Page 130 - Introduction to Transfer Phenomena in PEM Fuel Cells

In the absence of a proton concentration gradient in the membrane,
proton transport by diffusion also vanishes, and equation [3.73] is then
limited to Ohm’s law: Mass Transfer Phenomena 119
j =−δ + ⋅∇φ [3.74]
H
with (δ + ) always being the ionic conductivity of the membrane:
H

2
⋅
zD F ⋅ c
⋅
δ + = i i i [3.75]
H RT
Unlike the hydraulic model, the displacement of the charged solution
does not contribute to the current density. On the contrary, the flow of
protons induces a movement of water in the same direction: the electro-
osmotic flow. This flow results from interactions between protons and water
molecules. In the literature, an electro-osmotic coefficient (ξ) is defined as
the number of water molecules accompanying each proton during its
transport [OKA 98]:

n HO
ξ= 2 [3.76]
n +
H

The electro-osmotic flow (N Osmosis ) is directly proportional to the
HO
2
proton flow in the membrane and it is expressed by:

N Osmosis = ξ ⋅ j [3.77]
HO F
2
Water is an electrically neutral species (ZH 2 O = 0). According to equation
[3.73], the transport of water also occurs by diffusion and the diffusive flow
( N Diff ) is given by:
H 2 O

N Diff =− D m 2 ⋅∇ c HO [3.78]
H O
HO
2
2