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Impedance Analysis of Molten Carbonate Fuel Cell
Final Project Report
In partial requirement of ECHE 789B course
Nalini Subramanian
Course Instructor: Dr. Branko N. Popov
Date submitted: May 5,2002
Abstract
A three phase homogeneous model using averaging technique is developed to
study the impedance response of a molten carbonate fuel cell cathode, LiNiCoO2 in
particular. The simulation for different parameters in the model are presented along with
equivalent circuit fitting of the experimental data
Introduction
Molten carbonate fuel cell is an electrochemical device that continuously
transforms the chemical energy of the fuel and oxidant gas into electrical energy. The
porous electrodes of a fuel cell are under mixed control of electrode kinetics, mass
transfer and ionic conduction. AC impedance technique provides considerable insight
into electrode processes. The measurements from AC impedance can be used to analyze
the rate limiting process in an MCFC electrode. AC impedance can be combined with
other electrochemical techniques to get a broader picture of the electrode process.
Analyzing porous electrode data using equivalent circuit models is not feasible as in
planar electrodes. So a first principle model to study charge transfer and mass transfer
resistance and assess their importance relative to ohmic resistance in porous electrodes
under mixed control must be developed.
Literature Review
There are several models to describe the impedance in porous electrodes. Also
models for the characterization of MCFC cathode by AC impedance have been developed
by previous researchers (1-4). Yuh and Selman (1) developed an agglomerate model for
the analysis of fuel cell electrodes which incorporated the effects of mass transfer,
polarization level and electrode geometry. The porous electrode consists of agglomerates
of electrocatalyst particles, which contain the electrolyte in micropores formed by the
interstices between these particles. They agglomerates are themselves separated by the
gas filled macropores as shown in Fig.1. Two types of agglomerates were analyzed,
cylindrical and planar and analytical solutions obtained for each of them. Later Lee and
Selman (3) provided a theoretical analysis for planar electrode and analyzed the
experimental impedance results for a partially submerged smooth electrode using
equivalent circuits. Pins-Jansen el al. (2) used the three phase homogeneous model based
on an averaging theory of porous media. The combined the mass transfer in the liquid and
gas phases. In this paper we use this volume averaging technique to develop an
impedance model but consider transport in the gas and liquid phases separately.
Macropore
Gas Channel
z = L
z = 0
r = Rr = 0
Bulk Gas Flow Film δ
Electrolyte Tile
Agglomerate
Current Collector
Fig.1. Agglomerate model of a molten carbonate fuel cell cathode
Model development
In the molten carbonate fuel cell, oxygen and CO2 combine at the cathode to form
carbonate ions. At the anode hydrogen combines with the carbonate ions from the
cathode to form CO2 and water. The net reaction results in the formation of water with no
harmful side reactions. The system of interest to us is the cathode where reduction of
oxygen occurs. In order to overcome the difficulties associated with the agglomerate
approach, we start by considering a cross-section of the porous electrode as shown in Fig.
2. No difference is made between the macropores and micropores while deriving the
model equations. The primary reaction in the MCFC cathode is oxygen reduction, which
is given by:
−− →++ 23222
1 CO2eCOO (1)
The above reaction occurs at the interface between the NiO particle and the electrolyte.
We neglect any changes in the concentration of the carbonate ions and assume that the
concentration of the electrolyte does not change. Further, we assume that the system is at
steady state and neglect any changes in cathode due to corrosion. Finally, we neglect
changes in temperature in the cathode. Based on these assumptions we next derive the
volume-averaged equations describing transport and reaction in the MCFC cathode.
Concepts and Definitions of Volume Averaging
In this section, equations are derived for a porous electrode consisting of three
phases: solid, liquid and gas. Following De Vidts12, 13 we consider a small elemental
volume V. This volume should be small compared to the overall dimensions of the
porous electrode. But it should be large enough to contain all three phases (see Figure 2).
Also it should result in meaningful local average properties. This volume is so chosen
that adding pores around this volume does not result in a change in the local average
properties. We avoid the bimodal pore distribution where we consider macropores to be
filled with the gas and micropores to be occupied by the electrolyte. Rather pores of all
sizes are filled with both the electrolyte and the gas, which is more realistic. Some basic
definitions of volume averaging have to be presented before understanding the
development of the model equations.
Superficial volume average ψ and the intrinsic volume average ψ are defined as
( )
( ) 1
i
i
V
dVV
ψ ψ≡ ∫ (2)
( )
( )
( )
1
i
i
i V
dVV
ψ ≡ ∫ ψ (3)
Here the superscript i represents the phase. The superficial and intrinsic volume averages
are related by the porosity. ( ) ( )( )i iiψ ε ψ= (4)
Whenever volume averages of the gradients and the divergence appear they should be
replaced by the gradients and divergence of the volume averages as below. These are
referred to as the theorem of the local volume average of the gradient and the divergence.
14,15
lg l
( ) ( ) ( ) ( )(lg) (l )
1 1
s
l l ls
S S
n dS n dSV V
ψ ψ ψ ψ∇ = ∇ + +∫ ∫ l (5)
lg l
( ) ( ) ( ) ( )(lg) (l )
1 1
s
l l l ls
S S
n dS n dSV V
ψ ψ ψ ψ∇⋅ = ∇ ⋅ + ⋅ + ⋅∫ ∫ (6)
n(ls)
n(gs
n(gl)
Electrolyte Phase V(l)
Solid Phase V(s)
Gas Phase
V(g)
Matrix
x=L
x=0
Current Collector
n(lg)
Fig 2. Volume Averaging in Porous electrode
Mass transport equations
Mass transport occurs in the liquid and gas phases. Both oxygen and carbon
dioxide gas are fed to the MCFC cathode through the current collector. Both O2 and CO2
diffuse through the macropores in the cathode dissolve in the melt and are transferred by
diffusion to the surface of the NiO particles. The material balance in the liquid and gas
phases for any species i is given by
( )( )
20 ,l
lii
c N i COt
∂+∇⋅ = =
∂ 2O (7)
( )( ) 0
ggi
ic N
t∂
+∇ ⋅ =∂ (8)
There is no bulk reaction. All reactions are assumed to take place at the
electrolyte-electrode interface. This is denoted by the normal vector nls in Fig. 2. Gas
diffuses into the electrolyte at the normal interface ngl and reacts at the interface of the
electrolyte with the solid catalyst particles, nls. Hence the homogeneous reaction rate is
neglected. Fick’s law gives molar flux in the liquid and gas phases.
( ) ( ) ( ) ( )l l li i i iN D c c ∗= − ∇ + l v (9)
Binary diffusion is assumed in the gas phase. For a binary system the mass flux
relative to the mass average velocity is given by( )Aj 13
( ) ( ) ( ) ( ) ( )
2
A A B ABcj M M Dρ
= − ∇ Ax (10)
where A refers to O2 and B refers to CO2.
The relation between (molar flux relative to molar average velocity), ( )AJ ◊( )Aj◊ (mass flux
relative to molar average velocity) and for a binary system is given by ( )Aj
( )( )
( )
AA
A
jJ
M
◊◊ = (11)
( )( )
( )AB
MAj j
M◊ = (12)
The relation between (molar flux with respect to a fixed frame of reference) and ( )AN ( )AJ ◊
( ) ( ) ( )A A AJ N c◊ = − v◊
A
(13)
When convection is neglected
( ) ( )AN J ◊= (14)
Hence
( ) ( ) ( )A ABN cD x= − ∇ A (15)
( )( )A
A
cx
c= (16)
( )( )
( ) ( ) ( )A
A AB AB
cN D c D
c= ∇ − ∇ Ac (17)
In general for a binary gas the flux is given by,
( ) ( ) ( ) ( )( )
( )
gg g g g gi
i i i i g
cN D c Dc
= − ∇ + ∇
c (18)
Using the definitions of volume averaging we obtain the volume averaged flux in both
phases as, ( ) ( ) ( )( ) ( ) ( )( )1
bl ll l l
i i iN D cε ε−
= − ∇ (19)
( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( )
( )
( )( ) ( )( )1 1
g
b bg g gg g g g g gii i i i g
cN D c D c
cε ε ε ε
− −= − ∇ + ∇ (20)
Volume averaging Eqns. 7 and 8 and substituting the above definitions in Eqns. 19 and
20 gives the following volume averaged mass balance equations, ( )
( ) ( )lg0
llsi l
i iic N F Rt
∂+∇ ⋅ + − =
∂ (21)
( )( ) ( )lg
0g
gsi gi ii
c N F Rt
∂+∇⋅ − − =
∂ (22)
where ( )lgiF ,
lsiR and
gsiR are all derived from jump balances.
( )lgiF is the flux of species i
from the liquid to the gas phase, lsiR the rate of heterogeneous reaction at the liquid solid
interface and gsiR at the gas solid interface.
( ) ( ) ( )lg lg lgi iF a r= (23)
( ) ( )( )
( )lg lg
,
lgi
i i ie i
cr k c
K
= −
(24)
where for any species i, ki is the mass transfer coefficient and Ke,i is the distribution
coefficient. Rate of production of species i at the solid liquid interface is expressed in
terms of the local current density. Butler-Volmer kinetics is assumed for the reaction at
the electrode electrolyte interface.
( ) ( )( )
slls slik
i kk k
s aRn F
= − < >∑ j (25)
( )( )
( )
( )
( )
( )
( )
( )
( )
1 2 1 2
2 2 2 2
2 2 2 2
0* * * *
exp expl l l l
p p q ql l l l
sl CO CO CO COa ck
CO CO CO CO
c c c cF Fj iRT RTc c c c
α φ α − = −
φ
(26)
Here ( )slkj< >
16 is the local current density at the solid liquid interface and i and i are
the concentration dependent and concentration independent exchange current densities
respectively
000
13. The anodic and cathodic reaction orders p1, p2 and q1, q2 have values of –
2, 0, -1, 1/2 respectively.
( ) ( )1
2 2
0 * *0 0
r
CO Oi i p p= 2r (27)
where r1 and r2 have a value of –1.25 and 0.375 respectively for the peroxide mechanism.
These values will be different for other mechanisms3. At the gas-solid interface there is
no reaction. Hence, ( )
0gs
iR = (28)
Charge transfer equations
Since we neglect any changes in the concentration of , the effect of
migration need not be considered. Hence, Ohms’ law is valid in both the solid and liquid
phases.
−23CO
( ) ( )li κ φ= − ∇ l (29)
( ) ( )s si σ φ= − ∇ (30)
Volume averaging the current in the solid and liquid phases results in the following
equations.
( ) ( )( ) ( ) ( )( )1dl ll li κ ε ε φ−
= − ∇ (31)
( ) ( )( ) ( ) ( )(1ds )ss si σ ε ε φ−
= − ∇ (32)
The condition of electroneutrality applies everywhere within the electrode. This means
that the net sum of the solution and solid phase currents should be constant. ( ) ( )
( )l s
i i∇⋅ + = 0 (33)
Further, any current leaving the solid phase has to enter the liquid phase through the
electrochemical reaction. Applying a balance on the solution phase current gives,
( ) ( ) ( )( ) ( )( )s l
l slslk dli a j C
t
φ φ ∂ −∇ ⋅ = + ∂
(34)
In the above equation the gradient in the solution phase current is proportional to the
reaction rate at the solid-liquid interface. Substituting Eq. 34 into Eq. 33 we have,
( ) ( ) ( )( ) ( )( )s l
s slslk dli a j C
t
φ φ ∂ −∇ ⋅ = − + ∂
(35)
Next, we define the overpotential as ( ) ( )s lφ φ φ= − . Combining Eqns. 31 – 35 and
using the definition for overpotential results in,
( )( )( ) ( )( )
( )2
2
1 1sl sldl kd ds l
a Cx tφ φ
σ ε κ ε
∂ = + ∂ ∂
j∂
+ (36)
Method of Solution
Once the equations are been derived we look at how we can do impedance
analysis on the system. For this we follow the procedure adopted by Doyle et al. (6) for
Lithium Rechargeable batteries.
1. Linearize the equations
0 01 2 1 1 2 2
1 2
( , , ) ( ) ( ) ( )sF F FF c c F c c c cc c
φ φ φφ
∂ ∂ ∂= + − + − + −
∂ ∂ ∂0
2. Introduce deviation variables
0 01 1 2 2 1; ;c c c c c c φ φ φ= + = + = + 0
3. Convert into Laplace domain
1 21 2( ) ; ( ) ; ( )L c c L c c L φ φ= = =4. Express each Laplace variable as a sum of imaginary and real part
r the real and
imulation Results
d a qualitative agreement with the results obtained by the previous
1 21 1 ; 2 2 ;c c r c im c c r c im r imφ φ φ= + = + = +Then the set of resulting equations are solved using BAND(j) fo
imaginary parts of each variable. The open circuit condition is taken as the steady
state condition and hence all the steady state concentration gradients and the current
density become zero. The procedure used is to calculate the impedance is to solve the
set of linear equations at each frequency with the external current taken to be purely
real and unit magnitude. The magnitude chosen for current does not affect the
simulation results because of the explicit assumption of a linear response. In this case
the impedance is given by the relationship
S
The simulations showe
researchers. The impedance increased with the decrease of electrolyte conductivity (fig.
3), exchange current density (fig. 4) and the diffusion coefficients. Electrode conductivity
hardly had any effect on the total impedance. The gas compositions were varied and the
results showed that impedance increases for increasing CO2 partial pressure and
decreases for increasing O2 partial pressure (fig. 5), which is due to the negative and
positive dependence on CO2, and O2 partial pressure respectively
0.0
Fig.3 Effect of electrolyte conductivity
0.5 1.0 1.5 2.0 2.5Zreal (ohm)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Zim
(ohm
)
ke = 1.5e-1
ke = 4.5e-2
ke = 1.5
Fig.4 Effect of exchange current density
0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0
Zreal (ohm)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Zim
(ohm
)
i0=50 mA/cm2
i0=5 mA/cm2
i0=1 mA/cm2
Fig Effect of gas compositions
0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6
Zreal (ohm)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Zim
(ohm
)
30%CO2:14.7%O2
30%CO2:30%O215%CO2:45%O2
Increasing O2
.5
Experimental method
nce studies were done in a 3-cm2-lab cell. LiNi0.8Co0.2O2 was
used as
s
Half-cell performa
the working and counter electrodes. (Li0.62K0.38)2CO3 eutectic embedded in a
LiAlO2 matrix was used as the electrolyte. Polarization studies were done using an
oxidant gas composition of 70% air and 30% CO2. Two oxygen reference electrodes
(Au/CO2/O2) connected to the electrolyte tile with a salt bridge (50%(Li0.62K0.38)2CO3 +
50%LiAlO2) were used to monitor the polarization of cathode. Electrochemical
impedance spectroscopic studies were performed using a Model 1255 Schlumberger
Frequency Analyzer. The electrode was stable during the experiments and its open
circuit potential changed less than 1 mV. The impedance data generally covered a
frequency range of 1 mHz to 100kHz. A sinusoidal ac voltage signal varying by ± 5 mV
was applied in all cases. The results of the experiment are presented in the figure 6(a-f)
to follow.
0.0 1.1 2.2 3.3 4.40
1
2
3
40:60:162.5:60:1610:60:1620:60:1630:60:1645:60:1660:60:16
Real Z'(Ω)
-Im
agin
ary
Z"(Ω)
Increasing [CO2]
1 20.0
0.5
1.0
1.5
2.0
2.5
33:60:033:60:2.533:60:7.533:60:1533:60:22.533:60:30
Real Z'(Ω)
-Im
agin
ary
Z "(Ω
)
Increasing [O2]
3
Fig. 6(a) 650o C Fig. 6(b) 650o C
0 1 2 30.0
0.5
1.0
1.5
2.0
2.5
33:60:3033:60:22.533:60:1533:60:7.533:60:2.533:60:0
Real Z'(Ω)
-Im
agin
ary
Z"(Ω
)
Increasing [O2]
1 20.0
0.5
1.0
1.5
2.0
2.5
0:60:162.5:60:1610:60:1620:60:1630:60:1645:60:1660:60:16
Real Z'(Ω)
-Im
agin
ary
Z"(Ω
)
Increasing [CO2]
3
Fig. 6(c) 700o C Fig. 6(d) 700o C
0.6 1.2 1.8 2.40.0
0.5
1.0
1.5
2.0
0:60:1610:60:1620:60:1630:60:1645:60:1660:60:16
Real Z'(Ω)
-Im
agin
ary
Z"(Ω
)
Increasing [CO2]
0.7 1.2 1.7 2.20.0
0.5
1.0
1.5
2.033:60:033:60:2.533:60:7.533:60:1533:60:22.533:60:30
Real Z'(Ω)
-Im
agin
ary
Z"(Ω
)
Increasing [O2]
Fig. 6(e) 750o C Fig. 6(f) 750o C
Equivalent Circuit fitting of the experimental data
The experimental data for different compositions for three different temperatures
for LiNiCoO2 was fit to an equivalent circuit as shown in Fig.7. as a preliminary study.
The fit parameters are given in Table I for different CO2 compositions at 6500C for
LiNiCoO2
DE1
CA2 Rs CA1
DE2
RA2 RA1
Fig.7 Equivalent circuit used to fit the experimental data
Table I. Fit parameters using equivalent circuit for different
compositions of CO2 at 6500C
Parameter 0/120/32 5/120/32 20/120/32 40/120/32 90/120/32 120/120/32
Rohm 0.64071 0.65873 0.77663 0.64842 0.64891 0.64632
R1 2253 3.145 2.513 41.65 31.31 179.8
C1 3.378E-12 0.027249 3.6142E-8 0.02582 0.025409 0.023291
DE1-R 0.25923 0.71713 1.331 11.65 37.54 79.09
DE1-T 0.086228 0.064494 0.41379 98.79 1411 10416
DE1-P 0.80557 0.53048 0.57997 0.31042 0.30711 0.30116
DE1-U 0.1 0.1 0.1 0.1 0.1 0.1
R2 4.15 9.363 7.864 2.098 2.354 2.111
C2 0.31472 6.0682E-8 0.46935 3.273 4.785 6.231
DE2-R 2.677 5.065 7.22 5.924E+9 7.293E+13 7.293E+13
DE2-T 0.60695 1.404 1.449 9.690E-18 9.690E-18 4.773E-12
DE2-P 0.62767 0.80755 0.78503 0.60913 0.60913 0.60913
DE2-U 0.1 0.1 0.1 0.1 0.1 0.1
Conclusions
The model simulations showed a qualitative agreement with the results published earlier.
Electrolyte conductivity and exchange current density have a larger effect on the total
impedance. An equivalent circuit model was used to fit the experimental data of
LiNiCoO2 and the results are yet to be analyzed.
List of Symbols g
2CO Volume averaged concentration of CO2 in the gas phase, mol/cm3
g2O Volume averaged concentration of O2 in the gas phase, mol/cm3
l2CO Volume averaged concentration of CO2 in the liquid phase, mol/cm3
l2O Volume averaged concentration of O2 in the liquid phase, mol/cm3
( )*2
gCO Bulk concentration of CO2 in the gas phase, mol/cm3
( )*2
gO Bulk concentration of O2 in the gas phase, mol/cm3
( )*2
lCO Bulk concentration of CO2 in the liquid phase, mol/cm3
( )*2
lO Bulk concentration of O2 in the liquid phase, mol/cm3
(lg)a Specific surface area at the gas/liquid interface, cm2/cm3 (sl)a Specific surface area at the liquid/solid interface, cm2/cm3
b Correction for diffusion coefficient
c Total concentration, mol/cm3
( )ic Concentration of species i, mol/cm3
d Correction for conductivity
2
( ) gCOD Diffusion coefficient of CO2 in the gas phase, cm2/s
2
( ) gOD Diffusion coefficient of O2 in the gas phase, cm2/s
2
( ) lCOD Diffusion coefficient of CO2 in the liquid phase, cm2/s
2
( ) lOD Diffusion coefficient of O2 in the liquid phase, cm2/s
I Applied current, A/cm2 00i Concentration independent exchange current density, A/cm2
0i Concentration dependent exchange current density, A/cm2
( )li Current density in the electrolyte, A/cm2
( )si Current density in the solid, A/cm2
( )iJ ◊ Molar flux of species i relative to molar average velocity, mol/cm2s
( )ij◊ Mass flux of species i relative to molar average velocity, mol/cm2s
( )ij Mass flux of species i relative to mass average velocity, mol/cm2s
kj Average local current density due to reaction k taking place at the liquid/solid
interface, A/cm2
2,e COK Equilibrium constant relating the concentration of CO2 in the liquid and gas ( )l
phase,( )
( )2
2
*
*
l
CO
g
CO
c
c
2,e OK Equilibrium constant relating the concentration of O2 in the liquid and gas
phase,( )
( )2
2
*
*
l
O
g
O
c
c
2
(lg)COk Rate constant of molar flux of CO2 between the liquid and gas phase cm/s
2
(lg)Ok Rate constant of molar flux of O2 between the liquid and gas phase cm/s
L Thickness of the electrode, cm
( )iM Molecular weight of species i, gm/mol
iN Molar flux of species i with respect to a fixed frame of reference, mol/cm2s
giN Volume averaged molar flux of species i in the gas phase, mol/cm2s
liN Volume averaged molar flux of species i in the liquid phase, mol/cm2s
( )lgn Unit normal vector to the surface S(lg) pointing out of the liquid into the gas phase
( )l sn Unit normal vector to the surface S(ls) pointing out of the liquid into the gas phase
2
*COp Equilibrium partial pressure of CO2, atm.
2
*Op Equilibrium partial pressure of O2, atm.
( )lgS Surface that coincides with the liquid/gas interface inside volume V, cm2
( )l sS Surface that coincides with the liquid/solid interface inside volume V, cm2
V Volume of porous media, cm3
( )iV Volume of phase i in the porous media, cm3
( )ix Mole fraction of species i
φ Overpotential, V
( )lφ Liquid phase potential, V
( )sφ Solid phase potential, V
cα Cathodic transfer coefficient
aα Anodic transfer coefficient
(g)ε Gas porosity (l)ε Liquid porosity (s)ε Solid porosity
κ Electrolyte conductivity, S/cm
( )iρ Density of species i, gm/cm3
σ Electrode conductivity, S/cm
v Mass average velocity, cm/s
v◊ Molar average velocity, cm/s
Reference
1. C.Y. Yuh, J.R. Selman, AICHE journal, 34, 1949 (1988)
2. J.A. Prins Jansen, Joseph Fehribach, K. Hemmes, and J.H.W. de Wit,
J.Electrochem.Soc., 143, 1617(1996)
3. G.L. Lee and J.R. Selman, Electrochemica Acta, 38, 2281(1993)
4. J.A. Prins Jansen, G.A.J.M. Plevier, K. Hemmes and J.H.W. de Wit,
Electrochemica Acta, 41, 1323(1996)
5. R.C. Makkus, K. Hemmes, and J.H.W. de Wit, J.Electrochem.Soc., 141,
3429(1994)
6. Marc Doyle, Jeremy P. Meyers, and John Newman, J.Electrochem.Soc., 147,
99(2000)