Page 123 - Introduction to Transfer Phenomena in PEM Fuel Cells

112 Introduction to Transfer Phenomena in PEM Fuel Cells
where (n) is the refractive index of the solvent, (μ d) is the dipole moment of
the solvent and (k b) is the Boltzmann constant. Examination of this equation
shows that (ε r) decreases as the electrostatic field increases. The molecules
of the solvent are polarized, orientated towards the charged pore wall and
their mobility is reduced. The work of Paul and Paddison [PAU 04] and
Paddison [PAD 01] confirmed this evolution using statistical mechanics.
However, no experimental information is available in the literature concerning
the arrangement of water molecules under these conditions. The classical or
modified Poisson–Boltzmann theory has therefore often been used to describe
the spatial distribution of protons in the pore. The solvent is considered as an
incompressible continuous fluid whose displacement is governed by the
Navier–Stokes equation. According to Choi et al. [CHO 05, CHO 06], the ion
transport mechanisms at the atomic scale are well described in terms of
diffusion by the Nernst–Planck equation. Solving these equations makes it
possible to elucidate the transport in the ion exchange membrane. Assuming
a constant relative water permittivity (ε r) in the pore, Gross and Osterle have
linked the ion concentration and electrostatic potential profiles in the pore to
the transport coefficients of the theory of irreversible process
thermodynamics. Verbrugge and Hill [BOO 51] were the first to show the
relevance of the predictions of their model by comparing them with their
experimental data. Cwirko and Carbonell [CWI 92a, CWI 92b] have
confirmed the good model/experiment concordance based on the results of
Narebska [NAR 84]. More recently, Yang and Pintauro [YAN 04] developed
a model with a variable pore size. They show that the hydration forces
greatly influence the spatial distribution of the ions: the hydrated ions are
excluded from the areas close to the charged wall because of their real size.
In all cases, the numerical results of the models are compatible with the
experimental data recorded by the various authors.

Nevertheless, a model based on these equations (in particular, the
modified Poisson–Boltzmann equation) needs to be solved numerically and
it is difficult to introduce it into an overall fuel cell model [BOU 17].

3.6.4. Macroscopic scale

3.6.4.1. Modeling transport in a porous medium
This model is based on the description of Gierke [GIE 82]; it is also
called hydraulic model and it comes from the theoretical work of Pintauro
and Verbrugge [PIN 89].