Adjoint Method for the Optimization of the Catalyst Distribution in Proton Exchange Membrane Fuel Cells

Abstract

In this article we present an adjoint method for the optimization of the catalyst distribution in proton exchange membrane fuel cells (PEMFCs). By using the theory of functional analysis we derive analytical equations for the sensitivity functions of the cell voltage with respect to the catalyst distribution in a very general framework, independent on the transport model used to simulate the PEMFC. Then we present an efficient numerical algorithm to calculate the sensitivity functions using the adjoint method. The adjoint method has the advantage that it can be applied to the optimization of systems with a large (>10 4 ) number of optimization variables that are computed simultaneously and independently to maximize the objective function. Finally, we apply the method to the optimization of 2-D platinum distribution in PEMFCs. We show that the optimum platinum distribution varies with the operating conditions, position of landings and openings, cell geometry, and dimensions of the catalyst layers. The method presented in this work can be naturally extended to the optimization of other 2-D and 3-D field variables such as the porosity of catalyst and gas diffusion layers, particle size distribution, or microstructure of the cell. Proton exchange membrane fuel cells (PEMFCs) have attracted the attention of the research community because of their high power density, energy efficiency, and environmentally friendly characteristics. Operating at low temperatures, PEMFCs are suitable for a variety of applications ranging from automotive applications to stationary and portable applications. In the area of automotive applications, PEMFCs have already been introduced commercially or for demonstration purposes by all the major car manufacturing companies; in addition, they are increasingly being used in busses, motorcycles, boats, submarines, and airplanes.1,2 In the area of stationary and portable applications, PEMFCs are employed as emergency power systems, such as uninterrupted power supplies, and have the potential to be used for energy storage in electric grid systems.3 Since the technical feasibility of PEMFCs has already been established, a significant amount of current research focuses on increasing the durability and decreasing the total manufacturing cost, which, at large manufacturing volumes, is mainly given by the cost of the catalyst layers (CLs), bipolar plates, and membrane.4-8 (For instance, the US Department of Energy estimates that 45% of the cost of fuel cell stack fabricated at a volume of 500,000 systems/year is due to the cost of platinum, 27% to bipolar plates, 10% to the cost of membrane 9 ). In this article, we address the problem of decreasing the manufacturing cost of the CLs by presenting a large-scale optimization technique to optimize the platinum deposition in the CLs of PEMFCs. The technique can also be applied to the optimization of other two-dimensional (2-D) and three-dimensional (3-D) field variables, such as porosity distribution and ionomer content, in which the number of optimization p...

Analysis of Sensitivity of PEMFC Parameters to Non-Uniform Platinum Deposition

“…1. The model equations and the values of the model parameters are taken as in reference (18). Notice that, due to the symmetry of the problem, the catalyst distribution function is represented under half of the land and half of the opening.…”

“…In the computational electronics community the doping sensitivity functions have been applied for the analysis of random doping induced fluctuations in ultrasmall semiconductor devices (11)(12)(13)(14), for the optimization of doping profiles in nanoscale and power semiconductors (15,16), and to solve inverse problems in semiconductor materials (17). It is only recently that the concept of sensitivity functions was used in electrochemical community for the design and optimization of PEMFCs (18). A detailed mathematical introduction to sensitivity functions and the numerical computation of these functions using the adjoint method can be found in (5).…”

“…The adjoint method avoids both problems of the finite differences by using the perturbation technique and slightly changing the way the sensitivity functions are computed. As shown in (18), the discretized values of the catalyst sensitivity function (i.e. the values at each vertex) can be computed using ( )…”

Optimizing the Composition of the PEMFC Catalyst Layer

“…The platinum, nafion, and carbon sensitivity functions are extremely instrumental in the practical design and optimization of fuel cells. For a detailed description of how to compute these sensitivity functions we recommend (6,9).…”

“…In previous publications we have developed an optimization algorithm to compute the optimum platinum distribution by keeping the total mass (and, in this way, cost) of platinum constant (6)(7)(8). In particular, we have solved numerically, within the framework of finite element methods, the following mathematical optimization problem with constraints ( ) ( ), such that =constant [1] denotes the mass of platinum per unit volume of the catalyst layer, V is the voltage of the cell at a specified value of current density, and is the total mass of the platinum computed as…”