Page 126 - Introduction to Transfer Phenomena in PEM Fuel Cells

Mass Transfer Phenomena 115

– u is the microscopic velocity of water in the pores of the membrane, in
–1
(m.s );
– μ − + is the electrochemical potential of the protons.
H
This chemical potential is written for a charged particle in an electric
field:
μ − + = Fϕ + RTlnc + [3.64]
H H
If the concentration (c +) is assumed to be uniform in the membrane, only
H
an electrical potential gradient induces diffusion displacement of the charged
particles. We are talking about migration; equation [3.63] can be rewritten in
the following form:

⋅
Fc +
−
N + = c u D eff + ⋅ H ∇ϕ [3.65]
+
H H H O,H RT
2
Averaged over the membrane, Bernardi and Verbrugge [BER 92] wrote
the previous equation as follows:
⋅
Fc +
−
N + = c + u D eff + ⋅ H ∇ϕ [3.66]
H H H O,H RT
2
In their model, Bernardi and Verbrugge assume that the solvent velocity
is uniform over a straight section of a pore. Consequently, cu =
+
H
c + ⋅ u. In addition, electroneutrality is respected in the pore and it
H
becomes c + = C . It is then possible to rise to steady state; the average
H f
current density across the membrane j = F N given by the authors in
H +
[BER 91] is:

K  
j = − E ⋅∇ P − δ + ⋅∇ϕ [3.67]
μ b H