Mass/Charge Transfer in Mono-Block-Layer-Built-Type Solid-Oxide Fuel Cells

Transcription

1 Mass/Charge Transfer in Mono-Block-Layer-Built-Type Solid-Oxide Fuel Cells J. J. Hwang Professor Center for Advanced Science and Technology, Mingdao University, Peetou, Changhua 52345, Taiwan The mass/charge transfer characteristics in a simulated MOLB (mono-block-layer built)- type solid-oxide fuel cells have been studied numerically. The transport phenomena within a linear MOLB module, including flow channels, active porous electrodes, electrolyte, and interconnections, are simulated using the finite volume method. The gas flow in the porous electrodes is governed by the isotropic linear resistance model with constant porosity and permeability. The diffusions of reactant species in the porous electrodes are described by the Stefan-Maxwell relation. Effective diffusivities for porous layers follow the Bruggman model. Porous electrochemistry is depicted via surface reactions with a constant surface-to-volume ratio, tortuosity, and average pore size. Results of the cathode-supported cell and the anode-supported cell are obtained, discussed, and compared thereafter for the first time. DOI: / Keywords: Solid-Oxide Fuel Cell (SOFC), Anode-Supported Cell, Cathode-Supported Cell, Mono-Block-Layer Built (MOLB) Manuscript received November 21, 2004; revised January 31, Review conducted by: N. M. Sammes. Introduction The solid-oxide fuel cell SOFC is a promising alternative power source for distributed or residential power plants because of its high energy efficiency, low emission, and low noise. In addition, its ability in internal reforming allows direct use of natural gas as the fuel. Over the years, many ingenious designs of SOFCs have been devised, starting from thimbles and discs in the 1930s. Since the 1960s, most development has focused on planar and tubular designs. Recently, the planar-type SOFC has received much more interest over the tubular-type SOFC because of its easier fabrication and lower operating temperature. Figure 1 shows three typical designs of the self-supported planar-type SOFCs, namely, the electrolyte-supported cell, anode-supported cell, and cathodesupported cell, respectively. For the electrolyte-supported cell Fig. 1 a, both electrodes are very thin typically 50 m, and the electrolyte thickness is generally larger than 100 m. Under a low operating temperature 800 C, it has a high resistance due to poor ionic conductivity. Thus, higher operating temperatures are required to reduce the ohmic losses. Most mathematical models of the electrolyte-supported SOFC considered only the direction of fuel gas flow to determine the concentration and temperature profiles in the SOFC 1 5. The PEN positive electrode/ electrolyte/negative electrode layer is treated as a solid component with averaged properties. Such an approximation was adequate because only the main balances of a cell or stack were analyzed. Recently, the concept with a thin electrolyte layer supported on a thick substrate formed by one of the electrodes has gained much more interest Figs. 1 b and 1 c. In the electrodesupported cell configurations, the electrolyte is usually very thin i.e., 20 m, and either anode or cathode is thick enough to serve as the supporting substrate for cell fabrication. The development of the electrode-supporting substrate is aimed at a reduction of the operation temperature of the SOFC to below 800 C. The PEN layer is no longer treated as one solid component, and mass transport in the thick electrodes must be considered A long diffusion path for the gases from the gas channel to the threephase boundary TPB may cause the diffusion limitations. In addition, a thick anode substrate may lead to a spatial separation of the reforming reaction and the electrochemical reactions. To further enhance the power density and reduce the cost of SOFCs, Argonne National Laboratory ANL proposed the concept of monolithic cells in the mid-1980s 11. The development of monolithic cells has not been so successful due to the difficulty of simultaneous sintering of all cell components. Recently, Mitsubishi Heavy Industry, MHI and Chubu Electric Power Japan succeeded in constructing the mono-block-layer-built MOLB type SOFC 12. In one sense, it can be regarded as an advanced type of monolithic cell fabricated by multiple sintering steps instead one. In either ANL s monolithic SOFC or MHI s MOLBtype SOFCs, the PEN plate is molded unevenly in a trapezoidal or dimple shape. Figure 2 shows a cross-sectional view of corrugated-shape PEN plates of SOFCs, which simulate an MOLB-type SOFC of uniform geometry in the flow direction. The corrugated-shaped PEN plate provides a higher active area than the projected area, as well as a separation film, with the combined function of fuel and airflow paths. In the open literature, however, the research about the mass/charge transfer in SOFCs of corrugated PEN plates is rather sparse, both quantitatively and qualitatively. This motivates the present study to extend the author s previous works 6,10 to numerically investigate the mass/charge transfer characteristics in SOFCs of corrugated PEN plates. The numerical model involves solving conservation equations for mass, momentum, multicomponent species, and charge, along with electrochemical kinetics in the entire region of a single cell of the MOLB-type SOFC. Results of the anode-supported MOLB-type SOFC and the cathodesupported MOLB-type SOFC are obtained, discussed, and compared thereafter for the first time. Model Development SOFC Basics. Figure 2 shows a schematic drawing of the half sectional views of electrode-supported MOLB-type SOFCs. Air diffuses from the air channels into the porous cathode and reacts electrochemically at the three-phase boundary. The cathode is commonly an LSM layer chemically expressed as La 1 x Sr x MnO 3. The oxidant reduction reaction on the LSM surfaces can be expressed as O 2 +4e 2O / Vol. 2, AUGUST 2005 Copyright 2005 by ASME Transactions of the ASME

2 Fig. 1 Typical planar SOFC configurations proper account of electrochemical kinetics. The mass and momentum equations are numerically solved to obtain the flow field. The source terms in the momentum equations are added based on the Darcy s law, representing an additional drag force proportional to fluid viscosity and velocity, and inversely proportional to the permeability of a porous medium. The species concentration equations are solved to obtain hydrogen, carbon monoxide, carbon dioxide, water, and oxygen distributions, in which the source terms are implemented based on electrochemical kinetics. The mass flux in the species equation Eq. 7 is modeled by The oxygen ion is then migrated through the electrolyte and then into the porous anode, which is generally made of nickel/ yttria-stabilized zirconia Ni/YSZ. At the same time, the fuel gas flows through the fuel gas channel, and diffuses into the porous anode. Both H 2 and CO are oxidized with the oxygen ion at the three-phase boundary. The reactions are as follows: H 2 +O 2 H 2 O+2e 2 CO+O 2 CO 2 +2e 3 The above electrochemical reactions are limited to a single-step reaction in the present model, i.e., H 2 +CO+2O 2 H 2 O+CO 2 +4e 4 Governing Equations. In this model, ideal gas mixtures, incompressible and laminar flow due to small gas velocities, and pressure gradients are assumed. The electrodes have a homogeneous, isotropic structure and no gradients within the mechanical properties. Mass sources/sinks according to the electrochemical conversion arise in the entire region of the anode and cathode. The coordinate system is fixed for all simulations, where z is the direction of the gas flow in the channels; the x-y plane is perpendicular to the gas flow direction. Conservation equations of mass, momentum, chemical species, electron, and ion transport, as presented in Table 1 Eqs. 5 9 are numerically solved, with Fig. 2 Half cross-sectional views of the cathode-supported and anode-supported MOLB-type SOFCs J i = D i,eff Y i M D j,eff Y j 10 j where the first term on the right-hand side represents the Fickian diffusion due to concentration gradients, and the second term is the correction term to enforce the Stefan-Maxwell equation. Since diffusion occurs only within the pores, species face a larger resistance of diffusion as the porosity of the porous medium decreases. In addition, at a fixed porosity, resistance is higher for more tortuous pores. Thus, the effective diffusivity in the porous electrode is modeled by using the Bruggeman relation D i,eff = D i 11 In the solid phase of the PEN plate, the current passes through either the electrolyte or the catalyst. Thus, two charge conservation equations are solved inside the SOFC simultaneously. The electron transport equation Eq. 8 is solved in the electrodes and the interconnects, while the ion transport equation Eq. 9 is solved in the electrodes and the electrolyte. The corresponding source terms are attributed to the co-oxidation of the H 2 and CO, and the O 2 reduction reaction, respectively. These two current components interact through electrochemical reactions, during which electrons are either transferred to the catalyst from the electrolyte phase, or vice versa. This results in a source in one of the phases and a sink in the other phase. The source or sink during the electric or ionic conduction is represented by the transfer current density j T, which is a measure of the electrochemical reaction rate. It connects with the species transport kinetics by using the Butler-Volmer equation 13. Thus, the resulting transfer current densities in the anode and cathode are j T,A = j 0,A X H 2 1/4 X 1/4 CO exp AF X H2,ref X CO,ref X H2 O,ref 1/4 X 1/4 CO 2 X CO2,ref X H 2 O E RT C E exp 1 A F C RT j T,C = j 0,C X 1/4 O 2 X O2,ref exp CF RT C E where j 0,A and j 0,C are the anodic and cathodic exchange current densities, respectively, A and C are the anodic and cathodic Table 1 Governing equations and the corresponding source terms Source terms Conservation equations Anode Cathode Electrolyte Channels Interconnects Eq. Mass U =S m S m =0 S m =0 NA S m =0 NA 5 Momentum UU = p+ +S u S u = 2 U S u = 2 U NA S u =0 NA 6 Species UY i = J i +S i S i = a P i a R i j T,A F S i = a P i a R i j T,C F NA S i =0 NA 7 Electron C C +S e c =0 S e c =j T,A S i c =j T,C S i =0 NA 0 8 Ion E E +S i c =0 S i c = j T,A S i c = j T,C S i =0 NA NA 9 Journal of Fuel Cell Science and Technology AUGUST 2005, Vol. 2 / 165

3 Table 2 Electrochemical properties in the porous electrodes Description Value Anode exchange current density, J 0,A,Am Cathode exchange current density, J 0,C,Am Anode transfer coefficient, A 1.0 Cathode transfer coefficient, C 1.0 Anode Stoichiometric coefficient 1.5 Cathode Stoichiometric coefficient 3.0 Anode inlet pressure, atm 1.0 Cathode inlet pressure, atm 1.0 Stack temperature, K 973 Species mass fraction at the anode, % Species mass fraction at cathode, % Hydrogen, Y H2 40 Carbon monoxide, Y CO 35 Carbon dioxide, Y CO2 30 Water, Y H2 5 O Total 100 Oxygen, Y O2 21 Nitrogen, Y N2 79 Total 100 Tafel constants, respectively. In the electrodes, the potential difference between the catalyst C and the electrolyte E drives the transfer current j T,A or j T,C, keeping the electrochemical reaction continuous. The electrochemical and physical properties used in the calculation are given in Tables 2 and 3. The cell temperature is fixed at 973 K in the present calculation. The anode fuel gas has four species, i.e., H 2,CO,CO 2, and H 2 O. The cathode side feeds with ambient air, where N 2 is considered as an inert gas and serves as a diluent. The mass fraction of the fuel is 40%, 35%, 20%, and 5% for H 2,CO,CO 2, and H 2 O, respectively. Air and fuel delivered to the cathode and anode are kg/s and kg/s, respectively. Numerical Details The governing equations are numerically solved by the controlvolume-based finite difference method 14. The discretization procedure ensures conservation of mass, momentum, concentration, and charge over each control volume. Three momentum equations corresponding to three coordinates are solved, followed by a pressure correction equation that does the mass balance. Concentration transport equations are solved after the bulk flow calculation. Velocity control volumes are staggered with respect to the main control volumes, and coupling of the pressure and velocity fields is treated via the SIMPLER pressure correction algorithm In addition, a new voltage-to-current algorithm was implemented to solve the electrochemical characteristics 10. A unique feature of the iterative voltage-to-current algorithm is its capability for accurate calculation of the local activation overpotentials, which in turn results in improved prediction of the current density distribution. This algorithm is capable of predicting the cell current density based on a target cell operating voltage. The upwind difference scheme is used to treat the diffusion and convective terms. Obtaining new values for any desired variable, taking into account that the latest known estimated values of the variable from the neighboring nodes, solves the set of differential equations over the entire region. One iteration process is complete when, in the line-by-line technique, all lines in a direction have been accounted for. Because of the large variations in the source terms, underrelaxation is necessary for the dependent variables and the source terms to achieve convergence. Line inversion iteration with typical underrelaxation values of 0.1 for the velocity term and pressure correction terms are incorporated to facilitate the calculation. Solutions are considered to be converged at each test condition after the ratio of residual source including mass, momentum, and species to the maximum flux across a control surface drops below In the calculations, only half of both the channel and the innterconnect are considered because of the symmetry perpendicular to the direction of the gas flow. All computations are performed on X by Y by Z structured, orthogonal meshes in the present work. Additional runs for the coarser meshes, , and the finer meshes, , are taken for a check of grid independence. A comparison of the results of the two grid sizes, and , shows that the maximum discrepancies in the axial velocity and oxygen concentration profiles are 1.1% and 1.4%, respectively. In addition, results indicate a maximum change of 0.6% in current density distribution between the solutions of the and grids. These changes are so small that the accuracy of the solutions on grids is deemed satisfactory. A typical simulation requires about 300 min of central processing unit time on a Pentium IV 2.8 GHz PC. Table 3 Physical properties in the porous electrodes Description Anode Cathode Electrolyte Anode/cathode contact Interconnect Porosity, Permeability, /m S/V/m Pore size/m Electric conductivity, C / 1 m 1 Ionic conductivity, E / 1 m / Vol. 2, AUGUST 2005 Transactions of the ASME

4 Fig. 3 a Velocity distributions on several cutting surfaces of the channels and electrodes of the anode-supported MOLBtype SOFC, and b secondary flow structures in the electrodes Results and Discussion Velocity Distributions. Figure 3 a shows the velocity distribution on several selected sectional planes cutting across the anode-supported MOLB-type SOFC. The right half of each plane shows the velocity vectors, composed of three velocity components U, V, W, whereas the left half shows the axial velocity contours W. An additional magnification showing the secondaryflow vectors V, W in the electrodes is also provided in the bottom-right area of the plot Fig. 3 b. It is seen that the axial velocity W near the channel center gradually increases downstream due to the flow development. The axial velocity in the electrodes, however, is negligible. This is because a significant surface drag in the porous matrix retards the axial flow development. Attention is turned to the cross-vectors in the electrodes shown in Fig. 3 b. In the anode, the secondary flow is directed from the electrolyte toward the flow channel. This is because the gas bulk density in the anode is increased due to the co-oxidation reaction Eq. 4. Thus, the outflow should be higher than the inflow in the anode for mass conservation. In contrast, the oxygen reduction reaction in the cathode reduces the air bulk density Eq. 1. Thus, the fresh air flows continuously into the cathode from the channel to satisfy the continuity. It is noted that at the interface of the channel flows and the electrode, the secondary-flow velocity vectors are always directed toward the channel center due to the wall effect. Mass Fluxes. Figures 4 and 5 show the mass flux of the reactants in the anode side and the cathode side, respectively. Figures 4 a and 4 b compare the mass flux of carbon monoxide J CO between the anode-supported cell and the cathode-supported cell. Both figures reveal much higher mass fluxes in the anode than those in the channel. This is because the electrochemical reaction consumes CO, and thus a higher CO mass flow rate is required for conservation of the concentration. The maximum values of J CO occur at the corner formed by the inclined surface and the top surface. It is further seen that the vectors of J CO for the cathodesupported cell are larger than those for the anode-supported cell. This is because the thinner anode for the cathode-supported cell has a smaller reaction surface. Hence, the mass flux should be higher when keeping the same rate of electrochemical reaction. Comparing Figs. 4 b and 4 c reveals that the higher overpotential causes higher mass fluxes in the active anode. Figure 5 shows the oxygen mass fluxes J O2 in the cathode and in the air channel. Again, the oxygen mass fluxes in the cathode of Fig. 4 Comparison of mass flow flux of CO in the anode and anode channel, a cathode-supported cell, =0.5 V, b Anodesupported, =0.5 V, and c anode-supported cell, =0.1 V the anode-supported cell are higher than those of the cathodesupported cell due to the smaller reaction surfaces Figs. 5 a and 5 b. A higher overpotential has a higher J O2 in the cathode and the air channel Figs. 5 b and 5 c. Concentrations Distributions. The evolutions of O 2 and CO Journal of Fuel Cell Science and Technology AUGUST 2005, Vol. 2 / 167

5 Fig. 6 Comparison of the evolution of the species contraction along the channel direction between the cathode-supported and anode-supported SOFCs, =0.5, a cathode-supported cell and b anode-supported cell Fig. 5 Comparison of mass flux of O 2 in the cathode and cathode channel, a cathode-supported cell, =0.5 V, b anodesupported cell, =0.5 V, and c anode-supported cell, =0.1 V concentrations in the cathode-supported cell and the anodesupported cell are shown in Figs. 6 a and 6 b, respectively. The right half shows the several cross-sectional results and the left half shows the module surface results, including upper and lower surfaces and the symmetric planes. It is seen that both figures show that the concentrations of O 2 and CO decrease along the flow direction due to the electrochemical reaction. The oxygen concentration reduces significantly in the cathode, especially in the region in contact with the upper interconnect. The reduction rate is more significant for the anodesupported cell than for the cathode-supported cell. As for the CO concentration, two local minimums are observed. One is located at the corner of the anode-formed inclined plate and the top plate. The other local minimum is situated at the middle of the bottom plate in contact with the lower interconnect. The reduction rate of the CO concentration is higher for the cathode-supported cell than for the anode-supported cell. Figure 7 shows the distribution of species concentration in the electrodes of the MOLB-type SOFC. The curves displayed in the figure are the results cutting a cross the middle of the inclined PEN plate. In the anode, the O 2 concentration decreases form the channel/anode interface to the anode/electrolyte interface. In the cathode, concentrations of both H 2 and CO decrease from the channel/cathode interface to the cathode/electrolyte interface. The decrease rate of CO is higher than that of H 2 due to the lower effective diffusivity accompanied with CO. As for the products of the reaction, concentrations of both CO 2 and H 2 O increase from the channel/cathode interface to the cathode/electrolyte interface. Current Density Distribution. Figures 8 a and 8 b compare the current density distributions between the anode-supported cell and cathode-supported cell. The current is driven from the anode to cathode inside the SOFC, and distributes only on the electrode 168 / Vol. 2, AUGUST 2005 Transactions of the ASME

6 Fig. 7 Species concentration distributions in the electrodes of MOLB-type SOFCs along the middle of the inclined PEN plate, =0.5 a cathode-supported cell and b anode-supported cell Fig. 8 Current density distributions on the midchannel plane Z=0.05 of the MOLB-type SOFCs, =0.5, a cathodesupported cell and b anode-supported cell and the interconnects. The arrows indicate the direction as well as the magnitude of the currents. Qualitatively, these two plots have several similar points. First, the current density gradually decreases and increases along its direction in the anode and cathode, respectively. The above feature reflects the fact that the electron acts as the reactant in the oxygen reduction reaction Eq. 1, but as the product in the electrochemical oxidation reaction Eq. 4. Second, the thick electrode has a larger area for the current passage, and thus has a lower current density. The major difference between these two types of SOFCs is the magnitude of the current density. The anode-supported SOFC has a slightly higher current density than the cathode-supported SOFC, typically about 5%. The reason may be attributed to the higher effective electric conductivity for the anode-supported cell as compared to the cathodesupported cell Table 3. Conclusions A mathematical model has been developed to resolve the characteristics of mass/charge transfer in electrode-supported MOLBtype SOFCs. A complete set of conservation equations of mass, momentum, species, energy, and charge are numerically solved with proper account of electrochemical kinetics. A detailed comparison of the mass/charge transfer characteristics between the anode-supported and cathode-supported MOLB-type SOFCs is made. It is found that the mass fluxes of CO in the cathodesupported cell are significantly higher than those in the anodesupported cell. Conversely, the oxygen mass fluxes in the anodesupported cell are much higher than those in the cathodesupported cell. Moreover, the anode-supported MOLB-type SOFC has a slightly higher current density than the cathode-supported MOLB-type SOFC due to its higher effective electric conductivity. Acknowledgment This work was sponsored by the National Science Council of Taiwan under Contract No. NSC E Nomenclature a i P stoichiometric coefficient of the products a i R stoichiometric coefficient of the reactants D i diffusivity of species i, m 2 s 1 F Faraday s constant, C mol 1 i current density, A m 2 J diffusive flux kg s 1 m 2 j transfer current density A m 3 k thermal conductivity, W K 1 m 1 M molecular weight, kg mol 1 p pressure, Pa R universal gas constant, W mol 1 K 1 S source terms in the governing equations T temperature, K U velocity vectors, m 1 Journal of Fuel Cell Science and Technology AUGUST 2005, Vol. 2 / 169

7 U, V, W velocity components in X, Y, Z directions, respectively, ms 1 X, Y, Z coordinate system, Fig. 1 m X i fraction of molar concentration of species i Y i mass fraction of species of i Greek Symbols C catalyst potential, V E electrolyte potential, V symmetric factor electric conductivity, 1 m 1 porosity permeability, m 2 tortuosity density, kg m 3 overpotential, C E,V viscosity, ms 2 Superscript e electron i ion P products R reactants Subscript A anode C catalyst or cathode c charge eff effective E electrolyte F fluid i species m mass ref reference S solid T transfer current u momentum References 1 Achenbach, E., and Reus, U., 1999, The Effect of Mass-Flow Distribution on the Characteristics of a Solid Oxide Fuel Cell System, 6th Int. Symp. Solid Oxide Fuel Cells (SOFC VI), Hawaii. 2 Yuan, J., Rokni, M., and Sunden, B., 1999, The Development of Heat Transfer and Gas Flow Modeling in the Solid Oxide Fuel Cells, 6th Int. Symp. Solid Oxide Fuel Cells (SOFC-VI), Hawaii. 3 Hartvigsen, J., Khandkar, A., Elangovan, S., Rowley, D., Privette, R., and Tharp, M., 1999, Status and Progress in SOFCo s Planar SOFC Development, 6th Int. Symp. Solid Oxide Fuel Cells (SOFC-VI), Hawaii. 4 Yakabe, H., Ogiwara, T., Yasuda, I., and Hishinuma, M., Model Calculation for Planar SOFC Focusing on Internal Stresses, 6th Int. Symp. Solid Oxide Fuel Cells (SOFC-VI), Hawaii. 5 Khaleel, M. A., Recknagle, K. P., Lin, Z., Deibler, J. E., Chick, L. A., and Stevenson, J. W., 2001 Thermomechanical and Electrochemistry Modeling of Planar SOFC Stacks, 7th Int. Symp. Solid Oxide Fuel Cells (SOFC-VII), Tsukuba, Japan. 6 Hwang, J. J., Chen, C. K., and Lai, D. Y., 2005, Computational Analysis of Species Transport and Electrochemical Characteristics in an MOLB-type SOFC, J. Power Sources, 140, pp Donelson, R., Ratnaraj, R., and Foger, K., 1999, Demonstration of Anode Supported Cell Technology in kw Class Stack, 6th Int. Symp. Solid Oxide Fuel Cells (SOFC-VI), Hawaii. 8 Ackmann, T., de Haart, L. G. J., Lehnert, W., and Stolten, D., 2003, Modeling of Mass and Heat Transport in Planar Substrate Type SOFCs, J. Electrochem. Soc., 150, pp. A783 A789 9 Chung, B. W., Pham, A. Q., Haslam, J. J., and Glass, R. S., 2002, Influence of Electrode Configuration on the Performance of Electrode-Supported Solid Oxide Fuel Cells, J. Electrochem. Soc., 149, pp. A325 A Hwang, J. J., and Lai, D. Y., 2005, Detailed Characteristic Comparison Between Planar and MOLB-type SOFC, J. Power Sources, 143, pp Fee, D. C., Blackburn, P. E., Busch, D. E., Chaar, T. D., Dees, D. W., Dusek, J., Easler, T. E., Ellingson, W. A., Flandermeyer, B. K., Fousek, R. J., Mrazek, F. C., Picciolo, J. J., Poeppel, R. B., and Zwick, S. A., 1986, Monolithic Fuel Cell Development, Fuel Cell Seminar, Tuscon. 12 Sakaki, Y., Nakanyama, A., Hottori, M., Miyamoto, H., Aiki, H., and Kakenobu, K., 2001, Development of MOLB Type SOFC, SOFC VII, The Electrochemical Society, New York, pp Costamagna, P., and Honegger, K., 1998, Modeling of Solid Oxide Heat Exchanger Integrated Stacks and Simulation at High Fuel Utilization, J. Electrochem. Soc., 145, pp Patankar, S. V., 1980, Numerical Heat Transfer and Fluid Flow, Hemisphere, New York. 15 Van Doormaal, J. P., and Raithby, G. D., 1984, Enhancements of the SIMPLE Method for Predicting Incompressible Fluid Flows, Numer. Heat Transfer, 7, pp Hwang, J. J., and Lai, D. Y., 1998, Three-Dimensional Mixed Convection in a Rotating Multiple-Pass Square Channel, Int. J. Heat Mass Transfer, 41, pp Hwang, J. J., Chen, C. K., Savinell, R. F., Liu, C. C., and Wainright, J., A Three-Dimensional Numerical Simulation of the Transport Phenomena in the Cathodic Side of a PEMFC, J. Appl. Electrochem., 34, pp / Vol. 2, AUGUST 2005 Transactions of the ASME