Mathematical Modeling for a PEM Fuel Cell.

Mathematical Modeling for a PEM Fuel Cell. Mentor: Dr. Christopher Raymond, NJIT. Group Jutta Bikowski, Colorado State University Aranya Chakrabortty, RPI Kamyar Hazaveh, Georgia Institute of Technology Polina Jeglova, RPI Rajinder Mavi, RPI Joel Phillips, McGill University Veronica Respress, University of Dayton. Abstract The objective of this problem is the study of the mechanisms that take place in the cathode Gas Diffusion Layer (GDL) of a Hydrogen-Oxygen Proton Exchange Membrane (PEM) Fuel Cell. Within a Gas Diffusion Membrane, transport of liquid water, water vapor, oxygen, nitrogen and heat are the main processes taking place. This is modeled using convection-diffusion equations for both oxygen and water vapor in the layer supplemented with suitable boundary conditions. The goals in this work include (1) the numerical solution of the scaled two-dimensional problem, (2) the validation of the one-dimensional solutions, (3) an analysis of the anisotropic nature of the membrane, and (4) a better treatment of the temperature and pressure along the layer. The group was able to solve numerically the two-dimensional problem after the introduction of dimensionless variables. The solutions obtained using two different numerical schemes were compared. The non-dimensionalization introduces a small parameter which relates the aspect ratio between the height and the width of the layer. In the asymptotic case when this small parameter approaches zero the problem reduces to a set of two ODEs. The solution of these equations was compared against results previously obtained for the one-dimensional case. The convection-diffusion equations were generalized to consider the anisotropic nature of the layer. The resulting equations were solved numerically. For some particular choices, the distribution of oxygen along the layer appears to be more efficient as compared to the isotropic case. 1

1 Problem Description. The objective of this problem is the study of the mechanisms that take place in the cathode Gas Diffusion Layer (GDL) of a Hydrogen-Oxygen Proton Exchange Membrane (PEM) Fuel Cell, continuing the work done during the MPI 2004 workshop based on a problem proposed by a representative from W.L. Gore. In order to achieve this goal, a first approach to previous analysis is required to posteriorly add new features to the model. These features include a better treatment to the temperature and pressure fields, allow anisotropic permeability in the membrane, produce numerical solutions and compare them with the results from asymptotic analysis. Fuel Cells are considered one of the best approaches in the search for new energy sources. Among their advantages includes the low pollution emitted, their relatively high efficiency (80%) and quietness. A PEM fuel cell consists of several layers of basic cells. In these cells, a proton conducting membrane separates the anode and the cathode side, and each side has an electrode. On the anode side, hydrogen diffuses to the anode catalyst where it dissociates into protons and electrons. The protons are conducted through the membrane to the cathode, whereas the electrons provide electric energy in an external circuit. On the cathode catalyst, oxygen molecules react with the electrons closing the circuit and protons to form water. In the process, water/vapor, heat and electric energy are produced (see Figure 1). In the problem setup, the goal is to consider the physical and chemical processes involved to identify problem geometry, variables and parameters of interest. Once the model is identified, a restriction to a simplified version of the problem can be done keeping track of all assumptions. The processes of interest for the gas diffusion layer are the production of water and heat due to the electro-chemistry at the cathode and the consumption of oxygen. It can be assumed that the concentrations and temperatures at the boundary with the oxygen channel are specified. In the same way, it can be assumed that the temperature at the boundary with the shoulder is specified, otherwise a no flux boundary condition must be assumed (see Figure 2). Within the Gas Diffusion Membrane, transport of liquid water, water vapor, oxygen, nitrogen and heat are the main processes taking place and, in principle, evaporation/condensation between liquid water and water vapor must be allowed (see Figure 2). We can write mathematically these statements as [ ũ D u G [ ( G ) ṽ D v G G + k g µ ] ũ GR T ] = 0, + k g ṽ GR T = 0, µ G G T = 0, where ũ is the molar density of oxygen, ṽ is the molar density of water vapor, G is the total molar density (equal to u + v), T is the temperature and D p, p = u, v, is diffusion constant for oxygen and water vapor, respectively. The constant k g stands for the permeability of the 2

(a) (b) Figure 1: (a) Basic components of a fuel cell and (b) fuel cell composed of many repeated basic cells. Figure 2: Gas Diffusion Layer of a Fuel Cell. layer, µ is the viscosity of gas, and R is the universal gas constant. The boundary conditions consist of a source of oxygen (the channel) at the top of the layer, a non-penetration condition at the shoulder of the channel, periodic boundary conditions on each side of the layer, and 3

a Robin-like condition at the cathode. Mathematically these boundary condition are ũ = G = 0 0 x d, ỹ = h (shoulder), T = T 1 ũ = u 1 G = G 1 T = T 1 d x 2d, ỹ = h (channel), D u G ũ + k g G µ GR T ṽ D v G + k g GR T G µ D T T ũ x = G x = 0 T x = 0 = c u ũ, = 2c u ũ, = c T ũ, at x = 0, x = 2d, ỹ = 0 (cathode), where C p, p = u, v, is the Nusselt number for oxygen and water vapor, respectively. A first approach to this problem is to look for special solutions (steady state, etc.), asymptotic limits, simplified versions of the original problem and search for analytical solutions, if possible. Further analysis include numerical solutions and their validation against analytical results. The next step is the analysis of the results, addressing questions such as how good these solutions satisfy the expectations from the problem physics, what kind of insight does the results give (for instance, does it suggest how to choose experimentally adjustable parameters to optimize a quantity of interest). Finally, we may include a reconsideration of the assumptions made or some processes that were left out that could be relevant to the final answer. It is important to consider some characteristics of the cell during the analysis. The GDM material is hydrophobic (not hydrophilic), there are many recent and current research activities devoted to modeling dependence of permeability on water content. The GDM material is highly anisotropic and the aspect ratio between the height and the width of the layer (h/d in Figure 2) is small. Previous work (at MPI 2004) considered a formulation of an isothermal, steady model with no liquid water. The small aspect ratio allowed the derivation of quasi one-dimensional solutions valid under the channel and under the shoulder. 4

The main goals for future work involve numerical solutions of the scaled two-dimensional problem and the validation of the one-dimensional solutions. The GDM material is made of fibers, which mainly lie parallel to the cathode enhancing diffusion in this direction. Future work must allow for this anisotropic nature of the GDM and, depending on the size of this effect, it could be possible to salvage the small aspect ratio analysis previously done. It is necessary also to study the order of magnitude of the parameters considering the possibility of the lost of small ones. This might lead to the necessity of seeking a new small parameter or considering a different layer structure, for instance. Previous work assumed an isothermal layer, further analysis may include a better treatment of the temperature. In this case, one can still ask if it is still possible to make progress analytically. Since there is an actual drop of the pressure from channel to channel, a better treatment of the pressure must be done; again, one can ask if it is possible to make progress analytically. 2 Analysis. 2.1 Non-dimensionalized Equations. The first step in the analysis is the non-dimensionalization of the model. non-dimensional variables are introduced considering constant temperature The following s = G 2, r = ũ, x = x G 2 1 Gφ u d, y = ỹ h, where φ u is the volume fraction of oxygen. We also introduce the following parameters: ɛ = h d, P eg = k grt G 1 D u µ (Peclet number). This leads to the following system, [ Peg 1 ɛ 2 x + 1 2 ɛ 2 2 s x 2 + 2 s 2 = 0 s r + ( r r )] x [ ɛ 2 ( r s ) + ( r s )] x x = 0 in the layer, 5

with boundary conditions s = r } = 0 0 x 1, y = 1 (shoulder), ( ) s = mr Peg 1 r s = 1, r = 1} s x = r } x = 0 r = m + r 2 φ u 1 x 2, y = 1 (channel), at x = 0, x = 2, y = 0 (cathode). where m = C uφ u P eg. 2.2 Numerical Solutions. To solve this system numerically two schemes were used, a finite element method using the Matlab PDE toolbox and a finite difference method coded in Matlab. These two results were then compared for the same values of the aspect ratio ɛ. It was observed that among the characteristics of the finite element method we can mention as pros its relatively short computation time and the allowance of adaptive optimized mesh generation. As a disadvantage of the scheme we can point its strict format for input parameters. On the other hand, the finite difference method required relatively longer run time (perhaps due to inefficient code structuring in Matlab) and uniform mesh spacing, but possessed more flexibility in manipulating the parameters. A comparison of the numerical solution obtained using the two methods is shown in Figure 3. Now we turn our attention to the case when we consider a large aspect ratio (small ɛ). In this asymptotic limit the two-dimensional problem can be reduced to a pair of ODEs for the region under the channel. The solution to this problem is r = 1 φ + ( 1 + 1 φ ) exp (P eg (1 s)), The solution to the one-dimensional and two-dimensional problems are plotted in Figure 4 showing that the difference between these two responses is small (of the order 10 4 ). 2.3 Anisotropy. The GDL is composed of carbon fibers with certain disposition along the layer. Ordering of the fibers affects the permeability as well as the diffusivity of the layer and, ultimately, may 6

Figure 3: Numerical results using Finite element method (top) and Finite difference method (bottom), with ɛ = 0.1. Figure 4: Comparison between the asymptotic solution and the numerical two-dimensional solution using Finite differences. help us to design a layer such that the distribution of oxygen is uniform along the cathode. To model the anisotropic nature of the layer, the group considered changing the parameters D g corresponding to the diffusivity and k corresponding to the permeability of the layer from scalars to tensors. The simplest approach is to consider these tensors to be diagonal matrices with possibly different constant diagonal elements k x and k y, but with k 2 x + k 2 y = k 2 fixed. A similar choice is made for D g. This approach affects the normalized parameters, but it can 7

be incorporated to the finite difference model. A graph of the normalized concentration of oxygen is shown in Figure 5. Figure 5: Concentration of oxygen as a function of λ = kx k y and x. In Figure 6 we find the optimal value for λ = k x /k y such that the total amount of oxygen at the cathode is maximized. From this analysis the optimal value for λ is found to be 1.8231. However, the improvement over the isotropic case is only 0.5%. Nontheless, this suggests that looking for other forms of anisotropy could lead to a more significant improvement of the distribution of oxygen in the cathode over the isotropic case. Figure 6: Total concentration of oxygen at the cathode as a function of λ. Figure 7 shows the different anisotropies considered. In Figure 7(a), the result is an actual decrease in the efficiency of the distribution of the oxygen at the cathode respect the 8

(a) (b) (c) Figure 7: Different types of arrangement of the Carbon fibers in the layer. isotropic case. In Figure 7(b), using λ = 3, we find an increment of 4.3% in the efficiency, whereas for λ = 10 the eficiency increases 6.3%. Finally, for the case shown in Figure7(c), for λ = 3 the eficiency increases 10%, for λ = 10 the improvement is found to be 20%. 3 Conclusions. Two numerical schemes were used in order to solve the system of partial differential equations that model the distribution of oxygen in the GDL. These numerical schemes are the Matlab PDE toolbox and a finite difference method coded in Matlab. Also, a comparison between these two schemes was done using the same value for the parameter ɛ. The nondimensionalization of the problem allowed to solve analytically the problem in the limit case ɛ 0. This analytic solution was compared with the solution of the one-dimensional problem. The difference between these two solutions was found to be order 10 4. The allowance of anisotropy of the carbon fibers in the layer led to a more complicated problem, but it could be solved numerically using the finite difference scheme. The first approach was a special case of anisotropy where the permeability of the layer is just a diagonal matrix with constant coefficients. For this particular case, a numerical solution was found for different values of the ratio between the diagonal elements. The improvement of the efficiency in the distribution of oxygen in the layer compared with the isotropic case was not significant, but following this same idea, different types of orientation in the fibers were tested. In some cases the arrangement of fibers incremented the efficiency of the distribution of oxygen compared with the isotropic case, therefore, these results can be used to construct more efficient fuel cells. 9