Page 128 - Introduction to Transfer Phenomena in PEM Fuel Cells

Mass Transfer Phenomena 117
model for a membrane bathed in liquid water. This implies a continuous
liquid pressure at the interfaces [BOU 07].
In this case, the membranes are probably very hydrated. If the membranes are
less hydrated, Weber and Newman [WEB 04a] suggested to describe the
transport of the species in the membrane using the so-called diffusive model,
which is presented below.
For Onsager’s reciprocity relation, the two macroscopic transport laws
previously formulated for the transport of charge (Nernst–Planck) and matter
(Schlögl) in the membrane verify Onsager’s reciprocity relations [RAM 05]:

 K P K 
 u   μ E   ∇ P 
  =−  μ  ⋅  b  [3.71]
 j   K  ∇ϕ
   E δ H  +  
 μ 

On the contrary, the equations used by Bernardi and Verbrugge [BER 91]
do not respect these relations. Indeed, the authors started from writing the
Nernst–Planck relationship in the form:

N + = c + u − D + ⋅∇ μ − + [3.72]
H H H ,H O H
2
Whereas scaling methods require the expression of the convective
transport term as follows c + u ⋅ , with c ⋅ u ≠ c ⋅ u .
H H + H +
3.6.4.2. Phenomenological model

Unlike the hydraulic model, the diffusive model assumes the membrane
is formed of a single homogeneous phase. Several models are based on this
hypothesis including Springer [SPR 91], Fuller and Newman [FUL 93] and
Okada [OKA 98]. This theory was transformed into an equation for Nafion
membranes by Pintauro and Bennion [PIN 84].

The model of Fuller and Newman [FUL 93] is based on the theory of
concentrated solutions, and the Springer model [SPR 91] is based on the theory
of dilute solutions which considers only the interactions between dissolved
species (water and protons) and the solvent (membrane) [NEW 91].