Joumal of donghua University(Eng. Ed. )Vol. 25, No 4(2008) 463A Mathematic Model of Gas-diffusion Electrodes in Contact withLiquid ElectrolytesLI Jun(李俊)1·, XI Dan-i奚旦立), SHI Yong(石勇), wU Xi-hui吳錫慧)I Department of Chemical Engineering, Shanghai Institute of Technology, Shanghai 200235, ChinaInstitute of Environment Science and Engineering, Donghua University, Shanghai 201620, ChinaAbstract: a mathematic model is developed which is applied reactants decrease gradually and reaction rate falto analyze the main factors that affect electrode performance inchmeal if mass transfer rate is lower than chemical orand to account for the process of reaction and mass transfer electrochemical reaction rate. It is expected that smin gas-diffusion electrodes in contact with liquid difference of reactants concentration in the capillary canelectrolytes. Electrochemical Thiele modulus p and balance consumption for surface reaction if reaction rate iselectrochemical effectiveness factor "o are introduced to slow but diffusion coefficient is high enough, and thuselucidate the effects of diffusion on electrochemical reaction reactant concentration along the capillary is nearly equaland utilization of the gasdiffusion electrode. Profile of the In contrary, reactant in the capillary will be used up in areactant along axial direction k discussed, dependence of short distance from the entrance if reaction rate is quickelectrode potential V on current density J, are predicated by enough, so, concentration grads of reactants alongans of the newly developed mathematical modelcapillary have to be high for quick reaction. Moreover, themajority of electrochemical reaction completes in a thinelectrolyte gasdiffusion electrode; mathematical modelactive layer of the GDE for quick reaction, reactantCLC number: TM 911. 4Document code: aArticle l:1672-5220(2008)04-0463-05concentration in the rest of the gDe is very low, the gDecan not function fully, and efficiency of the gde will bevery poor in this case.IntroductionIt is necessary to study the aforementioned processes inorder to increase efficiency of the electrode and makeas-diffusion electrodes(GDEs)are the key devices in catalysts in the electrode work very well. Many models offuel cells (FC). In recent years, they have also beendifferent complexity have been proposed to describeincreasingly used to synthesis, separation processes and fuel diffusion and reaction in the electrodes. Some authorsfavccell membrane reactors (FCMR). FCMR can be used toco-generate electric energy and useful chemicals1-s3. heterogeneous moder s. 103. The primary models, whichDepending on the application, electrodes of FC or FCMRwere used for analysing the performance of electrode inare often made into porous structure or gas-diffusioncontact with liquid electrolyte, were heterogeneous modelelectrodes. There are complicatedtransfer and such as single-pore model, thin-film model, cross-porereaction in the electrodes. GDEs may contact with liquidmodel, etc. These heterogeneous models often containrolyte infor hydrogen peroxide generation, benzene oxidation for contains length, diameter of the hole, and pore numbersphenol production, and partial oxidation of cyclohexane per unit area. Due to the limited accuracy of experimentfor cyclohexanone and electricity co-generation. The mosdata, some of the model parameters have little statisticalimportant processes taking place in GDEs are as follows: significance. It is hard to obtain these parameters through(a)dissolution and diffusion of reactant:(b)adsorption experiment. In this paper, a macroscopic model will beand electrochemical reaction of the reactants atderived and be used to analyse diffusion and reactionelectrochemical active layer (gas-liquid-solid three phaprocess happened in electrodes in contact withsection):(c) desorption and diffusion of products.Internal diffusion and chemical reaction take placecollaterally in electrochemical active layer, in other words, 1 Mathematical Modelreactants diffuse along capillary in the electrode while theytake part in electrochemical reaction. Concentration ofH中國煤化工liquid electrolyte is22CNMHGReceived date: 2007This research is supported by Shanghai Education Committee(06-OZ-003)and Shanghai Key Subject(pl501)CorrespondenceshouldbeaddressedtoLiJun,Dr,E-mail:lijun@sit.edu.cn464Joumal of Donghua University(Eng. Ed. ) Vol 25, No 4(2008)divided into two sections, namely gas-solid section or gas.CA=HAPAGlid layer, and gas-liquid-solid or electrochemical active Where Pag is the partial pressure of the component A inlayer. See Fig. 1the gas bulk(5)The electrode is a good conductor, heat fromgas-solidelectrochemicalchemical reaction can move out in time, and temperatureactive laverof the electrode keeps constant on the whole. Otherwiseliquid electrolyte is considered to be nonvolatile(6)In order to decrease effects of ion concentrationand electrical resistance on chemical reactionliquidconcentration of reactants in electrolyte is high enoughreactants concentration in electrochemical active layer canbe thought to be constant.Diffusion and reaction in electrochemical active layerof the electrode are similar to that in porous catalysts if theabove-mentioned hypotheses are metInevitably, many parameters will be introduced in themathematical model of a GDE if considering mass transferFig 1 Schematic representation of a gas diffusion electrodein three dimensions, and it is often very difficult to get anHypothesesanalytical resolution of the model equation. Moreover(1) There is no chemical reaction in gas-solid layer ofDE is axially symmetric in general. Mass transfer in axialthe GDE. Moreover, gas diffusion rate in gas-solid layer is direction is decisive. Based on steady state materialmuch faster than that of gas dissolving and diffusion inbalance, The following diffusion-reaction equation (3)canliquid, Gas-solid layer is considered as pseudo-homogeneousbe developed if considering diffusion only in direction Zphase, that is to say, the crossing gas phase and solid phase(see Fig. 1)are seen as homogeneous in macroscopy.dCA(2) The conductivity of the electrode material is sohigh that resistance of the electrode may be neglected, and Where d, is effective diffusion coefficient, c, stands forthe electrode potential is thought to be identicalthe conceeverywhereD)Diffusion rates in direction Z(see Fig. 1)areelectrochemical reaction rate. Relationship between Ta andequal at the same cross section, the concentrations f i can be expressed as follows:reactants are uniform at the same cross section, andaelectrochemical reaction rates are equal too.(4)Diffusion rates of the reactants dissolved in liquid Where a is electrochemical active area per electrodeare much slower than that in gas, especially in capillary volume, i is electric current density or electric current perchannels of gas diffusion electrode. Components at the two unit electrochemical active area. n is electrons transferredsides of gas-liquid interface come to equilibrium. There is in electrochemical reaction for Imol reactant. F is Faradaythe following equationconstantCA=HAPAElectric current density i can be expressed as oJWhere CA is the concentration of component A at thei=nFK,Ca exliquid side of the gas-liquid interface, Ha is the solubilitycoefficient of component A, Pa is the partial pressure of Where Ka is a rate constant of the electrochemical reactioncomponent A at the gas side of the gas-liquid interface.at zero potential versus standard hydrogen electrode. VisDiffusion resistance of the gas in gas-solid layer may be electrode potential, also versus standard hydrogenneglected owing to the fact that mass transfer rates of the electrode.components are much faster in gas-solid layer than inThe following equation can be deduced from equationselectrochemical active layer or gas-liquid-solid three phase (3)section. The partial pressure Pa equals Pag, that is中國煤化工PM=PCNMHG DT) (6)thereforeWriting equation(6)in dimensionless form, we obtainJoumal of Donghua University(Eng. Ed. )Vol 25, No, 4(2008) 465CAd CA=ak, CmCAexy=0.5dc, iak,Cw"'ewhere C, and Z are called dimensionless concentration anddimensionless distance respectively. They are definedHere, a new dimensionless parameter group is introducedwhich allows a compact reformation of the above equationThe group is also called dimensionless number. It is define=c2cg(-)On the whole, dimensionless concentration CADdecreases with increase of dimensionless distance z asshown in Fig 2. Ca in electrochemical active layer is nearlyThe number f is as well called electrochemical Thiele uniform for the small B, this means the reaction occursmodulus. It can be seen from the above definition that f is throughout the electrochemical active layer, and the GDEaffected not only by reactant concentration but also by can be utilized effectively at small parameter p.electrode potential. Insertion of expression(7) into theDimensionless concentration Ca drops rapidly across thabove differentialelectrochemical active layer at bigger 2. This result=中cindicates that electrochemical reaction rate is greater tha(8)diffusion rate. effects of diffusionsignificant, and utilization of the electrode is poorz=0,CA=1(9)3 Eletrochemical Effectiveness FactordCZ=1,(10)Profile of reactant concentration will be nonuniform ifchemical reaction is affected by diffusion. The average2 Reactant Concentration in Electrodereaction rate in the electrode can be calculated as followsExplicit analytic solution of equation the (8)isn)=高2possible. For a first-order reaction, m= l, equation(8)isthen an ordinary differential equation. Its general solutionWhere A and B are constants. Application'of the∫: ak,CmCA exp((x)dzaforementioned boundary conditions then givesFor the first-order reaction, we can set m 1, thenintegration of the above expression gives the followingHence, the solution may be rewritten+e2-In order to characterize effect of diffusion on reaction in中國煤化工m, is introduceSome profile of dimensionless concentration computedCNMHGfrom equation (12) are plotted in Fig. 2 with 4 as466Joumal of Donghua Uhiversity( Eng. Ed. ) Vol 25, No 4(2008)Where (r)is the average electrochemical reaction rate in If diffusion rate is very slow compared with electrochemicathe electrode with diffusion effect, where Iw is the reaction rate, then i>>1, and the expression(15)can beelectrochemical reaction rate when the concentration of simplified as followsreactant A equals CA. Actually, Fa is an intrinsic reactionrate without diffusion effectThen, we obtainAccording to the above result, one can substitute sfor noTa=aKa Ca expin equation (16), and the following expression can beHence the final expression for electrochemical effectiveness deducedfactorno is only a function of f. The curve plotted according toaVF\CAequation ( 15) is shown inAs goes from Transformation of the above equation givesapproximate zero to more than ten, "o goes down sharply.In, =In[nF(aD, K, )f H PAoJ2RTVKEnF(aD,KA)2 HAOne can rewrite the above equation, and obtain0.6V=MIn(KPxg)-MIn/.sWe know from equation(18)that electrode poteV is related withFg3 Effects of∮omhReally, no reflects utilization of the electrode. Theall difference between(ra)and ra means the bigger npand the higher efficiency of the electrode. o decreaseswith increase of f, big s reflects big effect of diffusion onelectrochemical reaction. We know from the definition ofelectrochemical Thiele modulus that decrease of electrode5.6thickness will result in advantageous s and np. Moreover,increase of interspace fraction and diameter of the capillary fig 4 Compa (1) PaG=100 KPa, T=190c P/ nd model resultin the GDE leads to a high diffusion coefficient D,consequently a small s and a big p2) PAG=300 kPa, T=190C, P,/C:3)PA=600kPa,T=210℃,P/C;4 Relation between Electrode Potential and(4)PAG=800 kPa, T=190C, P,/C●,▲, V experimental- calculatedElectric CurrentDavid] had studied electrochemical reaction ofElectric current per unit cross-section area of the GDE oxygen in electrode in details, and obtained aJu can be associated with electric current density i, that is experiment data. In his experiment, phosphoric acid wasJ. =aoiused as electrolyte, and Pt was adopted as catalyst foroxygenaddition to David, there are manywhere i can be calculated according to equation(5). Thusother中國煤化工 ied electrochemicalwe get the following expressionCNMHGPhosphoric ac畫Joumal of Donghua University(Eng. Ed. ) Vol 25, No 4(2008) 467Reduction of oxygen on Pt-contained electrode (ra>- Average reaction rate in electrochemical active layercontacting with acidic electrolyte solution was confirmed tomo/(s·m3)be a first order reaction i. In order to test our modelsreduction of oxygen on electrode is taken as a model 2Distance from interface of gas-liquid, mreaction. Parameters in our model are calculated using the 2-Dimensionless distancePOWELL nonlinear, multivariate least-squares algorithmTotal thickness of electrochemical active layerComparison of experiment results with calculated data a- Charge transfer coefficientis presented in Fig 4. Difference between experimentalIpal efficient factorresults and the calculated is from 0. 2% to 1.%. By thip2- Electrochemical Thiele modulustoken, the calculated results coincide well with theexperimental results, so the developed model can be used toReferencesdescribe electrochemical processes in electrode contactingwith liquid electrolyte, and to provide instructions for LI] Li Jun, Zhao Ling, Zhang Xinsheng, Zhu Zhongnanelectrode desigProduction of Hydrogen Proxide in Fuel Cell Reactor-Ipering andTechnology, 2001, 17(3) 222-227(in Chines5 Conclusion[2] LiJun, Zhao Ling, Zhang Xinsheng, Zhu ZhongnanPreparation of Hydrogen Proxide in Fuel Cell Reactor-IIIn this paper, a quasi-homogeneous modelReaction Dynamics of Oxygen on Cathode[J]. Chemicalveloped and two useful parameters, electrochemReaction Engineering and Technology, 2001+ 17 (3)Thiele modulus y and effectiveness factor "p, were228-232(in Chinese)introduced for the first time. The derived model can be [3 Yuan xiao-zi, He qing-gang, Lin lin, Ma zi-fengapplied to analyse the structural parameters of the gDEElectrochemical Hydrogeneration of Allyl Alcohol Applyingsuch as thickness and interspace fraction, and their effectsPem Fuel Cell Reactor[J]. Chemical Reaction Engineeringon the performance of the electrode. Furthermore, theand Technology, 2002, 18(4): 328-333(in Chinese)model provides instructions for electrode design a[4J Yuan xiao-zi, Ma zi-feng. Electrogenerative hydrogenationmanufacture. The model also can be used to analyse effectsof some organic acids in PEMFC Fuel Cell Reactor [J]of diffusion on electrochemical reaction The currentChemical Reaction Engineering and Technology, 2004,potential characteristics of the gDE in contact with liquid20(2):151-156( in Chinese)electrolyte can be predicated with the newly developed[5] Stafford G R. The Eletrogenerative partial Oxidation ofPropylene [J]. Electrochem. 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The Mechanism of Operation of theJu- Electric current per unit cross-section area of electrode,Tefion-bounded Gas Diffusion Electrode: A MathematicalmA/cm?Model[J]. J. Electrochem. Soc., 1969, 116: 1125-1132.PA- Partial pressure of A at gas phase side of gas-liquid [11] Tarasevich M R. Comprehensive Treaties of Electrochemistry[M. New York: Plenum Press, 1983, 301.Ag- Partial pressure of a in gas body, Pa[12] David L. Hand Book of Batteries and Fuel Cells[M].NewIntrinsical reaction rate, mol/(S. m)中國煤化工4CNMHG