1. No category

Modeling Coupled Fluid Flow, Heat and Mass
Transfer with Thermo-Mechanical Process
Application to Cracked Solid Oxide Fuel Cell
Qian SHAO, L. Bouhala, A Makradi
and S. Belouettar
Background
ü  Coal
around 100 years
ü  Natural gas around 60 years
ü  Oil
around 50 years
BP Statistical Review 2014
2
Solid Oxide Fuel Cell
3
Why SOFC?
High energy conversion efficiency
•  Convention: thermo-mechanical based, 20%-30%
•  SOFC: electrochemical based, 45% or higher
Fuel flexibility
•  Internal reforming
•  Hydrogen, natural gas, biomass, coal gas, etc.
Low pollutant emission
•  No noise
•  Low greenhouse gas, no smog particulates
4
Why SOFC?
CHP
Propelling force
APU
Non-stationary
Stationary
5
Why SOFC?
Drawbacks:
•  High manufacturing costs
•  Low-reliability
6
How does SOFC work?
Electric current
Fuel in
Air in
ee-
e-
q  Cathode/Electrolyte interface
O2- e-
1
O2 +2e− → O 2−
2
H2
O2-
O2
(Oxygen reduction)
q  Anode/Electrolyte interface
H 2O
Η 2 + Ο 2− → Η 2 Ο + 2e −
(Hydrogen Oxidation)
Anode
Electrolyte
Cathode
7
Problem statement
High operating temperature
Residual thermal stresses
Enhance cell degradation rate
Crack nucleation and propagation
Affect the electrochemical
performance of SOFC
Penetrate into the electrolyte and
allow for fuel leakage
8
State of the Art
Experiments
Modeling
New materials
Processing parameters
Material processing
Fuel conversion efficiency
Porous microstructure
Temperature distribution
Supported configuration
Thermo-mechanical response
Lower temperature
Fracture prediction
9
State of the Art
2006, Ruiz-Morales et al.
•  Novel technique to control the porosity
•  Homogeneous porous medium improves electrochemical performance
2007, Joulaee et al.
•  Pores concentration and distribution affect the resistance to fracture in
porous material
2013, Mikdam et al.
•  Pores orientation direction influence the thermal and electrical
conductivities, as well as the pores connectivity
10
State of the Art
Ø  Commercial codes
Ø  Simplified in-house codes
•  Porous zones neglected
•  Flow field neglected
•  Pure diffusive, isothermal
Ø  No consideration of cell
degradation (failure)
11
Objective
v  Develop a general comprehensive numerical approach to model
the multi-physics phenomenon in a SOFC unit.
v  Study the effect of porous material properties on SOFC
fractures and electrochemical performance.
12
Multi-physics phenomenon
Fluid flow
Heat
transfer
Mass
transfer
Electrochemical
reactions
Thermomechanical
response
Crack
propagation
13
Basic assumptions
•  Incompressible, laminar flow
•  Saturated porous media of constant porosity
•  Permeable and adiabatic cracks
•  Cracks affect heat conduction in the solid phase
•  Radiation effect neglected
14
Basic models
DB
model
Fluid flow
Heat
transfer
XFEM
model
Thermomechanical
response
Mass
transfer
Crack
propagation
Electrochemical
reactions
EC
model
15
DB model
Darcy-Brinkman
ρf
⎧ ρ f ∂v µ
µ
+
v
+
v
.
∇
v
−
∇. (∇v ) + ∇P = 0
⎪
(
)
2
ε
ε
⎨ ε ∂t K
⎪∇.v = 0
⎩
Nonconforming Crouzeix-Raviart Finite Element Method
∂C
+ ∇.(qC ) − ∇.( D.∇C ) = Qs
∂t
Species conservation
ε
Energy conservation
∂ ( ( ρ c ) PM T )
∂t
+ ∇.( ρ f c f qT ) − ∇.( kPM ∇T ) = QT
Advective term -- Discontinuous Galerkin
Diffusive term – Multipoint Flux Approximation
16
Double-diffusive natural convection
Adiabatic, impermeable
Porous medium
T = Th
C = Ch
g
T = Tc
C = Cc
Adiabatic, impermeable
Boussinesq approximation
ρ (T , C ) = ρ 0 ⎡⎣1 − βT (T − T0 ) − β C (C − C0 )⎤⎦
17
Simulation results
Streamlines
Temperature
Concentration
18
Sensitivity to mesh
Level 1:
2500
Level 2:
10000
Temperature contours
Level 3:
40000
Level 4:
160000
Concentration contours
19
Sensitivity to mesh
Test case
Relative errors of Nusselt and Sherwood numbers between different grid levels
20
Fourier-Galerkin
Reformulate the governing equations
Expand the unknowns in Fourier series
Substitute the expansions in the equations
Solve the system with Powell hybrid method
21
Comparison of results
Da=10-1
Da=10-3
Da=10-5
Test case
Relative errors of Nusselt and Sherwood numbers between FG and FEM solutions
22
XFEM model
Heat conduction
Mechanical equilibrium
( ρ c ) PM
∂T
− ∇.( k s∇T ) = QTs
∂t
⎧ LT σ + b = 0
⎪
⎨ u = u
⎪
⎩σ ⋅ n = t
in Ω
on Γ u
on Γ t
eXtended Finite Element Method
Stress Intensity Factor
Crack propagation criterion
J-integral technique
Maximum principle stress
23
Benchmark 1: Isotropic cruciform plate
Boundary condition sets
BC
set
1
2
3
Temperature (°C)
A
0
0
-5
B
10
0
10
C
0
0
-5
D
-10
0
-10
A cracked cruciform plate
(Bouhala et al. 2012)
24
Traction
(Pa)
B
0
10
10
Benchmark 1: Isotropic cruciform plate
A cracked cruciform plate
(Bouhala et al. 2012)
Horizontal displacement
25
Benchmark 1: Isotropic cruciform plate
BC set 5
BC set 1
BC set 2
Present study
N.N.V. Prasad (1994)
26
Benchmark 2: Anisotropic rectangular plate
W
A rectangular plate made of glass/epoxy:
θ=θ0
Ø  Young’s modulus:
E11=55GPa
4W
a
Q
γ
E22=21GPa
Ø  Thermal expansion coefficient:
α11=6.3×10-6 K -1
P
α22=2×10-5 K -1
E22
Ø  Thermal conductivity ratio:
E11
λ11 λ22= 3.45 0.35
θ=0
27
Benchmark 2: Anisotropic rectangular plate
W
θ=θ0
4W
a
Q
γ
P
E22
E11
θ=0
Vertical displacement
Vertical stress
28
Benchmark 2: Anisotropic rectangular plate
*
Crack tip P
Crack tip Q
*ETM – Exact Transformation Method; SUP – Superposition approach; BEM – Boundary Element Method
References: Pasternak, 2012; Shiah et al., 2000
29
EC model
Nernst equation
Output voltage
Heat generation
Species consumption/
production
OCP
U TPB
=U H0 2
⎛
pH 2O,f
RT ⎜
−
ln
2 F ⎜ p
p
⎝ H 2 ,f O2 ,a
(
⎞
⎟
1 2 ⎟
⎠
)
OCP
U=U TPB
− ( ηOhm +ηconc +ηact )
T⋅j
Q&T =
2 SH 2O − 2 SH 2 − SO2 + j (ηOhm + ηact )
4F
(
j
Q&S ,H 2 = −
2F
)
j
Q&S ,O2 = −
4F
j
Q&S ,H2O =
2F
Powell hybrid method
30
Validation
P. Aguiar, 2004
Present study
31
Applications to SOFC
DB
model
Fluid flow
Heat
transfer
XFEM
model
Thermomechanical
response
Mass
transfer
Crack
propagation
Electrochemical
reactions
EC
model
32
DB-XFEM model
Ø  Energy conservation in porous media:
T n +1 − T n
( ρ c ) PM
+ ∇. ρ f c f qT − ∇. ( kPM ∇T ) = QT
Δt
(
)
Ø  Time splitting:
T n +1,* − T n
( ρ c ) PM
+ ∇. ρ f c f qT − ∇. k f ∇T = QTf
Δt
(
)
T n +1 − T n +1,*
( ρ c ) PM
− ∇. k s∇T = QTs
Δt
(
)
(
)
(DG and MPFA)
(XFEM)
33
DB-XFEM model
Start
DB model
T*
If K Ieq< K Ic
XFEM model
If K Ieq > K Ic
No
Extend the crack by
one increment
Threshold number
Yes
End
34
Heating process of SOFC
35
Simulation results
Flow channel
Porous anode
Electrolyte
Porous cathode
Flow channel
Pressure field
Temperature field
36
Simulation results
Enriched elements
Horizontal stress
37
Parametric studies
•  Pores volume fraction
•  Porous material anisotropy
38
Isotropic porous media
(a) Porosity ε=0.2
(c) Porosity ε=0.4
(b) Porosity ε=0.3
(d) Porosity ε=0.5
39
Study on onset of crack propagation
0.5
0.4
0.3
0.2
Evolution of SIFs on crack tip before the onset of crack propagation
KIc: material fracture toughness
40
Effect of porosity on crack growth
cathode/electrolyte interface
Crack propagation paths for different isotropic porous media
41
(along the cell height)
Anisotropic porous media
y
Anisotropy (a)
Anisotropy (b)
Anisotropy (c)
Anisotropy (d)
x
(along the cell length )
42
Study on the onset of crack growth
(a)
(b)
(c)
(d)
Evolution of SIFs on crack tip before the onset of crack propagation
43
Effect of anisotropy on crack growth
cathode/electrolyte interface
Crack propagation paths for different anisotropic porous media
44
Parametric studies
E1
•  Young’s Modulus ratio (E1/E2)
E2
•  Permeability (K)
•  Thermal Conductivity (λ)
45
Young’s modulus ratio effect
Porous/dense interface
Crack tip position in a three-layered structure for
different materials mismatch
46
Permeability and conductivity effect
Porous/dense interface
Different permeability values
of the porous layer
Porous/dense interface
Different thermal conductivities
of the solid
47
DB-EC model
Species concentration, temperature
DB model
EC model
Species, heat flux at boundaries
48
Simulation results
Fuel channel
(a) Temperature
Anode-electrolyte-cathode
Air channel
(b) Molar fraction of H2
(c) Molar fraction of H2O
(d) Molar fraction of O2
49
Validation
Comparison of I-V curves between model predictions and
experimental data (Fernández-González et al., 2014)
50
Temperature distribution
anode/electrolyte
51
I-V curves
Comparison of I-V curves between non-isothermal and isothermal cases
52
Conclusions
Numerical approach
•  DB model, XFEM model, EC model
•  Flow field, species concentration, temperature field
•  Thermo-mechanical response, crack propagation
•  Electrochemical performance
•  Benchmarks and validations: accurate, efficient
Study on SOFC
•  Porous material properties
•  Crack propagation path: complex dependence
•  Onset of crack propagation: anisotropy and porosity
•  Energy conversion performance: heat generation
53
Perspectives
EC model
•  Charge transport with and without the presence of a crack
DB-EC model
•  Material porosity and anisotropy effect on the electrochemical performance
DB-XFEM model
•  Interfacial crack propagation; Crack across the interface
•  Fatigue crack growth due to the heating and cooling process
DB-XFEM-EC model
•  Interaction between the crack and the energy conversion performance
54
More information
Ø  Q. Shao, R. Fernández-González, A. Mikdam, L. Bouhala, A. Younes, P. Núñez, S. Belouettar and A. Makradi (2014). “Influence of
heat transfer and fluid flow on crack growth in multi-layered porous/dense materials using XFEM: Application to Solid Oxide Fuel
Cell like material design”, International Journal of Solids and Structures, 51(21-22), pp.3557-3569.
Ø  Q. Shao, L. Bouhala, A. Younes, P. Núñez, A. Makradi and S. Belouettar (2014). “An XFEM model for cracked porous media:
effects of fluid flow and heat transfer”. International Journal of Fracture. 185(1-2): 155-169.
Ø  A. Younes, A. Makradi, A. Zidane, Q. Shao, L. Bouhala (2014). “A combination of Crouzeix-Raviart, Discontinuous Galerkin and
MPFA methods for buoyancy-driven flows”. International Journal of Numerical Methods for Heat and Fluid Flow. 24(3): 735-759.
Ø  L. Bouhala, Q. Shao, Y. Koutsawa, A. Younes, P. Núñez, A. Makradi, S. Belouettar (2013). “An XFEM crack-tip enrichment for a
crack terminating at a bi-material interface”. Engineering Fracture Mechanics. 102: 51-64.
Ø  Q. Shao, L. Bouhala, D. Fiorelli, A. Younes, P. Núñez, S. Belouettar and A. Makradi (2015). “Influence of fluid flow and heat
transfer on crack propagation in SOFC with anisotropic porous layers”, under review, International journal of Solids and
Structures.
Ø  Q. Shao, R. Fernández-González, J.C. Ruiz-Morales, L. Bouhala, A. Younes, P. Núñez, S. Belouettar, A. Makradi (2015). “An
advanced numerical model for energy conversion and crack growth predictions in Solid Oxide Fuel Cell units”, under review,
Journal of Power Sources.
55