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Microstructure Modeling and Optimization of Transport Properties of Gas Diffusion Layers in PEM Fuel Cells, Combining Graph Based Approaches and Full Field Computations

**Introduction**

The Gas Diffusion Layer (GDL) is one of the determining factors for the performance of PEM Fuel Cells (PEMFCs). Hence, in order to improve the performance of the PEMFC, optimizing the transport properties, i.e. the gas diffusivity, electrical conductivity and permeability of the GDL, is crucial. In this work, the modelling and the simulation of the transport through the GDL’s microstructure is presented. This enables the optimization of the microstructure of the GDLs with respect to its transport properties. Finally, microstructures, optimal to given requirements, are found using stochastic optimization.

**Geometric Modeling**

First, a stochastic model for the microstructure of the non-woven materials used for the GDL is defined, using the software GeoDict [1]. The GDL is modeled at the micrometer scale, as a fibrous porous medium, modeled by bent fibers randomly distributed and a binder added in the pore space. In addition, a Micro-Porous Layer (MPL) is placed below the GDL (see Figure 1), to account for the transport resistance of the MPL. The MPL is modeled as a homogeneous medium, since the porosity of the MPL cannot be resolved on the micrometer scale. Yet, additional microscale pores can be added to the MPL. The geometric model depends on several free parameters, such as: fiber radius, fiber volume fraction, curvature of individual fibers, preferred fiber direction and porosity of the MPL. While some parameters are known from the production process, such as fiber volume fraction and fiber diameter, parameters describing the curvature of the fibers and the preferred fiber directions have to be fitted to measurements.

**Model Fitting**

To determine the free parameters, the transport properties of the fiber network, i.e. electrical conductivity, and the pore space, i.e. diffusivity and permeability, of the modeled structures are computed. Then, the free model parameters were fixed by fitting the calculated the electrical conductivities to measurements taken on a reference GDL provided by Freudenberg FCCT. The model fitting was done by minimizing a distance function depending on the transport properties, using iterative stochastic optimization [2]. This fitting method accounts for the stochastic nature of the geometric model and the resulting stochastic nature of the effective properties. Due to the potentially high number of iterations, it requires a fast and yet accurate computation of the transport properties.

**Transport in Pore Space**

The diffusivity and permeability of the pore space of the microstructure were computed using GeoDict. For the computation, the generated microstructure was voxelized, and the transport equations were solved in the pore space of the model. The MPL was modeled as a homogeneous medium with an effective diffusivity and permeability.

**Transport in the Fiber Network**

Computation of the electrical conductivity of the fiber network using voxelized geometries not very accurate, due to the inability to model the fiber-fiber-contact resistance. Also, the large size of the representative volume element requires simulating a large domain, which leads to very high computational cost. This represents a major obstacle to the fitting and optimization, where a large number of computations is needed, to search the parameter space. The problem is solved by generating a graph form the fiber network. The result is a weighted graph with the fibers as edges and the contact points as nodes, with nodes added at contact points modeling the fiber-fiber-contact resistance, as well as the fiber-electrode contact resistance. The electrical conductivity can then be computed on the graph, using a graph-laplacian approach [3,4]. This model provides the computational speed and accuracy needed for the fitting and optimization.

**Optimization**

After the best-fit model parameters were determined by fitting and an agreement of the transport properties with measurements was reached, simulation studies were done assessing the sensitivity of the transport parameters to changes to the model parameters. Finally, the stochastic optimization is used again, to find optimal microstructures with transport properties, to meet requirements given by applications.

[1] www.geodict.de.

[2] J.C. Spall: Multivariate Stochastic Approximation Using a Simultaneous Perturbation Gradient Approximation, IEEE Transactions on Automatic Control, 37(3), (1992), pp. 332–341.

[3] O. Iliev, R. Lazarov, J. Willems: A Graph-Laplacian approach for calculating the effective thermal conductivity of complicated fiber geometries, Berichte des Fraunhofer ITWM, 142 (2008).

[4] J.-P. Vassal, L. Oregas, D. Favier, J.-L. Auriault: Upscaling the diffusion equation in particulate porous media made of highly conductive particles. II. Applications to fibrous materials, Physical Review E, 77:1 Pt 1, 2008