Random Oxide Thickness Research Articles

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

Recently, we have proposed a robust and computationally efficient numerical algorithm for the optimization of 2-D and 3 -D catalysts distributions in Proton Exchange Membrane Fuel Cells (PEMFCs) using a gradient-based technique [1-3]. The algorithm is based on the computation of the catalyst sensitivity functions of the cell current and cell voltage by using the adjoint method. This method was initially developed by the applied mathematics community to solve optimization problems in fluid dynamics, climate, and heat transfer problems and, later, by other research groups to solve optimization problems in magnetics, model parameter determination, and semiconductor devices [4-6]. It was only recently that COMSOL, a popular finite element solver, has introduced the adjoint method, which has started to be used by the electrochemical community for calculation of sensitivities of platinum distribution in PEMFCs [7,8]. In this presentation we analyze the cell current, cell voltage, and spectral electrochemical impendence properties of PEMFCs by using the catalyst sensitivity functions of cell voltage, γPt(r). These functions show how much the cell voltage is increasing by adding 1 unit amount of platinum at a given location inside the catalyst layer (and assuming the cell is operating at the same current density). For instance, the figure below shows the 2-D catalysis sensitivity functions of the cell voltage for two cells with membrane thicknesses of 60 mm and 25 mm, in which the catalyst is distributed uniformly inside the cell. It is apparent from these figures that in order to increase the cell voltage and consequently the power of the cell, one needs to increase the concentration of Pt particles in the regions under the oxygen openings. As it will be shown at the conference the optimum catalyst distribution also depends on the operating temperature, catalyst and gas diffusion layers porosities, Nafion content, and cell current. In addition to the above analysis, we will also introduce a technique for the analysis of variability of polarization curves induced by the randomness of pore sizes and pore shapes in the catalyst and gas diffusion layers in PEMFCs. The technique is based on the evaluation of sensitivity functions of the cell voltage and can also be extended to the analysis of variability induced by the random platinum distribution in the catalyst layers (which is assumed to have a uniform random distribution). More details about the method for the analysis of variability effects, as well as about the numerical implementation of the adjoint method in the case of finite element simulations and its computational efficiency will be presented at the conference. [1] J. Lamb, G. Mixon, and P. Andrei, “Optimization of polymer electrolyte membrane fuel cells with a large number of degrees of freedom”, ECS 231Meeting, New Orleans, LA, 2017. [2] J. Lamb, G. Mixon, and P. Andrei, “Mathematical optimization of the spatial distribution of platinum particles in the catalyst layer of PEMFCs”, 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] D. Cacuci, M. Ionescu-Bujor and M. N. Ionel, Sensitivity and Uncertainty Analysis, Volume 2: Applications to Large-Scale Systems, Chapman & Hall/CRC (2005) [5] W. K. Anderson and V. Venkatakrishnan, “Aerodynamic design optimization on unstructured grids with a continuous adjoint formulation”, Computers & Fluids, 28, 443 (1999). [6] 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). [7] COMSOL Multiphysics Software, https://www.comsol.com. [8] F. C. Cetinbas, S. G. Advani and A. K. Prasad, “Three dimensional proton exchange membrane fuel cell cathode model using a modified agglomerate approach based on discrete catalyst particles”, Electrochimica Acta, 174, 787 (2015). Figure 1

Mathematical Optimization of the Spatial Distribution of Platinum Particles in the Catalyst Layer of Pemfcs

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.

Optimization of Polymer Electrolyte Membrane Fuel Cells with a Large Number of Degrees of Freedom

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.

Quantum mechanical effects on random oxide thickness and random doping induced fluctuations in ultrasmall semiconductor devices

Quantum mechanical effects on fluctuations in ultrasmall semiconductor devices are studied. Quantum mechanical effects are included in the analysis by using the density gradient model, for which the main parameters (effective masses) are identified through the two-dimensional Schrödinger equation. A very fast technique for the computation of threshold voltage fluctuations induced by random oxide thickness and doping variations is proposed. This technique is based on linearization of the transport equations with respect to the fluctuating quantities. This technique is computationally very efficient because it avoids numerous simulations for various doping and oxide realizations (as in the case of Monte Carlo techniques). At the same time, it provides information about the sensitivity of threshold voltage to the fluctuations of oxide thicknesses and doping at different locations. Sample simulation results are reported and compared with those previously published and good agreement is observed.