Page 122 - Introduction to Transfer Phenomena in PEM Fuel Cells

Mass Transfer Phenomena 111
charged surface. In order to complete the description of the phenomena at
the pore scale, here are some authors: Koter [KOT 00, KOT 02], Cwirko and
Carbonell [CWI 92a, CWI 92b] and Pintauro et al.
The authors of the research [BON 94, GUZ 90, TAN 97, YAN 00, YAN
04] use a modified Boltzmann equation to determine the ion concentration.
This equation, developed by Gur et al. [GUR 78] and used by Verbrugge and
Hill [VER 88], takes into account the effects of dielectric saturation in the
vicinity of a charged surface [BOU 07]:

 zF⋅φ () r Δ m G () r − Δ b G 
b

c m () r = c ⋅ exp − i − i i   [3.59]

i
i
 RT RT 
where G i is the variation in Gibbs free energy of the ionic species considered to
be between a studied medium and the vacuum. The second term of the
exponential takes into account the hydration forces acting on the protons. These
forces exist because the interactions between the ions and the solvent are not
uniform in a charged pore. Theoretical studies to determine the hydration forces
were conducted by Bontha and Pintauro [BON 94] and by Eikerling and
Kornyshev [EIK 01]. Another modification of the classical Poisson–Boltzmann
theory is obtained by taking into account the confinement of water in regions
with sizes of only a few nanometers (typically, a Nafion membrane pore). High
electrostatic fields can significantly reduce the relative permittivity of water (ε r)
compared to that of the bulk (ε b) [CWI 92a, CWI 92b].

This is the case of the charged pore wall edges. This effect has been
considered by many authors [BON 94, CWI 92a, CWI 92b, GUZ 90, KOT
02, TAN 97, YAN 00, YAN 04] using the Booth equation [BOO 51]:
( ε− n 2 )  
0
1
1
2
ε= n + 3 β ⋅∇φ ⋅    tanh (β ⋅∇φ ) − β ⋅∇φ    [3.60]
r
B
B
B
with:
μ
d
2
β =⋅ 2k ⋅ T ⋅ ( n + 2 ) [3.61]
5
B
⋅
b