DEVELOPMENT OF A COMPREHENSIVE NUMERICAL MODEL FOR ANALYZING A TUBULAR-TYPE INDIRECT INTERNAL REFORMING SOFC

Takafumi Nishino1, Hajime Komori2, Hiroshi Iwai3, Kenjiro Suzuki4

1 Kyoto University, Kyoto, 606-8501, Japan; E-mail: nishino@t020908.mbox.media.kyoto-u.ac.jp 2 Kyoto University, Kyoto, 606-8501, Japan

3 Kyoto University, Kyoto, 606-8501, Japan; E-mail: iwai@mech.kyoto-u.ac.jp 4 Kyoto University, Kyoto, 606-8501, Japan; E-mail: ksuzuki@mech.kyoto-u.ac.jp

ABSTRACT A numerical model for analyzing a tubular-type Indirect Internal Reforming Solid Oxide Fuel Cell (IIR-SOFC), which is expected to become one of the most important power generators in the near future, was developed. The model simultaneously treats momentum, heat and mass transfer, fuel reforming, electrochemical phenomena and an electric circuit. Calculations for the gas flow fields inside and outside the cell tube are conducted with a two-dimensional cylindrical coordinate system adopting the axisymmetric assumption. At the same time, the electric current field in the cell tube is calculated with a quasi three-dimensional coordinate system in order to consider the ohmic loss properly. Activation overpotential is also considered using a temperature dependent model. As a consequence of the calculations, details of conditions in the cell and its power generation characteristics were revealed. Serious temperature gradients were generated in the cell under circumstances where the catalyst for reforming was distributed uniformly inside the feed tube. Complicated electric current fields that varied in both the axial and circumferential direction of the cell were observed. In addition, it became obvious that the temperature dependency of the activation overpotential could be the most significant factor governing the power generation characteristics. INTRODUCTION Fuel cells are high-efficiency and low-emission energy conversion devices which directly transform chemical energy into electricity using the effects of electrochemical reactions. In particular, Solid Oxide Fuel Cell (SOFC) has the advantage of being operated at high temperatures, and hybrid cycles fusing SOFC with Micro Gas Turbine (MGT) are said to be one of the most suitable units for establishing a distributed power supply system. The authors are presently studying a SOFC/MGT hybrid cycle (Suzuki et al., 2002).

There are several types of geometry proposed and tested for SOFC. Among them, the tubular-type SOFC, which consists of a number of tubular cells, is a very famous and promising one (George, 2000; Singhal, 2000). As shown in Fig.1, the tubular cell is composed of three ceramic layers, the anode, cathode and electrolyte. As the name “Solid Oxide” suggests, the solid electrolyte has oxide ion conductivity. Therefore, SOFC can generate electricity using not only hydrogen but hydrocarbons as fuel. Furthermore, steam reforming of hydrocarbons in the cell is possible because of the high operation temperature, 700~1000oC. Fuel reforming is a process of converting hydrocarbons, low-quality fuels for fuel cells, into hydrogen, the high-quality fuel. There are several methods for fuel reforming and the steam reforming is one of them. Steam reforming is an endothermic process and proceeds smoothly at high temperatures (Xu and Froment, 1989). Therefore, this process is extremely suitable for SOFC. From the point of internal reforming, there are two types of SOFC, the direct internal reforming (DIR) type and the indirect internal reforming (IIR) type. In the case of DIR-SOFC, it proceeds on the anode-fuel interface and so there is no need to put in any other catalysts. As for IIR-SOFC, it proceeds within the porous catalyst added in the fuel path away from the anode. Though the cell is a bit more complex in structure, it can be possible to control the cell temperature field depending on how the catalyst is allocated (Dicks, 1998). In analyzing an IIR-SOFC, momentum, heat and mass transfer, internal reforming and electrochemical phenomena and the electric potential field in the cell should be considered simultaneously because they are closely interrelated. In recent years, several numerical studies on IIR-SOFC were reported in literature such as Nagata et al. (2001) and Aguiar et al. (2002). They developed numerical models in which one-dimensional heat and mass transfer was considered, and conducted simulations under various conditions including changes in the gas inlet temperature and electric current density. However, the information obtained from those simulations is limited to some extent

due to the one-dimensional treatment. Hence, the present paper reports on a numerical model that solves the gas flow fields more accurately and on details of the calculation results obtained using this model. NUMERICAL PROCEDURE This study targets a tubular cell of an IIR-SOFC which consists of a cell tube and a feed tube as shown in Fig.1. Fuel, which is assumed to be a mixed gas composed of hydrogen, steam, carbon monoxide (CO), carbon dioxide (CO2) and methane, flows through the feed tube into the cell tube. The fuel is reformed inside the feed tube, changes flow direction at the end of the cell tube, reacts electrochemically on the anode, and then flows out of the cell tube. Meanwhile, air, which is also assumed to be a mixed gas composed of oxygen and nitrogen, flows outside the cell tube and reacts on the cathode. These gas flows are assumed to be laminar, steady and symmetric with respect to the cell axis. Calculations for momentum, heat and mass transfer around the cell are performed with a two-dimensional cylindrical coordinate system. The computational domain is shown in Fig.2 with a broken line. The cell tube is 500mm long with a 6.9mm inner radius and a 9.6mm outer radius. To simplify the problem, it is assumed that the air flows through an annular space confined by an adiabatic cylindrical surface. It is also assumed that the cell tube end is flat. Governing equations to be solved are the continuity, Navier-Stokes, energy and mass transfer equations.

0)(1)( =∂∂

+∂∂

rx vrrr

vx

ρρ (1)

∂∂

∂∂

+

∂∂

∂∂

+∂∂

−=

∂∂

+∂∂

rvr

rrxv

xxp

rvv

xvv xxx

rx

x µµρ1 (2)

21

rv

rvr

rrxv

xxp

rvv

xvv rrrr

rr

x µµµρ −

∂∂

∂∂

+

∂∂

∂∂

+∂∂

−=

∂∂

+∂∂ (3)

SrTr

rrxT

xrTv

xTvC rxp +

∂∂

∂∂

+

∂∂

∂∂

=

∂∂

+∂∂

λλρ1 (4)

SrYDr

rrxYD

xrYv

xYv j

jmj

jmj

rj

x +

∂∂

∂∂

+

∂∂

∂∂

=

∂∂

+∂∂

ρρρ1 (5)

Here, the gas properties are treated as variables dependent on local temperature and concentration of each chemical species. These governing equations are discretized using the finite volume method. The power-law scheme is adopted for the convection and diffusion terms of the equations. S in Eqs.(4) and (5) denote the source term generated by the chemical reactions described in detail later. SIMPLE algorithm is used for the computation of the pressure correction in the iteration procedure. The inlet and outlet conditions of the fuel and air are summarized in Table 1. Fuel mean velocity at the inlet is set to such a value that an average current density is capped at 5000 A/m2 (e.g. 0.923 m/s when the inlet temperature is 800oC). In contrast, air mean velocity at the inlet is consistently set at 2.0 m/s. Both inlet pressures are set at 1 atm. In regard to solid-fluid interfaces, the no-slip condition is applied. Heat conduction in the solid parts is considered, while the effect of radiation on the thermal field is excluded. The temperature of the closed end of the cell tube is set equal to that of the air inlet. In this study, the fuel reforming process is considered to be represented by the following chemical reactions.

COH 3OHCH 224 +↔+ (6) 222 COHOHCO +↔+ (7)

It is assumed that the steam reforming reaction (6) can occur only inside the feed tube and that the water-gas shift reaction (7) can occur across the fuel path. The local reaction rates, which govern S in Eqs.(4) and (5), are calculated as follows (Odegard et al., 1995).

Fig.2 Computational domain for the tubular cell of SOFC (inside of the broken line).

Fig.1 Schematic diagram of a tubular cell of SOFC.

Table 1. Gas inlet and outlet conditions.

Velocity Temperature Molar fraction

Fuel inlet Poiseuille flow Constant

H2 H2O CO CO2 CH4

0.200 0.500 0.020 0.030 0.250

Air inlet Plug flow Constant O2 N2

0.209 0.791

Outlet Neuman Neuman Neuman

−=

TRWpR

0cat

2.1CHst

57840exp75.1 4 (8)

222 COHshOHCOshsh ppkppkR −+ −= (9)

Here, the mass density of the catalyst for the steam reforming, Wcat, is set constant inside the feed tube in the present study. +

shk and −shk denote the velocity constants of

forward and backward water-gas shift reactions depending on the local temperature. Based on these reaction rates, the thermodynamic heat and production or consumption of each chemical species accompanied with the reactions are calculated, and then the source term S is calculated. Electrochemical reactions are assumed to occur on the cell tube surface and are represented as follows.

(cathode) O e 2O 1/2 22

−− →+ (10) (anode) e 2OH OH 2

22

−− +→+ (11) (anode) e 2CO OCO 2

2 −− +→+ (12)

Local current density, which also governs the source term S in Eqs.(4) and (5) in and on the cell tube, is calculated as described below. The electric current is assumed to flow in the circumferential (θ) direction in the electrodes and only radially in the electrolyte as shown in Fig.3. Dark and light coloring of the arrows imply large and small flows of electric current. The thickness of the interconnects is excluded in this study. Local electromotive force (EMF) is

calculated with the Nernst equation using local temperature and partial pressure of each chemical species near the cell tube surfaces. Electric and ionic resistivities of the electrodes and electrolyte are evaluated considering temperature dependency as suggested by Bessette (1994). Activation overpotential accompanied with each reaction is also evaluated as follows (Achenbach, 1994).

1

0

cathode0.25

OO

0COH)O( exp4 )( 2

222

−

−

+=

TRA

ppk

TRFiiActη (13)

CO) ,H( exp2 2

1

0

anode25.0

0)( =

−

=

−

jTR

Appk

TRFi j

jjjActη (14)

Here, iH2 and iCO are the current densities in the electrolyte arising from reaction (11) and (12), respectively. Acathode and Aanode denote the activation energies for each electrode. Using the Kirchhoff law with all the above components of the electric circuit, the electric potential and current density fields are calculated. In this process, the cell terminal voltage is given as a boundary condition and the average current density is obtained as a calculation result. The ohmic and thermodynamic heat and variation of each chemical species accompanied with the electrochemical reactions are

(a) Velocity

0.0 0.1 0.2 0.3 0.4 0.50.000

0.005

0.010

r [m

]

x [m] (b) Temperature

0.0 0.1 0.2 0.3 0.4 0.50.000

0.005

0.010

[oC]

600.0

750.0

900.0

1050

x [m]

r [m

]

Fig.4 Velocity and temperature fields around the cell for an average current density of 3990 A/m2.

Fig.3 Schematic diagram of an electric circuit and flow of electric current in the cell tube.

(a) O2 in the air path

0.0 0.1 0.2 0.3 0.4 0.50.0100.0120.014

0.1600

0.2100

x [m]

r [m

]

(b) H2 in the fuel path

0.0 0.1 0.2 0.3 0.4 0.50.000

0.005

0

0.5000

1.000

x [m]

r [m

]

(c) H2O in the fuel path

0.0 0.1 0.2 0.3 0.4 0.50.000

0.005

0

0.5000

1.000

x [m]

r [m

]

(d) CO in the fuel path

0.0 0.1 0.2 0.3 0.4 0.50.000

0.005

0

0.1250

0.2500

x [m]

r [m

]

(e) CO2 in the fuel path

0.0 0.1 0.2 0.3 0.4 0.50.000

0.005

0

0.1250

0.2500

x [m]r [

m]

(f) CH4 in the fuel path

0.0 0.1 0.2 0.3 0.4 0.50.000

0.005

0

0.1250

0.2500

x [m]

r [m

]

Fig.5 Molar fraction distributions of each chemical species for an average current density of 3990 A/m2.

also calculated based on the current density field. These values are averaged in the θ-direction and applied to the two-dimensional calculation of the gas flow fields. RESULTS AND DISCUSSION Details of Thermal and Concentration Fields Figure 4(a) shows velocity vectors inside and outside the tubular cell for an average current density of 3990 A/m2. In this case, the fuel utilization factor is 79.8% because the fuel flow rate is selected so that the utilization becomes 100% when the average current density is 5000 A/m2 as described above. Both fuel and air inlet temperatures are set at 800oC. This figure shows the typical flow pattern. The fuel gas flow accelerates inside the feed tube as it flows downstream because of an increase in the number of total moles caused by the steam reforming. The flow changes direction near the outlet of the feed tube and stagnates at the bottom of the cell tube. Figure 4(b) shows the temperature contours for the same case. Darker tone corresponds to the higher local temperature region. Owing to the effect of the steam reforming reaction, the fuel temperature goes down to about 650oC near the inlet of the feed tube. However, in the middle the cell tube temperature goes up to about 960oC because of the thermodynamic and ohmic heats caused by the electrochemical reactions. In consequence of this, steep temperature gradients occur within not just the gas fields but also the solid parts of the cell. These temperature gradients

can cause thermal crack failure of the cell. Hence, the details about the thermal field shown in Fig.4(b) are quite important for the design and operation of the SOFC. Figure 5 shows molar fraction contours of O2, H2, H2O, CO, CO2 and CH4 for the same case. Note that the color tone levels differ among each chemical species. As for the air side, oxygen is consumed by the electrochemical reaction and so its molar fraction decreases toward the outlet. However, variation of the molar fraction is not so large because plenty of air for the electrochemical reaction is pumped into the cell. In contrast, the distributions of the fuel side are a little more intricate as shown in Fig.5(b)~(f). Inside the feed tube, hydrogen and CO are produced while steam and methane are consumed in the process of the steam reforming. This reaction proceeds fastest near the fuel inlet, and the reaction rate decreases in the flow direction. Most of the methane is consumed within two-fifths of the feed tube in this case. At the same time, the water-gas shifting reaction also occurs inside the feed tube. Where the local temperature is relatively low, such as near the fuel inlet, the reaction proceeds in such a direction that hydrogen and CO2 are produced and steam and CO are consumed. Where the temperature is relatively high, the reaction proceeds to the opposite direction. Outside of the feed tube, hydrogen and CO are consumed and steam and CO2 are produced by the electrochemical reactions. The water-gas shifting reaction also occurs there. The reason for the considerable molar fraction change at the cell bottom is that the fuel gas flow stagnates there while the electrochemical reactions occur. These concentration fields highly depend on the density of the catalyst, Wcat, which is not optimized in the present study. The molar flow rate of each chemical species of the fuel inside and outside the feed tube is shown in Fig.6. These results are for the same case as Fig.4 and 5. Characteristics of the chemical change inside the fuel path described above also can be observed in this figure. Details of Electric Potential and Current Fields Distributions of local EMF, activation overpotential, ohmic loss and current density in the electrolyte layer are

(a) Inside the feed tube

(b) Outside the feed tube

Fig.6 Molar flow rate of each chemical species of fuel for an average current density of 3990 A/m2.

Fig.7 Local EMF, activation overpotential, ohmic loss and current density in the electrolyte for an average current density of 3990 A/m2.

shown in Fig.7. These results are for the same case as Fig.4, 5 and 6. The left vertical axis indicates the scale of the local current density and the right axis indicates the others. EMF has a peak at x=0.05m, where the outlet of the feed tube is located, and it becomes lower toward the cell outlet and also toward the cell bottom with the decrease in hydrogen and CO shown in Fig.5. However, the activation overpotential, ηAct, is basically small in the middle of the cell, where the local temperature is higher as shown in Fig.4(b). As a result, the current density has a peak in the middle of the cell rather than near the outlet of the feed tube. The magnitude of the ohmic loss in the electrolyte, ηOhm(electrolyte), is much smaller than that of the activation overpotential. To sum up the above, the current density tends to become larger where the temperature is higher because the activation overpotential is smaller there. At the same time, a large current density accompanies large thermodynamic and ohmic heats. This mechanism is the most fundamental factor causing the significant gradients of both current density and temperature. In the present calculation, activation overpotential, ohmic loss and current density in the electrolyte have distributions not just in the axial (x) direction of the cell but also in the circumferential (θ) direction (cf. Fig.3). This is because ohmic loss accompanied with the electric current which flows through the electrodes in the θ-direction is considered. In Fig.7, the term “local” is used for only the x-direction and all data except the EMF represent the average values for the θ-direction. In Fig.8 however, no averaging is done in the θ-direction and the actual local data of the electrode electric potential and the current density in the electrolyte are presented. Figure 8(a) shows the electric potential distributions of both anode and cathode at θ=0o, 45o, 90o, 135o and 180o. In regard to the anode side, the potential at θ=0o, where electric current flows into the cell tube from the interconnect, is set at 0 V. The anode potential drops toward θ=180o because of the ohmic loss caused by the circumferential electric current flowing there. Due to the decrease in the electric current, the rate of the potential drop also decreases toward θ=180o. As for the cathode side, the circumferential electric current increases toward θ=180o contrary to the anode side. Therefore, the rate of the potential drop also increases toward θ=180o. The cathode potential at θ=180o, where electric current flows out of the cell tube into the interconnect, is 0.47 V in this case. Taken as a whole, this figure shows that the ohmic loss in both electrodes has a peak in the middle of the cell as also occurs for the current density shown in Fig.7. In addition, the ohmic loss in the cathode is larger than that in the anode. The reason is that the resistance of the cathode to the circumferential electric current is larger than that of the anode, and this is one of the important factors which govern the electric current field. Figure 8(b) shows the electric potential difference between the electrodes. As a consequence of the matters described above, distributions of the potential difference become as shown in the figure. Figure 8(c) shows the current density in the electrolyte. By comparison of these

two figures, it is observed that the angles of large current density correspond to those of small potential difference. The reason why the current density at θ=180o is larger than at θ=0o is that the resistance of the cathode to the circumferential electric current is larger than that of the anode. If the resistances are equal, the results of these distributions theoretically could be symmetric with respect to θ=90o. In other words, it may be possible to control the electric current field to some degree by changing either resistivity or thickness of the electrodes. Effect of Average Current Density Figure 9 shows the temperature contours for an average current density of 0, 1999, 3095 and 3990 A/m2. In the case of 0 A/m2, no hot spot emerges in the cell tube because there are no thermodynamic and ohmic heats caused by the

(a) Electric potential in the electrodes

(b) Potential difference between the electrodes

(c) Current density in the electrolyte

Fig.8 Local electric potential and current density for an average current density of 3990 A/m2.

electrochemical reactions. Owing to the effect of the steam reforming reaction, the fuel temperature near the inlet goes down to about 620oC, which is a little lower than that for the other cases. When the average current density is 1999A/m2, a slight hot spot is observed rather near the bottom of the cell tube. As the average current density increases, the hot spot becomes more noticeable and its position gradually moves to the cell outlet direction. In this manner, the thermal field around the cell varies considerably depending on the average current density. Figure 10 shows the molar fraction contours of hydrogen for the same average current densities as Fig.9. In the case of 0 A/m2, the fuel reforming is nearly completed within three-fifths the feed tube. After this region, the molar fraction of each species changes very little because there are no electrochemical reactions and only the water-gas shift reaction occurs due to the local temperature difference. As the average current density increases, the consumption of hydrogen outside the feed tube increases. In addition, the fuel reforming velocity also increases because of the rise in the cell overall temperature. Distributions of local EMF and current density in the electrolyte for the same average current densities are shown in Fig.11. In this figure, the current density is averaged in the θ-direction. The larger an average current density becomes, the lower EMF becomes all over the cell. In the

case of 1999 and 3095 A/m2, the local EMF near the cell outlet is slightly higher than that in the middle despite that the local molar fraction of hydrogen is lower there. The reason is that the local temperature near the cell outlet is quite lower than that in the middle. As for the current density, the position of its peak moves with the hot spot shown in Fig. 9 depending on the average current density. Power Generation Characteristics Evaluation Figure 12(a) shows the power generation characteristics of a cell under the condition of both fuel and air inlet temperatures 800oC. Cell-averaged EMF, activation overpotential and ohmic loss in the electrolyte are shown in this figure. The cell terminal voltage, which is the potential difference between the anode at θ=0o and the cathode at θ=180o, is also plotted. The gap between the white circles and the black circles denote the cell-averaged ohmic loss in the electrodes. Due to the fixed condition of fuel flow rate,

(a) Av. current density : 0 A/m2

0.0 0.1 0.2 0.3 0.4 0.50.000

0.005

0.010

[oC]

600.0

750.0

900.0

1050

x [m]

r [m

]

(b) Av. current density : 1999 A/m2

0.0 0.1 0.2 0.3 0.4 0.50.000

0.005

0.010

[oC]

600.0

750.0

900.0

1050

x [m]

r [m

]

(c) Av. current density : 3095 A/m2

0.0 0.1 0.2 0.3 0.4 0.50.000

0.005

0.010

[oC]

600.0

750.0

900.0

1050

x [m]

r [m

]

(d) Av. current density : 3990 A/m2

0.0 0.1 0.2 0.3 0.4 0.50.000

0.005

0.010

[oC]

600.0

750.0

900.0

1050

x [m]

r [m

]

Fig.9 Temperature fields around the cell for various average current densities.

(a) Av. current density : 0 A/m2

0.0 0.1 0.2 0.3 0.4 0.50.000

0.005

0

0.5000

1.000

x [m]

r [m

]

(b) Av. current density : 1999 A/m2

0.0 0.1 0.2 0.3 0.4 0.50.000

0.005

0

0.5000

1.000

x [m]

r [m

]

(c) Av. current density : 3095 A/m2

0.0 0.1 0.2 0.3 0.4 0.50.000

0.005

0

0.5000

1.000

x [m]

r [m

]

(d) Av. current density : 3990 A/m2

0.0 0.1 0.2 0.3 0.4 0.50.000

0.005

0

0.5000

1.000

x [m]

r [m

]

Fig.10 Molar fraction distributions of hydrogen in the fuel path for various average current densities.

Fig.11 Local EMF and current density in the electrolyte for various average current densities.

the average current density is in proportion to the fuel utilization factor and that of 5000 A/m2 corresponds to 100%. As the average current density increases, which means the fuel utilization factor increases, the EMF gradually becomes lower. The activation overpotential, which is denoted by the gap between the crosses and the black squares, increases until the average current density reaches about 1500 A/m2. However, in contrary it decreases when the current density exceeds that magnitude. The ohmic loss in the electrolyte also increases until about 1500 A/m2, and becomes almost constant at larger current densities. These tendencies are attributable to temperature dependency of the activation overpotential and the ionic resistivity of the electrolyte. In contrast, the ohmic loss in the electrodes increases almost proportionally to the current density because electric resistivities of the electrodes are not so dependent on temperature.

Figure 12(b) shows the maximum and average temperature of the electrolyte for each average current density. Rise in the temperature accompanied with increase in the current density is confirmed by this figure. The output power and energy conversion efficiency of the cell are shown in Fig.12(c). The efficiency is calculated based on the lower heating value (LHV) of the fuel consumed in the cell. Along with the increase in current density, the output power becomes larger and the efficiency goes down. The increase rate of the output power becomes almost constant and the decline of the efficiency becomes modest when the current density exceeds about 1500 A/m2. This is primarily because the activation overpotential decreases in this range. Figure 13 shows the same kind of data about the cell performance under the condition of a fuel inlet temperature of 750oC and an air inlet temperature of 850oC. In the present study, the air inlet temperature has a great effect on

(a) Average EMF and voltage drops

(b) Electrolyte temperature

(c) Output power and efficiency

Fig.12 Power generation characteristics of a cell under the condition of fuel inlet 800oC and air inlet 800oC.

(a) Average EMF and voltage drops

(b) Electrolyte temperature

(c) Output power and efficiency

Fig.13 Power generation characteristics of a cell under the condition of fuel inlet 750oC and air inlet 850oC.

the overall cell temperature because the mass flow rate of air is much larger than that of fuel. As shown in Fig.13(a), the cell terminal voltage of this case tends to take a higher value than that of the previous case shown in Fig.12. The main factor is that the activation overpotential is relatively small, which is owing to the high cell temperature plotted in Fig.13(b), and then the higher output power and energy conversion efficiency of the cell are attained as shown in Fig.13(c). To sum up the matter, the power generation characteristics of the cell highly depend on the activation overpotential, which highly depends on the cell temperature. CONCLUSIONS A numerical model for an IIR-SOFC, which considers momentum transfer, heat and mass transfer, electrochemical phenomena and an electric circuit, was developed. By using this model, details of thermal and concentration fields for various average current densities were obtained. In the present study, the catalyst for the steam reforming was assumed to be distributed uniformly inside the feed tube. It follows that the reforming process was largely finished near the fuel inlet and noticeable temperature gradients occur in the cell. Details of electric potential and current fields were also obtained. These fields showed heterogeneous distributions in the circumferential direction of the cell due to the cell geometry and resistivity difference between the two electrodes. It seems to be possible to control these fields to some degree by changing either the resistivity or thickness of the electrodes. Regarding the power generation characteristics of the cell, a huge influence of the activation overpotential was revealed. Activation overpotential can be the biggest factor causing degradation in the cell performance when the cell temperature is rather low, however, it drastically decreases as the cell temperature rises. Meanwhile, increase in current density is one of the main causes for increases in temperature. For this reason, it is possible that the activation overpotential decreases as the current density increases. NOMENCLATURE A activation energy, J/mol Cp specific heat at constant pressure, J/kg K Djm effective diffusivity of chemical species j, m2/s F Faraday constant, C/mol ij electric current density arising from electrochemical reaction of species j, A/m2 kj pre-exponential factor of electrochemical reaction of species j, A/m2

+shk velocity constant of water-gas shift reaction,

(forward), mol/m3 Pa2 s −shk velocity constant of water-gas shift reaction,

(backward), mol/m3 Pa2 s

R0 universal gas constant, J/mol K Rsh reaction rate of water-gas shift reaction, mol/m3 s Rst reaction rate of steam reforming reaction, mol/m3 s Wcat catalyst mass density inside the feed tube, g/m3

Yj mass fraction of chemical species j, dimensionless ηAct activation overpotential, V ηOhm ohmic loss, V λ thermal conductivity, W/m K µ viscosity, Pa s ρ density, kg/m3 Subscripts cat catalyst sh water-gas shift reaction st steam reforming reaction ACKNOWLEDGMENT This article is written based on the research results for the CREST project “Micro Gas Turbine and Solid Oxide Fuel Cell Hybrid Cycle for Distributed Energy System” supported by the Japan Science and Technology Corporation (JST). REFERENCES Achenbach, E., 1994, Three-dimensional and time-dependent simulation of a planar solid oxide fuel cell stack, J. of Power Sources, Vol. 49, pp. 333-348. Aguiar, P., Chadwick, D., Kershenbaum, L., 2002, Modeling of an indirect internal reforming solid oxide fuel cell, Chemical Engineer Science, Vol. 57, pp. 1665-1677. Bessette, N.F., 1994, Modeling and Simulation for Solid Oxide Fuel Cell Power System, Ph.D. Thesis, Georgia Institute of Technology, Atlanta, GA. Dicks, A.L., 1998, Advances in catalysts for internal reforming in high temperature fuel cells, J. of Power Sources, Vol. 71, pp. 111-122. George, R.A., 2000, Status of tubular SOFC field unit demonstrations, J. of Power Sources, Vol. 86, pp. 134-139. Nagata, S., Momma, A., Kato, T., Kasuga, Y., 2001, Numerical analysis of output characteristics of tubular SOFC with internal reformer, J. of Power Sources, Vol. 101, pp. 60-71. Odegard, R., Jornsen, E., Karoliussen, H., 1995, Methane reforming on Ni/zirconia SOFC anodes, Proc. 4th International Symposium on Solid Oxide Fuel Cells (SOFC-IV), pp. 810-819. Singhal, S.C., 2000, Advances in solid oxide fuel cell technology, Solid State Ionics, Vol. 135, pp. 305-313. Suzuki, K., Iwai, H., Kim, J.H., Li, P.W., Teshima, K., 2002, Keynote paper: Solid Oxide Fuel Cell and Micro Gas Turbine Hybrid Cycle and related Fluid Flow and Heat Transfer, in Proc. 12th International Heat Transfer Conference. Xu, J., Froment, G.F., 1989, Methane Steam Reforming, Methanation and Water-Gas Shift: I. Intrinsic Kinetics, AIChE Journal, Vol. 35, pp. 88-96.