TRANSIENT HEAT TRANSFER MODELING OF A SOLID OXIDE

Proceedings of the International Conference on Hydrogen Production, June 16-18, 2010, Istanbul, Turkey

1

TRANSIENT HEAT TRANSFER MODELING OF A SOLID OXIDE FUEL CELL OPERATING WITH HUMIDIFIED HYDROGEN

C. Ozgur Colpan1, Feridun Hamdullahpur2, Ibrahim Dincer3

1 Mechanical and Aerospace Engineering Department, Carleton University 1125 Colonel by Drive, Ottawa, Ontario, Canada K1S 5B6

2 Mechanical and Mechatronics Engineering Department, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, Canada N2L 3G1

3 Faculty of Engineering and Applied Science, University of Ontario Institute of Technology, 2000 Simcoe Street North, Oshawa, Ontario, Canada L1H 7L7

ABSTRACT

This paper presents the development of a new transient heat transfer model of a planar solid oxide fuel cell (SOFC) operating with humidified hydrogen. The model is first validated with some benchmark test data and then used to simulate the transient behavior of the co- and counter-flow SOFCs at the heat-up and start-up stages. In addition, a parametric study including the effects of Reynolds number at the fuel channel inlet and excess air coefficient on the output parameters is conducted. The model predictions are found to be in very good agreement with the data published in the literature. The transient simulations show that counter-flow SOFC yields higher performance, e.g. power density and electrical efficiency, but it needs slightly more time to reach the steady-state condition. The results of the parametric study point out that taking the Reynolds number low and excess air coefficient high gives higher electrical efficiencies for both of the configurations. For the given input data, it is found that the counter-flow configuration has a higher electrical efficiency for low Reynolds numbers, e.g. 0.67 and all possible excess air coefficients.

Keywords: SOFC, solid oxide fuel cell, hydrogen, transient heat transfer, start-up, finite difference method, modeling

1. INTRODUCTION

The solid oxide fuel cell is one of the emerging energy technologies that is primarily used to generate electrical power from the movement of electrons produced by electrochemical reactions in the cell. In addition, the heating value of gas streams exiting the SOFC is high enough to be recovered using bottoming cycles to produce additional heat and/or electricity. SOFCs have advantages over other fuel cell types. These advantages include: a) no water management issues, since only solid and gas phases exist, b) cheaper materials used for manufacturing electrocatalysts, c) ability to utilize a variety of fuels including hydrocarbons, methanol and biomass produced gas, d) internal reforming of gases, and e) thermal integration with bottoming cycles, e.g. gas turbine and gasification systems. Possible disadvantages of the SOFC over other fuel cells are: a) challenges for construction and durability due to its high temperature, and b) a carbon deposition problem when a fuel such as methane, syngas or ammonia is used. The main application of the SOFC is stationary power and heat generation. Other areas include transportation, military and portable applications. SOFC models can be developed at cell, stack and system levels. When developing a SOFC model in cell and stack levels, different considerations may be taken into account according to the purpose and needs of the model. For example, 0-D, 1-D, 2-D and 3-D modeling approaches can be used depending on the necessity for knowledge of the output parameters, such as temperature and current density distributions [1-3]. Transient modeling is used when one, or a combination of the following stages, needs to be simulated: heat-up, start-up, shut-down and load change [4-7]. Thermomechanical modeling provides us with information to predict the stresses occurring inside the fuel cell [8]. The problems related to carbon deposition due to using a fuel containing carbon can also be analyzed through numerical studies [9-12]. In system level modeling, integrated SOFC


2

systems can be assessed using several methods including energy, exergy and thermoeconomic analyses [13-16]. More information on different types of SOFC models can be found in the paper by Colpan et al. [17]. In one of our previous studies [18], we modeled a direct internal reforming SOFC operating with a multi-gas mixture including CH4, CO, CO2, H2, H2O, and N2. The main emphasis was given to the effect of the reforming process on the performance of the cell. The main objective of the current study is to develop a comprehensive model for a planar SOFC operating with humidified hydrogen taking into account all three heat transfer mechanisms, (i.e. conduction, convection and radiation), and all polarization nodes, (i.e. ohmic, activation and concentration). This model is used for the simulation of the transient behavior of the co- and counter-flow SOFCs that operate using the humidified hydrogen. In addition, a parametric study is conducted to assess the effects of Reynolds number at the fuel channel inlet and excess air coefficient on the output parameters.

2. MODELING This section describes the modeling approach, the formulation of the SOFC including the continuity and heat transfer equations, and the validation of the model. 2.1 Modeling Approach The first step in the modeling of a SOFC is the formulation of the system considered together with the specification of the control volumes, and the coordinates. For this reason, the repeat element of a SOFC found in the middle of a stack, which is shown in Figure 1, is divided into five control volumes: anode interconnect, fuel channel, PEN (consisting of anode, electrolyte and cathode), air channel and cathode interconnect. Due to the symmetrical conditions of the considered repeat element in the stack, the solid structure has adiabatic boundary conditions at the exterior surfaces [8]. The Cartesian coordinate system is selected for all the control volumes given their specific geometry. Then, the general laws, e.g. conservation of mass, energy and momentum, and the particular laws, e.g. the relation between the cell voltage and polarizations, and the initial and boundary conditions are written for each of these control volumes. In modeling, instead of solving the conservation of momentum, some simplifications are made assuming fully developed laminar flow conditions. This assumption is well justified since the gases flow with low velocity, which is required to obtain a high fuel utilization ratio. Under these flow conditions, the Nusselt number becomes a single function of the aspect ratio for rectangular ducts. This derivation is based on solutions of the differential momentum and energy equations for different boundary conditions [19]. There is a discrepancy in the literature about how some of the input and output parameters of SOFC models are selected. Parameters such as average current density, fuel utilization ratio, cell voltage and mass flow rate of the channel inlets may be chosen as input or output according to the purpose of the model. In our model, the cell voltage, which is assumed to be equal at the top and bottom surfaces of the interconnect, the Reynolds number at the fuel channel inlet that controls the fuel mass flow rate, and the excess air coefficient that determines the mass flow rate at the air channel inlet are taken as input parameters. Other input parameters selected in this study are: the cell geometry, the properties of materials, the ambient temperature, the molar composition at the fuel and air channel inlets, the mass flow rate of air for the heat-up stage, and the cell pressure. The expected outcome parameters of the model are: the heat-up and start-up time, the fuel utilization ratio, the current density, the temperature and molar gas composition distributions, and the power output and electrical efficiency of the cell. In this study, the strategy followed for the modeling of the heat-up and start-up stages is as follows. In modeling the heat-up period, only the heat transfer equations are solved since there is no fuel flow taking place in the fuel channel. At this stage, the temperature of the air channel is controlled so as not to cause excessive thermomechanical stresses. The minimum solid temperature of the cell is calculated for each time step, and the air channel inlet temperature is set to Tmin,solid+100 °C for the subsequent time step [20]. In modeling the start-up period, the temperatures of the air and


3

fuel channel inlets are kept constant, and mass balances are first solved for air and fuel channels for each time step. The molar flow rates and the composition of the gas species, and current density distribution through the gas channels are determined after the mass balances are obtained. Using these data and the temperature distribution obtained from the previous time step, heat transfer equations are applied to each control volume. Considered in the heat transfer equations are conduction between PEN and interconnects, natural convection in the heat-up stage and forced convection in the start-up stage, surface-to-surface radiation between the PEN and interconnects. Hence, the temperature distribution in a given time step is calculated, and the iterations are repeated until the absolute temperature difference between the two consecutive time steps for each node becomes less than the threshold value. This value is chosen as 10-4 in this study. Among the different numerical solution methods, the finite difference method is used in this study because it is straightforward for orthogonal grids. In applying this method, spatial and temporal domains are divided into several sections, which is called meshing. After generating the mesh, finite-difference approximations are substituted for the derivatives to convert the partial differential equations to an algebraic form. Then, a computer code that is capable of solving the system of equations in an efficient way for different input parameters is developed. In this study, the code was developed in Matlab. The final step of modeling is validation. In this study, because of the lack of experimental results in the literature, the results of the SOFC benchmark test [21] and Braun’s model [22] are used for validating the model. 2.2 Formulation of the SOFC Continuity equations and transient heat transfer equations are applied to the control volumes enclosing the components of the cell, e.g. gas channels, PEN, and interconnects. These equations are shown below for the co-flow configuration. In the continuity equations, the source terms are derived from the reaction of oxidation of hydrogen. Based on this reaction, the continuity equations at the fuel channel are shown in Eqs. (1) and (2), and these equations at the air channel are shown in Eqs. (3) and (4).

fc

elH

t

r

dx

nd ′′−=

′′ &&2 (1)

fc

elOH

t

r

dx

nd ′′=

′′ &&2 (2)

ac

elO

t

r

dx

nd 2/2

′′−=

′′ &&

(3)

02 =′′

dx

nd N&

(4)

where the rate of conversion for electrochemical reaction becomes F

irel

2=′′& .

The local current density is found by solving the relation between the Nernst voltage and the polarizations using

conactohmN VVVVV −−−= (5)

where the Nernst voltage and the polarizations can be given as [23, 24]:


4

⋅⋅−∆−=

o

o

PPP

P

F

RT

F

gV

OH

OHrN

/ln

2222

2 , itVk

kkohm⋅

∑ ⋅= ρ ,

⋅+

⋅= −−

coao

acti

i

F

RT

i

i

F

RTV

,

1

,

1

2sinh

2sinh , and

⋅−−

+

⋅++

⋅−−=

iPVD

t

F

RTPPP

P

F

RT

iPVD

t

F

RT

F

RT i

PVD

t

F

RT

F

RTV

cvc

ccbO

bOs

bOHava

aas

bHava

aasconc

)(

)()(

4exp)(

ln4

21ln

221ln

2

2

2

22

τ

ττ

.

The electrical resistivity of the cell components, ρ , can be written as functions of the solid temperature as given in [17, 25]. The current density, the molar flow rate and molar composition of the gas species through the gas channels are found by solving the equations given above. The transient heat transfer equation for the air channel becomes

( ) ( ) ( ) ( )ac

gassolidOelaciacaPENac

NNOOacpact

wwhrTThTThhnhn

xt

Tc

⋅⋅′′−−+−=′′+′′

∂∂+

∂∂⋅⋅ 2,,

2222,

2/&&&ρ

(6) with the following boundary and initial conditions:

0=x ⇒up)-(Start TT

up)-(Heat tfT

acw _

)(

==

, 0=t ⇒ C100 o+= oTT .

The 2-D transient heat diffusion equation for the PEN is given as

( )t

T

tk

Vihhhr

y

T

x

T

PENPENPEN

cellOHOHel

∂∂⋅=

⋅⋅−−+⋅′′

+∂∂+

∂∂

α12/

222

2

2

2

2 &

(7)

with the following boundary and initial conditions:

Lx & x == 0 ⇒ 0=∂∂

x

T ,

acci tty += ⇒

[ ]ac

ciPENci

solid

gas

ciPENaraPENac

solid

gas

PENt

TTk

w

w TThTTh

w

w

y

Tk

)(1)()( ,,

−⋅⋅

−+−⋅+−⋅⋅=

∂∂⋅ ,

PENacci ttty ++= ⇒

[ ]fc

aiPENai

solid

gas

aiPENfrfPENfc

solid

gas

PENt

TTk

w

wTThTTh

w

w

y

Tk

)(1)()( ,,

−⋅⋅

−+−⋅+−⋅⋅=

∂∂⋅− ,

0=t ⇒ oTT = .


5

The transient heat transfer equation for the fuel channel is

( )( ) ( ) ( )[ ]

fc

gassolidOHHelfPENfcfaifc

OHOHHHfcpfc

t

wwhhrTThTTh

hnhnxt

Tc

/22,,

2222,

⋅+−⋅′′+−+−

=′′+′′∂∂+

∂∂⋅⋅

&

&&ρ (8)

with the following boundary and initial conditions: 0=x ⇒ up)-(Start _ fcwTT = , 0=t ⇒ oTT =

The 2-D transient heat diffusion equation for the cathode and anode interconnects is

t

T

y

T

x

T

i∂∂⋅=

∂∂+

∂∂

α1

2

2

2

2

(9)

with the following boundary and initial conditions:

Lxx == & 0 ⇒ 0=∂∂

x

T, aifcPENacci tttttyy ++++== & 0 ⇒ 0=

∂∂

y

T ,

fcPENaccici ttttyty +++== & ⇒

[ ]gc

PENi

i

solid

gas

PENigrgigc

solid

gas

it

TTk

w

wTThTTh

w

w

y

Tk

)(1)()( ,,

−⋅⋅

−+−⋅+−⋅⋅=

∂∂⋅− ,

0=t ⇒ oTT =

Here, g should be replaced with f and a; i should be replaced with ai and ci; and gc should be replaced with fc and ac in Eq. (9), when modeling the anode and cathode interconnects, respectively. In the above equations, hc is the heat transfer coefficient. It represents the natural convection and forced convection in the heat-up and start-up stages, respectively. The correlations for such heat transfer coefficients are available elsewhere [19, 26].

Dimensionless numbers: The Reynolds number for the fuel channel inlet based on the hydraulic diameter of the rectangular channel cross section is shown in Eq. (10). In the model, this number is considered as one of the input parameters. Hence, using this number, the mass flow rate of the gas mixture per cross section of the fuel channel at the inlet can be found.

( )( )gasfcmix

gasfcfi

Dwt

wtm

h +⋅⋅⋅⋅′′

=µ

2Re

&

(10)

which can also be written in terms of molar flow rate of the gas species at the fuel channel inlet as follows:

( )( )gasfcmixfik

gasfcmixfik

Dwtx

wtMn

h +⋅⋅⋅⋅⋅⋅′′

=µ,

, 2Re

&

(11)

where k denotes H2 and H2O. The excess air coefficient used as an input parameter in the model is defined as the amount of the oxygen in the inlet stream divided by the amount of oxygen that is needed for a stoichiometric reaction. This coefficient, as given in Eq. (12), is used to calculate the molar flow rate of the gas species per cross sectional area of the air channel at the inlet of the cell.

( ) fc

ac

fiH

aiO

airt

t

n

n⋅

′′′′

=2/,

,

2

2

&

&

λ (12)

Output parameters: The main output parameters of the model are: the fuel utilization ratio, the power density, the power output, and the electrical efficiency of the cell.


6

The fuel utilization ratio is defined as the amount of hydrogen that is electrochemically reacted to the amount of hydrogen in the inlet stream as follows:

( ) ( )

fcgasfiH

solid

m

iiel

Ftwn

wxr

U⋅⋅′′

⋅∆⋅∑ ′′= =

,

2

2&

&

(13)

where m denotes the number of nodes in flow direction. The power density and power output of the SOFC are given below:

cellavecSOFC ViW ⋅=′′,

& (14)

solidcellSOFCSOFC wLWW ⋅⋅′′= && (15)

The primary purpose of using SOFC is to generate electricity and its performance can be assessed through electrical efficiency of the cell as follows:

gasfck

fik

SOFCel

wtnLHV

W

⋅⋅∑ ′′⋅=

=

2

1,

&

&

η (16)

where k denotes the types of gases found at the fuel channel inlet, i.e. H2 and H2O. 2.3 Numerical Solution An implicit finite different scheme is used for the solution of the heat transfer equations. The finite difference equations for the boundary conditions are taken as second order accurate. These equations are derived by considering an imaginary node outside the control volume and eliminating this node between the general equation for interior nodes and the boundary equation. The details of this approach can be found in [27]. The set of equations are linearized by using the ‘lagging properties by one time step’ method [27], and solved using the Gauss elimination method. Iterations are repeated until the solid temperature reaches a certain value for the heat-up period, and the system reaches steady state for the start-up period. 2.4. Validation

The results of the benchmark test, which was conducted at a workshop organized by the International Energy Agency in 1994 [21], was used to validate the model. In this benchmark test, nine institutions modeled planar SOFCs with the same operating data. These institutions were: KFA-Julich (Germany), ISTIC, University of Genova (Italy), ECN Petten (Holland), Riso, National Laboratory (Denmark), Eniricerche (Italy), Dornier (Germany), Statoil (Norway), Ife-Kjeller (Norway) and Siemens (Germany). The main assumption used in the test was to accept each of the polarizations in the anode and cathode as equal to the ohmic loss of the electrolyte. These models were developed under steady-state conditions. The input data for the benchmark test are given in Table 1. In another study, Braun [22] developed a steady state model using the same input data and same assumptions as the benchmark test.

3. RESULTS AND DISCUSSION 3.1 Validation In this study two models, using different assumptions, have been developed for a co-flow and counter-flow SOFC. A transient heat transfer model was first developed using the same assumption for polarizations as the benchmark tests. This model is called Model-V1. In the second model, the assumption used in Model-V1 is altered in that different analytical equations are considered for ohmic, activation and concentration polarizations, as given in Eqs. (8)-(10). This model is called


7

Model-V2. There are some differences in the input parameters of this model and the benchmark test. Unlike the input parameters used in the benchmark test, fuel utilization and mean current density are taken as output parameters, but the cell voltage and Reynolds number are taken as input parameters in the present models, i.e. Model-V1 and Model-V2. Since the results of the benchmark tests are given in steady state condition, the model is validated for this condition.

The heat transfer model is simulated for the benchmark test-1 conditions. A nodal analysis is first carried out to find the number of nodes that will make the results independent from the grid size. It was found that taking 750 nodes in the flow direction, 15 nodes in the thickness direction and the time step as 1s is sufficient to obtain grid-independent results. For validating the present models, the input parameters were first calibrated. As discussed before, cell voltage is considered as an input parameter in the present models and not in the benchmark tests. The results for the cell voltage for the benchmark test are given in Table 2. From these results, we chose the cell voltage as 0.7 V for the co-flow and 0.71 V for the counter-flow case. Average current density and fuel utilization are input parameters in the benchmark tests and their values are given as 0.3 A/cm2 and 0.85, respectively. To get results closer to these values, the Reynolds number is found to be 0.67 in Model-V1. The same value for Reynolds number is used in Model-V2. Table 1: Input data used in the benchmark test. Variable Value Cell geometry Active area [mm2] Anode thickness [m] Cathode thickness [m] Electrolyte thickness [m] Channel width [mm] Channel height [mm] Rib width [mm] Total thickness (with ribs) [mm]

100×100 50×10-6 50×10-6

150×10-6

3 1 2.42 2.5

Material properties Density of PEN and interconnects [g/cm3] Specific heat of PEN and interconnects [J/g·K] Thermal conductivity of PEN and interconnects [W/cm·K]

6.6 0.4 0.02

Operating parameters Temperature at the fuel channel inlet [K] Temperature at the air channel inlet [K] Pressure of the cell [bar] Excess air coefficient Fuel utilization Mean current density [A/m2] Gas composition at the air channel inlet Gas composition at the fuel channel inlet

1173 1173 1 7 0.85 3000 21% O2,79% N2

90% H2, 10% H2O Maximum and minimum values for the current density, solid temperature and air and fuel channel outlet temperatures are given in Tables 3-5, respectively. From Table 3, it can be seen that the current density, found by different companies and institutions, is between 1020 A/m2 and 3956 A/m2 for the co-flow case, and 1080 A/m2 and 8970 A/m2 for the counter-flow case. It can be seen from this table that the results for Model-V1 are between these values. When we take the average of the maximum and minimum current densities found by the companies and institutions that participated in the benchmark test, and compare these average values with the results of Model-V1, it was found that the relative error for the maximum current density is 0.78% and 3.02%, and that for the minimum current density is 7.22% and 2.64% for co-flow and counter-flow cases, respectively. The


8

same procedure is followed for the solid temperature and air and fuel channel outlet temperatures, which are given in Tables 4 and 5, respectively. It was found that only the maximum solid temperature for the counter-flow case is not in the range given in Table 4. It is 0.57% lower than the bottom limit for the maximum solid temperature. This result is mainly due to the difference in modeling between Model-V1 and the benchmark test. In Model-V1 for counter-flow configuration, the outlet temperature for the fuel channel and the inlet temperature for the air channel are fixed to obtain a uniform temperature distribution. The inlet temperature of the fuel channel and the outlet temperature for the air channel were calculated. However, it is not clear how the inlet and outlet temperatures for the gas channels were calculated in the models by the companies and institutions that participated in the benchmark test. For Model-V1, it was found that the relative error for the maximum solid temperature is 1.74% and 2.00%, and that for the minimum solid temperature it is 2.29% and 0.59% for co-flow and counter-flow cases, respectively. For the same model, the results show that the relative error for the air channel outlet temperature is 1.40% and 2.26%, and that for the fuel channel outlet temperature is 1.58% and 1.14% for the co-flow and counter-flow cases, respectively. It should be noted in the comparison of air and fuel channel outlet temperatures with Model-V1, the results of Siemens are neglected. It is understood from Table 5 that Siemens chose inlet temperatures of air and fuel channels as 900 °C for the counter-flow case, which is not the case in the models developed by the author or the other institutions and companies. When we checked the results for Model-V2 from Tables 3-5, it was seen that except for the current density distribution, the results are comparable with the results of the benchmark test and Model-V1. The difference in the results for current density distribution between Model-V1 and Model-V2 is as expected since the models in the benchmark tests were developed using an assumption on polarizations, as discussed in Section 2.4. However, this assumption is not valid today. Detailed correlations have been published on the activation and concentration polarizations in the literature, [e.g. 23, 24]. However, the temperature distribution is still comparable between Model-V2 and the benchmark test-1. For example, for Model-V2, the relative error for the maximum solid temperature is 2.32 % and 2.19%, and for the minimum solid temperature it is 1.84% and 0.37% for co-flow and counter-flow cases, respectively. Also, for this model, the results show that the relative error for the air channel outlet temperature it is 1.98% and 2.26%, and that for the fuel channel outlet temperature is 1.97% and 1.14% for the co-flow and counter-flow cases, respectively. The distributions of current density, fuel channel temperature and molar hydrogen fraction in the fuel channel, found by using Model-V1 and Model-V2 for the co-flow case, are also validated with the data published by ECN, which is an institute that participated in the benchmark test. This validation is shown in Figures 2-4. The distributions for the counter-flow case, found by the companies participated in the benchmark test, are not available in the literature, but the distributions, found by using the present models, are added to these figures for comparison. As can be seen from Figure 2, current density trends for Model-V1, and the model developed by ECN, are similar except that the current density for Model-V1 is slightly higher at the first half of the cell. Model-V2 has a different trend for both co-flow and counter-flow cases because of the different correlations for activation and concentration polarizations in this model. However, when we calculate the average current densities for Model-V1 and Model-V2, it was found that the values are very close to the average current density of the model developed by ECN, which is 0.3 A/cm2. The average current densities for the co-flow case are 0.304 A/cm2 and 0.294 A/cm2 for the Model-V1 and Model-V2, respectively; whereas, those for the counter-flow case are 0.299 A/cm2 and 0.301 A/cm2 for the Model-V1 and Model-V2, respectively. When we compare the temperature distribution in the fuel channel found by Model-V1 and Model-V2 with the results of ECN, as shown in Figure 3, it can be seen that the trends are similar. The temperature at the fuel channel exit was found to be higher for ECN. However, when we check Table 5, it may be seen that this temperature is comparatively higher for ECN than for that of the other companies and institutions. From Figure 4, it can be seen that molar hydrogen fraction has almost the same trend as ECN.


9

Table 2: Cell voltage for the benchmark Test-I.

Company/Institution Co-flow [V] Counter-flow [V]

Dornier, D 0.684 0.689

ECN Petten, NL 0.704 N.A.

Eniricerche, I 0.722 0.730

Inst. For Energiteknikk Kjeller, N 0.71 0.71

KFA-Julich, D 0.706 0.712

Siemens, D 0.712 0.716

Statoil, N 0.702 0.709

Riso, DK 0.7034 0.7101 Table 3: Validation of maximum and minimum values of current density.

Company/Institution Co-flow (max/min)

(A/m2) Counter-flow

(max/min) (A/m2)

Dornier, D 3636/1686 7192/1297

ECN Petten, NL 3614/1211 N.A.

Eniricerche, I 3840/1020 8970/1080

Inst. For Energiteknikk Kjeller, N 3933/1191 7862/1113

KFA-Julich, D 3725/1237 7910/1163

Siemens, D 3863/1236 8513/1135

Statoil, N 3956/1366 7391/1235

Riso, DK 3739/1296 7107/1187

Braun's model 3799/1211 7393/1152

Model-V1 3760/1187 7564/1202

Model-V2 5175/1175 5530/1586 Table 4: Validation of maximum and minimum values of solid temperature.

Company/Institution Co-flow (max/min) (°C)

Counter-flow (max/min) (°C)

Dornier, D 1070/928 1085/914


Eniricerche, I 1069/916 1083/906


KFA-Julich, D 1059/913 1073/906

Siemens, D 1049/909 1062/904

Statoil, N 1098/970 1082/913

Riso, DK 1061/924 1075/910

Braun's model 1059/924 1073/910

Model-V1 1049/903 1056/904

Model-V2 1043/907 1054/906


10

Table 5: Validation of air and fuel channel outlet temperatures.

Company/Institution Co-flow

(air/fuel) (°C) Counter-flow (air/fuel) (°C)

Dornier, D 1068/1070 1080/914


Eniricerche, I 1068/1068 1080/906


KFA-Julich, D 1059/1059 1070/906

Siemens, D 1048/1048 1061/1064

Statoil, N 1067/1067 1082/914

Riso, DK 1059/1061 1070/910

Braun's model 1058/1059 1068/910

Model-V1 1048/1047 1051/900

Model-V2 1042/1043 1051/900

3.2 Transient behavior of the cell After validating the model, the transient behavior of the cell at the heat-up and start-up stages are simulated. These simulations give the change of temperature, fuel utilization, average current density, electrical efficiency, power density and molar fraction of hydrogen with time. These simulations are conducted for both co-flow and counter-flow cases. In Figure 5, temperature distributions for the co-flow case for Model-V2 are given for the heat-up and start-up stages. In the heat-up period, temperature at the air channel inlet is controlled due to thermomechanical considerations, as discussed in Section 2.1. This temperature increases by 100 °C more than the minimum solid temperature at each time step. At this stage, forced convection at the air channel, natural convection at the fuel channel, and radiation and conduction between the solid parts affect the temperature distribution. The heat-up period ends when the minimum solid temperature reaches a prescribed value, which was chosen as 700 °C in this study. At this temperature, the resistivity of the electrolyte and the ohmic polarization become low enough to produce meaningful amount of power. In Figure 5, we can see that the temperature drops, at the x and y directions at the end of the heat-up period, i.e. t=794 s, are approximately 5.5 °C/cm and 11.2 °C/cm for an air flow rate of 0.0712 g/s. In t he start-up period, the temperatures at the air and fuel channel inlets are fixed. There is a temperature rise through the channel length because of the heat generation due to polarizations; however some of this heat is carried away by the excess air sent through the air channel. The temperature gradients in the x and y directions at the end of the start-up period are approximately 13°C/cm and 2.9°C /cm, respectively. Figure 6 shows the temperature distribution of the counter-flow case for Model-V2 for the heat-up and start-up stages. In the counter-flow case, air enters the cell from the side opposite to that of the co-flow case. Hence, the temperature distributions for the heat-up stage, as shown in Figures 6a and 6b, are symmetrical to those shown in Figures 5a and 5b. The temperature gradients in the x and y directions, at the end of the start-up period, are approximately 14.6 °C/cm and 1.25 °C/cm, respectively. It follows from Figures 7 and 8 that the output parameters are zero in the heat-up period since there is no flow in the fuel channel. For the co-flow case, average current density, fuel utilization, power density and electrical efficiency increase from 0.19 to 0.3 A/cm2, 0.53 to 0.83, 0.13 to 0.21 W/cm2, and 0.29 to 0.47, respectively, during the start-up period. The molar flow rate of hydrogen at the exit of the fuel channel is higher at the beginning of the start-up period compared with the steady state


11

condition, as can be seen in Figure 9, because of the higher fuel utilization of hydrogen at the beginning of the start-up period due to the lower operating temperature. Figures 7-9 show that the transient behaviors for co- and counter-flow configurations do not differ significantly. They show a similar trend, but the counter-flow configuration yields higher performance and it takes slightly more time to reach the steady state condition. 3.3 Parametric studies

Reynolds number at the fuel channel inlet and excess air coefficient are the key input parameters affecting the performance of the cell. Due to this reason, the effects of these input parameters on the output parameters such as fuel utilization, average current density, electrical efficiency, and power density are investigated.

Figures 10 and 11 show the effect of the Reynolds number on the output parameters. Reynolds number is directly proportional to the mass flow rate of the fuel, which is shown on the second horizontal axis of these figures. As it can be seen from these figures, Reynolds number should be greater than a certain value to get any meaningful results. If we choose this number very low, the computer code gives us imaginary numbers as the output. From Figure 10, it is seen that as the Reynolds number increases, fuel utilization decreases, whereas average current density increases, which can be explained as follows: As the Reynolds number increase, both molar flow rate of hydrogen and molar flow rate of hydrogen that is utilized increase, which in turn increases the average current density. However, since the increase in molar flow rate of hydrogen is more than the molar flow rate of hydrogen utilized, fuel utilization decreases. Power density has the same trend with current density, as shown in Figure 11; because the cell voltage is assumed to be constant in the modeling. It can be easily shown that electrical efficiency is directly proportional to the fuel utilization; hence it has the same trend with fuel utilization as shown in this figure. These figures also show that counter-flow configuration has a better performance, e.g. electrical efficiency, for low Reynolds numbers that we obtain meaningful amount of fuel utilization, e.g. fuel utilization of 0.85. For example, for Reynolds number 0.67, electrical efficiency is 46.5% and 48.3%, for co-flow and counter-flow configurations, respectively.

Excess air coefficient, which controls the mass flow rate of air at the inlet of the air channel, is another important operating variable because it controls the current density and the temperature of the fuel cell, which in turn affects the performance of the cell. If low amounts of air is sent through the air channel, the temperature of the exit increases, as shown in Figure 12. Therefore, the excess air coefficient should be carefully selected not to cause a thermomechanical problem. It can be seen from Figures 13 and 14 that the excess air coefficient should be taken high enough to get a better performance from the cell, which can be explained as follows: As the excess air coefficient increases, temperature of the fuel cell decreases. This decrease causes an increase in the Nernst voltage, and decreases in the activation and concentration polarizations. Hence, the current density and the performance of the cell increase. However, the blower power requirement and the operation cost also increase with an increase in the excess air coefficient. In addition, higher exit temperature from the channels, which necessitates lower excess air coefficient, is generally required for the integrated SOFC systems. Hence, an optimum excess air coefficient should be selected depending on the application and taking into account the performance and economics. When we compare the co-flow and counter-flow configurations, Figure 14 shows that counter-flow configuration has a higher electrical efficiency than co-flow configuration for any given excess air coefficients. 4. CONCLUSIONS A new transient, 2D heat transfer model of a SOFC operating with humidified hydrogen has been developed. This model takes into account all the polarizations, (i.e. ohmic, activation and concentration), and heat transfer mechanisms, (i.e. conduction, convection and radiation). The


12

relations based on electrochemistry are coupled with the heat transfer equations. For a validation, a model using the same polarization assumption with the benchmark test is first developed. Then, the model is further improved by altering this assumption and using updated electrochemical relations on polarizations. It is found that the results are in very good agreement with those of the benchmark test. It is also shown that the activation and concentration polarizations affect the current density distribution significantly. After validation, transient behaviors of co- and counter-flow SOFC are simulated. It is found that counter-flow SOFC has a higher fuel utilization, average current density, power density and electrical efficiency when the system reaches steady-state condition. However, this type of SOFC needs slightly more time to reach the steady-state condition. The effects of Reynolds number at the fuel channel inlet and excess air coefficient on the output parameters are investigated. It is shown that the Reynolds number should be taken low; whereas the excess air coefficient should be taken high to get better electrical efficiencies. ACKNOWLEDGEMENT The financial and technical support of an Ontario Premier’s Research Excellence Award, the Natural Sciences and Engineering Research Council of Canada, Carleton University and University of Ontario and Institute of Technology is gratefully acknowledged. NOMENCLATURE cp specific heat at constant pressure, J/g-K D diffusivity, cm2/s Dh hydraulic diameter, m F Faraday constant, C h heat transfer coefficient, W/cm2-K

h specific molar enthalpy, J/mole

H& enthalpy flow rate, W i current density, A/cm2 io exchange current density, A/cm2 k thermal conductivity, W/cm-K L length of the cell, cm LHV lower heating value, J/mole M molecular weight, g/mole m& mass flow rate, g/s n& molar flow rate, mole/s P pressure, bar q& heat transfer rate, W

r& conversion rate, mole/s R universal gas constant, J/mole-K

hDRe Reynolds number in an internal flow

t time, s; thickness, cm T temperature, K UF fuel utilization ratio V voltage, V Vv Porosity w width, cm W& power output, W x molar concentration Greek letters ρ electrical resistivity of cell components, ohm-cm; mass density, g/cm3

elη electrical efficiency

λair excess air coefficient


13

τ tortuosity µ viscosity, g/s-cm α thermal diffusivity, cm2/s; aspect ratio Subscripts a anode; air ac air channel act activation ai anode interconnect ave average c cathode; convection ci cathode interconnect conc concentration e electrolyte el electrochemical; electrical fc fuel channel fi fuel channel inlet g gas gc gas channel i interconnect ohm ohmic mix mixture N Nernst PEN positive/electrolyte/negative r reaction; radiation s solid structure w wall Superscripts b bulk o standard state

REFERENCES

1. Colpan, C.O., Dincer, I., and Hamdullahpur, F., 2007, Thermodynamic modeling of direct internal reforming solid oxide fuel cells operating with syngas, International Journal of Hydrogen Energy, 32, pp. 787-795.

2. Campanari, S., Iora, P., 2005, Comparison of finite volume SOFC models for the simulation of a planar cell geometry, Fuel Cells, 5(1), pp. 34-51.

3. Mandin, P., Bernay, C., Tran-Dac, S., Broto, A., Abes, D., Cassir, M., 2006, SOFC modelling and numerical simulation of performances, Fuel Cells, 6(1), pp. 71-78.

4. Damm, D.L., and Fedorov, A.G, 2006, Reduced-order transient thermal modeling for SOFC heating and cooling, Journal of Power Sources, 159, pp.956-967.

5. Apfel, H., Rzepka, M., Tu, H., and Stimming, U., 2006, Thermal start-up behaviour and thermal management of SOFC’s, Journal of Power Sources, 154, pp.370-378.

6. Bhattacharyya, D., Rengaswamy, R., Finnerty, C., 2009, Dynamic modeling and validation studies of a tubular solid oxide fuel cell. Chemical Engineering Science. 64. pp. 2158 – 2172

7. Thorud, B., Stiller, C., Weydahl, T., Bolland, O., and Karoliussen, H., 2004, Part-load and load change simulation of tubular SOFC systems, Proc. The 6th European Solid Oxide Fuel Cell Forum, Lucerne, Switzerland, pp.716-729.


14

8. Yakabe, H., Ogiwara, T., Hishinuma, M., Yasuda, I., 2001, 3-D model calculation for planar SOFC, Journal of Power Sources, 102, pp.144-154

9. Sangtongkitcharoen, W., Assabumrungrat, S., Pavarajarn, V., Laosiripojana, N., Praserthdam, P., 2005, Comparison of carbon formation boundary in different modes of solid oxide fuel cells fueled by methane, Journal of Power Sources, 142, pp.75-80.

10. Singh, D., Hernandez-Pacheco, E., Hutton, P.N., Patel, N., Mann, M.D, 2005, Carbon deposition in an SOFC fueled by tar-laden biomass gas: a thermodynamic analysis. Journal of Power Sources, 142, pp.194-199.

11. Koh, J., Kang, B., Lim, C.H., Yoo, Y., 2001, Thermodynamic analysis of carbon deposition and electrochemical oxidation of methane for SOFC anodes. Electrochemical and Solid-State Letters, 4(2), pp.A12-A15.

12. Sasaki, K., Teraoka, Y., 2003, Equilibria in fuel cell gases I. Equilibrium compositions and reforming conditions, Journal of the Electrochemical Society 150(7), pp.A878-A884.

13. Winkler, W., Lorenz, H., 2002, The design of stationary and mobile solid oxide fuel cell - gas turbine systems, Journal of Power Sources, 105, pp.222-227.

14. Koyama, M., Kraines, S., Tanaka, K., Wallace D., Yamada K., Komiyama H., 2004, Integrated model framework for the evaluation of an SOFC/GT system as a centralized power source, International Journal of Energy Research, 28, pp.13-30.

15. Kuchonthara P., Bhattacharya S., Tsutsumi A., 2005, Combination of thermochemical recuperative coal gasification cycle and fuel cell for power generation, Fuel, 84, pp.1019-1021.

16. Ghosh. S., De. S., 2006, Energy analysis of a cogeneration plant using coal gasification and solid oxide fuel cell, Energy, 31, pp.345-363.

17. Colpan, C.O., Dincer, I., Hamdullahpur, F., 2008, A review on macro-level modeling of planar solid oxide fuel cells, International Journal of Energy Research, 32, pp. 336-355.

18. Colpan, C.O., Hamdullahpur, F., Dincer, I., 2010, Heat-up and start-up modeling of direct internal reforming solid oxide fuel cells, Journal of Power Sources, 195, pp.3579-3589.

19. Shah, R.K, 1978, Laminar flow forced convection in ducts: a source book for compact heat exchanger analytical data, New York, Academic Press

20. Selimovic A., Kemm, M., Torisson, T., Assadi, M., 2005, Steady state and transient thermal stress analysis in planar solid oxide fuel cells, Journal of Power Sources, 145, pp. 463-469.

21. Achenbach, E., 1996, SOFC stack modelling, Final Report of Activity A2, Annex II: Modelling and Evaluation of Advanced Solid Oxide Fuel Cells, International Energy Agency Programme on R, D&D on Advanced Fuel Cells, Juelich, Germany.

22. Braun, R.J., 2002, Optimal design and operation of solid oxide fuel cell systems for small-scale stationary applications, PhD thesis, University of Wisconsin-Madison

23. Kim, J., Virkar, A.V., Fung, K., Mehta, K., Singhal. S.C., 1999, Polarization effects in intermediate temperature, anode-supported solid oxide fuel cells, Journal of the Electrochemical Society, 146(1), pp. 69-78

24. Chan, S.H., Xia, Z.T., 2002, Polarization effects in electrolyte/electrode-supported solid oxide fuel cells, Journal of Applied Electrochemistry, 32, pp. 339-347.

25. Bossel, U.G., 1992, Final report on SOFC data facts and figures, Berne, CH:Swiss Federal Office of Energy.

26. Incropera, F.P., Dewitt, D.P., 1996. Fundamentals of heat and mass transfer, 4th ed., John Wiley& Sons.

27. Ozisik, N., 1994, Finite difference methods in heat transfer, CRC-Press, U.S.A.


15

28.

Figure 1: Schematic of the repeat element of a SOFC.

900

1,900

2,900

3,900

4,900

5,900

6,900

7,900

0 2 4 6 8 10

Cu

rre

nt

de

nsi

ty (

A/

m2)

Distance to inlet (cm)

ECN (co-flow)

V1 (co-flow)

V2 (co-flow)

V1 (counter-flow)

V2 (counter-flow)

Figure 2: Comparison of current density distribution found using the Model-V1 and Model-V2 with the benchmark test (ECN’s data).

L

Wsolid

Wgas

tfc

tPEN

tac

tai

tci

Anode interconnect

Cathode interconnect

PEN

Fuel channel

Air channel


16

880

900

920

940

960

980

1,000

1,020

1,040

1,060

1,080

0 2 4 6 8 10

Te

mp

era

ture

of

fue

l ch

an

ne

l [°

C]


ECN (co-flow)

V1 (co-flow)

V2 (co-flow)

V1 (counter-flow)

V2 (counter-flow)

Figure 3: Comparison of temperature distribution in the fuel channel found using Model-V1 and Model-V2 with the benchmark test (ECN’s data).

0.0

0.2

0.4

0.6

0.8

1.0

0 2 4 6 8 10

H2

fra

ctio

n [

%]


ECN (co-flow)V1 (co-flow)V2 (co-flow)V1 (counter-flow)V2 (counter-flow)

Figure 4: Comparison of molar hydrogen fraction distribution in the fuel channel found using the Model-V1 and Model-V2 with the benchmark test (ECN’s data).


17

(a) (b)

(c) (d) Figure 5: Temperature distribution for co-flow configuration at different time: (a) 397 s, (b) 794 s (end of heat-up stage), (c) 1503 s, (d) 4143 s (end of start-up)


18

(a) (b)

(c) (d) Figure 6: Temperature distribution for counter-flow configuration at different time: (a) 397 s, (b) 794 s (end of heat-up stage), (c) 1503 s, (d) 4143 s (end of start-up)

0

0.1

0.2

0.3

0.4

0.5

0

0.2

0.4

0.6

0.8

1

0 1000 2000 3000 4000 5000

Av

era

ge

cu

rre

nt

de

nsi

ty

[A/

cm2]

Fu

el u

tili

zati

on

Time [s]

co-flow

counter-flow

After this

point, start-

up stage

begins.

Fuel utilization

Average current density

Figure 7: Change of fuel utilization and current density with time for the SOFC fueled with humidified hydrogen.


19

0

0.05

0.1

0.15

0.2

0.25

0

0.10.2

0.3

0.40.5

0.6

0.70.8

0.9

1

0 1000 2000 3000 4000 5000

Po

we

r d

en

sity

[W

/cm

2]

Ele

ctri

cal

eff

icie

ncy

Time [s]

co-flow

counter-flow

After this

point, start-

up stage

begins.

Power density

Electrical efficiency

Figure 8: Change of electrical efficiency and power density with time for the SOFC fueled with humidified hydrogen.

(a)

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10

Mo

lar

fra

ctio

n o

f h

yd

rog

en

Distance from inlet [cm]

t=793 s

t=833 s

t=933 s

t=1023 s

t=1313 s

t=1853 s

t=2503 s

t=4143 s

time increases

co-flow

(b)

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10

Mo

lar

fra

ctio

n o

f h

yd

rog

en

Distance from inlet [cm]

t=793 s

t=883 s

t=983 s

t=1083 s

t=1313 s

t=1853 s

t=2503 s

t=4233 s

time increases

counter-flow

Figure 9: Change of molar fraction of hydrogen with time for the SOFC fueled with humidified hydrogen for (a) co-flow case, (b) counter-flow case.


20

0.00 1.97 3.94 5.91 7.88 9.86

0

0.050.10.15

0.20.25

0.30.350.4

0.450.5

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4 5

Mass flow rate per cross section [mg/s.cm2]

Av

era

ge

cu

rre

nt

de

nsi

ty [

A/

cm2]

Fu

el u

tili

zati

on

Reynolds number

co-flow

counter-flow

Average

current density

Fuel utilization

Figure 10: Effect of Reynolds number on the fuel utilization and average current density.

0.00 1.97 3.94 5.91 7.88 9.86

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0

0.1

0.2

0.3

0.4

0.5

0.6

0 1 2 3 4 5

Mass flow rate per cross section [mg/s.cm2]

Po

we

r d

en

sity

[W

/cm

2]

Ele

ctri

cal

eff

icie

ncy

Reynolds number

co-flow

counter-flow

Power

density

Electrical efficiency

Figure 11: Effect of Reynolds number on the electrical efficiency and power density.


21

950

1000

1050

1100

1150

1200

1250

1300

0 2 4 6 8 10 12 14 16

Air

ch

an

ne

l o

utl

et

tem

pe

ratu

re [°

C]

Excess air coefficient

co-flow

counter-flow

Figure 12: Effect of excess air coefficient on the air channel outlet temperature.

0.000 0.045 0.091 0.136 0.182 0.227 0.272 0.318 0.363

0.2

0.22

0.24

0.26

0.28

0.3

0.32

0.65

0.7

0.75

0.8

0.85

0.9

0 2 4 6 8 10 12 14 16

Mass flow rate per cross section [g/s.cm2]

Av

era

ge

cu

rre

nt

de

nsi

ty [

A/

cm2]

Fu

el u

tili

zati

on


co-flow (fuel util.)

counter-flow (fuel util.)

co-flow (curr. dens.)

counter-flow (curr. dens.)

Figure 13: Effect of excess air coefficient on the fuel utilization and average current density.


22

0.000 0.045 0.091 0.136 0.182 0.227 0.272 0.318 0.363

0.14

0.15

0.16

0.17

0.18

0.19

0.2

0.21

0.22

0.38

0.4

0.42

0.44

0.46

0.48

0.5

0 2 4 6 8 10 12 14 16

Mass flow rate per cross section [g/s.cm2]

Po

we

r d

en

sity

[W

/cm

2]

Ele

ctri

cal

eff

icie

ncy


co-flow (elec. eff.)

counter-flow (elec. eff.)

co-flow (power dens.)

counter-flow (power dens.)

Figure 14: Effect of excess air coefficient on the electrical efficiency and power density.

Documents

TRANSIENT HEAT TRANSFER MODELING OF A SOLID OXIDE