Dimensionless Model Analysis of PEFC Cathode
Authors: Kawase Motoaki , Yamaguchi Kazuhiro , Kageyama Miho , Sato Kazuyuki , Inoue Gen 
Published: February 2020

INTRODUCTION
Although extensive studies have been conducted on PEFC behavior and a lot of knowledge has been published, understanding the PEFC is still not easy since the system is considerably complicated. By investigating a dimensionless form of equations describing the cathode catalyst layer (CL), the cathode model was simplified and a few intrinsic moduli which control the dimensionless rate profile were proposed (1). The impact of convection on the oxygen reduction reaction (ORR) rate and some case studies are demonstrated.
DIMENSIONLESS MODEL
The ORR rate is proportional to the oxygen partial pressure at fixed cathode emf. The emf dependency follows a Tafel equation. Oxygen balance in cathode catalyst layer is expressed as follows:

d(N
g
y
O – C
g
D
eO
dy
O/dz) / dz = –r
vc[1]
where N
g is the total gas flux, y
O is the oxygen mole fraction, and D
eOis the effective oxygen diffusivity (2). Proton flux can be calculated by the reaction stoichiometry. Proton potential follows the Ohm’s law.
With the 1D model of cathode CL, 12 variables including boundary conditions have to be specified so as to determine the profiles of partial pressure, emf, reaction rate, etc. Dimensionless forms of model equations were derived, which include the following dimensionless moduli we propose:

M
O
(C)

m = δ
(C) (k
vc

m RT / D
eO)1/2[2]

M
p
(C)

m = δ
(C) {4F k
vc

m p
Oc
/ (σ
ep
b
c)}1/2 [3]
where σ
ep is the effective proton conductivity, b
c is the Tafel slope, δ
(C) is the CL thickness, k
vcm
is the ORR rate constant per unit volume of cathode CL at the PEM–CL boundary and p
Oc
is oxygen partial pressure at the CL–GDL boundary. k
vcm
is the greatest rate constant and p
Oc
is the highest partial pressure in the CL. Therefore, k
vcm

p
Oc
represents the intrinsic ORR rate without transport resistance. M
O
(C)

m
and M
p
(C)

m
respectively represent the ratio of reaction performance to oxygen transport performance and the ratio of reaction to proton transport.
A Peclet number which represents the ratio of convection to diffusion of oxygen is also required for formulation:

P
O
(C)

m = δ
(C)
N
A(M)
/(C
g D
eO) [4]
where N
A
(M)is the net water flux through the PEM, which equals the total gas flux at the PEM–CL boundary.
The dimensionless ORR rate profile is determined by only 4 dimensionless parameters, M
O
(C)

m
, M
p
(C)

m
, P
O
(C)

m
, and y
Oc
. The dimensionless system greatly reduces the complication of system.
RESULTS AND DISCUSSION
Define the cathode effectiveness factor F
ec as a ratio of the ORR rate to the intrinsic ORR rate without transport resistance. Figure 1 shows that increase in transport resistance reduces the effectiveness factor. Typical values of M
O
(C)

m
and M
p
(C)

m
in usual cells are around 1. When M
p
(C)

m
is high, proton transport is slow, which increases the cathode emf and reduces the ORR rate. If M
O
(C)

m
>> M
p
(C)

m
, oxygen transport is suppressed and ORR takes place near the oxygen inlet, the GDL side.
As the convective flow from PEM to GDL increases, oxygen distribution is shifted towards the GDL, reducing the effectiveness factor as shown in Fig. 2. Even at quite low M
O
(C)

m
and M
p
(C)

m
of 0.01, the convection can reduce the effectiveness factor. When M
O
(C)

m
and M
p
(C)

m
are high, the impact of convection is not remarkable since the reaction takes place only near the boundaries. When back diffusion of water in the PEM is dominant, P
O
(C)

m
can be negative. F
ec reaches 1 under reaction control.
Figure 3 shows a case study in which the CL thickness δ
(C) is varied with a fixed Pt loading per volume. As δ
(C) increases, the Pt amount increases proportionally but the transport resistance increases M
O
(C)

m
and M
p
(C)

m
and the effectiveness factor decreases as shown in Fig. 1. As a result, the increase in the current density is limited. Since the increase in δ
(C)raises both moduli, it overtakes the increase of Pt; a maximum current appears in Fig. 3.
Employing the proposed dimensionless moduli, the effects of dimensions and operating conditions can be estimated quantitatively without expensive simulation.

Acknowledgement This work is a part of the project by the New Energy and Industrial Technology Development Organization (NEDO), Japan.
REFERENCES
1. M. Kawase et al., 24th Int. Symp. Chem. React Eng. (Minneapolis, June 2016).
2. M. Kawase et al., ECS Trans. 16(2), 563–573 (2008).

Figure 1