AN INTEGRATED APPROACH TO MODELING AND MITIGATING SOFC FAILURE

Andrei Fedorov, Samuel Graham, Comas Haynes, Jianmin QuGeorgia Institute of Technology

DE-AC26-02NT41571Program Managers: Travis Shultz

National Energy Technology Laboratory

Project Overview and Objective

Electrochemical Reaction

Thermal Transport

Failure

Fracture

Cell/StackPerformance

Thermomechanical Damage

Through integrating structural, electrochemical and thermal transport analysis, we will develop numerical modeling and simulation tools for design analysis and reliability/durability predictions of SOFCs.

Overview of GA Tech-Developed Tools

Identify dominant physical mechanisms for structural failure

Computational algorithms for crack-tip parameters (K, G, J)

Computational algorithms for thermal/fluid transport

Computational algorithms for evaluating the effects of cracks onelectrochemical reactions

Constitutive laws for SOFC materials under operating environment

Micromechanics modeling of inhomogeneous SOFC materials (anode, cathode and possibly seals)

Failure criteria and damage accumulation models and associated MARC subroutines

Utility: User defined subroutines to be integrated into the SECA design tools

Some Major Technical Accomplishments To DateDeveloped a simplified two-flux approximation for radiative heat transfer calculations in SOFC cells, resulting in a ten-fold reduction in the required computational time as compared to the standard discrete ordinate method (1/03).

Energy release rates for both the edge delamination and buckling-driven blister delamination of SOFC cells were obtained, which can be used to assess cell fracture. Developed criteria for estimating maximum tolerable fabrication defects based upon fracture analysis (11/02).

Developed a model for the spalling phenomenon and thermal expansion-induced stress during thermal transients and shock. The model relates the rate of heat generation in the cell to microcrack initiation and may be used predict the maximum allowable heat generation before microcracks are initiated (2/03).

Developed methods for accurately calculating the stress intensity factors and the energy release rate in SOFC cells. Developed an advanced theoretical methodology for modeling gas flow, mass, and heat transfer in the porous electrodes (3/03).

A global/local analysis scheme was developed and illustrated on a 3D co-flow cell model that allows the integration of thermal/fluid simulation results directly combined with local stress analysis (6/03).

Developed a domain integration formulation to evaluate crack tip parameters for fracture analysis (9/03).

Some Major Technical Accomplishments To Date

It was experimentally determined using FTIR spectrometer that the electrode (anode made of 40 vol% Ni; 60 vol% 8YSZ and cathode made of Sr-doped Lanthanum ferrite) samples appear to be opaque over the entire near and mid infrared spectra (10/03).

Developed a general formulation (on a spectral basis) of the radiative heat transfer in the optically thin electrolyte of the planer SOFC, and wrote and validated a code for implementation of the formulation (2/04).

Compiled a database of radiative properties of SOFC materials (3/04).

Developed a computational algorithm to include creep deformation in the electrodes (4/04).

It was determined, based on certain subjective assumptions, that in typical SOFCs, the local thermal equilibrium assumption holds.

Major Structural Failure Modes and Mechanism

• Warpage• Cracks/leak in seals• Cracks in electrodes• Cracks in electrolyte• Delamination of interfaces• Creep/fatigue of interconnects• ?? (industry inputs)

• Thermal mismatch• Thermal gradient (spatial)• Thermal shock (temporal)• Thermal diffusion• Mass diffusion• Cyclic Redox

Potential SOFC Cell Mechanical Failure Mechanisms

Thermoelastic Deformation: coefficient of thermal expansion, elastic modulus, tensile strength.Elastic-Plastic Deformation: yield criterion and strength, hardening rules, flow rules.

Fracture: fracture toughness.

Fatigue: S-N curve, da/dN curve.Creep: creep exponents.Migration: diffusivity.

Common Thermomechanical Failure Mechanisms at the Material Level

Computing Stresses

Design against Initial Failure

Design against Degradation

Modeling Methodologies

Cell Structure(L > 10-3 m)

PEN Structure(10-5 m < L < 10-3 m)

Material StructureL < 10-5 m

x

za a

• Warpage• Seal failure• Seal design• Residual stresses

• Plate and laminate theories

5 µm

anod

eca

thod

eel

ectro

lyte

5 µm

anod

eca

thod

eel

ectro

lyteelectrolyte

cathode

anode

A CB

F

D

GEelectrolyte

cathode

anode

A CB

F

D

GE

• Cracks growth• Delamination• Spalling

• Fracture mechanics• Finite element method

• Crack initiation• Plasticity• Creep

• Micromechanics• Damage mechanics

0 50 100 150 200 250 300

0.0

0.2

0.4

0.6

0.8uniform anodegraded anode

sE

(GPa)

(GPa)

xσ

Max. stress in anode

0 50 100 150 200 250 300100

101

102

103

104

uniform anodegraded anode

Max

. Def

lect

ion

(µm

)

Es (GPa)

Design Criteria Against Fracture (graded anode)

0 50 100 150 200 250 300-0.2

0.0

0.2

0.4

0.6

0.8uniform anodegraded anode

sE

(GPa)

(GPa)

xσ

Max. stress in electrolyte

anode

cathode

electrolyte

anode

cathode

electrolytePorosity in anode

10%

50%

Design Criteria Against Fracture (Flaw Tolerance)

2a

h

anode

electrolyte h

0 1 2 3 40

5

10

15

20

25

30

35

Flaw Size (mm)

Ener

gy R

elea

se R

ate

(J/m

^2)

h = 5 µm

h = 20 µm

h = 10 µm

0 200 400 600 800 1000 1200 1400 1600 18000.0

0.5

1.0

1.5

2.0

2.5

3.00 100 200 300 400 500 600 700 800

h = 5 µm

h = 10 µm

h = 20 µm

h = 30 µm

h = 40 µm

h = 50 µm

Temperature Change (oC)

Compressive Stress in the Electrolyte Layer (MPa)

Crit

ical

Fla

w S

ize

(mm

)

0 5 10 15 20 25 300

100

200

300

400

500

5%20%30%

Fracture Toughness 2(J/m )cG

(J sec)q

Thermal Shock Induced Microcrack Initiation

22sU Nbπ γ=

Temp Distribution

Surface Energy

Strain Energy

( ) 0s bd U Uda+

=Griffith Fracture Criterion

q = rate of heat generation (J/sec)Gc = Fracture toughness of the materialb = crack sizeN = number of cracks per unit volumek = Thermal conductivityα = Coefficient of linear thermal expansionr0 = A length parameter characterizes the spatial non-uniformity of the heat source.

2 2 30

0

3 (1 )16(1 )12 9(1 2 ) (1 )

ckr GNbqE b

π π ννα ν ν

−−= + − +

2 2

2 2 3 20

(1 2 )6 (1 )bq EUk rα ν

ν π−

=−

020 0

( , ) erf erf4 4

q r rT r t Trk r r tπ κ

= − + +

( )2

3 200

( ) expq rfrc rρ π

= −

rHeating Source

( )2

3 2 23 2 200

( , ) exp44

T t q rTt r tc r t κρ π κ

• ∂ −= = ∂ ++

x

)exp(RTQA cn

c −= σε&

A cn Q Ni/YSZ 2.0e-6 1.2 550kJ/mol

Effect of Creep on Stress Evolution During Operation

YSZ (Tm ~ 24000C): no creep

Ni (Tm ~ 14500C): power law creep

LSM (Tm ~ ??): no creep

cε&σnc = stress exponent for creep

Q = is activation energy,

R = the universal gas constant

A = a parameter that depends on the material and test conditions (e.g. oxygen partial pressure in the case of oxides).

Steady-State Creep

= strain rate

= effective stress

Stress in Anode

-1

0

1

2

3

4

5

6

0 200 400 600 800 1000

Time (hour)

Stre

ss (M

Pa)

Von Mises StressS11

Stress in Electrolyte

0

20

40

60

80

100

120

140

0 200 400 600 800 1000

Time (hour)

Stre

ss (M

Pa)

Von Mises StressS11

Anode Supported Cells

anode

cathode

electrolyte

anode

cathode

electrolyte

CTE ~ 11

CTE ~ 10

CTE ~ 12

The anode sheltered the electrolyte. As creep progresses, such sheltering decreases!

0

20

40

60

80

100

120

0 100 200 300 400 500 600 700 800

Delta Temperature (ºC)

Tens

ile In

-Pla

ne S

tress

Sigm

a xx

(MPa

)

1-Constant Moduli

2-Temp DependentElectrolyte3-Temp DependentElectrolyte and Anode

Temperature Dependant Material Properties

( ) ( .058 196)[1 1.96 ]

( ) ( .021 76)[1 1.93 ]

a

a

E T T p

G T T p

= − + −

= − + −

( ) .051 233eE T T= − +

25mol%Ni/75mol%YSZ 8mol % YSZ

0.3ν =

Current Work for Mechanical Failure Prediction

T[(P q) ( P ) q]V

I tr dV= − ∇ + ∇∫v v

Calculating KI, KII, and KIII using the crack-tip interaction integral

Crack grows typically in the direction where KII = 0

Currently, no commercial code is capable of doing this! We are writing the code in MatLab so it is portable to MARC, ANSYS, ABAQUS, etc.

Domain Integral Formulation and ANSYS/MatLab Codes for Calculating Crack-Tip Fracture Parameters

{ }( )V

G J tr W dV = = − − ∇ ∇ ∫ I u σ qv v

( )( )

cL

GG sa s ds

=∆∫

Input •Element connectivity •Nodal coordinates •Nodal displacements •Nodes on crack tip

Select Node S •Select volume of elements •Calculate Unit Outward Normal •Transform coordinates •Transform displacements

Loop through elements

Begin Gaussian Quadrature by looping through integration points.

Calculate components of integrand •Strain energy density •Stress tensor •Derivatives displacement •Derivatives test function

Calculate integrand and add to previous component.

Go to next integration point

Go to next element

Calculate pointwise value of domain integral

Sample

Collimated Beam

Sample

Collimated Beam

Light Source IR detector

Sample

Collimated Beam

Sample

Collimated Beam

Light Source IR detector

A 10-degree specular reflectance accessory fitted to an FTIR spectrometer

θ1

θ2 θ2

θ3

d

Medium 1 (n=1)

Medium 2 (k, n)

Incident beam

Medium 3 (n=1)

θ1

θ2 θ2

θ3

d

Medium 1 (n=1)

Medium 2 (k, n)

Incident beam

Medium 3 (n=1)

Reflectance and Transmittance

-1.0

1.0

3.0

5.0

7.0

2.0 4.0 6.0 8.0 10.0Wavelength [um]

Perc

ent [

%]

Reflectance and Transmittance

-1.0

0.0

1.0

2.0

3.0

2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0Wavelength [um]

Perc

ent [

%]

Tr

R

Tr

R

Reflectance and Transmittance

-1.0

1.0

3.0

5.0

7.0

2.0 4.0 6.0 8.0 10.0Wavelength [um]

Perc

ent [

%]

Reflectance and Transmittance

-1.0

0.0

1.0

2.0

3.0

2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0Wavelength [um]

Perc

ent [

%]

Reflectance and Transmittance

-1.0

1.0

3.0

5.0

7.0

2.0 4.0 6.0 8.0 10.0Wavelength [um]

Perc

ent [

%]

Reflectance and Transmittance

-1.0

0.0

1.0

2.0

3.0

2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0Wavelength [um]

Perc

ent [

%]

Tr

R

Tr

R

Reflectance and Transmittance of YSZ

0%

5%

10%

15%

20%

25%

30%

2 4 6 8 10Wavelength [um]

Perc

ent [

%]

R

Tr

Reflectance and Transmittance of YSZ

0%

5%

10%

15%

20%

25%

30%

2 4 6 8 10Wavelength [um]

Perc

ent [

%]

R

Tr

(La1-xSrx)FeO3-δ Zr1-xYxO2-δNi-(Zr1-xYxO2-δ)

Experimental Determination of Optical Properties of YSZ, Ni/YSZ and LSF using FTIR Spectroscopy

200 µm

200 µm330 µm

Radiation Modeling – Spectral 2-flux FormulationValidation of spectral 2-flux User Defined Function (UDF) for FLUENT

1-D, plane-parallel medium; isothermal boundaries

1

1

1

160 ; 0.0 3.5

110 ; 3.5 5.0

50 ; 5.0

cm

cm

cm

λ

β λ

λ

−

−

−

< < = ≤ <

≤ < ∞

3-band model approximates measured spectral variation of YSZ extinction coefficient

T (top)

T (bottom)

T (centerline)

y

T (top)

T (bottom)

T (centerline)

y

3-band Model

800

850

900

950

1000

1050

1100

1150

1200

0 0.002 0.004 0.006 0.008 0.01

y [m]

Tem

p [K

]

DO methodUDF 2-flux

Validation vs. Discrete Ordinates method built-in FLUENT

Porous Electrodes – Non-Equilibrium Analysis

( ) ( ) ( )( ) ( )

,

,

- - (Gas phase)

0 - (Solid phase)

p g g eff g v g s

s eff s v g s gen

V c T k T h T T

k T h T T Q

ρ∇ ⋅ = ∇ ⋅ ∇

′′′= ∇ ⋅ ∇ + + ∑

r

&

Assumption of local thermal equilibrium (LTE) is questionable when:a) Difference in solid and fluid thermal properties is significant (√ in SOFC)b) Significant heat generation in porous media – existence of hot spots (√ in SOFC)c) Low Reynolds number or flow velocities through porous media (√ in SOFC)

Performed Scaling Analysis of Solid Phase energy equation

1. Volumetric solid-to-gas phase heat transfer coefficient order of magnitude:

2. Volumetric heat generation due to exothermic reaction and Ohmic heating:

11310v g s s

Wh h a m− =

9310gen

WQ m K ′′′ ⋅

& Global (cell-level) estimate!

Key Assumptions• Depend on nature of porous microstructure: approximately spherical particles (0.5–1.5 µm average diameter) & 30-40% electrode porosity.

• Average (global) current density used in analysis: local current density on microscale level might be several orders of magnitude greater leading to much higher heat generation and expected solid-gas temperature difference!

•Validity of LTE depends on validity of these assumptions!

Porous Electrodes – Non-Equilibrium Analysis

( ) ( ), - s eff s g s c g sQ k T h a T T−′′′ = ∇ ⋅ ∇ +&

Assume negligible

( ) ( ) 210g s c g s g sQ h a T T T T T K−−′′′→ = − → ∆ = −&

Consider Volumetric Heat Generation Within the Solid Phase

Negligible – LTE is Valid!!!

Porous Electrodes –Microscale Analysis, Local Current Density

Three Phase Boundary Ni / YSZ / gas

Solid Electrolyte (YSZ) (Ionic Conducting)

Porous Electrode (Ni-doped YSZ)

H2O

H2 2H+ + 2e-

(Surface Adsorption)

O-2

e-

H2

2H+ + O-2 H2O

Characteristic Length

Future Work

Material failure modeling and MARC integration

Transient heating effects during start-up/shut-down

FLUENT and MARC Integration of two-flux approximation for radiative transfer

Effects of mechanical damage on cell stack's electrical performance

Elastic-Plastic Deformation: plasticity can be neglected below 9000C

Fracture: Sub-critical crack growth (transgranular)

0

0

nft

tσσ

=

n ~ 10

Steady-State Creep: diffusion creep (cation transport along grain boundaries)

expcn

c q

A Qd T RTσε = −

& ~ 0.7cn ~ 320 J molQ k

Fatigue: Sub-critical crack growth (transgranular)

2~ 3.5J mcG

nda dN AG= n = 8 ~12

Migration: ??

0 0

( ) expm

sVP VV

σσ

= −

Tensile strength: The probability of a sample of volume V can survive a given stress σ is given by

Electrolyte (8 mol% YSZ)

Elastic-Plastic Deformation: After reduction (Ni/YSZ) deforms plastically

Anode (NiO/YSZ, Ni/YSZ)

0 02( )( )3ij ij ij ij yσ σ δ σ σ δ σ− − =

Yield condition for Ni

10 ~ 100MPayσ =

Steady-State Creep: cavitation coalescence expcn

c q

A Qd T RTσε = −

&

1 ~ 6cn =~ 44 J molQ k

Mechanical Failure Prediction

Creep Laws for Ni/YSZ Cermet expn

q

A Qd T RTσε = −

&

Ni5 ~ 8Nin =

YSZ0.5 ~ 1YSZn =

Ni/YSZ

/ ?Ni YSZn =

Micromechanics: A self-consistent approach to obtain

/ ( , , , , )Ni YSZ Ni YSZ Ni YSZ Voidsn F n n V V V=

This will be coded into a user defined subroutine in MARC

Mechanical Failure Prediction

Digital Microscopy

( ) ( )( )

1

1( )N r

kk

r I rN r

φ=

= ∑

( ) ( )2

d rAg rrN dr

φπ

=Radial

distribution

Extract the statistical

features of distribution

realizations

Monte Carlo

MARC

0( ) ( )G g r G r dr

∞= ∫

0( ) ( )K g r K r dr

∞= ∫

FEMMonte Carlo

simulation

Mean crack-tip parameters

0

1

G

Transient Heating During Start-Up/Shut-Down

QUESTION: how fast can SOFC be heated without thermomechanical failure?- Analytical solutions for transient temperature distribution with the SOFC are possible for the simplified quasi 1-D case in the limit of the thermally thin cell- Numerical simulations will be used to analyze more complex and realistic scenarios involving combined convective-conductive-radiative heating

⇓Develop a simple, yet technically sound transient thermal model which, when combined with failure analysis, could be used by the SECA industrial teams for preliminary design calculations and selection of envelope of “safe” operating conditions

Effects of Mechanical Damage on Cell Stack's Electrical Performance

Electrochemical degradation sensitive to effective losses in electroactive area and current paths, impact upon surface phenomena, possible reactants crossover, etc.

fuel

air

Electrochemical Impact of Fracture

Reactants Crossover/ “Leak

Current”

Substantial increase in resistance

Electrolyte

Changes in TPBs/Electroactive

Area

Changes in TPBs/Electroactive

Area

Interlayers

Smaller impactCharge/masstransport

redirection

Bulk electrode layer

Vertical Crack Impact

Parallel Crack Impact

Component Layer

Which cracks/ crack modes does industry deem to be the more prevalent/influential upon cell performance??

Crack-Electrochemistry Interaction {Conventional Current Sign Convention}

Cathode

Anode

Electrolyte

Figure 4(a): Normal, undamaged operation

Cathode Anode

Figure 4(b): Induced sheet resistance within the electrolyte due to cohesive crack

Cathode Anode

Electrolyte

Figure 4(c): Induced contact and sheet resistance about and within the electrolyte via delamin./blistering

“Masking” Approximation: Deactivated ZonesAxial Current Distribution for Different Crack Positions

0.00E+00

1.00E-02

2.00E-02

3.00E-02

4.00E-02

5.00E-02

6.00E-02

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

Slice Number

curr

ent

(mA

) no crack10-1002010-21004010-41006010-61008010-8100

Pros: • Most conservative “safety factor” approach• Readily implemented within engineering code such as PNNL MARC

developmentPrimary Consideration: Resolving the threshold ratio of electrolyte thickness-to-crack characteristic length (e.g., delamination radius) below which masking approximation is plausible --- potential flow analyses

Summary

Quantify the Importance of a

Relevant Mechanism

Ignore It and Move on

Integrate Existing Simulation CodesOr Develop New

Ones

Validate the New Codes

Integrate into MARC

yes

no

e.g., Local Thermal

Equilibrium

e.g., Radiation

CreepFracture

e.g., Domain IntegralTwo-Flux

Self-Consistent Law