Page 124 - Introduction to Transfer Phenomena in PEM Fuel Cells

Mass Transfer Phenomena 113
The authors of this model propose a quantitative description of the
equilibrium of a membrane immersed in an acidic solution. Indeed, the
membrane is considered to be made up of channels with impermeable
negatively charged walls in which liquid water and protons circulate. Liquid
water and protons form a charged solution that is subject to electrical
potential and hydraulic pressure gradients.
The principle of this modeling is to link the electrical potential
distribution to ion concentration profiles in the pores of the membrane
[CWI 92b].

The distribution of the electric potential and the transport of ions and
water are respectively described by the Poisson–Boltzmann, Nernst–Planck
and Navier–Stokes equations. The transport coefficients, hydraulic
permeability, electrokinetic permeability and ionic conductivity are usually
derived from experimental measurements or can be evaluated from data such
as swell, ion exchange capacity, pore size and physical properties of the
solution that surrounds the membrane [KOT 02]. The validity range of this
type of description is set at a pore size of around 10 times the size of a water
molecule, that is, approximately 3 nm.

This approach, based on a microscopic description of pore transport
phenomena (double layer type), allows the modeling of macroscopic
transport mechanisms by the application of a scaling method, such as taking
an average, for example. This type of model has been used for many years
for transport in clays and membranes, a synthesis of which is presented in
work by Lemaire [LEM 04], and it is recently applied to the case of a Nafion
membrane, particularly by Bernardi and Verbrugge [BER 92], and Singh and
Djilali [SIN 99].

For the transport of water, as formulated by Bernardi and Verbrugge, the
hydraulic model assumes the coexistence of a solid phase (membrane) and a
liquid phase (water + protons). The solid phase is supposed to be inert; the
displacement of the charged solution in the membrane is the result of the
action of an electric field and a pressure gradient. The proton water solution
in the membrane has a low flow velocity. To describe the transport of water
in the membrane, we can then use the Stokes equations where the
gravitational forces are negligible compared to the electric forces.