Cathode and interdigitated air distributor geometry optimization in polymer electrolyte membrane (PEM) fuel cells

Materials Science and Engineering B 108 (2004) 241 252 Cathode and interdigitated air distributor geometry optimization in polymer electrolyte membrane (PEM) fuel cells M. Grujicic, C.L. Zhao, K.M. Chittajallu, J.M. Ochterbeck Department of Mechanical Engineering, Clemson University, Clemson, SC 29634-0921, USA Received 31 May 2003; accepted 5 January 2004 Abstract A steady-state single-phase three-dimensional electro-chemical model is combined with a nonlinear constrained optimization procedure to maximize the performance of the cathode and the interdigitated air distributor in a polymer electrolyte membrane (PEM) fuel cell. The cathode and the interdigitated air distributor design parameters considered include: the cathode thickness, the thickness of the interdigitated air distributor channels and the width of the interdigitated air distributor channels. A statistical sensitivity analysis is used to determine robustness of the optimal PEM fuel cell design. The results of the optimization analysis show that higher current densities at the membrane/cathode interface are obtained in the PEM cathode and the interdigitated air distributor geometries that promote convective oxygen transport to the membrane/cathode interface and reduce the thickness of the boundary diffusion layer at the same interface. The statistical sensitivity analysis results show that, while the predicted average current density at the membrane/cathode interface is affected by uncertainties in a number of model parameters, the optimal designs of the PEM cathode and the interdigitated air distributor are quite robust. 2004 Elsevier B.V. All rights reserved. Keywords: Polymer electrolyte membrane (PEM) fuel cells; Design; Optimization; Robustness 1. Introduction Due to their potential for reducing the environmental impact and the dependence on fossil fuels, fuel cells have emerged as an attractive alternative to the internal combustion engines. In a fuel cell, fuel (e.g. hydrogen gas) and an oxidant (e.g. oxygen gas from the air) are used to generate electricity, while heat and water are typical byproducts. As the hydrogen gas flows into the fuel cell on the anode side, a platinum catalyst facilitates fuel oxidation which produces protons (hydrogen ions) and electrons, Fig. 1. Protons diffuse through a membrane (the center of the fuel cell which separates the anode and the cathode) and, with the help of a platinum catalyst, combine with oxygen and electrons on the cathode side, producing water. The electrons produced at the anode side cannot pass through the membrane and flow from the anode to the cathode through an external circuit containing an electrical motor. The resulting voltage from one single fuel cell is typically around 1.0 V. This voltage Corresponding author. Tel.: +1-864-656-5639; fax: +1-864-656-4435. E-mail address: mica.grujicic@ces.clemson.edu (M. Grujicic). is generally increased by stacking the fuel cells in series, in which case the operating voltage of the stack is simply equal to the product of the operating voltage of a single cell and the number of cells in the stack. Fuel cells are typically classified according to the type of membrane (polymer electrolyte membrane fuel cells, solid oxide fuel cells, molten carbonate fuel cells, etc.) they use. One of the most promising fuel cells are the so-called polymer electrolytic membrane or proton exchange membrane fuel cells (PEMFCs). The polymer electrolyte membrane is a solid, organic polymer, usually poly[perfluorosulfonic] acid. The most frequently used PEM is made of Nafion TM produced by DuPont, which consists of Teflon-like chains with a fluorocarbon backbone and sulfonic acid ions, SO 3, permanently attached to the side chains. When the membrane is hydrated by absorbing water, protons attached to the SO 3 ions combine with water molecules to form hydronium ions. Hydronium ions are quite mobile and hop from one SO 3 site to another within the membrane making the hydrated solid electrolytes like Nafion TM excellent conductors of the hydrogen ions. The anode and the cathode (the electrodes) in a PEM fuel cell are made of an electrically conductive porous mate- 0921-5107/$ see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.mseb.2004.01.005

242 M. Grujicic et al. / Materials Science and Engineering B 108 (2004) 241 252 Nomenclature N i molar flux of species (kg/m 2 /s) p pressure (Pa) u gas velocity (m/s) w mass fraction φ potential (V) ρ density (kg/m 3 ) Subscripts H 2 O water-related quantity oxygen-related quantity O 2 Superscripts eff effective quantity sym symmetry x molar fraction rial, typically carbon. The faces of the electrodes in contact with the membrane (generally referred to as the active layers) contain, in addition to carbon, polymer electrolyte and a platinum-based catalyst. Each active layer is denoted by a thick vertical line in Fig. 1. As also indicated in Fig. 1, oxidation and reduction fuel-cell half reactions take place in the anode and the cathode active layers, respectively. The PEM electrodes are of gas-diffusion type and generally designed for maximum surface area per unit material volume (the specific surface area) available for the reactions, for minimum transport resistance of the hydrogen and the oxygen to the active layers, for an easy removal of the water from the cathodic active layer and for the minimum transport resistance Fig. 1. A schematic of a polymer electrolyte membrane (PEM) fuel cell. of the protons from the active sites in the anodic layer to the active sites in the cathodic active layer. As shown in Fig. 1, a PEM fuel cell also typically contains an interdigitated fuel distributor on the anode side and an interdigitated air distributor on the cathode side. The use of the interdigitated fuel/air distributors imposes a pressure gradient between the inlet and the outlet channels, forcing the convective flow of the gaseous species through the electrodes. Consequently, a 50 100% increase in the fuel-cell performance is typically obtained as a result of the use of interdigitated fuel/air distributors. The regions of the interdigitated fuel/air distributors separating the inlet and the outlet channels, generally referred to as the shoulders, serve as the anode and cathode electric current collectors. Due to their high-energy efficiency, a low temperature (333 353 K) operation, a pollution-free character, and a relatively simple design, PEM fuel cells are currently being considered as an alternative source of power in the electric vehicles. However, further improvements in the efficiency and the cost are needed before the PEM fuel cells can begin to successfully compete with the internal combustion engines. The development of the PEM fuel cells is generally quite costly and the use of mathematical modeling and simulations has become an important tool in the fuel-cell development. Over the last decades a number of fuel-cell models have been developed. Some of these models are single-phase (e.g. [1,2]) while the others are two-phase (e.g. [3]), i.e., they consider the effect of the liquid water supplied to the anode and the one formed in the cathodic active layer. Due to the slow kinetics of oxygen reduction, some of these models focus only on the cathode side of the fuel cell (e.g. [1,3]) while the others deal with the entire fuel cell (e.g. [2]). Most of the models like the ones cited above are used to carry out parametric studies of the effect of various fuel-cell design parameters (such as the cathodic and anodic thicknesses, the geometrical parameters of the interdigitated fuel/air distributors, etc.). However, a comprehensive optimization analysis of the PEM fuel cell design is still lacking. Hence, the objective of the present work is to modify the steady-state single-phase three-dimensional PEM fuel-cell cathode model associated with a U-shaped air distribution system [4] to include the effect of the interdigitated air distributor and to combine it with an optimization procedure and a statistical sensitivity analysis in order to identify the optimum geometry of the PEMFC cathode and the interdigitated air distributor. The organization of the paper is as following: in Section 2, the modification of the steady-state single-phase threedimensional model for a PEM fuel-cell cathode presented in [4] and a solution method for the resulting set of partial differential equations are briefly discussed. An overview of the optimization and the statistical sensitivity methods is presented in Section 3. The main results obtained in the present work are presented and discussed in Section 4. The main conclusions resulting from the present work are summarized in Section 5.

M. Grujicic et al. / Materials Science and Engineering B 108 (2004) 241 252 243 2. The model As indicated in Fig. 1, the PEM fuel cell works on the principle of separation of the oxidation of hydrogen (taking place in the anodic active layer) and the reduction of oxygen (taking place in the cathodic active layer). The oxidation and the reduction half-reactions are given in Fig. 1. Electrons liberated in the anodic active layer via the oxidation halfreaction travel through the anode, an anodic current collector, an outer circuit (containing an external load, typically a power conditioner connected to an electric motor), a cathodic current collector and the cathode until they reach the cathodic active layer. Simultaneously, protons (H + ) generated in the anodic active layer diffuse through the polymer electrolyte membrane until they reach the cathodic active layer, where the oxygen reduction half-reaction takes place. Due to a relatively slow rate of the oxygen reduction reaction, the design of the fuel-cell cathode and the associated interdigitated air distributor are considered as critical for achieving a high performance of the PEMFCs. Therefore, the work presented here focused on the cathode side of the fuel cell. 2.1. Assumptions and simplifications In this section, a simple steady-state single-phase threedimensional model for a PEM fuel-cell cathode developed in [4] is modified to include the effect of an interdigitated air distributor. The model is developed under the following simplifications and assumptions: The computational domain, denoted with dashed lines in Fig. 1, consists of a porous cathode which is in contact with an interdigitated air distributor. A three-dimensional view of the computational domain is shown in Fig. 2. The membrane/cathode interface is considered as a zerothickness cathodic active layer. The portions of the cathode surface which are not in contact with the interdigitated air distributor are designated as the current collector surface. Humidified air of fixed composition and pressure is supplied at the cathode inlet. Liquid water entering the cathode from the membrane side and the one generated during the oxygen reduction reaction in the cathodic active layer are assumed to be of a negligible volume (i.e., no two-phase flow in the cathode is considered). It should be noted that this simplification could be critical since liquid water residing in pores of the cathode can significantly reduces its effective permeability. In fact, when the liquid water saturation level reaches 100% (the phenomenon known as cathode flooding ), the cathode becomes impervious to the gases. The inlet/outlet pressure difference specified in Table 1 has been selected in such a way that a compromise is achieved between an effective removal of water droplets Fig. 2. Schematic of the cathode and interdigitated air distributor in a PEM fuel cell. Table 1 General parameters and reference-case cathode and interdigitated air distributors parameters used for 3D modeling the PEM fuel cell Parameter Symbol SI units Value Faraday s constant F A s/mol 96,487 Universal gas constant R J/mol K 8.314 Temperature T K 353 Atmospheric (reference) p 0 Pa 1.013 10 5 pressure Gas viscosity µ kg/m s 2.0 10 5 Pressure differential p Pa p 0 2.5 10 1 Activation overpotential η V 0.5 Molecular weight of oxygen M O2 kg/mol 32 10 3 Molecular weight of water M H2 O kg/mol 18 10 3 Molecular weight of M N2 kg/mol 28 10 3 nitrogen Molar diffusion volume of v O2 m 3 /mol 16.6 10 6 oxygen Molar diffusion volume of v H2 O m 3 /mol 12.7 10 6 water Molar diffusion volume of v N2 m 3 /mol 17.9 10 6 nitrogen Porosity of cathode ε No units 0.5 Inlet mass fraction of oxygen w O2,0 No units 0.1447 Inlet mass fraction of oxygen w H2 O,0 No units 0.3789 Inlet mass fraction of oxygen w N2,0 No units 0.4764 Empirical value k MS Nm 2 kg 1/2 / 99.9 10 8 K 1.75 mol 7/6 Specific surface area S a m 2 /m 3 1.0 10 8 Thickness of the active δ m 1.0 10 5 layer Number of electrons n No units 4.0 Exchange current density i 0 A/m 2 4.0 10 3 Cathode conductivity k S/m 1.0 10 5

244 M. Grujicic et al. / Materials Science and Engineering B 108 (2004) 241 252 Fig. 3. Dependent variables, governing equations and boundary conditions in the interdigitated air distributor. from the cathode and the retention of structural integrity of the cathode. The behavior of the O 2 +N 2 +H 2 O gas mixture in the cathode is considered to be governed by the ideal gas law while their diffusivities are described by the Maxwell Stefan diffusion model. The cathode is considered as a homogeneous medium in which the porosity and permeability are distributed uniformly. The kinetics of the electro-chemical oxygen reduction half-reactions taking place in the cathode active layer is assumed to be governed by the Tafel equation. 2.2. The dependent variables As indicated in Fig. 2, there are two sub-domains in the present model: (a) the interdigitated air distributor and (b) the porous cathode. As indicated in Figs. 3 and 4, the following dependent variables are used in the present model: In the interdigitated air distributor: (a) the gas-phase velocity, u; (b) the gas-phase pressure, p; (c) the oxygen mass fraction in the gas-phase, w O2 ; and (d) the watervapor mass fraction in the gas phase, w H2 O; In the cathode: (a) the gas-phase velocity, u; (b) the gasphase pressure, p; (c) the oxygen mass fraction in the gasphase, w O2 ; (d) the water-vapor mass fraction in the gas phase, w H2 O; and (e) the electronic potential, φ. 2.3. The governing equations The governing equations for the interdigitated air distributor and the cathode are given in Figs. 3 and 4, respectively. They include: For the interdigitated air distributor: (a) a momentum conservation equation; (b) a continuity equation; (c) an oxygen mass-balance equation; and (d) a water-vapor mass-balance equation. For the cathode: (a) a momentum conservation equation; (b) a continuity equation; (c) an oxygen mass-balance equation; (d) a water-vapor mass-balance equation; and (e) an electric charge conservation equation. 2.4. The boundary conditions The boundary conditions used are also listed in Figs. 3 and 4. The boundary conditions can be briefly summarized as following: 2.4.1. For the interdigitated air distributor (a) The composition of the air at the inlet to the interdigitated air distributor is set equal to that of steam-saturated air at 353 K. The air pressure head is set to a fixed value; (b) At the interdigitated air distributor outlet, a straight-out flow is assumed, the pressure is fixed and the mass transfer is taken to be dominated by convection;

M. Grujicic et al. / Materials Science and Engineering B 108 (2004) 241 252 245 Fig. 4. Dependent variables, governing equations and boundary conditions in the porous PEMFC cathode. (c) At the remaining boundaries, no-slip impervious boundary conditions are assumed. The molar density of the gas mixture is defined using the ideal gas law as: 2.4.2. For the cathode ρ = p RT (2) (a) At the portions of the top surface of the cathode, which are not in contact with the interdigitated air distributor, no-slip, impervious, and zero electric potential boundary conditions are assumed; (b) At the portions of the top surface of the cathode which are in contact with the interdigitated air distributor which is a boundary only with respect to the electric charge conservation equation, a zero electric current boundary condition is assumed; (c) At the membrane/cathode interface, no-slip boundary conditions for the gas-phase velocity are assumed while the electric current and the oxygen and the water vapor fluxes are given by the Tafel equation; (d) At the remaining surfaces, no-slip and zero-flux boundary conditions for the electric potential, oxygen and water vapor mass fractions are prescribed. 2.5. Closure relationships To express the governing equations in terms of the dependent variables, the following closure relations are used: The molar mass of the O 2 + N 2 + H 2 O gas mixture is defined as: M = M O2 w O2 + M H2 Ow H2 O + M N2 w N2 (1) The symmetric binary diffusivities appearing in the mass balance equations are dependent on the gas-phase composition and are given as: D sym ii = ((w j + w k ) 2 /x i D jk ) + (w 2 j /x jd ik ) + (w 2 k /x kd ij ) (x i /D ij D ik ) + (x j /D ij D jk ) + (x k /D ik D jk ) (i, j = O 2, H 2 O, N 2 ) (3) D sym ij = (w i (w j + w k )/x i D jk ) + (w j (w i + w k )/x j D ik ) (w 2 k /x kd ij ) (x i /D ij D ik ) + (x j /D ij D jk ) + (x k /D ik D jk ) (i, j = O 2, H 2 O, N 2 ) (4) The Maxwell Stefan diffusivities used in Eqs. (3) and (4) are defined as: T 1.75 [ 1 D ij = k MS p(v 1/3 i + v 1/3 + 1 ] 12 j ) 2 M i M j (i, j = O 2, H 2 O, N 2 ) (5) The effective binary diffusivities within the porous cathode are given as: D eff ij = D ij ε 1.5 (i, j = O 2, H 2 O, N 2 ) (6)

246 M. Grujicic et al. / Materials Science and Engineering B 108 (2004) 241 252 The Tafel coefficient which relates the oxygen and the water vapor fluxes with the local current density at the membrane/cathode interface is defined as: k Tafel = S aδi 0 nf ( ) 0.5Fη exp RT 2.6. Computational method (i, j = O 2, H 2 O, N 2 ) (7) The steady-state, nonlinear three-dimensional system of governing partial differential equations (discussed in Section 2.3 and in Figs. 3 and 4) subjected to the boundary conditions (discussed in Section 2.4 and in Figs. 3 and 4) are implemented in the commercial mathematical package FEM- LAB [5] and solved (for the dependent variables discussed in Section 2.2 and in Figs. 3 and 4) using the finite element method. The FEMLAB provides a powerful interactive environment for modeling various scientific and engineering problems and for obtaining the solution for the associated (stationary and transient, both linear and nonlinear) systems of governing partial differential equations. The FEMLAB is fully integrated with the MATLAB, a commercial mathematical and visualization package [6]. As a result, the models developed in the FEMLAB can be saved as MATLAB programs for parametric studies or iterative design optimization. 3. Design optimization and robustness There are many PEMFC cathode and the interdigitated gas distributor parameters which affect the performance of a PEM fuel cell. Some of these parameters such as the cathode permeability and porosity are controlled by the microstructure of the cathode porous material. Since these microstructure sensitive parameters are mutually interdependent in a complex and currently not well-understood way, they will not be treated as design parameters within the PEMFC cathode optimization procedure described below. Instead, geometrical parameters (e.g. the cathode thickness, the width of the interdigitated air distributor channels, etc.) will be considered. As explained earlier, the kinetics of reduction half-reaction in the cathodic active layer is very slow and, it is generally recognized that significant improvements in the fuel cell performance can be obtained by optimizing the design of its cathode side. This approach is adopted in the present work and, consequently, the following three cathode and interdigitated air distributor design parameters have been identified as the most important: the cathode thickness (0.0001 m); the thickness of the interdigitated air distributor channels (0.0001 m); and the width of the interdigitated air distributor channels (0.0001 m). The numbers given within the parentheses correspond to the values of the three design parameters in the initial (reference) design of the PEMFC cathode and the interdigitated air distributor. A schematic of the cathode is given in Fig. 5 to explain the three design parameters (denoted as x(i), i = 1 3) defined above. The width and the length of the cathode are kept constant at 0.0009 and 0.0013 m, respectively. The objective function f[x(1), x(2), x(3)] is next defined as the average electric current density at a typical value of the cell voltage of 0.7 V. The average electric current density can be considered as a measure of the degree of utilization of the expensive Pt-based catalyst in the cathode active layer. Thus, the fuel-cell design optimization problem can be defined as: minimize 1/f[x(1), x(2), x(3)] with respect to x(1), x(2), and x(3) subject to: 0.00002 m x(1) 0.0005 m 0.00002 m x(2) 0.005m, and 0.000025 m x(3) 0.000175 m The upper limits for the three design parameters are chosen based on considerations of the size constraints for a PEMFC stack. The lower limits for the three design parameters, on the other hand, are selected based on the consideration of minimal feature sizes which can be attained using the current PEMFC cathode and the interdigitated air-distributor manufacturing processes. 3.1. Optimization The optimization problem formulated above is solved using the MATLAB optimization toolbox [6] which contains an extensive library of computational algorithms for solving different optimization problems such as: unconstrained and constrained nonlinear minimization, quadratic and linear programming and the constrained linear least-squares method. The problem under consideration in the present work belongs to the class of multidimensional constrained nonlinear minimization problems which can be solved using the MATLAB fmincon() optimization function of the following syntax: fmincon(fun,x 0,A,B,A eq,b eq, LB, UB, confun, options); (8) where fun denotes the scalar objective function of a multidimensional design vector x, while confun contains nonlinear non-equality (c(x) 0) and equality (c eq (x) = 0) constrained functions. The matrix A and the vector b are used to define linear non-equality constrains of the type Ax b, while the matrix A eq and the vector b eq are used to define linear equality constraining equations of the type A eq x = b eq. LB and UB are vectors containing the lower and the upper bounds of the design valuables and x 0 is the initial design point. The MATLAB fmincon() function implements the sequential quadratic programming (SQP) method [7] within which the original problem is approximated with a quadratic programming subproblem which is then solved successively

M. Grujicic et al. / Materials Science and Engineering B 108 (2004) 241 252 247 Fig. 5. Definition of the three cathode and the interdigitated air distributor-based design parameters used in the PEMFC optimization procedure. until convergence is achieved for the original problem. The method has the advantage of finding the optimum design from an arbitrary initial design point and typically requires fewer function and gradient evaluations compared to other methods for constrained nonlinear optimization. The main disadvantages of this method are that it can be used only in problems in which both the objective function, func, and the constrained equations, confun, are continuous and, for a given initial (reference) design of the PEMFC cathode and the interdigitated air distributor, the method can find only a local minimum in the vicinity of the initial design. 3.2. Statistical sensitivity analysis Once the optimum combination of the design parameter is found using the optimization procedure described above, it is important to determine how sensitive is the optimal fuelcell design to the variation in the model parameters whose magnitudes are associated with considerable uncertainty. In the field of mechanical design, these parameters are generally referred to as factors, and this term will be used throughout this paper. To determine the robustness of the optimum design with respect to variations in the factors the method commonly referred to as the statistical sensitivity analysis [8] will be used in the present work. The first step in the statistical sensitivity analysis is to identify the factors and their ranges of variation. The selection of the factors and their ranges is subjective and depends on engineering experience and judgment and it is essential for proper formulation of the problem. Typically two to four values (generally referred to as levels) are selected for each factor. The next step in the statistical sensitivity analysis is to identify the analyses (finite element computational analyses in the present work) which need to be performed in order to quantify the effect of the selected factors. In general, a factorial design approach can be used in which all possible combinations of the factor levels are used. However, the number of the analyses to be carried out can quickly become unacceptably large as the number of factors and levels increases. To overcome this problem, i.e. to reduce the number of analyses which should be performed, the orthogonal matrix method [9] will be used. The orthogonal matrix method contains a column for each factor, while each row represents a particular combination of the levels for each factor to be used in an analysis. Thus, the number of analyses which needs to be performed is equal to the number of rows of the corresponding orthogonal matrix. The columns of the matrix are mutually orthogonal, that is, for any pair of columns, all combinations of the levels of the two fac-

248 M. Grujicic et al. / Materials Science and Engineering B 108 (2004) 241 252 tors appear and each combination appear an equal number of times. A limited number of standard orthogonal matrices [9] is available to accommodate specific numbers of the factors with various number of levels per factor. A computational finite element analysis is next performed for each combination of the factor levels as defined in the appropriate row of the orthogonal matrix. The results of all the analyses are next tabulated and the overall mean value of the objective function calculated. The mean values of the objective function associated with each level of each factor are also calculated. As discussed earlier, each level of a factor appears an equal number of times within its column in the orthogonal matrix. The objective function results associated with each level of a factor are averaged to obtain the associated mean values. The effect of a level of a factor is then defined as the deviation it causes from the overall mean value and is thus obtained by subtracting the overall mean value from the mean value associated with the particular level of that factor. This process of estimating the effect of factor levels is generally referred to as the analysis of means (ANOM). The ANOM allows determination of the main effect of each factor. However using this procedure it is not possible to identify possible interactions between the factors. In other words, the ANOM is based on the principle of linear superposition according to which the system response η (the objective function in the present case) is given by: h = overall mean + (factor effect) + error (9) where error denotes the error associated with the linear superposition approximation. To obtain a more accurate indication of the relative importance of the factors and their interactions, the analysis of variance (ANOVA) can be used. The ANOVA allows determination of the contribution of each factor to the total variation from the overall mean value. This contribution is computed in the following way: first, the sum of squares of the differences from the mean value for all the levels of each factor is calculated. The percentage that this sum for a given factor contributes to the cumulative sum for all factors is a measure of the relative importance of that factor. The ANOVA also allows estimation of the error associated with the linear superposition assumption. The method used for the error estimation generally depends on the number of factors and factor levels as well as on the type of the orthogonal matrix used in the statistical sensitivity analysis. The method described below is generally referred to as the sum of squares method. The sum of squares due to the error, SS error, can be calculated using the following relationship: SS error = SS grand SS mean SS factors (10) where SS grand is the sum of the squares of the results of all the analyses, SS mean value is equal to the overall mean squared multiplied by the number of analyses and SS factors is equal to the sum of squares of all the factor effects. Each quantity in Eq. (10) is associated with a specific number of degrees of freedom. The number of degrees of freedom for the grand total sum of squares, DOF grand, is equal to the number of analysis (i.e. the number of rows in the orthogonal matrix). The number of degrees of freedom associated with the mean value, DOF mean, is one. The number of degrees of freedom for each factor, DOF factor, is one less than the number of levels for that factor. The number of degrees of freedom for the error can hence be calculated as: DOF error = DOF grand 1 (DOF factor ) (11) For Eq. (10) to be applicable, the number of degrees of freedom for the error must be greater than zero. If the number of degrees of freedom for the error is zero, a different method must be used to estimate the linear superposition error. An approximate estimate of the sum of the squares due to the error can be obtained using the sum of squares and the corresponding number of degrees of freedom associated with the half of the factors with the lowest mean square. Once the sum of squares due to the error and the corresponding number of degrees of freedom for the error have been calculated, the error variance, VAR error, and the variance ratio, F, can be computed as: VAR error = SS error DOF error, and (12) F = (MEAN factor) 2 (13) VAR error where MEAN factor is a mean value of the objective function for a given factor. The varaince ratio, F, is used to quantify the relative magnitude of the effect of each factor. A value of F less than one normally implies that the effect of the corresponding given factor is smaller than the error associated with the linear superposition approximation and hence can be ignored. A value of F above four, on the other hand, generally suggests that the effect of the factor at hand is significant. 4. Results and discussion 4.1. The reference case Contour plots of the current density and the oxygen mole fraction at the membrane/cathode interface in the case of the reference design of the PEMFC cathode and the interdigitated air distributor are shown in Fig. 6a and b, respectively. Based on the results displayed in Fig. 6a and b, the following main observations can be made: The variation of the current density over the membrane/cathode interface is significant ( 50%) and the largest values of the current density are found in the region adjacent to the inlet of the interdigitated air distributor. The average current density at the membrane/cathode interface is 10,600 A/m 2.

M. Grujicic et al. / Materials Science and Engineering B 108 (2004) 241 252 249 Fig. 6. Variations of: (a) the current density and (b) the oxygen mole fraction over the membrane/cathode interface in the reference design of the PEMFC cathode and the interdigitated air distributor. The variation of the oxygen mole fraction over the membrane/cathode interface is also significant ( 50%) and the largest values of the oxygen mole fraction are also found in the region adjacent to the inlet of the interdigitated air distributor. However, the lowest levels of the oxygen mole fraction are not found in the region adjacent to the outlet of the interdigitated air distributor but further away from it. A careful examination of the diffusion flux fields revealed oxygen back-diffusion at the outlet of the interdigitated air distributor which results in higher oxygen mole fractions in this region. There is a clear correlation between the current density and the local oxygen mole fraction at the membrane/cathode interface. The variations of the oxygen mole fraction over three planes normal to the membrane/cathode interface and parallel with the fuel-cell width are shown in Fig. 7a-c. In the first plane which bisects the inlet channel of the interdigitated air distributor, Fig. 7a, the values of the oxygen mole fraction are the highest and the peak val- ues are found in the vicinity of the air entrance. In the third plane which bisects the outlet channel of the interdigitated air distributor, Fig. 7c, the oxygen mole fractions are the lowest. However, due to the effect of oxygen back-diffusion, the oxygen mole fractions are slightly increased in the region near the air exit. In the second plane which is located halfway between the other two planes, Fig. 7b, intermediate values of the oxygen mole fraction are found. 4.2. Fuel-cell design optimization The optimization procedure described in Section 3.1 yielded the following optimal PEMFC cathode and the interdigitated air distributor geometric parameters: the cathode thickness: 0.000037 m; the channel thickness of the interdigitated air distributor: 0.0005 m (the upper bound); and the channel width of the interdigitated air distributor: 0.000175 m (the upper bound). Fig. 7. Variations of the oxygen mole fraction throughout the PEMFC cathode thickness at three different locations relative to the fuel cell front. Please see text for details.

250 M. Grujicic et al. / Materials Science and Engineering B 108 (2004) 241 252 The results of the optimization analysis presented above show that the optimal design of the cathode side of a PEM fuel cell is associated with the upper bounds of the interdigitated air distributor channel thickness and the channel width. The optimal value of the cathode thickness, on the other hand, is quite low but larger than its lower bound. A brief explanation of these findings is given below. As the cathode thickness increases, the air begins to take the shortest route between the inlet and the outlet and to flow mainly near the cathode/current-collector interface. This causes the thickness of the diffusion boundary layer adjacent to the membrane/cathode interface to increase and, in turn, gives rise to a decrease in the rate of oxygen transport to the cathodic active layer. This causes a decrease in the local current density at the membrane/cathode interface. It is hence justified that the optimal PEMFC cathode design is associated with a low value of the cathode thickness. However, if the gas flow through the cathode is analyzed using an analogy with the gas flow through a pipe, then a decrease in the cathode thickness is equivalent to a decrease in the pipe diameter. Since as the pipe diameter decreases, the resistance to the gas flow increases, one should expect that below a critical cathode thickness, the electric current would begin to decrease with a further decrease in the cathode thickness. Hence, the intermediate optimal value of the cathode thickness ( 0.000037 m) found in the present work, is justified. A similar finding was obtained in our recent two-dimensional analysis of the effect of cathode thickness on the average current density on the membrane/cathode interface [2]. The effect of the interdigitated air distributor channel thickness can also be explained using the analogy with gas flow through a pipe. The pressure drop between the interdigitated air distributor inlet and the outlet is fixed and equal to p. As the channel cross section area increases with an increase in the channel thickness, the pressure drop inside the channel decreases. Consequently, a larger pressure difference exists between the surfaces at which the air enters the cathode and the surfaces at which the air leaves the cathode. The resulting higher air velocity promotes convective oxygen transport to the membrane/cathode interface. However, the effect of the interdigitated air distributor channel width is found to be quite small and if the channel width is changed from the optimal (upper bound) value to the lower bound value the magnitude of the average current density is decreased by only 0.8%. The effect of the interdigitated air distributor channel width can be explained using the same argument as the one used to explain the effect of the interdigitated air distributor channel thickness. However, as discussed above, this effect is found to be relatively small. A careful examination of the oxygen mole fraction and the air velocity fields revealed that as the interdigitated air distributor channel width increases, a larger fraction of the membrane/cathode interface is benefiting from the convective oxygen transport and this is believed to represent the major contribution that the air distributor channel width has on the average current density. This is consistent with the fact that the effect of the air distributor channel width is substantially larger than that associated with the air distributor channel thickness. In fact, changing the air distributor channel width from its upper bound to its lower bound value causes the average current density to decrease by 4.7%. 4.3. Statistical sensitivity analysis Six model parameters (the factors) whose values are associated with the largest uncertainty along with their three levels (one of which corresponds to the reference value) are listed in Table 2. The non-reference levels are arbitrarily selected to be 10% below and 10% above their corresponding reference values. The L 18 (3 6 ) orthogonal matrix [9] whose rows define the 18 finite element computational analyses which are carried out as a part of the statistical sensitivity analysis is given in Table 3. The values 1, 2 or 3 in this table correspond respectively to the three levels of the corresponding factor as defined in Table 2. It should be noted that the reference case corresponds to the analysis 2 in Table 3. The values of the objective function (the averaged current density at the membrane/cathode interface in A/m 2 ) obtained in the 18 analyses are given in the last column in Table 3. The results of the statistical sensitivity analysis are displayed in Table 4. The results presented in this table are obtained in the following way. In the first column, the factors are listed in the decreasing order of the variance ratio, F, (given in the last column of the same table). The difference from the mean of the objective function associated Table 2 PEM fuel cell factors and levels used in the statistical sensitivity analysis Factor Symbol Units Designation Levels 1 2 3 Empirical value k MS Nm 2 kg 1/2 /K 1.75 mol 7/6 A 89.9 10 8 99.9 10 8 109.9 10 8 Thickness of the active layer δ m B 0.9 10 5 1.0 10 5 1.1 10 5 Molar diffusion volume of oxygen v O2 m 3 /mol C 14.9 10 6 16.6 10 6 18.3 10 6 Molar diffusion volume of water v H2 O m 3 /mol D 11.4 10 6 12.7 10 6 14.0 10 6 Molar diffusion volume of nitrogen v N2 m 3 /mol E 16.1 10 6 17.9 10 6 19.7 10 6 Porosity of cathode ε No units F 0.45 0.50 0.55

M. Grujicic et al. / Materials Science and Engineering B 108 (2004) 241 252 251 Table 3 L 18 (3 6 ) Orthogonal matrix used in the statistical sensitivity analysis Analysis number Factors A B C D E F Average current density (A/m 2 ) 1 1 1 1 1 1 1 11,093 2 2 2 2 2 2 2 11,880 3 3 3 3 3 3 3 13,084 4 1 1 2 2 3 3 11,185 5 2 2 3 3 1 1 11,933 6 3 3 1 1 2 2 12,777 7 1 2 1 3 2 3 11,935 8 2 3 2 1 3 1 12,811 9 3 1 3 2 1 2 11,135 10 1 3 3 2 2 1 13,123 11 2 1 1 3 3 2 11,087 12 3 2 2 1 1 3 11,897 13 1 2 3 1 3 2 11,961 14 2 3 1 2 1 3 13,043 15 3 1 2 3 2 1 11,068 16 1 3 2 3 1 2 13,211 17 2 1 3 1 2 3 11,189 18 3 2 1 2 3 1 11,664 Overall mean of the average current density (A/m 2 ) 12,004 factor B and its level 1 is obtained by first calculating the mean of the objective function obtained in the analyses 1, 4, 9, 11, 15 and 17 in which the level 1 of factor B is used. The overall mean of the objective function (12,004 A/m 2, the bottom row, the rightmost column in Table 3) isnext subtracted from the mean associated with the level 1 of factor B to yield the value 0.8781 A/m 2. The same procedure is next used to obtain the remaining values in columns 2 4, Table 4. The sums of squares associated with each factor are listed in column 5, Table 4. They are obtained by first squaring the difference from the mean for each level of each factor (columns 2 4, Table 4) and then by multiplying these with 6 (the number of times each level appears in each column of the orthogonal matrix). The values obtained for different levels of the same factor are next summed to obtain the sum of squares associated with that factor. The number of degrees of freedom associated with each factor (column 7, Table 4) is one less than the number of levels of that factor. The number of degrees of freedom associated with the error is obtained using Eq. (11) to yield: 18 1 6(2) = 5. Since the number of degrees of freedom associated with the error is nonzero, the sum of squares associated with the error is calculated using Eq. (10), where the SS grand term is obtained by summing the squares of the values given in the last column of Table 3, the SS mean term is computed as 18 (12,004 A/m 2 ) 2 and the SS factor term is obtained by summing the sum of squares values associated with the six factors (column 5, Table 4). The percent contribution of each factor and the error to the total sum of the sum of squares is next calculated and listed in column 6, Table 4. The mean sum of squares given in column 8, Table 4 is obtained by dividing the values given in column 5, Table 4 with the number of degrees of freedom, column 7, Table 4. Finally, the variance ratio, F, for each factor (column 9, Table 4) is obtained by dividing its mean sum of squares (column 8, Table 4) by the mean sum of squares associated with the error (=0.0032 A 2 /m 4, Table 4). The results displayed in the last column in Table 3 show that an uncertainty in the value of the thickness of the active layer has by far the largest effect on the predicted average current density at the membrane/cathode interface. The effect of other factors is comparably smaller, but since their values of the variance ratio, F, are either above or near 4, the effect of these factors is also significant. The results displayed in Table 4 show that the levels of the six factors associated with analysis 16 gives rise to the largest deviation from the reference case, analysis 2. To test the robustness of the optimal design of the PEMFC cathode and the interdigitated air distributor discussed in the previous section, the optimization procedure is repeated but for the factor levels corresponding to analysis 16. The optimization results obtained show that the optimal values of the three fuel-cell design parameters are almost identical to the ones for the reference case. This finding suggest that while uncertainties in the values of various model parameters can have a major effect on the predicted average current density at the membrane/cathode interface, the optimal design of a PEM fuel-cell is not significantly affected by such uncertainties (Fig. 8). Therefore, despite some uncertainties in the Table 4 Statistical sensitivity analysis of the optimal design of the PEM fuel cell Factor Difference from mean (A/m 2 ) Sum of squares (A/m 2 ) 2 Level 1 Level 2 Level 3 Percent of sum of squares Number of DOF Mean sum of squares (A/m 2 ) 2 Variance ratio F A 804 137 667 0.0667 0.6064 2 0.0333 10.37 B 8781 1259 10,039 10.7684 97.9406 2 0.5384 167.55 C 711 44 666 0.0570 0.5187 2 0.0285 8.87 D 496 8 488 0.0290 0.2639 2 0.0145 4.51 E 478 89 389 0.0232 0.2114 2 0.0116 3.62 F 556 43 513 0.0344 0.3129 2 0.0172 5.35 Error 0.0161 0.1461 5 0.0032 Total 10.9948 100.00

252 M. Grujicic et al. / Materials Science and Engineering B 108 (2004) 241 252 Fig. 8. Variations of: (a) the current density and (b) the oxygen mole fraction over the membrane/cathode interface in the optimal design of the PEMFC cathode and the interdigitated air distributor. values of a number of model parameters, the optimal design of the PEMFC cathode and the interdigitated air distributor is quite robust. 5. Conclusions Based on the results obtained in the present work, the following conclusions can be drawn: Optimization of the PEMFC cathode and the interdigitated air distributor designs can be carried out by combining a multi-physics model consisting of the continuity, momentum, mass and electric charge conservation equations with a nonlinear constrained optimization algorithm. The optimum design of the cathode side of a PEM fuel cell is found to be associated with the cathode and the interdigitated air distributor geometrical parameters which promote the role of convective oxygen transport to the membrane/cathode interface and reduce the thickness of the boundary diffusion layer at the same interface. The predicted average current density at the membrane/cathode interface of PEM fuel cells is highly dependent on the magnitude of several model parameters associated with the mass transport and the oxygen reduction half-reaction. However, the optimal design of the PEMFC cathode and the interdigitated air distributor is essentially unaffected by a ±10% variation in the values of these parameters. References [1] J.S. Yi, T.V. Nguyen, Multi-component transport in porous electrodes in proton exchange membrane fuel cells using the interdigitated gas distributors, J. Electrochem. Soc. 146 (1999) 38. [2] M. Grujicic, K.M. Chittajallu, Design and optimization of polymer electrolyte membrane (PEM) fuel cells, Appl. Surf. Sci., 2003, submitted for publication. [3] M. Grujicic, K.M. Chittajallu, Geometrical optimization of the cathode in polymer electrolyte membrane (PEM) fuel cells, Appl. Surf. Sci., 2003, submitted for publication. [4] Three-dimensional Model of a Proton Exchange Membrane Fuel Cell Cathode, The Chemical Engineering Module, FEMLAB 2.3a, COMSOL Inc., Burlington, MA, 2003, pp. 2-218 2-230. [5] http://www.comsol.com, FEMLAB 2.3a, COMSOL Inc., Burlington, MA, 2003. [6] MATLAB, sixth ed., The Language of Technical Computing, The MathWorks Inc., Natick, MA, 2000. [7] M.C. Biggs, Constrained minimization using recursive quadratic programming, in: L.C.W. Dixon, G.P. Szergo (Eds.), Towards Global Optimization, North-Holland, 1975, pp. 341 349. [8] M.S. Phadke, Quality Engineering Using Robust Design, Prentice Hall, Englewood Cliffs, NJ, 1989. [9] P.J. Ross, Taguchi Techniques for Quality Engineering: Loss Function, Orthogonal Experiments, Parameter and Tolerance Design, second ed., McGraw-Hill, New York, 1996.