Mathematical Optimization of the Spatial Distribution of Platinum Particles in the Catalyst Layer of Pemfcs
Authors: Lamb James , Mixon Grayson , Andrei Petru 
Published: February 2020

The cost of platinum accounts for approximately half of the price of PEMFCs when the fuel cells are produced in large volume (over 500,000 systems/year) [1]. In order to reduce the cost of fuel cells, one needs to reduce the content of platinum used in each cell, which can be achieved by using low platinum loadings but placing the platinum particles optimally, at the most electrochemically active locations inside the catalyst layers. Mathematically, this optimization problem can be formulated as how to compute the optimum catalyst distribution function inside the catalyst layer, ρ(r), that maximizes the power density of the fuel cell, where ρ is the amount of platinum per unit volume of the catalyst layer (expressed in gPt/cm3) and r is the 3-dimensional position coordinate. Solving this problem requires finding the values of function ρ at each location r in the catalyst layers. For instance, suppose we simplify the problem by dividing the catalyst region in 10 × 10 × 10 cells and try to evaluate the optimum amount of platinum in each of these cells. In this case, the optimization problem has 1,000 independent parameters, which need to be optimized numerically or experimentally. Such problems require a large number of trial-and-error simulations or actual device realizations, which is practically impossible to accomplish numerically or experimentally. For instance, in the case numerical optimization, if we try to use traditional heuristic techniques such as genetic algorithms, swarm-optimization, or other evolutionary algorithms to optimize 1,000 independent parameters, we need to perform more than 1010device simulations, which is unmanageable even on large computer clusters (the typical simulation time for one fuel cell is of the order of 1 min to 1 hour for a regular fuel cell).
Recently, we have proposed an adjoint method to numerically optimize electrochemical systems with a large number of degrees of freedom [2,3]. This method was initially developed by the applied mathematics community to solve optimization problems in fluid dynamics, climate, and heat transfer problems but, more recently, the method was used by other research communities to solve optimization problems in magnetics, model parameter determination, and semiconductor devices [4]. In the case of electrochemical systems, the adjoint method can be used to evaluate the catalyst sensitivity functions ΥPt(r), which show how sensitive the parameters of the cell are to the platinum density at location r. For instance, the catalyst sensitivity function of the cell voltage, ΥPt,V(r), shows how much the cell voltage is increasing by adding 1 mg of platinum at a given location inside the catalyst layer (by operating at the same current density). Since the catalyst sensitivity functions measure the sensitivity of cell parameters to the local platinum content, they are instrumental in the design of PEFMC.
In this presentation we formulate the optimization problem in matrix form, appropriate for 2-D and 3-D finite element simulations of PEMFCs, in which the number of optimization parameters is over 106. Optimization results for low and large platinum loadings will be presented for unconstrained and constrained optimizations (when the amount of platinum is fixed). Our numerical results show that the optimum platinum distribution function, ρ(r), depends on operating current, temperature, and humidity. In general, at low operating currents, the platinum can be distributed approximately uniformly inside the cathode catalyst layer, however, at large operating currents, it is more efficient to increase the platinum content in the region close to the membrane and under the oxygen channels. The technique can also be extended to the optimization of GDL porosity ε(r), particle size, and ionomer content. More details about the numerical implementation of the method in the case of multiscale transport models and the computational efficiency will be presented in the full paper.
[1] J. Spendelow, J. Marcinkoski, A. Wilson, and D. Papageorgopoulos “Fuel Cell System Cost - 2015”, DOE Hydrogen and Fuel Cells Program, https://www.hydrogen.energy.gov/pdfs/15015_fuel_cell_system_cost_2015.pdf
[2] J. Lamb, G. Mixon, and P. Andrei, "Optimization of polymer electrolyte membrane fuel cells with a large number of degrees of freedom," PRIME 2016 Meeting, Honolulu, HI, 2016.
[3] P. Andrei, G. Mixon, M. Mehta, and V. Bevara, "Design of the catalyst layers in PEMFCs using an adjoint sensitivity analysis approach," ECS Transactions 66(8), 91-128, 2015.
[4] 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, 7163-7172, 2003.