A simple model of a high temperature PEM fuel cell

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1 international ournal of hydrogen energy 35 (2) Available at ournal homepage: A simple model of a high temperature PEM fuel cell O. Shamardina a, *, A. Chertovich a, A.A. Kulikovsky b, A.R. Khokhlov a a Moscow State University, Faculty of Physics, 999 Moscow, Russian Federation b Institute for Energy Research d Fuel Cells (IEF 3) Research Centre Jülich D Jülich, Germany article info Article history: Received 2 June 29 Received in revised form 25 October 29 Accepted 2 November 29 Available online 5 December 29 abstract We develop a simple analytical model of a high temperature hydrogen fuel cell with proton exchange membrane. The model is validated against experimental results obtained in our group. The model is pseudo two dimensional, steady-state and isothermal, it accounts for the crossover of reactant gases through the membrane and it can be solved analytically. The role of the crossover is considered in detail. ª 29 Professor T. Neat Veziroglu. Published by Elsevier Ltd. All rights reserved. Keywords: PEM fuel cell PBI Simulation Crossover. Introduction Fuel cells as electrochemical current generators attract increasing attention of scientific community, industry and consumers. Fuel cells produce low environmental pollution, they have high efficiency and they are silent. Apparently fuel cells will play the key role in the future hydrogen economy []. In a proton exchange membrane (PEM) fuel cell the current is produced due to a space separation of two electrochemical reactions: anode : H 2 2H þ þ2e cathode : 4H þ þ4e þ 2H 2 O total reaction : 2H 2 þ 2H 2 O The space separation is achieved due to special properties of an electrolyte membrane. The subect of this work is high temperature PEM fuel cell (HT PEMFC) with polymer membrane. Working temperature of these cells is about 6 C (433 K), which is rather high for polymer materials. The design of a typical fuel cell is shown in Fig.. () HT PEM fuel cells have obvious advantages over conventional PEMFCs with Nafion-type polymer electrolyte membranes. HT PEMFC tolerate a high level of poisoning molecules in feed gases and they do not require water management. However, these cells still have several unresolved engineering problems. Modeling can help to understand the processes and effects occurring under different operating conditions. There is still lack of models for HT PEMFCs, compared to conventional PEM fuel cells. Some of the fuel cell characteristics and properties cannot be measured in situ in the fuel cell environment and their value may differ from those measured ex situ. Besides, complex experimental diagnostic procedures and tools are often needed for such measurements [2,3]. The reasonable model should help to estimate the cell properties. Furthermore, it is much easier to make a simulation experiment than a real one to study the response of the system to changes in some parameters or conditions. A vast maority of HT PEMFC models are either phenomenological, or numerical. A parametric D numerical model [4] fails to predict a mass transport limitation effect. 2D numerical * Corresponding author. Tel.: þ ; fax: þ addresses: shamardina@polly.phys.msu.ru (O. Shamardina), chertov@polly.phys.msu.ru (A. Chertovich), A.Kulikovsky@fzuelich.de (A.A. Kulikovsky), khokhlov@polly.phys.msu.ru (A.R. Khokhlov) /$ see front matter ª 29 Professor T. Neat Veziroglu. Published by Elsevier Ltd. All rights reserved. doi:.6/.ihydene.29..2

2 international ournal of hydrogen energy 35 (2) Nomenclature marks dimensionless variables A h Auxiliary dimensionless variable b Tafel slope on the cathode side (V) D eff Effective diffusivity of oxygen in the GDL (cm 2 s ) O2 c Oxygen concentration in the MEA (mol cm 3 ) c Constant hydrogen concentration (mol cm 3 ) H2 c h Oxygen concentration at the inlet (mol cm 3 ) c h Local oxygen concentration in the cathode channel (mol cm 3 ) c ref Reference concentration (mol cm 3 ) c t Oxygen concentration in the cathode catalyst layer (mol cm 3 ) F Faraday constant (C mol ) H Channel height (cm) J Average current density (A cm 2 ) J fix The required stoichiometric flow is achieved at this current density (A cm 2 ) J Local current density (A cm 2 ) * Exchange current density (A cm 2 ) lim Limiting current density (A cm 2 ) Pseudo current density of the crossover of H H2 2 in the membrane (A cm 2 ) Pseudo current density of the crossover of O O2 2 in the membrane (A cm 2 ) L Channel length (cm) l d l m Thickness of the GDL (cm) Thickness of the membrane (cm) N i Molar flux of gas species i (mol cm 2 s ) P Total pressure (Pa) p i Partial pressure of species i (Pa) R Gas constant (J K mol ) R U Resistance of the cell (U cm 2 ) T Temperature (K) V cell Cell potential (V) V oc Open-circuit voltage (V) v Air flow velocity (cm s ) w Channel width (cm) x Distance in the through-plane direction (cm) z Distance along the channel (cm) Greek a Transfer coefficient of the ORR d Inlet nitrogen oxygen mole ratio e Porosity of the GDL h Cathode overpotential (V) k Membrane conductivity (S cm ) l Stoichiometric flow of the cathode inlet gas l fix Required stoichiometric flow at J fix s GDL conductivity (S cm ) i Permeation coefficient of species i through the membrane (mol cm cm 2 s Pa ) models [5] predict the temperature and concentration profiles taking into account rib effects and variation of transport properties along the gas channels. A 3D numerical model [6] takes into account all the transport phenomena in a fuel cell and predicts the oxygen depletion under the ribs and temperature increase along the channel flow. The two-phase 2D numerical model [7] accounts for the dissolution of gases into the electrolyte. The model equations [4 7] are solved by the finite element method using the commercially-available software. In [8,9] the authors develop a D model and solve it numerically and analytically. The account for the cathode concentration overpotential in non-equilibrium conditions is based on the Nernst equation. All the models above are steady-state. A phenomenological control-oriented D transient model of HT PEMFC is developed in []. The model treats the crossover current as a real current; the transient effects are accounted for using a pseudocapacitance. The effect of catalyst poisoning by carbon monoxide is considered in []. The numerical model [] is transient, D and it considers only the membrane and catalysts layers. A single-phase steady-state [2] (D isothermal), [3] (3D nonisothermal), and transient [4] numerical models of HT PEMFC are based on commercially-available software. Our goal was to develop a simple analytical model which could help in understanding the properties of HT PEMFC and its structural layers. The model should show which parameters affect the cell efficiency and how the efficiency can be improved. 2. Experimental Fig. A typical fuel cell and its components: d bipolar plates, 2 d gas diffusion layers, 3 d anode catalyst, 4 d electrolyte membrane, 5 d cathode catalyst. The experiments for model verification were carried out in our group, all the components of the test membrane electrode assemblies (MEAs), except graphite paper and catalyst powder, were self-made. Gas diffusion electrodes (the gas diffusion layers with applied catalyst layers) were identical on the cathode and anode sides. We used Johnson Matthey HiSpec 3 catalyst powder (2% Pt/C). The catalyst layers were prepared by spraying the aqueous dispersion with 2% by weight PTFE powder

3 9956 international ournal of hydrogen energy 35 (2) Model N N H O onto the gas diffusion layers (GDLs) using an airbrush. After that the electrodes were baked at 35 C (623 K) for 2 min. The resulting platinum loading was approximately mg/cm 2.We used phosphoric acid doped PBI O PhT membranes with different additives as electrolyte (Fig. 2). These polymer membranes were prepared in A.N. Nesmeyanov Institute of Organoelement Compounds of Russian Academy of Sciences according to the patent [5]. The MEA size was 5 cm 2 and the geometry of the flow field was serpentine. The experimental conditions and setup parameters are listed in Table. The polarization curves were measured at the cell temperature of 6 C, i.e., 433 K (except few experiments at the other temperatures) and ambient pressure. The reactant gases were pure dry hydrogen and dry air (except one experiment with pure oxygen). The cell was operated at a fixed flow rate of gases so that at.4 A cm 2 the stoichiometry of the reactants had the required value, typically 5. We used potentiostat Autolab 32 with a frequency response analysis module and 2 A booster to measure polarization curves, MEA resistance and hydrogen crossover permeation coefficient. Table Parameters and properties for the base case. Parameter Value N N H Fig. 2 Monomer unit of PBI O PhT. Gas constant, R 8.3 J K mol Faraday constant, F C mol Cell temperature, T 433 K Total pressure, p 5 Pa Reference concentration, c ref 2.8$ 5 mol cm 3 Required stoichiometry 5 of cathode inlet gas at J fix ¼.4 A cm 2, l fix Thickness of the GDL, l d 4$ 2 cm Thickness of the membrane, l m 6$ 3 cm Channel length, L 2.24 cm Inlet nitrogen oxygen.79/.2 mole ratio, d Porosity of the GDL, e.7 Effective diffusivity 2.4$ cm 2 s of oxygen in the GDL, D eff O2 Electrical conductivity of the 2.2 S cm gas diffusion electrode, s Hydrogen permeation 3.$ 6 mol cm cm 2 s Pa coefficient, H2 Oxygen permeation 7.8$ 7 mol cm cm 2 s Pa coefficient, O2 O O n The schematic diagram of the modeled system is presented in Fig. 3. The developed model is pseudo two dimensional and it can be solved analytically [6]. The crossover of gases through the membrane has been taken into account in an attempt to explain the initial drop in the cell voltage [7]. At the temperature of 433 K the theoretical open-circuit voltage is about.3 V, while the measured OCV hardly reaches V. The voltage loss due to the crossover can be considerable in the case of production defects or incorrect membrane conditioning. Moreover, this loss is observable at a low stoichiometric flow of the reactant gases. The main assumptions for the model are: The temperature and total pressure are constant. The water in the system is in the gas phase. The oxygen reduction reaction (ORR) is described by the Tafel law. The rate of the ORR is uniform across the cathode catalyst. Polarization voltage on the cathode side is constant along the channel. Air flow in the channel is a plug flow with the constant velocity. Anode overpotential can be neglected. Transport losses on the anode side are neglected. The water in the anode region is neglected, i.e. the partial pressure of the hydrogen on the anode side is equal to the total pressure. For the sake of simplicity we do not consider temperature and total pressure variation within the cell. The cell temperature is high and the effects of liquid water (which may exist in small pores) are neglected. We use the Tafel law for the ORR since the Butler Volmer equation is more accurate only in a very small region near the OCV. We processed currentvoltage curves starting from the point where the Tafel approximation becomes valid. The assumption of uniform distribution of the ORR rate across the cathode catalyst layer means that the local current generated in the layer is a product of the reaction rate and the catalyst layer thickness. The Fig. 3 Schematic diagram of the modeled system.

4 international ournal of hydrogen energy 35 (2) polarization voltage on the cathode side is considered to be constant along the channel, though this assumption is possibly violated in the case of low stoichiometries of reactant gases. The assumption of a plug flow with the constant velocity in the cathode channel allows us to take into account the effect of oxygen depletion along the channel. The anode voltage loss is ignored since the anode reaction kinetics is very fast. The local current generated in the cathode catalyst layer can be expressed by the Tafel equation: ct h ¼ exp c ref b where * is the exchange current density, c t and c ref are the oxygen concentration in the cathode catalyst layer and the reference concentration respectively, h is the cathode overpotential and b is the Tafel slope b ¼ RT af Here R and F are the gas and Faraday constants, respectively, T is the cell temperature and a is the transfer coefficient of the ORR. The molar flux of the ith gas through the membrane is N cross i ¼ i Vp i (4) where i stands for hydrogen or oxygen, i is the membrane permeation coefficient [8] and p i is the respective partial pressure. We assume that the gas permeated through the membrane is fully consumed in an opposite electrode due to a direct catalytic combustion. It is convenient to use concentrations instead of partial pressures, thus equation (4) yields: (2) (3) ~x ¼ x l d ; ~ ¼ lim ; ~c ¼ c c h; ~h ¼ h b With these variables Eq. (6) can be written as: v~c v~x ¼ ~ þ ~ ~c t þ ~ H 2 ~c ~x¼ ¼ ~c h () The solution to Eq. () gives the oxygen concentration in the cathode catalyst layer: ~c t ¼ ~c h ~ ~ H 2 þ ~ () With (9) and () Eq. (2) takes the form ~ ¼ Ah ~c h ~ H 2 where ~ exp~h ~c ref A h ¼ þ ~ ~ þ exp~h ~c ref (9) (2) (3) is independent of ~x. To determine ~c h consider the flow along the channel in the z-direction (Fig. 3). The oxygen mass balance equation in the flow gives: v vc hðzþ ¼ vz h c h z¼ ¼ c h ðzþ 4F þ 4F! c t ðzþ þ H 2 c h 4F (4) N cross H 2 ¼ H RTc 2 H 2 lm N cross ¼ RTct lm (5) where v is the flow velocity and h is the channel height.with dimensionless variables where c H 2 is the constant concentration of hydrogen on the anode side and l m is the membrane thickness. On the cathode side, part of the oxygen flux is spent to produce useful current, the other part is wasted to react with the hydrogen permeated through the membrane and the rest permeates to the anode and reacts with the hydrogen: D eff vc vx ¼ 4F þ Ncross H 2 2 þ N cross (6) Here D eff and c are the effective oxygen diffusivity and concentration in the GDL, respectively. It is convenient to introduce the equivalent pseudo-crossover current densities: ~z ¼ z L ; ~v ¼ l dhv D eff L ¼ 4Fhvc h lim L Eq. (4) transforms to ~v v~c h v~z ¼ ~ þ ~ ~c t þ ~ H 2 ~c h ~z¼ ¼ (5) (6) Substituting for ~c t from () and solving the resulting equation we arrive at ~c h ~z ~ H þ 2 ð A h Þ Ah þ A h þ ~ A exp@ ~ O 2 ~z ~v þ ~ A ~ H 2 ð A h Þ A h þ ~ H 2 ¼ 4FRT H 2 c H 2 2lm ¼ 4FRT c h lm and the limiting current density: (7) (7) The z-dependence of the local current density ~ ð~zþ can be obtained from Eq. (2) with ~c h ð~zþ (7). By definition, the average cell current ~ J is lim ¼ 4FDeff c h l d (8) where l d is the GDL thickness, c h is the oxygen concentration in the channel and the superscript marks the values at the channel inlet. To simplify calculations we use dimensionless variables ~ J ¼ Z ~ ~z d~z (8) It can be shown that ~v ¼ l ~ J ¼ l fix ~ Jfix where l is the oxygen stoichiometry:

5 9958 international ournal of hydrogen energy 35 (2) l ¼ 4F v c h wh J Lw (9) and w is the channel width. The subscript fix means that the flow rate of the inlet gas is fixed so that at the current density ~ Jfix a required stoichiometric flow is achieved. Calculating the integral (8) we get: þ ~ J ¼ lfix ~ ~ ~ O Jfix A 2 H h A h þ þ 2 ð A h Þ A h þ ~ A exp@ h þ ~ ~ O 2 l fix ~ Jfix þ ~ AA H 2 A h ð A h Þ A h þ ~ A h ~ H 2 (2) O 2 Thus, an explicit expression for ~ Jð~hÞ is obtained. The dependence ~hð ~ JÞ required for the cell polarization curve can be found by iterations. Note that in the limit of zero crossover currents, Eq. (2) recovers the result [9]. The cell polarization curve is V cell ¼ V oc h R U J (2) Here V cell is the cell voltage, V oc is the open-circuit voltage, the cathode overpotential h is defined by [2] and R U is the cell resistance determined by the membrane proton conductivity k and the electrode electric conductivity s: R U ¼ l m k þ 2l d s (22) Using the Nernst equation, appropriate empirical correlations and data from [,2], the cell open-circuit voltage can be written as! V oc ¼ :8 2:3, 4 ðt 298Þþ RT 2! 4F ln c H 2 c h c ref c ref (23) Therefore, the simple pseudo 2D model of a high temperature PEM fuel cell is constructed. 4. Results Experimental conditions and parameters used in the simulations are listed in Table. By default these base case values have been used, the exceptions are explicitly specified below. The effective oxygen diffusivity was calculated for a given pressure and temperature by empirical correlations [2,22] with the account for the GDL porosity e according to the Bruggeman expression: D eff ¼ e :5 D p;t The crossover of hydrogen was measured experimentally by a method similar to the one described in [23], the resulting permeation coefficient of hydrogen was assumed to be four times greater than the oxygen permeation coefficient [24,25]. These permeation coefficients are comparable to the other literature data for similar membranes [25,26]. The MEA resistance can be measured independently by an impedance spectrum analysis. The membrane resistance gives the main contribution to the total resistance. The membrane conductivity can be estimated under the assumption that the membrane thickness remains the same after the fuel cell assembling. However, the membrane conductivity was considered as a fitting parameter, since it is not always possible to use a time-consuming impedance technique. The membrane conductivities estimated from the impedance analysis are in good agreement with those resulted from fitting. The ORR transfer coefficient and exchange current density were also considered as fitting parameters. The fitting procedure is quite straightforward and it is based on the least squares technique. The transfer coefficient can be reliably estimated only if the region of very low current densities is measured carefully. From our experiments we got a ¼ :8 which is close to the literature data for a similar system [27]. We fixed this transfer coefficient for all simulations, while the exchange current and membrane conductivity were variable fitting parameters. We have only two independent fitting parameters, therefore there is no need to use complicated fitting techniques, like genetic algorithms or particle swarm optimization [28]. The exchange current density and membrane conductivity may vary considerably depending on the MEA design, assembly process and slight changes in materials. The values obtained (see below) are comparable to the literature data [25,27]. In [27] the exchange current densities were measured for nearly flat platinum electrodes. The exchange current densities obtained below should be scaled by the factor of about 3 for comparison with the results [27] due to the roughness of our electrodes [7]. According to the information from the catalyst powder producer and our measurements, the platinum surface of our catalyst layers was about 3 cm 2 /mg. 4.. Voltage losses in a fuel cell The results of fitting are shown by solid line in Fig. 4, the parameters are: ¼ 2:5$ 5 Acm 2 ; k ¼ 3:8$ 2 Scm As can be seen, the model fits the experimental data reasonably well experiment simulation current density / A cm-2 Fig. 4 Current-voltage curves for the base case conditions.

6 international ournal of hydrogen energy 35 (2) cell potential activation on the cathode side ohmic in the membrane.5 l m l m /2 l m /4 voltage loss / V voltage loss / V Fig. 5 The maor voltage losses in a fuel cell. The model allows to distinguish different contributions to the voltage loss in a fuel cell. Various contributions to the current-voltage curve in Fig. 4 are shown in Figs. 5 and 6. Obviously, the largest part of the voltage loss is due to the cathode activation and membrane resistance. The cell efficiency can be improved if the exchange current density is increased or the cell resistance is decreased. It is clearly seen (Fig. 6) that for typical conditions the effect of crossover is very small and it can be neglected. However, in the case of low stoichiometries the crossover effect is more noticeable. When the cell operates at a fixed flow rate, in the region of high current densities the oxygen stoichiometry becomes close to and the voltage loss due to the crossover substantially increases. In this region almost all oxygen is consumed by the ORR. Under the lack of oxygen, the oxygen loss due to crossover noticeably increases the voltage loss. Fig. 7 depicts this effect for the membranes of different thickness. The thinner membrane has lower resistance but the crossover effect for this membrane is higher Fig. 7 The voltage loss due to crossover of gases though the membranes of different thickness. ¼ 2:$ 5 Acm 2 ; k ¼ 7:6$ 2 Scm The result of fitting is shown in Fig. 8 (solid line). After the base case experiment, the fuel cell was operated on pure oxygen (squares in Fig. 8). The short dashed line represents the model curve obtained for pure oxygen with the parameters * and k indicated above. It is seen that the model correctly describes the change in the oxygen concentration, though in the case of pure oxygen the model is not very accurate at high current densities. Indeed, when a fuel cell is fed with pure oxygen, the assumption of a constant flow velocity fails as l /. In these experiments, the data in the region of low current densities were measured very precisely, as evident from the logarithmic current-voltage curves in Fig. 9. We have used these data to determine the ORR transfer coefficient, as mentioned above Variation of the composition of gases on the cathode side A good model should be able to predict the system response to the changes in working conditions. Fitting the experiment with another MEA under the base case conditions we obtained voltage loss / V. due to crossover transport on the cathode side ohmic in the GDL 4.3. Variation of the membrane conductivity For this series of three experiments we used the membranes doped with different additives while the gas diffusion experiment, air simulation, air experiment, oxygen simulation, oxygen Fig. 6 The minor voltage losses in a fuel cell Fig. 8 Current-voltage curves for the same MEA and different compositions of the cathode inlet gas.

7 996 international ournal of hydrogen energy 35 (2) experiment, air simulation, air experiment, oxygen simulation, oxygen.. log( ) Fig. 9 Logarithmic current-voltage curves for the different compositions of the cathode inlet gas. electrodes and the MEA construction technique were the same. Therefore, the exchange current density was fixed while the membrane conductivity varied. The results are presented in Fig.. In the first experiment, the data in the region of low current densities were measured most accurately. This experiment was used to determine both parameters: ¼ 3:9$ 5 Acm 2 ; k ¼ 6:9$ 2 Scm The resulting exchange current density was fixed and for the two remaining experiments only the membrane conductivities were fitted. Fig. shows that the model correctly estimates * and k. ¼ 3:9$ 5 Acm 2 ; k ¼ 5:6$ 2 Scm The lines corresponding to the two other regimes were obtained varying only the inlet flow velocity; these lines show good agreement with the experiment (Fig. ). Thus, the model predicts the system response to the variation of the cathode flow rate very accurately Variation of the working temperature To describe the dependence of the fitting parameters on the working temperature we have used the Arrhenius equation. The temperature dependence of the exchange current density and conductivity of the membrane can be expressed as follows: ðtþ ¼ exp E A R T! T kðtþ ¼k exp Ek A R T T where E A and E A k are the activation energies and T was taken to be 43 K. The open-circuit voltage and the oxygen diffusivity in the GDL are the other temperature-dependent parameters of the system, they were altered appropriately. To estimate the parameters in the Arrhenius equations above, we fitted two experimental curves: 43 K : ¼ :9, 5 Acm 2 ; k ¼ 3:2, 2 Scm 433 K : ¼ 4:, 5 Acm 2 ; k ¼ 4:3, 2 Scm 4.4. Variation of the flow rate Another MEA was used to measure current-voltage curves for the three constant flow rates of the cathode gas. Fitting the experiment with the highest flow rate (circles in Fig. ), we got experiment simulation, κ=6.9 S m - experiment 2 simulation 2, κ=5.6 S m - experiment 3 simulation 3, κ=3.8 S m - These parameters allowed us to estimate the activation energies and thus to calculate the exchange current density and membrane conductivity at any temperature. The modeled polarization curves for the two other working temperatures are in good agreement with the experiments (Fig. 2) experiment, λ=6. simulation, λ=6. experiment 2, λ=3. simulation 2, λ=3. experiment 3, λ=.5 simulation 3, λ= Fig. Current-voltage curves for different membranes in the otherwise identical MEAs. The data from the first experiment (circles) is used to determine the exchange current. For the approximation of the other two experiments only the membrane conductivity was fitted Fig. Current-voltage curves for different inlet flow rates on the cathode side. The stoichiometries specified correspond to the current density of.4 A cm L2.

8 international ournal of hydrogen energy 35 (2) experiment, 43 K simulation, 43 K experiment, 433 K simulation, 433 K experiment, 43 K simulation, 43 K experiment, 453 K simulation, 453 K Acknowledgments The work was done with the partial support of Faculty for the Future program of Schlumberger Foundation. We thank Dr. I. Ponomarev for providing us with the PBI O PhT membranes used in the experiments. We would also like to thank Dr. W. Lehnert and Dr. C. Wannek (Research Centre Jülich, Germany) for helpful discussions Fig. 2 Current-voltage curves for the same MEA operating at different temperatures. The explanations are given in the main text. 5. Conclusions An analytical model of a high temperature PEM fuel cell is developed. In spite of its simplicity, the model is a valuable tool for the processing of experimental data. The model accounts for the crossover of gases through the membrane and shows that the crossover effect is considerable only at a low stoichiometric flow of oxygen; in the experiments reported this occurred near the limiting current density. A more accurate model which takes into account the processes in the catalyst layer is needed to explain the initial drop in the cell voltage (see e.g. [29]). The model has clear physical and mathematical basis. It can be used to estimate properties of an HT PEMFC from its current-voltage curve. The model is applicable to a high temperature PEMFC fed with dry gases (hydrogen and air) at a constant pressure and flow rate. It has been demonstrated that the model correctly captures the effects due to the variation of working conditions (these conditions are summarized in Table 2). The model is computationally inexpensive: the fitting procedure takes only several minutes. Simulation of the system response to changes in conditions takes ust a few seconds on a modern PC. Table 2 The operation conditions tested. Parameter Values Operation conditions: Gas composition on the cathode side Required stoichiometry of cathode inlet gas at J fix ¼.4 A cm 2, l fix Cell temperature, T Obtained parameters range at the temperature of 433 K: Membrane conductivity, k Exchange current density, * air, oxygen K 3.8$ 2 7.6$ 2 Scm 2.$ 5 4.$ 5 Acm 2 references [] Vielstich W, Lamm A, Gasteiger HA, editors. Handbook of fuel cells: fundamentals, technology, applications, vol.. John Wiley & Sons Ltd; 23. [2] Wu J, Yuan XZ, Wang H, Blanco M, Martin JJ, Zhang J. Diagnostic tools in PEM fuel cell research: part I electrochemical techniques. International Journal of Hydrogen Energy 28;33: [3] Wu J, Yuan XZ, Wang H, Blanco M, Martin JJ, Zhang J. Diagnostic tools in PEM fuel cell research: part II physical/ chemical methods. International Journal of Hydrogen Energy 28;33: [4] Cheddie D, Munroe N. Parametric model of an intermediate temperature PEMFC. Journal of Power Sources 26;56: [5] Cheddie DF, Munroe ND. Two dimensional phenomena in intermediate temperature PEMFCs. International Journal of Transport Phenomena 26;32: [6] Cheddie DF, Munroe ND. Three dimensional modeling of high temperature PEM fuel cells. Journal of Power Sources 26;6: [7] Cheddie DF, Munroe ND. A two phase model of an intermediate temperature PEM fuel cell. International Journal of Hydrogen Energy 27;32: [8] Cheddie D, Munroe N. Mathematical model of a PEMFC using a PBI membrane. Energy Conversion and Management 26; 47( 2): [9] Cheddie D, Munroe N. Analytical correlations for intermediate temperature PEM fuel cells. Journal of Power Sources 26;6: [] Zenith F, Seland F, Kongstein O, Borresen B, Tunold R, Skogestad S. Control oriented modelling and experimental study of the transient response of a high temperature polymer fuel cell. Journal of Power Sources 26;62(): [] Wang C-P, Chu H-S, Yan Y-Y, Hsueh K-L. Transient evolution of carbon monoxide poisoning effect of PBI membrane fuel cells. Journal of Power Sources 27;7(2): [2] Scott K, Pilditch S, Mamlouk M. Modelling and experimental validation of a high temperature polymer electrolyte fuel cell. Journal of Applied Electrochemistry 27;37(): [3] Peng J, Lee SJ. Numerical simulation of proton exchange membrane fuel cells at high operating temperature. Journal of Power Sources 26;62:82 9. [4] Peng J, Shin J, Song T. Transient response of high temperature PEM fuel cell. Journal of Power Sources 28; 79:22 3. [5] Ponomarev J, Rybkin Y, Volkova Y, Razorenov D. Method for producing polybenzimidazoles based on 4,4 diphenylphthalide dicarboxylic acid, Patent WO/28/ 366, 28. [6] Chertovich A, Khokhlov A, Kulikovsky A, Shamardina O. Analytical model for a high temperature proton exchange membrane fuel cell, in: Book of abstracts of the First

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