Ind. Eng. Chem. Res. 1996,34, 3632-3639
3632
Steady-State Model for an Oxygen Fuel Cell Electrode with an Aqueous Carbonate Electrolyte Kathryn k Stdebel,* Frank R McLarnon, and Elton J. Cairns Energy and Environment Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720
A mathematical model has been developed to describe the steady-state constant-current operation of a n oxygen cathode in aqueous carbonate electrolyte, as might be used a s a component of a direct-methanol fuel cell. The diffision and reaction of oxygen in PTFE-bonded Pt-catalyzed porous gas diffusion electrode agglomerates as well as Ohmic and migration effects over the catalyst layer were taken into account. The model accurately predicts the shape and the oxygen pressure dependence of cathode polarization data for 2 and 4 M KzCO3 electrolyte, using no adjustable parameters. Transport of OH- ions out of the catalyst layer is shown to be a limiting factor in the operation of these modern fuel cell cathodes, which have been generally optimized for maximum oxygen transport in the electrolyte. The model indicates that higher catalyst layer porosities and thinner electrodes would yield better cathode performance in this electrolyte.
Introduction Direct-methanol fuel cells would provide significant advantages over the more common Ha-based fuel cells in an electric vehicle, largely due to system simplicity, i.e., avoidance of the reformer hardware required to convert hydrocarbon fuels to hydrogen on board the vehicle. However, a practical direct-methanol fuel cell will necessarily operate with a COz-invariant electrolyte. This requirement limits the choices to acidelectrolyte cells, which udortunately diminish the performance of the air electrode, or to carbonateelectrolyte cells, which will also reject COz. Work has been proceeding in our laboratory to investigate various aspects of direct-methanol fuel cells with Cs2CO3 and KzCO3 electrolytes. Previously, we present& a comparison between 0 2 reduction kinetics on smooth platinum electrodes in concentrated KzCO3 and KOH electrolytes (Striebel et al., 1990). Comparisons of mass-transfer-corrected current densities for the two electrolytes at the same concentration showed higher currents in the KOH electrolyte, due to the much lower solubility of 0 2 in the carbonate electrolyte. However, when normalized to the same 02 concentration, activity of Pt for oxygen reduction (OR) is about 2 s greater in the carbonate electrolyte. This inter result was partly attributed to a lower degree of Pt oxidation and lower peroxide-production rates in the lower-pH carbonate electrolytes. The polarization behavior of F’t-catalyzed gas-diffision electrodes was studied in the same electrolytes (Striebel et al., 1990). The cathode in carbonate electrolyte exhibited a transition to non-first-order dependence on oxygen pressure and Tafel slopes twice that predicted from kinetic measurements a t extremely low overpotentials. These anomalies were attributed to slow OH-ion transport. This behavior is similar mathematically to the “resistance polarization” or migration effects discussed by Vetter (1967) in general and by Ross (1986) with regard to a gas-diffusion cathode in phosphoric acid electrolyte. Mathematical modeling af the steady-state behavior of fuel cell electrodes has been carried out since the 1960s. Varying degrees of sophistication have been *Author to whom all correspondence should be addressed.
used t o describe the envisioned structure and behavior of the three-phase reaction zone necessary to accommodate the electrocatalytic reduction of oxygen (Grens et al., 1964; Bockris and Cahan, 1969; Bennion and Tobias, 1966; Darby, 1965). Modern polytetraflouroethylene (PTFE)-bonded porous gas-diffusion electrodes (GDEs) have a three-dimensional active region, and have been modeled as a series of flooded catalystcontaining agglomerates connected by hydrophobic gas pores produced by the F‘TFE binder. The cylindrical (Grens et al., 1964; Giner and Hunter, 1969)or spherical (Iczkowski and Cutlip, 1980; Cutlip et al., 1986) agglomerates are typically considered to have small dimensions, compared to the thickness of the active layer. Each agglomerate is therefore assumed to be at uniform electrolyte composition and potential, yielding an analytic solution to the parallel diffision and electrochemical reduction of oxygen. Extensions of this model have included consideration of gas-phase oxygen transport and Ohmic losses in the solid and electrolyte phases across the thickness of the electrode; however the ionic composition of the electrolyte is usually considered to be uniform. This last assumption is poor when the electrolyte is a weak acid (concentrated phosphoric acid) or weak base (aqueous carbonate). In this paper we present a steady-state model of the reduction of oxygen at a gas-diffision electrode in contact with KzCO3 electrolyte based on the flooded-agglomerate model. The effects of ionic migration and the homogeneous acidbase reactions occurring in aqueous carbonate electrolytes are included. Model predictions are then compared with previously reported experimental results to yield suggestions on redesign of GDEs for use with carbonate electrolytes.
Model Development A schematic diagram of the cathode we propose to model is shown in Figure 1. We assume that oxygen diffuses freely through the large PTFE-bonded gas pores to each flooded agglomerate, where it dissolves and diffises through the electrolyte to a reaction site within the agglomerate. Oxygen solubility in concentrated electrolytes is small and can be described by Henry’s law. We assume that the agglomerates are fully wetted by the electrolyte and that the radius of the spherical agglomerate is small compared to the active layer
This article not subject to US.Copyright. Published 1995 by the American Chemical Society
Ind. Eng. Chem. Res., Vol. 34, No. 10,1995 3633 I
agglomerate to the next in the x-direction. For a firstorder reaction with respect to 0 2 concentration so2 = 1. The radial diffision equation (eq 2) can be simplified by defining some parameters:
I
4s-I
where
Porous catalyst partides saturated with electrdyt
/
\
/
/
\
\
and written in the familiar form for diffusion and reaction inside a porous catalyst particle (Bird et al., 1960)
This equation has an analytic solution with the boundary conditions of constant composition at the surface of the particle and zero flux a t its center Figure 1. Schematic diagram of the flooded-agglomeratemodel of the FTFE-bonded porous gas diffusion electrode.
thickness so that the electrolyte ionic composition is uniform within the agglomerate. The potentials in the liquid and solid phases are also taken to be uniform within each agglomerate. Oxygen is reduced via a fourelectron mechanism to hydroxide ion without accumulation of peroxide species in the carbonate electrolyte (Striebel et al., 1990), according to
0,
+ 2H20 + 4e- %!
40H-
a t r = R,,
Co2= Cop, at r = 0, VCo2= 0 (8) (9)
The volumetric current to the GDE is equal to the product of the 0 2 flux a t the surface of an agglomerate and the number density of agglomerates in the electrode:
(1)
Np=
The simultaneous diffision and reaction of 0 2 within a spherical agglomerate takes the form
3(1 - ee) 4nR;
(10)
Equation 9 can be integrated over the surface of the agglomerate to calculate the interfacial current per unit volume of electrode (J)produced by the reduction of 02 where ea and t a are the agglomerate microporosity and tortuosity factors, (1 - €e) is the volume fraction of agglomerates in the electrode, 6 is the thickness of the active layer, and a. is the Pt area per unit area of the electrode. The second term on the right side of eq 2 is the calulated area of platinum per unit volume of agglomerate. The current density ilocal is based on the area of Pt and is assumed to exhibit Tafel-like behavior
YFl
ilocal = nFk~Co2)s02 exp - -
[
(3)
where E is the potential of the solid catalyst with respect to a reference electrode just outside the boundary layer, E = as - Elocd Elocd can be referred to a reference electrode just outside the porous electrode at x = 0, E&,, where
where
Equation 11 takes on the limiting cases of single and double Tafel slope polarization behavior as shown by Giner and Hunter (1969) for small and large values of the overpotential, respectively. Ion transport across the thickness of the active layer is complicated by the dynamic equilibria among the three anions in a carbonate electrolyte according to HC0,-
(4)
We can define @LO = E b H E and assume that COH- is constant within each agglomerate, but varies from one
+ OH- 2 C0:- + H 2 0
(13)
Steady-state material balances for the three anions, including production and consumption terms from reactions 1 and 13 can be combined to eliminate the rate of the equilibrium reaction, as suggested by Hseuh and
3634 Ind. Eng. Chem. Res., Vol. 34, No. 10, 1995 Table 1. Dimensionless Form for Steady-State Model of Oxygen GDE Cathode in K&Os Electrolyte' dimensionless variables dimensionless parameters
20, d, =D6
c
G
e3 = - e4 = 2q' 2q'
e7 = 2G
VF RT
y*=-=
(@s-@L)F RT
Electrolyte Parameters Model parameter 2 M Kzco3 qfp (cm3/mol) 7.71 x 10, q2(umol/cm3) 0.226 DO, (cm2/s) 0.585 x n 4 a, (cm2of wetted FW 172 cm2 of electrode) case 1:6.05 x lo-" io = kynFCE2 (Mcm2) case 2: 1.36 x b = 2303RTIaF case 1: 57 case 2: 62.5 WQH-JDCO~~10.82 W H C O ~ - / D C O ~ ~ - 2.65 WK+lDCO324.25 u = 0.971 (Q-cm)-' 6 = 0.012 cm ea = 0.5
ioaoGF 2, = uRT ' 2, =
Table 2. Parameters for GDE Carbonate Electrolyte
ioa,GFta
C,KRT
d,e,[V(C,*V In C,*,
+ V2C$] + 2 V ( C p In C?) + V2C$+ -(@*coth 4* - 1)= t6
d,e,[V(C,YJ In CT)
are used t o describe the ionic fluxes. This approach is simplistic in the view of the electrolyte concentrations of interest. However, it greatly simplifies the calculations and yields an instructive phenomenological explanation of the cathode behavior. For dilute electrolytes without convection, Ni is expressed as
+
The material balance on K+ ions (V-NK+= 0) can be solved immediately for the potential gradient in the liquid phase, making use of the Nernst-Einstein relation according to Newman (1973)
Because the interfacial current (eq 11)depends only on the difference in potential between the liquid and solid phases, simplification is realized by defining
Y = QL - Q, or V2Y = V2QL- V2Qs (18)
+ Cg - e7C? = 0
Differentiated forms of Ohm's law (in the solid phase) and eq 18 along with the expression for conservation of current in the electrode (J = Vi,) give
boundary conditions
at x* = 0, electrodelelectrolyte interface:
c;=c*=c*-c*4 67-1
(19)
+ e4d4VC4+ 2VC$ - e,d7VC?) = ZLZ*
at x* = 1, electrodelgas-supply interface:
where u is the conductivity of the solid matrix. Finally, the expressions for carbonate equilibrium
w*= -Zd * , v c g = vc;= v q = v q = 0 a
3
= OH-,4
(16)
+ V2C$ + 2 V ( C p In C,*, + V2C,*= 0
e3C$ e4Cf
V Y * + f(e,d3VCg
11.65 2.85 4.25
(*'I3
ct = C3V$
t
2.13 x lo-'' 2.13 x lo-'' 51
Electrode Parameters ta= 5 R, = cm E 2.1 mg/cm2 Wc = 4.5 mg/cm2 W ~ F = = 0.49
Ni = -ziuiFCiVQL - DiVCi dimensionless equations
4 M Kzco3 10.75 x lo6 0.04 0.36 x 10-5 4 100
HC03-, 6
(20)
= cos2-,7 =K+.
and electroneutrality
Newman (1971) for H2SOdHS04(14)
We assume that water is supplied by rapid condensation from the gas phase so that only ionic concentrations are varying in the electrolyte. Dilute-solution expressions, with concentration-independent diffusion coefficients,
and substitution of eq 16 for each ion and eq 11 into eqs 14 and 15 give us a total five equations (eqs 14,15, COH-, 19-21) in five unknowns: CK+,cco32-, CHCO~-, and Y. For controlled-current operation of the cathode, the current will be carried by the electrolyte at the interface x = 0 and by the solid phase at the current collector x = 6. For a well-mixed electrolyte phase (the case for the experimental data) the boundary conditions
Ind. Eng. Chem. Res., Vol. 34, No. 10, 1995 3635 1000
1
E
I
1
i
1
..
loo0
c
4
100
100 :
n^ E
-e< E
10
10;
-
L c
L
Qc 8c c
s
d
1:
€
1
€ B
u
0.1
0.1 :
t 0 0.01
I
.
1
t I
0.01 000
i 1
L
1
700
000
000
I
I
1000
1100
Potontirl VI RHE (mV) Figure 2. Comparison between steady-state model calculations and observed performance behavior for a Prototech electrode in KzCO~ experiment; -, model with io from kinetic measurements (Striebel et al., electrolyte with pure oxygen. (a) 2 M KzCO3; (b) 4 M KzCO3, 1990). - - -, model with io calculated from low current density experimental data. *e*,
can be written as
where K ’ is the effective conductivity of the electrolyte in the agglomerates and I is the current density applied to the electrode. These equations were rendered dimensionless, linearized about a trial solution, and solved with the finitedifference technique utilizing the BAND program (Newman, 1973). The dimensionless parameters and the resulting equations and boundary conditions are summarized in Table 1. One hundred mesh points were used and convergence usually occurred in two to three iterations to a tolerance of lo+.
model, we used the value of a, measured immediately before the polarization measurements. Electrolyte properties for concentrated aqueous carbonate electrolytes were taken from the literature as follows and were assumed to be independent of position x . Oxygen solubilities were estimated using the salting-out expression given by Schumpe, Adler and Deckwer (1978)
where HK+= -0.596 and Hco32- = 0.485. Oxygen diffisivities were calculated using the estimated oxygen solubility along with the previously measured value of the Levich slope from rotating disk experiments in the same electrolytes (Striebel et al., 1990). Ionic and water activities (a, and fi), reported by Roy et al. (1984), were used to calculate the apparent carbonate dissociation constant, Gfp,which can be written as
Parameters Table 2 summarizes the electrolyte and kinetic parameters used to model the polarization behavior of the Prototech GDE with 100% 0 2 in 2 M and 4 M %COS electrolyte. The exchange current density (io) and the Tafel slope ( b ) were previously measured for these electrolytes via rotating-disk electrode experiments on smooth Pt electrodes (Striebel et al., 1990). The area of F’t catalyst wetted by the electrolyte (a,)was calculated from the charge due to hydrogen desorption as determined with cyclic voltammetry, immediately before and &r the polarization measurements (Striebel et al., 1990). The value of a, after a polarization measurement was always bigger than that measured before a polarization measurement in the concentrated carbonate electrolytes. This was due to a n increase in catalyst wettability with increasing electrolyte pH. In the
Ionic diffisivities were estimated following the suggestions of Lin and Winnick (1974) from conductance and viscosity data for C s ~ C 0 3electrolyte. The uncertainty in the transport parameters led us to invoke the Nernst-Einstein relation and use ody dfisivity ratios, DjDco3z- for i = OH-, HC03-, and K+. These values were then scaled by measuring bulk electrolyte conductivity with a Wayne-Ken- conductivity bridge. The electrode structural parameters 0,z, and R, were taken from various sources. The constant 0 for this type of electrode was reported by Izckowski and Cutlip (1980) from fits of their version of the flooded-agglomerate model to 0 2 cathode data for phosphoric acid electrolyte. The tortuosity factor for carbon black has been suggested t o fall in the range of 4-6 (Ross, 1986). A value of 5 was used in this work. Cutlip et al. (1986)measured
3636 Ind. Eng. Chem. Res., Vol. 34,No. 10, 1995 10
........................................................ 1
-------°
0.01 I
0 0.2 electrolyte side
0.4
0.6
I
0.2
0.4
I
I
0.6
0.8
i
I 1
0.8
gas side
XI8
I
........................................................
0.001
0 electrolyte side
1
XI8
........................................................
--------
11
°
i
1
O.Ol*
0.001
0.0001
0.0001
0 0.2 electrolyte side
0.4
0.6 XI8
1
0.8
gas side
0.00001I 0 electrolyte side
I 0.2
I 0.6
I 0.4
XI8
I 0.8
I 1
gas side
Figure 3. Predicted ionic concentration profiles for 2 M &CO3 electrolyte and pure oxygen at several current densities: (a) 0.1, (b) 1, CK*. (c) 10, and (d) 100 mAhm2. (- A) COH-;(- - - x ) CHCO~-; (- -0)Cco32-;
an agglomerate radius of about cm for PTFEbonded carbon particles by SEM examination. From geometric considerations we have the additional relation
Wc -+-
WFTFE
where Wc,@e,WPTFE, and @PTFE are the loading and density of carbon and PTFE, respectively, and 6 is the measured thickness of the active layer. The porosity of the catalyst layer, ce, and the porosity of the agglomerate particles, ca, were taken to be 0.5 and 0.49, respectively, from a fit between experimental data and a similar model for pure 6.9 M KOH electrolyte (Striebel, 1987). There were no adjustable parameters, therefore, in the application of the model.
Results and Discussion The model was used to calculate electrode potentials for input current densities matching experimental
values of oxygen electrode performance curves from Striebel et al. (1990). At each current density, electrolyte concentration profiles within the active layer of the electrode were generated, along with local potential profiles, current distribution profiles, and effective electrolyte conductivity profiles. The calculated performance curves are compared with half-cell (IR-corrected) polarization data in Figure 2 for 2.0 and 4.0 M K&O3 electrolyte. Two different values of io were used for the 2 M KzCO3. The value measured on a smooth Pt electrode (case 1)resulted in predicted potentials that were higher than the measured potentials. The lower value (case 2) was estimated from the low current data from the supported electrode. The model predicts a transition t o the double-Tafel slope at current densities of about 1&cm2 as compared with about 100 &cm2 in pure KOH (Striebel et al., 1990). The deviation between model and experiment for 4 M KzCO3 was probably caused by the greater variability in the wetting of the electrode by this electrolyte. Calculated ionic concentration profiles a t several current densities are shown for the 2 M K&03 electro-
Ind. Eng. Chem. Res., Vol. 34,No. 10,1995 3837 10
0 8 7 C
.-c0
= e . I c L. 0
6
6
c
c 0
t
a
4
3 2
1 01 0
I
I
0.8
1
I
I
I
0.2
0.4
0.8
I
0
0.2
I
1
I
0.8
1
I
0.8
0.4 X0
X0
Figure 4. Predicted dimensionless current distribution (Jd/i) profiles for (a) 2 M and (b) 4 M KzC03 electrolyte, pure oxygen. (-A) i = 0.1 mA/cm2; (- - - x ) i = 1.0 mA/cm2;(- -0) i = 10 mA/cm2; (**a) i = 100 mA/cm2. l0O0
1:
0.1
The OH- species is diffusing through a relatively constant concentration of KzC03, but never attains a concentration greater than about 0.2 M , except a t the highest current density. Current distribution profiles are shown in Figure 4 for the two electrolytes at several values of applied current density. Except for the very front of the electrode, the current distribution is fairly uniform. These curves suggest that electrode structure could be optimized with a thinner active layer, effectively producing a shorter diffision path in the liquid phase. Polarization curves were calculated for the 2 M electrolyte at three values of the 0 2 pressure (see Figure 5). The dependenceof current density onpOzat constant electrode potential switches 6.om first order to half-order as shown in the inset. The general behavior of the model for three of the dimensionless parameters, Zs,ZL, and [*2 was investigated for 2 M KzCO3. These parameters can be considered as ratios of kinetic to Ohmic resistance in the solid phase, kinetic t o Ohmic resistance in the liquid phase, and electrode kinetics to the diffision rate of oxygen in the agglomerates, respectively. Figure 6 shows predictions for "base case", high values and low values of these parameters. Increases in the electrolyte conductance (decrease in ZL) show the greatest influence on electrode performance, as expected for an ionic masstransfer-limited process.
23.
1
Wl
0.1
I
QlrrrnrnW)
0.01 600
,
t
600
700
I
I
BOO BOO Potentirl VI RHE (mV)
1
1000
1100
Figure 5. Calculated performance curves for variable oxygen pressure in 2 M KzC03 electrolyte. The inset shows the dependence of current density onpo, at E = 0.7 and E = 0.97 V. -A,PO, = 1.0 atm;- - -x, PO, = 0.21 atm; - -0, PO, = 0.04 atm.
lyte in Figure 3. The OH- ion concentration a t the gassupply side reaches a value of 132 times the bulk value of 0.017 M at a current density of 100 mA/cm2. The average OH- ion concentration (across the electrode) at this current is 95x or 1.6 M. If changes in the activity coefficients are neglected, this corresponds to a pH change of 2 units or 120 mV! This compares well with the 160 mV shift in the R cyclic voltammogram in this electrolyte before and after the polarization measurement (Striebel et al., 1990). It can be noted that our use of dilute-solution flux expressions in the model may be justified in light of the calculations shown in Figure
Conclusions and Recommendations We have a constructed steady-state model for the 0 2 cathode in KzCO3 electrolyte, accounting for oxygen diffusion and kinetics in the catalyst particles, ionic transport in the electrolyte, and Ohmic conduction in the solid phase. The model successfully predicts the low-current transition t o the double-Tafel slope region and the half-order dependence on 0 2 partial pressure that are observed experimentally. Calculated performance curves agree remarkably well with the data for
3638 Ind. Eng. Chem. Res., Vol. 34, No. 10, 1995
is = current density in solid phase (A/cm2) J = interfacial current per unit volume (A/cm3) K1laPp = apparent carbonate dissociation constant (mol/cm3) k, = chemical rate constant (cm/s) mi = molality (moVkg) n = number of electrons N = flux (moY(cm2.s)) Np = number density of agglomerates in active layer (number of agglomerates/cm3) p = partial pressure (atm) r = radial coordinate R = gas constant (J4mol.K)) Rp = particle radius (cm) s1 = reaction order t = time (s) T = temperature (K) u = ionic mobility (cm2*moY(J*s)) WC = carbon loading (mg/cm3) Wpt = R loading (mg/cm3) W ~ = EPTFE loading (mg/cm3) x = distance coordinate ZL= dimensionless liquid conductivity ratio 2s = dimensionless solid conductivity ratio z = multiple of electron charge Greek Symbols
a = transfer coefficient -24
-22
-20
-18
-16
-14
-12
-10
E,'
Figure 6. Sensitivity of calculated dimensionless performance curves for 2 M K&03 electrolyte. For the dimensionless parambase case = 5.635 eters, (a)Zs,base case = 1.126 x lo-'; (b) ZL, -A, base case; -x, x ( c )
