Transient Heat Transfer Modeling of a Solid Oxide Fuel Cell

8/2/2019 Transient Heat Transfer Modeling of a Solid Oxide Fuel Cell

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Transient heat transfer modeling of a solid oxide fuel cell

operating with humidified hydrogen

C. Ozgur Colpan a,*, Feridun Hamdullahpur b, Ibrahim Dincer c

aMechanical and Aerospace Engineering Department, Carleton University, 1125 Colonel By Drive, Ottawa, Ontario, Canada K1S 5B6b Mechanical and Mechatronics Engineering Department, University of Waterloo, 200 University Avenue West, Waterloo, Ontario,

Canada N2L 3G1cFaculty of Engineering and Applied Science, University of Ontario Institute of Technology, 2000 Simcoe Street North, Oshawa,

Ontario, Canada L1H 7L7

a r t i c l e i n f o

Article history:

Received 16 August 2010

Received in revised form

6 November 2010

Accepted 26 November 2010

Available online 3 January 2011

Keywords:

Efficiency

Solid oxide fuel cell

Hydrogen

Transient heat transfer

Fuel utilization

Finite difference method

a b s t r a c t

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 conditions. 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.

Copyright 2010, Hydrogen Energy Publications, LLC. Published by Elsevier Ltd. All rights

reserved.

1. Introduction

The solid oxide fuel cell is one of the emerging energy tech-nologies 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/orelectricity.

SOFCs have advantages over other fuel cell types. These

advantages include [1]: 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 gasi-fication systems. Possible disadvantages of the SOFC over

other fuel cells are: a) challenges for construction and dura-

bility due to its high temperature, and b) a carbon deposition

problem when a fuel such as methane or syngas 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,

* Corresponding author.E-mail address: [email protected] (C.O. Colpan).

A v a i l a b l e a t w w w . s c i e n c e d i r e c t . c o m

j o u r n a l h o m e p a g e: w w w . e l s e v i e r . c om / l o c a t e / h e

i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 6 ( 2 0 1 1 ) 1 1 4 8 8 e1 1 4 9 9

0360-3199/$ e see front matter Copyright 2010, Hydrogen Energy Publications, LLC. Published by Elsevier Ltd. All rights reserved.doi:10.1016/j.ijhydene.2010.11.127


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different considerations may be taken into account according

tothepurposeandneedsofthemodel.Forexample,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 [2e4]. Transient

modeling is used when one, or a combination of the followingstages, needs to be simulated: heat-up, start-up, shut-down

and load change [5e8]. Thermomechanical modeling provides

us with information to predict thestresses occurringinside the

fuel cell [9]. The problems related to carbon deposition due to

using a fuel containing carbon can also be analyzed through

numerical studies [10e13]. In system level modeling, inte-

grated SOFC systems can be assessed using several methods

including energy, exergy and thermoeconomic analyses

[14e17]. More information on different types of SOFC models

can be found in the paper by Colpan et al. [1].

The transient behavior of the cell during the heat-up and

start-up stages plays an important role in the successful

operation of the fuel cell. In the literature, a few studies thatmodel these stages to find the characteristics of a SOFC can be

found. For example, Colpan et al. [18] modeled a direct

internal reforming SOFC operating with a multi-gas mixture

including CH4, CO, CO2, H2, H2O, and N2. The effects of the

reforming process on the performance of the cell were

included in the model. The heat-up, start-up, and steady state

behaviors of the cell were investigated. In addition, the first

principal thermal stresses were calculated to find the proba-

bility of failure of the cell during its operation. Ki and Kim [19]

developed a model to predict the thermal dynamics of planar

SOFCs. Two methods for stack-heating (hot air through

cathode channels and electric heating inside a furnace) were

investigated. Barzi et al. [20] developed a 2-D transient modelto analyze the start-up behavior of a tubular SOFC. Their

model gives the cell voltage, the electromotive force, and the

variables such as pressure, temperature and species concen-

tration during the start-up as the output. In addition, the cell

heat-up rate for hot inlet gases as well as the start-up time of

the SOFC were calculated.

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 radia-

tion), and all polarization nodes, (i.e. ohmic, activation and

concentration). This study differs from the other transient

modeling studies in the literature in terms of the modeling

equations and strategy used in the heat-up and start-up

stages, and the selection of the input and output parameters.

This model is usedfor 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 con-

ducted 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 formula-

tion 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 themiddle of a stack,whichis shown in Fig. 1, is divided into five control volumes: anode

interconnect, fuel channel, PEN (consisting of anode, electro-

lyte andcathode), airchanneland cathodeinterconnect. Dueto

the symmetrical conditions of the considered repeat element

in the stack, the solid structure has adiabatic boundary

conditions at the exterior surfaces [9]. The Cartesian coordi-

nate 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 conser-

vation of momentum, some simplifications are madeassuming fully developed laminar flow conditions. This

assumption is well justified since the gases flow with low

velocity,whichis requiredto obtaina high fuel utilizationratio.

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 [21].

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 thechannel inletsmay

be chosen as input or output according to the purpose of the

model. In ourmodel, the cell voltage, which is assumed to beequal 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 deter-

mines 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 parame-

ters 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

Fig. 1 e Schematic of the repeat element of a SOFC.

i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 6 ( 2 0 1 1 ) 1 1 4 8 8 e1 1 4 9 9 11489


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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 toTmin,solid 100 C for the subsequent time step [22]. In

modeling the start-up period, the temperatures of the air and

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 convec-

tion in the heat-up stage and forced convection in the start-up

stage, surface-to-surface radiation between the PEN andinterconnects. 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 104 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 ofsolving 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 theSOFC benchmark test [23] and Brauns model

[24] 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 componentsof 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 and neglecting the transient terms, the continuity

equations at thefuel channel are shown in Eqs. (1) and (2), and

theseequations at theair channelare shown in Eqs. (3) and (4).

d _n00H2dx

_r00eltfc

(1)

d _n00H2 Odx

_r00eltfc

(2)

d _n00O2dx

_r00el=2

tac(3)

d _n00N2dx

0 (4)

where the rate of conversion for electrochemical reaction

becomes _r00el i

2F.

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

V VN Vohm Vact Vcon (5)

where the Nernst voltage and the polarizations can be given as

[25,26]:

VN Dg

r

2F

RT

2F$ln

PH2O

PH2$ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

PO2 =P+

p!

Vohm

Xk

rel;k$tk

!$i

Vact RTF $sinh1

i2io;a

RTF $sinh

1

i2io;c

Vconc RTs2F

ln

1

RT

2F$

sataDaVvaPbH2

i

!

RTs2F

ln

1

RT

2F$

sataDaVvaPbH2O

i

!

RTs4F

ln

26664 P

bO2

P

P PbO2

exp

RT4F$

sctcDcVvcP

i

37775

The electrical resistivity of the cell components, r, can be

written as functions of the solid temperature as given in [27].

The current density, the molar flow rate and molar composi-

tion 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

rac$cp;ac$vT

vtv

vx

_n00O2 hO2 _n

00N2

hN2

hc;aTPEN Ta hc;aTci Ta

_r00el=2

$hO2$wsolid=wgas

tac(6)

with the following boundary and initial conditions:

x 00T ft Heat up

T Tw ac Start up; t 00T To 100

oC

Here, it should be noted that we have used a 1-D modeling

approach for the air and fuel channels. Hence, the convectionheat transfer and enthalpy inflow/outflow terms appear as

a source term in these equations.

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

given as

v2T

vx2v

2T

vy2

_r00el$

hH2 hO2=2 hH2 O

i$Vcell

kPEN$tPEN

1aPEN

$

vT

vt(7)


x 0 and x L0vT

vx 0

y tci tac0kPEN$vT

vy

wgas

wsolid$hc;a$TPEN Ta hr;a$TPEN Tci

1

wgaswsolid

$kci$

TPEN Tci

tac



4/12

y tci tac tPEN0 kPEN$vT

vy

wgaswsolid

$

hc;f$

TPEN Tf

hr;f$TPEN Tai

1

wgaswsolid

$kai$

TPEN Tai

tfc

t 00T To

It should be noted that we have neglected the effects of the

flow in the porous media in the equation shown above.

The transient heat transfer equation for the fuel channel is

rfc$cp;fc$vT

vtv

vx

_n00H2 hH2 _n

00H2 O

hH2O

hc;f

Tai Tf

hc;f

TPEN Tf

_r00el$

hH2 hH2O

$wsolid=wgas

tfc

(8)with the following boundary and initial conditions:

x 00T Tw fcStart up; t 00T To

The 2-D transient heat diffusion equation for the cathode

and anode interconnects is

v2T

vx2v

2T

vy2

1ai$

vT

vt(9)


x 0 and x L0vT

vx 0; y 0 and y tci tac tPEN tfc tai

0

vT

vy 0

y tci and y tci tac tPEN tfc0 ki$vT

vy

wgaswsolid

$

hc;g$

Ti Tg

hr;g$Ti TPEN

1 wgas

wsolid

$ki$T

i TPENtgc

t 00T To

Here, g should be replaced with fand 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 avail-

able elsewhere [21,28].

Dimensionless numbers: The Reynolds number for the fuelchannel inlet based on the hydraulic diameter of the rectan-

gular channel cross section is shown in Eq. (10). In the model,

this number is considered as one of the input parameters.

Hence,usingthisnumber,themassflowrateofthegasmixture

per cross section of the fuel channel at the inlet can be found.

ReDh

_m00fi$

2$tfc$wgasmmix

$

tfc wgas

(10)Eq. (10) can also bewritten interms ofmolarflow rate ofthe

gas species at the fuel channel inlet as follows:

ReDh _n00k;fi$Mmix$

2$tfc$wgas

xk;fi$mmix$

tfc wgas

(11)where k denotes H2 and H2O.

The excessair coefficient used as an input parameterin 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.

lair _n00O2 ;ai

_n00H2;fi=2$tac

tfc(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.

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:

UF

Pmi2

_r00el

i$Dx$wsolid

_n00H2;fi$wgas$tfc

(13)

where m denotes the number of nodes in flow direction.

(1,1)

(1,p)(1,p+1)

(1,p+2)

(1,p+3)

(1,r)(1,r+1)

(1,r+2)(1,r+3)

(1,s)

(m,1)

(m,p)

(m,r)

(m,s)

(m,p+1)

(m,p+2)

(m,p+3)

(m,r+1)

(m,r+2)(m,r+3)

Cathode Interconnect

Air channel

PEN

Fuel channel

Anode Interconnect

Lcelly

x

Fig. 2 e Numbering scheme for the finite difference

solution of the repeat element of a SOFC.

Table 1 e Input data used in the benchmark test.

Variable Value

Cell geometry

Active area [mm2] 100 100

Anode thickness [m] 50 106

Cathode thickness [m] 50 106

Electrolyte thickness [m] 150 106

Channel width [mm] 3

Channel height [mm] 1Rib width [mm] 2.42

Total thickness (with ribs) [mm] 2.5

Material properties

Density of PEN and interconnects [g/cm3] 6.6

Specific heat of PEN and interconnects [J/gK] 0.4

Thermal conductivity of PEN

and interconnects [W/cmK]

0.02

Operating parameters

Temperature at the fuel channel inlet [K] 1173

Temperature at the air channel inlet [K] 1173

Pressure of the cell [bar] 1

Excess air coefficient 7

Fuel utilization 0.85

Mean current density [A/m2] 3000Gas composition at the air channel inlet 21% O2, 79% N2Gas composition at the fuel channel inlet 90% H2, 10% H2O



5/12

The power density and power output of the SOFC are given

below:

_W00

SOFC ic;ave$Vcell (14)

_WSOFC _W00

SOFC$Lcell$wsolid (15)

The primary purpose of using SOFC is to generate electricity

and its performance can be assessed through electrical effi-

ciency of the cell as follows:

hel _WSOFC

LHV$P2

k1 _n00

k;fi$tfc$wgas(16)

where k denotes the types of gases found at the fuel channel

inlet, i.e. H2 and H2O.

2.3. Numerical solution

The schematic of the 2-D cross section of a repeat element andthe numbering scheme for the nodes used in the numerical

solution are shown in Fig. 2. Ascanbe seenfrom the figure,the

repeatelementhas m s nodes. The length of the cell,and the

thicknesses of the cathode interconnect, PEN, and anode

interconnect aredividedintom-1,p-1, r-p-3,ands-r-3parts. An

implicit finite different scheme is used for the solution of the

heat transfer equations. The finite difference equations forthe

boundaryconditionsare takenas secondorder 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 boundaryequation. The details of thisapproach canbe found in [29].The

set of equations are linearized by using the lagging properties

by one time step method [29], and solved using the Gauss

elimination method. Iterations are repeated until the solid

temperaturereachesacertainvaluefortheheat-upperiod,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 [23], was used to validate the model. In this benchmark

test, nine institutions modeled planar SOFCs with the sameoperating 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 thetest 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 bench-

mark test are given in Table 1. In another study, Braun [24]

developed a steady state model using the same input data

Table 2 e 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 0.730Inst. For Energiteknikk Kjeller, N 0.710 0.710

KFA-Julich, D 0.706 0.712

Siemens, D 0.712 0.716

Statoil, N 0.702 0.709

Riso, DK 0.703 0.710

Table 3 e Validation of maximum and minimum valuesof 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

Brauns model 3799/1211 7393/1152Model-V1 3760/1187 7564/1202

Model-V2 5175/1175 5530/1586

Table 4 e Validation of maximum and minimum valuesof solid temperature.

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

Counter-flow(max/min) (C)

Dornier, D 1070/928 1085/914ECN Petten, NL 1082/899 N.A.

Eniricerche, I 1069/916 1083/906


Kjeller, N

1058/930 1084/912

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

Brauns model 1059/924 1073/910

Model-V1 1049/903 1056/904

Model-V2 1043/907 1054/906

Table 5 e Validation of air and fuel channel outlettemperatures.

Company/Institution Co-flow (air/fuel)(C)

Counter-flow(air/fuel) (C)

Dornier, D 1068/1070 1080/914

ECN Petten, NL 1082/1082 N.A.

Eniricerche, I 1068/1068 1080/906


Kjeller, N

1055/1058 1073/912

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

Brauns model 1058/1059 1068/910Model-V1 1048/1047 1051/900

Model-V2 1042/1043 1051/900



6/12

and same assumptions as the benchmark test. In addition to

the data given in Table 1, we have considered the data neededfor calculating the activation and concentration polarizations

in the model as follows. Exchange current densities of anode

and cathode are taken as 0.53 A/cm2 and 0.2 A/cm2, respec-

tively; diffusivity of the anode and cathode are taken as

0.91 cm2/s and 0.22 cm2/s, respectively; porosity of the anode

and cathode are both taken as 0.5; and tortuosity of the anode

and cathode are both taken as 4.

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 tran-

sient heat transfer model was first developed using the same

assumption for polarizations as the benchmark tests (i.e.

activation and concentration polarizations were included

with a simplification). 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 Section

2.2. This model is called Model-V2. There are some differences

in theinput 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 aretaken 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 outto findthe

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 andthe time step

as 1 s 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 andnot 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 utili-

zation 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.

Maximum and minimum values for the current density,

solid temperature and air and fuel channel outlet tempera-

tures are given in Tables 3e5, respectively. From Table 3,itcan

be seenthat the currentdensity, found by different companiesand 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

l

l

l

l

l

l

Fig. 3 e Comparison of current density distribution found

using the Model-V1 and Model-V2 with the benchmark test

(ECNs data).

l

fl

l

l fl

fl

fl

fl

Fig. 4 e Comparison of temperature distribution in the fuel channel found using Model-V1 and Model-V2 with the

benchmark test (ECNs data).



7/12

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 same procedure isfollowed 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 forthe 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 benchmarktest. In Model-V1 for counter-flow configuration, the outlet

temperature forthe fuel channeland the inlet temperature for

the air channel are fixed to obtain a uniform temperature

distribution. The inlet temperature of the fuel channeland 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 bench-

mark test. The same procedure has also been followed for the

maximum and minimum solid temperature, and air and fuel

channeloutlettemperature. It is found the relative error is less

than 2.3% for all cases.

When we checked the results for Model-V2 from Tables3e5, it was seen that except for the current density

l

l

l

l

l

l

Fig. 5 e Comparison of molar hydrogen fraction

distribution in the fuel channel found using the Model-V1

and Model-V2 with the benchmark test (ECNs data).

Fig. 6 e 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).



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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, thisassumption is not valid

today. Detailed correlations have been published on the acti-

vation and concentration polarizations in the literature, [e.g.

[25,26]]. However, the temperature distribution is still

comparable between Model-V2 and the benchmark test-1. For

this model, we have also checked the relative error for the

maximum and minimum solid temperature, and air and fuel

channel temperature. It is found that this error is less than

2.4% for all cases.

The distributions of current density, fuel channel temper-

ature 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 insti-

tute that participated in the benchmark test. This validation is

shown in Figs. 3e5. 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 Fig. 3, current density

trends for Model-V1, and the model developed by ECN, are

Fig. 7 e 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).

I I

I

I

I

I

Fig. 8 e Change of fuel utilization and current density with

time for the SOFC fueled with humidified hydrogen.



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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 thedifferent correlations for activation and concentration polar-

izations 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 Fig. 4, itcan beseen thatthe trends

are similar. The temperature at the fuel channel exit wasfound 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 Fig. 5, it can be seen that molar hydrogen fraction has

almost the same trend as ECN.

3.2. Transient behavior of the cell

After validating the model, the transient behaviors 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 Fig. 6, temperature distributions for the co-flow case forModel-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 dis-

cussed 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 tempera-

ture reaches a prescribed value, which was chosen as 700 C in

this study. At this temperature, the resistivity of the electro-

lyte and the ohmic polarization become low enough to

produce meaningful amount of power. In Fig. 6, we can see

Fig. 9 e Change of electrical efficiency and power density

with time for the SOFC fueled with humidified hydrogen.

a

b

Fig. 10 e Change of molar fraction of hydrogen with time

for the SOFC fueled with humidified hydrogen for

(a) co-flow case, (b) counter-flow case.

Fig. 11 e Effect of Reynolds number on the fuel utilization

and average current density.

Fig. 12 e Effect of Reynolds number on the electrical

efficiency and power density.



10/12

thatthe 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 thestart-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 polari-

zations; 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, respec-

tively. We compared these values with the results of our

previous study in which a multi-gas mixture was used as

a fuel in a direct internal reforming SOFC (DIR-SOFC) (15.6 C/

cm in the x direction, and 1.03 C/cm in the y direction [18]).

The results show that the temperature gradients are close to

each other for both cases.Fig. 7 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 Fig. 7a and b,

are symmetrical to those shown in Fig. 6a and b. The temp-

erature 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. Whenthese values arecompared with the results

of the counter-flow DIR-SOFC model (7.48 C/cm in the

x direction and 1.01 C/cm in the y direction [18]), the

temperature gradient in the x direction for humidified

hydrogen case is found to be higher but that in the y directionis found to be comparable for both cases.

It follows from Figs. 8 and 9 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 condition, as can be seen in Fig. 10, because of

the higher fuel utilization of hydrogen at the beginning of the

start-up period due to the lower operating temperature.

Figs. 8e10 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

The Reynolds number at the fuel channel inlet and excess air

coefficient are the key input parameters affecting the perfor-

mance of the cell. Due to this reason, the effectsof theseinput

parameters on the output parameters such as fuel utilization,

average current density, electrical efficiency, and power

density are investigated.

Figs. 11 and 12 show the effect of the Reynolds number on

the output parameters. Reynolds number is directly propor-

tional 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 getany meaningful results. If we choose this

number very low, the computer code gives us imaginary

numbers as the output. From Fig. 11, 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 Fig. 12; because

the cell voltage is assumed to be constant in the modeling. It

can be easily shown that electrical efficiency is directly

Fig. 13 e Effect of excess air coefficient on the air channel

outlet temperature.

Fig. 14 e Effect of excess air coefficient on the fuel

utilization and average current density.

Fig. 15 e Effect of excess air coefficient on the electrical

efficiency and power density.



11/12

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 utiliza-

tion, e.g. fuel utilization of 0.85. For example, for Reynoldsnumber 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 oper-

ating variable because it controls the current density and the

temperature of the fuel cell, which in turn affects the perfor-

mance of the cell. If low amounts of air is sent through the air

channel, the temperature of the exit increases, as shown in

Fig. 13. Therefore, the excess aircoefficient should be carefully

selected not to cause a thermomechanical problem. It can be

seen from Figs. 14 and 15 that the excess air coefficient should

be taken high enough to get a betterperformance from the cell,

which can be explainedas follows: As the excess aircoefficientincreases,temperatureof thefuel cell decreases.This decrease

causes an increase in the Nernst voltage, and decrease 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

aircoefficientshouldbe selected depending on the application

and taking into account the performance and economics.

When we compare the co-flow and counter-flow configura-

tions, Fig. 15 shows that counter-flow configuration hasa higher electricalefficiencythan co-flowconfiguration forany

given excess air coefficients.

4. Conclusions

A new transient, 2-D 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 relations based on

electrochemistry are coupled with the heat transfer equa-

tions. 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 distri-

bution 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 Premiers

Research Excellence Award, the Natural Sciences and Engi-

neering 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/gK

D diffusivity, cm2/s

Dh hydraulic diameter, mF Faraday constant, C

h heat transfer coefficient, W/cm2K

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 per cross section, mole/cm2s

P pressure, bar_q heat transfer rate, W

_r} conversion rate, mole/cm2s

R universal gas constant, J/moleK

ReDh 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_W

}

power density, W/cm2

x molar concentration

Greek letters

r mass density, g/cm3

rel electrical resistivity of cell components, U cm

hel electrical efficiency

lair excess air coefficient

s tortuosity

m viscosity, g/s-cm

a thermal diffusivity, cm2/s

Subscripts

a anode; air

ac air channel

act activation

ai anode interconnect

ave average



12/12

c cathode; convection

ci cathode interconnect

conc concentration

e electrolyte

el electrochemical; electrical

fc fuel channelfi 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

Superscriptsb bulk

o standard state

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Transient Heat Transfer Modeling of a Solid Oxide Fuel Cell

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