Depth Averaged Analytic Solution for a Laminar Flow Fuel Cell with Electric Double Layer Effects

Abstract

A comprehensive multidimensional analysis is presented for a laminar flow fuel cell with electric double layer (EDL) dependent kinetics in a planar microdevice. The EDL is described with the Stern model, and a generalized Frumkin--Butler--Volmer (gFBV) equation is used to describe the EDL dependent kinetics. The liquid electrolyte is modeled with the Poisson--Nernst--Planck (PNP) equations and the incompressible Navier--Stokes (NS) equations. For planar microchannel applications, the three-dimensional model is reduced to an in-plane depth averaged set of equations through an asymptotic analysis. The diffuse layers are resolved in the thin double layer limit through asymptotic matching by considering the Debye length to channel width ratio as a smallness parameter. This yields an outer problem for the bulk electrolyte and an inner problem for the anode and cathode diffuse regions. Fuel cell performance is then evaluated by introducing several specified local current density profiles. The resulting approximate analytic expressions, based on the proposed specified current density profiles, are validated against results from a numerical solution of the full in-plane PNP--NS--gFBV model, in which a priori current profile approximations are not required. We demonstrate that simple current density profiles yield physically unrealistic electrode potential distributions despite producing reasonably accurate overall device performance results. We also present an appropriate current density profile which yields accurate spatial distributions for the continuum fields and electrode potential distributions.