Optimization of the cathode geometry in polymer electrolyte membrane (PEM) fuel cells

Transcription

1 Chemical Engineering Science 59 (2004) Optimization of the cathode geometry in polymer electrolyte membrane (PEM) fuel cells M. Grujicic, K.M. Chittajallu Department of Mechanical Engineering, Clemson University, 241 Fluor Daniel, Clemson SC , USA Available online 26 August 2004 Abstract A nonlinear constrained optimization procedure is used in the cathode design in order to maximize the average current density at a fixed voltage in a polymer electrolyte membrane (PEM) fuel cell with interdigitated fuel/air distributors. The operation of the PEM fuel cell is studied using a steady-state, two-phase, two-dimensional electro-chemical model. The following geometrical parameters of the cathode are considered: the thickness, and length per one shoulder of the interdigitated air distributor and the length of the shoulder. The optimization results obtained show that within manufacturability controlled lower and the space-limitation controlled upper bounds of these parameters, the optimal-cathode design corresponds to the lower bounds in the cathode length per one shoulder of the interdigitated air distributor and the fraction of the length associated with the shoulders and at a low (but larger than the lower bound) value of the cathode thickness. These findings are explained using an analogy with the effect of pipe dimensions on the fluid flow through a pipe and by considering the role of forced convection on the oxygen transport to the membrane/cathode interface Elsevier Ltd. All rights reserved. PACS: a; Gh Keywords: Polymer electrolyte membrane (PEM) fuel cells; Optimization 1. Introduction Fuel cells are electrical energy generation device in which the efficiency of converting fuel to power is typically two to three times that in an internal combustion engine. A fuel cell works on the principle of separation of the oxidation of the fuel (e.g. hydrogen gas) and reduction of an oxidant (e.g. oxygen gas from the air), while heat and water are typical byproducts of the fuel-cell operation. As the hydrogen gas flows into the fuel cell on the anode side of a fuel cell, a platinum catalyst facilitates oxidation of the hydrogen gas which produces protons (hydrogen ions) and electrons, Fig. 1. The protons diffuse through a membrane (the center of the fuel cell which separates the anode and the cathode) and, again with the helpof a platinum catalyst, combine Corresponding author. Tel.: ; fax: address: mica.grujicic@ces.clemson.edu (M. Grujicic). with oxygen and electrons on the cathode side, producing water. The electrons, which cannot pass through the electrically insulating membrane, flow from the anode to the cathode through an external electrical circuit containing a motor or another electrical load, which consumes the power generated by the cell. The resulting voltage from one single fuel cell is typically around 1.0 V. This voltage can be 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 generally classified according to the type of membrane (polymer electrolyte membrane 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 (PEMFC). The polymer electrolyte membrane is a solid, organic polymer, usually poly [perfluorosulfonic] acid. The most frequently used PEM is made of Nafion TM produced /$ - see front matter 2004 Elsevier Ltd. All rights reserved. doi: /j.ces

2 5884 M. Grujicic, K.M. Chittajallu / Chemical Engineering Science 59 (2004) e - H 2 INTERDIGITATED FUEL DISTRIBUTOR H 2 H 2 H 2 e - ANODE H + POLYMER ELECTROLYTE MEMBRANE CATHODE O 2 O 2 O 2 INTERDIGITATED AIR DISTRIBUTOR O 2 H H + 2 = 2 + 2e O2 + 4H + 4e = 2H 2O ( liquid) Oxidation Reduction Half-Reaction Half-Reaction e - e - Load Fig. 1. A schematic of a polymer electrolyte membrane (PEM) fuel cell. by DuPont, whose chemical structure consists of three regions: (1) a Teflon-like, fluorocarbon backbone containing hundreds of repeating CF 2 CF CF 2 units; (2) O CF 2 CF O CF 2 CF 2 side chains which connect the molecular backbone to the third region; and (3) ionic clusters consisting of sulfonic acid ions, SO 3 H+. The negative SO 3 ions are permanently attached to the side chains and are immobile. On the other hand, when the membrane is hydrated by absorbing water, the hydrogen ions combine with water molecules to form hydronium ions which are quite mobile. Hydronium ions hopfrom one SO 3 site to another within the membrane and, thus, give rise to the diffusion of protons making the hydrated solid electrolytes like Nafion TM excellent conductors of hydrogen ions. The anode and the cathode (the electrodes) in a PEM fuel cell are porous and made of an electrically conductive material, 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 heavy 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 layer, respectively. The electrodes in PEM fuel cells 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 cathode and for the minimum transport e - e - resistance 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 % increase in the fuel-cell performance is typically obtained as a result of the use of serpentine or interdigitated fuel/air distributors (Yi and Nguyen, 1999). 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 ( K) operation, a pollution-free character, and a relatively simple design, the 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 traditional 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 fuelcell development. Over the last decades a number of fuelcell models have been developed. Some of these models are single-phase (e.g. Yi and Nguyen, 1999; FEMLAB 2.3a, 2003a) while the others are two-phase (e.g. He et al., 2000), 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. Yi and Nguyen, 1999; He et al., 2000) while the others deal with the entire fuel cell (e.g. FEMLAB 2.3a, 2003a). Most of the models like the ones cited above, and the more recent ones (Kulikovsky et al., 1999; Wang et al., 2001; Natarajan and Nguyen, 2001; Grujicic et al., 2004; Grujicic and Chittajallu, 2004), are used to carry out parametric studies of the effect of various fuel-cell design parameter (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 combine the two-phase twodimensional PEM fuel-cell model as presented in He et al. (2000), with a mechanical design optimization and design robustness analysis in order to suggest the optimum fuel-cell design. The organization of the paper is as following: in Section 2, the two-phase, two-dimensional model for a PEM fuel cell proposed by He et al. (2000) is briefly discussed and a solution method for the resulting set of partial differential equations presented. 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

3 M. Grujicic, K.M. Chittajallu / Chemical Engineering Science 59 (2004) and discussed in Section 4. The main conclusions resulting from the present work are summarized in Section The model As indicated in Fig. 1, a 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 listed in Fig. 1. Electrons liberated in the anodic active layer via the oxidation half-reaction 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, 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. It is generally recognized that due to a slow kinetics of the oxygen reduction half-reaction, the cathode is perhaps one of the most critical components of the PEM fuel cells. Therefore, the present model focuses on the cathode side of a PEM fuel cell Assumptions and simplifications In this section, the two-phase, two-dimensional steadystate model of the cathode side of a PEM fuel cell proposed by He et al. (2000) is presented. The model is developed under the following simplifications and assumptions: The computational domain denoted with dashed lines in Fig. 1 is located within the porous cathode. The membrane/cathode interface is considered as a zerothickness cathodic active layer. A portion of the right edge of the cathode which is in contact with the shoulder of the interdigitated air distributor is designated as the cathode current collector surface. Dry (vapor-free) air of a fixed chemical composition and fixed pressure is supplied at the cathode inlet. The liquid water present in the fuel supplied to the anode is transported by electro-osmosis through the membrane and enters the porous cathode at the membrane/cathode interface. In addition, as indicated in Fig. 1, water is produced only as the liquid phase by the oxygen reduction half-reaction. However, evaporation of the liquid water creates water vapor which is then carried by the air flowing through 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. The cathode is considered as homogeneous media in which the porosity and permeability are distributed uniformly. Within the cathode s pores, the gas and the liquid phase are considered as continuous phases and, hence, their momentum conservation equations can be represented using the Darcy s law. The electro-chemical oxygen-reduction half-reaction taking place in the cathodic active layer is assumed to be represented by the Tafel equation which relates the local current density, I, with the local oxygen concentration at the membrane/cathode interface, C g y H2 O, and the electrode overpotential, η, as ε g C g ( ) kf I = I 0 exp ε 0 C 0,ref RT η. (1) The mass flux of oxygen and a portion of the mass flux of liquid water at the membrane/cathode interface are related to the local current density. The second component to the mass flux of liquid water at the membrane/cathode interface arises from the liquid water crossing from the membrane into the cathode The dependent variables The following dependent variables are used in the present model: (a) The gas-phase pressure, P; (b) the oxygen mole fraction in the gas phase, y O2 ; (c) the water vapor mole fraction in the gas phase, y H2 O; and (d) the saturation level of the liquid water, i.e. the fraction of the cathode porosity occupied by the liquid phase, s The governing equations The governing equations for the cathode are given in Fig. 2 and they include: (a) A gas-phase continuity equation; (b) an oxygen mass balance equation; (c) a water vapor mass balance equation; and (d) a liquid water mass balance equation The boundary conditions The boundary conditions used are also given in Fig. 2 and can be briefly summarized as following: (a) The pressure and the chemical composition of the O 2 + N 2 + H 2 O air and a (zero) saturation level of the liquid water are prescribed at the cathode input. (b) The pressure and zero-gradients in the oxygen, water vapor and the saturation level are defined at the cathode output. (c) At the cathode/shoulder interface, zero-gradient conditions are prescribed for all four dependent variables. (d) At the topand the bottom portion of a cathode segment associated with one shoulder of the interdigitated air distributor, the symmetry (zero-gradient)

4 5886 M. Grujicic, K.M. Chittajallu / Chemical Engineering Science 59 (2004) Fig. 2. Dependent variables, governing equations and boundary conditions for the cathode in a PEM fuel cell. Please see the text for explanation of the symbols. boundary conditions are applied for all four dependent variables. (e) At the membrane/cathodic interface, zero-gradient boundary conditions are applied for the pressure gradient and for the mole fraction of the water vapor while the oxygen flux and the liquid water flux are related with the cathodic current density as indicated in Fig Closure relations To express the governing equations in terms of the dependent variables, the following closure relations as originally proposed by Yi and Nguyen (1999) are used: The relationshipbetween the molar concentration of the gas phase and the pressure is defined as C g = P RT. (2) The water vapor source/sink term, r H2 O, is defined by the following interfacial rate-transfer equation: ε g y H2 O r H2 O=k c RT ε 0 sρ l H +k 2 O v M H2 O ( yh2 OP P sat H 2 O) q ( yh2 OP P sat H 2 O) (1 q), (3) where q is a condensation/evaporation switching function defined as { = 1 for yh2 OP P sat q H 2 O > 0, = 0 for y H2 OP PH sat 2 O < 0. (4) The first term on the right-hand side of Eq. (3) represents the condensation rate while the second-term represents the evaporation rate. The effective binary diffusivity of a gaseous component in the porous cathode, Di e(i = O 2 N 2, H 2 O N 2 ) is related to the corresponding binary diffusion coefficients, D i,r as ( ) T bi Di e = D i,r [ε 0 (1 s)] 1.5. (5) T i,r

5 M. Grujicic, K.M. Chittajallu / Chemical Engineering Science 59 (2004) Table 1 General parameters used for modeling the PEM fuel cell Parameter Symbol SI units Value Faraday s constant F As/mole 96,487 Universal gas constant R J/mole/K Temperature T K 333 Atmospheric (reference) pressure P ref Pa Reference temperature T ref K 298 The gas-phase and the liquid phase velocities are related to the respective pressure gradients via the Darcy s law as v g = K 0(1 s) P (6) μ and v l = Kl μ l P l. (7) The pressures of the liquid and the gas are related by the capillary pressure as P l = P P c. (8) After application of the gradient operator to Eq. (8) and multiplication with K l /μ l, and rearranging, one obtains v l = fv g D c s, (9) where the gas/liquid interfacial drag coefficient, f, is defined as f = Kl μ g μ l K g (10) while the capillary diffusion coefficient, D c,as D c = Kl μ l dp c ds. (11) While both f and D c are assumed to be constant in the present work, sensitivity of the optimal design of the fuel cell to variations in these two parameters is analyzed in Section 3.2. Eq. (9) shows that the transport of the liquid water within the porous cathode is controlled by an interfacial shear force exerted by the gas flow and by a saturation-gradient driven capillary force. The values of all the model parameters used in the present work are listed in Tables 1 and Computational method The stationary (steady-state), nonlinear two-dimensional system of governing partial differential equations (discussed in Section 2.3 and in Fig. 2) subjected to the boundary conditions (discussed in Section 2.4 and in Fig. 2) are implemented in the commercial mathematical package FEMLAB (FEMLAB 2.3a, 2003b) and solved (for the dependent variables discussed in Section 2.2 and in Fig. 2) 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 (MATLAB, 2000). As a result, the models developed in the FEMLAB can be saved as MATLAB programs for parametric studies or iterative design optimization. 3. Optimization and design robustness There are many cathode parameters which affect the performance of PEM fuel cells. Some of these parameters such as gas and liquid permeabilities and effective gas and liquid diffusivities are controlled by the porous microstructure of the cathode 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 fuel cell optimization procedure described below. Instead, the following geometrical parameters of the cathode will be considered: (a) the cathode thickness ( m); (b) the cathode length per one shoulder of the interdigitated air distributor (0.002 m); and (c) the fraction of cathode length associated with the shoulders of the interdigitated gas distributor (0.5). The numbers given within the parentheses correspond to the values of the three design parameters in the initial (reference) design of the PEMFC cathode. A schematic of the PEMFC cathode is given in Fig. 3 to explain the three design parameters (denoted as x(i),i=1 3) defined above. The objective function f [x(1), x(2), x(3)] is next defined as the average current density at a typical fuel cell voltage of 0.7 V. The electric current per unit fuel-cell depth is determined as an integral of the current density over a distance equal to the cathode length associated with one shoulder of the interdigitated air supply system along the cathode/membrane interface. 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 m x(1) m, m x(2) m, 0.25 x(3) The upper limits for the three design parameters are selected based on the parametric study of PEM fuel cells

6 5888 M. Grujicic, K.M. Chittajallu / Chemical Engineering Science 59 (2004) Table 2 Reference-case values of the PEM fuel cell parameters at 333 K (60 C) Parameter Symbol SI units Value Gas permeability of the cathode K g m Dry porosity of the cathode ε Inlet mole fraction of oxygen y O2,in 0.21 Inlet mole fraction of water y H2 O,in 0.0 Inlet mole fraction of nitrogen y N2,in 0.79 Inlet pressure P in Pa Outlet pressure P out Pa Gas viscosity (at K) μ g kg/m/s Binary diffusivity of oxygen D O2 m 2 /s (at T O2,r = 273.0K) Binary diffusivity of water D H2 O m 2 /s (at T H2 O,r = 307 K) Capillary diffusion coefficient of liquid water D c m 2 /s Interfacial drag coefficient f Condensation rate constant k c s Vaporization rate constant k v Pa 1 /s Exchange current density (at K) I 0 A/m Transfer coefficient of oxygen reduction reaction k 0.5 Cathode overpotential η V 0.5 Net Water transport coefficient of the membrane α 0.5 Saturation pressure of water P sat H 2 O Pa Exponent used to calculate diffusion for oxygen b O Exponent used to calculate diffusion for water b H2 O Density of liquid water ρ l H 2 O kg/m Molecular weight of water M H2 O kg/mol L h CATHODE Design Parameters x (1) = h x (2) = L x (3) = l / L Fig. 3. Definition of the design parameters used in the cathode optimization procedure. carried out in the work of He et al. (2000). The lower limits for the three design parameters are selected based on the consideration of minimal feature sizes which can be attained INLET SHOULDER OUTLET l using the current PEMFC cathode and the interdigitated airdistributor manufacturing processes Optimization The optimization problem formulated above is solved using the MATLAB optimization toolbox (MATLAB, 2000) 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), (12) 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.lband UB are vectors containing the lower and the upper bounds of the design valuables and x 0 is a vector containing initial values of the design parameters. The MATLAB fmincon()

7 M. Grujicic, K.M. Chittajallu / Chemical Engineering Science 59 (2004) function implements the Sequential Quadratic Programming (SQP) method (Biggs, 1975) within which the original problem is approximated with a quadratic programming subproblem which is then solved successively until convergence is achieved for the original problem. The method has the advantage of finding the optimum design from an arbitrary initial design point using 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 design point, the method can find only a local minimum in the vicinity of the initial design point. Cell Potential, V Experimental [3] Uncorrected Corrected 3.2. Statistical sensitivity analysis Once the optimum PEMFC cathode design is obtained using the optimization procedure described above, it is important to determine how sensitive is the optimal design to potential variation in the values of a number of model parameters (e.g. capillary diffusion coefficient, interfacial drag coefficient, etc.) whose values are associated with considerable uncertainty. Within the mechanical design community, such parameters are generally referred to as factors and this term will be adopted here. The sensitivity of the optimal design to potential variations in the factors will be analyzed here using the statistical sensitivity analysis method (Phadke, 1989). Application of this model to PEM fuel cell design based on a steady-state, single-phase two-dimensional method has been recently discussed by us (Grujicic et al., 2004; Grujicic and Chittajallu, 2004) and, hence, will be reviewed here only briefly. The main steps associated with the statistical sensitivity analysis can be summarized as follows: Identify the factors, their range of variation and for each factor select (typically) 2 4 values (generally referred to as levels) within its range. Identify the analyses (finite element computational analyses in the present case) which need to be carried out in order to quantify the effect of variations of the factors on the optimal design. A factorial design approach in which all possible combinations of the factor levels are used is generally too computationally intensive, and a design of experiments method (e.g. the orthogonal matrix method, Ross, 1996) can be used to reduce the number of analyses which have to be carried out. Sensitivity of the optimal design to variations in the factors is next quantified by applying the analysis of variance (ANOVA) to the results of the analyses carried out. The finite outcome of the ANOVA analysis are the values of the variance ratio, F, for each factor. A value for F of less than one implies that the effect of the corresponding factor is statistically insignificant. A value for F above four Average Current Density, A/m 2 Fig. 4. Comparison of the predicted polarization curve (before and after the corrections) with its experimental counterpart (He et al., 2000). suggests that the design is quite sensitive to the variations of the factor at hand. 4. Results and discussion 4.1. The reference case The voltage vs. the current density polarization curve for the reference design of the PEMFC cathode is shown in Fig. 4 (the curve denoted as Uncorrected ). The polarization curve is next corrected to take into account ohmic losses associated with the cathode catalyst layer and the membrane as well as the anode overpotential. The data used for this correction are listed in Table 3. The resulting polarization curve (denoted as Corrected ) is displayed in Fig. 4 along with the experimental data of He et al. (2000). The agreement between the corrected polarization curve and its experimental counterpart can be judged as quite good. The variation of the current density along the cathode/membrane interface in the cathode-length direction as a function of the distance from the oxygen outlet at the cell voltage of 0.7V is shown in Fig. 5 (the curve denoted as Reference ). The results displayed in Fig. 5 are as expected and show that the highest current densities are obtained near the oxygen inlet where the concentration of the oxygen in the air (and hence at the cathode/membrane interface) is the highest. Contour plots for the mole fraction of oxygen and water vapor, the liquid-water saturation level and the pressure for the reference PEMFC cathode design are displayed in Figs. 6(a) (d). The most important finding shown by the results displayed in Figs. 6(a) (d) is that oxygen concentration along the cathode/membrane interface rapidly decreases

8 5890 M. Grujicic, K.M. Chittajallu / Chemical Engineering Science 59 (2004) Table 3 Parameters used for correction of the polarization curve Component Type of overpotential Thickness (m) Conductivity (S/m) Overpotential (V)/ current density (A/m2 ) Cathode catalyst layer Membrane Anode Ohmic Ohmic Activation N/A 11 N/A Current D ensity, A/m Optimal Reference Distance from the Center of Oxygen Outlet, m Fig. 5. Current density distribution along the membrane/cathode interface in the cathode-length direction in the reference and the optimal PEMFC cathode designs with a distance from the oxygen inlet. In addition, the liquidwater saturation in the low oxygen-concentration region is quite high increasing the possibility for cathode flooding. These findings are in excellent agreement with the result displayed in Fig. 5 which shows that the current density is quite low in the region of low oxygen concentration. The vector plot of the air velocity for the reference PEMFC cathode design is shown in Fig. 7(a). It is seen that the component of the air velocity normal to the cathode/membrane interface which gives rise to an important contribution of the convective mass transport to this interface is small and that the air flow, for the most part, is parallel with the interface. The parameters listed in Table 2, in particular, the condensation rate constant, kc, the vaporization rate constant, kv, the interfacial drag coefficient, f, and the capillary diffusion coefficient of liquid water, Dc, are not known with a high certainty. It is hence critical to assess sensitivity of the present model to potential variations in these four parameters. The values for the condensation rate constant, kc, and for the vaporization rate constant, kv, were taken from He et al. (2000) in which they were estimated in such a way that the rates of both reactions under the reference operat (a) (b) (c) (d) Fig. 6. Contour plots for the: (a) mole fraction of oxygen; (b) mole fraction of water vapor; (c) saturation level of liquid water; and (d) pressure (atm) in the reference PEMFC cathode design. ing conditions for a PEMFC are consistent with experimental observations. The value for the interfacial drag coefficient, f, have also been taken from He et al. (2000) where they were calculated using Eq. (10) and the gas viscosity at 333 K, g = kg/m/s, the viscosity of water, l = kg/m/s, and the permeabilities for the gas and the water, K g = m2 and K l = m2. The value for the capillary diffusion coefficient of liquid water, Dc, was also taken from He et al. (2000), were it was assessed in such a way that the model predicted polarization curve for the reference conditions of a PEMFC analyzed agrees with its experimental counterpart. Since these four parameters are subjected to the largest uncertainty

9 M. Grujicic, K.M. Chittajallu / Chemical Engineering Science 59 (2004) (a) (b) Fig. 7. Vector plot of the gas velocity in the: (a) reference and (b) optimal PEMFC cathode designs. (a) (b) (c) (d) regarding their values, a brief sensitivity analysis is carried out. Within this analysis, each parameter mentioned above is allowed to vary by ±30% from its values listed in Table 2. The results obtained show that, the largest deviation of the current density in the polarization curve relative to the reference case is approximately 7% with the root mean square error being 3.2%. Based on these findings, it was concluded that the PEMFC model proposed in the present work is not very sensitive to variations in the four model parameters listed above, at least within the range of values for these parameters explored Fuel-cell design optimization The optimization procedure described in Section 3.1 resulted the following optimal fuel-cell design parameters. (a) the cathode thickness m; (b) the cathode length per one shoulder segment of the interdigitated air distributor m (the lower bound); and (c) the fraction of cathode length associated with the shoulder of the interdigitated gas distributor 0.25 (the lower bound). The variation in the current density along the cathode/membrane interface for the optimal PEMFC cathode Fig. 8. Contour plots for the: (a) mole fraction of oxygen; (b) mole fraction of water vapor; (c) saturation level of liquid water; and (d) pressure (atm) in the optimal PEMFC cathode design. design is compared in Fig. 5 with its reference PEMFC cathode design counterpart. It is seen that substantially higher current density levels are attained in the case of the optimal PEMFC cathode design. In addition, it is seen that only about one half of the available membrane/cathode surface area is utilized for the reduction of oxygen. This implies that major temperature variations may exist over the membrane/cathode interface resulting in potential hot spots, which may endanger the integrity of the polymer electrolyte membrane. Furthermore, since many parameters used in the present model depend on temperature, it is evident that in order to properly optimize the design of a PEMFC, one must include the effects of temperature. A more detailed discussion of the potential effect of temperature on the optimum design of a PEMFC is given in Section 4.4. Contour plots for the mole fraction of oxygen and water vapor, the liquid-water saturation level and the pressure for the optimal PEMFC cathode design are displayed in Figs. 8(a) (d). To improve clarity of the contour plots, the horizontal axis is scaled upby a factor of 4. A comparison of the results displayed in Figs. 8(a) (d) with the corresponding results displayed in Figs. 6(a) (d) shows that oxygen concentration along the cathode/membrane interface is significantly larger for the optimal PEMFC cathode design. In

10 5892 M. Grujicic, K.M. Chittajallu / Chemical Engineering Science 59 (2004) addition, the levels of the liquid-water saturation are lower for the optimal PEMFC cathode design reducing the possibility for cathode flooding. These findings are in excellent agreement with the result displayed in Fig. 5 which show that the current density levels are significantly higher for the optimal PEMFC cathode design. The vector plot of the air velocity for the optimal PEMFC cathode design is shown in Fig. 7(b). A comparison of the vector plots displayed in Figs. 7(a) and (b) shows that the component of the air velocity normal to the cathode/membrane interface which gives rise to an important contribution of the convective mass transport to this interface is quite large for the optimal PEMFC cathode design in the region near the oxygen inlet. Furthermore, a comparison of the results displayed in Fig. 7(b) and in Fig. 5 (curve labeled Optimal ) reveals the important role the air velocity normal to the cathode/membrane interface plays in promoting the convective transport of oxygen to the membrane/cathode interface and, hence, in increasing the rate of oxygen reduction. Consequently, the largest current densities along the membrane/cathode interface are found in the region facing the oxygen inlet. The results of the optimization analysis presented above show that the optimal PEMFC cathode design is associated with the lower bounds of the cathode length per one shoulder of the interdigitated air distributor and the fraction of the cathode length associated with shoulders of the interdigitated gas distributor. 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/shoulder 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. 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 ( m) found in the present work, is justified. The effect of cathode length per one shoulder of the interdigitated air distributor can also be explained using the gas flow through a pipe analogy. Hence, a reduction of the cathode length per one shoulder of the interdigitated gas distributor is equivalent to a reduction in the pipe length and, due to a fixed pressure drop over a shorter pipe length, a higher gas-flow rate is attained. Consequently, in the absence of a consideration of electronic resistance of the gas diffusion layer (as is the case in the present work), the optimal PEMFC cathode design is expected to be found at the lower bound of the cathode length per one shoulder of the interdigitated gas distributor. The effect of the fraction of the cathode length associated with shoulders of the interdigitated gas distributor appears to be closely related to the effect of the air flow normal to the membrane/cathode interface near the oxygen inlet, Fig. 5 (curve labeled Optimal ) and Fig. 7(b). This was confirmed by showing that further increases in the average current density can be obtained by holding the fraction of the cathode length associated with shoulders of the interdigitated gas distributor fixed while increasing the size of the oxygen inlet at the expense of the size of the oxygen outlet Statistical sensitivity analysis Six PEMFC cathode-based model parameters (the factors) whose values are associated with the largest uncertainty along with their three levels (level 2 corresponds to the reference value) are listed in Table 4. The non-reference levels are arbitrarily selected to be 10% below (level 1) and 10% above (level 3) the corresponding reference values. The L 18 (3 6 ) orthogonal matrix (Ross, 1996) whose rows define 18 finite element computational analyses which are carried out as a part of the statistical sensitivity analysis is given in Table 5. The values 1, 2 or 3 in this table correspond respectively to the three levels of the corresponding factor as defined in Table 4. It should be noted that the reference case corresponds to analysis 2 in Table 5. The values of the objective function (the averaged current density in A/m 2 ) obtained in the 18 analyses are given in next to the last column in Table 5. The results listed in the next to the last column in Table 5 are next treated using the statistical sensitivity analysis following the same steps as the ones described in our recent work (Grujicic et al., 2004; Grujicic and Chittajallu, 2004) and, hence, the details of this procedure will not be discussed here. The variance ratio, F, results obtained are listed in the last column in Table 5. The results displayed in the last column in Table 5 show that an uncertainty in the value of the exchange current density has the largest effect on the predicted average current density at the cell voltage of 0.7V. The effect of the net water transport coefficient from the membrane is also significant since the value of its variance ratio, F, is above 4. The effect of the remaining factors can be judged as statistically not very significant. The results displayed in the last column in Table 5 show that the levels of the six factors associated with analyses 12 and 18 give rise to the largest deviation from the reference case, analysis 2. To test the robustness of the optimal design of the PEMFC cathode discussed in the previous section, the optimization procedure is repeated but for the factor levels corresponding to analyses 12 and 18. The

11 M. Grujicic, K.M. Chittajallu / Chemical Engineering Science 59 (2004) Table 4 PEM fuel cell factors and levels used in the statistical sensitivity analysis Factor Symbol Units Designation Levels F Capillary diffusion D c m 2 /s A coefficient of liquid water Interfacial drag coefficient f B Condensation rate constant k c s 1 C Vaporization rate constant k v Pa 1 /s D Exchange current density (at K) I 0 A/m 2 E Net water transport coefficient of the membrane α F coefficient of the membrane Table 5 L 18 (3 6 ) orthogonal matrix used in the statistical sensitivity analysis Analysis number Factors Average current density A B C D E F (A/m 2 ) Overall mean of the average current density, A/m optimization results obtained show that the optimal values of the three PEMFC cathode design parameters are essentially identical to the ones associated with the reference case. This finding suggest that while uncertainties in the values of various model parameters can have a significant effect on the predicted average current density, the optimal design of the fuel-cell is not greatly affected by such uncertainties The effect of temperature on the optimal PEMFC design As discussed in Section 4.2, significant temperature variations may exist over the membrane/cathode interface in the optimal design of the PEMFC. Such variations may create hot spots endangering the integrity of the polymer electrolyte membrane. In addition, since many parameters used in the present model depend on temperature, it is clear that one consider the effects of temperature when analyzing the optimal PEMFC design. While a rigorous treatment of temperature, as an operating parameter, which includes the use of an energy conservation equation and temperature-dependent model parameters is beyond the scope of the present work, a simplified approach is utilized to test the sensitivity of the optimal design of the PEMFC to potential variations in temperature of along the length of the fuel cell (the x-axis in Fig. 5). In accordance with the polarization curve for the optimal design shown in Fig. 5, the upper half of the length of the fuel cell is associated with a high level of the current density and, hence, will be generally subjected to higher temperatures. To assess the effect of these higher temperatures on the morphology and the performance of the optimal PEMFC design,

12 5894 M. Grujicic, K.M. Chittajallu / Chemical Engineering Science 59 (2004) K 333K/353K curve of the PEMFC while they are not expected to have a major effect on the optimal design of the PEMFC. Current Density, A/m K Distance from the Center of Oxygen Outlet, m Fig. 9. The effect of temperature on the current density distribution along the membrane/cathode interface in the cathode-length direction in the optimal PEMFC cathode designs. Please see Section 4.4 for an explanation of the labels. the following two approaches have been explored: (a) the temperature of the fuel cell is assumed to be uniform, but is raised to 353 K (80 C) and; (b) the bottom portion of the fuel cell is kept at 333 K (60 C) while the topportion of the fuel cell is raised to 353 K (80 C). A maximum temperature level of 353 K (80 C) is selected since the standard fuel cell cooling technologies are capable of maintaining the temperature below this value (e.g. Grujicic and Chittajallu, 2004). All temperature-dependent parameters listed in Table 2 are then recalculated at 353 K (80 C) and the PEMFC design optimization procedure repeated. The optimization results obtained are identical to those reported in Section 4.2, i.e.: (d) the cathode thickness m, (e) the cathode length per one shoulder segment of the interdigitated air distributor m (the lower bound); and (f) the fraction of cathode length associated with the shoulder of the interdigitated gas distributor 0.25 (the lower bound). The polarization curves, however, are found to be different than the one obtained in the case of model parameters at 333 K (60 C). This is depicted in Fig. 9. It is seen that both an increase in temperature in the upper half of the fuel cell length, the curve labeled 333 K/353 K, and the one where the temperature of the entire fuel cell length is increased to 353 K, the curve labeled 353 K, lead to higher current density levels. Nevertheless, the variation of the current density along the fuel cell length is quite similar in all three cases. All these finding suggest that, at least within the simplified approach utilized here, temperature variations along the membrane/cathode interface may affect the polarization 5. Conclusions Based on the results obtained in the present work, the following conclusions can be drawn: (a) Optimization of the PEMFC cathode design can be carried out by combining a multiphysics model consisting of the continuity, momentum and mass conservation equations with a nonlinear constrained optimization algorithm. (b) The optimum PEMFC cathode design is found to be associated with the cathode 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 while lowering the possibility for cathode flooding. (c) The predicted electrical response of PEM fuel cells is highly dependent on the magnitude of several parameters associated with the oxygen transport and the oxygen reduction half-reaction. However, the optimal design of the PEMFC cathode is essentially unaffected by a +/ 10% variation in the values of these parameters. Notation C total molar concentration of the gas phase (mole/m 3 ) Di e effective diffusivity of component i in the porous cathode (m 2 /s) ε porosity of the cathode r gas/liquid interfacial mass-transfer rate (mole/m 3 /s) I local current density (A/m 2 ) P pressure (Pa) s saturation level of liquid water v velocity (m/s) y mole fraction Subscripts c capillary quantity H 2 O water related quantity in inlet quantity 0 dry electrode quantity O 2 oxygen related quantity out outlet quantity r binary diffusion coefficient reference condition ref Tafel-equation reference state

13 M. Grujicic, K.M. Chittajallu / Chemical Engineering Science 59 (2004) Superscripts e effective quantity g gas-state quantity l liquid-state quantity sat saturation-state quantity The remaining quantities are defined in Tables 1 and 2 (He et al., 2000). References Biggs, M.C., In: Dixon, L.C.W., Szergo, G.P. (Eds.), Constrained minimization using recursive quadratic programming, Towards Global Optimization. North-Holland, Amsterdam, pp FEMLAB 2.3a, 2003a. The Proton Exchange Membrane Fuel Cell. The Chemical Engineering Module. COMSOL Inc., Burlington, MA FEMLAB 2.3a, 2003b. COMSOL Inc., Burlington, MA 01803, Grujicic, M., Chittajallu, K.M., Design and optimization of polymer electrolyte membrane (PEM) fuel cells. Applied Surface Science 227, 56. Grujicic, M., Zhao, C.L., Chittajallu, K.M., Ochterbeck, J.M., Cathode and interdigitated air distributor optimization in polymer electrolyte membrane (PEM) fuel cells. Materials Science and Engineering B 108, 241. He, W., Yi, J.S., Nguyen, T.V., Two-phase flow model of the cathode of PEM fuel cells using interdigitated flow fields. A.I.Ch.E. Journal 46, Kulikovsky, A.A., Divisek, J., Kornyshev, A.A., Modeling the cathode compartment of polymer electrolyte fuel cells: dead and active reaction zones. Journal of Electrochemical Society 146, MATLAB, The Language of Technical Computing. The MathWorks Inc., 24 Prime Park Way, Natick, MA, Natarajan, D., Nguyen, T.V., A two-dimensional, two-phase, multicomponent, transient model for the cathode of a proton exchange membrane fuel cell using conventional gas distributors. Journal of Electrochemical Society 148, A1324. Phadke, M.S., Quality Engineering Using Robust Design. Prentice- Hall, Englewood Cliffs, NJ. Ross, P.J., Taguchi Techniques for Quality Engineering: Loss Function, Orthogonal Experiments, Parameter and Tolerance Design. second ed. McGraw-Hill, New York. Wang, Z.H., Wang, C.Y., Chen, K.S., Two-phase flow and transport in the air cathode of proton exchange membrane fuel cells. Journal of Power Sources 94, 40. Yi, J.S., Nguyen, T.V., Multi-component transport in porous electrodes in proton exchange membrane fuel cells using the interdigitated gas distributors. Journal of Electrochemical Society 146, 38.