CONDENSED MATTER FINITE ELEMENT METHOD MODELLING OF A HIGH TEMPERATURE PEM FUEL CELL V. IONESCU1 1 Department of Physics and Electronics, Ovidius University, Constanta, 900527, Romania, E-mail: [email protected] Received July 31, 2013 A proton exchange membrane fuel cell, also known as polymer electrolyte membrane fuel cell (PEMFC), becomes in the last decade a prime candidate for applications in power systems, submarines and aerospace. The single-cell PEMFC consists of a carbon plate, a gas diffusion layer (GDL), a catalyst layer for each of the anode and the cathode sides, as well as a PEM membrane at the center. This model of the fuel cell, created from Electrochemistry Module of Comsol Multiphysics software package uses current balances, mass transport equations (Maxwell-Stefan diffusion for reactants, water and nitrogen gas) and momentum transport (Darcy’s law for the gas flows) to simulate a PEM fuel cell’s behavior. Darcy’s velocity field at GDL channels, water diffusive flux, current density distributions at anode catalyst electrode and membrane electrolyte potential distribution across the length of the cell for three models of PEM cell models (different from the point of view of membrane width) were computed in order to investigate the cell model performance. Key words: diffusive flux, electrolyte potential, current density. 1. INTRODUCTION Proton exchange membrane fuel cells (PEMFC) [1,2] are highly efficient power generators, achieving up to 50–60% conversion efficiency, even at sizes of a few kilowatts. PEMFCs have zero pollutant emissions when fueled directly with hydrogen, and near zero emissions when coupled to reformers. These attributes make them potentially attractive for a variety of applications, especially in stationary and mobile power generators and electric vehicles . There are several compelling technological and commercial reasons for operating H2/air PEM fuel cells at temperatures above 100°C. Rates of electrochemical kinetics are enhanced, water management and cooling is simplified, useful waste heat can be recovered, and lower quality reformed hydrogen may be used as the fuel. The performance of the fuel cell system is characterized by current-voltage curve (i.e. polarization curve). The difference between the open circuit potential of the electrochemical reaction and cell voltage occurs from the losses associated with the operation. The corresponding voltage drop is generally classified in three parts: Rom. Journ. Phys., Vol. 59, Nos. 3–4, P. 285–294, Bucharest, 2014 286 V. Ionescu 2 i) activation over-potential caused by the electrochemical reactions, ii) ohmic drop across the polymer electrolyte, iii) mass transfer limitations of reactants. These associated losses dominate over different current density ranges. For low current densities, the activation over-potential is dominant. For high current densities, which are of particular interest for vehicle applications because of higher power density, the mass transfer limitations dominates the losses. For moderate current densities, the ohmic drop across the polymer membrane dominates. Moreover, for high current densities, water starts to exist in liquid form leading to a two-phase transport of reactants to reaction site, which is an additional transport phenomenon of PEM Fuel Cell operation. In the last years, numerical modeling of PEFCs has received much attention. Many two- and three-dimensional models have been developed in which the computational fluid dynamics (CFD) method has been rigorously coupled with electrochemical phenomena [4–7]. Electron transport and heat-transfer phenomena have also been incorporated [8–10]. Chiang and Chu  investigated the effects of transport components on the transport phenomena and performance of PEM fuel cells by using a threedimensional model. The impacts of channel aspect ratio (AR) and GDL thickness were examined. It was found that a flat channel with a small AR or a thin GDL generates more current at low cell voltage due to the merits of better reactant gas transport and liquid water delivery. The goal of the present work is to create a 2D isothermal, steady state PEM fuel cell model in perpendicular to the gas flow direction to investigate the performance of fuel cell from the point of view of activation over-potential caused by the electrochemical reactions at cathode, ohmic drop across the electrolyte membrane and water diffusive flux distribution at anode for three different geometries of the PEM cell model. 2. MODEL SET-UP 2.1. COMPUTATIONAL DOMAIN The modeled section of the fuel cell consists of three domains: an anode (Ωa), a proton exchange membrane (Ωm), and a cathode (Ωc) as indicated in Fig.1.a. Each of the electrodes (gas diffusion layers) is in contact with an interdigitated gas distributor, which has an inlet channel (∂Ωa,inlet), a current collector (∂Ωa,cc), and an outlet channel (∂Ωa,outlet). The same notation is used for the cathode side. The 2D modeling geometry is shown in Fig.1.b representing a cross-section along the parallel channels of an along-the-channel test fuel cell. It is assumed that the channels on the anode and cathode side appear in the same cross-section. 3 Finite element method modelling of a high temperature PEM fuel cell 287 Oxygen (O2) or air and hydrogen (H2) are supplied via gas channels in counter-flow and diffuse through the gas diffusion layers to the reactive layers where the following oxidation/reduction reactions take place. a) b) Fig. 1 – The 2D modeling geometry: a) sub-domain and boundary labels and b) model geometry. On the anode, hydrogen oxidation occurs according to: H 2 ⇒ 2 H + + 2e − (1) (no water molecules are assumed to be involved in the proton transport). At the cathode, oxygen reacts together with the protons to form water according to: O2 + 4 H + + 4e − ⇒ 2 H 2O (2) Protons reach the cathode directly through the membrane. Beside water at the cathode side, heat is produced due to strong exothermic reaction. 2.2. GOVERNING EQUATIONS Mass conservation or continuity equation tells that the change of mass in a unit volume must be equal to the sum of all species entering or exiting the volume in a given time period. This law applies to the flow field plates, GDL and the catalyst layer. Momentum conservation relates net rate of change of momentum per unit volume due to convection, pressure, viscous friction and pore structure. This law applies to the flow field plates, GDL and the catalyst layer. Species conservation relates the 288 V. Ionescu 4 net rate of species mass change due to convection, diffusion and electrochemical reaction. The most commonly used one is the Stefan-Maxwell diffusion equation . Charge conservation corresponds to the continuity of current in a conducting material. This is applied to the GDL, catalyst layer and the membrane. 2.3. ASSUMPTIONS In this model set-up the following assumption are used: i) steady-state operating conditions, ii) as the cell is operated at higher temperatures single-phase water-flow is assumed, iii) since the velocity of the gases is low, laminar flow is assumed, iv) all gases are treated as ideal, v) all electrochemical reactions are gaseous phase reactions and there is no crossover of gases through the membrane, vi) all material parameters are isotropic and homogenous. 2.4. NUMERICAL PROCEDURE COMSOL Multiphysics, which is a commercial solver based on the finite element technique, is used to solve the governing equations. The stationary nonlinear solver is used since the source terms of the current conservation equation make the problem non-linear. Furthermore, the convergence behavior of this nonlinear solver is highly sensitive to the initial estimation of the solution. To accelerate the convergence, the following procedures are adopted. The Conductive Media DC module is firstly solved based on the initial setting. Secondly, Darcy’s Law and Incompressible Navier-Stokes modules are solved together using the results from the previous calculation as initial conditions. After the previous two modules converge, all the coupled equations including Maxwell-Stefan Diffusion and Convection module are solved simultaneously until the convergence is obtained. 2.5. BOUNDARY CONDITIONS The boundary conditions for the model in this study are as follows: i) continuity at all internal boundaries, ii) no slip boundary condition for all channel walls, iii) all initial values set to zero, iv) velocity and temperature defined at channel inlet, step function used for these two parameters in time dependent study, v) no backpressure at channel outlet, convective flux boundary conditions applied, vi) constrain outer edges set to zero for both inlet and outlet, vii) bipolar plates on the most side of the cell set to electric ground and cell operation potential, viii) HTPEM fuel cell is insulated from environment. 5 Finite element method modelling of a high temperature PEM fuel cell 289 3. RESULTS AND DISCUSSIONS Using the aforementioned numerical procedures, the 2D model geometry as described in Fig. 1.b was simulated at a single voltage of 0.7V, for different width dimensions of PEM membrane: 50µm, 100µm and 200µm. From this point of discussions, we will refer at three PEM cell models for witch the simulations are computed, named CM1, CM2 and CM3 for 50µm, 100µm and 200µm PEM membrane width cell geometries. The cell operated at a constant temperature of 120°C, and the reference pressure pref was 1 atm. Nafion type membrane was modeled in this case, with the chosen value of 3 for electro-osmotic drag coefficient H2O/H+, determined previously at different temperatures [13, 14]. The cell models dimensions are listed in Table 1. The physicochemical parameters values that were used in this study are presented in Table 2. Table 1 Cell geometry Parameter name Cell length Current collectors height Inlet and Outlet channels width Inlet and Outlet channels height GDL layers width PEM membrane width Value 2[mm] 1[mm] 0.1[mm] 0.5[mm] 0.25[mm] 50[µm] 100[µm] 200[µm] Table 2 Physicochemical parameters Parameter name GDL conductivity GDL permeability Nafion membrane conductivity Fluid viscosity Anode inlet pressure Cathode inlet pressure Exchange current density, anode Exchange current density, cathode Specific surface area Effective binary diffusivity, H2_H2O Effective binary diffusivity, O2_N2 Effective binary diffusivity, H2O_N2 Effective binary diffusivity, O2_H2O Inlet weight fraction H2 Value 1000[S/m] 10-13[m2] 9[S/m] 2.1·10-5[Pa·s] 1.1·pref [Pa] 1.1·pref [Pa] 1·105[A/m2] 1[A/m2] 1·107[m-1] 3.351·10-5[m2/s] 8.6369·10-6[m2/s] 9.3573·10-6[m2/s] 1.0278·10-5[m2/s] 0.1 290 V. Ionescu 6 Table 2 (continued) Inlet weight fraction O2 Cathode inlet weight fraction, H2O Molar mass, H2 Molar mass, O2 Molar mass, H2O Molar mass, N2 Inlet molar fraction, H2 Reference concentration, H2 Reference concentration, O2 Inlet molar fraction, O2 0.168 0.2 0.002[kg/mol] 0.0032[kg/mol] 0.018[kg/mol] 0.028[kg/mol] 0.5 1.299[mol/m3] 0.42699[mol/m3] 0.13483 It is very important to know how reactants are transported into the membrane. Figure 2 shows the velocity fields of hydrogen and oxygen at the anode and cathode sides of flow channel at the operation condition of 0.7 V for the cell model with 50µm PEM membrane width (CM1). These fields indicated the mass transport at each section of the fuel cell very well. Figure 2 indicated that the flow-velocity magnitude attains its highest values at the current collector corners; an almost constant distribution of the field (at high values) are located close to the membrane sides in the Y direction, which meant that hydrogen and oxygen were supplied to the membrane by the diffusion mechanism. Fig. 2 – Typically component of Darcy’s velocity field (m/s) at anode (left side) and cathode (right side) GDL channels for CM1 cell model. To ensure a fully hydrated membrane, fuel and oxidant (air) streams are fully or partially humidified before entering the fuel cell. The ionic conductivity of the proton-conducting membrane is strongly dependent on its degree of humidification, or water content, with high ionic conductivities at maximum humidification. 7 Finite element method modelling of a high temperature PEM fuel cell a) b) 291 c) 2 Fig. 3 – Water diffusive flux distribution (Kg/m s) at the anode catalyst layer for the following cell models: a) CM1, b) CM2 and c) CM3. In Figure 3 it was presented water diffusive flux distribution at the anode electrode to show the degree of humidification of the PEM membrane in the current collector area. From Fig. 3.a we can see that CM1 cell model had the largest and most compact area for the maximum diffusive flux distribution in the current collector zone, presenting in this way the highest level of humidification at the PEM membrane interface. Spikes with the highest flux distribution were present at the corners of the anode current collector area for all the cell models. Figure 4 shows the plots of the current density distribution at the anodic catalyst layer as a function of cell length for all the cell models. Fig. 4 – Current density distribution plots at the anode electrode for the cell models CM1, CM2 and CM3. 292 V. Ionescu 8 The current density was uneven with the highest density in the cell’s upper region, the cell model CM1 presenting the maximum variation domain of the current density along the cell length, of about 1085 A/m2. This means that the oxygen-reduction reaction rate in the cathode determines the current-density distribution. From Figure 4 we can see that the maximum current density arises close to the air inlet, probably because of the high concentration of hydrogen and oxygen and high electro-osmotic mass flux at the inlet region. Figure 5 shows the distribution of electrolyte membrane potentials across the membrane length (y–coordinate). An obvious potential drop should be found within the membrane due to the theoretical low proton conductivity of membrane. From Figure 5 we observed that ohmic (resistance) losses are very small on both anode and cathode electrodes in the case of cell model CM1, conclusion validated by the smallest voltage drop (of 6.9 mV) across the electrolyte membrane for this model. The activation potential is the potential difference above the equilibrium value required to produce a current which depends on the activation energy of the redox event. Reaction overpotential is an activation overpotential that specifically relates to chemical reactions that precede electron transfer. This overpotential can be reduced or eliminated with the use of homogeneous or heterogeneous electrocatalysts. The electrochemical reaction rate and related current density is dictated by the kinetics of the electrocatalyst and substrate concentration. A high overpotential should be found on the cathode side of the PEM fuel cell due to the slow kinetic reaction of oxygen reduction. Fig. 5 – Membrane electrolyte potential distribution across the length of the cell (y direction) for the cell models CM1, CM2 and CM3. 9 Finite element method modelling of a high temperature PEM fuel cell 293 As we can see from Fig.6, the cell model CM1 presented the smallest overpotential drop of 7 mV along the cell length, suggesting a fastest electrochemical reaction rate of the oxygen in this particular case. Fig. 6 – Reaction overpotential distribution at the catalyst layer in cathode side of the cell for the cell models CM1, CM2 and CM3. 4. CONCLUSIONS A 2D numerical model simulating the performance of a PEM fuel cell operating at 120oC has been successfully applied in this study, for three different values of membrane width. From Darcy’s velocity field distribution at the anode and cathode GDL channels it was showed that hydrogen and oxygen were supplied to the PEM membrane by the diffusion mechanism. The most proper water diffusive flux distribution at the anode catalyst electrode for a highest level of humidification at the PEM membrane interface was established for CM1 cell model, with 50 µm membrane width. The current density distribution at the anode electrode was determined by the oxygen-reduction reaction rate, CM1 model presenting the maximum variation domain of the current density along the cell length. CM1 cell model showed the smallest ohmic loses by registering smallest voltage drop (of 6.9mV) across the electrolyte membrane and also the smallest overpotential drop along the cell length. 294 V. Ionescu 10 REFERENCES 1. B. Krishnamurthy, S. Deepalochani, Int. J. Electrochem. Sci. 4, 386 (2009). 2. S. Ou, L.E.K. Achenie, J. Power Sources 140, 31 (2005). 3. F. Laurencelle, R. Chahine, J. 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