 13.

In this validation case, bubbles are created and injectedsince cell number 20 which has the position:y = 20 cm. Bub-bles relaxation time is small, about 1 ms, for smaller diameterand terminal velocities are obtained quickly. The discretephase modelling programmed in Fluent has been validatedusing manual calculation of the bubble velocity evolution.The terminal velocity using the Morsi and Alexander[25]analytical calculation is given with the three following equa-tions:

us2 =

(1 − ρp

ρ

)(4g dp)

(3CD)

with:

CD = a1 + a2/Rep + a3

Rep2

(drag coefficient)

and:

Rep = ρ(u1 − up)dp

loc-i dera thatt

A

ou-pu ne

w

K

TT exande[23

D Numvelo

n

1 0.571 0.111 0.00

gasfree bubble electrolyte

-)(1) liquid phase-)ij i component-)av average value-)p,b relative to particle or bubble-)a relative to anode-)c relative to cathode-)s relative to a surface

ppendix B. Morsi and Alexander [25] coefficientsed for spherical rigid particles

D = a1+a2/Rep + a3/Rep2 with Rep =ρ(ul − up)/�l

1 = 0, a2 = 24,a3 = 0 for Rep < 0.1

1 = 3.69,a2 = 22.73,a3 = 0.0903 for 0.1 <Rep < 1.0

1 = 1.222,a2 = 29.1667,a3 =−3.8889 for 1.0 <Rep < 10.0

1 = 0.6167,a2 = 46.5,a3 =−116.67 for 10.0 <Rep < 100.0

1 = 0.3644,a2 = 98.33,a3 =−2778 for 100.0 <Rep < 1000.0

1 = 0.357,a2 = 148.62,a3 =−4.75× 104 for 1000.0 <Rep < 5000.0

1 = 0.46,a2 =−490.546,a3 = 57.87× 104 for 5000.0 <Rep < 10000.0

able 3erminal velocities calculated with the theoretical law of Morsi and Al

iameter (m) Theoretical terminalvelocity (m s−1)

Theoretical ReynoldsnumberRep

0−2 0.5773 5773.50−3 0.1135 113.510−4 0.005265 0.5265

µ1

Table 3shows a good agreement between terminal veties calculated with theoretical law of Morsi and Alexannd numerically calculated values. This is a confirmation

he virtual mass force negligible assumption is realistic.Table 4

ppendix D. Analytical solution (Tobias [13])

The analytical expression of the Tobias modelling cling [13] for the vertical electrodes (seeFig. 14), is givensing Eq.(C.1), (C.2), and(C.3)under the zero polarisatiolectrode assumption:

j(y)

jav= 8.(K + 2)2(K + 4)−1

(Ky

ymax + 2

)−3

(C.1)

ith

= 2

[1 −

(1 − 1

Cymaxjav

)−1]0.5

− 2 (C.2)

r] (seeAppendix A)

erical terminalcity (m s−1)

Numerical residencetime (s)

Numerical relaxatiotime (s)

73 0.5 0.0135 2 0.015264 45 0.001


Fig. 13. Relaxation time and terminal velocity for the three explorer bubble diameters.

Table 4Calculated resistance according to Eq.(E.1), (E.2), (E.3), withn = 10,ε = 0.7, and�x = 0.003 m,κ0 = 60 S m−1

k = l k = 2 k = 3 k = 4 k = 5 k = 6 k = 7 k = 8 k = 9

Rk 7.54e-4 5.91e-4 5.73e-4 5.67e-4 5.63e-4 5.61e-4 5.60e-4 5.59e-4 5.58e-4Rn 5.575e-4 – – – – – – – –Rk/Rn 1.35 1.06 1.028 1.017 1.0101 1.006 1.004 1.0027 1.0009

Fig. 14. Electrochemical cell geometry with a vertical electrode studied byTobias[13].

and

C = M(g)

(upρpnF hg)(C.3)

whereY = y/ymax is the normalized height of the electrode. Inthis expression, the velocityup of the particles is supposed tobe constant in the section, and the void fraction is assumed tobe independent of horizontal componentx (bubbles homoge-nously distributed in the width of the cell).

Appendix E. Resistance evolution with layerthickness

The local conductivity coefficient is calculated withBruggeman’s formula:

κ = κ0(1 − ε)3/2 (2)

The total resistance of the electrochemical cell is:

Rk = 1

κ0

[k�x

(1 − ε/k)3/2+ (n − k) �x]

](D.1)


Fig. 15. Distributed void fraction in the width of the cell (ID).

whereκ0 is the conductivity of the electrolyte free of bub-bles,ε is the void fraction when the bubbles are confinedin the first column (k = 1), k is the number of computationalcells occupied by bubbles (seeFig. 15) and n is the totalnumber of cells which cover the cell widthhg. If the bub-bles are distributed in the entire width of the cell (Tobiashypothesis), and not in a near electrode bubble layer, Eq.(D.1)becomes:

Rn = �x

κ0

[n

(1 − ε/n)3/2

](D.2)

And the ratio between (D.1) and (D.2) is:

Rk

Rn=

[k/(1 − ε/k)3/2 + (n − k)

][n/1 − ε/n)3/2] (D.3)

Results for different values of n and varying parameterkare plotted inFig. 16. As showFig. 16, it appears an increasingresistance when the bubbles are situated in a near electrodebubble layer. This resistance becomes constant when thesame amounts of bubbles is used and occupy the total width(Tobias assumption).

Appendix F. Diffusion terms

puta-t atedh g the

Fig. 16. Resistance evolution with bubble distribution in the electrochemicalwidth. Calculated according to Eq.(E.1), (E.2), (E.3), withn = 10, ε = 0.7,and�x = 0.003 m,κ0 = 60 S m−1.

conductivity of the electrolyte to be defined by the followingmathematical function:

κ = κ0[0.25 tanh(β(x − hg/2) + 0.75)] (E.1)

where slopeβ is an amplifying coefficient which controls thesharpness of the conductivity profile. It is then possible totest numerical robustness for an a priori given bubble layerthickness.

Fig. 17shows the analytical profile of the current density(constant) over the cell width. TheFig. 18shows the numeri-cally calculated current density. For a small amplifying coef-ficient (β = 10+2), the current density profilej calculated withEq.(8)exhibits a fairly constant value ofj in good accordancewith analytical value (seeFig. 17right) if the mesh is suffi-ciently refined. For sharper profiles (β = 10+3), an oscillationappears in current density profile for coarse mesh due to anabrupt variation of conductivityκ from 30 to 60. When meshis refined, there are more computational points in the transi-tion gap and therefore accuracy is enhanced and oscillationsare dumped. When the conductance boundary layer thicknessis reduced again (β = 10+4), abrupt transition remains and the

F nducti dc

The consequences on the current density profile comion due to the unrealistic dispersion force model is illustrere, using a mono dimensional problem and considerin

ig. 17. Electrical properties evolution with coordinatex; left: electrical courrent densityj according with electrical conductivity evolution.

vity for three different values ofβ(=102, 103 or 104); right: constant relate


Fig. 18. Numerical robustness with conductivity sharpness: current density profile across the ID cell for three different slope values with four homogenoussuccessively refined meshes. Top:β = 102; R = 0.000698, centre:β = 103; R = 0.000747 and bottom:β = 104; R = 0.000749.

oscillation in the current density profile remains too with theexplored meshing hypothesis.

References

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1095.[3] R. Wuthrich, Spark Assisted Engraving: a Stochastic Modelling

Approach, PhD thesis, EPF Lausanne, 2003.[4] F. Hine, K. Murakami, J. Electrochem. Soc. 127 (2) (1980) 292.[5] A.A. Dahlkild, J. Fluid Mech. 428 (2001) 249.[6] C. Schneider, Gas fraction, velocity and bubble size distributions in a

model of alkaline chlorine electrolysis cells. Rossendorf ForschungsZentrum, Annual Report.

[7] P. Boissonneau, P. Byrne, J. Appl. Electrochem. 30 (2000) 767.[8] C. Gabrielli, F. Huet, M. Keddam, A. Sahar, J. Appl. Electrochem.

19 (1989) 683.

[9] V. Fascio, H.H. Langen, H. Bleuler, Ch. Comminelis, Electrochem.Commun. 5 (2003) 203.

[10] H. Wuethrich, Bleuler, Electrochim. Acta 49 (2004) 1547.[11] H. Vogt, Electrochim. Acta 26 (9) (1981) 1311.[12] J.E. Funk, J.F. Thorpe, J. Electrochem. Soc. (1969) 48.[13] C.W. Tobias, J. Electrochem. Soc. 106 (9) (1959) 833.[14] V. Fascio, R. Wuthrich, H. Bleuler, Electrochim. Acta 49 (2004)

3997.[15] H. Vogt, R.J. Balzer, Electrochim. Acta 50 (2005) 2073.[16] R. Wuthrich, Ch. Comninellis, H. Bleuler, “Electrochemistry under

extreme current densities-Application to electrode effects” 55th ISEmeeting, 19–24 September 2004 Thessaloniki, Greece.

[17] D. Ziegler, J.W. Evans, J. Electrochem. Soc. 133 (3) (1986) 567.[18] P. Byrne, PhD thesis, Faxen Laboratoriet, Sweden, 2000.[19] R. Wedin, A. Cartellier, L. Davoust, P. Byrne, Experiments and mod-

elling on electro chemically generated bubbly flows, in: Proceedingsof the 4th I.C.M.F, New Orleans, 2001.

[20] R. Wetind, PhD thesis, chapter 2: two-phase flow in gas-evolvingelectrochemical reactions, Department of Mechanics, Faxen Labora-toriet, Sweden, October 2001.


[21] K. Bech, S.T. Johansen, A. Solheim og T. Haarberg, in: J.L. Anjier(Eds.), Coupled current distribution and convection simulator forelectrolysis cells, light metals 2001. Miner. Met. Mater. Soc. (2001).

[22] Fluent®, documentation and websitehttp://www.fluent.com.[23] S.V. Patankar, Numerical Heat Transfer and Fluid Flow, Taylor and

Francis, 1980.[24] Ph. Mandin, Th. Pauporte, Ph. Fanouillere, D. Lincot, J. Electroanal.

Chem. 565 (2004) 159.

[25] S.A. Morsi, A.J. Alexander, J. Fluid Mech. 55 (2) (1972) 193.[26] S.P. Antal, R.T. Lahey, J.E. Flaherty, Int. J. Multiphase Flow 17 (5)

(1991) 635.[27] E. Krepper, CFD Simulations of a Bubbly Flow in a Vertical

Pipe, Rossendorf Forschungs Zentrum, Institute of Safety Research,Annual Report 1999.

[28] M. Ishii, N. Zuber, AIChE J. 25 (5) (1979) 843.[29] H. Kawamoto, J. Appl. Electrochem. 22 (1992) 1113.

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