Optimization of Polymer Electrolyte Membrane Fuel Cells with a Large Number of Degrees of Freedom
Authors: Andrei Petru , Lamb James , Mixon Grayson 
Published: February 2020

There is much work in the literature related to the mathematical optimization of low-temperature polymer electrolyte membrane fuel cells (PEMFCs). In general, the optimization of PEMFCs involves (1) geometry optimization, which focuses on finding the optimum channel geometry and dimensions; and (2) microstructure optimization, which focuses on finding the optimum distributions of the porosity, catalyst, and electrolyte that maximize the power density of the cell. Depending on the complexity of the mathematical model used to describe the PEMFC, the number of variables that need to be optimized can become very large. For instance, let us consider a fuel cell that is discretized using a 3-D finite element mesh with 106 nodes. If we would like to find the optimum values of the porosity, catalyst concentration, and electrolyte density at each mesh point, the total number of design parameters is 3 ×106. Because the number of design parameters is so large, traditional heuristic techniques such as genetic algorithms, swarm-optimization, or other evolutionary algorithms are practically impossible to use even when implemented on massive computer clusters. (Notice that the number of optimization parameters in non-heuristic techniques is usually between 1-10, which is a few orders of magnitude smaller than the total number of design parameters for the PEMFC presented above.) Hence, so far, to make the problem computationally tractable, heuristic optimization techniques have been applied to optimize only a few number of PEMFC parameters.
In this presentation we develop a gradient-based technique for the optimization of PEMFCs, when the number of degrees of freedom, which is defined as the number of optimization parameters, is very large (the number of degrees of freedom in our work should not be confused with the number of nodes or elements in the discretization of the fuel cell) [1]. The technique is based on the computation of the sensitivity functions of the parameters that need to be optimized and is using these sensitivity functions to optimize the cell. The sensitivity functions of the parameters of interest are computed using an adjoint space method initially developed by the applied mathematics community to solve 1-D optimization problems in fluid dynamics, climate, and heat transfer problems [2]. Our group has also used a similar adjoint space technique to analyze the variability and optimize the doping profiles in 2-D and 3-D semiconductor devices [3]. The computational cost required to compute the sensitivity functions using the adjoint space method is relatively small since this method requires solving only one sparse system of linear equations instead of performing multiple fuel cell simulations.
Using the proposed technique we are able to predict the optimum 3-D distribution of platinum particles and porosity profile that maximizes the power density of the cell at different operating current densities. The optimum distributions and porosity profiles depends on the positions of the landings and openings, and on the geometry and dimensions of the layers. In agreement with existing experimental data and previous theoretical estimations [4], we obtain that, at large current densities, the catalyst density should be distributed non-uniformly inside the cell in order to increase the power density of the cell. At low operating current densities the optimum catalyst distribution is more or less uniform distributed inside the catalyst layer. In the case of the porosity distribution, at large operating current, it is more efficient to increase the porosity of the catalyst layer towards the gas diffusion layer side in order to increase the flow of the oxygen and water vapors. At low operating current densities the optimum porosity is uniformly distributed inside the catalyst and gas diffusion layers. More details about the technique, the numerical implementation, and a number of fully optimized structures will be presented at the meeting.
[1] P. Andrei and M. Mehta, "Large-scale optimization of polymer electrolyte membrane fuel cells," 227th ECS Meeting, Chicago, IL, 2015.
[2] D. Cacuci, Sensitivity and Uncertainty Analysis, Volume 1: Theory, Chapman & Hall/CRC, 2003.
[3] P. Andrei, I. Mayergoyz, “Quantum mechanical effects on random oxide thickness and random doping induced fluctuations in ultrasmall semiconductor devices”, Journal of Applied Physics, 94 (2003) 7163-7172.
[4] P. Andrei, G. Mixon, M. Mehta, and V. Bevara, "Design of the catalyst layers in PEMFCs using an adjoint sensitivity analysis approach," 227th ECS Meeting, Chicago, IL, 2015.