On the existence of a weak solution of a half-cell model for PEM fuel cells.(Research Article)(Report)

1. Introduction

The modelling of fuel cells has been an attractive topic in the field of electrochemical theory. In the last decade, models for proton exchange membrane (PEM) fuel cells have been formulated by many scientists (see, e.g., [1]). Among these models, some complicated systems of partial differential equations (PDEs) were constructed from principles of fluid mechanics, electrostatics, and heat transfers; however, most of them were solved by numerical simulations only. We are interested in the mathematical analysis of the system of differential equations and the discussion is restricted on the transport phenomenon of a single-phase model given by [2]. The more complicated two-phase models, like those mentioned in [1, 3], are not in the scope of this paper.

In [4], by reducing space variables to one dimension and making several assumptions, a system of PDEs in [5] was simplified to a boundary value problem (BVP) for a linear system of decoupled ordinary differential equations (ODEs), and an exact solution was constructed. In [6], a 1D half-cell model reduced from [5] is considered; that model is a BVP for a nonlinear system of three ODEs of second order which are no longer decoupled and it seems to be hard to find an exact solution. By Schaefer's fixed point theorem, the study in [6] is able to show the existence of a solution in the space of continuously twice differentiable functions. In this paper, motivated by [4, 6], we will derive a 1D half-cell model from the 3D model of [2]; it is still a BVP for a nonlinear system of three ODEs of second order; however, the nonlinearity is different from that of [6] and an alternative strategy will be applied; namely, a weak formulation of the BVP will be considered. In this weak formulation, the function space is replaced by the Sobolev space [H.sup.1] and an iteration process associated with Schauder's fixed point theorem will be adopted. The result of this paper indicates a direction of attacking the complicated system of PDEs for the modelling of PEM fuel cells.

Now, we briefly describe the contents of this paper. In Section 2, the governed equations and boundary conditions in the cathode catalyst layer for the 1D half-cell model of PEM fuel cells are derived. In Section 3, the weak form of a linear generalized Neumann problem is described. Existence and uniqueness of the generalized Neumann problem is guaranteed by the Lax-Milgram theorem and it will be shown that the solution for the linear problem has an a priori bound. In Section 4, Schauder's fixed point theorem is applied to prove the existence of an [H.sup.1] solution for the nonlinear system of ODEs.

2. The Model

In this section, we will reduce a 3D model of Zhou and Liu [2, 7] to a 1D half-cell model. This 3D model was a modification of the 2D model given by Gurau et al. [5], so the derivation of the 1D model is quite the same with what we did in [6], we describe the derivation here for the reader's convenience.

Recall (e.g., see [8]) the species equations are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.1)

where [Y.sub.k] is the concentration of kth component gas mixture, and [D.sup.eff.sub.k] is the effective diffusivity of the kth component in the gas mixture, which is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.2)

At the cathode, the mass generation source terms [S.sub.k] for oxygen, water, and protons are …