Heinz Nabielekpbzforschungszentrum Jülich, 23 December 1999

Heinz NabielekPBZForschungszentrum Jülich, 23 December 1999

Christian GindorfIWV2

Rainer von SeggernZAM

A Quantitative Model of Chromium Transport
through the Air Channel of SOFC Stacks

Chromium from the interconnect arriving at the cathode/ electrolyte interface contributes significantly to cell degradation during SOFC stack operation by reducing its electrocatalytic activity. Chromium, if released from the interconnect, can take two pathways:

• Via interconnect rib and contact paste: experiments have shown that Cr passes largely unhindered through LSM contact paste leading to degradation rates in the range 20–50% per 1000h. On the other hand, the older LC contact paste delays the Cr transport significantly.
• Chromium is transported via evaporation from air channel surfaces and subsequent gaseous transport of Cr6+ species, mostly CrO2(OH)2. When arriving at the cathode, reduction to Cr3+ and immobilisation at the electrolyte-near area of the cathode take place with resulting degradation rates around 5% per 1000h.

In response to considerations of Cr removal by air flow in the second case[1], a diffusion model for chromium transport and convective air flow in the gas channels has been developed. The model predicts the fraction of chromium arriving at the cathode as 1-{1-exp[-F/(1-F)]}/{F/(1-F)} where F=(1+2 sinh /)/3cosh and 2=vh2/LD. Here, v is air velocity, D is the diffusion coefficient of CrO2(OH)2 species in air, h is channel width and height, and L is channel length.

For 2cell1010 cm2 SOFC stacks of Jülich design B1002 or C1002, and 800°C operation under usual test conditions, we get =0.703 and a chromium deposition rate fraction of 81% as part of all Cr evaporated from air channel interconnect surfaces. With an unrealistic tenfold air throughput, only 23% chromium gets to the cathode via the air channel.

The conclusion is that convective air flow does not reduce chromium transport to the cathode significantly.

To effectively improve SOFC lifetimes, it will be necessary to minimise Cr evaporation from the interconnect surfaces by either application of a protective layer or by developing a suitable interconnect alloy with reduced Cr evaporation rates.

Problem Definition

During a recent discussion[2] in Jülich, it has been observed that (i) microprobe measurements show massively Cr in the cathode area near the interface to the electrolyte when using LSM contact paste, but (ii) very little Cr in the cathode when using LC contact paste. Microprobe measurements after stack operation were done on:
FZJ SOFC stack C1002-4LSMU/LC1044h21.5%/1000h
FZJ SOFC stack B1002-8LC3302h6.6%/1000h

It was then speculated by the Stack-Postoperation-Examination group that most Cr evaporated from interconnect air channel surface walls will be carried away by the air stream and appropriate measurements should be performed to guarantee the Cr removal. In anticipation of the results of the present study, it appears that—under usual Jülich SOFC stack operating conditions—little chromium is removed by air flow. Nevertheless, experiments should be done to confirm the model predictions.

The geometry of gas channels in a SOFC stack is shown below.

In a test of duration T, the general description of the transport of Cr6+ species in the air channel of width w, height h and length L is given by the solution of the diffusion equation

Eq. 1

where

is the concentration of gaseous species in the air channel

is the diffusion coefficient of CrO2(OH)2 in air
is the air velocity

Boundary conditions for the solution of the diffusion equation (Eq. 1) are

• Initial
• Start of air channel
• The boundary condition for z=L is difficult to formulate mathematically. One possibility is to define the problem for , but let the channel wall boundary conditions (detailed below) act only for .
• Zero concentration of Cr6+ species at the cathode surface because of the reduction to Cr3+:
• For the length of the air channel , constant evaporation rate from air channel side walls
  ; ;
  and the rear wall
  .

where co is the concentration of Cr in the interconnect and is the evaporation rate derived from measurements under SOFC operating conditions.

Since the general problem in x, y, z, t cannot be solved easily, we will proceed stepwise by various simplifications and demonstrate a sufficiently concise solution.

The solution of interest is the amount deposited onto the cathode surface

.

It should be noted that – within the framework of the model – the amount of chromium arriving at the cathode will be

• strictly proportional to the evaporation coefficient ,
• strictly proportional to stack operation time T.

Data Collection

Considering the usual 2cell1010 cm2 stack, we have w= h= 1.5 mm, L= 10 cm and 30 equidistant air channels on the cathode side of the interconnect plate. Thus, the standard X10CrAl18ferritic steel interconnect (1.4742, KTN) has a volume of 46.5 cm3 and – assuming density 7.7 g/cm3 – weight 356 g. The effective air-channel-side surface is 135 cm2. With 17.5% Cr in the interconnect, the Cr density in the bipolar plate is co= 1.35 g/cm3. For evaluation of the evaporation rates from the ODS alloy Cr5Fe1Y2O3, we assume 94% Cr content and density 7.7 g/cm3.

Chromium evaporation rates from metallic interconnects have been measured by Gindorf under carefully controlled laboratory conditions simulating SOFC operation on samples of 830.5 cm3 volume. The measured[3] activation energy is 69 kJ/mol in good agreement with an earlier prediction by Hilpert[4]. Evaporation data used in this study are:

Gindorf[5],[6]measurement of evaporation rates from
83.5 cm3 plates / Evaporation coefficient [m s-1] at 800°C
assuming co= 1.35 g/cm3 for 1.4742,
co= 7.24 g/cm3 for ODS
Unprotected std. IC surface / 1.12 µg/h at 850°C / 2.7510-14
Protected IC surface on ODS / 0.08 µg/h at 950°C / 1.9710-16

Cr6+ evaporation coefficients are:

,

.

In 1000 hours of operation at 800°C and 1010 cm2 interconnect/ cathode area, 1802 µg chromium is available for transport to the cathode, 69 µg in the case of ODS with surface protection and 13 µg in the hypothetical case of ferritic steel with the Siemens[7] type of surface protection La0.9Sr0.1CrO3.

The diffusion coefficient of CrO2(OH)2 as the most abundant species in humid air is derived by converting the value at atmospheric pressure[8] with the kinetic gas theory approach[9] to

.

Operational data are:

• operating temperature 800°C,
• air velocity:

The usual air throughput[10] in a 2cell1010 cm2 short stack is 4 litres per minute giving an average velocity in the air channel of

.

Case 0 – Time Dependence

Here, we assume an average concentration in the whole of the air channel and approximate the diffusional flux into the cathode by

The balance equation is leading to

Eq. 2

with the solution for w=h:

,

where

.

The amount deposited on the cathode surface in the t-model is

is the interconnect Cr evaporation rate per unit length. The chromium fraction removed by the air flow is .

For our data set, the deposition rate fraction is 80.2%. The time constant is completely negligible and we will, therefore, research steady-state solutions only.

Case 1 – z-Dependence

Following a proposal by Elmar Achenbach[11], the balance equation for an average concentration in a cross section of the air channel with differential depth is

leading in the steady-state case and w=h to

Eq. 3

for with the solution .

The “running-in distance” is is 2.47 cm with current parameters. The amount deposited onto the cathode surface is , where the parenthesis describe the deposited fraction which, here, is 76%. Chromium removed by air flow is .

Case 2 – xy- Air Channel Cross Section Model

Neglecting momentarily variations along the gas channel in the z-axis, we consider the details across the xy-cross section of the channel. Because of the uniaxial air flow in the z-direction, we can integrate all necessary properties later.

Fig. 2: Cross section through Jülich SOFC stack showing geometry of xy-model.

The description of the transport of Cr6+ species in the air channel starts with the consideration of mass balance in a volume element of width x, height y and depth L with the two diffusion terms in x, y direction and the convection term in z direction:

yielding the steady-state form

Eq. 4

where

is the concentration of Cr6+ species in the air channel

; length of air channel

(height of cathode contact paste is neglected)

is the diffusion coefficient of Cr6+ species in air

is the [constant] air velocity

with the boundary conditions

at the cathode surface,

at the gas channel side walls,

at the gas channel rear wall.

where co is the concentration of Cr in the interconnect and is the Cr evaporation rate from the interconnect.

Of interest is the fraction F deposited onto the cathode surface expressed as a rate relative to the evaporation rate from the three channel walls:

.

As shown in Appendix A, equation 4 can be solved analytically with the result
, where and

with air velocity v, diffusion coefficient D of CrO2(OH)2 in air, air channel height and width h, and length L.

The result for F can be generalised (not shown here) for cases where the
evaporation from side walls is different from the
evaporation from the rear wall giving

.

Reverting to w=h, s=r= for 2cell1010 cm2 SOFC stacks of Jülich design B1002 or C1002, we get =0.703 and a deposition rate fraction of 84%. Variations in air throughput and in the diffusion coefficient are shown in Tab. 1 below.

Tab. 1: Deposition rate fraction F for various air velocities in gas channel (equivalent to 0.4, 4, and 40 l/min)
and a variation of diffusion coefficients by a factor 2 in the xy-model.
diffusion coefficient [m2 s-1] / 0.0494 / 0.494 / 4.94 / m s-1 air velocity
D=1.13E-05 / 96.3% / 72.6% / 24.0%
D=2.25E-05 / 98.1% / 84.0% / 36.5%
D=4.50E-05 / 99.0% / 91.3% / 52.2%

In the xy-model, the steady-state amount of Cr in the air channel is



such that GL= 5.4110-16 kg would be the steady-state amount of chromium in the gas channel for the reference case.

Case 3: xyz-Model with Full Analytical Steady-State Solution of the 3d Problem

Now that we have solved the diffusion equation in the xy-model, we can – due to the uniaxial air flow – define a shape factor such that the concentration profile along the z-axis is given by .

To obtain the solution for , we are writing the balance equation

Eq. 5

with the solution .

See Fig. 3.

Fractional Cr deposition along air channel
Fig. 3:Prediction of Cr deposition rate profile Ff(z)=1-exp[-Fz/(1-F)L] onto the cathode surface in three-dimensional xyz-model. Variation of diffusion coefficient by a
factor 2 and predictions for usual air velocity v and an unrealistic value of 10v.

Integrated over channel width, the complete analytical solution is

.

The total chromium deposition rate onto the cathode is now given by

,

square brackets representing the deposited fraction which is 81.1% (Tab. 2 and Fig. 4).

The fraction of the chromium removed from the air channel is

Tab. 2: Final result on the deposition rate fraction for various air velocities in gas channel (equivalent to
0.4, 4, and 40 l/min) and a variation of diffusion coefficients by a factor 2 in the xyz-model.
diffusion coefficient [m2 s-1] / 0.0494 / 0.494 / 4.94 / m s-1 air velocity
D=1.13E-05 / 96.2% / 65.0% / 14.3%
D=2.25E-05 / 98.1% / 81.1% / 23.9%
D=4.50E-05 / 99.0% / 90.4% / 39.2%
Fraction of evaporated chromium arriving at cathode
Fig. 4:The prediction of the total Cr cathode deposition rate fraction as functions of v and D show that only in the case of unrealistically high air throughput and low diffusion coefficients will chromium be massively removed.

When applied to a 1000h stack test at 800°C, the amounts of chromium predicted in the cathode are compiled in Tab. 3 for the standard ferritic steel interconnect and in Tab. 4 for the case of the Siemens type protection layer applied to our standard interconnect material.

Tab. 3: Standard interconnect: final result on the deposition of Cr mass onto the 1010 cm2 cathode
after 1000 hours of operation at 800°C for various air velocities and diffusion coefficients.
diffusion coefficient [m2 s-1] / 0.0494 / 0.494 / 4.94 / m s-1 air velocity
D=1.13E-05 / 1733 µg / 1171 µg / 257 µg
D=2.25E-05 / 1767 µg / 1461 µg / 431 µg
D=4.50E-05 / 1785 µg / 1630 µg / 706 µg
Tab. 4: Interconnect with surface protection: final result on the deposition of Cr mass onto the 1010 cm2 cathode after 1000 hours of operation at 800°C for various air velocities and diffusion coefficients.
diffusion coefficient [m2 s-1] / 0.0494 / 0.494 / 4.94 / m s-1 air velocity
D=1.13E-05 / 12.5 µg / 8.4 µg / 1.9 µg
D=2.25E-05 / 12.8 µg / 10.5 µg / 3.1 µg
D=4.50E-05 / 12.9 µg / 11.8 µg / 5.1 µg

Model Comparison Summary

The goal of this study was the modelling of Cr6+ transport in the air channel of the metallic SOFC interconnect. Source of Cr6+ is the evaporation from the channel surfaces. Inside of the channel, the transport represents a competition process between diffusion onto the cathode surface as perfect sink and the removal of Cr6+ by convective air transport.

Predicted results are summarised in Tab. 5 for:

Case 0 – t-model with one fixed Cr6+ concentration value in the channel;

Case 1 – z-model with one fixed Cr6+ concentration value at a certain position along the channel;

Case 2 – xy-model assuming everything constant along the channel, but detailed two-dimensional treatment across channel cross section; and, finally,

Case 3 – xyz-model with a full analytical steady-state solution of the three-dimensional problem.

Tab. 5:Chromium deposition and removal rate fractions predicted for 2cell1010 cm2 stacks of FZJ design B1002 and C1002 with 1.5 mm high and wide air channels of 10 cm length, air
velocity 0.494 ms-1, and CrO2(OH)2 diffusion coefficient in air 2.2510-5 m2s-1.
Fraction of chromium evaporation rate
deposited on cathode surface / removed from
air channel
t-model
zero-dimensional / 80.2% / 19.8%
z-model
one-dimensional / 75.7% / 24.3%
xy-model
two-dimensional / 84.0% / 16.0%
xyz-model
three-dimensional / 81.1% / 18.9%

cosine series[12]

In a finite volume type of approach typical for SOFC and PEFC modelling[13], the full balance equation for a volume element in the interior of the Overrelaxation[14] (SOR)

References:

By using the numerical simulation code CrSOR, we demonstrated that a two-dimensional parabolic velocity profile v(x,y) gives practically identical results as the constant velocity model (Appendix B).

.

[1]Notes from Meeting SK Grosse Runde, 10 Nov 1999

[2]Notes from 13. Meeting Arbeitsgruppe Test/ Autopsie, 28 Oct 1999

[3]C Gindorf and K Hilpert “Untersuchung zur Chromfreisetzung aus Interkonnektmaterialien für die SOFC”, FZJ IWE – TN 17/98, October 1998

[4]K Hilpert, D Das, M Miller, D H Peck and R Weiss “Chromium Vapour Species over Solid Oxide Fuel Cell Interconnect Materials and Their Potential for Degradation Processes”, J Electrochem Soc 143/11 (1996), 3642

[5]Notes from 14. Meeting Arbeitsgruppe Alterung/Langzeitstabilität, 1 Sep. 1999

[6]C Gindorf, K Hilpert, H Nabielek and L Singheiser “Chromium Release from Metallic Interconnects with and without Coatings”, in preparation
[4th European SOFC Forum, Lucerne 10-14 July 2000]

[7]C Günther, H-J Beie, P Greil and F Richter
“Parameters Influencing the Long-term Stability of the SOFC”
2nd European SOFC Forum, Oslo, 6-10 May 1996

[8]H C Graham and H H Davis “Oxidation/Vaporization Kinetics of Cr2O3”
J Amer Cer Soc 54 (1971), 89

[9]S Chapman, T G Cowling, The Mathematical Theory of Non-Uniform Gases, Cambridge University Press 1970

[10]I C Vinke, private communication, Jülich, 12 Nov 1999

[11]E Achenbach, private communication, Jülich, 1 Dec 1999

[12]H Groemer, Geometric Applications of Fourier Series and Spherical Harmonics, New York: Cambridge University Press, 1996

[13]J Divisek “Verfahrenstechnische Analyse von elektrochemischen Energieumwandlungssystemen” Forschungszentrum Jülich Report Jül-3469, Dec 1997

[14]The relaxation method was first used in 1935 by Sir Richard Southwell at Oxford University. As a numeric technique invented before electronic computers, it is simultaneously simple, intuitive and powerful (from BYTE, January 1987).