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they haveto operate at relatively low temperatures (400600C)
because selectivity of the electrodes decreases with increase in
temperature. This low temperature operation enhances the
thermal and mechanical shock resistance (much safer from
a structural point of view), thereby enabling rapid start-up and
shut-down. Not only this, the SC-SOFCs can be made much
lighter because the heavy bipolar plates (separate flow channels
for fuel and air) are not required[2].The research work on SC-SOFCs is mainly experimentally
driven with a very few reports on modelling of SC-SOFCs[2].
These experiments give information about the overall cell
performance, a deeper understanding of the hydrodynamic
and electrochemical interaction is lacking. Modelling can be
a best tool to study all insights but unfortunately there is
a very little work in this direction. The models developed so
far are mainly limited to two-dimensional analysis, showing
that the theoretical studies in this area are at an early stage
yet [37]. The only developed three-dimensional model by
Chung et al. [8] does not provide any information on flow/
species distribution inside the gas-chamber. This model is
also lacking temperature distribution due to isothermalassumption. The only two active groups (Hao et al., USA and
Chung et al., Korea) have reported initial numerical studies
mainly focused on understanding the causes of low perfor-
mance in SC-SOFCs. Very recently, Hao et al. [6] studied the
effect of various parameters such as flow rate, mixing ratio,
flow geometry, stack and balance gas on the performance of
conventionalplanar type SC-SOFC. Their main emphasis
was to study the causes of low fuel utilization (
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2.2. Model assumptions
The flow is steady.
The electrodes are assumed to be ideally selective for the
respective electrochemical reactions, therefore, electro-chemical oxidation of fuel and reduction of oxygen are
taking place in the anode and cathode catalyst layers,
respectively[8].
Ohmic heating (in the porous electrodes and catalyst layers)
due to electrical current transport is neglected because of
high electrical conductivity as compared to the ionic
conductivity[13].
The electrolyte is a non-porous (dense, solid) material.
The effect of radiative heat exchange between the cell and
furnace walls can be neglected because of negligible
temperature gradients. This assumption is made on the
basis that the real catalytic chemistry is not known due to
experimental limitations and the electrodes are assumed tobe ideally selective with reversible/irreversible loss as the
only source of heat.
The conduction heat transfer is dominant in the cell as
compared to the heat transport by convection because of
the low gas speed in the porous electrodes.
The outer surfaces of the anode and cathode diffusion
electrodes are used as current collectors, therefore the
effects of interconnects are neglected.
In following sections, detailed modelling strategy is pre-
sented for each sub-domain.
2.3. Computational domain
2.3.1. Gas-chamber
The gas-chamber consists of a rectangular duct with
a membrane electrode assembly also called positive electrode,
electrolyte and negative electrode (PEN) located in the center
of the duct [Fig. 1]. The PEN element has a hole in the center to
allow gaseous mixture to pass through the axial direction
therebyrelieving the pressure that could be built-up in front of
the cell. The applicable equations are:
Continuity equation:
V$ru 0 (3)
Momentum equation:
ru$Vu Vp mV2u13mVV$u (4)
Species conservation equation:
V$ruYi V$ji (5)
Energy conservation equation:
V$rCpuT
V$kVT (6)
The density of the mixture is calculated using[14]:
r 1PN
i1Yi=ri(7)
The density of each species, riis obtained from the perfect
gas law relation[14]:
ri pMi
RT (8)
Concentration of each species is calculated by:
ci pXiRT
(9)
where Xiisthe mole fraction of the ith species which is related
to the mass fractionYiby the following relation:
Xi Yi
M
Mi
(10)
and
M XNi1
XiMi (11)
In equation (5) the multicomponent diffusive mass flux
vector (ji) is described by the generalized Ficks law [15,16]:
ji XN1
j1
rDijVYj (12)
Table 1 Geometry dimensions.
Dimensions Values (mm)
Chamber length (x-axis) 150
Chamber height (y-axis) 25
Chamber width (z-axis) 25
PEN width 20
PEN height 20Anode thickness 70 103
Cathode thickness 50 103
Electrolyte thickness 20 103
Anode catalyst
layer thickness
5 103
Cathode catalyst
layer thickness
5 103
Hole diameter 2.6
Table 2 Diffusion volumes in FuellerSchettlerGiddingscorrelation parameters.
Molecule Diffusion volume (cm3/mole)
H2 7.07
O2 16.6
H2O 12.7
N2 17.9
Table 3 MaxwellStefan diffusion coefficients calculatedusing values given inTable 2.
Molecular pair Dij(m2/s)
H2N2 4.0176 104
H2O2 4.1178 104
N2O2 1.0982 104
H2H2O 4.6541 10
4
N2H2O 1.3972 104
O2H2O 1.3992 104
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Table 4 Input parameters used in the model.
Property Symbol Value Units References
Working electrical
potential at anode
4s,a 0 V [28]
Working electrical
potential at cathode
4s,c 0.7 V [28]
Anodic pre-exponential coefficient ga 1.6e9
[13]Cathodic pre-exponential coefficient gc 3.9e9 [13]
Anodic activation energy Eact,a 120 J mole1 [13]
Cathodic activation energy Eact,c 120 J mole1 [13]
Effective anode
ionic conductivity
seff
ea 0.29 S m1 [27]
Effective cathode
ionic conductivity
seff
ec 0.24 S m1 [27]
Effective anode
electronic conductivity
seff
sa 4800 S m1 [27]
Effective cathode
electronic conductivity
seff
sc 1600 S m1 [27]
Anode electronic conductivity ssa 2.0 106 S m1 [32]
Electrolyte ionic conductivity se 3.34 104 exp (10,350/T) S m1 [32]
Cathode electronic conductivity ssc (42 106/T) exp (1150/T) S m1 [32]
Inlet temperature T0 773 K []Anodic anodic
charge transfer coefficient
aaa 2 [27]
Anodic cathodic
charge transfer coefficient
aac 1 [27]
Cathodic anodic
charge transfer coefficient
aca 1.5 [27]
Cathodic cathodic
charge transfer coefficient
acc 0.5 [27]
Faradays constant F 96487 C mole1 [24]
Universal gas constant R 8.314 J mole1 K1 [24]
Porosity e 0.3 [13]
Tortuosity s 3.80 [24]
Electrochemically active
surface area
As 102,500 m1 [33]
Permeability k 1.0 1013
m2
[28]Hydrogen inlet
mass fraction
(of 4%)
YH2in 0.5 [5]
Oxygen inlet
mass fraction
(of 4%)
YO2in 0.105 [5]
Nitrogen inlet
mass fraction
(of 4%)
YN2in 0.38 [5]
Water inlet
mass fraction
(of 4%)
YH2Oin 0.015 [5]
Operating pressure po 1.013 105 N m2 [23]
Inlet velocity uin 0.1 m s1 []
Average pore diameter dp 1.0 mm [23]Hydrogen viscosity mH2 6.162e
6 1.145e8 T Pa s [13]
Oxygen viscosity mO2 1.668e5 3.108e8 T Pa s [13]
Water viscosity mH2O 4.567e6 2.209e8 T Pa s [13]
Nitrogen viscosity mN2 1.435e5 2.642e8 T Pa s [13]
Hydrogen specific heat Cp,H2 13960 0.950 T J kg1 K1 [13]
Oxygen specific heat Cp,O2 876.80 0.217 T J kg1 K1 [13]
Water specific heat Cp,H2O 1639.2 0.641 T J kg1 K1 [13]
Nitrogen specific heat Cp,N2 935.6 0.232 T J kg1 K1 [13]
Hydrogen thermal conductivity kH2 0.08525 2.964e4 T Wm1 K1 [13]
Oxygen thermal conductivity kO2 0.01569 5.690e5 T Wm1 K1 [13]
Water thermal conductivity kH2O 0.01430 9.782e5 T Wm1 K1 [13]
Nitrogen thermal conductivity kN2 0.01258 5.444e5 T Wm1 K1 [1]
Anode thermal conductivity ka 3 Wm1 K1 [13]
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where, N total number of species in the mixture. Dij in
equation (12) is the multicomponent diffusion coefficient
which in general is not symmetric (DijsDji). Also, the multi-
component Dijdoes not have the physical significance of the
binary Fick diffusivity in that the Dij do not reflect the ij
interactions [17]. Multicomponent diffusion coefficients Dij are
interrelated with MaxwellStefan diffusion coefficients Dijthrough matrixB, such that[17]:
D B1G (13)
For ideal gases the thermodynamic matrix G reduces to the
identity matrix and equation(13)becomes:
D B1 (14)
where D is the multicomponent diffusion coefficient matrix
andB is a square matrix of orderN 1 with elements given by:
Bii YiM
MiDiN
XNk1;isk
YkM
MkDik(15)
Bij YiM
Mi
1Dij
1DiN
; isj (16)
D in equations(15) and (16) is the MaxwellStefan diffusion
coefficient for binary pairs and is dependent on both
temperature and pressure[18]. For gas pressures up to about
10 atm at moderate to high temperatures, the diffusion coef-
ficient for a binary mixture of gases i and j may be estimated
from the Fueller, Schettler and Giddings relation [19]:
Dij
T1:751=Mi 1=Mj1=2p
V1=3i V1=3
j
2 10
7
(17)
where Dijis the MaxwellStefan diffusion coefficient in m2 s1,
Tis the temperature in kelvin (K), p is the pressure in atmo-
spheres (atm), Mi is the molecular weight of molecules in
gmol1,and Vi is the moleculardiffusion volumein cm3 mol1.
The values ofVifor different molecules are tabulated in [20].
Typical values ofDijm2 s1 for molecules commonly used in
fuel cells are calculated using equation(17)and are given in
Table 3 ata pressure of 1 atm and at anaverage temperature of
773 K, which is assumed to be the operating condition for the
single-chamber solid oxide fuel cell in this study.
Note that equation (17) is dimensionally inhomogeneous
(i.e. it is only valid for the stated unit system, and not for any
other unit system, unlike all other equations) and is a result of
regression analysis of 340 experimentally determined binary
Fig. 1 Gas-chamber with PEN element inside (not to scale).
Table 4 (continued)
Property Symbol Value Units References
Cathode thermal conductivity kc 3 Wm1 K1 [13]
Electrolyte thermal conductivity ke 2 Wm1 K1 [13]
Anode specific heat Ca 595 J kg 1 K1 [1]
Cathode specific heat Cc 573 J kg 1 K1 [1]
Electrolyte specific heat Ce 606 J kg
1
K
1
[1]Anode density ra 6870 kg m3 [1]
Cathode density rc 6570 kg m3 [1]
Electrolyte density re 5900 kg m3 [1]
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diffusion coefficients [20]. Also, this equation can be easily
applied for pressure and temperature ranges used in a typical
fuel cell operation.
2.3.2. Gas diffusion electrodes
Gas diffusion electrodes consist of an anode and a cathode
which are porous media. The following equations model gas
diffusion electrodes:
Continuity equation:
V$ru Sm (18)
Smis the source/sink term for the production/consumption
of gas molecules and is zero for gas diffusion electrodes
because the electrochemical reaction is assumed to take
place only in the catalyst layers. The reason for introducing
the source/sink term (given in Ref. [21]), is that sometimes
the reactive layers can extend into the electrodes.
Momentum equation:
In porous media flow where viscous forces dominate
convective ones, the momentum equation in the porousmedia may be modified from the NavierStokes equation to
the Brinkman equation. In order to do so, the convective
term has been neglected and an additional term forpressure
drop in porous media, given by Darcys law, has been added.
This describes flow in porous media with a pressure
gradient as the only driving force[22].
Vp m
ku (19)
By inserting this term in the NavierStokes equation, we
have the Brinkman equation as:m
k
u Vp mV2u1
3
mVV$u (20)
Species conservation equation:
V$ruYi V$ji Si (21)
where, Si is the source/sink term for the production/
consumption of species, set to zero for gas diffusion elec-
trodes because of the above mentioned reason.
ji is the multicomponent diffusive mass flux vector in
porous media, given by:
ji XN1j1
rDeffDGVYj (22)
whereDeffDGis the effective dusty gas diffusivity.Diffusion in porous media is usually described by
a molecular (particle-particle collision) and/or a Knudsen
(particlewall collision) diffusion mechanism[23]. In order
to account for a detailed diffusion mechanism, both modes
have been considered by implementing the Dusty Gas
Model (DGM). The DGM is derived by considering the solid
matrix as large stationary spheres suspended in the gas
mixture as one of the species present. The DGM diffusivity is
then given by[23]:
DDGi;j DijDk;iDij Dk;i
(23)
where
Dk;i 13
dp
ffiffiffiffiffiffiffiffiffi8RTpMi
s (24)
The values ofDijare calculated using equation (17) and the
DGM diffusitivities are further corrected using the following
expression[23]:
DeffDGi;j e
s
DDGi;j (25)
Finally, the value ofDeffDGi;j is used to calculate the matrix
DeffDGi;j , having elements:
BeffDGii YiM
MiDeffDGi;N
XN
k1;isk
YkM
MkDeffDGi;k
(26)
BeffDGij YiM
Mi
1DeffDGi;j
1
DeffDGi;N
; isj (27)
Charge conservation equation:Electrical current transport is described by a governing
equation for conservation of charge[13,23]:
V$ssVfs AsSfs (28)
Sfs , electrical current source term which is kept zero in the
anode and cathode gas diffusion electrodes.
Energy conservation equation:
Energy transport in gas diffusion electrodes is modelled
by considering porous nature of the electrodes. Instead of
using the thermal conductivity (ks), density (rs) and specific
heat (Cp,s) of the solid matrix, effective thermal conductivity
(keff), effective density (reff) and effective specific heat (Cp,eff)
have been used in the model[24].
V$
rCp
eff
uT V$
keffVT
Se (29)
Effective properties of the porous media are given by
[24]:rCp
eff 1 ersCp;s erCp (30)
keff 2ks
e
2ks k
1 e3ks
1(31)
In equation(29),Seis the heat source term which is zero.
2.3.3. Catalyst layers
An additional ionic charge conservation equation in the
catalyst layer will represent the ionic current, and is given by
[13,23]:
V$ seffe Vfe
AsSfe (32)
Sfe , ionic current source term defined below.
Other transport equations in the catalyst layers are the
same as in the electrodes, except the source/sink terms are
activated in the conservation equations and the electrical
conductivity term (ss) in equation(28)is changed to effective
electrical conductivity (seffs) in order to account for mixed
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ionicelectronic conductivity in the catalyst layers. The
source/sink terms in mass and species conservation equation
are given below[25,26].
At the anode side, hydrogen is consumed and water is
produced, so that:
Sm Si SH2
MH2
2Fia (33)
Sm Si SH2 O MH2O
2F ia (34)
At the cathode side, oxygen is consumed:
Sm Si SO2 MO24F
ic (35)
Source/sink terms in the electrical and ionic charge
conservation equation are given below[27].
In the anode catalyst layer: electrical current density is
a sink term,
Sfs ia (36)
In the anode catalyst layer: ionic current density is a source
term,
Sfe ia (37)
In the cathode catalyst layer: electrical current density is
a sink term,
Sfs ic (38)
In the cathode catalyst layer: ionic current density is
a source term,
Sfe ic (39)
whereia anodic current density given by the ButlerVolmer
equation[27]:
ia i0;a
exp
aaaFhacta
RT
exp
aac Fhacta
RT
(40)
ic cathodic current density given by[27]:
ic i0;c
exp
acaFhactc
RT
exp
accFhactc
RT
(41)
The exchange current densities,i0,aandi0,care expressed as
a function of local partial pressure of the species [13]:
i0;a ga
pH2pref
pH2Opref
0:5exp
Eact;a
RT
(42)
i0;c gc
pO2pref
0:25exp
Eact;c
RT
(43)
where pref is the reference pressure in the gas-chamber, i.e.
the total pressure of 1 atm. ga and gc are the anodic and
cathodic pre-exponential coefficients, Eact,aand Eact,care the
anodic and cathodic activation energies, respectively.
The anode and cathode side activation overpotentials are
calculated by[28]:
hacta 4s=a;cl 4e=a;cl anode side activation overpotential:
hactc 4s=c;cl 4e=c;cl Voc
cathode side activation overpotential:
where 4s/a,cland 4e/a,clare respectively the solid phase (elec-
tronic) and electrolyte phase (ionic) potential in the anode
catalyst layer, 4s/c,cl and 4e/c,cl are the electronic and ionicpotential in the cathode catalyst layer, and Voc is the open
circuit (Nernst) voltage, as expressed by[29]:
Voc Vo
RT
2Fln
pH2
pH2O
!
RT
4FlnpO2
(44)
Since the model is solved using Finite Element Method
(FEM), all unknown variables are calculated at each node. For
example, the operating voltage is defined at the cathode outer
surface and a zero voltage is set at the anode outer surface. By
implementing the charge conservation equation (i.e. Ohms
law), the ohmic loss is embedded while calculating the current
at each node (since at each node, all equations are solved in
FEM). The pressure distribution is calculated via mass trans-
port model, and, at each node, current is calculated corre-
sponding to that concentration. Therefore, there is no needfor
explicit expressions for ohmic and concentration over-
potentials in FEM, rather theyare an integral part of the model
being solved numerically.
The energy equation in the catalyst layers remains the
same as was in gas diffusion electrodes (equation(29)), except
the heat source term, Se, is activated because of chemical/
electrochemical heat generation in the catalyst layers. For
anode this term is given by[13]:
Se qreva qirra qohmea (45)
where reversible heat generation in the anode [13,25]:
qreva Tdsia=zF (46)
dsz 23:328 0:0042T (47)
irreversible heat generation in the anode[13]:
qirra hacta ia (48)
ohmic heat due to ionic resistance[13]:
qohmea seffea Vfe$Vfe (49)
Heat source term for the cathode is given by[13]:
Se qirrc qohmec (50)
irreversible heat generation in the cathode[13]:
qirrc hactc ic (51)
ohmic heat due to ionic resistance[13]:
qohmec seffecVfe$Vfe (52)
2.3.4. Electrolyte
The electrolyte is impermeable to gases and allows only ionic
charge transfer, so that:
V$seVfe 0 (53)
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Since there is no generation of ionic or electrical current
inside the electrolyte, the right hand side of the above equa-
tion is zero. Furthermore, the electrolyte is a dense, non-
porous material and thus only conduction is possible. The
only heat source term in electrolyte is the ohmic resistance
due to ionic current transfer[13,28].
V$
ksVT Se 0 (54)
Se seVfe$fe (55)
2.4. Boundary conditions
The boundary conditions for each layer are given below:
2.4.1. Gas-chamber
2.4.1.1. Inlet. At the inlet, the velocity in the x-direction is
prescribed to be:
u uin (56)
At the inlet, the mass fraction is defined as:
Yi Yiin (57)
At the inlet, operating temperature is defined[13]:
T T0 (58)
2.4.1.2. Walls. At the walls no-slip boundary condition is
applied. A no-slip condition means that the fluids velocity is
equal to the boundary velocity, which is zero in the case of
a fixed wall.
u 0 (59)
A mass insulation boundary condition is applied at the
walls, meaning that no mass flux is allowed to cross these
boundaries.
n$ rDijVYi ruYi
0 (60)
All of the four side walls (i.e. top, bottom, side and left) are
assumed to be at furnace temperature which is the operating
temperature.
T T0 (61)
2.4.1.3. Outlet.The outflow boundary condition is prescribedas:
p po (62)
The convective flux boundary condition is applied at the
outlet, meaning that at the outlet boundary, diffusion term is
negligible.
n$ rDijVYi
0 (63)
At the outlet, the heat transport is convection dominated.
Convective heat flux boundary condition ensures that at the
outlet boundary the only heat transport is by convection and
thus conduction heat transfer is negligible at this boundary
[13].
n$ kVT 0 (64)
2.4.2. Gas diffusion electrodes
2.4.2.1. Anode electrode. At all exterior surfaces of the anode
and at an interface between the anode electrode and anode
catalyst layer, continuity of flow is applied. This puts no
constraints on the velocity and mass flux. The current on theanode side is collected from the outer surface which faces
inlet flow direction. Therefore, a zero voltage (ground)
boundary condition is applied at the anode outer surface.
fs 0 (65)
All other outer surfaces of the anode electrode are insulated
to the electrical current, hence it is assumed that there is no
current flow across these boundaries.
n$ ssVfs 0 (66)
At the interface between the anode electrode and anode
catalyst layer, continuity of electrical current is maintained.
2.4.2.2. Cathode electrode. The same boundary conditions
which are described above for the anode will also hold for the
cathode, except the electrical potential boundary condition for
the cathode electrode surface which faces the outlet flow
direction has been modified to:
fs Vc (67)
This boundary condition defines the operating cell voltage
and closes the current circuit. At all exterior and interior
boundaries of the gas diffusion electrodes, continuity of heat
flux is maintained. This boundary condition specifies that the
normal heat flux inside of the boundary is equal to the normal
heat flux outside of the boundary[13].
n$ksVTin n$ksVTout (68)
2.4.3. Catalyst layers
2.4.3.1. Anode catalyst layer. At all exterior surfaces of the
anode catalyst layer continuity of flow is applied, meaning
putting no constraints on the velocity and mass flux.
However, at the interface between the anode catalyst layer
and the electrolyte, gases are not allowed to enter into the
electrolyte because of its non-porous nature. Hence, normal to
the electrolyte boundary, both the velocity and mass flux are
zero.
n$u 0 (69)
n$ rDeffDGi;jVYi ruYi
0 (70)
At all exterior surfaces of the anode catalyst layer and at
interface between the anode catalyst layer and the electro-
lyte, electrical insulation boundary condition has been
applied.
n$ seffs Vfs
0 (71)
Ionic current cannot flow out of the catalyst layers. All
exterior surfaces of the anode catalyst layer and at interface
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between the anode electrode and the anode catalyst layer,
ionic current is insulated by applying ionic current insulation
boundary condition.
n$ seffe Vfe
0 (72)
2.4.3.2. Cathode catalyst layer.The same boundary conditions
as described above for the anode catalyst layer will also holdfor the cathode catalyst layer. Furthermore, continuity of heat
flux is maintained on all surfaces of the catalyst layers.
n$ksVTin n$ksVTout (73)
2.4.4. Electrolyte
Since the electrolyte is impermeable to gases, both the mass
flux and velocity normal to all surfaces of the electrolyte are
zero.
n$u 0 (74)
n$ rDeffDGi;jVYi ruYi 0 (75)Continuity of ionic current is maintained at interfaces
between the anode/cathode catalyst layers and the electro-
lyte. All outer surfaces of the electrolyte are insulated to ionic
current by applying an ionic insulation boundary condition.
n$ seVfe 0 (76)
Continuity of heat flux is maintained at all exterior surfaces
of the electrolyte, i.e.
n$ksVTin n$ksVTout (77)
The hole in the PEN allows free flow in a similar way as in
the gas-chamber, therefore continuity of flow is maintained in
the hole. However, we consider the electrolyte surfaceboundaries consisting of a hole not to allow the gases to
permeate into the electrolyte, therefore insulation boundary
conditions have been applied at these boundaries.
3. Numerical implementation
The model equations are solved using COMSOL Multi-
physics 3.4, a commercial Finite Element Method (FEM)
based software package. The computations were performed
on a 32-node Linux cluster; 32 dual 3 GHz Intel Xeon Sun
Fire V60 servers each with 4 GB memory. The mesh consistsof 9398 triangular elements of good quality and is shown in
Fig. 2. In order to ensure convergence and accuracy of the
results in three-dimensional domain, the following were
employed:
Use of lower element order to reduce the number of
unknowns.
The mesh was kept more refined in the cell element where
higher resolution was needed to capture large gradients.
A direct solver such as UMFPACK is very stable but not good
for large problems, since it requires too much memory.
Therefore, an iterative solver (GMRES) with a preconditioner
(Incomplete LU) and tolerance of 0.005 were used.
The system of equations was solved iteratively, i.e. first the
charge balance, then the NavierStokes equation, then the
mass/species balance, while at each stage the solution was
stored and used as an initial condition for the subsequent
stage. The total computing time for all stages (with 65, 192
degrees of freedom) was approximately 13.5 min for one value
of voltage scan. The grid independency test was performed
using different mesh sizes.Fig. 3shows that the velocity fieldbecomes grid independent as we move above 7154 elements.
In order to keep balance between accuracy and computational
time, a mesh with 9398 elements was opted.
4. Results and discussion
The values of the electrochemical/hydrodynamic transport
parameters for the operating conditions are listed in Table 4.
The governing equations are summarized in Table 5. InFigs.
413, the results shown are based on parameters listed in
Table 4, whereas for the results shown in Figs. 1417, the value
of porosity has been changed to 0.4 in order to see the effect ofelectrochemistry and hydrodynamics, with all other param-
eters remain the same as in Table 4. Figs. 1820 show
temperature,IVand IPcharacteristics based on parameters
listed inTable 4, respectively.
InFig. 4, the velocity field in the gas-chamber is shown. It
can be seen that as soon as the flow reaches the anode elec-
trode surface, some of the flow is diverted through the edges
of the electrode surface. It shows that the cell is an obstacle to
the fluid flow with fluid traveling a tortuous path in the gas
diffusion electrode leading to a reduced velocity in the cell.
Due to the diversion near the cell edges, an increased velocity
of the flow is observed because of the free passage available
for the flow as compared to the limited passage available inthe cell. Just behind the cell, the velocity is reduced and the
flow regains its center velocity after a distance of approxi-
mately 0.03 m from the anode electrode surface. This distance
is an important parameter to be considered in a case where
one plans to investigate several cells downstream.
Fig. 5shows pressure drop in the gas-chamber. It repre-
sents another parameter to observe the flow behavior. The
pressure drop along the flow direction is responsible for
accelerating the flow. Moreover, near the anode electrode
surface there is a large pressure drop (due to the presence of
cell as an obstacle) followed by a Darcy pressure drop in the
gas diffusion electrodes which is strongly dependent on the
permeability of porous electrodes, that in turn depends uponporosity and tortuosity of the porous media. The pressure
drop behind the cell is responsible for further acceleration of
the fluid downstream.
Fig. 6 shows the hydrogen concentration distribution in the
gas-chamber. As can be seen, the hydrogen concentration
remains constant until the gaseous mixture reachesthe anode
electrode surface as there is no electrochemical reaction
upstream in front of the anode electrode. As soon as the flow
reaches the anode catalyst layer, it starts reacting electro-
chemically (according to equation(1)). Since concentration is
pressure dependent, one observes a slight increase near the
cell front edges due to the increase of pressure, as discussed
before. Just behind the cell, hydrogen concentration is low but
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quickly mixes with diverted (non-reacted) hydrogen coming
through the cell edges via convection and diffusion. Just after
approximately, 0.02 m from the cathode last edge, hydrogen
concentration attains another lower constant value of
4.765 mole m3. Again, this parameter is of interest when one
plans to put several cells downstream. Also, it is important to
note that the hydrogen available for the downstream cell may
be depleted partly due to consumption at the anode electrode
of the upstream cell and partly due to mixing with the by-product water vapor. Therefore, additional supply of hydrogen
is suggested for uniform cell performance in a stack of several
cells. Alternatively, one could use a distributed feed forall cells
in a stack e.g. in a parallel cell arrangement rather than serial.
Fig. 7shows the oxygen concentration in the gas-chamber
where it can be seen that the oxygen due to electrochemical
reaction (according to equation (2)) starts consuming in the
cathode catalyst layer. The large concentration gradient is
along the cell due to electrochemical reduction of oxygen at
the cathode catalyst layer. Just behind the cell a lower oxygen
concentration is observed and attains another lower constantvalue (0.195 mole m3) at approximately0.038 m from cathode
last edge. This distance is approximately double what is
Fig. 2 The computational mesh (9398 elements).
Fig. 3 The grid independency test.
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needed for hydrogen mixing and the reason could be that
hydrogen has a high diffusion coefficient as compared to
oxygen, therefore hydrogen diffuses quicker than oxygen.Also, hydrogen is the lightest molecule and the same
momentum can give a high velocity to a light molecule as
compared to heavy molecule like oxygen. Therefore mixing by
convection/diffusion for hydrogen is quicker than oxygen.
Fig. 8 represents water vapor concentration due to theelectrochemical reaction (according to equation(1)), water is
produced at the anode catalyst layer and this effect is clearly
Fig. 4 Velocity field in gas-chamber.
Table 5 Computational domain and governing equations.
Domain Equations solved
UGC,h V$(ru) 0
ru$Vu Vp mV2u 13mVV$u
V$(ruYi)V$jiV$(rCpuT) V$(kVT)
Ua V$
(ru) 0mku Vp mV2u 13mVV$u
V$(ruYi)V$jiV$(ssV4s) 0
V$((rCp)effuT) V$(keffVT)
Uc V$(ru) 0m
ku Vp mV2u 13mVV$u
V$(ruYi)V$jiV$(ssV4s) 0
V$((rCp)effuT) V$(keffVT)
Ua,cl V$(ru) Sm,Sm SH2 SH2 O MH2
2Fia MH2 O
2F iamku Vp mV2u 13mVV$u
V$(ruYi)V$ji Si,Si SH2 MH2
2Fia,SH2 O MH2 O
2F iaV$seffs Vfs AsSfsV$seffe Vfe AsSfe
V$((rCp)effuT) V$(keffVT) Se,Se qrev
a qirra qohmeaUc,cl V$(ru) Sm,Sm SO2
MO24Fic
mku Vp mV2u 13mVV$u
V$(ruYi)V$ji Si,Si SO2 MO2
4FicV$seffs Vfs AsSfsV$seffe Vfe AsSfeV$((rCp)effuT) V$(keffVT) Se,Se qrevc qirrc qohmec
Ue V$(seV4e) 0
V$(ksVT) Se 0,Se seV4e$V4e
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observed near the anode electrode surface. The by-product
water mixes with hydrogen and oxygen and thus dilutes the
gaseous mixture downstream.
InFig. 9, the legend description corresponding toFigs. 10
17is shown. As can be seen the variables plotted in Figs. 1017
are along the channel length with varying values of the y-
coordinate. Thez-coordinate is kept fixed in the center of the
PEN in all these figures.
Fig.10 shows the velocity field plot in thegas-chamber. Blueand black curves are almost on top of each other showing the
same trends in velocity profile, these lines are plotted at
a distance of 2 mm below and above the cell in y-direction.
Initially the velocity is constant until it reaches the cell. Then,
due to diversion of theflow (due to thepresence of thecell) the
velocity starts increasing (up to approximately 0.24 m s1) and
then settles behind the cell at a value of approximately
0.0996 m s1. Lavender and light green curves show the
velocity plot inside the cell at a distance of 3 mm from top and
bottom edge of the cell. Due to be on same distances they are
also on top of each other and show that the velocity remainsconstant until it reachesthe cell. In close vicinity of thecell, the
flow feels the presence of the obstacle (the cell), therefore the
Fig. 5 Pressure drop in gas-chamber.
Fig. 6 Hydrogen concentration in gas-chamber.
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velocity decreases and becomes almost zero in the PEN. This
suggeststhattheflowinthePENisnomoreconvectivebutonly
diffusive. Just behind the cell the flow again accelerates and
attains a velocity of approximately 0.18 m s1. The red curve
shows the center linevelocityand ascan beseenfromthe plot,
thecenterline velocity hasthe maximum value just beforeand
after thecell. The center line velocity also feelsthe effectof the
obstacle and therefore drops to a value of 0.025 m s1 at the
anode electrode surface andthen dueto thepresence of hole in
the center of the PEN, the flow sees a free flow (accelerates
again). Due to the radial convection/diffusion in the hole
domain the velocity fluctuates for a while and then the flow
comesoutoftheholewithagreater(nozzlelike)push.Finallyit
regains its maximum center line velocity at a distance of
approximately 0.03 m from the cathode electrode.
Fig. 11shows the hydrogen mass fraction plot and it can be
seen that the mass fraction remains constant until the reac-
tants reach the cell, where the electrochemical reaction is
taking place. A mass fraction drop (though very little due to
high degree of dilution in the mixture) is observed in the cell
(light green, lavender and red curves), precisely at the anode
catalyst layer due to the consumption of hydrogen. Also, the
Fig. 7 Oxygen concentration in gas-chamber.
Fig. 8 Water vapor concentration in gas-chamber.
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hole is not a completely non-reactive domain and there is
some consumption along radial direction in the hole. Only the
electrolyte is impermeable and; although it is possibly hard to
see in the plot the light green and lavender curves showa discontinuity of mass fraction in the electrolyte.
Fig. 12shows the oxygen mass fraction in the gas-chamber
and as discussed before for hydrogen consumption, the same
trend is seen for oxygen consumption but this time in the
cathode catalyst layer. Also, a discontinuity of the curves
(light green and especially of lavender) is clearly seen at the
electrolyte due to its impermeable nature.
Fig. 13shows the water vapor mass fraction and it can be
seen that due to the water generated at the anode catalyst
layer, a sharp increase in the mass fraction is observed.
Clearly, light green and lavender curves show a discontinuity
as discussed before.
Figs. 1417 are the repetition of simulations with anincreased value of porosity (e 0.4).In contrast to Fig. 10, there
is some convective flow (Fig. 14) inthe cellelementdueto ease
of the Darcy flow through the porous media. Increase in
porosity reduces the resistance in the fluid flow and thus both
convective/diffusive fluxes exist in the porous electrodes.
Although, it looks like that the increase in porosity will havea better effect on hydrodynamics of the flow butFigs. 15 and
16show a less utilization of reactants (compared with Figs. 11
and 12). This effect suggests that increase in porosity will
reduce the available active area for the electrochemical reac-
tion, possibly due to more open pores rather than connected.
Fig. 17 is a plot of the water vapor distribution and shows
a reduced value of water produced as compared with Fig. 13.
Therefore increasing the porosity could have an effect on
electrochemical performance of the cell.
Fig.18 shows temperatureprofile(forvalueslistedin Table4).
As can be seen, the anode has a higher temperature than
cathode and the reason is reversible heat generation in the
anode catalyst layer dueto electrochemicalproductionof water.Although theincrease in temperatureis not marginal, thisis due
to thefact that theonlyheat sourceis due to inherent reversible
and irreversible effects. Furthermore, the full combustion of
hydrogen is neglected by assuming ideally selective electrodes
Fig. 9 Description of the legend used inFigs. 1017.
Fig. 10 Velocity along the channel length at differenty-
coordinates for e[0.3. For interpretation of the references
to colour in this figure legend, the reader is referred to the
web version of this article.
Fig. 11 Hydrogen mass fraction along the channel length
at differenty-coordinates for e[0.3.
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and diluted gas mixture. It should be noted that the real elec-
trodes are not perfectly selective, and, one could observe full
combustion of hydrogen near the cell inlet if pure hydrogen/
oxygen mixture is used. It is therefore strongly recommended to
never use pure hydrogen/oxygenmixtures in an SC-SOFCdue to
the above mentioned reason.
Figs. 19 and 20show the calculated IVand IPcharacter-
istics curves at 500 C for hydrogen/oxygen/nitrogen mixture
(with input parameters given inTable 4), respectively. As can
be seen the maximum power density obtained is218.5 W m2(i.e. 21.85 mW cm2). The observed low current is
due to the low operating temperature and dilution effect.
Furthermore, the power density is quite low as compared to
the dual-chamber configuration, this is because of extremely
low fuel utilization due to the presence of large amount of
inert gas in the mixture.
5. Safety issues
The safe operation is the prime issue for any device, not only
limited to SC-SOFCs. However, in case of SC-SOFCs the level of
safety is of much more concern because of the explosive
nature of fuel/oxidant mixtures. At this stage, it would be
appropriate to consider most commonly used fuels such ashydrogen or hydrocarbons because of their well-known
application history in SOFCs. For instance, hydrogen/oxygen
mixtures could be the right candidates in terms of being less
problematic for commonly used nickel anodes (no coking
Fig. 12 Oxygen mass fraction along the channel length at
differenty-coordinates for e[ 0.3.
Fig. 13 Water mass fraction along the channel length at
differenty-coordinates for e[ 0.3.
Fig. 14 Velocity along the channel length at differenty-
coordinates for e[0.4.
Fig. 15 Hydrogen mass fraction along the channel length
at differenty-coordinates for e[ 0.4.
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problem), however their safety concerns are exceptionally
very high. Hydrogen is considered as explosive in the flam-
mability range of 475% by volume in air at standard condi-
tions (ambient pressure and temperature) and this limit
widens as the operating temperature increases. Moreover, the
auto-ignition temperature of a combustible hydrogen/oxygen
mixture is 585 C, far too low than a typical operating condi-
tion for an SOFC. For hydrogen/oxygen mixtures to be prac-
tical for SC-SOFCs, their safety window of operation is quite
narrow, for example either below 2% or above 93% hydrogen
(by volume in air) may be considered as safe at temperatures
up to 200300 C. Unfortunately, there is no practical
study available for hydrogen/oxygen mixture at temperatures
necessary for SC-SOFC operation. Even though, if we assume
that the mixture is safe at higher temperatures, it is still
impractical due to: 1) the use of rich fuels (>93% hydrogen)
may result in extremely low fuel utilization, because all the
fuel passed through the cathode will be nearly unutilized; 2)
the use of lean fuels (
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possible in order to extinguish the flame in case of fire.
Flashback arrestors and back flow preventing valves willincrease the systems safety level.
We must point out here that to our knowledge, only Isaiah
et al. [30] have reported diluted mixed gas (argon/hydrogen/
oxygen mixture) behavior for dual-chamber SOFC, thus simu-
lating the single-chamber conditions on individual electrodes.
The heat release (bydirect combustion) in a concentrated fuel/
oxidant gas mixture will be much more higher than a diluted
fuel/oxidant mixture. Therefore, one should never use
concentrated hydrogen/oxygen mixture to be employed in an
SC-SOFC, as the presence of nickel on the anode side could
promote full combustion directly at the anode inlet causing
severe local overheating. This effect is observed by many
researchers when using hydrocarbon/air mixtures. Ourexperimental studies[2,31]on methane/air mixture operated
SC-SOFC show a temperature rise of as high as 93 C, which
scales-up with flow rate and temperature. If hydrogen would
be used instead of methane, then for the same power output,
atleast 34 times higher flow rate of hydrogen will be required
(this can easilybe calculated on the basis on electron produced
per electrochemical reaction in each case). Now by knowing
the fact that the hydrogen hasapprox. 2.3 times higher specific
energy (kJ g1) than methane, the local heat produced by full
combustion of hydrogen could be much higher than the
methane/airmixtures.Therefore, it is not recommendedto use
concentrated hydrogen/oxygen mixtures in an SC-SOFC for
safety reasons, also, neither they are practical in terms ofpower production capacity, nor offering an acceptable elec-
trical efficiency. Furthermore, 96% inert gas diluted hydrogen/
oxygenmixture could be safe butstillthey arenot practicaldue
to very low OCV and power production capacity mainly
because of strong dilution effect.
6. Conclusions
A three-dimensional numerical model of a single-chamber
solid oxide fuel cell was developed. The model accounts for all
transport phenomena and cell potential. Results show that
varying one parameter (associated with the hydrodynamics)
can affect another parameter (associated with the electro-
chemical performance). Increase of porosity in the catalystlayers can reduce the available active area for electrochemical
reaction, conversely, fine pores can cause a large pressure
drop and subsequently lead to a hydrodynamic problem. The
best design could be to stay with fine porosity of the electrodes
and optimize the hydrodynamic problem by reducing the
tortuous path of the flow inside the porous electrodes. This
could be done by varying the permeability of the porous
electrodes with a fixed fine porosity. Instead of using dense
solid (non-porous) electrolyte, use of a fully porous cell
(including the porosity of the electrolyte) can give ease in flow
with reduced manufacturing cost. Another idea could be
instead of having holes inside the PEN element, the holes can
be provided at the circumference/edges of the interconnect(or separator layer). This would not reduce the active area but
still provides better hydrodynamics. Distributed feed to
a number of cells placed in a stack can give a uniform reactant
utilization in each cell, therefore parallel feed with branches
at each cell could give a better performance both hydrody-
namically and electrochemically.
Although, experimental workon SC-SOFCsusing hydrogen/
oxygen mixtures (at temperatures necessary for SOFC opera-
tion) is never reported, this study would help the reader to
understand what are the major restrictions in doing so.
Acknowledgments
The authors would like to thank E.ON-UK for funding Mr.
Naveed Akhtar through Dorothy Hodgkin Postgraduate Award
(DHPA) scheme.
Fig. 19 IVcharacteristic curve for SC-SOFC. Fig. 20 IPcharacteristic curve for SC-SOFC.
Nomenclature
n normal vector
u velocity vector
p pressure
Yi mass fraction of theith species
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Xi mole fraction of theith species
N total number of species in the mixture
Dij binary diffusion coefficient for pairij
ji mass diffusion flux of theith species
x,y,z Cartesian coordinates inx,y and z direction,
respectively
M average (mixture) molecular weight
Mi molecular weight of theith speciesR universal gas constant
T temperature
ci concentration of theith species
B1 matrix function of inverted binary diffusion
coefficients
Bii diagonal elements of inverted binary diffusion
coefficient matrix
Bij non-diagonal elements of inverted binary diffusion
coefficient matrix
D matrix of Fick diffusion coefficients
Dij MaxwellStefan diffusivity for pairij
Vi molecular diffusion volume
H2 hydrogenO2 oxygen
H2O water
N2 nitrogen
e1 electron
O2 oxygen ion
Sm mass source term
Si species source term
DeffDG Fick effective dusty gas diffusivity matrix
DeffDGi;j Fick effective dusty gas diffusivity for pair ij
DDGi;j MaxwellStefan dusty gas diffusivity for pairij
DeffDGi;j MaxwellStefan effective dusty gas diffusivity for
pairij
Dk;i Knudsen diffusivityBeffDGii diagonal elements of inverted binary effective dusty
diffusion coefficient matrix
BeffDGij non-diagonal elements of inverted binary effective
dusty diffusion coefficient matrix
dp average pore diameter
As electrochemically active surface area of the medium
per unit volume
S4 current source term
ia anodic current density
ic cathodic current density
i0,a anodic exchange current density
i0,c cathodic exchange current density
Vc cell voltageVoc open circuit (Nernst) voltage
Vo ideal (standard) voltage
z number of electrons participating per
electrochemical reaction
F Faradays constant
exp exponent
ks thermal conductivity of solid (i.e. cell components)
Cp,s specific heat of solid (i.e. cell components)
keff effective thermal conductivity of solid and gas phase
reff effective density of solid and gas phase
Cp,eff effective specific heat of solid and gas phase
k thermal conductivity of gas
Cp specific heat of gas
q heat generation
ds change in entropy generation
E activation energy
Greek letters
V differential operator
V2 Laplace operator
fs solid phase (electronic) potentialfe electrolyte phase (ionic) potential
m dynamic viscosity
r average (mixture) gas density
ri density of theith species
rs density of solid (i.e. cell components)
G thermodynamics matrix
k permeability
e porosity
s tortuosity
p constant (3.14159)
s conductivity
a charge transfer coefficient
g pre-exponential coefficienth overpotential
Subscripts
GC gas-chamber
h hole
a anode
e electrolyte
c cathode
cl catalyst layer
rev reversible
irr irreversible
ohm ohmic
act activation
eff effectives solid phase (electronic)
ref reference
Superscripts
a anodic
c cathodic
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http://dx.doi.org/doi:10.1016/j.jpowsour.2009.04.078http://dx.doi.org/doi:10.1016/j.jpowsour.2009.04.078http://dx.doi.org/doi:10.1016/j.jpowsour.2009.04.078http://dx.doi.org/doi:10.1016/j.jpowsour.2009.04.078