Numerical Study of a Planar Solid Oxide Fuel Cell during Heat-up and

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Ind. Eng. Chem. Res. 2011, 50, 1360–1368

Numerical Study of a Planar Solid Oxide Fuel Cell during Heat-up and Start-up Operation YoungHwang Kim, MyungSuk Son, and In-Beum Lee* Department of Chemical Engineering, POSTECH, San 31 Hyoja Dong, Pohang, 790-784, Korea

Solid oxide fuel cells (SOFCs) consist of ceramic materials. Because of the brittle nature of ceramics, durability decrease of a SOFC system or mechanical failure of cells can be caused by transient behavior, that is, sudden temperature variation and axial temperature gradient. Therefore, it is important to understand the transient behavior of a SOFC. To study the transient behavior of a direct internal reforming (DIR) planar solid oxide fuel cell (SOFC), a one-dimensional dynamic model is presented. This model is modified to predict the heatup and start-up behavior. The heat-up time and start-up time are calculated from the model. The heat-up time can be adjusted by manipulating air velocity and temperature. During the start-up mode, the effects of initial temperature of the PEN (positive electrode/electrolyte/negative electrode) structure and air temperature are investigated. The fuel cell characteristics such as cell voltage, current density distribution, and temperature distribution can be calculated from the modified dynamic model. Consequently, this model can be useful to investigate the transient behavior during heat-up and start-up modes. 1. Introduction Solid oxide fuel cells (SOFCs) are environmental-friendly power generation devices with low pollutant emission and high efficiencies. Because of their high operating temperature (873-1273 K), they have several characteristics including high overall efficiency. SOFCs generate electricity and heat by electrochemical reactions between fuel and oxygen ions. Hydrogen, carbon monoxide, methane, and other hydrocarbons can be used as fuel of SOFCs. A SOFC is made up of four layers; anode, cathode, electrolyte, and interconnect. In the anode layer, water is produced and two electrons are released by the electrochemical reaction between a hydrogen molecule and an oxygen ion migrated from the electrolyte. The released electrons pass through the external circuit to the cathode. In the cathode, an oxygen molecule reacts with those electrons to produce oxygen ions. It is necessary to convert methane or other hydrocarbons into hydrogen-rich gases for the electrochemical reaction of hydrogen. There are several common technologies for hydrogen production and one of them, the steam reforming reaction, is a well-established commercial process. Steam reforming reactions of gaseous hydrocarbons are endothermic and can be performed externally in a catalytic steam reformer. The waste heat with high temperature, due to the operating temperature of SOFCs, can be used for steam reforming reactions. Thus, the steam reforming reaction can be performed both in the anode (direct internal reforming) and in the separated reactor attached to the stack (indirect internal reforming).1 SOFCs are used for building and power plant applications with a power range from a few kilowatts to a few megawatts. As mentioned above, because high quality waste heat is produced in SOFCs, it can be combined with a gas turbine (GT) or a combined heat and power (CHP) system. The efficiency of these hybrid systems, like SOFC/GT and SOFC/CHP, is higher than a general stand-alone SOFC. In other words, the stand-alone SOFC efficiency can reach up to 55%, while the efficiency of a SOFC/GT hybrid system can be theoretically obtained up to 70%.10 * To whom correspondence should be addressed. Tel.: +82-54-2792274. E-mail: [email protected].

SOFCs are usually made of ceramic materials. The electrolyte is usually yttria-stabilized zirconia (YSZ) which has high ion conductivity at the operating temperature of cells. Typically, the anode is a nickel/zirconia cermet which provides high electrochemical performance. The cathode is a strontium-doped lanthanum manganite which has high electronic and ionic conductivities.1,2,15 The cell is weak from the thermal stress due to the brittle nature of ceramic materials. There are several studies which investigated the thermal stresses in SOFCs.3-7 The thermal stresses by temporal temperature gradients (sudden temperature variations), axial temperature gradients and differences of thermal expansion coefficients between components cause microcrackings in the cell layer and the performance degradations of the system.5 Two steps are required to generate electricity in the SOFC. At first, the cell is heated by an external heat source, from ambient temperature (298 K) to higher temperature above 873 K. This step is called the heat-up mode. When the temperature of the electrolyte reaches above 873 K, the electrolyte can transfer oxygen ions from the cathode to the anode. After the heat-up mode, fuel is fed to the anode channel, and the electrochemical reaction takes place at the triple phase boundary (TPB) where the anode/electrolyte interface or cathode/ electrolyte interface is. The second step is the start-up mode. The start-up time takes from a few minutes to an hour, while the heat-up takes at least a few hours.3,7 During heat-up and start-up modes, the cell may be damaged by a sudden temperature variation or an axial temperature gradient. Therefore it is important to understand transient behavior during the heat-up and start-up modes. There are few studies3,4,7 for heat-up and start-up modes. Selimovic et al.3 and Apfel et al.4 investigated a planar SOFC for heat-up and start-up modes (Barzi et al.7 investigated a tubular SOFC). The work of refs 3 and 4 did not analyze effects of important variables such as air velocity, air temperature, and initial temperature of PEN (positive electrode/ electrolyte/negative electrode) structure. These variables affect not only the heat-up and start-up time but also temperature variations. Therefore, it is important to know how these variables affect heat-up and start-up modes.

10.1021/ie100783g  2011 American Chemical Society Published on Web 08/26/2010

Ind. Eng. Chem. Res., Vol. 50, No. 3, 2011

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CH4 + H2O T CO + 3H2

(1)

CO + H2O T CO2 + H2

(2)

A hydrogen molecule is electro-oxidized by an oxygen ion at the anode, and an oxygen molecule is reduced to two oxygen ions at the cathode. The overall reaction is summarized as follows: H2 + O2- f H2O + 2e-

(3)

1 O + 2e- f O22 2

(4)

1 H2 + O2 f H2O 2

(5)

Figure 1. Schematic view of a coflow planar SOFC.

To understand the transient behavior during the heat-up and start-up modes, it is necessary to develop a dynamic SOFC model. There are various dynamic models from a simple lumped (zero-dimensional) model to a complex three-dimensional one. The lumped model is suitable for real time simulation and process control. A lumped model8-14 is simple, because it is formulated with a group of ordinary differential equations (ODEs). However, the lumped model is inaccurate for high current density.11 To improve this, some authors used detailed models (lumped model)11,14 or one-dimensional dynamic models.19,20 Complex models are presented by partial differential equations (PDEs). The complex dynamic models3-7,15-21 are suitable to design cell components. The complex models are generally used when detailed information within the cell is. The purpose of this paper is to study the transient behavior described as a temporal temperature gradient and an axial temperature gradient during heat-up and start-up periods. The lumped model only predicts a temporal temperature gradient, while the one-dimensional dynamic model can predict not only a temporal temperature gradient but also a axial temperature gradient. Therefore in this study, the one-dimensional dynamic model is introduced and modified. Section 2 shows the onedimensional dynamic model of the planar SOFC. The proposed model is adopted to a case study, as we will describe in section 3. In section 4, we investigate the effect of variables (air velocity, air temperature, initial temperature of PEN structure) and also discuss simulation results of the heat-up mode, start-up mode, and steady-state. Finally, conclusions are presented in section 5.

The one-dimensional dynamic model of a planar SOFC is based on following assumptions:15 (i) the gases are ideal gas; (ii) pressure drop along the channel is neglected; (iii) the cell voltage is constant throughout the cell; (iv) the repeating single cell is considered to be in the center of a large stack; (v) only hydrogen is electrochemically oxidized. Mass and Energy Balance. The mass balances are considered for the fuel channel and the air channel. Five gas species, H2, H2O, CH4, CO, and CO2 are considered at the fuel channel, and two species, N2, and O2, are considered at the air channel. Three reactions are included: a stream reforming reaction, a water-gas shift reaction, and an electrochemical reaction. The reaction rate for each reaction can be seen in Table 1. The mass balance for each species of the fuel channel is written as follow:15,17 ∂Ci,f ∂Ci,f 1 + ) -uf νi,kRk ∂t ∂z t f k∈{(1),(2),(5)} i ∈ {H2, H2O, CH4, CO, CO2}

∑

(6)

where the variable z is the fuel and air flow direction. In the air channel, the oxygen reduction reaction is considered. The mass balance can be obtained from the conservation law that is similar to the one for the fuel channel. The mass balance is represented as follow:15,17 ∂Ci,a ∂Ci,a 1 + νi,(5)R(5) ) -ua ∂t ∂z ta i ∈ {O2, N2}

(7)

2. Mathematical Model In this study, a direct internal reforming (DIR) intermediate temperature planar SOFC fuelled as methane is considered. A one-dimensional dynamic model15 is introduced and modified to study the transient behavior during the heat-up and start-up modes. There are three flow configurations; coflow, counterflow, and crossflow. Figure 1 shows a schematic view of a coflow planar SOFC. Conventional high temperature SOFCs generally operate between 1073 and 1273 K. Each part of an SOFC is made of ceramic materials. Intermediate temperature (IT) SOFCs operate between 823 and 1023 K, allowing a range of materials and cost-effective SOFC fabrication. IT-SOFCs can use metal-ceramic or stainless steel as the material of interconnect.15 The research of Selimovic et al.3 shows that thermal stress was much lower when the ceramic interconnect was replaced by the metallic one. The methane is reformed at the inside of anode, that is, internal reforming. The steam reforming reaction and the watergas shift reaction are summarized as follows:

Energy balances are considered for fuel channel, air channel, interconnect (IC), and PEN structure. The convective heat transfer is considered between gas stream and solid component. The heat conduction is considered for solid components. Due to high temperature operation, the radiation between the PEN and interconnect is considered in energy Table 1. Reaction Rates for the Reactions 1, 2, and 5 reaction

reaction rate

( )

1 CH4 + H2O T CO + 3H2

R(1) ) K0PCH4 exp -

2 CO + H2O T CO2 + H2

R(2) ) KwgPCO 1 -

1 5 H2 + O2 f H2O 2

R(5) )

(

J 2F

EA RT

PCO2PH2 /PCOPH2O Keq,(2)

)

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balances. The energy balance for each component can be represented as follows:15,17 Fuel channel ∂Tf ∂Tf 1 ) -ufFfcp,f + hf(TPEN - Tf) + Ffcp,f ∂t ∂z tf 1 1 (-∆H)kRk hf(TIC - Tf) + tf t f k∈{(1),(2)}

∑

(8)

Air channel ∂Ta ∂Ta 1 1 ) -uaFacp,a + ha(TPEN - Ta) + ha(TIC - Tf) Facp,a ∂t ∂z ta ta (9) Interconnect ∂TIC ∂2TIC 1 ) kIC 2 - hf(TIC - Tf) FICcp,IC ∂t tIC ∂z ha(TIC

FU )

]

[

σ(TIC4 - TPEN4) 1 1 - T a) - 1 tIC /εIC + 1/εPEN - 1 tIC

[

4 4 σ(TIC - TPEN )

+

1 /εPEN

]

1 t - 1 PEN

(11)

The thermal conductivity of the gases has been found in the literature,24 and the heat transfer coefficient can be obtained from the following equation: h ) Nu

k Dh

(12)

where Nu is the Nusselt number and Dh is hydraulic diameter. The heat capacity of the gases has been found in the literature.25 Additional information of the one-dimensional dynamic model can be found in the literature.15,17-20 Electrochemical Model. The open circuit voltage (OCV) represents the voltage of a unit cell, when there is no current flow. The OCV is given by the Nernst equation. When current is drawn from the fuel cell, cell voltage decreases due to irreversible overpotentials. The actual voltage is represented by following relationships:1 V ) VOCV - ηact - ηohm - ηconc VOCV )

( )

pH2O RT -∆G0 ln 2F 2F pH2pO20.5

(13)

(14)

The ohmic overpotential due to the resistivity of components is given by Ohm’s law. The activation overpotential is related with the electrochemical reaction kinetics. The governing equation of activation overpotential is obtained from the ButlerVolmer equation. The concentration overpotential appears when the reactant inlet flux and the product outlet flux from an electrode are slower than the flux corresponding to the discharged current.15 There are many studies about the electrochemical reaction,1,12,15,19,20,22,23 and it is briefly introduced in this paper.

in (4xCH 4

jj LW/2F in + xHin2 + xCO )Nfin

AR )

(10)

PEN structure ∂TPEN ∂2TPEN 1 ) kPEN FPENcp,PEN - hf(TPEN - Tf) ∂t tPEN ∂z2 1 1 ha(TPEN - Ta) + (-∆H(5)R(5) - jV) + tPEN tPEN 1 /εIC

Solution Strategy. The mass balance at the fuel channel and the air channel, the energy balance for each structure component, electrochemical model, and equivalent electrochemical model are solved using gPROMS ModelBuilder 3.2.0 (Process Systems Enterprise Ltd.) which is suitable for not only the steady-state simulation but also the dynamic simulation. During the heat-up mode, there is no reaction (no electrochemical and chemical reaction). Therefore reaction rates are zero in the mass balance and the energy balance and the electrochemical model is also omitted. There are three degrees of freedom for the complete solution of the start-up mode. One option usually used in the fuel cell field is to specify the values of average current density, fuel utilization, and air ratio.12,15,19,20 The fuel utilization and the air ratio are represented as follows:

xOin2Nin a jj LW/4F

(15)

(16)

During the start-up mode, the cell voltage must be predicted at every time step. To predict the cell voltage, an equivalent electrochemical model is used. The equivalent model uses the average current density, the average values of temperature of the PEN, and partial pressure of H2O, H2, O2. In the electrochemical model, local values of temperature, partial pressure, and other properties are used. A local current density (current density distribution along the cell) is calculated using predicted voltage obtained by the equivalent electrochemical model. In here only the coflow configuration is considered. The boundary conditions of mass balances for seven species can be written as follows: 0 Ci,f | z)0 ) Ci,f ,

Ci,a | z)0 ) C0i,a

(17)

The boundary conditions of energy balances for the fuel stream and air stream can be written as follows: Tf | z)0 ) Tf0,

Ta | z)0 ) T0a

(18)

The both ends of the solid components are assumed to be insulated.19,20 The boundary conditions of energy balance for solid components can be written as follows: ∂T ) 0, ∂z z)0

∂T )0 ∂z z)L

(19)

3. Case Study In this paper, the heat-up and start-up modes are considered. To study those behaviors, the following operation modes are defined.3,7 Heat-up. To enable the start of the electrochemical reaction within a SOFC, the cell must be heated up by an external heat source from ambient temperature (298 K) to start-up temperature (1023 K). A hot air stream is used as that external heat source in this study; however, the exhaust gas from an after-burner will be used in a practical SOFC system. For the heat-up simulation, the hot air stream was fed into the air channel. Three cases are considered for the heat-up mode: Case 1. Air temperature is constant. (uair ) 3 m/s, Tair ) 1023 K). Case 2. Air temperature is constant. (uair ) 1.5 m/s, Tair ) 1023 K). Case 3. Air temperature is subjected to the minimum temperature of the PEN (uair ) 3 m/s).


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Table 3. Parameters of the Electrochemical Model kanode kcathode Eanode Ecathode anode diffusion coefficient cathode diffusion coefficient anode electrical conductivity electrolyte electrical conductivity cathode electrical conductivity

6.54 × 1011 Ω-1 m-2 2.35 × 1011 Ω-1 m-2 140 kJ/mol 137 kJ/mol 3.66 × 10-5 m2/s 1.37 × 10-5 m2/s (9.5 × 107)/(T) · exp((- 1150)/(T)) Ω-1 m-1 (3.34 × 104) · exp((- 10300)/(T)) Ω-1 m-1 (4.2 × 107)/(T) · exp((- 1200)/(T)) Ω-1 m-1

4. Simulation Results

Figure 2. Molar fraction distributions along the cell at the steady-state.

Figure 3. Axial temperature distributions along the cell at the steady-state.

Start-up. After the heat-up mode, fuel is fed into the fuel channel for the beginning of the electrochemical reaction. Those fuel and air streams of 1023 K are fed into each channel. Three cases are considered for the start-up mode. Case 4. Initial temperature of the PEN is 1023 K. Case 5. Initial temperature of the PEN is 973 K. Case 6. Air temperature is varied from 1023 to 1073 K (other parameters are same as case 4).

To validate the model used in this paper, steady-state simulation results are compared with the previous results of Aguiar et al.15 Figure 2 shows molar fraction distributions, which are similar to results in the literature15 at the steady-state. Figure 3 presents temperature distributions along the cell at the steadystate. Solid lines are our simulation results, and dot points are results of Aguiar et al.15 As can be seen in Figure 3, results are in good agreement with the data of Aguiar et al.15 Figures 2 and 3 are explained in the later subsection; Start-up and SteadyState Simulation. The model parameters, which are used in this paper, are listed in Tables 2 and 3.1,15,26 Heat-up Simulation. During the heat-up mode, temperatures of cell components and hot stream vary from the ambient temperature (298 K) to the start-up temperature (1023 K). Assuming that the specific heat capacity and the heat transfer coefficient are constant, there might be a significant error because of the broad operating temperature range of the heatup mode. To simulate the heat-up mode with more accuracy, we employed the values of specific heat capacities and heat transfer coefficients as the functions of temperature. Figure 4 shows axial temperature distributions along the cell (case 1). The hot air stream is fed into air channel with air velocity of 3 m/s and air temperature of 1023K. Figure 4 (a) shows an initial temperature of PEN, IC, and air stream. At one end (where is close to the air inlet) of the solid components, a heat is transferred from the hot air stream to the solid components by a convection. Therefore the temperatures of the solid components are increased with heat accumulation at the air inlet, whereas the temperature of the air stream is decreased. As can be seen in Figure 4b-f, the temperatures of middle part and opposite end (close to the air outlet) of the solid components are gradually increased with conduction and convection. A large

Table 2. Model Parameters and Operating Conditions parameter

value

parameter

value

Physical Properties PEN PEN PEN PEN

density heat capacity emissivity thermal conductivity

5900 kg/m3 500 J/(kg · K) 0.8 2 J/(m · s · K)

IC IC IC IC

density heat capacity emissivity thermal conductivity

8000 kg/m3 500 J/(kg · K) 0.1 25 J/(m · s · K)

Dimensions of the Cell Elements cell length cell width fuel channel height air channel height

0.4 m 0.1 m 1 mm 1 mm

fuel utilization average current density air ratio air composition

75% 5000 A/m2 8.5 21% O2, 79% N2

anode thickness cathode thickness electrolyte thickness IC thickness

500 µm 50 µm 20 µm 500 µm

pressure fuel inlet temperature air inlet temperature

1 bar 1023 K 1023 K

Operating Conditions

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Figure 4. Temperature distributions along the cell during the heat-up mode (case 1): time ) (a) 0, (b) 60, (c) 360, (d) 1000, (e) 2000, (e) 4000 s.

axial temperature gradient observed in Figure 4 can cause a thermal shock. Figure 5 shows temperature variations with respect to the time of the PEN structure for three axial positions; two ends and the middle part. A sudden temperature variation of the PEN structure at the air inlet can be seen in the early time in Figure 5. The sudden temperature variation can also cause thermal shock. The simulation result shows that the heatup time is about 3200 s (53 min). In case 2, the velocity is changed to 1.5 m/s from the previous value of 3 m/s to study the effect of air velocity. Other parameters are same with case 1. Figure 6 shows temperature variations of the PEN structure at two ends for both case 1 and case 2. As can be seen in Figure 6a,b, the temperature variation rates of case 2 are slower than results of case 1. The heat-up time is about 7500 s (125 min). The heat-up time increased from 53 to 125 min. To reduce a sudden temperature variation, the heat-up rate can be adjusted by reducing the velocity of hot air stream. However, it takes a lot of time for the cell heat-up. In the previous two cases, the constant air temperature is considered. To understand the effect of hot air stream temperature, the input air temperature was adjusted in case 3. The hot air stream is fed into air channel with an air velocity of 3 m/s

Figure 5. PEN temperature variations with the time during the heat-up mode.

(same as case 1). However, the air temperature is subjected to the minimum temperature of PEN structure (As can be seen in


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Figure 8. PEN temperature variations with the time during the heat-up mode (case 3).

Figure 6. PEN temperature variations with the time at the inlet and the outlet for case 1 and case 2: (a) temperature variations at the inlet, (b) temperature variations at the outlet.

Figure 9. PEN temperature distributions along the cell during the start-up mode.

Figure 7. PEN temperature distributions along the cell during the heat-up mode (case 3).

Figure 4, the temperature of the PEN at the air outlet will be lower than other places) in order to relax the axial temperature gradient. The temperature of the inlet air stream is adjusted by the following equation: Tin a ) TPEN,min + ∆T

(18)

The ∆T of 100 K is used for case 3. Figure 7 shows axial temperature distributions for each time. These axial temperature gradients are slower than for the results of case 1. Figure 8 shows the temperature variation with respect to the time for three points; two ends and middle part. These temperature variation rates about the time are also slower than results of case 1. However, the heat-up time increased to 8500 s (141

min) compared with 3200 s at case1. The sudden temperature variation and large axial temperature gradient which cause a thermal shock to cells can be avoided by controlling hot air temperature. The heat-up time can be properly reduced with increasing the value of ∆T. Start-up and Steady-State Simulation. After the heat-up mode, fuel is fed into the fuel channel to enable the beginning of electrochemical reaction. Contrary to the heat-up mode, the constant values on the specific heat capacity and the heat transfer coefficient are used for the start-up mode. The fuel and the air are fed into each channel at 1023 K. The average current density, fuel utilization, and air ratio is specified as 5000 A/m2, 0.75, and 8.5, respectively. The parameters of the electrochemical model are listed in Table 3. Figure 9 shows axial temperature distributions for each time (case 4). Figure 10 presents temperature variations for three points. The start-up time is about 650 s (11 min). Figure 11 shows the predicted voltage from the equivalent electrochemical model. As mentioned above, the average values of PEN temperature and partial pressure are used in the equivalent electrochemical model, and, as such, a similar trend between voltage and average temperature of PEN is observed in Figures 10 and 11. The predicted voltage is 0.665 (V) at the steadystate. The result is similar to that of Aguiar et al.15 (0.664 V). Figure 12 presents the current density distribution along the cell at a steady-state. Steady-state simulation results of case 4 are presented in Figures 2 and 3. Most of the methane is consumed by the endothermic steam reforming reaction near the inlet. For

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Figure 13. Predicted voltage during the start-up mode for case 4 and case 5. Figure 10. PEN temperature variations with the time during the start-up mode.

Figure 14. Temperature variations with the time for case 4 and case 5.

Figure 11. Predicted voltage during the start-up mode.

Figure 12. Current density distribution along the cell at the steady-state.

this reason, the molar fraction of water is decreased and the molar fraction of hydrogen is increased from the inlet to the axial position of 0.12 m. The molar fraction of hydrogen is decreased and the molar fraction of water is increased from the axial position of 0.12 m to the outlet by the electrochemical reaction. The axial temperature distribution can be explained by molar fraction profiles. From the inlet to the axial position of about 0.1 m, the temperature of the PEN is decreased because the endothermic steam reforming reaction is dominant in that area. However, from the axial position of about 0.1 m to outlet,

the temperature of the PEN is gradually increased on account of the exothermic dominant reaction, that is, the electrochemical reaction. To investigate the influence of an initial temperature of the PEN structure, the simulation of case 5 is executed at the initial PEN temperature of 923 K, whereas it is at 1023 K in case 4. Figure 13 presents the predicted voltage with respect to time in both cases. Due to a lower initial temperature of the PEN structure, an initial voltage is lower than that of case 4. The predicted voltage is 0.665 V which is same as result of case 4. The start-up time is about 730 s (12 min). The start-up time difference between case 4 and case 5 is about 1 min. Figure 14 shows the temperature variations with respect to the at two ends of the PEN. From Figure 14, the temperature variation is higher than in the result of case 4 at the air inlet. This sudden temperature variation is from the temperature difference between the air temperature and initial temperature of the PEN. The air temperature is changed from 1023 to 1073 K in case 6 to analyze the effect of an air temperature. Figure 15 shows average temperature of the PEN structure with respect to time. On account of increasing the air temperature, the temperature of PEN structure is increased. As can be seen in Figure 16, the predicted voltage is also increased because of an increase in the average temperature of the PEN structure. Figure 17 shows axial temperature distributions at the steady-state. The axial temperature gradient is slower than that of case 4. 5. Conclusion A one-dimensional dynamic model was introduced and modified for the purpose of simulating the heat-up and start-up


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temperature of the PEN structure are investigated for the startup mode. In conclusion, this dynamic model can be useful to investigate not only the transient behavior during heat-up and start-up modes but also the steady-state. Acknowledgment This research was financially supported by the Ministry of Knowledge Economy (MKE), Korea Institute for Advancement of Technology (KIAT), and Dae-Gyeong Leading Industry Office through the Leading Industry Development for Economic Region. Nomenclature Figure 15. Average temperature variations of the PEN structure with time for case 4 and case 6.

Figure 16. Predicted voltage with time for case 4 and case 6.

AR ) air ratio cp ) specific heat capacity (J/(g · K)) Ci ) molar concentration of species i (mol/m3) Dh ) hydraulic diameter (m) Ea ) activation energy for the steam reforming reaction (J/mol) Eelectrode ) activation energy of the exchange current density (J/mol) F ) Faraday’s constant (C/mol) FU ) fuel utilization h ) heat transfer coefficient (J/(m2 · s · K)) j, jj ) local and average current density (A/m2) k ) thermal conductivity (J/(m · s · K)) k0 ) steam reforming reaction constant (4274 mol/(s · m2 · bar)) kwg ) water-gas shift reaction constant kelectrode ) pre-exponential factor of the exchange current density (A/ m2) Keq,(2) ) equilibrium constant of the water-gas shift reaction L ) length of the cell N ) molar flow rate (mol/s) Nu ) Nusselt number p ) pressure or partial pressure (bar) R ) universal gas constant (8.314 J/(mol · K)) Rk ) rate of reaction k (mol/(m2 · s)) t ) thickness of solid component (m) T ) temperature (K) ua ) air velocity (m/s) uf ) fuel velocity (m/s) V ) voltage (V) W ) width of the cell x ) molar fraction z ) flow direction (m) Greek Letters ) emissivity η ) overpotential (V) σ ) Stefan-Bolzmann constant (W/(m2 · K4)) F ) density (g/m3) νi,k ) stoichiometric coefficient of species i in reaction k

Figure 17. Axial temperature distributions at the steady-state.

modes. The model was validated with literature data15 because of the lack of experimental data. The simulation results are in good agreement with the data. For steady state, maximum temperature difference between the literature data15 and our result is 5 K. The heat-up and start-up behavior were investigated by the modified model. The effect of the air velocity and the temperature were studied for the heat-up mode. A sudden temperature variation and large axial temperature gradient could be avoided by adjusting the air velocity and temperature. However, the heat-up time was increased. For the start-up mode, the voltage could be predicted by the model. The model could also calculate other variables such as temperature, molar fraction, and current density. The effects of air temperature and initial

Subscripts act ) activation air ) air conc ) concentration fuel ) fuel i ) species IC ) interconnect k ) reaction PEN ) PEN structure

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ReceiVed for reView April 1, 2010 ReVised manuscript receiVed July 20, 2010 Accepted August 9, 2010 IE100783G

Numerical Study of a Planar Solid Oxide Fuel Cell during Heat-up and

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