Page 121 - Introduction to Transfer Phenomena in PEM Fuel Cells

110 Introduction to Transfer Phenomena in PEM Fuel Cells
However, equations of motion can only be solved by means of complex
numerical methods [COL 08].
3.6.3. Mesoscopic scale
Mesoscopic modeling is the application of macroscopic transport laws to
the microscopic structure of the membrane. This kind of modeling uses the
Poisson–Boltzmann theory and it has been the subject of great attention.

The Poisson–Boltzmann theory is used by Gross and Osterle [GRO 68] to
describe the interactions between the electrostatic field and the ion
concentration in the pore.

They have thus developed a model of coupled transfers in a charged
cylindrical pore with a uniform charge surface density.

Knowing the charge density (ρ e), the Poisson equation calculates the
electrostatic potential (ϕ) in the pore [BOU 07]:

  ρ
∇ε ⋅∇φ ) = − e [3.56]
(
r
ε 0
with:

⋅
ρ = F  z c m [3.57]
e
i
i
i
where c m is the concentration of the mobile ionic species (i) with charge

i
(z ) in the pore, it is given by the statistic distribution law of Boltzmann:
i
 zFφ () r 
⋅
b
c m () r = c ⋅ exp − i  [3.58]

i
i
 RT 
with ( c ), the concentration of the ionic species in an external solution or in
b
i
a bulk is the point of the solution where the electroneutrality relation is
verified: the effects of the positive and negative charges cancel each other
out; the electrostatic potential of the double layer is zero. This equation takes
into account the attraction/repulsion forces between the protons and the