Electrical coupling in proton exchange membrane fuel cell stacks: mathematical and computational modelling

Transcription

1 IMA Journal of Applied Mathematics (2006) 71, doi: /imamat/hxh092 Advance Access publication on May 4, 2005 Electrical coupling in proton exchange membrane fuel cell stacks: mathematical and computational modelling PETER BERG Faculty of Science, UOIT, Oshawa, Ontario, Canada L1H 7K4 ATIFE CAGLAR Mathematics Department, UW Green Bay, Green Bay, WI 54311, USA KEITH PROMISLOW Mathematics Department MSU, East Lansing, MI 48824, USA JEAN ST-PIERRE Ballard Power Systems, 4343 North Frasier Way, Burnaby BC, Canada V5J 5J9 AND BRIAN WETTON Mathematics Department, UBC, Vancouver BC, Canada V6T 1Z2 [Received on 7 April 2004; accepted on 29 March 2005] A mathematical model describing the effects of electrical coupling of proton exchange membrane unit fuel cells through shared bipolar plates is developed. Here, the unit cells are described by simple, steadystate, 1D models appropriate for straight reactant gas channel designs. A linear asymptotic version of the model is used to give analytic insight into the effect of the coupling, including estimates of the extent of the coupling in terms of the number of adjacent cells affected. An efficient numerical method is developed to solve the non-linear coupled system. Numerical results showing the effects on stack voltage due to a single cell with anomalous oxidant flow rate are given. The effects on stack performance due to end plate effects are also given. It is shown that electrical coupling has a significant effect on fuel cell performance. Keywords: fuel cell stack; polymer electrolyte fuel cell; wagner number. 1. Introduction The development of proton exchange membrane (PEM) fuel cells for automotive, stationary and portable electrical power has received considerable attention in recent years. High-level computational design tools have been developed, recently including coupled mass and thermal transport in 3D (Berning et al., 2002; Dutta et al., 2001; Natarajan & Van Nguyen, 2003; Mazumder & Cole 2003a,b). These models strive to describe all significant processes affecting performance of a unit cell, but neglect the electrical Corresponding author. wetton@math.ubc.ca Peter.Berg@uoit.ca caglara@uwgb.edu kpromisl@math.msu.edu jean.st-pierre@ballard.com c The Author Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

2 242 P. BERG ET AL. coupling of unit cells in a stack environment. In this work, the authors present the first attempt at evaluating the character and importance of these effects. In our stack model, the performance of unit cells under locally varying voltages is approximated by a simplified version of the model developed by the authors in Berg et al. (2004). This model couples along the channel transport of reactant gases with the electrochemistry of the catalyst layer. It assumes straight reactant gas channels, for which it is reasonable to average over the cross-channel direction (z in Fig. 1). Performance (local current density) varies down the length of the channel (x), generally worsening from cathode inlet to outlet as oxygen concentration in the oxidant channels diminishes due to consumption. We consider here a stack environment where many cells in series are electrically coupled through bipolar plate resistances. For identical cells operating under identical conditions, the current density distributions along the channel are the same, and the coupling is not evident. However, if there is a variation in cell performance, or operating conditions, from cell to cell, then the local performance of the individual cells must compete against the non-local voltage equations which govern the motion of current through the stack. Through scaling arguments we will show that the stacks are electrically dominated and that anomalous behaviours propagate a significant distance in the stack and estimate the anomalous sensitivities of the stack performance to perturbation in individual cell performance. Anomalous sensitivities are computed for two types of anomalies: cathode understoich of a single cell and the effects of the bus plate at the stack ends. In isolation, a cell faced with a (negative) anomaly will operate at a lower voltage to maintain the same current. Its local current density down the channel will also be different from the base case. However, in the electrically coupled stack setting, differences in local current densities between adjacent cells must result in currents in the bipolar plates. If the scaled bipolar plate resistance is large, differences in local current densities between adjacent cells are penalized: it is expected that the voltage drop due to the anomalous cell will be higher and the effects of the anomalous cell will extend further to more adjacent cells. Both these effects are shown in the simulations below. FIG. 1. PEM unit cell schematic.

3 ELECTRICAL COUPLING IN PROTON EXCHANGE MEMBRANE FUEL CELL STACKS 243 The exposition of this work is targeted to applied mathematicians, highlighting the modelling, analysis and numerical approach. A brief introduction to the fuel cell application and terminology is given in Section 2 below. This is followed by a section on the simple unit cell model used in this study, followed by the model of the electrical coupling in Section 4. Some insight into the effect of the coupling can be gained by the analysis of a simple version of the model corresponding to a high reactant flow rate limit. This is shown in Section 5. The computational method for the non-linear problem based on approximate Newton iteration on current densities is described in Section 6 with results in Section 7. A section summarizing the nomenclature and parameter values is given at the end of the paper in Section Fuel cell basics and terminology The basic components of a PEM fuel cell are a membrane (typically, Nafion or other good protonic conductor, electronic insulator) with oxidant (air in this study) flow channels on one side (cathode) and fuel (hydrogen) channels on the other (anode). On either side of the membrane are platinum catalyst layers, and gas diffusion electrodes (GDE), often made of teflonated carbon fibre paper. The purpose of the electrodes is to allow reactant gases to the catalyst sites and to carry away the products (current, water vapour and liquid water). The membrane, catalyst layers and electrodes together are called the membrane electrode assembly (MEA). The reactant gas channels are formed in plates on either side of the MEA that conduct the current further. A device consisting of anode and cathode plates with gas channels and an MEA is called a unit cell. We consider unit cells in this study with straight channels (such as the Ballard Mk9 hardware). A schematic of such a device is shown in Fig. 1. In a complete device, unit cells are often combined in series. The anode plate of one cell is pressed against the cathode plate of the next. The two combined plates are known as a bipolar plate. The combined unit cells are called a stack. In practice, stacks of the order of 100 unit cells are not uncommon. At the end of the stack are current collector plates (often made of copper or other good conductors) and connections are made on the collector plate to cables to the external circuit. Since the unit cells are in series, they have the same total current I T in amperes flowing through them. Because of this, it is typical in the field to consider fuel cell operation parametrized by total current, rather than voltage. Despite the uniformity of the total through-mea current, cell-to-cell variations lead to differing along-the-channel current densities which produce in-plane (x) currents in plates. A schematic of a fuel cell stack is shown in Fig. 2. In this study, we consider mainly the effect of channel oxygen concentration on performance. Inlet oxidant flow rates are often expressed as dimensionless ratios (stoichiometric flow rates or stoich, s) of the molar flux of oxygen at inlet Q for the unit cell divided by the molar consumption of oxygen to produce the target current: s = Q4F, I T where F is Faraday s constant (C/mol). There are several aspects of fuel cell performance that are neglected in this study. Chief among them are the effects of water: membrane hydration and its relationship to proton conductivity, liquid water blockage effects in the GDE and channels. The model is isothermal and the catalyst layer is treated as an interface. 3. Unit cell model The steady-state unit cell model developed in Berg et al. (2004) considered counterflowing gases along the cathode and anode channels coupled to each other and to the electrochemistry through a

4 244 P. BERG ET AL. FIG. 2. PEM fuel cell stack schematic. semi-analytic model of the through-mea transport. All quantities are averaged over the cross-channel (z) direction. Anode electrochemical losses are neglected. In this work, we simplify the model, assuming that the cathode gas channels are saturated along their entire length, and that the anode is run at a reasonably high hydrogen stoich. We neglect water crossover between the anode and the cathode and assume that the cell is run at isothermal conditions. Under these assumptions, the membrane electrical resistivity R is constant along the channel and the influence of the anode channels on cell performance is negligible. The anode flow can be removed from the computation. This reduction avoids the need for an iterative process necessary for the counterflowing gases, and faster computation for the stack model can be achieved. Several physical and experimentally fitted empirical constants are needed in the models. They are summarized in Section 9 at the end of the paper. Many fitted quantities come from previous work (Berg et al., 2004) and correspond to Ballard Mk9 architecture and standard operating conditions. 3.1 Unit cell equations A simple unit cell model involves the determination of the cell voltage V (taken to be constant along the length of the cell) and the current density i(x), oxygen flux Q o (x) and oxygen concentration C o (x) profiles. Here, x is the coordinate of the channel from x = 0 (cathode inlet) to x = L c (cathode outlet). The cathode stoichiometric flow rate s > 1 is varied in the computations described below. It determines the cathode inlet fluxes: Q o (0): si d L c /(4F), Q n : Q o (0) (0.79)/(0.21), where i d is the target average current density and (0.79)/(0.21) appears as the assumed ratio of nitrogen to oxygen in the inlet stream. Other trace gases in air are neglected. Oxygen is consumed to generate current. The following mass balance results dq o dx = i(x) 4F, (1)

5 ELECTRICAL COUPLING IN PROTON EXCHANGE MEMBRANE FUEL CELL STACKS 245 where Q o is the oxygen flux in gas channels per unit width (z) and i is the local current density. Using the ideal gas law, the fact that the nitrogen is unreactive and so the flux is constant from inlet to outlet, an assumption that oversaturated vapour immediately condenses, and that all gaseous species are transported by channel convection only with a common average velocity, we obtain the algebraic relationship C o = (P c P v )Q o (Q o + Q n )RT, (2) between Q o (x) and C o (x). See Berg et al. (2004) for details. We begin with the following modified Butler Volmer relationship, relating local oxygen channel concentration (C o ), sufficiently large current density (i) and cell voltage (V ) with fitted parameters R, α c, C ref, δ and i ref listed in Section 9: V = E 0 ir RT ic ref ln Fα c i 0 (C o δi). (3) At very low current densities, there are additional terms that modify the logarithm to give the open circuit E 0 in the limit. Here, C o δi approximates the oxygen concentration at catalyst sites, lower than the channel average because the oxygen must diffuse through the GDE and catalyst layer to the active sites. The concentration reduction should be proportional to the flux, which is proportional to the local current, leading to the given form. Relationship (3) should theoretically hold as long as i < i max = C o /δ (the maximum current allowed locally by mass transport). In the limit i i max, the voltage tends to. The relationship (3) cannot provide voltages for i > i max. In the iterative procedure described below for the coupled cell problem, the constraint that i < i max is not preserved at intermediate steps (at final solutions, all local relationships are physically reasonable). In order to capture the important features of the modified Butler Volmer relationship while extending it to allow any current to be given, we introduce the computational parameter C ɛ, a transition oxygen molar concentration at the catalyst layer. We consider the local transition current density i = C o C ɛ. δ If i < i, we use (3) to produce the local voltage. Otherwise, we use V = E 0 i R RT ln i ( C ref R + RT { 1 + δ Fα c i 0 C ɛ Fα c i C ɛ }) (i i ). (4) The above formula is just the tangent line extension of (3) at i. Graphs of the original (3) and new Butler Volmer extension are shown in Fig. 3. Equations (1), (2), (3) with extension (4), inlet fluxes given above and the requirement that the cell draws the given average current density: completely specify the unit cell model. 1 L c Lc 3.2 V -formulation and computational results 0 i(x) dx = i d, (5) If V is given, then (3) can be solved implicitly for i for a given channel oxygen concentration, which can be related to oxygen flux using (2). Thus, (1) can be viewed as an implicitly defined ODE that can be

6 246 P. BERG ET AL. FIG. 3. Original modified Butler Volmer relationship (left) and new relationship with extension to all currents (right, using the computational parameter C ɛ = 0.02 mol/m 3 ). solved from inlet to outlet using standard numerical techniques. An approximate local current density i(x) is obtained in this process. The value of V can be chosen iteratively until the target current condition (5) is met. The extension of the V -formulation in the case of electrically coupled unit cells described below was found to be numerically ill-conditioned for physical values of bipolar plate resistivities. An alternate formulation was developed in this case, described in Section 4. Computational current densities for oxidant stoichiometric flow rates of 1.8 and 1.2 are shown in Fig. 4, compared with experimental results from the Ballard Mk9 architecture (the experiments are described in more detail in Berg et al., 2004). We also show the corresponding cathode channel molar concentrations from the model in Fig. 5. There are no corresponding experimental measurements for this quantity. Note that with the cell voltage V constant in (3), the local current densities will decrease with the channel oxygen concentrations as seen in Figs 4 and 5. The agreement to experimental current densities is reasonable, and is not much improved when the membrane resistance variation with temperature and hydration is taken into account as in Berg et al. (2004), although these effects can be used to predict experimentally observed effects due to low inlet humidity at the cathode. We attribute much of the discrepancy in current density profiles to experimental observations in Fig. 4 to the neglect of the impact of liquid water in the catalyst layer. The results highlight the need for the electrically coupled model: changes in cell operating conditions can lead to changes in local current density. In a stack environment, adjacent cells with different local current densities lead to currents in the bipolar plates which will be shown below to have a significant effect on performance. We review here the basic structure of the model and the changes in the structure that occur when electrical coupling between cells is introduced in the next section. With oxygen concentration determined non-locally by current density profiles through (1) and (2) and the cell voltage V determined by the constraint of given average current density (5), the unit cell problem can be thought of as a non-local problem for the current density profile i(x). In the coupled cell model, currents in the bipolar plates lead to voltage variations in the plates. The model can still be viewed as a non-local problem for the current density profiles but the voltage couples not in an algebraic way but as a system of boundary value problems.

7 ELECTRICAL COUPLING IN PROTON EXCHANGE MEMBRANE FUEL CELL STACKS 247 FIG. 4. Comparison of experimental unit cell current density measurements with the model at two cathode stoichiometric flow rates: stoich 1.8 solid line (model, voltage ) and circles (experiment, voltage ); stoich 1.2 dotted line (model, voltage ) and pluses (experiment, voltage ). FIG. 5. Model predictions of cathode channel molar oxygen concentrations in a unit cell at two cathode stoichiometric flow rates: stoich 1.8 (solid line) and stoich 1.2 (dotted line). 4. Coupled cell model Consider now a stack of cells, indexed by a superscript j. We consider stacks of 2M +1 cells, numbered M,...,0, 1,...,M. The unknowns to be determined for each cell are V j (x) (cell voltage, now dependent on channel position because of the effects of the bipolar plate currents on voltage), i j (x) (local current density), Q j o(x) (cathode channel oxygen flux), C j o (x) (channel oxygen concentration) and Q j n (channel nitrogen flux). Fluxes and concentrations obey (1) and (2) in each cell and the voltage must satisfy (3) at each point in each channel. Each cell must meet the target average current density (5).

8 248 P. BERG ET AL. Additional material parameters needed in the coupled cell setting are λ: Length-specific resistivity of the bipolar plates. λ = ρ s, L t where ρ s is the volume-specific resistivity of the plate material and L t is its thickness. We take ρ s = m (graphite (Hodgman, 1959)) and L t 2mmtogive λ , (6) as a realistic value for current fuel cell designs. Although the units of λ are in, it should be understood that λ measures the resistance on the plates per unit length to currents per unit width of the cell. In the computations below, the effect of varying λ is considered. λ e : Length-specific resistivity of the end plates. We use λ e = m m , using formula (6) considering the end plates to be made of copper (Hodgman, 1959) of 2 mm thickness. The coupling between the cells appears due to voltage drops due to plate currents. We denote the current in the plate between cells j and j + 1byI j/j+1 (x). These plates currents are given per unit orthogonal z distance and are integrals (in y) of the in-plane current density. The structure of the in-plane current density can be neglected due to the large aspect ratio of the plates. If cell j is an interior cell, the current I j/j+1 in the plate between cell j and cell j + 1 satisfies di j/j+1 = i j (x) i j 1 (x). (7) dx This equation represents conservation of current and can be verified using the aid of Fig. 6, keeping in mind that plate currents are given per unit orthogonal length and cell currents are given per unit area. The plate currents lead to voltage drops in the plates. The difference in voltage between plate j + 1 and j is V j (x) so dv j dx = λ( I j/j+1 + I j 1/j ). (8) This can also be understood with the aid of Fig. 6. Differentiating (8) and using (7), the plate currents can be eliminated giving d 2 V j dx 2 λ(i j 1 2i j + i j+1 ) = 0, (9) which we name the fundamental voltage equation. Boundary conditions for each interior cell are dv j (0) = 0, (10) dx dv j dx (L c) = 0. (11) These correspond to the conditions that the plate currents must be zero at inlet and outlet using (7).

9 ELECTRICAL COUPLING IN PROTON EXCHANGE MEMBRANE FUEL CELL STACKS 249 FIG. 6. Diagram showing the derivation of (7) and (8). 4.1 End plates The equations at the end cells j =±M are slightly different. At j = M, the voltage satisfies dv M = λ e I M + λi M 1/M. (12) dx Here, I M is the upper end plate current that satisfies di M dx = i M (x), (13) except at the bus connection point if it is not at one of the two ends. If the bus connection is made at inlet (x = 0), then I M (0) = i d L c, (14) otherwise, I M (0) = 0. Equation (14) is just a statement that the total current through the stack is drawn off at inlet from the collector plate. If the connection is made at outlet, then I M (L c ) = i d L c, otherwise, I M (L c ) = 0. If the connection is made at the midpoint of the cell, then (13) holds except at x = L c /2. There, (13) has a discontinuity I M (L c /2 + ) = I M (L c /2 ) i d L c. (15) Differentiating (12) and applying (13) leads to a modified version of the voltage equation (9) for the end cell. In the case of a midpoint bus connection, (15) leads to a delta-function source term in the resulting equation at x = L c /2.

10 250 P. BERG ET AL. Note: The bus connections modelled above correspond physically to a line connection across the plate parallel to the z axis. In actuality, connections are made to discrete cables. For the Ballard Mk9 architecture, three cables are attached at channel midpoint, across a relatively narrow (5 cm) width. Our model is a reasonable approximation in this case. 4.2 The i-formulation The currents i j (x) are considered to be the unknowns. For given currents, (1) can be integrated to give Qo(x) j and then Co j (x) recovered from (2). Voltages V j (x) can then be recovered from (3). Now (9) (and modifications at the end cells) can be considered a system of non-local equations for the vector of functions i j (x). We were motivated to consider this formulation by the work in Freunberger et al. (2002). Note that this system of equations and boundary conditions has a one-parameter family of solutions. The addition of the condition that one cell has the given target total current (5) leads to a unique solution (the equations force all other cells to run at this same total current). Note that if unit cells are given symmetric conditions about cell j = 0, then the resulting currents satisfy i j (x) = i j (x) for all x and j > 0. This symmetry is used to reduce the number of unknowns in the computations described below. A computational approach to this problem is described in Section 6. First, the structure of the system is investigated analytically below using an asymptotic version of the model. 5. Linear analysis of cell interaction In this section, some analytic insight into electrical cell interactions is gained. First, we nondimensionalize the equations: We scale the oxygen by the constant non-vapour gas concentration C nv = (P c P v) /(RT ), the current by the mean current density i d, the voltage by the open circuit voltage and the along-the-channel distance by the channel length, C o = C o, x = x, ĩ = i, Ṽ = V. (16) C nv L c i d E 0 In the analysis below, it is convenient to derive a differential equation for the oxygen concentration. We take the x derivative of (2) and use the mass balance law (1) to obtain dc o dx = i(x) 4F Q n C nv (C nv C o (x)) 2. (17) We re-scale the governing equations (5), (17), (3) and (9) and drop the tilde notation, obtaining 1 0 i j (x) dx = 1, (18) dco j dx = ρ 0 s i j (1 Co j ) 2, (19) ( ) V j = 1 η R i j i j b ln Co j, (20) δ i j d 2 V j dx 2 = λ (i j 1 2i j + i j+1 ). (21)

11 ELECTRICAL COUPLING IN PROTON EXCHANGE MEMBRANE FUEL CELL STACKS 251 TABLE 1 Dimensionless parameters C o (0) ρ 0 C n (0) R s 4F Q o (0) b s 1 b = i d L c E 0 i d R E δ δi d C nv η b ln RT Fα c E ( ) 0 id C ref i 0 C nv λ λi d L 2 c E Recall also that the voltage has Neumann boundary conditions for (10) and (11). The dimensionless parameters are given in Table 1. For an infinite array of identical cells operating under uniform conditions, each cell will have the same oxygen profile, C o (x), current distribution, i(x), and voltage, V. The scaled fundamental voltage equation (21) forces the voltage to be constant. Further, we consider the high-stoich limit, s 1, in which the oxygen flux at the inlet dominates the total current. In this regime, the oxygen equation (19) implies that the oxygen concentration is constant to leading order, prescribed by its inlet value. The voltage balance equation forces a constant current profile, which by the scaled current target equation (18) yields i = 1. The voltage is then prescribed by the voltage balance equation ( ) 1 V = 1 η R b ln. (22) C o δ We investigate the sensitivity of this constant solution of the coupled cell equations to the introduction of an anomalous cell at j = 0, which operates at a prescribed current distribution i + ɛi 0 (x).for j 0, we expand, Co j = C o + ɛco j + O(ɛ 2 ), (23) i j = i + ɛi j + O(ɛ 2 ), (24) V j = V + ɛv j + O(ɛ 2 ), (25) with some abuse of notation using C j o, i j and V j to represent both the full solution and the perturbation from the base case. We substitute the expansion into the coupled cell equations, neglecting terms at O(ɛ 2 ) and O(s 1 ). We obtain the linearized equations, for j 0, 1 0 i j (x) dx = 0, (26) dco j = 0, (27) dx V j = 1 η c 1 i j c 2 Co j, (28) d 2 V j dx 2 = λ (i j 1 2i j + i j+1 ). (29) Here, the coefficients c 1 = R + C ob is a measure of the effective resistivity of the unit cell and (C o δ ) c 2 = b measures the impact of electrochemical limitations on cell voltage. These two terms arise C o δ

12 252 P. BERG ET AL. from the linearization of the voltage balance equation. Taking two derivatives of (28) and substituting into (29), we eliminate the voltage and obtain the current disturbance relationship λ λ(i j+1 2i j + i j 1 ) + d2 i j = 0. (30) c 1 dx2 Taking the x derivative of (28), and using the Neumann boundary conditions for V j, we obtain Neumann conditions for the current as well. Solutions take the form i j (x) = n=1 A n G j n φ n (x). (31) For each n, the cell interaction damping factors G n satisfy G n < 1. Without loss of generality we enumerate the solutions so that G n > G n+1. The coefficients {A n } are determined by the anomalous disturbance i 0 (x). The quantity of interest is the principal cell interaction damping factor, G 1, and its eigensolution, φ 1, which is the disturbance which affects the largest number of cells about the anomalous one. To determine the damping factors G and the eigensolutions φ, we substitute into (30) and obtain φ + µφ = 0, (32) with the boundary conditions φ (0) = φ (1) = 0. The eigenvalue µ relates to G through µ = 1 (G 2 + 1/G), (33) W where the key parameter W = c 1, λ is the ratio of the effective unit cell through-mea resistivity to the scaled bipolar plate resistivity. The parameter W is analogous to the Wagner number (Newman, 1973) which was derived to describe similar competition between effective electrochemical conductivity and electrolyte conductivity. In the present situation, the conductivity of the electrolyte (the membrane) is negligible but the bipolar plates play a similar role in current redistribution. It follows immediately from (32) and its boundary conditions that µ n = n 2 π 2 and φ n (x) = cos nπ x. The damping factors G n come in reciprocal pairs, of which we take the smaller ( G n = (2 + n 2 π 2 W ) )/ (2 + n 2 π 2 W ) The coefficients {A n } in (31) are the cosine series expansion for the anomalous disturbance i 0 (x). The principal interaction damping factor, G 1, is real and positive which precludes oscillatory phenomena in the stack direction. As W 0, the plate resistance dominates the unit cell resistance and G 1 1, while as W, the plate resistance is dominated by the cell resistance and G 1 0. Small values of W correspond to stacks where cells are strongly coupled and voltage losses associated with cell anomalies will be accentuated. Large values of W correspond to stacks whose cells are only weakly coupled: anomalous losses would not be significantly increased and the effects on neighbouring cells are weak. Using the parameters from Sections 9 and 4, we find C o = , c 1 = and W = 1/127.

13 ELECTRICAL COUPLING IN PROTON EXCHANGE MEMBRANE FUEL CELL STACKS 253 This leads to a principal cell interaction damping factor G 1 = It is clear that anomalous effects will spread to several adjacent cells. The analysis above gives insight into the way that the effects of anomalous cells spread. However, the analysis is restricted to the high stoichiometric flow rate limit and many fuel cells are operated at only moderate oxidant stoichiometric flow rates (around 2 is typical). Also, the theory predicts the extent an anomaly spreads, but not its size. To get quantitative information on realistic regimes, we turn to computational methods described below. 6. Coupled cell computational model We consider a symmetric stack as discussed in Section 4.2, so only cells j = 0, 1,...,M need be considered explicitly. The local current densities are approximated at N channel locations giving N(M + 1) discrete unknowns I. These discrete current densities are located at the centres of a uniform subdivision of the channel (into lengths h = L c /N). We use subscripts to denote spatial grid location. Oxygen fluxes at grid subinterval ends can be recovered from the given local currents using a naturally centred approximation of (1). Oxygen concentrations at grid subinterval ends are computed using the algebraic relationship (2). Oxygen concentrations at subinterval centres can be approximated using the average of these values at the interval ends. The local current and the oxygen concentration at cell centres can be used to obtain a local voltage at cell-centres using the algebraic relationship (3) with the extension described in Section 3.1. The vector of these cell-centred voltages is denoted by V (I ). A finite difference discretization of the system of equations (9) and boundary conditions (10), (11) is given below: (BV (I ) λc I ) = 0, (34) where B is an N(M+1) N(M+1) matrix representing the discrete approximation of second derivatives down the channel with homogeneous Neumann boundary conditions and C is a matrix describing the cell-to-cell coupling. Specifically, we have (BV ) j l = V j l 1 2V j l + V j l+1 h 2, if l 1, N and (BV ) j 1 = V j 2 V j 1 h 2, and a similar form at l = N. The matrix C is given explicitly for cells j 0, M by (C I ) j l = i j 1 l 2i j l + i j+1 l, with modifications corresponding to the symmetry at j = 0 and the collector plates at j = M. As for the continuous case, the system (34) is rank deficient of order one and we must add the additional approximate conditions for target current density (5): 1 N N l=1 I j l = i d, (35) for any one cell j. We use the centre cell j = 0 and this equation replaces the equation in (34) at the end of the zeroth cell (the l = N equation), although computational results do not depend on the choice

14 254 P. BERG ET AL. of cell j or location l (the discrete equations are exactly consistent). We denote the resulting non-linear discrete system for I as N (I ) = 0. (36) The equations and conditions have all been approximated to second order in h. We observe the secondorder convergence in h in the computations below. The number of cells 2M + 1 is not a computational parameter, it should be considered as a given parameter from the fuel cell stack architecture being modelled. 6.1 Iterative solution of the non-linear system Equation (36) is solved iteratively I (k) with k being the iteration number. We use Newton s method I (k+1) = I (k) δ (k), (37) where δ (k) solves where Gδ (k) = N (I (k) ), (38) G = I N (I (k) ) = (BS(I (k) ) λc) (39) is the N(M +1) N(M +1) Jacobian matrix. The systems are small enough in this application; they can be solved with a direct matrix solver. While it is not explicitly represented in (39), one line of the matrix is modified so that the update (37) preserves the constraint that all cells have the target average current density i d. S is the sensitivity matrix of the vector voltages to changes in the local currents. While S is not a diagonal matrix (changes in local currents change the oxygen concentration and so the voltages down-channel), it is reasonable to approximate it by the diagonal terms, which are the derivatives with respect to i of the modified Butler Volmer voltages (with the computational regularization described above) at the given local current and channel oxygen concentration. We obtain convergence with this technique in all cases with less than 10 iterations. Actually, the i-iteration method is quite effective in the single cell setting (faster than the old iterative process on cell voltage at least for this reduced unit cell model). In the single cell at base conditions, we begin iteration with a constant current density (the target average i d ). For coupled cell computations with one anomalous cell, we use this base local current density for the initial local current density profile for all cells. It is unexpected that the iterations were convergent in all cases, even extreme anomalies. A continuation procedure could be used in situations where convergence was not achieved from our starting vectors. We note that the diagonal approximation of S has all negative entries (local voltage always decreases with local current if other conditions are held fixed). Thus, (39) is positive definite with the structure of a discretization of a non-linear, 2D elliptic operator. This structure could be exploited to develop a fast iterative solver, e.g. based on multi-grid principles. 7. Numerical results We take the base operating conditions to be cathode stoich flow 1.8 and the other conditions listed in Section 9. We consider a 13-cell stack (M = 6). The centre cell (#0) is given anomalous conditions. We compute using a discretization of N = 64 channel locations except in Section 7.3, where the expected accuracy of the method is verified.

15 ELECTRICAL COUPLING IN PROTON EXCHANGE MEMBRANE FUEL CELL STACKS 255 TABLE 2 Stack voltage losses for various anomalous cathode stoichs and plate resistances λ Stoich λ = Stack losses Since the effects of the anomalous cell extend to all cells of the stack, the negative impact of this cell cannot be measured using only that cell. We compute stack losses as the change in voltage due to the anomaly as follows: V 0 (x) + 2 M V j (x) (2M + 1)V b, (40) j=1 where V b is the voltage of an isolated unit cell running at the base conditions. The last term of (40) is the voltage of the stack undisturbed by anomalous conditions in cell #0 or end plates. The first two terms are the sum of the voltages of the disturbed cells (the doubling of the sum reflects the symmetry about the anomaly). In (40), the x value must be the location of the bus plate connection. In the case where bus plate effects are neglected (λ e is set to zero), it can be shown that the expression does not depend on x. This corresponds physically to the fact that if the end plates are infinitely conductive, they are at a constant voltage. 7.2 Anomalous cathode understoich In Table 2, we show stack voltage losses as a function of plate resistivity λ and anomalous cathode stoich. Recall that the estimated resistivity for the Ballard Mk9 plates is λ = The effect of resistivities a decade larger and smaller are shown. Note that the λ = 0 entry represents uncoupled cells and the voltage change here is confined to the anomalous cell. In this section, base plate effects are neglected (we set λ e = 0). It is clear that plate resistivity has little effect for mild anomalies, but a significant effect for extreme ones, when the anomalous cell would run at a very different current distribution from the base if it were not coupled. In the last row of Table 2, note that the stack voltage loss due to the anomalous cell is a factor three times larger for an anomalous cell stoich of 1.2 and physical resistivity. Clearly, electrical interaction of cells in a stack can have a large influence on the size of losses due to unit cell variation. Graphs showing the local current densities, plate currents, cell voltages and cathode oxygen concentration profiles are given in Figs 7, 8, 9 and 10, respectively. These graphs show the effects of the cathode stoich 1.2 anomaly for λ = Inlet and outlet voltages are shown in Fig. 11. The experimentally observed volcano effect in the outlet voltages, where the anomalous cell has a severe drop but neighbouring cells a slight increase, is clearly seen.

16 256 P. BERG ET AL. FIG. 7. Local current densities for anomalous cathode stoich of 1.2. Pluses denote cell 0 (the anomalous cell), crosses cell 1 and circles cell 2. The solid line is the base local current density. These results are for λ = FIG. 8. Plate currents for anomalous cathode stoich of 1.2. Pluses denote plate 0/1, crosses plate 1/2 and circles plate 2/3. These results are for λ = (the Mk9 estimated plate resistivity). 7.3 Convergence study As discussed in Section 6, we expect second-order convergence in N, the number of channel points. We use the current density in the centre cell with anomalous cathode stoich of 1.2, with λ = and no end plate effects in the following convergence study. We can estimate the errors using E(N) = max I l 0 l 0 (N) I l (2N), where the in the expression above denotes the averaging of adjacent grid values to obtain a vector of length N of values centred at the locations of the currents for the N-point discretization. This is

17 ELECTRICAL COUPLING IN PROTON EXCHANGE MEMBRANE FUEL CELL STACKS 257 FIG. 9. Cell voltages for anomalous cathode stoich of 1.2. Pluses denote cell 0 (the anomalous cell), crosses cell 1 and circles cell 2. The solid line is the base voltage. These results are for λ = (the Mk9 estimated plate resistivity). FIG. 10. Cathode channel molar oxygen channel concentrations for anomalous cathode stoich of 1.2. Pluses denote cell 0 (the anomalous cell), crosses cell 1 and circles cell 2. These results are for λ = (the Mk9 estimated plate resistivity). necessary due to the use of the cell-centred discretization. An estimate of the convergence rate ρ is obtained using ρ(n) = log 2 (E(N/2)/E(N)). Estimated errors and convergence rates are shown in Table 3. Second-order convergence in maximum norm is clearly obtained. Recall that the size of the currents being computed are of the order 10 4,sothe relative estimated errors are small.

18 258 P. BERG ET AL. FIG. 11. Inlet (circles) and outlet (pluses) voltages for the stack with anomalous cathode stoich of 1.2 in cell 0. The base voltage is shown as the solid line. TABLE 3 Estimated errors E and convergence rates ρ N E ρ End plate effects In this study, we consider all cells to be running at base conditions but introduce end collector plate effects. In Table 4, we show stack voltage losses as a function of end plate resistivities λ e and connection location (inlet, outlet and mid-length). Note that the plate resistance has little effect on these losses, although the effects do extend to adjoining cells. The losses are almost linear in λ e, suggesting that there is little interplay between plate losses and changes in cell performance due to the voltage changes induced by the end plates. Graphs showing the cell voltages profiles for inlet, mid-length and outlet connections are shown in Figs 12, 13 and 14, respectively. These graphs show the effects of the end plate for λ = and λ e = Conclusions A model and an efficient and robust numerical method for the electrical coupling of unit fuel cells in a stack have been developed. Two important dimensionless quantities have been identified and their role in a simple constant coefficient analysis has been explored. It is shown that the effects of electrical coupling on stack performance are quite significant. Since the preparation of this manuscript, some experimental data to validate the model has been obtained (Kim et al., 2005). The agreement is quite good with only experimental determination of the

19 ELECTRICAL COUPLING IN PROTON EXCHANGE MEMBRANE FUEL CELL STACKS 259 TABLE 4 Stack voltage losses for end plate as a function of collector plate resistance λ e and connection location Connection λ e = λ e = Inlet Mid-length Outlet FIG. 12. Cell voltages with end plate effects, inlet connection. Pluses denote cell 6 (next to plate), crosses cell 5 and circles cell 4. The solid line is the base voltage. FIG. 13. Cell voltages with end plate effects, mid-length connection. Pluses denote cell 6 (next to plate), crosses cell 5 and circles cell 4. The solid line is the base voltage.

20 260 P. BERG ET AL. FIG. 14. Cell voltages with end plate effects, outlet connection. Pluses denote cell 6 (next to plate), crosses cell 5 and circles cell 4. The solid line is the base voltage. bipolar plate resistivity (λ) and no additional fitting. This is a significant point and indicates that local relationships such as (3), for which it is possible and appropriate to fit extensively to experimental data, can be combined with more conventional models of electrical networks and possibly also thermal transport to lead to accurate computational models of stack level performance. 9. Nomenclature and parameter values C o (x): Average channel oxygen concentration (mol/m 3 ). C ref : A reference oxygen concentration, taken to be that of pure O 2 at standard conditions (40.9 mol/m 3 ). C ɛ : Computational parameter, a minimal catalyst layer oxygen concentration. Taken to be 0.02 mol/m 3. F: Faraday constant (96485 C/mol). i(x): Local current density in A/m 2. Indexed by cell in coupled model. I j+1/j (x): Current in the bipolar plate between cells j and j +1 per unit orthogonal (z) distance (A/m). i 0 : Fitted (Berg et al., 2004) exchange current density (64 A/m 2 ). i d : Average current density A/m 2 (1 A/cm 2 ). E 0 : Open circuit voltage V (unit cell voltage when no current is drawn). L c : Cell length 0.67 m. R: Ideal gas constant J/K mol. P c : Cathode channel pressure Pa (approximately 3 atm). P v : Saturation pressure (34.8 kpa) at T below, from empirical fit to experimental data. We use the cubic fit to log P v found in Springer et al. (1991). Q o (x): Cathode channel oxygen flux, per unit z direction in mol/s m. Q n : Cathode channel nitrogen flux, per unit z direction in mol/s m.

21 ELECTRICAL COUPLING IN PROTON EXCHANGE MEMBRANE FUEL CELL STACKS 261 R: Membrane area-specific resistivity ( m 2 ), from the fitted model developed in Berg et al. (2004) under saturated conditions at T given below. s: Stoichiometric flow rate, dimensionless. T : Cell temperature 348 K (approximately 75 C). V : Cell voltage, constant for idealized unit cell setting, dependent on x and indexed by cell in coupled model. x: Down-channel coordinate (m). y: Through-MEA coordinate (m). z: Cross-channel coordinate (m). α c : Cathode transfer factor, taken to be 1. δ: Fitted (Berg et al., 2004) mass transport loss parameter ( mol/a m). λ: Length-specific resistivity of the bipolar plates, ( m). λ e : Length-specific resistivity of the end plates ( m). Acknowledgements All authors acknowledge the financial support of the MITACS NCE and Ballard Power Systems. BW and KP would also like to thank the support of NSERC. Many thanks to Gwang-Soo Kim at Ballard Power Systems for several insightful comments on this work. REFERENCES BERG, P., PROMISLOW, K., ST.-PIERRE, J., STUMPER, J.& WETTON, B. (2004) Water management in PEM fuel cells. J. Electrochem. Soc., 151, A341 A353. BERNING, T.,LU, D.M.&DJILALI, N. (2002) Three-dimensional computational analysis of transport phenomena in a PEM fuel cell. J. Power Sources, 106, 284. DUTTA, S., SHIMPALEE, S.& ZEE, J. W. (2001) Numerical prediction of mass-exchange between cathode and anode channels in a PEM fuel cell. Int. J. Heat Mass Transf., 44, FREUNBERGER, S., TSUKADA, A., FAFILEK, G.&BUECHI, F. N. (2002) 1+1dimensional model of a PE fuel cell of technical size. Paul Scherrer Institut Scientific Report 2002, vol. 5, article #94. Available to order from HODGMAN, C. (ed.) (1959) Handbook of Chemistry and Physics, 41st edn. Cleveland, OH: Chemical Rubber Publishing Company. KIM, G.-S., ST.-PIERRE, J., PROMISLOW, K.& WETTON, B. (2005) Electrical coupling in proton exchange membrane fuel cell stacks. J. Power Sources to appear. MAZUMDER, S.& COLE, J. V. (2003a) Rigorous three-dimensional mathematical modeling of proton exchange membrane fuel cells. Part 1: model predictions without liquid water transport. J. Electrochem. Soc., 150, A1503 A1509. MAZUMDER, S.& COLE, J. V. (2003b) Rigorous three-dimensional mathematical modeling of proton exchange membrane fuel cells. Part 2: model predictions with liquid water transport. J. Electrochem. Soc., 150, A1510 A1517. NATARAJAN, D.&VAN NGUYEN, T. (2003) Three-dimensional effects of liquid water flooding in the cathode of a PEM fuel cell. J. Power Sources, 115, NEWMAN, J. S. (1973) Electrochemical Systems. Englewood Cliffs, NJ: Prentice-Hall. SPRINGER, T. E., ZAWODZINSKI, T. A. & GOTTESFELD, S. (1991) Polymer electrolyte fuel cell model. J. Electrochem. Soc., 138, 2334.