A three-dimensional full-cell CFD model used to investigate the effects of different flow channel designs on PEMFC performance

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1 International Journal of Hydrogen Energy 32 (2007) A three-dimensional full-cell CFD model used to investigate the effects of different flow channel designs on PEMFC performance Yuh Ming Ferng a,,aysu b a Fuel Cell Center, 135 Yuan-Tung Road., Nei-Li, Chung-Li, Taoyuan, Taiwan 32026, ROC b Department of Mechanical Engineering, Yuan Ze University, 135 Yuan-Tung Road., Nei-Li, Chung-Li, Taiwan 320, ROC Received 21 November 2006; received in revised form 10 April 2007; accepted 10 May 2007 Available online 27 June 2007 Abstract A three-dimensional full-cell computational fluid dynamics (CFD) model is proposed in this paper to investigate the effects of different flow channel designs on the performance of proton exchange membrane fuel cells (PEMFC). The flow channel designs selected in this work include the parallel and serpentine flow channels, single-path and multi-path flow channels, and uniform depth and step-wise depth flow channels. This model is validated by the experiments conducted in the fuel cell center of Yuan Ze University, showing that the present model can investigate the characteristics of flow channel for the PEMFC and assist in the optima designs of flow channels. The effects of different flow channel designs on the PEMFC performance obtained by the model predictions agree well with those obtained by experiments. Based on the simulation results, which are also confirmed by the experimental data, the parallel flow channel with the step-wise depth design significantly promotes the PEMFC performance. However, the performance of PEMFC with the serpentine flow channel is insensitive to these different depth designs. In addition, the distribution characteristics of fuel gases and current density for the PEMFC with different flow channels can be also reasonably captured by the present model International Association for Hydrogen Energy. Published by Elsevier Ltd. All rights reserved. Keywords: PEMFC; Flow channel design; CFD 1. Introduction Being lightweight, ensuring environmental safety, and having high power density and immediate response to power demands, the proton exchange membrane fuel cell (PEMFC) is one of the best candidates for power sources for commercial applications. These applications include the remote power supplies, vehicle power systems, and stationary power generations, etc. Therefore, many experiments and simulations have been conducted to investigate the PEMFC. A PEMFC typically consists of gas flow channels, gas diffusion layers, catalyst layers dispersed with platinum, and a polymer membrane, which is schematically shown in Fig. 1. The operating mechanisms for a PEMFC are typically characterized as: Gaseous hydrogen fuel is supplied to the anode flow channel, travels through the porous gas diffusion layer, and then Corresponding author. Tel.: address: ymferng@ess.nthu.edu.tw (Y.M. Ferng). reaches the catalyst layer where it dissociates into protons and electrons. The protons migrate across the membrane to the cathode side and the electrons pass through the outer circuit to generate electricity. Gaseous reactant oxygen or air is forced into the cathode flow channel, travels through the porous gas diffusion layer, and dissolves into the catalyst layer. The dissolved oxygen reacts with these protons and electrons to produce water. There were many experiments [1 10] and simulation models [11 33] to investigate the PEMFC during the past decade, including the measurement of water transfer rate and proton diffusion coefficients, the experimental and analytical studies of humidity effect, the investigation of water uptake and transport phenomena, the experimental and analytical investigation of operating parameter effects, the experimental and analytical studies of water and heat management, the optimization of operating parameters, etc. However, several works were recently /$ - see front matter 2007 International Association for Hydrogen Energy. Published by Elsevier Ltd. All rights reserved. doi: /j.ijhydene

2 Y.M. Ferng, A. Su / International Journal of Hydrogen Energy 32 (2007) Nomenclature a,a normalized stoichiometric coefficients of reactions and products D i effective mass diffusion coefficient of ith species in the porous media, m 2 /s D io original mass diffusion coefficient of ith species, m 2 /s d porous bead diameter, m F Faraday constant, 96,487 C/mol F I inertial coefficient for porous media F Darcy Darcy drag, N/m 3 F Forch Forchheimer drag, N/m 3 K porosity area tensor g gravitational acceleration, m/s 2 i current density, A/m 2 J i diffusion flux of ith species, kg/m 2 s j o,j exchange current density for jth reaction step, A/m 2 j T transfer current, A/m 2 j T,j transfer current for jth reaction step, A/m 2 M molecular weight, kg/mol N step number of electrochemical reaction step P pressure, N/m 2 R ( gas constant, J/mol K SV ) effective surface-to-volume ratio of catalyst, eff m 1 U velocity vector, m/s u axial velocity, m/s V cell cell voltage, V V oc open circuit voltage, V v, w radial velocity, m/s Y species mass fraction Y in,k mass fraction of kth species at the electrolyte/catalyst interface Y p,i Greek symbol α a,j α c,j mass fraction of ith species in the pore fluid in the catalyst region anode transfer coefficient for jth reaction step cathode transfer coefficient for jth reaction step porosity of gas diffusion layer ε δ unit tensor δ diffusion length scale κ permeability for porous media η cathode overpotential, V μ viscosity, kg/m s potential, V σ electrical conductivity, 1/Ωm ρ density, kg/m 3 [Λ k ] average molar concentration of kth species, kmol/m 3 ω i production rate of ith species, kg/m 3 s τ tortuosity P porous extra pressure drop, N/m 2 Subscript F i S in interface eq outlet electrolyte phase (or ionic phase) ith species solid phase (or electronic phase) inlet condition boundary value at the catalyst/cathode interface equilibrium state outlet condition Fig. 1. Schematic of a PEMFC.

3 4468 Y.M. Ferng, A. Su / International Journal of Hydrogen Energy 32 (2007) carried out to study the effect of flow channel on the PEMFC performance [34 37]. The flow channels in the PEMFC are essentially used to distribute the fuel gases on the electrode surface and remove the by-product water. Therefore, the types and dimensions of flow channels play an essential role on the performance of PEMFC. Without considering the membrane electrode assembly (MEA), a computational fluid dynamics (CFD) model proposed by Hontanon et al. [34] is used to analyze the flow distribution systems of different designs. A threedimensional half-cell model is proposed by Kumar and Reddy [36] to concentrate on the improvement of PEMFC performance through optimization of the channel dimensions and shape in the flow channels. Soler et al. [37] experimentally investigated the effect of both the permeability of electrodes and the configuration of the gas flow distributor on the PEMFC performance. A three-dimensional full-cell CFD model is proposed in this paper to investigate the effects of different flow channel designs on the PEMFC performance. The flow channel designs selected in this study include the parallel and serpentine flow channels, single-path and multi-path flow channels, and uniform depth and step-wise depth flow channels. The effects of different flow channel designs on the PEMFC performance predicted by this model correspond well with the experimental data. The calculated performance of PEMFC with the serpentine flow channel is shown to be higher than that of PEMFC with the parallel flow channel, which is confirmed by the experiments. The parallel flow channel with step-wise depth design will enhance the performance of PEMFC, which is confirmed by both the experiments and predictions. Although, for the serpentine flow channel, the PEMFC performance is insensitive to the flow channels with different depths. It must be noted that these conclusions are applicable to the flow channel designs of the same kinds, such as geometry, shape, and direction of fuel fed into channel, etc. In addition, distribution characteristics of fuel gases and current densities in the PEMFC are reasonably captured by the present model. These correspondences between the experiments and predictions clearly reveal that the three-dimensional full-cell CFD model proposed in this paper can be used to assist in the optimal design of flow channels for the PEMFC. 2. Mathematical models A three-dimensional full-cell PEMFC model proposed herein should consider the three-dimensional hydraulic transport, electrochemical reactions, and current transfer, etc. Therefore, this model consists of both the hydraulic equations and electrochemical models. Several assumptions are needed for this model and are presented as follows: The PEMFC flow system modeled by this paper essentially belongs to the laminar flow system. A PEMFC is assumed to be steady state and isothermal. The energy equation is not needed in the flow equations. All gases within a PEMFC can be considered as the ideal gases. Gravitational effect is neglected. Stefan Maxwell equation is used to describe the species diffusion. Butler Volmer equation is adopted to model the electrochemical reactions in the catalyst layer. Nerst Planck equation is adopted to model the transport of protons through the membrane Hydraulic equations Continuity equation (ρ K U)= 0. (1) An isotropic porous medium is assumed in the porous electrode, then K = ε δ, (2) ρ = i Y i ρ i. (3) Momentum equation [ρ( K U) U]= (μ K U) ε P + ερ g ε P porous, (4) where P porous represents the extra pressure drop that is caused by the existence of solid particles in the porous media of diffusion layer. This term includes the Darcy drag and Forchheimer drag and is needed in the porous diffusion layer. The Darcy Forchheimer drag force can be evaluated via following formulas [38,39]: P porous = F Darcy + F Forch, (5) F Darcy = Darcy drag = μ U κ, (6) F Forch = Forchheimer drag = ρf I κ U U. (7) The permeability κ and inertial coefficient F I can be defined as [40] ε 3 d 2 κ = 150(1 ε) 2, (8) F I = ε 1.5. (9) Species transport equation [ρ U K)Y i ]= J i + ω i, (10) where J i = ρd i Y i + ρy i M D i M ρy i ρy i ΔM M j D j Y j D j Y j [41], (11) j D i = D i,o ε τ, (12)

4 Y.M. Ferng, A. Su / International Journal of Hydrogen Energy 32 (2007) where τ is tortuosity and is commonly set as 1.5 [11 13,20,21] ω i = (a i a i )j T F, (13) j T = N step 1 j T,j. (14) The first term on the right-hand side of Eq. (11) is the Fickian diffusion due to concentration gradients. The last three terms are used for the Stefan Maxwell equations that are applied in the multi-component species systems. The sum of all mass fractions equals to one, thus Y i = 1. (15) i 2.2. Electrochemical models Based on the electro-neutrality, current conservation is written as i = 0 and i = i F + i S (16) where i F = part of current through the fluid phase (or ionic phase), i S = part of current through the solid phase (or electronic phase). The two current components shown above are completely independent and can interact only through the electrochemical reactions. Then, i S = i F = j T ( S V and ) eff ( ) N S steps = j T,j (17) V eff 1 i S = σ S S, i F = σ F F, (18) where (S/V ) eff is the effective surface-to-volume ratio that is related to the catalyst loading. j T,j is the transfer current that is a measure of electrochemical reaction and can be obtained from the Bolter Volmer equation in the general form [42]. j T,j = j o,j Nk=1 [Λ k,ref ] α k,j [ ( αa,j F exp ( exp α )] c,j F RT η ) RT η N [Λ k ] α k,j, (19) k=1 [Λ k ]=ρy in,k /M k, (20) η = S F. (21) Based on the balance between the reaction flux and the diffusion flux, the following equation can be derived by Jaouen et al. [43]: N steps j=1 M i (a ij a ij )j T,j F = ρd i Y i. (22) The right-hand side of above equation can be approximated in the following discrete form. Y in,i Y p,i ρd i Y i ρd i. (23) δ Coupled with Eqs. (17), (18), (20) and (22), the mass fraction at the interface (Y in,i ) can be obtained and the source of ith species ( ω i ) in Eq. (9) can be solved via ( ) Y in,i Y P,i S ω i = ρd i δ V 2.3. Boundary conditions. (24) eff The governing equations described above are essentially the partial differential equations. Appropriate boundary conditions are needed in order to solve the whole problem. The inlet boundary conditions are specified based on the test conditions. Velocity (U in ) and mass fractions of species (Y H2 O,in,Y O2,in,Y H2,in) are set at the inlet. The values of these parameters can be derived from the given total inlet flow rate and relative humidity. For an open calculation domain, no special boundary condition is needed for the CFD calculation, except for outlet pressure (P out ). The boundary conditions needed for simulations can be summarized as follows: Inlet boundary conditions: U inlet = U in, (25) Y H 2O inlet = Y H 2 O,in,Y O 2 inlet = Y O 2,in,Y H 2 inlet = Y H 2,in. (26) Outlet boundary conditions: P outlet = P out,v outlet = w outlet = 0, (27) Axial gradient of (u, Y ) outlet = 0. (28) Wall boundary conditions. According to the assumption of laminar flow in the PEMFC, no-slip boundary condition is used at the wall. 3. Numerical treatment Using a control volume approach, the differential governing equations can be discretized to the finite-difference forms for numerical calculations. The coupled equations for both the velocity and pressure are solved by the so-called SIMPLE scheme [44]. The procedure for simulating a whole PEMFC can be described as follows: 1. Set the boundary conditions based on the test conditions, including the inlet flow velocity, species mass fraction, and pressure, etc.

5 4470 Y.M. Ferng, A. Su / International Journal of Hydrogen Energy 32 (2007) Preset the cell voltage. 3. Set the initial guess for all the parameters in the computational domain. 4. Solve the momentum equation (Eq. (4)) to determine the three-dimensional characteristics of flow field within a PEMFC. 5. Solve the pressure correction equation derived from the continuity equation (Eq. (1)) to eliminate the mass conservation error and then apply appropriate corrections [30] to update the velocity and pressure field. 6. Solve both the pore phase and solid phase potential using Eqs. (16) (18) and then calculate electrode overpotential. 7. Solve the coupled equations, including Eqs. (19), (20), (22), and (23), to obtain the mass fraction at the interface and subsequently calculate the generation rate for ith species. Fig. 2. Photographs of flow channels used in this work: (a) serpentine and (b) parallel. 8. Solve the species transport equation (Eq. (10)) to determine the distribution of ith species. 9. Repeat steps 4 8 until the convergence criteria are satisfied. Then, the localized characteristics of flow fields and electrochemical parameters are obtained. 10. Average current density for the PEMFC under the preset cell voltage is obtained by averaging the calculated local current density over the cross section of current collector. 11. Repeat steps 2 10, the relationship between the cell voltage and current density (i.e. performance curve) for the PEMFC is then obtained. The convergence criteria for all governing equations in the calculations are set as that summation of the relative residual in every control volumes is smaller than 1.0E 3. The flow channels adopted in this study essentially include the serpentine and parallel flow channels, respectively, as shown in Fig. 2. The corresponding grid model is schematically shown in the plot (c) of Fig. 3 that displays the top view of the threedimensional grid model. The flow directions for the fuel gases are indicated in this figure, which are denoted as anode inlet, anode outlet, cathode inlet, and cathode outlet, respectively. The space coordinate is indicated in the left portion. There are 30 grids in the X-direction. Finer grid distributions are adopted in the two-side regions where these are the elbow regions of flow channel and the fluid will flow reverse there. The 205 grids are uniformly distributed in the Y-direction. The grid numbers in the Z-direction typically includes 5, 8, 5, 5, and 5 grids for the collector, flow channel, diffusion layer, catalyst layer, and membrane, respectively. Grid independent calculations are performed. The comparisons show that the results presented in this paper are grid independent. All of the simulation works are Fig. 3. Schematic of flow channels and grid model.

6 Y.M. Ferng, A. Su / International Journal of Hydrogen Energy 32 (2007) performed using the CFD code-cfdrc on a PC with Pentium GHz CPU. This CFD code was developed by the CFD Research Corporation [45]. 4. Results and discussion A CFD model is proposed herein to investigate the effects of flow channel designs on the performance of PEMFC. The Table 1 Dimensions of the test PEMFCs Length of flow channel (mm) 50 Width of flow channel (mm) 1.2 Depth of flow channel (mm) Type A 1 Type B 1.2 Type C 1.8 Type D 1.4, 1.2, 1 Type E 1.8, 1.4, 1 Type F 1.4, 1.2, 1.1, 1 Type G 1, 1.1, 1.2, 1.4 Width of rib (mm) 1.2 Thickness of collector (mm) 1 Thickness of diffusion layer (mm) 0.2 Thickness of membrane (mm) Thickness of anode catalyst layer (mm) Thickness of cathode catalyst layer (mm) Reaction area of PEMFC (cm Table 2 Typical test conditions H 2 flow rate (kg/s) O 2 flow rate (kg/s) Humidity 100% Operating temperature (K) 323 Operating pressure (atm) 1.0 related input data for the simulation works is shown in Table 3. In order to validate this model, the corresponding experiments are conducted by the staff of fuel cell center in the Yuan Ze University. Graphite plates for the test PEMFC are used as current collectors with flow channels and Gore 5620 is used as an MEA. Hydrogen (H 2 ) and pure oxygen (O 2 ) are used as the fuel and reactant gases, respectively. The flow direction of fuel gas in the anode channel is opposite to that of reactant gas in the cathode channel. The related dimensions for the test PEMFCs are indicated in Table 1 and the test conditions are typically indicated in Table 2. Seven types of flow channel designs are used in the experiments, which include uniform depth type (Type A Type C), three channels with step-wise depth type (Type D, Type E), and four channels with step-wise depth type (Type F, Type G). The test facility and control system are schematically shown in Fig. 4. The test system for the fuel cell is developed by Scribner Associates and its control system is made by Globe Tech. Computer Cell GT. Figs. 5 and 6 show the voltage and power density vs. current density plots for the PEMFC with serpentine and parallel flow channels, respectively. Both the flow channels are the Type A flow channels indicated in Table 1. In these two figures, the symbols represent the predicted results and the lines are the experimental data. Left and right longitude coordinates indicate the voltage and power density of PEMFC, respectively. Both figures clearly reveal that the performance curves (i.e. voltage vs. current density curve) predicted by the present CFD model agree well with those obtained by the experiments. The same results are indicated in the corresponding power density vs. current density. Discrepancies between the experiments and the calculations are essentially caused by neglecting the effects Fig. 4. Schematic of test facility and control system.

7 4472 Y.M. Ferng, A. Su / International Journal of Hydrogen Energy 32 (2007) Table 3 Input data for simulations Diffusion and catalyst layers porosity 0.4 Diffusion and catalyst layers permeability (m 2 ) Effective diffusivity for diffusion and catalyst layers 1.5 Membrane porosity 0.28 Membrane permeability (m 2 ) Effective diffusivity for membrane 7 Transfer coefficient at anode 0.5 Transfer coefficient at cathode 1.5 Concentration dependence at anode 0.5 Concentration dependence at cathode 1.0 Reference current density at anode (A/m 3 ) (m 3 /kg mol H 2 ) 1/2 Reference current density at cathode (A/m 3 ) (m 3 /kg mol O 2 ) 1/2 Diffusion layer conductivity (1/Ωm) 100 Catalyst layer conductivity (1/Ωm ) 100 Fig. 5. Comparisons of voltage and power density vs. current density between experiments and predictions for the PEMFC with serpentine flow channels (Type A). of contact resistance, temperature variations, and liquid water effects, etc. It can be clearly seen in the comparison of Figs. 5 and 6 that the performance for the PEMFC with the serpentine flow channel is superior to that of the parallel flow channel. Both the experiments and predictions confirm this result. Fig. 7 shows the comparison of oxygen concentration distributions between the serpentine (a) and parallel (b) flow channels (Type A) at the cell voltage of 0.6V. In the cathode compartment, the oxygen will deplete as the reactant gas fuel passes downstream along the flow channel since part of oxygen will diffuse toward the catalyst layer and react with protons. Both the figures demonstrate this depletion phenomenon. In addition, a large region in the cathode compartment of PEMFC with parallel flow channel in which the oxygen is almost completely depleted is shown in Fig. 7(b). This excessive non-uniform distribution of oxygen will induce the concentration polarization (or mass limitation), causing that the performance of PEMFC with the parallel flow channel is inferior to that of PEMFC with the serpentine flow channel. This mass limitation phenomenon is also revealed in Fig. 8. This figure shows the current density distributions in the current collector for both the serpentine (a) and parallel (b) flow channels under the cell voltage of 0.2V. As demonstrated in Fig. 8(b), the current density in the middle region is near zero for the PEMFC with parallel flow channel, which renders lower cell performance. However, the mass limitation caused by the excessive non-uniform fuel distributions can be improved by the appropriate design of geometric shape for the parallel flow channel [46]. Fig. 9 shows the comparison of voltage and power density vs. current density for the PEMFC with parallel flow channel between the experiments (lines) and predictions (symbols). This flow channel is a Type F one that belongs to a four path flow channel with step-wise depths. These four paths form one group in which the dimensions for these four paths are 1.4, 1.2, 1.1, 1 mm, respectively. There are five groups (20 paths) in the parallel flow channel with step-wise depths, which are used for experiments and simulations. Reasonable correspondence between the experiments and predictions is also revealed in this Fig. 6. Comparisons of voltage and power density vs. current density between experiments and predictions for the PEMFC with parallel flow channels (Type A). figure. Using the parallel pattern, the effects of different flow channel designs on the PEMFC performance is demonstrated in Fig. 10 by comparing the experimental results of Type A and Type F. This figure shows that the performance of PEMFC with flow channel of step-wise depths (Type F) is higher than that of uniform depth (Type A). Similar results are precisely captured by the calculations, which are demonstrated in Fig. 11. These phenomena can be also demonstrated in the current density distribution in the current collector, as displayed in Fig. 12. Using model predictions, this figure shows the comparison of current density distribution in the PEMFC under cell voltage of 0.2V. Plot (a) is the PEMFC with parallel flow channel of uniform depth (Type A) and plot (b) is that of stepwise depths (Type F). It can be clearly seen in this figure that the region of near zero current density for Type A flow channel design is much larger than that for Type F flow channel design. This interpretation for the lower cell performance can

8 Y.M. Ferng, A. Su / International Journal of Hydrogen Energy 32 (2007) Fig. 7. O 2 concentration in the cathode flow channel: (a) serpentine and (b) parallel (cell voltage = 0.6, other conditions in Table 2). Fig. 8. Current density distribution in the collector: (a) serpentine and (b) parallel (cell voltage = 0.2, other conditions in Table 2). Fig. 9. Comparisons of voltage and power density vs. current density between experiments and predictions for the PEMFC with parallel flow channels (Type F). Fig. 10. Experimental comparisons of voltage and power density vs. current density between Type A and Type F parallel flow channels for the PEMFC. be only obtained by the calculated results, which demonstrate the merits of simulation model for the PEMFC. However, for the serpentine flow channel, the PEMFC performance is insensitive to the different depths of flow channel designs. The calculated results shown in Fig. 13 confirm this conclusion, which also displays a comparison of voltage and power density vs. current density between the experiments and predictions. In this figure, the symbols represent the predicted results for the PEMFC with parallel flow channel of step-wise depths (Type F) and the lines are those of uniform depth (Type A). Similarly, this phenomenon can also be revealed in the current density distributions for both cases, as shown in Fig. 14.

9 4474 Y.M. Ferng, A. Su / International Journal of Hydrogen Energy 32 (2007) The region of lower current density for the flow channel of Type A is similar to that for the flow channel of Type F, showing the performance of PEMFC with the serpentine flow channel is same for both the types of flow channels. 5. Conclusions This paper presents a three-dimensional CFD model in order to investigate the effects of different flow channel designs on the PEMFC performance. This model is validated by the experiments. All of the experiments were conducted in the fuel cell center of Yuan Ze University. The flow channel designs used in this study include the parallel and serpentine flow channels, single-path and multi-path flow channels, and uniform depth and step-wise depth flow channels. Compared to the experimental results, the predicted effects of different flow channel designs on the PEMFC performance demonstrate good agreement. The performance of PEMFC with the serpentine flow channel is superior to that of PEMFC with the parallel flow channel, which is shown in both the experiments and the calculations. In addition, based on the calculation results, the performance of PEMFC with the parallel flow channel is significantly influenced by the different designs of flow channels. The performance of PEMFC with flow channels of step-wise depths (Type F) is superior to that of uniform depth (Type A), which is confirmed by the experimental data. However, the performance of PEMFC with the serpentine flow channel is insensitive to the flow channels with different depth designs. The calculated results of fuel gases and current density distributions can reasonably explain these phenomena, which demonstrate the merits of simulation model for the PEMFC. However, it must be noted that these conclusions are applicable to the flow channel designs of the same kinds, such as geometry, shape, and direction of fuel fed into channel, etc. These correspondences between the experiments and predictions clearly reveal that the three-dimensional full-cell CFD model proposed in this paper can effectively help the optimal design of flow channels for the PEMFC. Fig. 11. Computational comparisons of voltage and power density vs. current density between Type A and Type F parallel flow channels for the PEMFC. Fig. 13. Computational comparisons of voltage and power density vs. current density between Type A and Type F serpentine flow channels for the PEMFC. Fig. 12. Current density distribution in the collector for the PEMFC with parallel flow channel: (a) Type A and (b) Type F (cell voltage = 0.2, other conditions in Table 2).

10 Y.M. Ferng, A. Su / International Journal of Hydrogen Energy 32 (2007) Fig. 14. Current density distribution in the collector for the PEMFC with serpentine flow channel: (a) Type A and (b) Type F (cell voltage = 0.2, other conditions in Table 2). Acknowledgment The authors wish to thank their working partners in the fuel center of Yuan Ze University for supporting the current investigation work and helping them to carry out the experiments presented in this paper. References [1] Verbrugge MW, Hill RF. Ion and solvent transport in ion-exchange membranes. J Electrochem Soc 1990;137: [2] Verbrugge MW, Hill RF. Transport phenomena in perfluorosulfonic acid membranes during the passage of current. J Electrochem Soc 1990;137: [3] Fuller TF, Newman J. Experimental determination of the transport number of water in Nafion 117 membrane. J Electrochem Soc 1992;139: [4] Zawodzinski Jr. et al.. Determination of water diffusion coefficients in perfluorosulfonate ionometric membranes. 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