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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


aac 1 [27]

Cathodic anodic


aca 1.5 [27]

Cathodic cathodic


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.


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.


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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r e f e r e n c e s

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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

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