EFFECT OF HUMIDITY ON PEM FUEL CELL PERFORMANCE PART II - NUMERICAL SIMULATION

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1 Proceedings of AME IMECE, Nashville, TN, HTD 364-1, pp (1999 EFFECT OF HUMIDITY ON PEM FUEL CELL PERFORMANCE PART II - NUMERICAL IMULATION. himpalee and. Dutta Department of Mechanical Engineering W. K. Lee and. W. Van Zee Department of Chemical Engineering University of outh Carolina Columbia, C 908 ABTRACT Experiments have shown that the inlet humidity has a significant influence on the performance of a polymer electrolyte membrane (PEM fuel cell, and theory indicates that the ionic resistivity of the electrolyte membrane is dependent on the activity of water at the membrane surface. Water flux and activities change along the flow field direction. To understand the inner flow and mass transfer processes, a numerical model is developed to predict the flow inside a single fuel cell. Detailed velocity fields, pressure profiles, and current density distributions are obtained and predictions from the full-cell model are compared with the experimental data. Predictions indicate that flow inter-linkage between side-by-side flow channels occurs through the porous diffusion layer. Results also indicate that the diffusion of hydrogen is aided by the flow toward the membrane in the anode side and diffusion of oxygen is opposed by the flow direction present in the cathode side. NOMENCLATURE a K activity of water A cv specific face area of the control volume (c.v., m -1 C wk concentration of water at K membrane interface, mol/m 3 D w diffusion coefficient of water, m /s F Faraday constant, C/gm-equivalent I local current density, A/m I o exchange current density for the oxygen reaction, A/m diffusion mass flux, kg/m -s m mass fraction of the species M molecular weight, kg/kmol equivalent weight of a dry membrane, kg/kmole M m,dry P sat w,k vapor pressure of water in k channel, atm P total pressure, atm P O partial pressure of oxygen, atm R gas constant, /mol/k t m membrane thickness, m T temperature T s surface temperature, K u i velocity in i direction, m/s V oc cell open-circuit voltage, V V cell cell voltage, V x channel length measured from anode inlet, m X w,k mole fraction of water in K stream dynamic viscosity, kg-s/m α net water flux per proton flux β permeability, m n d electro-osmotic drag coefficient (number of water molecules carried per proton η overpotential for oxygen reaction, V σ m membrane conductivity, 1/ohm/m ρ m,d density of a dry membrane, kg/m 3 ρ density of the mixture, kg/m 3 ubscripts and superscripts a anode c cathode cr critical H hydrogen i x,y,z components K anode or cathode

2 l O w sat dummy variables for species oxygen water saturated INTRODUCTION A three-dimensional flow simulation is created to understand the physics inside a Polymer Electrolyte Membrane (PEM fuel cell. The complete three-dimensional Navier-tokes equations are solved to obtain the velocity and pressure distributions along the flow channels. These mass flow conditions and corresponding water activities in both cathode and anode are coupled to the computation of the net water flux per proton, the electro-osmotic drag coefficient, and the membrane conductivity; and they are used to predict the local current density for optimum operation of a fuel cell. Reacting gases are transported from the main flow channel to the membrane surface through anode and cathode diffusion layers. This three-dimensional PEM fuel cell analysis gives details of current density and velocity profiles in the cross-flow plane that cannot be obtained by simplistic onedimensional and two-dimensional simulations. The electrochemical reactions of hydrogen and oxygen are modeled, and bulk flow is computed based on a control volume approach. In this work, the spatial source terms are developed and these terms can be included in any three-dimensional flow solver. The general concept of fuel cell operation is characterized as gasmixture transport and transformation of species by electrochemical reactions. The hydrogen from anode flow channel is transported through the diffusion layer toward the Membrane Electrode Assembly (MEA surface. Hydrogen molecules are dissociated to protons and electrons in the catalyst as: H H e The water that impregnates the MEA hydrates the protons. Electroosmosis and diffusion transport the water in the MEA. The air mixture in cathode channel is transported through the diffusion layer toward the MEA where oxygen reacts with protons as: 1 + O + H + e HO In previous studies, most models focused on the one-dimensional flow. Only a few models included two-dimensional transport of the reactants and products in the flow channels and across the membrane. Main difficulty in increasing the spatial dimension is that the source code becomes significantly complicated in three dimensions. The model presented by pringer et al. (1991 laid the foundation for flow simulations. They considered a pseudo one-dimensional model in which flow channels were treated as being perfectly well mixed. Their work also provided the key properties of the membrane required for a numerical model. Later, Fuller and Newman (1993, and Nguyen and White (1993 developed two-dimensional heat and water transport models that accounted for variation in temperature and membrane hydration conditions along the flow channels. Fuller and Newman (1993 included the diffusion layer for water transport, but Nguyen and White (1993 did not incorporate the diffusion layer. However, Nguyen and White (1993 investigated the effectiveness of various humidification systems in maintaining high membrane hydration and performance for PEM fuel cells. Recently, Yi and Nguyen (1998 modified the previous models to describe mass and thermal conditions in both the liquid and gas phase along the flow path of both anode and cathode sides of a PEM fuel cell. Furthermore, Gurau et al. (1998 developed a two-dimensional model for PEM fuel cell. However, the + + Inlet The channels (white line are machined into the graphite Outlet Figure 1. The picture shows actual flow-field plate with the gas channel. There are twenty straight channels connected in a serpentine fashion. Anode side and cathode side flow channels are symmetric and placed properly aligned (nonstaggered on top of each other. density change due to species transformation was not accounted in their two-dimensional model. Moreover, the connectivity of the main channel and diffusion layer involved change of primary variables. Another recent numerical prediction by Yi and Nguyen (1999 analyzed a two-dimensional hydrodynamics in the cathode of a PEM fuel cell. In this simulation, a steady state, three-dimensional, isothermal, single-phase, and multi species PEM fuel cell model is developed. The model regions consist of two flow channels (anode and cathode separated by the MEA. There are two diffusion layers, which are made of porous materials and are placed in between the flow channels and the MEA. This work is the first prediction for a fuel cell using the three dimensional aspects of a full-cell model. The governing electrochemical equations used are similar to the analysis based on Nguyen and White (1993 and Yi and Nguyen (1998. Present model converts governing electrochemical differential equations to source terms that can be used for control volume analysis as shown by Dutta et al.(1999 for a straight channel. A commercial flow solver, FLUENT, solves the complete Navier-tokes flow equations with our source terms. The effects of porosity in the diffusion layers on the local current density, velocity fields, and pressure profiles are studies with different inlet humidity, which is varied based on experimental work (see Part 1 of this paper. MODEL DEVELOPMENT This numerical simulation is based on a steady state, isothermal, and three-dimensional mass transfer model of a full-cell PEM fuel cell. The actual flow path consists of a serpentine gas channel that has 0 passes as shown in Figure 1. Figure shows the channel geometry and associated coordinate system. A thin membrane-electrode-assembly (MEA is sandwiched between anode and cathode diffusion layers. Figure 3 shows more details of the computational domain, which is consisted of the anode flow channel, anode diffusion layer, MEA, cathode diffusion layer, and cathode flow channel. Figure 3 also shows different z-locations that are used in defining source terms (see Table 1. As mentioned earlier, activities in the membrane are simulated with source terms in the control volumes in contact with the membrane.

3 z x y Width Figure. The geometrical model of the complete fuel-cell A straight section of a flow channel Typical cross-section of a flow channel Anode diffusion layer Cathode diffusion layer z 0 z 5 Anode flow channel z 4 z 3 z z 1 MEA Cathode flow channel Model Equations A commercial three-dimensional flow solver, FLUENT, is modified to include the electrochemical activities in a PEM fuel cell. Governing equations and respective source terms are listed in Table 1. Note that source terms are zero in most of the computation domain. Non-zero source terms are defined in selected regions as listed in Table 1 and these modeled source terms affect the flow physics near the MEA of a PEM fuel cell. The source term m is the mass sources/sinks to the continuous phase for each control volume in the domain. This term is calculated by the consumption of hydrogen and water in the anode, and by the consumption of oxygen and production of water in the cathode. In this work, there are four species: hydrogen, oxygen, nitrogen, and water vapor. Mass sources and sinks are calculated from the total of species sources and sinks in the cells as shown in the species transport equations. The momentum transport equation has a sink term for the porous media ui used for flows through diffusion layer. This term is βi developed based on Darcy s Law. The addition of this source term effectively converts the momentum equation in the x-direction to P = u β x x in the porous layer because β x is very small and thus the other terms become negligible. A pressure drop is created in the porous layer that is proportional to the mixture velocity in the cell. The diffusion mass flux, i,l, in the species transport equation is defined as ml i,l = ρ Dl (13 i where, D l is the binary diffusion coefficient of species l with the gasmixture, j,defined as (Bird et al., 1966: PD ( P cr j * P 1 / 3 cr l.(t T. j,l cr j cr l 5 / 1.( 1 M j + 1 M = 3.64x10 1 / l 8 ( cell cr j T. cr l T T.334 (14 The diffusion coefficient of each species to the mixture is reduced by 50 percent in the diffusion layer to include the effect of porosity and pore-tortuosity. The equation for water transport coefficient, α, is given as: F ( Cwc ( ( ( x, y Cwa ( x, y α ( x, y = nd x, y DW x, y (15 I( x, y tm where, n d is the electro-osmotic drag coefficient that is a function of the activity of the water in the anode side, a a, of the MEA. The electroosmotic drag coefficient is calculated as: ( x,y nd = aa 4.53aa aa ;aa 1 = ( a 1 ;a > 1 a a 3 (16 where, a a is the water activity in the anode. The water diffusion coefficient, D W, used in this work is given by Nguyen and White (1993 as: D W = η d 5.5 x10 exp 416 x ( Ts Figure 3. The detail of computational domain and grid arrangement used in this model (Dutta et al.,

4 Table 1. Governing equations and source terms Governing Equations Mathematical expressions Non-zero volumetric source terms and location of application (see Fig. Conservation of mass ρu ρv ρw m = H + + = m (1 + aw at z = z3 = + at z = z (7 Momentum transport ρ u ρu ρu P u u u u + v + w = x y z x x y ρ v ρv ρv P v v v u + v + w = x y z y x y ρ w ρw ρw P w w w u + v + w = pz x y z z x y ( Hydrogen transport (anode mh ρmh ρmh x,h y,h z,h u + v + w = H side Water transport (anode side px py m px py pz O 1 cw u = ; β x v = ; β y w = β at z z z ρ (3 I( x, y ρ maw ρmaw ρmaw x,aw y,aw z,aw + v + w = + + (4 ( x, y aw aw u + Oxygen transport (cathode ρmo ρmo ρmo x,o y,o z,o u + v + w = O side Water transport (cathode mcw ρmcw ρmcw x,cw y,cw y,cw u + v + w = cw side z 4 (8 H = M H A cv at z = z3 (9 F ( x, y M H A O cv at z z3 = α I = F (10 (5 I( x, y O = M A O cv at z = z (11 4F ρ (6 1+ α( x, y cw = I( x, y M H O Acv at z = z Table. Different flow conditions used for different inlet humidity conditions Flow condition Very low humidity Low humidity High humidity Very high humidity Anode channel inlet Gas-mixture velocity Mole fraction of H Mole fraction of H O Cathode channel inlet Gas-mixture velocity Mole fraction of O Mole fraction of N Mole fraction of H O F (1 The expression for water concentration of the anode and cathode sides, C w,a and C w,c, are calculated as: ρm,dry 3 CwK ( x,y = ( aK 39.8aK aK ;ak 1 M m,dry (18 ρm,dry = ( ( ak 1 ;ak > 1 M m,dry where, K in the subscript, represents either the anode or cathode, and ρ m,dry and M m,dry are the density and the equivalent weight of a dry PEM. The activity of water is defined as follows: X w,k ( x, y P( x, y a K ( x, y = (19 sat Pw,K The mole fraction for each species used in these equations are linearly extrapolated to the membrane surface. This linear extrapolation achieves a grid independent solution. To calculate the local current density in the fuel cell model, the equation is given as I( x, y σ m( x, y = { Voc Vcell η( x, y } (0 t m where, σ m is a membrane conductivity and calculated as a function of water content on the membrane surface at the anode interface. The equation is given as (pringer et al, 1991: M m,dry 1 1 σ m( x, y = Cwa( x, y 6.exp168 x10 m,dry 303 T ρ s (1 and the cell over-potential, η, can be calculated from the following equation (pringer et al, 1991: RTs I( x,y. η ( x,y = ln( 0.5F I.P ( x,y ( o O Where I o is the exchange current density at one atmosphere of oxygen, and PO ( x, y is the local partial pressure of the oxygen in the cathode. NUMERICAL PROCEDURE To solve the coupled flow and species transport equations, a control volume technique is used. Commercial flow solver, FLUENT (version 4.48, is used in this simulation. ince this software cannot compute the electrochemical reactions occurring in the fuel cell, new subroutines are written to incorporate the modeled source terms. 4

5 According to the Figure 3 as mentioned earlier, the geometry of fuel cell system simulated in this work consists of two flow channels (upper is anode and lower is cathode separated by diffusion layers and MEA. There are twenty serpentine passes in the flow path, so that the flow path is approximately sixty centimeters long in the axial direction with 0.1 (height x 0.08 (width cm cross-section flow area. Each diffusion layer has dimension of 0.05 (height x 3.0 (width x 3.0 (length cm 3. The total grid size of this fuel cell simulation is 34x00x8 uniform grid cells. A separated grid independence test is performed by increasing and decreasing the number of the grid cells on a straight channel. The number of grid cell is decreased and increased by 50 percent of the base case, and predicted results are compared with the base result. The results are less than percent different from each other. Therefore, the numerical grid density that is used for this work is grid independent. The transport of water and proton is simulated by source terms in control volumes in contact with the membrane. The operating pressure is one atmosphere and cell temperature is constant at 70 o C. The membrane thickness used in this simulation is 50 micron and the open circuit cell voltage is set to 0.6 volt. The solution procedure used in this commercial flow solver is based on a IMPLE algorithm. Three momentum equations corresponding to three coordinates are solved, followed by a pressure correction equation that does the mass balance. pecies transport equations are solved after the bulk flow calculation. The mixture properties at each control volume are calculated based on the local species content. The anode side gas mixture contains hydrogen and water vapor. On the other hand, the cathode side gas mixture contains oxygen, water vapor, and nitrogen. Therefore, the density and viscosity of the two flow channels are different and vary from one location to the other. Furthermore, the four different inlet humidity conditions are considered in this study based on experimental data presented in part 1 of this paper. This analysis considers variations in the permeability of the diffusion layers in each case, which are.0x10-10 m for high permeability, 1x10-1 m for medium permeability, and 1x10-15 m for low permeability. Table shows different flow conditions used in this work based on experimental operating conditions. There are four different inlet humidity conditions, which are very low humidity, low humidity, high humidity, and very high humidity. REULT AND DICUION The results are divided into two parts. The effect of permeability of diffusion layers on the velocity distributions and pressure contours of the mixture are described in part one. The second part discusses the effect of inlet humidity on the predicted current densities. The effect of permeability of diffusion layers on the velocity distributions and pressure contours of the mixture As mentioned previously, the porosity of the diffusion layers is varied for each inlet humidity condition, and therefore there are three different numbers for permeability, which are 1.0x10-15 m (low permeability, 1.0x10-1 m (medium permeability, and.0x10-10 m (high permeability respectively. The high humidity inlet condition is selected to study the secondary velocity vectors in the cross flow planes (y-z plane at the center of the fuel cell model (0.013m< y < 0.017m, 0.005m< z <m, and x = m. Figure 4 shows the effect of low permeability on the secondary velocity vectors at both side of the channel. At the anode side (top channel, both hydrogen and water vapor are consumed on the membrane surface. Therefore, the vectors are pointing downward Vector plot in the anode channel reference vector(0.005m/s Figure 4. The velocity vectors at selected cross-flow planes for high humidity with low permeability Vector plot in the cathode channel reference vector(0.005m/s Vector plot in the anode channel Vector plot in the cathode channel reference vector(0.01m/s reference vector(0.01m/s Figure 5. The velocity vectors at selected cross-flow planes for high humidity with medium permeability from the channel to the diffusion layer. According to the low permeability in the diffusion layer, there is a high flow restriction in all direction. Consequently, the velocity vectors are spreading to the side of the interface of the diffusion layer and pointing down to the membrane surface. In the cathode side (bottom channel, oxygen is consumed on the membrane surface while the water vapor is produced from the membrane. The velocity vectors are pointing from the membrane surface to the cathode flow channel through the diffusion layer because more water is produced than the oxygen consumed. The velocity in the axial direction in the cathode flow 5

6 Vector plot in the anode channel Vector plot in the cathode channel reference vector(0.5m/s reference vector(.0m/s Figure 6. The velocity vectors at selected cross-flow planes for high humidity with high permeability channel is much greater than the velocity in anode flow channel (see Table ; and therefore, the vorticity caused by sharp turns in the flow channel are observed in the cathode channel. The overall vector patterns in this figure indicate the total mass coming in or going out of the membrane. The vectors at the anode side show that there is a higher rate of mixture transport compared to the cathode side. Note that hydrogen is lighter than oxygen and therefore, volumetric flow of hydrogen caused by electrochemical mass-consumption is greater than oxygen. There is a change in the velocity vectors when the permeability of diffusion layers is changed to 1.0x10-1 m (medium permeability. Figure 5 shows the effect of medium permeability on both sides of fuel cell model. At the anode side, The overall velocity vectors pattern is similar to Figure 4 but the magnitude of the velocity in diffusion layers is greater. This is because of the smaller restriction in the diffusion layer compared to the low permeability case. The mass of the mixture can easily be transported from the channel to the membrane surface through the diffusion layer. In the cathode side, there is a significant change in the pattern of the velocity vectors compared to Figure 4. There is a flow driven from left to right in the channel width by pressure gradient. The cathode main flow channel shows the vorticity created by turns. Note that a reduction of flow restrictions in diffusion layers increases flow-coupling between side-by-side flow channels. Figure 6 shows the velocity vectors when high permeability (.0x10-10 m is used in both anode and cathode. The flow pattern is similar in both sides of the channel and results indicate a strong coupling between flow channels. The flow in both cathode and anode is driven from left to right by the pressure gradient but the magnitude of velocity in cathode side is greater than anode side as a result of higher inlet velocity in the cathode side compared to the anode. The pressure contour plots of different permeability are shown in Figures 7, 8, and 9. These figures present the pressure contours of the center channel of Figures 4, 5, and 6 respectively. Figure 7 shows the effect of low permeability on the pressure contours at both sides of the channel. The average inlet pressure in the anode side is Avg. Density = 0.084kg/m 3 velocity = 0.78m/s Figure 7. Pressure (Pa contour plots at selected crossflow planes for high humidity with low permeability Pa and that for the cathode side is 6570 Pa. The exit pressure for all simulations is set to 0 Pa. Therefore, these inlet pressures indicate the pressure drop in the flow channels. Near the flow inlet the mixture density in the anode side is 0.1 kg/m 3 and for the cathode side, the mixture density is 0.84 kg/m 3. In the anode side, there is a pressure gradient from the flow channel to the membrane surface through the diffusion layer due to the consumption of both hydrogen and water vapor at the membrane surface. In the cathode side, the pressure gradient is varied from the membrane surface to the flow channel through the diffusion layer. This is because of the total mass going out of the cathode side membrane surface. There is a high-pressure difference between flow channel and diffusion layer at both sides of the channel. This is because of high flow restriction caused by low permeability in the diffusion layers. Average density of anode channel decreases due to migration of water from this side of the fuel cell to the cathode side. Whereas, the average density of the cathode side decreases due to dilution by added water vapor. Therefore, both cathode and anode channels show a decrease in density compared to corresponding inlet values. Figure 8 shows the pressure contour plots when medium permeability is used in the diffusion layer at both cathode and anode. The inlet pressure for anode side is 155Pa and that for the cathode side is 1900 Pa. In the anode side, there is the pressure gradient from the flow channel to the membrane surface through the diffusion layer. As a result of the lower flow restriction in the diffusion layer, the pressure difference between the flow channel and diffusion layer is Avg. Density = 0.78kg/m 3 velocity = 4.01m/s

7 Avg. Density = kg/m 3 velocity = 0.71m/s Avg. Density = 0.088kg/m 3 velocity = 0.55m/s Avg. Density = kg/m 3 velocity = 3.94m/s Avg. Density = 0.80kg/m 3 velocity =.8m/s Figure 8. Pressure (Pa contour plots at selected cross-flow planes for high humidity with medium permeability lower than that as shown in Figure 7. In the cathode side, the pressure gradient is varied from the left side to the right side of the channel width. Furthermore, the pressure contour plots as illustrated in Figure 9 are shown when the high permeability is used in the diffusion layer at both cathode and anode. The inlet pressure for this case is 16 pa and 1558 Pa for anode and cathode channels respectively. Note that these pressures are significantly lower than low permeability and therefore, indicate a lower pressure drop. In both cathode and anode, the pressure gradient is varied from the left side to the right side of the channel width but the pressure magnitude of cathode side is greater than anode side. This is because of the higher inlet velocity in the cathode side compared to the anode side. Therefore, the pressure contours are consistent with velocity vectors for each case. The effect of inlet humidity on the average current density Figure 10 illustrates the ratio of averaged current densities of entire fuel cell model for different inlet humidity conditions. Current densities are presented as a ratio because the experimental data is based on atm and numerical predictions are based on 1 atm operating pressures. In the numerical results, it has been shown that the averaged current density is increasing when the humidity of the inlet is increased. This is because of the current density is a function of the activity of anode water, which is calculated from the mole fraction of anode water. Moreover, the effect of the permeability on the local current density of the fuel cell model is not significant due to the sufficient availability (no starvation of reactants in the gas-mixtures. Figure 9. Pressure (Pa contour plots at selected crossflow planes for high humidity with high permeability Hence, the averaged current densities of each condition for three types of permeability are similar. It is anticipated that this similarity will not hold at reduced mass flow rates (i.e., near end of cell stack. In the experimental results, the averaged current densities are greater than numerical results for very low humidity, low humidity, and high humidity conditions. This could be because of the operating pressure of experimental is greater than numerical calculations. However, the experimental current density is increasing along with the increased inlet condition up to high humidity. In the very high humidity condition, the averaged current density of the experimental results is lower than high humidity condition. It can be argued that there are condensation occurred at some sections of fuel cell; and the liquid water blocks the pores in the diffusion layer. There is no reaction in those sections and no local current density occurred. Figure 11 shows the numerical prediction of local current densities along the flow path of the fuel cell for medium permeability of each inlet conditions. The current densities are decreasing along the channel path because of the progressively reduced activity of the anode water. The local current density contours of high humidity condition with medium permeability are shown in Figure 1. This figure presents the local current densities at each location on the membrane surface, which can not be provided in the one-dimensional or two-dimensional simulations. Results indicate that there are significant spatial variations in the local current density values. 7

8 cell-average current density / I (high humidity very low humidity low humidity high humidity very high humidity Different inlet humidity conditions experimental low-permeability med-permeability high-permeability Figure 10. The cell-average current density ratio dependence on the inlet humidity. A comparison of numerical predictions with experiment Channel length(m inlet Figure 1. The local current density (A/m contours on the membrane surface for high humidity with medium permeability outlet Current density(a/m veryhigh humid. high humid. low humid. verylow humid. ACKNOWLEDGMENT This project was supported by the Department of Energy through Cooperative Agreement Number DE-FG0-91ER REFERENCE Bird, R., tewart, W., and Lightfoot, E., 1966, "Transport Phenomena," 7 th edition, ohn Wiley & ons, Inc. Dutta,., himpalee,., and Van Zee,. W.,1999, Threedimensional numerical simulation of straight channel PEM fuel cells, submitted to. Applied Electrochem Distance(m from inlet along the serpentine channel Figure 11. Local current density plots along the flow path for medium permeability for different inlet humidity conditions CONCLUION The three-dimensional flow simulation of PEM fuel cells has been developed in which the addition of diffusion layers in both anode and cathode shows the mass transportation behavior caused by inlet velocity, chemical reactions, and permeability of the diffusion layers. The velocity distributions and pressure contours in cross-flow planes are analyzed. ignificant flow interactions between side-by-side flow channels are observed. Furthermore, the effect of inlet humidity conditions on the current density is presented in this work and averaged current densities of each condition are calculated and compared to the experimental results. It is concluded that changing the permeability of diffusion layer can significantly affect velocity fields and pressure contours. It is also concluded that in absence of condensation, an increase in the inlet humidity increases the averaged current density of the fuel cell. It can be argued that the experiment has a condensation effect with very high humidity inlet condition and therefore, measured current density is lower than the predicted values. Hence, the performance of a PEM fuel cell drops when the membrane is flooded. Fuller, F. and Newman,., 1993, "Water and Thermal Management in old-polymer-electrolyte Fuel Cells,". Electrochem. oc., Vol. 140, pp Gurau, V., Liu, H., and Kakac,., 1998, "Two-Dimensional Model for Proton Exchange Membrane Fuel Cells," AIChE ournal, Vol. 44, pp Nguyen, T. and White, R., 1993, "A Water and Heat Management Model for Proton-Exchange-Membrane Fuel Cells,". Electrochem. oc., Vol. 140, pp pringer, T., Zawodzinski, T., and Gottesfeld,., 1991, "Polymer Electrolyte Fuel Cell Model,". Electrochem. oc., Vol. 138, pp Yi,. and Nguyen, T., 1999, "Multicomponent Transport in Porous Electrodes of Proton Exchange Membrane Fuel Cells Using the Inerdigitated Gas Distributors,". Electrochem. oc., Vol. 146, pp Yi,. and Nguyen, T., 1998, "An Along-the Channel Model for Proton Exchange Membrane Fuel Cells,". Electrochem. oc., Vol. 145, pp

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