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Finite-Difference Simulation

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2732J. Electrochem. Soc., Vol. 144, No. 8, August 1997 The Electrochemical Society, Inc.Plenum Press, New York (1977).65. T. Kenjo and K. Wada, Solid State lonics, 67, 249(1994).66. A. M. Svensson and K. Niancioglu, To be published.67. N. L. Robertson and J. N. Michaels, AIChE Symp.Series 254, 83, 56 (1987).68. R. J. Brook, W. L. Pelzmann, and F. A. Kroger, ThisJournal, 118, 185 (1971).Finite-Difference Simulation of Multi-Ion ElecfrochemicalSystems Governed by Diffusion, Migration, and ConvectionImplementation in Parallel-Plate Electrochemical Reactor andBackward-Facing Step GeometriesMaria Georgiadou*Department of Electrical Engineering, Vrije Universiteit Brussel, B-i 050 Brussels, BelgiumABSTRACTA finite-difference method for predicting the current-density distribution in multi-ion electrolytes was developed,then implemented to actual experimental systems and compared with other numerical methods, analytic solutions forlimiting conditions, and experimental measurements in order to be tested. The method accounted for diffusion, ionicmigration, laminar convection, and heterogeneous electrochemical reactions following Butler-Volmer kinetics in two-dimensional geometries under steady-state conditions, and is applicable at high velocities as well. The finite-differencemethod was applied to the acidic copper sulfate electrolyte with composition 0.01 M CuSO, + 0.1 M H2S04 and the alka-line potassium ferri/ferrocyanide electrolyte with composition 0.005 M K,Fe(CN)6 + 0.01 M K4Fe(CN)6 + 0.5 M NaOH,placed between two equally sized parallel-plate electrodes fixed in a channel's walls at a separating distance being 3%of their length (PPER), under conditions of laminar parabolic flow. It was also applied to the ferri/ferrocyanide systemplaced in a similar configuration in which a backward facing step with height being 1.5% of the electrode length wasintroduced just before the cathode in the flow channel. The model computed concentration, potential, and current densi-ty distributions at Reynolds numbers Re =230and 1200 and applied cell voltage in the range —0.5 V V—0.03 Vfor the first system; also at Re =55,180, 300, and 2100 in the PPER and Re =55,180, and 300 in the backward-facing-step geometry under limiting-current conditions for the second. The computed local cathodic current density was com-pared to experimental data, multidimensional-upwind-method predictions, and the Léveque analytic solutionwhereapplicable. Very good agreement was demonstrated, thus verifying the method's applicability to real cases.IntroductionTheprediction of current density in electrochemicalapplications is an essential step in the rational design andscale-up of electrochemical reactors and plays a centralrole in the engineering analysis of electrochemical pro-cesses. In this work a finite-difference method (FD) wasdeveloped for predicting mass and charge transport andcurrent-density distributions in multi—ion electrolytesgoverned by diffusion, migration, and convection. Thismethod was then applied to the experimental tertiaryacidic copper sulfate system placed in a parallel-plateelectrochemical reactor (PPER) with copper electrodesunder hydrodynamic conditions of laminar parabolic flowand then to the alkaline potassium ferri/ferrocyanide elec-trolyte placed in two geometries: (i) a PPER with nickelcathode where reduction of ferricyanide ion took place atlimiting-current conditions, and (ii) the same PPER con-figuration fixed in the walls of a flow channel in which abackward facing step (BFS) was placed just upstream ofthe cathode.The calculation of the current-density distribution inelectrochemical systems has been a frequent subject ofstudy,"2 but mathematical models have been reported forsimplified cases only, ignoring either migration or convec-tion.34 For the PPER geometry there are models based onthe thin_diffusion-layer approach,'' and on the mass-transfer control assumption.8 More complicated modelswere based on the dilute solution theory including migra-tion in the transport equations, which were simplified byignoring axial diffusion and axial migration9 and did notaccount for obtaining a solution at high velocities.9"0Finite-difference modeling has been reported'1 in investi-gating anisotropic chemical pattern etching of copper.Recently, a new numerical method originating from fluidmechanics, the multidimensional upwind method(IvIDIJM), has been developed for current-density predic-tion; this method includes diffusion, migration, and con-vection, resembles the finite-element method, and is analternative approach to finite-difference, finite-element,and finite-volume methods. The MDUM method is dis-cussed in detail elsewhere."The FD method computed the local cathodic currentdensity for the two systems and results were compared toexperimental measurements for the copper ion'4 and theferricyanide ion reduction in both geometries," predic-tions by the MDUM," and the Lévêque analytic solutionfor limiting current conditions where applicable.'6MathematicFormulationBasicassumptions included steady-state, two-dimen-sional geometries, validity of the dilute solution theory,constant and concentration-independent transport, andphysical properties. The governing equations and theirboundary conditions were made dimensionless prior tonumerical solution to avoid numerical problems, i.e.,instabilities, with the solution algorithm arising frommagnitude differences between dimensional terms, Thecharacteristic quantities chosen to nondimensionalize themass-transport equations and the dimensionless quanti-ties are defined in the List of Symbols. In the PPER geom-etry the characteristic length used was the channel height,h, and the characteristic velocity, U, was six times theaverage inlet channel velocity. In the BFS geometrythesequantities were the step height, h/2, and the average inletchannel velocity, respectively. In terms of mathematicalmodeling, the dimensionless mass transport equation forion i was* ElectrochemicalSociety Active Member.N =0[1]J. Electrochem. Soc., Vol. 144, No. 8, August 1997 The Electrochemical Society, Inc.2733andthe electroneutrality conditionE =E°—E° —RTlnHhf2ltikk kREEz,C=O[2]where N is the dimensionless flux of species i defined as+ —1!.inJJJ(2as.][16]flREF=_D7V*C7—z1D7ct VS+ Pe V"C7[3]Finally, the exchange current density, i,6, depended on con-The Péclet number, Pe, was defined bycentration according to the equationULPe =—[4]'6 =z'ii (:J[17]DThe determination


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