Page 118 - Introduction to Transfer Phenomena in PEM Fuel Cells

Mass Transfer Phenomena 107
The transport of water in the membrane is described by the
phenomenological model given by the references [BER 92, BER 91, NGU 93,
SPR 91, SPR 93]. It is the combination of two movements: diffusion, created
by the gradients of water concentration in the membrane, and the other is
electro-osmotic, generated by the clusters of water molecules carried by
+
protons (H 2O) nH when they cross the membrane from the anode towards the
cathode.
The number of water molecules carried by a proton (see [3.76]), denoted
(ξ), is known as the water transport coefficient by the electro-osmotic effect.

The molar flow density of the water transported in the membrane is
–1
–2
defined in [mol.m .s ] and it is written as:

 m −σ + ⋅∇φ 
N HO =ξ⋅ H F m − D m 2 ⋅∇ c HO [3.54]
HO
2
2
where:

– ϕ is the membrane potential [V];
m
–1
– σ + is the ionic conductivity of the membrane [S.m ];
H
–1
– D m 2 is the diffusion coefficient of water in the membrane in [m².s ].
HO
These last two parameters strongly depend on the water content; the
conservation equation of the amount of water is written as:

c ∂ HO   m
2
t ∂ +∇⋅ N HO = 0 [3.55]
2
3.6.2. Microscopic scale

In the literature, we find modeling of the water transport in the membrane
according to three scales: microscopic, mesoscopic and macroscopic. The
different models found in the literature are detailed. At the microscopic
scale, where the studied systems are smaller than 100 atoms, techniques
based on statistical mechanics and molecular dynamics are used to describe
the transfer mechanisms. These techniques consider all the atoms of a
molecular system and only a pore fragment is studied.