1.5 Fuel Cell Analysis
1.5.2 Cell Voltage Under Load
The OCV seems to be of less practical use. It is much more useful for a fuel cell to work under a load, with power being delivered to the users. With the electric energy output, the fuel cell voltage will decrease as the electric current load increases. A popular way to evaluate a fuel cell is to measure its polarization curve (abbreviated as “I–V curve”), which is a plot showing the cell voltage change with current or current density. Figure 1.20 shows a typical polarization curve obtained with a PEMFC.
In the polarization curve, three parts can be observed: kinetic, ohmic, and mass transfer. In the kinetic part, the cell voltage drop is due to the charge-transfer kinetics, i.e., the O2 reduction and H2 oxidation rate at the electrode surface, which is dominated by the kinetic I–η equation (Equation 1.37). In the ohmic part, the cell voltage drop is mainly due to the internal resistance of the fuel cell, including electrolyte membrane resistance, catalyst layer resistance, and contact resistance.
In the mass transfer part, the voltage drop is due to the transfer speed of H2 and O2
to the electrode surface.
32 X-Z. Yuan, C. Song, H. Wang and J. Zhang
Figure 1.20. A typical polarization curve in a fuel cell [3]. (From Larminie J, Dicks A. Fuel cell systems explained. ©2003 John Wiley & Sons Limited. Reproduced with permission from the publisher and the authors.)
1.5.2.1 Cell Voltage Drop Due to Reaction Kinetics [20]
When a fuel cell is under operation, the electric current passes through both anode and cathode sides, and the electrode potentials of both deviate from their equilibrium values. These deviations are called anode and cathode overpotentials, respectively. At both sides, Equation 1.37 holds. Thus, at the anode side,
2
(1 )
0 ( α α )
α η − −α η
= H H a − H H a
n F n F
H RT RT
Ia i e e (1.65)
where i0H2is the exchange current of the HOR, naH is the electron number at the rate-determining step, aH is the electron transfer coefficient, ηa is the anode overpotential, while R and T have their usual meanings.
In Equation 1.65, the first part in the parentheses of the equation represents the forward reaction rate, indicating hydrogen oxidation; the second part represents the backward reaction rate, indicating the hydrogen evolution reaction. The net hydrogen oxidation rate equals the forward reaction rate minus the backward reaction rate.
In the literature, the cathode Pt electrode surface of a fuel cell is considered to be pure Pt. However, from the above-mentioned OCV argument it can be seen that
Current density / mA.cm–2
the electrode surface status is actually related to the electrode potential. In our recent paper investigating PEMFC reaction kinetics, two cathode surfaces were considered. If the Pt electrode potential is higher than ~ 0.8 V, Pt oxidation occurs and the electrode surface will be a mixture of Pt/PtOx. Otherwise, the Pt electrode will be a pure Pt surface. For the ORR at low current densities, the electrode potential is high, the electrode surface is mixed, and the observed kinetics is the ORR on Pt/PtOx. At high current densities, the potential is low and the electrode surface is pure Pt; therefore, the ORR kinetics is different from that at low current densities. Two sets of kinetic parameters, including two Tafel slopes and two exchange current densities, should be obtained. This has been reported extensively in the literature on half-cell investigations of the ORR on Pt electrodes.
At low current densities, the previous anode reaction equation also holds true for the cathode, and the equation becomes
/ / / /
2
(1 )
/
0 ( α α )
α η α η
− − − − − −
= − nO Pt PtO O Pt PtOF c − nO Pt PtO O Pt PtO Fc
O Pt PtO RT RT
Ic i e e (1.66)
The parameters in this equation have similar meanings to those for the hydrogen oxidation reaction, as applied to the cathode.
In a fuel cell, the overpotential for the HOR is defined as ηa =Ea−Ea0, while the overpotential for the ORR is defined as ηc=Ec0−Ec; here, Ea0=0 and
Ec0=1.23 under standard conditions.
In a fuel cell, Ia should equal Ic, and then the fuel cell voltage (Vcell) can be expressed as
Vcell=EOCV −ηc−ηa−IcellRel (1.67)
where EOCVis the open circuit voltage, Icell is the fuel cell current density, and Rel is the polymer electrolyte membrane resistance.
At high current densities, the backward reactions for both anode and cathode can be omitted, and the mass transfer process in both reactions should be considered. The cathode and anode current densities can be expressed as Equations 1.68 and 1.69:
02[(1 ) α ]
α η
= − O O c
n F
O c RT
c f
dc
I i I e
I (1.68)
02[(1 ) α ]
α η
= H − a nHRTHF a
a f
da
I i I e
I (1.69)
The relationship between the cell voltage and the cell current density can be simply written as the following equation if the fuel cell polarization is larger than 60 mV:
34 X-Z. Yuan, C. Song, H. Wang and J. Zhang
2 2
0 0
ln( ) ln( )
0.001678 0.5
ln( ) ln( )
0.001678 0.5
α α
α α
= + +
− − −
− −
O H
cell OCV
O H
f f
cell dc cell da
cell el
f f
O dc cell H da cell
RT RT
V E i i
Tn F n F
I I I I
RT RT
Tn F I I n F I I I R
(1.70)
where i0O2is the apparent exchange current density for the cathodic ORR on a pure Pt surface, which should be a different value from that on a Pt/PtO surface. The electron number (naO) should accordingly be taken as 1.0 rather than 2.0. Ida and Idc
are the diffusion limiting current densities of the anode and cathode reactions, respectively, and 0.001678T indicates the temperature-dependent transfer coefficient for the ORR. It is believed that i0H2 should have the same value as that obtained at low current densities because in both low and high current density ranges only the pure Pt surface is involved.
In a very rough treatment, the statuses of the cathode electrode surface at low and high current densities are considered to be the same, as reported in most of the literature. A simple empirical expression can then be given:
Vcell=EOCV−blogIcell−IcellRm−mexp(Icelln) (1.71) where b is the Tafel slope, and m and n are transfer coefficient related parameters,
as discussed below.