Analytical Pore Scale Modeling of the Reactive Regions of Polymer Electrolyte Fuel Cells L. Pisani, z M. Valentini, and G. Murgia Center for Advanced Studies, Research and Development in Sardinia, 09010 Uta (Cagliari), Sardinia, Italy This paper analyzes the effects of the catalyst layer porous structure on the performances of polymer electrolyte membrane fuel cells. Comparing the characteristic lengths of the porous structure with the characteristic lengths of the diffusion phenomena shows that the oxygen and hydrogen concentrations in the electrolyte phase change significantly at the pore scale level; therefore, the related diffusion phenomena need a nonhomogeneous description. These rapidly varying concentrations are coupled to the cell potentials through the reaction rate expression, i.e., the Butler-Volmer equation. Thus, to employ a macrohomogeneous description of the fuel cell without loss of accuracy, it is necessary to find an effective expression for the reaction rate which does not depend explicitly on the rapidly varying concentrations. This is done here through an analytical averaging procedure and results in an effective Butler-Volmer expression that includes implicitly the effects of nonhomogeneity of the porous structure. This expression is compared with the ordinary Butler-Volmer expression and with the agglomerate models in the literature. The former turns out to be valid only in the limit of low current densities, and the latter only in the high porosity limit. Finally, the effective Butler-Volmer expression is inserted in the framework of macrohomogeneous models. From the analysis of the model results, one can conclude that the effects of the porous structure on the cell performances are crucial for the correct description of the cell concentration polarization and the estimation of the effective Tafel slope at high current densities. © 2003 The Electrochemical Society. DOI: 10.1149/1.1621876 All rights reserved. Manuscript submitted December 16, 2002; revised manuscript received May 5, 2003. Available electronically October 9, 2003. The main goal of fuel cell modeling is the description of the device performances starting from the underlying physical phenom- ena, material parameters, and operating conditions. The models should be as simple as possible to reduce the numerical complexity but accurate enough to describe correctly the fuel cell operation. Some assumptions are often used to simplify the mechanistic mod- els such as: 1 one-dimensional 1D geometry, constant gas porosity, fully hydrated membrane, isothermal conditions, steady-state opera- tion, and homogeneity of the media. Qualitative considerations can be used to estimate the applicability range of such assumptions. For example, the 1D geometry approximation can be applied when the gas concentrations do not vary too much along the flow channels, as in the case of high stoichiometric flow ratio, and when channels and ribs are sufficiently thin to render homogeneous the delivery of elec- trons and reactants. Where the approximations are no longer appli- cable, the underlying assumptions must be relaxed, and the models become more complex. In the literature, several models have been presented with the aim of going beyond the following approxima- tions: 1D geometry, 2-4 constant gas porosity in the diffusive region, 4-6 fully hydrated electrolyte membrane, 7,8 isothermal conditions, 8,2 and steady-state operation. 9,4 Although these exten- sions are straightforward, relaxation of the homogeneity approxima- tion requires a clever strategy to handle the complex porous struc- ture of the media. The reactive region of a polymer electrolyte membrane fuel cell PEMFC has a complicated structure: 10-12 a matrix of electronically conductive 20-40 nm carbon grains forms agglomerates of 200-300 nm with platinum islands of 2-3 nm supported on them. This solid porous structure has a bimodal pore size distribution. Smaller, 20-40 nm pores exist inside the agglomerates between the carbon grains, and larger pores 40-200 nm constitute the void space between agglomerates. The ionic conductive electrolyte fills part of the larger pores, possibly together with Teflon, which can be added as a hy- drophobizing agent. The smaller pores are available for the transport of the gas species when they are not flooded with water. In the literature, several models have been published based on simplified descriptions of the porous reactive region: Giner and Hunter 13 and Iczkowski and Cutlip 14 consider cylindrical agglomer- ates consisting of a homogeneous mixture of carbon, platinum, and electrolyte, surrounded by gas pores; Perry et al. 15 consider spheric agglomerates, and Gloaguen et al. 16 slab geometry agglomerates. Despite the various geometries, the effects of the nonhomogeneity on the fuel cell performances, as described by these models, can be summarized by a value of the Tafel slope at high current densities twice the value of the Tafel slope at lower current densities. Some authors, such as Giner and Hunter, 13 Broka and Ekdunge, 17 and Jaouen et al., 18 consider three-phase reactive regions made by solid agglomerates covered by an electrolyte layer and separated by gas pores. The presence of the electrolyte layer limits the maximum current density achievable. In a previous work, 6 we considered cylindrical gas pores sepa- rated by a homogeneous mixture of carbon, platinum, and electro- lyte, and we have found a very good agreement with the experimen- tal results. The models from all the preceding papers have been presented with insufficient analysis on the influence of the employed porous structure geometry on the model results. This lack of analysis re- stricts the model reliability. The main goal of this paper is to elimi- nate this shortcoming and, consequently, to reach a deeper under- standing of the phenomenology associated with the diffusion reaction on a two-phase nonhomogeneous medium. To reach this aim, we first ascertain which phenomena need a pore scale description; subsequently, we apply a volume averaging procedure to decouple the variables varying at the pore level from the constant ones. This allows the preservation of a macrohomoge- neous level description of the fuel cell without loss of accuracy. By considering five different simplified geometrical descriptions of the porous structure, the volume averaging leads to five analytical ex- pressions for the effective reaction rate. The five expressions are compared among themselves and with the expressions used within the homogeneous and agglomerate models to get a better under- standing of the phenomenology associated with reactive transport within a porous structure. In the macrohomogeneous model section, the use of these effective reaction rate expressions in the framework of macrohomogeneous models is discussed. In the results section, we show the effects of the porous structure on the polarization curve of air and methanol electrodes. Pore Scale Model The relevant transport phenomena inside the reactive region of a PEMFC are proton transport in the electrolyte phase, electron trans- port in the solid carbon phase, and reactant diffusion in the gas, liquid, and electrolyte phases. To decide the extent of details re- quired for the description of the phenomena, we must compare the characteristic lengths of the region i.e., thickness of the region and pore-agglomerate lengths with the diffusion lengths i.e., the dis- tances over which the physical variables related to the transport z E-mail: pisani@crs4.it Journal of The Electrochemical Society, 150 12 A1558-A1568 2003 0013-4651/2003/15012/A1558/11/$7.00 © The Electrochemical Society, Inc. A1558