International Journal of Heat and Mass Transfer 131 (2019) 1032–1051
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer
journal homepage: www.elsevier.com/locate/ijhmt
A numerical analysis of unsteady transport phenomena in a Direct
Internal Reforming Solid Oxide Fuel Cell
Maciej Chalusiak a, Michal Wrobel a, Marcin Mozdzierz a, Katarzyna Berent b, Janusz S. Szmyd a,
Shinji Kimijima c, Grzegorz Brus a,⇑
a
b
c
AGH University of Science and Technology, Department of Energy and Fuels, 30 Mickiewicza Av., 30-059 Krakow, Poland
AGH University of Science and Technology, Academic Centre for Materials and Nanotechnology, 30 Mickiewicza Av., 30-059 Krakow, Poland
Shibaura Institute of Technology, Department of Machinery and Control Systems, 307 Fukasaku, Minuma-ku, 337-8570 Saitama City, Japan
a r t i c l e
i n f o
Article history:
Received 22 August 2018
Received in revised form 14 November 2018
Accepted 22 November 2018
Available online 30 November 2018
Keywords:
Solid oxide fuel cell
Internal reforming kinetics
Dynamic model
FIB-SEM
Microstructure
a b s t r a c t
In this paper, a transient microstructure-oriented numerical simulation of a planar Direct Internal
Reforming Solid Oxide Fuel Cell (DIR-SOFC) is delivered. The performance criteria in a direct steam
reforming for a fuel starvation scenario are analyzed in order to optimize the underlying process. The proposed two-dimensional multiscale model takes into account mass and heat transport, electrochemistry,
as well as the intrinsic steam-reforming kinetics. In the paper, the methane/steam reforming process over
the Ni/YSZ catalyst is experimentally investigated to verify the used chemical reaction model. A threedimensional digital microstructure representation of the commercial anode is analyzed using a
Focused Ion Beam-Scanning Electron Microscope (FIB-SEM) and the nickel-pore contact surface is calculated to relate the reforming reaction rate to the catalyst’s active area. Based on the complete DIR-SOFC
model, a parametric study is carried out, to simulate the dynamic response of a fuel cell for different
design and operating conditions. The results prove the dominant impact of inlet fluid temperature and
methane content on the calculated distribution of hydrogen across the channel, while the collected current density was found to be a less important factor. The simulations indicate, that in the case of the
direct reforming, fuel starvation is likely to occur in the upstream of the anode channel, where the
reforming reaction does not keep up with producing hydrogen. The obtained results provide a significant
insight into safe and efficient control strategies for Solid Oxide Fuel Cells.
Ó 2018 Elsevier Ltd. All rights reserved.
1. Introduction
Growing popularity and rapid development of Solid Oxide Fuel
Cells (SOFCs) stem for their potential to become a gamechanger in
the field of clean power generation technologies. Due to a low level
of pollutants’ emission, high operating temperature (600–1000 °C)
and a wide range of utilized fuels, SOFC’s can play a key role in supporting conventional power systems [1,2]. A distinguishing feature
of SOFCs with nickel-based anodes (e.g. Ni/YSZ) is the ability to
internally reform a wide range of hydrocarbons, like methane.
Together with increased tolerance to most of the impurities, it represents a significant advantage of this type of fuel cell. Direct internal reforming of methane on the DIR-SOFC anode prominently
increases the system efficiency by recuperating waste heat from
the electrochemical reactions to supply strongly endothermic
reforming reaction, at the same time reducing the complexity, size
⇑ Corresponding author.
E-mail address: brus@agh.edu.pl (G. Brus).
https://doi.org/10.1016/j.ijheatmasstransfer.2018.11.113
0017-9310/Ó 2018 Elsevier Ltd. All rights reserved.
and cost of the system by elimination of the external reformer [3,4].
This beneficial configuration of stack, however, engages various
phenomena occurring simultaneously in one place and may be
found highly challenging for thermal and substantial management.
Endothermic steam reforming reaction may cause severe temperature gradients in the cell, leading to high thermal stresses, which, in
the long run, may damage the cell’s ceramic components [5,6].
What is more, mismatching thermal effects in the DIR-SOFC may
result in an anode cooling effect or insufficient hydrogen production. These issues were widely addressed in recent studies in terms
of a steady state operation [7,8]. However, stages in which the cell’s
operating conditions change in time, e.g. start-up or external load
connection, are inevitable in its lifetime and it is extremely difficult
to control the cell’s behaviour in such transient states. Therefore, it
is crucial for optimization and management of the cell’s performance to understand the dynamic behaviour of DIR-SOFCs [9].
Yet, experimental studies of DIR-SOFCs under transient conditions is difficult, due to obstacles in non-invasive measurements
of physical fields within the cell unit. Thus, numerical simulations
M. Chalusiak et al. / International Journal of Heat and Mass Transfer 131 (2019) 1032–1051
is a reasonable and cost-effective way to study DIR-SOFCs performance. With the introduction of Computational Fluid Dynamics
codes, a much deeper understanding of the processes inside any
fuel cell can be provided [8,10].
Dynamic behavior and the modeling of DIR-SOFC was of interest to several research groups in the past [11–14]. A dynamic
model of DIR-SOFC stack was recently presented in the paper of
Kupecki et al. [14]. The simulations of the stack operation took a
complete zero-dimensional reforming model into account and
the results confirmed the impact of reforming reaction kinetics
on the thermal balance of the cell [14]. A highly endothermic
reforming process may locally reduce the temperature, which
leads to higher thermal stresses. The simulations indicate that
temperature changes at the outlet are mostly affected by the high
current values [14]. Aguiar et al. [11] investigated a timedependent response of an anode-supported DIR-SOFC to a step
load change. The proposed one-dimensional model took into
account mass and energy balances, as well as electrochemical
and chemical reactions. In the model, a shift in power demands
was represented by the current density step-change. The dynamic
simulations indicate, that the overall SOFC temperature rises after
positive load step-change, and drops in an opposite case. Additionally, the load increase was accompanied by an instant growth in
hydrogen consumption. Huangfu et al. [15] developed a onedimensional SOFC model and performed its sensitivity analysis.
In their study, they provided a transient distribution of temperature and anode gas in the channel and it shows, that the transition
time due to the cells thermal capacity is in a range of seconds. An
interesting study was also presented by Kang et al. [16], where the
authors develop a one-dimensional dynamic model of a planar
DIR-SOFC. Their results prove the dominance of the internal
reforming process on the distributions of the anode gas composition and temperature, while various operating parameters change
such as load current and fuel flow were introduced to the model.
Nevertheless, as Bae et al. [17] noticed while analyzing the transient transport phenomena in SOFCs, more experimental and modeling studies should be performed to understand the effect of cells
microstructure on transport processes and the overall dynamic
response of SOFC thermodynamic variables. A limited amount of
research, however, has been conducted to study the microstructure
dependency on SOFCs performance, while recent studies by Kishimoto et al. [18] indicate that the Ni/YSZ anode microstructure
affects the catalytic reaction rates and should be taken into account
in CFD models.
In the viewpoint of fuelling DIR-SOFC a substantial management appears as an important matter. Due to limitations originating from the kinetics of the methane reforming reaction,
insufficient amounts of hydrogen may be produced, causing a fuel
starvation phenomenon [19]. The authors observed, that during
hydrogen starvation, acceleration of the degradation of the anode
occurs mainly due to the mechanical stresses and disconnection
of nickel particles.
A large interest in developing the DIR-SOFC technology and a
lack of current reports on three-dimensional dynamic models
which include microstructure, inclines to a thorough research.
Commercial SOFC cells’ manufacturers overcome the critical effects
in the stagnation points, such as edges or corners, by using cassettes, in which the electrode does not cover the entire support.
However, some problems in the longitudinal direction still need
a solution. This work, therefore, aims to fill the void in the
literature and to investigate the behaviour of DIR-SOFCs by
time-dependent numerical simulations of a simplified twodimensional model that account for the anode microstructure,
which reflects the qualitative characteristics in the core of the fluid
flow. A complete model requires a detailed information about the
methane steam reforming kinetics. Therefore, following the results
1033
from the literature, the utilized reforming reaction rate was verified using experimental data. Moreover, the anode microstructural
parameters obtained with the FIB-SEM electron tomography were
included in the rate equation to best reflect the real conditions. The
empirical rate was then employed in the proposed numerical
model. The computational scheme was built based on the Finite
Volume Method. The scheme was thoroughly verified against the
analytical benchmark solution and a good agreement was found.
Having verified the accuracy of the computations we numerically
analyzed the influence of the anode microstructure, load changes,
fuel inlet flow, inlet temperature and fuel composition, on the
DIR-SOFC performance. The parameters mentioned above, were
presented in the numerical model by boundary and operating conditions. The calculated cross-sectional temperature and species
distributions are shown as the results of the calculations and discussed with reference to the fuel starvation, in order to seek the
properties of DIR-SOFC using which electrochemical and chemical
reactions can be governed.
2. Mathematical model
The computational domain refers to a 2D cross-section of the
DIR-SOFC anode (fuel) channel. Its geometrical configuration is
presented in Fig. 1.
In the developed model, SOFC is fueled with methane (CH4 ) and
a corresponding amount of steam (H2 O) to satisfy the set steam-tocarbon ratio (SC). Moreover, the hydrogen (H2), carbon monoxide
(CO), carbon dioxide (CO2) and temperature (T) distributions are
analyzed in the model. The following assumptions are made to
describe the anode channel [1,20–22]:
– flow in the fuel channel is unidirectional, laminar and time –
independent;
– atmospheric pressure is assumed in the calculation domain;
– all fluids are incompressible, ideal gases;
– heat transfer by radiation and fuel leakage is neglected;
– all chemical and electrochemical reactions occur at the interface
between the fuel channel and the anode;
– the influence of current collectors, sealings, isolations, etc. are
neglected in the model;
– the anode is completely and fully penetrated by the gases;
– hydrogen is assumed to be a primary fuel for the cell;
– the thermal and substantial effects of the reforming reaction are
considered as if they occurred in the entire volume of the anode.
The fluid flow model is described in detail in Section 2.2.
Fig. 1. Schematic configuration of the system.
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M. Chalusiak et al. / International Journal of Heat and Mass Transfer 131 (2019) 1032–1051
@u
¼ 0;
@x
2.1. Heat and mass transfer model
In the present model, the transport equations are derived by the
volume-averaging method. Heat transfer inside the anode channel
is governed by the following differential equation [21,23]:
@
ðqC P T Þ þ $ ðquC P T Þ ¼ $ ðk$T Þ;
@t
ð1Þ
3
where q stands for density in [kg m ], C P is heat capacity, in
[J kg1 K1], T stands for fluid temperature, in [K], k is the thermal
conductivity of the mixture of the gases, in [W m1 K1], and u is
the velocity vector, in [m s1]. The concentration of chemical species in the anode channel is governed by the mass conservation
equation [23]:
@
ðqY i Þ þ $ ðquY i Þ ¼ $ ðqDi $Y i Þ;
@t
ð2Þ
ð3Þ
Moreover, they are assumed to depend on the mixture temperature and composition. The thermophysical properties in Eqs. (3)
are calculated locally according to the mixing laws, taken from
[24]. The respective laws are described in detail in Appendix A.
In the proposed model, the system of mass transport equations is
solved with respect to the species mass fractions and the results
are converted into partial pressures or molar fraction during
post-processing e.g. for use in the steam reforming model, which
is described in Section 2.3.
2.2. Fluid flow model
In the general case, the time-independent fluid flow in the
channel is governed by [23,25]:
the continuity equation
$ ðquÞ ¼ 0;
where u ¼ uðyÞ is the only non-zero component of the velocity vector. The system of Navier-Stokes Eq. (5) yields:
@ 2 u 1 @p
;
¼
@y2 l @y
@p
< 0:
@x
The above system of equations describes the Poiseuille-type flow.
For the negative pressure gradient @p
< 0, a positive value of velocity
@x
u is obtained. We assume that the fluid flows in the channel at a
known rate N_ [m3 s1], defined as:
N_ ¼ d
Z
H
ð9Þ
uðyÞdy;
ð4Þ
where d in [m] is the width of the unit cell in the MSTB test bench
[26,22]. One can prove that the solution to the above system of
equations is:
uðyÞ ¼
6N_ Hy y2 :
3
dH
The methane/steam reforming process is widely known as a
conventional process for producing hydrogen. The process consists
of a set of numerous elementary reactions. However, two of them
are proved to be largely dominant in the whole process [1,27–29]:
the steam reforming reaction:
1
CH4 þ H2 O ! 3H2 þ CO DH0 ¼ 206:2 kJ mol ;
CO þ H2 O
CO2 þ H2
Rsh ¼ ksh pCO pH2 O þ ksh pH2 pCO2 ;
ð6Þ
p is the fluid pressure, in [Pa], while q, in [kg m3], and l, in [Pa s],
are the fluid density and viscosity, respectively.
In the presented model, a unidirectional, incompressible
(q = const) and steady flow was assumed in the channel. As a
result, the continuity Eq. (4) reduces to:
ð12Þ
The rates of the methane/steam reforming and water-gas shift
reactions are taken from Sciazko et al. [31,32], where they were
derived experimentally:
where u is the fluid velocity vector, in [m s1], s defines the shearstress tensor, in [Pa] (here we assume a form for the Newtonian
fluid):
s¼l
1
DH0 ¼ 41:2 kJ mol :
Since the reaction described by Eq. (11) is strongly endothermic,
the temperature drops within the reaction area, if no heat source is
present. Therefore a supply of thermal energy is needed. Moreover,
the steam reforming reaction is slow, thus one needs to describe it
by a pertinent rate equation. The water-gas shift reaction,
described by Eq. (12), is fast and weakly exothermic and can be
assumed to be in equilibrium at the reforming temperature
[1,21,30].
ð5Þ
2
$u þ $uT $u ;
3
ð11Þ
the Water-Gas-Shift reaction:
_ cat 6:47 103 exp Rst ¼ w
ð10Þ
2.3. Steam methane reforming model
the system of Navier-Stokes equations
$ ðquuÞ ¼ $p þ $s;
ð8Þ
0
where Y i stands for the dimensionless mass fraction of species i and
Di denotes the mass diffusion coefficient of species i in the mixture
of gases, in [m2 s1]. The energy and mass conservation reflect the
unsteady heat and mass transfer by both diffusion and convection
in a single, gaseous, continuous gas mixture. In the studied system,
all chemical and electrochemical reactions take place inside the
anode, which, as a separate subdomain, is neglected in the model.
Instead, the mass and heat effects of those reactions are introduced
to the balances in the channel via the mass and heat fluxes. Equations, which describe the fluxes, are shown in Section 2.4.
The physical parameters in Eqs. (1) and (2), the density q, the
specific heat C P and the thermal conductivity k refer to the gas mixture, thus:
q ¼ qmix ; C P ¼ C P;mix ; k ¼ kmix :
ð7Þ
þ
121 103
RT
!
0:083
p0:88
CH4 pH2 O ;
ð13Þ
ð14Þ
where
Eact
kst ¼ Ast exp :
RT
ð15Þ
The water-gas shift reaction rate, given by Eq. (14), reaches equilibrium rapidly, hence CO2 ; H2 ; CO oraz H2 O have to satisfy the equilibrium equation as follows [21,33]:
K sh
!
þ
ksh pCO2 pH2
DG0sh
:
¼ ¼
¼ exp ksh pCO pH2 O
RT
ð16Þ
M. Chalusiak et al. / International Journal of Heat and Mass Transfer 131 (2019) 1032–1051
_ cat stands for the catalyst density, in [g m3], ast and bst
In Eq. (13), w
stand for the steam reforming reaction orders towards methane and
steam, respectively. The species concentrations in the reaction site
are expressed by their partial pressures, pCH4 and pH2 O , in [Pa]. The
reforming rate constant kst , in [mol s1 g1 Paðast þbst Þ ], is described
by Arrhenius constant Ast , in [mol s1 g1 Paðast þbst Þ ], activation
energy Eact , in [J mol1], universal gas constant R, in [J kg1 K1],
þ
and temperature T, in [K]. In Eqs. (14) and (16), ksh and ksh are the
rate constants of the forward/backward water-gas shift reaction,
in [mol s1 m3 Pa2], DG0sh is the change in Gibbs free energy, in
[J mol1].
Anode microstructure plays an important role in fuel cell operation, especially during the internal reforming process. It provides
chemical and electrochemical reaction sites, as well as pathways
for the migration of the reaction components. In this study the
Ni/YSZ cermet anode is used, in which nickel serves as a reforming
reaction catalyst. The effectiveness of this reaction depends
strongly on the accessibility of the reaction sites to the methane
penetrating the anode through the pores; in other words, nickel
needs to be well exposed to the flowing methane and steam. Its
exposure can be estimated by two parameters, which account for
the local steam reforming reaction rate: the SNi-Pore – nickel-pore
Ni
contact area density, in [m2cat m3 ], and the A – Ni/YSZ specific
surface area, in [m2cat g1 ]. The methodology for the evaluation of
these parameters is described in Section 4.
2.4. Boundary and initial conditions
The boundary conditions, employed in the model are shown
schematically in Fig. 2. The anodic reactions are introduced
through the dedicated non-local boundary conditions imposed at
the channel/anode interface. The respective formulae for the
boundary conditions are collected in Table 2. Their implementation
assumes firstly calculating the overall effects for the whole volume
of the anode V ano , in [m3], and the subsequent evaluation of the
fluxes exchanged by the interface Aano , in [m2].
The heat generated by the methane/steam reforming reaction in
Eq. (11) and Water-Gas Shift reaction in Eq. (12) is as follows [1]:
Q st ¼ DH0st Rst V ano ;
Q sh ¼
DH0sh Rsh V ano ;
DH0st
DH0sh
ð17Þ
ð18Þ
where
and
stand for the enthalpy change in the methane/
steam reforming reaction and with the water-gas shift reaction,
respectively, in [J mol1]. Hydrogen, the product of the reforming
process, is utilized in the exothermic oxidation reaction:
Fig. 2. Boundary conditions of the calculations’ domain.
H2 þ O2
2 ! H2 O þ 2e ;
1035
ð19Þ
which supplies the fuel cell with the electric charge. Heat from this
reaction is generated as follows:
Q form ¼
I
DHfH2 O ;
2F
ð20Þ
where DHfH2 O is the enthalpy change in the water formation reaction, in [J mol1]. The electric charge flows through the anode generating Joule’s heat, according to:
Q Joule ¼ I2 Rcell ;
ð21Þ
where I stands for the overall current collected from the unit cell, in
[A], which is calculated as the current density iden , in [A cm2], multiplied by the cell area, Rcell stands for the unit cell resistance, in [X],
and F denotes Faraday’s constant, in [C mol1]. The total heat flux
Q total , in [W m2], through the anode surface is calculated as
follows:
1 Q_ total ¼
Q þ Q st þ Q form þ Q Joule ;
Aano st
ð22Þ
and applied in accordance with Table 2.
The mass fluxes in the model are calculated in the same manner
as the heat fluxes. Mass is produced/consumed in both chemical
and electrochemical reactions only in the anode volume. The
methane and steam entering the fuel channel take part in the
reforming process, during which hydrogen, carbon monoxide and
carbon dioxide are produced. Simulatenously to the reforming process, go the electrochemical reactions. The total sources/sinks of
species are shown in Table 1. For example, the hydrogen generation flux due to the reforming process in the anode volume can
be calculated as follows:
J ref;H2 ¼ M H2 ð3Rst þ Rsh ÞV ano :
ð23Þ
Additionally, hydrogen is electrochemically oxidized, according to
Eq. (19). As a result, high temperature steam is produced. The rate
of the respective species production in the electrochemical reactions can be calculated according to Faraday’s law. For example,
for H2 one obtains:
J F;H2 ¼
I
M H2 O :
2F
ð24Þ
In the proposed model, hydrogen and carbon monoxide are utilized
as fuels. The total species flux density through the anode surface is
as follows:
i
1 X
J_ total;i ¼
J ref;i þ J F;i ;
Aano
ð25Þ
where i ¼ hH2 ; H2 O; CO; CO2 ; CH4 i:
At the external boundaries of the domain, constant temperature
and predefined mass fluxes are imposed, as presented in Fig. 2.
On the inlet constant values of gas composition, fluid temperature and volumetric flow are applied. In this paper, the model sensitivity analysis is carried out inter alia, by introducing various
Dirichlet boundary conditions to the domain, depending on the
case (species or temperature). The upper boundary is assumed to
be a gas-tight, adiabatic partition, which isolates the anode channel, so that no heat or mass is transferred through it. On the outlet
a zero-gradient assumption is made both for the temperature and
mass fraction gradients [25]. On the lower wall (the anode surface)
steam reforming reaction, Q st , water-gas shift reaction, Q sh , Joule’s
heat, Q Joule and electrochemical reactions, Q form , thermal effects are
introduced, together with the mass flux of species i, J i . The inlet,
outlet and upper boundary conditions are time independent in
the model. The anode surface boundary condition is calculated
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M. Chalusiak et al. / International Journal of Heat and Mass Transfer 131 (2019) 1032–1051
Table 1
Mass sources/sinks in the model.
Species
Mass source/sink due to methane
steam reforming
Mass source/sink due to the water-gas
shift reaction
Mass source/sink due to
electrochemical reactions
Total mass source/sink
2FI M H2
3Rst MH2 þ Rsh MH2 2FI M H2
2FI M CO
Rst M CO Rsh M CO 2FI MCO
MCO2
Rsh MCO2 +2FI M CO2
Rst M CH4
M H2 O
Rst M H2 O Rsh M H2 O +2FI M H2 O
H2
3Rst MH2
Rsh MH2
CO
Rst M CO
Rsh MCO
CO2
0
Rsh MCO2
CH4
H2 O
Rst M CH4
Rst M H2 O
I
2F
0
Rsh MH2 O
0
I
2F
Table 2
Boundary conditions.
Position
u
Inlet
@u
@x
@u
@x
Outlet
Table 3
Catalyst properties.
v
T
Yi
Type
Nickel content
Particle size
Specific surface area, Acat
¼0
v=0
T = Tinlet
Yi = Yinlet
i
Ni/YSZ
60%
0.85 lm
5.2 m2 g1
¼0
@v
@x
@T
@x
¼0
@T
@y
@T
@y
¼ Q total
k
@Y i
@x
@Y i
@y
@Y i
@y
¼0
Bottom
u=0
v=0
Top
u=0
v=0
_
¼0
¼0
_
¼ qJDi i
¼0
iteratively at every time step, based on the conditions from the
previous iteration.
Initially (t = 0 s), the mass fractions for hydrogen, methane,
steam, carbon monoxide, carbon dioxide for all studied cases are
0. Moreover, it is assumed that nitrogen fills the anode channel
completely at the initial time. The initial temperature of the gases
in the fuel channel is set as 1023 K.
3. Experimental analysis
Type
Bed height
Radius
Length
Stainless steel
1 mm
25.4 mm
450 mm
were placed in the experimental set-up as shown in Fig. 3 (marked
as T). All measurements presented in this paper were performed at
atmospheric pressure. The geometrical configuration of the reactor
is summarized in Table 4.
4. Relation between reforming reaction rate and microstructure
morphology
A reforming process experiment over the Ni/YSZ catalyst was
carried out to verify the utilized chemical reaction model. A schematic view of the experimental setup is shown in Fig. 3. A stainless
steel reformer was located in an electrical furnace, which can be
heated up to 800 C. High purity methane was the fuel used in
the experiment. It was supplied to the reformer via a flow controller and evaporator which was also used as a pre-heater. Water
was fed to the system with a pump. The gas composition after the
reforming process was analyzed by gas chromatography prior to
which the steam had been separated by cooling down the gas mixture to 2 C. The reforming reaction tube was filled by a nickel supported on the yttria stabilized zirconia. The catalyst material was
the industrial catalyst provided by AGC Seimi Chemical Co. Ltd.
The properties of the catalyst material are shown in Table 3.
Before feeding methane to the system, the catalyst material was
treated for 2 h at an elevated temperature of 600 C by a mixture of
1
nitrogen (supplied at a rate of 150 ml min ) and hydrogen
1
Table 4
Reactor properties.
(100 ml min ), to reduce NiO to metallic Ni. In order to avoid large
temperature gradients in the reformer and a cooling effect of entering fluid, the reformer was partially filled with Al2 O3 balls. To control the thermal conditions of the experiment, four thermocouples
In this study, the methane/steam reforming reaction rate on a
conventional Ni/YSZ anode is investigated under varying temperature and gas conditions. To derive the reforming kinetics, the powder catalyst was used. The original rate (see Eq. (13)) was derived
from a catalyst weight. However, recent studies by [19,18] using
Focused Ion Beam-Scanning Electron Microscopy (FIB-SEM) indicate, that the reaction rate constant (see Eq. (15)) depends on the
anode microstructure, as the underlying process takes place only
at the nickel-pore interface. Thus, a respective rate equation is
applicable only to a very specific anode. To overcome this obstacle,
a generalized model was introduced [18]. It takes into account two
parameters, the nickel-pore interface area density within the catalyst, SNi-Pore and Ni/YSZ specific surface area, ANi as a substitute for
_ cat . By modifying the reaction rate equation
the catalyst density w
(see Eq. (13)), one obtains a universal formula suitable for any
anode.
Microstructural morphology and its influence on the reforming
reaction rate was evaluated for a commercial state-of-the-art Ni/
YSZ anode provided by a leading SOFC stacks manufacturer,
SOLD-POWER S.p.A [26,22]. In order to examine the anode
Fig. 3. Schematic view on the experimental setup.
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M. Chalusiak et al. / International Journal of Heat and Mass Transfer 131 (2019) 1032–1051
microstructure, the 0.5 cm2 anode sample was extracted and
impregnated with epoxy resin under vacuum conditions (Marumoto Struers KK). This procedure is essential for pore region recognition, as the resin fills the pores completely and yields black spots
during SEM observation.
A FIB-SEM tomography (Focused Ion Beam Scanning Electron
Microscopy) was used to closely investigate the anode microstructure. The measurement procedure scheme is presented in Fig. 4a. A
”cut and see” cycle using FIB (cutting) and SEM (cross-section capturing) was repeated with nanoscale resolution to obtain a series of
sample cross-sections. The images were then manually segmented
in AVIZO software, where nickel, YSZ and pores were identified,
and the 3D reconstructed microstructure was generated (see
Fig. 4b) [26,22].
In-house codes were employed to calculate parameters, which
describe the fabricated the anode microstructure (see Table 5).
Based on the values of surface area of species, the nickel-pore
interface area density SNi-Pore was calculated, according to the
formula:
SNi-Pore ¼
1 Ni
S þ SPore SYSZ :
2
ð26Þ
Hence, the methane/steam reforming reaction rate for a unit Ni/YSZ
reaction area can be formulated, as follows:
SNi-Pore
Eact
st
Rst ¼ cat Ast exp pbHst2 O :
paCH
4
RT
A
ð27Þ
5. Numerical model
The system of the governing differential equations given in Section 2 can be presented in the following general linearized form:
wt
@/
@/
@/
@
@/
@
@/
þ wx
þ wy
¼
h
þ
h
:
@t
@x
@y @x
@x
@y
@y
ð28Þ
The values of the respective coefficients from Eq. (28) are presented in Table 6.
The numerical solution to the system will be sought in the
framework of the Finite Volume Method [23]. The spatial discretization is based on the introduction of the control volumes over
which the transport equations are integrated (details of the discretization scheme can be found in [23,25]). As for the approximation of the temporal derivative, we accept the formula:
@/ðiþ1Þ
/ðiþ1Þ /ðiÞ @/ðiÞ
¼2
;
@t
Dt
@t
ð29Þ
where /ðiÞ and /ðiþ1Þ represent the pertinent depending variables at
time t and t þ Dt, respectively. It provides a parabolic distribution of
/ between two consecutive time instants [34]. The boundary condi-
Table 5
Microstructure parameters.
Connectivity [–]
Grain size [lm]
Tortuosity [–]
Specific surface area, Sa [lm2 lm3]
Nickel
YSZ
Pore
0.82
0.94
7.16
7.05
0.98
0.58
4.45
11.64
0.99
1.09
2.11
8.63
Table 6
Coefficients in Eq. (28) for the discretized governing equations.
Eq.
/
wt
wx
wy
h
(1)
(2)
T
Yi
qCP
q
qCPu
qu
qCPv
qv
k
qDi
tions of the second type are introduced by assuming a second order
approximation of the dependent variables in the proximity of the
respective boundaries. Regular spatial and temporal meshes are
used in the computations. The linear system of algebraic equations
resulting from discretization of (28) is solved by means of the
Gauss-Seidel method [25].
6. Results
6.1. Verification of the proposed models
6.1.1. Verification using benchmarking function
As the proposed algorithm of the solution relies on consecutive
solving the linearized PDEs in the form (28), we need to identify
the accuracy of the computations provided by our solver for each
of such steps. To this end we shall introduce an auxiliary problem
with a known analytical solution which preserves all the essential
features of the original formulation.
First, remark that the governing Eqs. (1) and (2) in their linearized versions can be represented by the following general
formula:
@
ða1 /Þ þ $ ða2 u/Þ ¼ $ ða3 $/Þ þ S;
@t
ð30Þ
where / ¼ /ðt; x; yÞ stands for the respective dependent variable
(temperature or mass fraction of species), coefficients ai refer to
the material properties, while S is a source term. In the original
problem S is identically zero, however we assume here that it can
take an arbitrary value.
Now, let the solution to the PDE (30) be of the form:
^ yÞ;
/ðt; x; yÞ ¼ ect /ðx;
provided that:
Fig. 4. (On the left) The procedure scheme for the FIB-SEM tomography, (on the right) the 3D reconstructed sample using AVIZO.
ð31Þ
1038
M. Chalusiak et al. / International Journal of Heat and Mass Transfer 131 (2019) 1032–1051
Sðt; x; yÞ ¼ ect ^Sðx; yÞ;
ð32Þ
where c is some parameter. In the framework of this self-similar
formulation, the original initial boundary value problem (described
by PDE (30) together with the pertinent initial and boundary conditions) can be reduced to a boundary value problem, defined by the
following PDE:
ca1 /^ þ $ a2 u/^ ¼ $ a3 $/^ þ ^S;
ð33Þ
and the respective boundary conditions. Now, having an analytical
solution to the Eq. (33) we can immediately extend it to its timedependent counterpart by relations (31) and (32). In this way, a full
time-dependent analytical benchmark solution is obtained. It can
be used to verify the accuracy and efficiency of the computations.
In the following analysis a dedicated strategy to built the
benchmark solution will be adopted. Namely, we shall assume
^ B ðx; yÞ, which presome form of potential solution to the Eq. (33), /
serves all the essential features and boundary conditions of the
^ B ðx; yÞ to (33) and simoriginal problem. Then, on substitution of /
ple algebra, we will arrive at the expression for the source term, ^
S,
for which the equation is identically satisfied. In other words,
instead of looking for some example of solution for (33), we accept
a predefined benchmark function and then we find the form of the
governing PDE for which the latter is fulfilled.
Let the benchmark solution be:
^ B ðx; yÞ ¼ b þ exp b ð1 x=LÞ2 þ b ð1 y=HÞ2 ;
/
1
2
3
ð34Þ
where L and H are the constants defining the spatial extent of the
domain (in x and y directions, correspondingly), while
bj ðj ¼ 1; 2; 3Þ are some parameters to be taken as convenient. Note
that, expression (34) preserves the respective natural boundary
conditions (see Table 2) at x ¼ L and y ¼ H, while the normal derivative (and thus the flux) at y ¼ 0 is different from zero. Moreover, the
^ B , can be controlled
spatial distribution of the benchmark function, /
by the proper handling of the coefficients bj and powers of the x and
y related terms (provided however that the respective powers are
greater than 2). The highly non-linear form of the function (34)
makes it a demanding benchmark for the computational algorithm,
^ B does not directly comply with most of
as the spatial behaviour of /
the standard discretization schemes based on the polynomial
representation.
The corresponding source term is defined as:
^
^
^ þ $ a2 u/ $ a3 $/ ;
Sðx; yÞ ¼ SB ðx; yÞ ¼ ca1 /
B
B
B
ð35Þ
where the velocity function u is taken from Section 2.2 [10].
The respective parameters of the benchmark solution used in
our analysis are collected in Table 7.
As a comprehensive investigation of the solver performance
goes beyond the scope of this paper, we will identify and signalize
only the most important trends. Let us start by presenting the spatial distributions of errors obtained for two different densities of a
regular mesh (we assume that the number of points, N, is the same
in both directions, x and y). In Fig. 5 we show the relative error of
^ for t ¼ 50 s. Two different mesh densities
solution, d/,
(N ¼ f20; 70g) were used in the computations for two different
Table 7
Benchmark parameters.
c
a1
a2
a3
b1
b2
b3
L
H
0.1
150
150
2
1
0.1
0.1
1
1
magnitudes of the time step, Dt (Dt ¼ f0:01; 0:1g s). As can be seen
in the figure, the error level is very low regardless of the computational parameters in use (it does not exceed the value of 6 105 in
^ is observed. The location of
any case). No sharp magnification of d/
the maximal error remains the same (x ¼ L; y ¼ H) for all considered variants. Furthermore, the error distributions are very similar
^ are different. It shows in the
to each other, even if the levels of d/
figure that, when comparing the results obtained for the coarse
spatial mesh, no significant improvement of the solution accuracy
is observed while reducing the time step. However, the situation is
different for the finer mesh. Here, by taking Dt ¼ 0:01 s instead
Dt ¼ 0:1 s one decreases the error level by one order of magnitude.
Thus, it is very important for the accuracy and efficiency of the
computations to keep the proper balance between the density of
spatial and temporal meshing. In other words, the overall error
of solution depends on the interplay between the component
errors introduced by the approximations of the temporal and the
spatial derivatives. If, for some value of N, the latter is an accuracy
limiting factor, then no improvement of the solution can be
achieved by taking smaller time steps (and conversely, if the time
step is sufficiently large, no error reduction can be obtained by
finer spatial meshing). This trend can be also observed in Fig. 6,
where the temporal evolutions of maximal and mean relative
^ B are shown. Notably, although the overall error is
errors of /
always greater for the coarse spatial meshing, the stabilization
(saturation level) is achieved here faster. The rate of error convergence for different values of N is depicted in Fig. 7, were the mean
errors are shown at time t ¼ 50s. The errors can be approximated
by the exponential-type expression to give:
for Dt ¼ 0:1 s
^ ðmeanÞ ¼ 6:32 104 þ 0:0112 expð0:08532NÞ;
d/
ð36Þ
for Dt ¼ 0:01 s
^ ðmeanÞ ¼ 1:34 104 þ 0:0102 expð0:08384NÞ:
d/
ð37Þ
The above expressions provide estimations for the maximal
achievable accuracy which yields: 6:32 104 % for Dt ¼ 0:1 s and
1:34 104 % for Dt ¼ 0:01 s. We would like to emphasize, that a
solution of a very good quality (sufficient for any engineering
application) can be obtained even for larger time steps.
The presented analysis proves that our solver is an efficient and
very accurate computational tool. Thus, the credibility of numerical results described in the following subsections can be fully
substantiated.
6.1.2. Chemical reaction model verification
The chemical reaction model (see Eq. (13)) was verified in this
section. It was used to predict the outlet gas composition of the
reformer (reactor) from Fig. 3. The methane conversion rate xcr
defined in Eq. (38) was chosen as a testing parameter, as it requires
calculating the outlet content of all spiecies [33]:
xcr ¼
xCO þ xCO2
;
xCO þ xCO2 þ xCH4
ð38Þ
where xj denotes molar fractions of species j ¼ hCO; CO2 ; CH4 i at the
reformer outlet. The calculated conversion rates were compared to
the corresponding experimental measurements from [33]. The
results are presented in Fig. 8. The dashed line represents the perfect correlation between the measured and the calculated conversion rates. A good agreement observed in Fig. 8 implies, that the
chemical reaction model was elaborated correctly and accurately
predicts the composition at the outlet of the reformer.
M. Chalusiak et al. / International Journal of Heat and Mass Transfer 131 (2019) 1032–1051
1039
Fig. 5. Solver benchmark verification results tor time t = 50 s. The influence of mesh density and time step: (a) N = 20, Dt = 0.01 s; (b) N = 20, Dt = 0.1 s; (c) N = 70, Dt = 0.01 s;
(d) N = 70, Dt = 0.1 s.
6.2. Numerical simulation results
6.3. Reforming reaction rate
In this section, a parametric study is carried out to simulate the
performance of Direct Internal Reforming Solid Oxide Fuel Cell. We
employed the mathematical and numerical models developed in
Sections 2 and 5 to analyze the performance of DIR-SOFC.
Furthermore, we selected the most important parameters for
DIR-SOFC control to be investigated, namely: the nickel-pore inter-
The reforming reaction rate Rst is a parameter, which introduces
the microstructural properties to the model and prominently
affects the DIR-SOFC operation. Its complete form in Eq. (27)
depends on a number of parameters and influences the distributions of all species and temperature, which makes it an informative
parameter to trace. Fig. 9 depicts a series of Rst plots after stabilization (20 s), showing how the investigated parameters (nickel-to-
face area density SNi-Pore , the collected current density iden , the
inlet
and
methane inlet mass fraction Y inlet
CH4 , the inlet temperature T
inlet
the volumetric flow of inlet gas mixture N_
. The values of these
parameters as well as some general parameters, which describe
the model, are summarized in Table 8.
Calculations are carried out for the time-independent boundary
conditions (see Table 8) over the time interval sufficient to reach
the steady state. The system response was analyzed on the basis
of the temperature and mass fraction distributions of the respective species. Also, the averaged-by-height values of temperature
and mass fractions are collected for the conducted studies. Numerical simulations are made for the 8 cm per 0.1 cm rectangular fuel
channel for which a 100 30 uniform mesh was applied. In order
to preserve the qualitative character of the elaborated model, the
influences of the aforementioned parameters are studied in the
core of the fluid flow and reflect the behaviour of the gases and
the temperature in the DIR-SOFCs longitudinal direction.
pore surface density SNi-Pore , steam-to-carbon ratio SC and methane
mass fraction at inlet Y inlet
CH4 ) influenced the reforming reaction rate,
directly related to mass and heat source terms. A dominant impact
of SNi-Pore can be observed in Fig. 9. An almost proportional increase
in the reaction rate towards increasing nickel exposure in Fig. 9a is
in agreement with the Eqs. (13) and (27), describing the reforming
reaction rate, that account for the anode microstructure. In Eq.
(13), the reaction order towards steam is low, thus negligible
changes on Rst in Fig. 9b can be observed, when subjected to different SC ratios. For a similar reason, a high order of the reaction
towards methane makes the Rst sensitive to the methane content
changes, what was shown in Fig. 9c. Apart from the sensitivity
analysis conducted for the stabilized operation of DIR-SOFC, a transient behaviour of the reforming reaction rate is presented in
Fig. 10. One can notice that after the initial state, in which the reaction is completely suppressed due to the absence of methane and
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M. Chalusiak et al. / International Journal of Heat and Mass Transfer 131 (2019) 1032–1051
Fig. 6. Maximal and mean relative errors of solution for: (a) and (c) – Dt = 0.1 s, (b) and (d) – Dt = 0.01 s.
Table 8
Parameters used in simulation.
Fig. 7. Mesh sensitivity analysis, mean error dependence on mesh density.
Parameter
Unit
Values
Fuel channel length
Fuel channel width
Initial H2 content
Initial CO content
Initial CO2 content
Initial N2 content
Initial temperature
Steam-to-carbon ratio SC
m
m
–
–
–
–
K
–
–
0.08
0.001
0
0
0
1
1023/1073
4.0
Inlet carbon monoxide Y inlet
CO
Inlet steam Y inlet
H2 O
–
SC Y inlet
CH4
0
Inlet carbon dioxide Y inlet
CO2
–
0
hydrogen Y inlet
H2
nitrogen Y inlet
N2
–
0
Inlet
Inlet
Ni-Pore
Ni-Pore interface S
Current density iden
Inlet CH4 content Y inlet
CH4
Inlet temperature T inlet
Inlet gas flux N_
–
inlet
1 Y inlet
H2 O Y CH4
2
3
m m
1 106/2.5 106/5.0 106
A cm2
–
0.15/0.35/0.55
0.05/0.1/0.15
K
1023/1073/1123
L min1
0.1/0.5/1/1.5
6.4. Transient analysis of the mass fractions and temperature
distributions
Fig. 8. Correlation plots of the measured against the calculated fractional conversions of methane during the experiment.
steam, Rst grows rapidly as the gases fill the anode channel. After
stabilization it gets a descending profile towards the outlet.
These figures give a closer look at the intensity of the reforming
reaction rate, revealing its scale and providing information for the
interpretation of the further outcomes.
This section presents the computation results of the transient
evolution of the height-averaged mass fractions and temperature
profiles from the initial state, up to the steady state. These profiles
are presented in Fig. 11, in which temperature, hydrogen, methane
and steam are included. Within the stabilization time, rapid
changes occur in the anode channel of DIR-SOFC, as the reforming
reaction is progressing. The temperature equalization at the beginning of the simulation (Fig. 11a) finishes quickly after about 4s as
the endothermic reforming reaction starts to dominate the heat
balance. The temperature drops drastically causing a large temperature gradient along the cell after stabilization. In Fig. 11b, after the
initial absence of hydrogen, it gradually fills the channel, however
the pace of the increase is bigger when the temperature reaches its
M. Chalusiak et al. / International Journal of Heat and Mass Transfer 131 (2019) 1032–1051
1041
highest value. It is worth noting that the transient trend in hydrogen appearing in the channel, according to Eq. (23), is directly
related to the transient trend of Rst and in fact follows it tightly.
However, due to the fact that the reforming reaction is considered
to be slow [1,20,21,27,33,35], it does not keep up with producing
hydrogen what results in a severe deficiency and the cell undergoes a fuel starvation in this region. Fig. 11c and d show the
methane and steam average distributions, respectively. One can
notice, that due to the dominant convective term, methane and
steam spread the fastest in the channel and affect the reforming
reaction rate according to Eq. (27). It is worth noting, that the profiles on all figures reach the steady state at similar time, this is
around 20 s.
6.5. The effect of the anode microstructure
Fig. 9. Methane steam reforming reaction rate profiles as a result of applying
various: (a) SNi-Pore , (b) SC, (c) Y inlet
CH4 values.
The nickel-pore interface area SNi-Pore is one of the parameters
which describe the anode mictrostructure. The area of the nickel
exposed to the gases penetrating the anode is a measure of its ability to reform methane into hydrogen. In this section, the sensitivity
analysis of the aforementioned parameter was carried out to
address the problem of the anode microstructure impact on the
fuel starvation and anode cooling effect. By extracting only one
parameter to be changed, a deeper analysis of the numerical solution and finding the key factors is possible. Moreover, this section
presents the transient distributions of species and temperature
inside the DIR-SOFC with the extracted crucial microstructural
parameter, Nickel-to-Pore contact area density SNi-Pore , which
determines the rate of the reforming reaction in the cell. Figs. 12
and 13 present the results of the numerical simulations for the
influence of SNi-Pore on the hydrogen and methane distributions in
the anode channel, respectively. The first phenomenon to be
observed is a severe hydrogen starvation shown in Fig. 12a. Only
a negligible amount of hydrogen is produced on the anode exhibiting 106 m2 m3 of the nickel-pore interface area. This figure shows
a state, when hydrogen consumption outbalances its production
causing a prominent deficiency of fuel in the fuel channel. An
anode with over twice as much reforming reaction sites,
Fig. 10. Methane steam reforming reaction rate profiles in a function of time.
inlet
_
Conditions: SNi-Pore = 2.5e6 m2 m3, SC = 4.0, Y inlet
= 1073 K.
CH4 = 0.1, N = 0.5 L/min, T
2:5 106 m2 m3 , presented in Fig. 12b produces enough hydrogen
to prevent the starvation state in almost the entire channel. However, due to the forced convection and the fact that the reforming
reaction is slow, even the anode with improved microstructural
properties, shown in Fig. 12b as well as in Fig. 12c, is not able to
Fig. 11. Height-averaged profiles of the respective parameters’ distributions throughout time: (a) hydrogen, (b) temperature, (c) methane, (d) steam; simulation conditions:
inlet
SNi-Pore = 5.0e6, N_ = 0.1 L/min, SC = 4.0, iden=0.15 A cm2, Y inlet
= 1073 K.
CH4 = 0.1, T
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M. Chalusiak et al. / International Journal of Heat and Mass Transfer 131 (2019) 1032–1051
Fig. 12. Distribution of hydrogen for the different nickel exposure coefficients in the anode microstructure SNi-Pore : (a) SNi-Pore = 1e6 m2 m3, (b) SNi-Pore = 2.5e6 m2 m3, (c)
inlet
inlet
= 1073 K.
SNi-Pore = 5.0e6 m2 m3. Conditions: N_ = 0.5 L min1, SC = 4.0, iden = 0.15 A cm2, Y inlet
CH4 = 0.1, Y H2 = 0, T
Fig. 13. Distribution of methane for the different nickel exposure coefficients in theanode microstructure SNi-Pore : (a) SNi-Pore = 1e6 m2 m3, (b) SNi-Pore = 2.5e6 m2 m3, (c)
inlet
inlet
SNi-Pore = 5.0e6 m2 m3. Conditions: N_ = 0.5 L min1, SC = 4.0, iden = 0.15 A cm2, Y inlet
= 1073 K.
CH4 = 0.1, Y H2 = 0, T
completely eliminate the hydrogen insufficiency from the fuel
channel. One can notice, that despite producing sufficient amounts
of hydrogen at the outlet section, the inlet section still suffers from
hydrogen starvation. The results presented in this figure suggest
utilizing certain pre-reforming methods to provide any hydrogen
to the upstream section of the cell.
The hydrogen production process in the DIR-SOFC is always
related to methane consumption in the reforming process. Fig. 13
presents the methane distribution in the anode channel for the
various nickel exposures in the anode’s microstructure. It can be
observed, that methane consumption rises for larger SNi-Pore values
and the region of methane depletion responds to the region of the
intensified hydrogen production. Despite higher methane consumption it is worth noticing in Fig. 13c, that only a small amount
of methane is converted, about 13% at the outlet, while the rest can
still be utilized.
The methane steam reforming is a strongly endothermic process and the amount of heat transferred in this process depends
mostly on the reforming reaction rate Rst . Increasing the rate of
the reforming by increasing the SNi-Pore thus results in large heat
consumption, what is reported in Fig. 14. One can observe a drastic
change in temperature distribution with the SNi-Pore . The temperature for the SNi-Pore ¼ 1 106 m2 m3 in Fig. 14a rises for about
10 K while for the SNi-Pore ¼ 5 106 m2 m3 in Fig. 14c it drops for
Fig. 14. Distribution of temperature for the different nickel exposure coefficients in the anode microstructure SNi-Pore : (a) SNi-Pore = 1e6 m2 m3, (b) SNi-Pore = 2.5e6 m2 m3, (c)
inlet
inlet
SNi-Pore = 5.0e6 m2 m3. Conditions: N_ = 0.5 L min1, SC = 4.0, iden = 0.15 A cm2, Y inlet
= 1073 K.
CH4 = 0.1, Y H2 = 0, T
M. Chalusiak et al. / International Journal of Heat and Mass Transfer 131 (2019) 1032–1051
about 5 K causing an anode cooling effect. In order to sustain as
high temperature as possible, other performance control mechanisms need to be used. However, appropriate microstructural
parameters of the used anode, as presented in Fig. 14b, can provide
a desired uniform temperature distribution and the reduction of
thermal stresses in the cell.
The effect of an enhanced nickel exposure in the anode
microstructure on DIR-SOFC performance can be also seen in
Fig. 15, where the temperature and species distributions are averaged over the cell height. This figure presents their trends in terms
of changing microstructural parameters of the anode. The numerical simulations for various SNi-Pore values confirm the existence of
fuel starvation regions in the fuel channel. The most vulnerable
turns out to be the inlet section, where no hydrogen is supplied
(see Section 2.4) and not enough methane is converted in the
reforming process.
To obtain a complete study of the behaviour of DIR-SOFC
towards the microstructural changes, transient plots of the studied
variables are required. An investigation of the distributions especially near the anode surface is essential and therefore in Figs. 16
and 17, the distributions of mass fraction and temperature as a
function of time was presented in the vicinity of the anode surface.
Fig. 16 presents the transient distributions of temperature, hydrogen and methane for two simulated SNi-Pore values: 1 106 m2 m3
and 2:5 106 m2 m3 . A rapid temperature growth during the first
4 s is in agreement with the results from Fig. 11a, however the
lower SNi-Pore applied here give two different trends for further temperature profile evolution. In Fig. 16a there are fewer reaction sites,
so that the elctrochemical reaction balances the reforming reaction
and the cell temperature stabilizes quickly. It also exhibits a rising
profile towards the channel outlet, which is consistent with a
descending profile of the reforming reaction rate from Fig. 9a. In
Fig. 16b, in turn, there is enough reforming reaction sites for the
chemical reaction to outbalance the exothermic electrochemical
reactions, and after 4s the temperature drops, resulting in a more
uniform temperature distribution when a steady state is reached.
Fig. 16c and d present the results of hydrogen transient distribution when subjected to the analogical conditions as for Fig. 16a
1043
and b. One can observe, that both distributions exhibit similar
trends in time with a distinguishing feature of the enhanced production of hydrogen after the initiatory temperature equalization.
What is more, an increased temperature does not directly influence
the hydrogen production but the heat can only be consumed in the
reforming process, which is more intense when the higher SNi-Pore
are imposed. As a result, there is an almost two times lower hydrogen production in Fig. 16c than in Fig. 16d and it also can be
observed in Fig. 12b. A close relation between methane distribution and the reforming reaction rate, can be observed in Fig. 16e
and f. As a result of the reforming reaction rate towards the
methane close to 1 (see Eq. (13)), the methane transient distribution follows the Rst profile. What is more, due to proportionality
between the Rst and the SNi-Pore , the increasing nickel exposure
causes adequate methane consumption increase.
Due to the intense changes in the temperature transient distribution observed in Fig. 16a and b a closer analysis of the phenomena inside an anode channel was carried out and shown in Fig. 17.
In order to eliminate the impact of the temperature difference
between the initial and inlet conditions, a similar simulation was
made for identical temperatures of 1073 K and the initial time
and inlet. A fine contribution of nickel exposure SNi-Pore on the
anode cooling effect can be observed. An analysis of this Figure indicates the existence of the optimal microstructure, for
which a uniform distribution in a steady state can be achieved
and the initiatory temperature rise can be suppressed.
6.6. The effect of current density
The reforming process in the DIR-SOFC aims to provide fuel for
the cell, it is then responsible for hydrogen and carbon monoxide
production. These species are then consumed in order to satisfy
the external load applied to the cell, which in this model is represented by current density iden . In the developed model, the current
density is assumed constant along the cell, which reflects the perfect conditions for the charge transport. Fig. 18 presents the results
of the numerical calculations for the different current density values, ranging from 0.0 A cm2, which simulates the situation with
Fig. 15. Height-averaged distributions of species and temperature for the different nickel exposure coefficient in anode microstructure SNi-Pore : (a) hydrogen, (b) temperature,
inlet
inlet
(c) methane, (d) carbon monoxide, (e) carbon dioxide. Conditions: N_ = 0.5 L min1, SC = 4.0, iden = 0.15 A cm2, Y inlet
= 1073 K.
CH4 = 0.1, Y H2 = 0, T
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M. Chalusiak et al. / International Journal of Heat and Mass Transfer 131 (2019) 1032–1051
Fig. 16. Transient distributions of temperature (first row), hydrogen (second row) and methane (third row) in the vicinity of the anode for different SNi-Pore : (a), (c), (e)
inlet
inlet
SNi-Pore = 1e6 m2 m3, (b), (d), (f) SNi-Pore = 2.5e6 m2 m3. Conditions: N_ = 0.1 L min1, SC = 4.0, iden = 0.15 A cm2, Y inlet
= 1073 K, T initial = 1023 K.
CH4 = 0.2, Y H2 = 0, T
no load, up to 0.55 A cm2. It shows a prominent impact of iden on
the temperature distribution. One can observe an intensive cooling
in Fig. 18a, which suggests a considerable mismatch between the
heating and cooling processes in the cell, as a result of which a
large temperature drop is reported. Nevertheless, increasing iden
significantly intensifies Joule’s heat generation according to Eq.
(21) as well as the heat released during water formation, according
to Eq. (20). As a result, in Fig. 18 as well as in Fig. 19a, a considerable temperature rise is observed near the cell’s outlet, especially
at the beginning of the simulation time. The heat generation
increase following the enhanced iden increases the temperature in
the vicinity of the anode so much that the hydrogen production
due to the reforming process almost outbalances its enlarged consumption, what can be observed in Fig. 19b, in which the amount
of hydrogen in the fuel channel decreases just slightly, for ca. 30%.
None of the studied cases in this section, however, eliminates the
starvation state at the inlet. Fig. 19b also implies, that supplying
no hydrogen to the cell by the inlet may cause its starvation as
the reforming process does not keep up with producing it. The
result from Fig. 18a in particular shows a crucial insight into the
transient behaviour of DIR-SOFC and reveals a large temperature
gradient along the cell after 2s, which diminishes in time.
M. Chalusiak et al. / International Journal of Heat and Mass Transfer 131 (2019) 1032–1051
1045
Fig. 17. Transient distributions of gas temperature (identical initial and inlet boundary conditions), in the vicinity of the anode for different SNi-Pore : (a) SNi-Pore = 1e6 m2 m3,
inlet
inlet
(b) SNi-Pore = 2.5e6 m2 m3, (c) SNi-Pore = 5.0e6 m2 m3. Conditions: N_ = 0.1 L min1, SC = 4.0, iden = 0.15 A cm2, Y inlet
= 1073 K, T initial = 1073 K.
CH4 = 0.1, Y H2 = 0, T
Fig. 18. Distribution of temperature for the different collected current densities iden : (a) iden = 0.0 A cm2, (b) iden = 0.35 A cm2, (c) iden = 0.55 A cm2. Conditions:
inlet
inlet
N_ = 0.5 L min1, SC = 4.0, SNi-Pore = 1e6 m2 m3, Y inlet
= 1073 K.
CH4 = 0.1, Y H2 = 0, T
6.7. The effect of methane inlet content
Methane is directly converted into hydrogen and carbon
monoxide in the reforming process and additionally is a major
component of the reforming reaction rate, given by Eq. (27).
Adjusting inlet composition proved to be an effective way to control SOFC systems performance. The numerical results for the different methane inlet content Y inlet
CH4 are presented in Figs. 20 and
21. In our model methane is not electrochemically oxidized, but
chemically converted to hydrogen in a strongly endothermic reaction. The changes in methane concentration directly influence the
reforming reaction rate which grows together with methane content Y CH4 at the inlet. This explains the significant temperature
drop accompanied by a boosted hydrogen production when more
methane is supplied, which can be observed in Figs. 20 and 21.
Moreover, a strong anode cooling effect is observed in Fig. 20c,
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M. Chalusiak et al. / International Journal of Heat and Mass Transfer 131 (2019) 1032–1051
methane inlet content Y inlet
CH4 increase can be observed in Fig. 22,
where height-averaged plots of temperature, hydrogen, as well as
carbon monoxide and carbon dioxide are shown. An average temperature tends to drop when more methane is supplied, which is a
conformation of the results from previous figures. The amount of
all gases in the figure as a result of the reforming process tends
to rise with Y inlet
CH4 increase.
One can see in Fig. 21c, that despite the hydrogen insufficiency
at the inlet section, an appreciable amount of H2 is present at the
outlet and which can be further utilized. This suggests recuperating the outlet hydrogen and redirecting it back to the inlet to cover
the areas suffering from its absence.
6.8. The effect of inlet temperature
Fig. 19. Height-averaged distributions of species and temperature for the different
collected current densities iden : (a) temperature, (b) hydrogen. Conditions:
inlet
inlet
N_ = 0.5 L min1, SC = 4.0, SNi-Pore = 1e6 m2 m3, Y inlet
= 1073 K.
CH4 = 0.1, Y H2 = 0, T
while a contrary tendency to heat up the cell is observed when less
methane is supplied (Fig. 20a) as a result of a domination of electrochemical reactions in the cell. Similarly to Figs. 14b and 18b, a
very uniform distribution of temperature can be observed in
Fig. 20b, which suggests the optimal conditions for the operation
of the DIR-SOFC cell. In Fig. 21 it can be seen, that as a result of
the Rst growth, the hydrogen starvation region is diminished when
more methane is supplied to the cell. It is not, however, eliminated
and the hydrogen starvation region still occupies some part of the
channel volume. The overall trends as a response to various
Figs. 23 and 24 show the results of the numerical computations
for hydrogen and methane for the different inlet temperatures
T inlet , which varied from 1023 K to 1073 K. In Fig. 23 an increase
in hydrogen production can be observed while the inlet temperature grows. In fact, this is the result of the boosted methane conversion due to the higher temperature in the reaction site
according to Eq. (27), which can be observed in Fig. 24. It is worth
to notice, that for all cases, fuel starvation was not eliminated from
the fuel channel and the deficiency of hydrogen can still be
observed. However, hydrogen starvation is reduced prominently
when a gas mixture of a higher temperature is provided to the system. Moreover, increasing the inlet temperature by 50 K can intensify hydrogen production two times and rise methane conversion
by 5%, while an 100 K increase gives three times higher H2 production and a 13% higher CH4 conversion rate (see Fig. 25).
inlet
inlet
inlet
1
_
Fig. 20. Distribution of temperature for the different methane content at inlet Y inlet
, SC = 4.0,
CH4 : (a) Y CH4 = 0.05, (b) Y CH4 = 0.1, (c) Y CH4 = 0.15. Conditions: N = 0.5 L min
inlet
SNi-Pore = 1e6 m2 m3, iden = 0.15 A cm2, Y inlet
= 1073 K.
H2 = 0, T
inlet
inlet
inlet
1
_
Fig. 21. Distribution of hydrogen for the different methane content at inlet Y inlet
, SC = 4.0,
CH4 : (a) Y CH4 = 0.05, (b) Y CH4 = 0.1, (c) Y CH4 = 0.15. Conditions: N = 0.5 L min
inlet
SNi-Pore = 1e6 m2 m3, iden = 0.15 A cm2, Y inlet
= 1073 K.
H2 = 0, T
M. Chalusiak et al. / International Journal of Heat and Mass Transfer 131 (2019) 1032–1051
1047
Fig. 22. Height-averaged distributions of species and temperature for the different methane content at inlet Y inlet
CH4 : (a) temperature, (b) hydrogen, (c) carbin monoxide, (d)
inlet
carbon dioxide. Conditions: N_ = 0.5 L min1, SC = 4.0, SNi-Pore = 1e6 m2 m3, iden = 0.15 A cm2, Y inlet
= 1073 K.
H2 = 0, T
Fig. 23. Distribution of hydrogen for the different inlet temperature T inlet : (a) T inlet = 1023 K, (b) T inlet = 1073 K, (c) T inlet = 1123 K. Conditions: N_ = 0.5 L min1, SC = 4.0,
inlet
SNi-Pore = 1e6 m2 m3, iden = 0.15 A cm2, Y inlet
H2 = 0, Y CH4 = 0.1.
Fig. 24. Distribution of methane for the different inlet temperature T inlet : (a) T inlet = 1023 K, (b) T inlet = 1073 K, (c) T inlet = 1123 K. Conditions: N_ = 0.5 L min1, SC = 4.0,
inlet
SNi-Pore = 1e6 m2 m3, iden = 0.15 A cm2, Y inlet
H2 = 0, Y CH4 = 0.1.
Inlet temperature T inlet proves to be an effective means to control the DIR-SOFC work. Its increase has all of the positive results
on the distribution of species, increases the hydrogen content,
reducing hydrogen starvation at the inlet and increasing methane
conversion. Higher temperature thus enables larger loads and
when using hydrogen recuperation, it can completely eliminate
its insufficiency from the cell.
6.9. The effect of gas inlet velocity
Changing fluid flow velocity entering the cell is the basic control
strategy for chemical reactors, including SOFC [12,36,37]. Not only
can it influence the fluid velocity in the cell but also the temperature and the rate at which the reaction takes place inside the reactor. In this study DIR-SOFC system response was simulated for
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M. Chalusiak et al. / International Journal of Heat and Mass Transfer 131 (2019) 1032–1051
Fig. 25. Height-averaged distributions of species and temperature for the different inlet temperature T inlet : (a) hydrogen, (b) methane, (c) steam. Conditions: N_ = 0.5 L min1,
inlet
SC = 4.0, SNi-Pore = 1e6 m2 m3, iden = 0.15 A cm2, Y inlet
H2 = 0, Y CH4 = 0.1.
three inlet volumetric flows: 0.5 L min1, 1 L min1 and
1.5 L min1. A time-independent flow was assumed, according to
the profile from Section 6.9. The results of the conducted simulations for hydrogen and temperature were presented in Figs. 26
and 27, respectively. A remarkable growth of the hydrogen starvation region is noticed when the volumetric flow rises to
1.5 L min1, as it occupied more than 60% of the channel. Notably,
for the smallest flow the fuel starvation region occupied about 15%
of the channel and a significant amount of hydrogen could be produced, as it was not flushed (see Figs. 26a and 28a).
From the viewpoint of fluid flow influence, similar trend was
reported for the temperature distribution, presented in the twodimensional distribution in Figs. 27 and 28b as the heightaveraged plot. From the previous simulation results one can notice
that a cooling effect is dominant for these conditions (see Figs. 14c
or 15b). Hence, as shown in Fig. 27a, the slower the gas mixture
goes through the channel, the more time the reforming process
can take place and the stronger the cooling effect, as the gases
reach the inlet temperature instantly (see Section 2.4). However,
the larger the volumetric flow N_ inlet , the higher the temperature
Fig. 26. Distribution of hydrogen for the different inlet volumetric fluid flow N_ inlet : (a) N_ inlet = 0.5 L min1, (b) N_ inlet = 1.0 L min1, (c) N_ inlet = 1.5 L min1. Conditions: SC = 4.0,
inlet
inlet
SNi-Pore = 1e6 m2 m3, iden = 0.15 A cm2, Y inlet
= 1073 K.
H2 = 0, Y CH4 = 0.1, T
Fig. 27. Distribution of temperature for different inlet volumetric fluid flow N_ inlet : (a) N_ inlet = 0.5 L min1, (b) N_ inlet = 1.0 L min1, (c) N_ inlet = 1.5 L min1. Conditions: SC = 4.0,
inlet
inlet
SNi-Pore = 1e6 m2 m3, iden = 0.15 A cm2, Y inlet
= 1073 K.
H2 = 0, Y CH4 = 0.1, T
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M. Chalusiak et al. / International Journal of Heat and Mass Transfer 131 (2019) 1032–1051
Fig. 28. Height-averaged distributions of species and temperature for the different inlet volumetric fluid flow N_ inlet : (a) hydrogen, (b) temperature, (c) methane. Conditions:
inlet
inlet
SC = 4.0, SNi-Pore = 1e6 m2 m3, iden = 0.15 A cm2, Y inlet
= 1073 K.
H2 = 0, Y CH4 = 0.1, T
at the outlet and the more uniform distribution is obtained, what
inclines to a careful gas flow adjustment.
Foundation for Polish Science, co-funded by the European Union
under the European Regional Development Fund.
7. General conclusions
Appendix A. Thermodynamic and transport properties of gas
mixtures
The dynamic behaviour of Direct Internal Reforming Solid Oxide
Fuel Cell anode channel was analyzed using the microstructureoriented two-dimensional numerical model. The accuracy of the
numerical solver was thoroughly verified using the benchmarking
function and a conformal agreement at a level of 0.005% of relative
error was reported. As the aim of this work was to reflect the
impact of the Ni/YSZ anode microstructure, the reforming reaction
rate was modified to include the actual accessibility of the material
to reform methane to hydrogen. The FIB-SEM computer tomography was used to evaluate the essential quantitative parameters
of the anode’s microstructure. A parametric study of a complete
DIR-SOFC anode channel model was carried out to identify the system transient response to various boundary and operating conditions. The conducted parametric studies lead to the conclusion
that the convective nature of the flow in the channel, as well as a
non-equilibrium methane reforming, which occurs at a limited
rate, are the key factors contributing to the existence of the hydrogen starvation in the vicinity of the channel inlet. Modifying specific process parameters, such as reducing the volumetric fluid flow
or increasing the inlet gas temperature, matched to the anode’s
methane reforming effectiveness, contributes to the diminishing
of the hydrogen starvation state significantly. As it was indicated
in the paper, the elaborated Nickel-to-Pore area has a dominant
effect of the DIR-SOFC dynamic behaviour. Therefore, designing
anode microstructure to balance the contribution of chemical
and electrochemical reactions seems a promising strategy for
future. Yet, in order to eliminate the hydrogen starvation completely, a pre-reforming process or recuperating of the waste
hydrogen is required.
Density
The gas mixture density, applied in transport equations (Eq. (1),
(2)), is calculated with the ideal gas equation of state:
qmix ¼
ð39Þ
where Mmix stands for the molar mass of mixture and is a weighed
sum of molar mass species:
Mmix ¼
X
xi M i :
ð40Þ
i
Specific heat
The specific heat of gas mixture is estimated with empirical
relationship by Todd and Young [38], utilizing the least squares
method. Experimentally obtained, molar specific heat is temperature and composition dependent over the temperature interval
from 273 K to 1473 K. Its value for species i is calculated as follows:
C P;i ¼
6
X
ak rk ; where r ¼ T=1000
k ¼ 0 . . . 6:
ð41Þ
k¼0
The values of coefficients a0 ; . . . ; ak for all species are presented in
Table 9.
Thermal conductivity
In order to calculate the thermal conductivity, the Wassiljewa
equation was used [24]:
1
0
kmix
Conflict of interest
PMmix
;
RT
C
n B
X
B xi k i C
C;
B
¼
n
C
BX
A
i¼1 @
xj Aij
ð42Þ
j¼1
The authors declare no conflict of interest.
Acknowledgements
The presented research is a part of ‘Easy-to-assemble Stack
Type (EAST): Development of solid oxide fuel cell stack for innovation in polish energy sector’ project, carried out within the FIRST
TEAM programme (project number First TEAM/2016-1/3) of the
with the Mason and Saxena modification employed to compute Aij
[24]:
Aij ¼
12 14 2
Mi
1 þ lli
M
j
j
12
i
8 1þM
Mj
;
for i – j:
ð43Þ
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M. Chalusiak et al. / International Journal of Heat and Mass Transfer 131 (2019) 1032–1051
Table 9
Specific heat coefficients from Eq. (41) [38].
Species, i
a0
a1
a2
a3
a4
a5
a6
CH4
H2 O
CO
CO2
H2
N2
47.964
37.373
30.429
4.3669
21.157
29.027
178.59
41.205
8.1781
204.6
56.036
4.8987
712.55
146.01
5.206
471.33
150.55
38.040
1068.7
217.08
41.974
657.88
199.29
105.17
856.93
118.54
66.346
519.9
136.15
113.56
358.75
79.409
37.756
214.58
46.903
55.554
61.321
14.015
7.6539
35.992
6.4725
10.35
Table 10
Parameters of gas mixture components [24].
Species
Molar mass [g mol1]
Van der Vaals radius
CH4
H2 O
H2
CO
CO2
N2
16.04
18.015
2.016
28.01
44.01
28.155
3.758
2.641
2.827
3.69
3.941
3.798
The dynamic viscosity of species
follows:
r [Å]
li in Eq. (43), is calculated as
1
li ¼ 16:64
M 2i T
:
ð=kB Þr2
ð44Þ
The values of parameters
in Table 10.
r; and kB used in Eq. (44) are presented
Diffusion coefficient
The diffusion coefficient was calculated using Fuller’s method
[24], beginning with the binary diffusion coefficient estimated
with Eq. (45):
0:00143T 1:75
Dij ¼ pffiffiffiffiffiffiffih
i2 ;
P M ij ðRÞi1=3 þ ðRÞj1=3
ð45Þ
where Ri denotes the atomic diffusion coefficient of species i and M ij
are the harmonic means, which were obtained with Eq. (46):
M ij ¼ 2
1
Mi
þ
1
Mj
1
:
ð46Þ
Next, according to Blanc’s law, the diffusion coefficient of species
(denoted as i) in the gas mixture was calculated [24]:
Dmix;i ¼
n
X
Xi
D
ij
i¼1;i–j
!1
:
ð47Þ
Anode conductivity
The electrical conductivity of the anode depends mainly on the
electrical conductivity of nickel. In this paper the temperature
dependence was applied, according to the following formula [39]:
rel ¼
4:5 105
1200
:
exp T
T
ð48Þ
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