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)