Thesis for degree: Licentiate of Engineering

SOFC modeling from micro to macroscale:
transport processes and chemical reactions
Hedvig Paradis
Thesis for the degree of Licentiate of Engineering, 2011
Division of Heat Transfer
Department of Energy Sciences
Faculty of Engineering (LTH)
Lund University
www.energy.lth.se

Copyright © Eva Hedvig Charlotta Paradis, 2011
Division of Heat Transfer
Department of Energy Sciences
Faculty of Engineering (LTH)
Lund University
Box 118, SE-221 00, Lund, Sweden
ISRN LUTMDN/TMHP-11/7075 – SE
ISSN 0282-1990

Abstract
The purpose of this work is to investigate the interaction between transport processes and
chemical reactions, with special emphasis on modeling mass transport by the Lattice
Boltzmann method (LBM) at microscale of the anode of a solid oxide fuel cell (SOFC). In
order to improve the performance of an SOFC, it is important to determine the microstructural
effect embedded within the physical and chemical processes, which usually are modeled
macroscopically. Without detailed knowledge of the transport processes and the chemical
reactions at microscale it can be difficult to capture their effect and to justify assumptions for
the macroscopic models with regard to the source terms and various properties in the porous
electrodes. The advantage of an anode-supported SOFC structure is that the thickness of the
electrolyte can be reduced, while still providing an internal reforming environment. For this
configuration with an enlarged anode, more detailed knowledge of the porous domain in
terms of the physical processes at microscale is called for.
In the first part of this study, the current literature on the modeling of transport processes and
chemical reactions mechanisms at microstructural scales is reviewed with special focus on the
LBM followed by a report on the emphasis to couple conventional CFD to LBM. In the
second part, two models are described. The first model is developed at microscale by LBM
for the anode of an SOFC in MATLAB. In the LB approach, the main point is to carefully
model the diffusion and convection at microscale in the porous region close to the threephase-boundary
(TPB). The porous structure is reconstructed from digital images, and
processed by Python. The second model is developed at macroscale for the whole unit cell.
For the macroscale model the kinetic model is evaluated at smaller scales to investigate if any
severe limiting effects on the heat and mass transfer occur.
LBM has been found to be an alternative method for modeling at microscale and can handle
complex geometries easily. However, there is still a need for a supercomputer to solve models
with several physical processes and components for a larger domain. The result of the
macroscale model shows that the three reaction rate models are fast and vary in magnitude.
The pre-exponential values, in relation to the partial pressures, and the activation energy
affect the reaction rate. The variation in amount of methane content and steam-to-fuel ratio
reveals that the composition needs a high inlet temperature to enable the reforming process
and to keep a constant current-density distribution. As experiments with the same chemical
compositions can be conducted on a cell or a reformer, the effect of the chosen kinetic model
on the heat and mass transfer was checked so that no severe limitation are caused on the
processes at microscale for an SOFC.
For future work, macroscale and microscale models will be connected for the design of a
multiscale model. Multiscale modeling will increase the understanding of detailed transport
phenomena and it will optimize the specific design and control of operating conditions. This
can offer crucial knowledge for SOFCs and the potential for a breakthrough in their
commercialization.
Keywords: mass transport, diffusion, microscale, porous media, kinetics, LBM, CFD, anode
multicomponent, MATLAB.
2

Populärvetenskaplig beskrivning på svenska
Bränsleceller kan bidra till ett mer hållbart och miljövänligt samhälle ur ett
energiutvinningsperspektiv genom hög energieffektivitet och väldigt låga utsläpp av
koldioxid, kväveoxider och hälsoskadliga partiklar. Bränsleceller, speciellt vissa
högtemperaturceller, kan arbeta med en rad olika bränslen förutom väte. I den här studien
har både naturgas och väte använts som bränsle men andra kompatibla kandidater är
etanol, metanol, biogas, och ammoniak. För att det skall leda till en mer miljövänlig
energiproduktion måste bränslena tas fram på ett miljövänligt vis.
Det var först omkring 1950 när bränsleceller användes som en kraftkälla i rymdraketer,
som de blev mer allmänt kända och kompletta bränslecellssystem konstruerades.
Bränslecellernas utveckling har tagit fart bland annat för att energipriserna ständigt ökar
och likaså oron kring växthuseffektens påverkan på jordens klimat. Det är nu möjligt att
tillämpa bränsleceller i en rad olika system i olika storlekar från mobiltelefoner till stora
kraftverk med olika bränslen. Eftersom de kan byggas i olika storlekar, har bränsleceller
en stor potential inom flera områden. De största hindren för en kommersialisering i stor
skala är den höga tillverkningskostnaden, korta livslängden och avsaknaden av en
funktionell infrastruktur för vätgas och biogas/naturgas.
Uppbyggnad av en bränslecell
Det finns olika typer av bränsleceller med olika kemiska processer och material. För att
beskriva en typ av bränslecell; fastoxidbränslecellen som används i den här studien, väljs
den med väte som bränsle i det här fallet. I en sådan bränslecell reagerar syre och väte
med varandra och bildar vatten. En bränslecell är uppbyggd av en anod, en katod och en
elektrolyt. En anod är den del i en elektrolytisk cell som är förbunden med strömkällans
positiva pol, och katoden är sammanbunden med dess negativa pol. Det gasformiga
bränslet transporteras till anoden där det reagerar i elektrokemiska reaktioner med
syrejoner. Syrejonerna produceras i katoden där syre reagerar med elektroner till jonform.
Syrejonerna transporteras igenom elektrolyten för att nå bränslet i anoden. Elektronerna
släpps inte igenom elektrolyten, vilket gör att en spänning uppstår. Den specifika
bränslecellen i den här studien har en hög arbetstemperatur och elektrolyten, bestående av
en fastoxid, är utformad för att endast släppa igenom syrejoner från katoden till anoden.
Skillnaden mellan olika typer av bränsleceller är främst vilken typ av elektrolyt som
används och bränslecellens arbetstemperatur.
Övergångsfas
Bränsleceller producerar elektricitet och värme direkt från kemiska och elektrokemiska
reaktioner mellan bränsle och syret i luften, dvs. inget mekaniskt arbete. När ren vätgas
eller biogas används blir det inga nettoutsläpp av koldioxid, hälsoskadliga partiklar eller
kväveoxider, vilket gör processen helt miljövänlig och koldioxidneutral. För
bränsleceller, såsom fastoxidbränslecellen, där arbetstemperaturen är mellan 600 och 800
°C är det möjligt att använda utöver vätgas mer komplexa bränslen, som naturgas, biogas
eller etanol. Då sker en omvandling (reformering) av bränslet, antingen i en separat enhet
som bränslet får passera innan det kommer till bränslecellen, eller inne i bränslecellens
anod. Det material som vanligtvis används i anoden har visat sig vara lämpligt för
3

katalytisk omvandling av naturgas och biogas till vätgas och kolmonoxid. Dessa reagerar
sedan i bränslecellens anod genom de elektrokemiska reaktionerna med syrejoner.
Dagens forskning om bränsleceller med hög arbetstemperatur fokuseras till stor del till att
öka förståelsen kring den porösa mikrostrukturens inverkan och både de kemiska och
elektrokemiska reaktionernas inverkan på de fysikaliska processerna. Mer realistiska
numeriska modeller som fångar upp dessa mikroskaliga processers inverkan på de
närliggande fysikaliska processerna kan bidra till en enorm förbättring i bränslecellens
effektivitet. Målet för dagens forskning är att kunna simulera de fysikaliska processerna
på alla nivåer. Det är även viktigt att kunna jämföra experimentella och numeriska
analyser speciellt på mikronivå då det finns ett kunskapsglapp där. De parametrar som har
störst inverkan på de minsta nivåerna kan kopplas till de större nivåerna med de
konventionella parametrarna såsom hastighet och temperatur. Med ökad förståelse om vid
vilken nivå en process har störst inverkan, skapas möjligheten att bidra till förbättrad
prestanda vilket i sin tur kan leda till att kostnaden kan sänkas. För att möjliggöra denna
simulering ställs höga krav på den tillgängliga datorkapaciteten och i takt med att
datorkapaciteten ökar kan mer komplexa simuleringar utföras.
Några av problemen och utmaningarna med dagens energisystem, både globalt och lokalt,
är utsläpp av bland annat koldioxid, hälsoskadliga partiklar och kväveoxider. Det
diskuteras hur länge mänskligheten kan fortsätta att utvinna fossila bränslen i samma takt
som idag då det verkar finnas en begränsad mängd att tillgå. Möjligheten för en ren,
miljövänlig och energieffektiv bränsleanvändning driver utvecklingen av
bränslecellssystem framåt i allt snabbare takt. Fastoxidbränslecellen kan fungera
framförallt som en övergång från den konventionella teknologin för energiutvinning med
möjligheten att internt hantera kolvätebränslen till en mer hållbar och miljövänlig
energiproduktion. Det som kommer att bestämma tillväxten inom bränslecellsområdet är
hur mycket tillverkningskostnaden kan sänkas, och livslängden ökas på kort tid.
4

Acknowledgements
I am very grateful to my supervisors Professor Bengt Sundén and Docent Jinliang Yuan for their
support and guidance during the past two years. Many thanks also to Professor Sundén who made my
forthcoming research visit at University of California, Berkeley possible. I also want to extend my
thanks to Docent Christoffer Norberg for encouraging me to go on to doctoral studies and for being a
teacher role model.
To Martin Andersson, I owe particular thanks for knowledgeable guidance and constant
encouragement. I want to thank Olof Ekedahl for help and encouragement in general and for all the
support with work on Python in particular.
The study was made possible through financial support from the Swedish Research Council
(Vetenskapsrådet, VR) and the European Research Council (ERC).
i

List of publications
Journal publications:
I. H. Paradis, M. Andersson, J. Yuan, B. Sundén, CFD Modeling: Different Kinetic Approaches
for Internal Reforming Reactions in an Anode-Supported SOFC, ASME Journal of Fuel Cell
Science and Technology, 8, 031014, 2011
II.
III.
IV.
H. Paradis, M. Andersson, J. Yuan, B. Sundén, Simulation Analysis of Different Alternative
Fuels for Potential Utilization in SOFCs, International Journal of Energy Research, 35, DOI:
10.1002/er.1862, 2011
M. Andersson, H. Paradis, J. Yuan, B. Sundén, Modeling Analysis of Different Renewable
Fuels in an Anode-Supported SOFC, ASME Journal of Fuel Cell Science and Technology, 8,
031013, 2011
M. Andersson, H. Paradis, J. Yuan B. Sundén, Review of Catalyst Materials and Catalytic
Steam Reforming Reactions in SOFC anodes, International Journal of Energy Research, 35,
DOI: 10.1002/er.1875, 2011
Conference publications:
I. H. Paradis, M. Andersson, J. Yuan, B. Sundén, The Kinetics Effect in SOFCs on Heat and
Mass Transfer: Interparticle, Interphase, and Intraparticle Transport, submitted to: ASME 9 th
Fuel Cell Science Conference, ESFuelCell2011-54015, 2011
II.
III.
IV.
H. Paradis, M. Andersson, J. Yuan, B. Sundén, CFD Modeling Concerning Different Kinetic
Models For Internal Reforming Reactions in an Anode-Supported SOFC, ASME 8 th Fuel Cell
Science Conference, FuelCell2010-33045; pp. 55-64, 2010
H. Paradis, M. Andersson, J. Yuan, B. Sundén, Review of Different Renewable Fuels for
Potential Utilization in SOFCs, 5 th International Green Energy Conference, 2010
M. Andersson, H. Paradis, J. Yuan, B. Sundén, Modeling Analysis of Different Renewable
Fuels in an Anode-Supported SOFC, ASME 8 th Fuel Cell Science Conference, FuelCell2010-
33044; pp. 43-54, 2010
V. M. Andersson, H. Paradis, J. Yuan, B. Sundén, Catalysts Materials and Catalytic Steam
Reforming Reactions in SOFC Anodes, 5 th International Green Energy Conference 2010
ii

Table of contents
Abstract ................................................................................................................................................... 2
Populärvetenskaplig beskrivning på svenska .......................................................................................... 3
Acknowledgements .................................................................................................................................. i
List of publications .................................................................................................................................. ii
Table of contents .................................................................................................................................... iii
Nomenclature .......................................................................................................................................... v
1 Introduction .......................................................................................................................................... 1
1.1 Research objectives ....................................................................................................................... 2
1.2 Methodology ................................................................................................................................. 2
1.3 Thesis outline ................................................................................................................................ 2
2 SOFC modeling at smaller scales ......................................................................................................... 3
2.1 Solid Oxide Fuel Cells .................................................................................................................. 3
2.2 SOFC modeling development ....................................................................................................... 5
2.2.1 Multiscale and multiphysics modeling .................................................................................. 5
2.2.2 Lattice Boltzmann concept .................................................................................................... 7
2.2.3 Monte Carlo method and Molecular dynamics ...................................................................... 7
2.2.4 Modeling integration issues ................................................................................................... 8
2.3 Lattice Boltzmann method ............................................................................................................ 9
2.3.1 Boundary conditions in LBM .............................................................................................. 12
2.3.2 Choice of units in LBM ....................................................................................................... 12
2.3.3 Mass diffusion in LBM ........................................................................................................ 14
2.3.4 Chemical reactions in LBM ................................................................................................. 15
2.3.5 General comments on CFD and LBM coupling .................................................................. 15
2.4 Previous case studies of SOFCs in LBM .................................................................................... 17
2.5 Electrode microstructure remarks ............................................................................................... 19
3 Mathematical models ......................................................................................................................... 22
3.1 Model visualization for the microscale model ............................................................................ 22
iii

3.2 Governing equations for the macroscale model .......................................................................... 24
3.2.1 Mass transport ...................................................................................................................... 25
3.2.2 Heat transport....................................................................................................................... 26
3.2.3 Momentum transport ........................................................................................................... 27
3.2.4 Electrochemical reactions .................................................................................................... 27
3.2.5 Internal reforming reactions ................................................................................................ 28
3.3 Heat and mass transfer limitations of the kinetic model ............................................................. 30
3.3.1 Interparticle transport ........................................................................................................... 31
3.3.2 Interphase transport ............................................................................................................. 32
3.3.3 Intraparticle transport ........................................................................................................... 33
4 Results and discussion ........................................................................................................................ 36
4.1 Microscale model by LBM ......................................................................................................... 36
4.2 Evaluation of kinetics at smaller scales ...................................................................................... 40
4.3 Macroscale model by CFD ......................................................................................................... 43
4.3.1 Case study: Internal reforming reaction rates ...................................................................... 43
4.3.2 Case study: Methane content and steam-to-fuel ratio .......................................................... 46
5 Conclusions ........................................................................................................................................ 51
6 Future work ........................................................................................................................................ 52
7 References .......................................................................................................................................... 53
iv

Nomenclature
a lattice direction, –
AV surface area-to-volume ration, m 2 /m 3
Bi Biot number, –
c lattice speed of sound, lu/ts
c T total concentration, mol/m 3
c p specific heat at constant pressure, J/(kg·K)
C concentration, mol/m 3
Da Darcy number, –
D ij Maxwell-Stefan binary diffusion coefficient, m 2 /s
D thermal diffusion coefficient, kg/(m·s)
E activation energy, kJ/mol
e lattice speed, lu/ts
f a particle density distribution function, –
f eq equilibrium distribution function, –
F force vector, N/m 3
F Faraday constant, 96485 C/mol
F gravitational force, mu·lu/ts 2
h enthalpy, kJ/mol
h heat transfer coefficient, W/(m 2·K)
h v volume heat transfer coefficient, W/(m 3·K)
i current density, A/cm 2
i 0 exchange current density, A/cm 2
J mole flux, mol/(m 2 s 1 )
J* dimensionless mole flux, –
k thermal conductivity, W/(m·K)
k reaction rate constant, mol/(m
3·bar
2·s)
k c mass transfer coefficient, m/s
k´´ pre-exponential factor, 1/(·m 2 )
K e equilibrium constant, Pa 2 or dimensionless
M j molecular weight of species j, kg/mol
N number of grid points, –
N iter number of iterations, –
Nu Nusselt number, –
n e number of electrons transferred per reaction, –
P pressure, bar
p partial pressure, Pa
v

Q source term (heat), W/m3
q heat flux, W/m 2
r reaction rate, mol/(m 3·s), mol/(m 2·s)
r mean pore radius, m
R gas constant, 8.314 J/(mol·K)
Re Reynolds number, –
S i source term for the reaction rate, kg/(m 3·s)
T temperature, K
T viscous stress tensor, N/m 2
t time step, –
u, v velocity, m/s
w a weight factor, –
w i mass fraction of species i, kg/kg
x, y coordinate system, m
x mole fraction, –
z pixel point, –
Greek symbols
β dimensionless maximum temperature rise, –
γ dimensionless activation energy, –
δ reference step (time or length)
ε porosity, –
η overpotential, V
κ permeability, m 2
κ dv deviation from thermodynamic equilibrium, Pa·s
μ dynamic viscosity, Pa·s
ρ density, kg/m 3
ζ ionic/electronic conductivity, Ω -1 m -1
η tortuosity, –
η relaxation time
υ kinematic viscosity, m 2 /s
χ Damköhler number for the heat transfer, –
ω Damköhler number for the mass transfer, –
Ω D diffusion collision integral, –
Φ phase function, –
Subscripts
0 reference state or initial state
a lattice direction in LBM
act activation polarization
D dimensionless system
e electrode, ea,c
eff
el
effective
electrolyte
vi

g
i
j
LB
P
r
s
w
gas phase
molecule i
molecule j
discrete system, lattice Boltzmann
physical system
steam reforming reaction
solid phase or water-gas shift reaction
wall
Abbreviations
AFL anode functional layer
CFD computational fluid dynamics
CFL cathode functional layer
FDM finite difference method
FEM finite element method
FVM finite volume method
IEA International Energy Agency
IT intermediate temperature
LBM lattice Boltzmann method
LTNE local temperature non-equilibrium
MC Monte Carlo method
MD molecular dynamics
SEM scanning electron microscopy
SF steam-to-fuel ratio
SMM Stefan-Maxwell model
SOFC solid oxide fuel cell
TPB three-phase boundary
YSZ yttria-stabilized zirconia
XCT x-ray computer tomography
Chemical
CH 4 methane
CO carbon monoxide
CO 2 carbon dioxide
H 2 hydrogen
H 2 O water
Ni nickel
O 2 oxygen
vii

1 Introduction
After some time of decreasing interest, fuel cell research is now receiving a lot of attention.
Fuel cells are considered a promising future resource for both stationary and distributed
electric power stations because of their high performance and high reliability [1, 2]. In order
to explore these aspects in greater depth, there is a need for both multiphysics and multiscale
modeling. The approach is by solving the equations for momentum, charge, heat and mass
transport and chemical reactions at the same time at corresponding scales. At this point,
simulation of mass diffusion and convection as well as chemical and electrochemical
reactions at microscale will offer crucial insight to improve the performance of the fuel cell
[3]. Furthermore, future prospects for fuel cell research are to connect microscale to
macroscale and obtain stable solutions for multiscale modeling.
Solid oxide fuel cells (SOFCs) are particularly interesting because it operates at high
temperature and can therefore handle the reforming of hydrocarbon fuels directly within the
cell. SOFCs have a number of advantages, e.g., high conversion efficiency, high quality
exhaust heat and flexibility of fuel input. However, as expected, SOFCs also have
disadvantages. For instance, because they operate at very high temperatures (800–1100C
[1]), material performance and manufacturing costs are currently of concern. Recently,
attempts have been made to lower the operating temperature of SOFCs by adopting a porous,
anode-supported structure to reduce the thickness of the electrolyte.
It is important to model all physical processes and chemical reactions simultaneously since
the mass and charge transport depends on the multifunctional material structure in the porous
electrodes, the chemical reactions, the temperature distribution and the species concentrations.
The fluid flow depends on the chemical reactions, temperature and fluid characteristics. The
heat transport depends on the polarization losses, the chemical reactions, the fluid flow and
the material structure. The reforming reactions depend on temperature, concentration and
amount of catalyst available. The interaction issues are numerous and here the microscopic
contributions are important to include. Also it is necessary to model SOFCs into detail and
explore them at a microscopic level, in order to fully understand how different parameters
affect the performance, by connecting different physical phenomena at different scales [2, 4].
The purpose of this study is to develop a microscale model of an anode of an SOFC for the
transport processes and the chemical reactions to get a deeper understanding of the effect on
the different physical processes at multiple scales. The lattice Boltzmann method (LBM) is
used to model the mass transport at microscale for a limited part of the porous anode. Also an
evaluation of a macroscale model for the whole unit cell is carried out.
1

1.1 Research objectives
The knowledge of the effect of different processes at microscale, such as mass diffusion and
electrochemical reactions, on the unit cell in whole is expected to be clarified when the model
includes microscale phenomena within the electrodes and electrolyte. When these processes
are studied, the effect on the overall performance of the cell can provide useful information
for improvement [5].
The aim is two-fold. Firstly, it provides a description of current research on modeling of
transport processes and kinetics effect on transfer processes in the electrodes of SOFCs.
Focus is put on LBM to model the microstructural phenomena. Further, other modeling
methods and equations are briefly reviewed as well as the coupling of LBM to conventional
CFD methods. Secondly, it reveals a microscale model of an anode of an SOFC using the
LBM to carefully investigate the physical and chemical processes at smaller scales which
simulates the mass diffusion and momentum transport for a small part of the anode close to
the electrolyte. A macroscale model of an anode-supported SOFC is also developed where the
equations for mass, heat and momentum transport are solved simultaneously. Different
internal reforming reaction rates are tested to examine the effect on the cell. Also the
parameters inlet methane content and steam-to-fuel ratio are tested for different ranges. The
kinetics for the macroscale model are carefully investigated to check so no severe heat and
mass transfer limitations occur at microscale. Further, the future step is to model all processes
at microscale where microstructural effects are considered to affect the performance and also
integrate the LB model and the macroscale CFD model, i.e., coupling different physical
models at different scales, to form a multiscale model which can reproduce even more
realistic simulation results. More precisely, the objectives here are:
To identify whether the LBM can function as a method for SOFCs at microscale to
investigate the transport processes and chemical reactions, in terms of capabilities and
limitations.
To capture and study the microscale effect of mass diffusion and momentum
transport within the anode close to the active area.
To identify whether diffusion and chemical reactions will significantly affect the cell
performance by modeling them at both macro- and microscale.
1.2 Methodology
In order to analyze the transport processes and chemical reactions in SOFCs in detail, an LB
model for the porous region close to the three phase boundary (TPB) is developed in
MATLAB. The LB model uses the single relaxation time BGK (Bhatnagar-Gross-Krook)
method and is developed for a D2Q9 (two dimensional domain with eight interconnected
directions and nine interconnected speeds) [3]. The model focuses solely on the mass
diffusion and momentum transport at microscopic level in this region. The simulation
procedure is divided into stepwise cases, from a simple channel to a more complex porous
media. The macroscale model of an SOFC is developed in COMSOL where the equations for
mass, heat and momentum transport are solved simultaneously. Finally, a study of limiting
effects on the heat and mass transfer by the kinetics is also performed.
1.3 Thesis outline
Chapter 1 contains a short presentation of the thesis. Chapter 2 gives a general description of
SOFCs, and an overview of the relevant literature of the LB methodology. A detailed
description of the mathematical model of the LBM is presented in Chapter 3 with governing
equations and boundary conditions. The results are presented in Chapter 4 while Chapter 5
provides conclusions drawn from the results. Finally, Chapter 6 gives reflections over future
work.
2

2 SOFC
modeling at smaller scales
This chapter gives a short description of SOFCs and fuel cell modeling at different scales is
described with focus on LBM.
2.1 Solid Oxide Fuel Cells
Fuel cells directly convert the free energy of a chemical reactant to electricity and heat. This
is different from a conventional thermal power plant, where the fuel is oxidized in a
combustion process and subsequently a conversion process (thermal-mechanical-electrical
energy) occurs. Fuel cells have high energy conversion efficiency due to the direct
conversion. If pure hydrogen is used, there is no destructive environmental pollution, because
the output from the fuel cells is electricity, heat and water.
Among various types of fuel cells, the SOFC has attracted significant interest thanks to its
high efficiency and low emissions of pollutants like carbon dioxide and hazardous gases. The
SOFC’s high operating temperature offers many advantages, such as high electrochemical
reaction rates, flexibility in choice of fuel and high tolerances for impurities. However, fuel
cell systems are still immature technologies, as can be noted in the lack of a dominant design,
low number of commercial systems and a low market demand. The creation of strategic niche
markets is of a vital importance for further development [4].
The International Energy Agency (IEA) has concluded in many reports that the fuel cell will
be a key component in a future sustainable energy system. About 80 percent of the energy
resources traded today are fossil fuels (coal, oil and natural gas) [1] and these resources are
considered limited. Here the SOFC can be a key component facilitating the transition towards
more environmentally friendly energy generation. It is possible to use conventional fuels as
natural gas to ease the transition to hydrogen based power generation without emissions of
pollutants. During recent years, another promoter of interest in using fuel cells as auxiliary
power units (APUs) in on-board transport applications, for example in luxury passenger
vehicles, military vehicles and leisure boats has increased immensely [1].
SOFCs can function with a variety of fuels, e.g., hydrogen, carbon monoxide, methane and
combinations of these [5]. Because SOFCs operate at high temperature, they supply a
sufficiently good environment to internally reform the hydrocarbon-based fuel within the cell.
The fuel flexibility gives the SOFCs a major advantage over pure hydrogen, which is highly
flammable and volatile and therefore problematic to handle. Also, hydrogen has low density,
which makes storage costly [4]. It should also be mentioned that pure hydrogen is difficult to
obtain because it has to be extracted from another source, most commonly natural gas or
through electrolysis. Within the cell several reactions may take place and vary depending on
which fuel is used. The overall global reactions for methane are stated below. More detailed
surface reactions can be found in the literature [5, 6].
3

(2.1)
(2.2)
(2.3)
(2.4)
(2.5)
Equation (2.1) is the reduction of oxygen in the cathode. Equations (2.2) and (2.3) are the
electrochemical reactions at the anodic three phase boundary (TPB). TPB is the region where
the ionic, electronic and gas phases meet close to the electrolyte and the electrode interface.
Equation (2.4) is the steam reforming of methane, which needs to be carried out prior to the
electrochemical reactions, and is usually called catalytic steam reforming reaction. Carbon
monoxide can be oxidized as in equation (2.3) or react with water as in equation (2.5).
Equation (2.5) is often called water-gas shift reaction. Note that methane is not participating
in the electrochemical reactions at the anodic TPB. It is catalytically converted, within the
anode, into carbon monoxide and hydrogen, which are used as fuel in the electrochemical
reactions [5-7]. These reactions can be viewed in the schematic illustration of an SOFC in
Figure 2.1.
Figure 2.1: Schematic SOFC structure.
The configuration of an SOFC can be formed in different ways; electrolyte or electrode
supported cells, shaped in a planar or tubular manner, co or counter directional flow. An
electrolyte supported SOFC has thin anodes and cathodes (~50 m), and the thickness of the
electrolyte is more than 100 m. An electrolyte supported SOFC works preferably at
temperatures around 1000 °C. In an electrode-supported SOFC, either the anode or the
cathode is thick enough to serve as the supporting substrate for cell fabrication, normally
4

etween 0.3 and 1.5 mm. In this configuration the electrolyte is thin (could be as thin as 10
m) [5-7].
SOFC research in the last years has focused on the electrode-supported configuration to lower
the operating temperature. Positive outcomes of development in this direction are a decrease
in start-up and shut-down time and, simplified design and cell material requirement. Electrode
supported design makes it possible to have a very thin electrolyte, i.e., the ohmic losses
decrease and the temperature can be lowered to 600-800 °C. Fuel cells working at those
temperatures are classified as intermediate temperature ones if compared to conventional
SOFCs that operate at 800-1000 °C [6]. Corrosion rates are significantly reduced and stack
lifetime is extended. Lowering the operating temperature to an intermediate range will cause
an increase of both ohmic- and polarization losses in the electrodes. This requires the
development of a highly active electrolyte that has low polarization loss at intermediate
temperatures. Possible electrolyte materials could be doped ceria or doped lanthanum gallate
[7-8]. The SOFC in this case is built up by an electrolyte containing yttria-stabilized zirconia
(YSZ) and a cathode containing strontium doped lanthanium manganite (LSM), and finally,
the anode nickel/YSZ [1, 7].
2.2 SOFC modeling development
Before designing and constructing a model, it is important to specify what is needed and why.
The selection of computational methods must come from a clear understanding of both the
information being computed and the actual physical processes implemented. In order to solve
some of the remaining issues for the understanding of the detailed physical phenomena of
SOFC, the computational modeling is the crucial factor. The microstructure is one of the least
understood areas of research of the SOFC. As increased computational capability opens up
for more detailed research, today’s fuel cell research challenge is to fully understand
microscale and nanoscale transport phenomena in the active electrochemical material and
further, connect these to macroscale to form a multiscale model.
2.2.1 Multiscale and multiphysics modeling
While multiphysics modeling involves the coupling and interaction between two or more
physical disciplines, the multiscale approach involves the connection of specific physical
processes, based on the different levels of scale: micro, meso and macro. The division and
ranges of scales varies slightly in the literature but an attempt is made here to divide the scales
to suit the modeling perspective in this study. The microscale model corresponds to the
particle level (~10 -6 – 10 -9 m). Also even smaller scales, nanoscales, may be integrated but are
not further studied in this work. Here the mesoscale and macroscale stretch from a scale
larger than a particle to the global flow field (~10 -5 – 10 1 m) [1, 6]. To understand the
multiscale concept, one needs first to understand the different scales involved. Not only
proper length scales need to be considered, but also different time scales. Convective
transport appears in 10 -1 s, cell heating and anode thermal diffusion are in 10 3 s and cathode
thermal diffusion appears in 10 4 s [5]. A relation between time- and length scales with proper
modeling methods can be seen in Figure 2.2. Only the microscale and mesoscale are shown
here but the division can be done differently or in more intermediate steps.
5

Figure 2.2: Characteristic time and length scales for various modeling methods. Data is taken from [1].
At macroscale, homogeneity is assumed throughout the model, which subsequently can cause
errors during the loop of the modeling algorithm. Several models are based on the assumption
that the porous structure is isotropic and can be described by a few experimentally estimated
parameters [4, 8]. Porosity, tortuosity, and surface area to volume ratio are examples of
parameters that are affected by assumptions concerning homogeneity at all scales. Note that
these microstructural parameters are known to have a significant influence on the cell
performance and durability [9]. As the available computational power increases, it opens up
for a more sophisticated and deeper understanding of the physical processes and effects of
chemical reactions within the porous microstructure. This makes it possible to address the
microstructural uncertainties to improve cell performance, as these are limiting the SOFC
progress [10].
Multiphysics modeling takes into account the interaction between several physical processes,
which can be described by partial differential equations. A good computational design
considers the physical processes and the system at both a microscale and a macroscale level.
Some of the limitations are the lack of material structure and test data in the literature to
validate the models [11, 12]. The results of a numerical simulation cannot guarantee of how
well the cell actually will operate in reality. Because of the numerical approximations and
arbitrary unknowns implemented in the model, there will most likely be a number of errors
and inaccurate results [11]. Still, the use of numerical modeling as a predictive tool can be
validated through careful consideration of results and comparisons of numerical and
experimental data. A great deal of computational modeling research, where the results are
obtained from numerical codes, has achieved sufficient accuracy both in comparison with
other different numerical modeling approaches and with experimental data [10-11].
Fuel cell modeling is complicated due to the interaction of physical and chemical processes,
such as multicomponent gas flow with heat and mass transfer, electrochemical and reforming
reactions [10-11]. To model SOFCs, it is common to handle the governing equations in
differential forms by deriving them in forms of discretized equations. These equations are
solved numerically by the Gaussian-elimination method or the Tri-diagonal matrix algorithm.
There are several approaches to solve these by numerical methods. For macroscale, in a
simplified manner, one may say that the methods differ in the sense of how the flow variables
are approximated. The commercial software which is currently available is mainly based on
the Finite Difference Method (FDM), the Finite Element Method (FEM) and the Finite
6

Volume Method (FVM) which adopt the macroscopic structure [11-12]. Examples of
commercial software are FLUENT and STAR-CD, based on FVM, and COMSOL
Multiphysics, based on FEM. Through computational modeling, the output can provide
details of the processes, such as the fuel cell species distribution, flow patterns, current
density, temperature distribution and pressure drop, etc. [11]. The simulation environment in
commercial software facilitates all steps in the modeling process. It is easy to define the
geometry and the mesh as well as specify the physics for the domain. As fuel cell testing is
expensive and time consuming, a careful simulation study before testing can lower the cost
for research activities. For this reason, numerical modeling of SOFCs is necessary.
2.2.2 Lattice Boltzmann concept
Advances in micro-modeling have been made thanks to better availability of computational
power, where lattice Boltzmann mass diffusion models have been found to predict and
visualize the phenomena well in the microstructure and the capability to simulate not only
single phase or multiphase flow but also both of these in complex geometries [3]. Based on
the Boltzmann equation, the model is considered to be an alternative to the traditional CFD
based on the Navier-Stokes equations, without any empirical modifications. The idea of the
LB model is that it is viewed from a particle perspective and based on a statistical approach to
track a large number of particles. The framework is built upon interaction between the
particles, e.g., collision, either particle-to-particle or particle-to-surface interaction, and
streaming [3].
LBM has served as a numerical information bank with detailed simulation results for a large
number of physical processes. In most cases, previous work on LBM in both the continuum
and non-continuum flow regime has focused on single component flow problem. However,
there is still a lack of results by applying the LBM with more than one species especially at
high Kn in complex geometries. This is the case for the porous electrodes in SOFCs where H 2
and H 2 O diffuse or internal reforming of hydrocarbons occurs [11, 14-16].
Lattice gas cellular automation models have acted as the forerunners of the LBM. Like them
the LBM is based on a concept with an algorithmic entity at a position connected to its
neighbors. The next step in the development of LBM was when the basic idea of Boltzmann’s
work was included. The idea is based on a gas, composed of interacting particles as described
by classical mechanics [3]. The LBM simplifies Boltzmann’s original view by reducing the
number of participating and possible spatial particle positions. LBM tracks the statistical
behavior of the molecules at each lattice point. Distinct steps have been developed for the
microscopic momentum and distribution paths. The spatial positions are confined to the
lattice nodes, and the variations of momentum due to velocity changes, are reduced to 8
directions, 3 magnitudes and a single mass in the 2D case [3]. LBM is built up on lattice units
which need to be converted to actual physical properties after the simulation [14]. This is
often handled through the mole fraction when the mass diffusion is the main focus. The
electrochemical reactions and also the reforming reactions are then coupled with the mass
diffusion to LBM via mole flux boundary conditions at the active surface. To obtain these
dimensionless values, the methodology for the LBM needs to be described in detail which is
carried out in a latter section (see section 2.3.2).
2.2.3 Monte Carlo method and Molecular dynamics
A number of nano- and microscale models have been developed to simulate microscopic
transport phenomena in SOFCs. The atomistic model refers to a broad group of algorithms
and can provide detail information to make realistic boundary and interface definitions for a
larger scale model [17]. Monte Carlo (MC) methods are based on algorithms which through
repeated random steps can be used in simulating physical and mathematical systems [18]. The
system is propagated through time by stochastically establishing the coming event based on a
relative probability of each possible event [18]. The probability is often determined by the
7

intrinsic rate constant of each event. These methods are most suited for computational
calculations when it is unfeasible to compute an exact result with a deterministic algorithm
[19]. The MC methods is especially useful for simulating systems with many coupled
parameters, such as fluids and disordered materials, but unfortunately also tend to have a high
computational cost. MC methods vary, but tend to follow a particular pattern: Define a
domain of possible inputs, generate inputs randomly from a probability distribution over the
domain, perform a deterministic computation on the inputs and aggregate the results.
MC is a modeling method for the dynamic behavior of molecules by comparing the rates of
individual steps with random numbers. The method is used to investigate non-equilibrium
systems or the time evolution of some specific processes occurring in nature, typically
processes that occur with a known rate. The probability needs to be defined prior to the
simulation [19]. MC was proven functional by a study by Lau et al. [19], conducted on a
cathode of an SOFC for the oxygen reduction reaction. Lau et al. [19] found that the
temperature had a great effect throughout the simulation which in this case was on the ionic
current density.
One difference between Monte Carlo and Molecular dynamics (MD) modeling is the different
time scales. MC can handle longer time scales, typically seconds, whereas MD handles time
scales around microseconds or even smaller [19]. To make statistically valid conclusions from
the simulations, the time span simulated should match the kinetics of the natural process. The
modeling should allow for different reaction pathways [17]. The relevant parameters are
possible to be captured by both MC and MD. However, one would prefer to use the models
for different processes depending on the needed elapsed time.
MD simulates physical movements of atoms and molecules by computer modeling based on
statistical mechanics. MD allows insight into molecular motion on an atomic scale and
detailed time and space resolution into representative behavior for carefully selected systems
[20]. MD can be used to model local structures at elevated temperatures where it is
challenging to perform experiments with conclusive outcomes. MD has been used to perform
simulations on diffusion and ion transport with successful outcome [20].
2.2.4 Modeling integration issues
Physical problems can often be described with a set of partial differential equations. The
coupled partial differential equations can be solved simultaneously in physical domains for
corresponding physical phenomena. The integration issues occur because the physical and
chemical processes are linked to each other and so also the equations in the model [11]. The
mass, heat, momentum transport as well as ionic/electronic transport and the reaction rate are
dependent on each other. The fluid properties and the momentum transport (flow field)
depend on the temperature and concentrations and so does the chemical reactions. The
chemical reactions generate and consume heat, i.e., the temperature distribution depends on
the chemical reaction rate, as well as on the solid and the gas properties, for example the
specific heat and the thermal conductivity [1, 11].
Until now, multicomponent gas diffusion has been modeled in the continuum flow regime for
the porous electrodes using the Stefan-Maxwell equations by several research groups. Some
of the equations used in our CFD-model, e.g., the Stefan-Maxwell model (SMM), are
described based on a few empirical parameters, which are difficult to measure [14]. To
enhance the knowledge of the impact of the transport process on the performance within the
electrodes, the microstructure needs to be modeled in detail. Recent advances have made it
possible to evaluate the microstructure using LBM without any modification of empirical
parameters. Models have been developed for all the three spatial dimensions, but so far the
main focus has been on particular parts of the fuel cell, e.g., anode, and also simple
geometries [14-16].
8

2.3 Lattice Boltzmann method
LBM has shown promising simulation results of fluid flows and mass diffusion through
complex geometries and this is an attractive characteristic for fuel cell modeling.
Conventional CFD methods use fluid density, velocity and pressure as their primary
variables, while the LBM uses a more fundamental approach with a so-called particle velocity
distribution function (PDF) [16]. The PDF is here denoted f a and originates from the basic
Boltzmann gas concepts where a derived simplified form of the Boltzmann equation is
described by classical mechanics with statistical treatment based on the high number of
particles. The distribution is described by the coordinates of the position and momentum
vectors, and the time step [3].
The LBM framework is built on lattice points, which are given locations placed all over the
solution domain. The lattice unit (lu) is a fundamental measure of length and the time step (ts)
is the measure of the time unit. The neighboring particles are connected to the main focused
particle at the time with the velocity magnitude e a , schematically described in Figure 2.3 from
a 2D point of view. This type of structural framework is called D2Q9 and stands for two
dimensions with nine velocities marked as e a , where a represents the direction (= 0, 1, 2... 8)
[3, 16]. For the 2D case, there exist two appropriate choices of system structure, namely
D2Q5 and D2Q9 with 5 and 9 directional velocities, respectively. The D2Q9 framework has
the possibility to capture more information but will be computationally heavier than the
smaller one D2Q5.
Figure 2.3: Schematic structural framework for the D2Q9 lattice and velocities.
The PDF is defined as the number of particles of the same species travelling along a particular
direction with a particular velocity. The single-particle distribution function f a can essentially
be seen as a histogram representing a frequency of occurrence. The frequencies can be
considered to be direction specific fluid densities. The LBM is described by two different
actions taking part at each lattice point (site); namely streaming and collision. Streaming
describes the movement of the particles of each species and the collision describes the
interactions between the particles of the same or different species. Furthermore, these actions
are combined in the LB equation called the distribution function [3, 14-16].
9

(2.6)
where f a is the PDF, e a the velocity and a the collision term at any spatial location x and time
t along the direction a. The time is increased by the time step Δt. The macroscopic fluid
density is [3]:
(2.7)
The macroscopic velocity u is evaluated by the microscopic velocities e a and the PDF f a and
divided by the macroscopic fluid density ρ as [3]:
(2.8)
This allows the LBM to recover the continuum macroscopic parameters from the discrete
microscopic ones, in this case by the velocities. The distribution function presented in
equation (2.6), called the single relaxation time BGK (Bhatnagar-Gross-Krook) LBM, is one
of the simplest models [3, 14]. The BGK is described by using one relaxation time for the
collision term. The collision term consists of the present PDF and the relaxation toward the
local equilibrium. The collision term Ω a and the D2Q9 equilibrium distribution function f a
eq
are defined as [3]:
(2.9)
(2.10)
where w a is 4/9 for the particle a = 0, 1/9 for a = 1, 2, 3, 4 and 1/36 for a = 5, 6, 7, 8, and τ is
the relaxation number [14-16]. In the simplest implementation the basic speed on the lattice c,
which is also called the lattice speed of sound, is 1 lu/ts [16].
When the mass diffusion is modeled in LBM, two approaches are often used; pure diffusion
or advection-diffusion (also called convection-diffusion). Both pure diffusion and advectiondiffusion
is simulated by another equilibrium distribution f ζ,a eq which is very much alike the
normal fluid distributions function but with a simpler equilibrium equation. For the first case
with pure diffusion only the equilibrium function is defined as [3]:
(2.11)
In the second case, advection-diffusion, which is applied here, the equilibrium function will
include a second term to handle the convective velocity. The equilibrium function is defined
as [3]:
(2.12)
The mixing due to density variations and buoyant effects in porous media can here be handled
as advective and diffusive components rather than an input parameter (such as porosity). For a
porous media, the collision term is considered as a second intermediate step after the
10

streaming [3]. The concentration ρ ζ is defined similarly as the fluid density in equation (2.7)
[3]:
(2.13)
To avoid numerical instability, it is recommended to keep the relaxation times at the order of
unity. This would mean that the diffusivity values are adjusted by scaling them up to ensure
relaxation times of unity [14-16].
The LB equation can be extended to include several components or species. The equations
remain the same for each species but the interaction and combination of the streaming and
collision action for species i is defined as [3, 14-16].
(2.14)
where f i a is the PDF, e i a the velocity and
time t along the direction a for species i.
a i the collision term at any spatial location x and
Streaming is described in two steps. The PDFs, for example for species 1, are streamed from
one lattice point to the adjacent lattice points, while PDFs for the remaining species, i.e.,
species 2 and 3, are streamed from one lattice point to off-lattice points in the first step. Offlattice
points are defined as sites which are not related to the current start point. In the second
step, the PDF values for species 2 and 3 are determined by interpolation at lattice points. The
collision concept is divided into self-collision, i.e., collision between particles of the same
species and cross-collision, i.e., collision between particles of different species when the
relative velocity between particles of the different species is non-zero [14-16].
For the porous media, the collision term is considered as a second intermediate step after the
streaming [3]. Finally if there are multiple species, the LB model is adjusted by upgrading the
distribution function by also including a composite velocity:
(2.15)
where τ is the relaxation time and i represents the different components involved [3].
To form a realistic model, it is possible to include an external force, e.g., gravitational force,
on the particles or interaction forces between the particles. This force is incorporated in a
velocity term which is added to the velocity calculation. These parts are defined as [3]:
(2.16)
(2.17)
where F is the force acting on the particle, m the mass and a the acceleration of the particle.
Further, u is the velocity, ρ the density and τ the relaxation time.
11

2.3.1 Boundary conditions in LBM
A common choice of boundary conditions for the mole fractions are specified at x = 0 and the
mole flux is specified at x = L. The boundary conditions for velocity and density need to be
described indirectly by the PDFs in the LBM. This differs from the Navier-Stokes equations
in conventional CFD as the boundary conditions are prescribed directly for velocity and
density [15]. For the implementation of the electrochemical activity in the mass transfer
framework of LBM, the specific boundary nodes need to be specified as discrete Neumann or
Dirichlet boundary conditions. The Dirichlet boundary condition is associated with
concentrations specified at the pores along the inlet of the domain. The Neumann boundary
condition or flux boundaries are employed similarly to the Dirichlet boundaries. Instead of
defining the concentration, the particle velocities are explicitly defined at the boundary nodes.
The electrochemical kinetic mechanism can be defined as active and assigned a unique flux at
each node. In turn this affects the particle velocity as a function of location within the domain
[14, 21-23].
To treat the electrode structure realistically, it will be built up of open space and solid
obstacles where the obstacles are treated like impermeable solid surfaces. The velocities at the
solid obstacles need to be reset to zero at each time step for all species The obstacles are
suited for three different boundary conditions for the velocities; no-slip, free-slip and diffuse
reflection. The no-slip condition is simply a bounce-back definition, as the particles are
reflected back in the same direction as they arrive. For the free-slip, the particles are reflected
back in the angle of reflection, which is set to be the same as the angle of incidence. For the
diffuse reflection, the particles are reflected with the same probability in every direction [14].
Joshi et al. [15] showed that the implementation of the three different boundary conditions
obtained approximately the same results. LBM makes no conceptual distinction between the
transport of single species over an obstacle or multiple species over the same object in
opposite directions. The only difference will be the impact of the motion of species by the
collision [15].
Finally to summarize the whole procedure, the mole fractions are initially assigned specific
values over the whole domain. The velocities of the species in the perpendicular direction to
the flow direction (y-direction) are initially set to zero, and are always set to zero for the
boundary of the obstacle. The following algorithm is repeated until a steady solution has been
reached. The calculation of the equilibrium functions is followed by that of the collision terms
and subsequently by streaming and interpolation of the PDF values. The unknown values at x
= 0 and x = L are calculated and followed by a calculation of the values at the boundaries of
the obstacles. Finally, all the physical parameters are calculated, such as density, velocity etc.
When the total concentration has been checked and a steady solution has been reached, the
modeling is done. Obviously, all the restriction and constraints applied to the model need to
be fulfilled [21, 23-24].
2.3.2 Choice of units in LBM
As the LBM parameters are not in physical units when simulated but in so-called lattice units,
this part of the algorithm procedure needs extra attention. The approach here is divided into
two steps. First, the physical system is converted to a dimensionless one. This system is
independent of both the physical scales and the modeling parameters. Second, this system is
converted into a discrete modeling system.
To exemplify, the incompressible Navier-Stokes equation depends only on a single
dimensionless parameter, namely the Reynolds number Re. Consequently, both the physical
and the dimensionless system will be connected to have the same Re. But there will still be a
need for a transition between the two systems, and this is done by a characteristic length scale
l 0 and a characteristic time scale t 0 . The transition from the dimensionless to the discrete
system is done by implementing a discrete length step δ x and a discrete time step δ t [25]. To
12

ease the correspondence between the three systems, the physical system is titled P, the
dimensionless D and the discrete LB. One could also go directly from the physical system (P)
to the discrete system (LB). But the variables δ x and δ t are important for the accuracy and
stability of the numerical simulation, and information regarding them may be lost if the direct
approach is used. The procedure for the transition is schematically summarized in Figure 2.4.
Figure 2.4: Transition scheme between the three systems [25].
To illustrate the process of choosing units, an example for a flow of an incompressible fluid
developed by Latt [25] is outlined here. For an incompressible fluid, the density can be
assumed to stay constant; does not vary in time and space, throughout the domain. The
Navier-Stokes equation is often chosen for describing the motion of the fluid and is governed
by the conservation of mass and momentum. The conservation of mass is here written as:
(2.18)
where u is the velocity of the fluid and the index P stands for the physical system. The
conservation of momentum is written as:
(2.19)
where p is the pressure, ρ the density and υ the kinematic viscosity [25]. The next step is to
convert the system into a dimensionless one. Further, two scales are introduced; a length scale
l 0 , e.g., approximate size of an obstacle, and a time scale t 0 , e.g., time to pass the obstacle.
These are used to scale the variables between the systems.
(2.20)
(2.21)
where x is a position vector in a numerical modeling environment [25]. Similarly, the other
variables are scaled and then replaced in the equation for conservation of mass and
momentum.
(2.22)
(2.23)
where the Reynolds number Re is defined as [25]:
13

(2.24)
where v is the viscosity. For two flows with equivalent geometry and same Re, these will
obey Reynolds transport theorem and provide equivalent solutions which can be converted
from one flow to the other. Finally, the dimensionless system can now be transformed into the
discrete system by defining a reference length step δ x and time step δ t . If the reference
variables, i.e., δ xD and δ tD , in the dimensionless system are defined, t 0D and x 0D will turn out to
have the value of unity, respectively. Then the reference variables in the discrete system are
defined as:
(2.25)
(2.26)
where N is the number of cells and N iter the number of iterations [25]. Now, the conversion is
easily handled through dimensionless analysis and the variables are defined as.
(2.27)
(2.28)
Both δ x and δ t are attached with some constraints in LBM. The value of u LB may not be larger
than the basic lattice speed of sound c even if the fluid is possibly compressible because LBM
does not support supersonic flows [25]. This leads to the relationship of the reference
variables as follows:
(2.29)
The LBM is a quasi-compressible solver which means it enters a slightly compressible regime
when solving the pressure equation without any significant impact on the numerical accuracy.
To specify how to choose δ t , it is important to note that the LB model is of second-order
accuracy. This means that the lattice error ε(δ x ) ~ δ x 2 . The compressibility error would take
over if the lattice error is reduced and therefore the errors should be kept at the same order,
i.e., ε(Ma) ~ ε(δ x ). This leads to a more specific relationship between the reference variables,
namely δ t ~ δ x 2 [25].
2.3.3 Mass diffusion in LBM
The next step here is to show the connection between variables for the mass diffusion. It
should be mentioned that other physical processes can also be handled by LBM. For example
thermal flow would be handled by setting the temperature as T ≡ ρ in equation (2.7).
Mass diffusion in LBM can be divided into two cases, namely; pure diffusion (u = 0) or
advection-diffusion (u ≠ 0) [26]. For LBM, this means that the equilibrium distribution
function will differ by containing the velocity or not in the equation (see equations (2.11) and
(2.12)). The LBM parameters in an equivalent system of lattice units should be such that
diffusion fluxes are in the same ratio as the actual ones, only larger in magnitude [15-16].
Mass diffusion in a mixture can be described by Fick’s law of diffusion. As Fick’s law is only
14

applicable for a mixture of two species the equation for the mass diffusion is better described
by the SMM [23-24]. The SMM is defined as:
(2.30)
where J i is the mole flux of species i, c T the total mole concentration, x i the mole fraction of
species i and D ij is the mass diffusivity between species i and j. Equation (2.31) is often
difficult to solve as it is, and therefore simplified forms are often applied to obtain an
analytical or numerical solution. In this case, the connection to the physical spectra is made
by the dimensionless diffusivity ratios, which are still in the same range as the actual
diffusivity but of a larger magnitude [14-16]. The other parameters in the LBM, such as J i and
X i are adjusted so that the same value of J* is maintained. For species i and j the
dimensionless mole flux J* is defined in equation (2.15) which is used as the main parameter
when the system is evaluated in the continuum regime [21, 23].
(2.31)
where the physical meaning of J* can be described as the mole fraction drop along the length
L and D ij is the mass diffusivity between species i and j.
2.3.4 Chemical reactions in LBM
Mass transport and chemical reactions play a significant role in modeling the anode of a fuel
cell. Here an approach is presented to solve the transport of a passive scalar reacting chemical
species connected to mass diffusion. They will be assigned separate particle density
distributions with different values of relaxation time and then they are coupled via the flow
velocity. Only a passive scalar transport is used so there will be no feedback of the species
distribution on the flow field [28]. A surface catalytic heterogeneous chemical reaction
between two species A and B is considered with a reaction rate proportional to the
concentrations of the species, C A and C B as:
(2.32)
(2.33)
where k is the reaction rate constant. This chemical reaction takes place only on the surface of
the porous media. The reaction coefficient is then only space-dependent in the LBM. The
differential equation for the reaction above is implemented to modify the local distribution
functions after the relaxation process [28]. The surface reactions are handled similarly as the
gravity force term in the particle density distribution by source term. However, by this
procedure it is only possible to simulate the reaction at the solid surface with a specified rate.
Other processes such as adsorption and desorption have been modeled by LBM with good
results by using simple local rules and explicit discretisation of the porous geometry.
2.3.5 General comments on CFD and LBM coupling
For fluid flows the CFD approaches, such as FDM, FVM and FEM, are solvers based on a
macroscopic discrete representation of the Navier-Stokes equation and at a mesoscale to
microscale level the LBM has evolved. For this example FVM is coupled with LBM by
connecting the boundaries but is can also be performed with FEM or FDM. The importance
15

lies to correctly set up the interface conditions and also to manage the conversion from the
lattice to the macroscopic variables or vice versa [28].
The positioning of the variables for the LB model and FV model is presented in Figure 2.5.
The FV model uses a staggered grid where the scheme is explicit for the velocity and implicit
for the pressure [25]. The choice of a staggered grid for the FV model is to prevent possible
pressure oscillation. The LB variables are evaluated at the corner positions for all lattice
variables at the same positions. For the FVM, the velocity variable is evaluated at the
interface of the grid cells between two corner positions and the pressure is evaluated at the
center of the grid cell [26]. Note that only a 2D domain is discussed here.
Figure 2.5: The indexes and grid positions for the LB and FV models variables.
This leads to the question of how to couple the two models at their interacting boundary. The
set of boundary nodes for the two models are connected by an overlap layer of nodes so linear
interpolation can be performed to find the effective boundary condition [26]. This overlap is
illustrated in Figure 2.6. Note that the overlap for the interface is about one and a half grid
cell to capture the physical processes at the boundary. This can be chosen arbitrarily
depending on the specific need of detailed information and high resolution, and the access to
computational power.
16

Figure 2.6: Effective boundary arrangement for the coupling of FVM and LBM.
The FV model is formulated with adimensionless system and so LBM should also be
converted to a dimensionless system to meet this constraint. The approach to connect the FV
model and LB model at the boundary is by linear interpolation because the variables are not
defined at the same positions in the domain. This gives the following relationships for the
velocity.
(2.34)
(2.35)
(2.36)
(2.37)
The procedure would start off, for the inner nodes, so that the incoming particle distribution
function f a at time t is used to compute the local density ρ and velocity u or v. For the
boundary nodes all three variables are obtained from the variables of the FV model at time t
[26, 28]. Next, all nodes are subject to the collision step. Finally, the inner nodes will perform
the streaming step.
2.4 Previous case studies of SOFCs in LBM
Although LBM is a relatively new actor among the numerical modeling methods, some work
has already been carried out on SOFCs. There are some limitations connected to the LBM
that need to be highlighted. These have been detected through previous studies. LBM has
only recently been used as a numerical method for transport processes in SOFC and compared
with conventional methods such as FDM, FEM and FVM [24]. According to Joshi et al., there
is still a need for a supercomputer to perform the LBM simulations. The method is described
in detail by Joshi et al. [14-16, 23] and a comprehensive discussion of the method and various
terms in the LB equation are offered. Hence, the reader is referred to the work of Joshi et al.
for further discussion.
17

A method at microscale needs detailed geometric information, and the size of computation
domain cannot be too large due to limited computer resources. Because each pore should
contain several lattice nodes in LBM and if the mass diffusion occurs in a large domain, the
method may be unsuitable. When the LBM is applied to solve the mass diffusion in the pores,
the gas velocity cannot be too high, due to the low Mach number limit for LBM [29]. In less
porous media with low pore velocity, the density gradient can be very large. Further problems
can occur when the resulting velocity field is applied to the simulation of the transport
process, because a non-physical process and a system which is not in chemical equilibrium
for the reaction process may be obtained [29]. The LBM has been shown to be less efficient
for steady-state problems. However, this is expected because it is an explicit time-marching
method that solves steady-state as the asymptotic solution in time. Accurate and quantitative
comparisons between numerical methods are complicated because different methods have
different underlying approaches so making statements on the relative merits of the numerical
methods must be done with care [29-30].
Table 2.1 is a summary of works on SOFCs done by different groups. For each work it is
highlighted how the reconstruction of the microstructure was carried out, which modeling
approach was used and at which dimension the work was studied. Further, Table 2.1 is
divided into what analysis level the work was conducted on and which equation methodology
was used in the microscale model. The last column contains some notes of conclusions made
in each work to summarize the current situation of previous work. It can be concluded from
Table 2.1 that there are basically three different approaches for the evaluation and
reconstruction of the microstructure of the porous electrode. Furthermore, these can be
grouped into a stochastic model where the structure is formed by a random statistical method
or grouped into a computer aided scanning model such as X-ray computed tomography
(XCT) and scanning electron microscope (SEM) [31-32]. The different studies are carried out
for all three dimensions, but are mainly focused on the porous anode or a simpler geometry
such as a channel. Because of the microscale modeling complexity, the focus is on mass
diffusion, charge transport and electrochemical reaction because their behavior dominates at
the microscopic level. The conclusions from previous studies relate mainly to the possibility
to actually perform a realistic reconstruction of the microstructure, to perform modeling at
microscale as well as to validate the model against other models. Overall, the LBM has shown
good agreement with both SMM and DGM for 2D-cases of simple geometries. Improvements
of the homogeneity of the anode microstructure as well as a better understanding of ionicelectronic
phenomena at the active surface are viewed as the next interesting area of study
[31]. To summarize, the LBM has proven to be a reliable microscale model for the complex
porous anode.
18

Table 2.1: Comparison of previous works of microscale models for SOFCs.
Author
Microstructure
reconstruction
Modeling
Dimension
Analysis level
Equation
analysis
Concluding comments
Suzue et al.
(2008)[27]
Stochastic
model
LBM
3D
Anode
(electrode
performance)
Mass diff.
Charge tr.
Electrochem.
Decrease in temp. → Current
concentration closer to TPB and
reactive anode becomes thinner.
Grew et al.
(2010)[14]
Not in article
LBM
2D
Anode
Mass diff.
Charge tr.
Electrochem.
Constant overpotential acts
limiting → Large resistivity and
unable to account for the coupled
transport processes
Joshi et al.
(2007)[15]
Stochastic
model
LBM, DGM
2D
Channel
Mass diff. (Kn)
LBM suitable for a wide range of
Kn in non- and continuum
regime. DGM not suitable for
porous geometry.
Joshi et al.
(2007)[16]
Stochastic
model
LBM, SMM
1D, 2D
Channel (straight,
tortuous, forked)
Porous geometry
Mass diff.
Good agreement for LBM and
SMM. LBM possible for porous
geometry without empirical
modification.
Joshi et al.
(2010)[24]
XCT
LBM
3D
Anode
Mass diff.
Good agreement for XCT and 2D
stochastic reconstruction.
Sohn et al.
(2010)[22]
Stochastic
model
LBM, RW
3D
Anode (micro)
Cell (macro)
Electrochem.
Energy (macro)
Mass
diff.(macro)
Nu and Sh similar behavior →
heat and mass transport coupling
needed. Micro/macro model
serves as a good modeling
approach.
Chiu et al.
(2009)[21]
XCT
LBM, SMM
2D
Anode
Mass diff.
Electrochem.
Int. reforming
Good agreement for LBM and
SMM with methane reforming
and electrochemical activity at
TPB.
Asinari et al.
(2007)[32]
SEM
LBM
3D
Anode
(electrode
performance)
Mass diff.
Microstructural reconstruction for
micro-modeling serves well for
the macro performance.
2.5 Electrode microstructure remarks
The particle size in SOFCs is in the range of microscale, and the TPBs are in nanoscale. The
morphology and the properties of these scales are important for the performance of the fuel
cell, because they control how much of the Gibbs free energy is available for use. The science
at nano and microscale is critical to the performance at a system scale, but it is problematic to
find a suitable and reliable kinetic model within a simplified framework which still fully
describes the interactions between the particles [12]. For example the kinetic model can be
constructed by combining chemical values for each species with computed activation energies
and transition-state properties. An important aspect of generation of steam reforming kinetics
19

data on YSZ-supported and Ni catalyst anodes is that they are prone to carbon deposition.
Small changes during the process of manufacturing can have an effect on the catalytic
characteristics. The structural features, for example the particle size distribution, have a strong
influence on the anodes catalytic and electrochemical characteristics [33]. Structural
parameters and conditions of the experimental cell are not always clearly stated in the
literature, which makes it hard to reproduce results numerically. Another issue is that the
original kinetic data is often taken from a variety of different catalysis studies which makes
the mechanism thermodynamically inconsistent. Due to this issue, some of the original kinetic
parameters are often modified to ensure the overall consistency of the enthalpy and entropy.
Averaged structural parameters, such as porosity and tortuosity, may have the same value for
many different microstructure topologies but the material structure and path ways may differ
significantly [32]. The correlations between the large numbers of operating conditions in
combination with the simplified structural parameters make it difficult to verify the diffusion
in practical ranges [14]. In summary, at the time of writing, no macroscopic parameters can
properly describe the microscopic physical behavior. One of these macroscopic parameters is
the tortuosity. The tortuosity factor is used to handle the discrepancy of the pressure loss for
the complex path ways in porous media and should include both shear forces and elongation
forces of the fluid elements in the flow. But even the meaning of tortuosity goes apart, where
in some models it represents the effect of additional pathways and in some models just a
numerical parameter to fit the experimental data [28]. Microscale models may enhance the
knowledge and bridge over this gap.
Another important remark is that the functional electrode structures are known to work in
favor of electrochemical reaction, mass and charge transport. Several earlier microscale
models have adopted a structural schematic spherical approach to reconstruct the porous
media [22]. The approaches have until recently been based upon statistical parameters from
the random spherical particles to create the model-connected physical parameters. The
structure can be created by a random statistical simulation, by size and location, or by
computer tomography X-ray absorption contrast (XCT), or similarly, by scanning electron
microscope (SEM) of a real existing microstructure of a SOFC electrode. The SEM requires a
great deal of effort in terms of measurement and data processing, and still the effect of
microstructures on the electrochemical activity remains unclear [27]. According to Asinari et
al. [32], the XCT does not provide good enough reconstruction for the microscale and the
only viable resolution for SOFCs is provided by SEM, but still XCT is cheaper, faster and
demands less effort. XCT can be improved by statistical regression, which is often used when
a 3D structure needs to be obtained by a 2D structure [32, 34].
To visualize in a basic schematic way on the microscale structure, it is schematically
described in Figure 2.6 by randomly situated spherical particles and transport of the different
species when hydrogen is fed. The electron flow is represented and the ionic and electronic
feature is represented by the binary particles. Part of the cell is divided into three microscopic
regions which are named as cathode functional layer (CFL), electrolyte and anode functional
layer (AFL). The functional layer defines the part of the anode or the cathode closest to the
electrolyte where most of the active reactions take place. A TPB location is enlarged to show
the different material components of an SOFC.
20

Figure 2.7: Schematic illustration of the microstructure components and the main processes.
The electrode performance can be increased if there is sufficient porosity so that gas transfer
is not limiting and so that the TPB needs to be maximized. While a fine microstructure and a
high surface area are clearly desirable, this can lead to low mechanical strength [35]. The
anode is often used as the mechanical support for the cell and a change of the anode structure
can be problematic. Because the active region in the anode where the electrochemical
reactions takes place, extends less than approximately 10 μm from the anode–electrolyte
interface, a graded porosity (like functional layers) is sometimes used to maximize the
amount of TPB in the active region while still maintaining a high mechanical strength for the
rest of the anode which is used primarily as the cell support [35].
Another effect captured by microstructure considerations is that the cell performance can be
permanently affected by the electric field. The pore formation in the material can be affected
by oxygen potential gradient at the cathode/electrolyte interface and nickel agglomeration at
the TPB. This will cause a degradation of the performance by the electrochemical reactions at
the TPB due to the sintering of metal particles, which causes a decrease in specific contact
area of Ni particles [34, 36].
21

3 Mathematical
models
This chapter presents the LB model visualization and equation methodology for the transport
processes. Finally, a validity check of the kinetic effects on the transport processes is
presented for interparticle, interphase and intraparticle transport within the microscopic range.
3.1 Model visualization for the microscale model
To visualize the complex geometry flow, the model discretisation is created from a digital
image. The digital image is created by a 3D computer tomography from a real object and is
shown in Figure 3.1. The complex structure of the Ni/YSZ anode was printed in grayscale
(256 colors) and through data conversion functional voxel information was obtained. The
conversion process was conducted in Python where the voxel data, numerical information
about the color at specific positions, was transferred to a matrix functional for MATLAB. To
distinguish the two phases (pore/solid) a phase function is defined at each pixel (z) as:
(3.1)
Figure 3.1: Digital image of a Ni/YSZ anode.
This border can be adjusted to obtain different values for the porosity which offered the
possibility to vary the porosity in the LBM simulation. To visualize the physical quantities
(velocity, concentration and pressure) these are obtained by the Lattice Boltzmann particle
distribution function and updated every iteration loop. In Figure 3.2, the image for the LB
22

model is presented after the data conversion from the original digital image (Figure 3.1) and
the choice of border between pore and solid, i.e., white and black, was chosen to obtain a
porosity of 40%.
Figure 3.2: Image for LBM by two colors, black (solid) and white (pore), with a porosity of 40%.
The conversion process for three colors, namely white, grey and black, was also performed in
Python to create a matrix functional for MATLAB. To distinguish the three phases for the
pore and the two solid types, a phase function is defined at each pixel (z) as:
(3.2)
In Figure 3.3 the real image of the XCT scan is shown. In Figure 3.4 and 3.5 the three-colorconversion
is implemented where the choice of border between pore and solid (both grey and
black) was chosen to obtain a porosity of 40% and 60%, respectively. The black colored
patches represents YSZ, the grey Ni and the white represents the pores.
Figure 3.3: Digital image of a Ni/YSZ anode.
23

Figure 3.4: Image for LBM by three colors (black, grey and white) with a porosity of 40%.
Figure 3.5: Image for LBM by three colors (black, grey and white) with a porosity of 60%.
3.2 Governing equations for the macroscale model
It is essential to connect the different transport processes when modeling SOFCs. The transfer
of fuel gases to the active surface for the electrochemical reactions is governed by different
parameters, such as the porous microstructure, the gas consumption/generation, the pressure
gradient between the fuel flow duct and the porous anode, and finally the inlet conditions [9].
The gas molecules diffuse to the TPB, where the electrochemical reactions take place. The
hydrogen concentration depends on the transport within the porous anode and the
heterogeneous reforming reaction chemistry [1]. In the following sections the main transport
processes are briefly described.
24

3.2.1 Mass transport
In the electrodes, mass transfer is dominated by gas diffusion and the transport takes place in
the gas phase, which is influenced by the electrochemical reactions at the solid surface at the
active TPB [33]. The transport phenomena can be classified into some general categories
based on the Knudsen number. For the porous layer, continuum phenomena are predominant
for the case with large pores, whose size is much bigger than the free path of the diffusion gas
molecules [11]. The Knudsen diffusion is used when the pores are small in comparison to the
mean free path of the gas. For Knudsen diffusion, molecules collide more often with the pore
walls than with other molecules [1].
Mass transport can be calculated using Fick’s law, which is the simplest diffusion model. But
for a multicomponent system, the Stefan-Maxwell model is often implemented to calculate
the diffusion [37-38]. Furthermore, when extended with the Knudsen diffusion term to predict
the collision effect by the molecules it is usually called the Dusty Gas model [5, 37, 39]. The
Stefan-Maxwell equation is a simplified equation of the Dusty Gas Model. The Knudsen term
is neglected because the collision between the gas molecules and the porous medium is not
considered. The Stefan-Maxwell equation is defined for the electrodes, the fuel and air
channels, as below [5, 22]:
(3.3)
(3.4)
(3.5)
where w is the mass fraction, D ij the Stefan-Maxwell binary diffusion coefficient, x the mole
fraction, D i T the thermal diffusion coefficient and S i the source term. S i is, in this case, zero
because the electrochemical reactions are assumed to take place at the interfaces between the
electrolyte and electrodes. Therefore, they are defined as an interface condition and not as a
source term. The diffusion coefficient in the electrodes is calculated as [40]:
(3.6)
where is the tortuosity. Moreover, D ij is calculated as:
(3.7)
(3.8)
(3.9)
(3.10)
25

where T is the temperature, P the pressure, σ AB the characteristic length and D the
dimensionless collision integral. The averaged molecular weight M AB between substance A
and B is defined as [40]:
(3.11)
Note that the pressure P is in bar in equation (3.7) and in our case P is set to 1 bar. Also note
that the parameters in this equation are not all in SI units. The ones which differ from the SI
unit standard here is M AB [g/mol] and σ AB [Å].
However, if the non-continuum regime is present, additional dimensionless parameters need
to be introduced in terms of the Knudsen diffusivity defined as [21]:
(3.12)
where is the pore mean radius, T the temperature, M i the molecular mass. Furthermore, the
dimensionless Knudsen diffusivity for species i in relation to species j is obtained by [19]:
(3.13)
3.2.2 Heat transport
The heat transfer within the whole unit cell consists of convection between the solid surface
and the gas flow, conduction in the solid and the porous parts, and heat
generation/consumption occurs due to the electrochemical reactions at the TPB as well as the
internal reforming reactions. The temperature distribution in this study uses a local thermal
non-equilibrium (LTNE) approach due to the low Reynolds number and large differences in
thermal conductivities between the gas and solid phases. The temperature distribution is
calculated separately for the gas and the solid phases. The general heat conduction equation is
used to calculate the temperature distribution for the solid medium in the porous electrodes [8,
22]:
(3.14)
where is thermal conductivity for the solid media, the temperature in the solid phase and
the heat source (heat transfer between the gas and solid phases, the heat generation due to
the ohmic polarization and due to the internal reforming reactions). The temperature for the
gas phase in the fuel gas and air channels and the porous electrodes are calculated according
to [22]:
(3.15)
where T g is the gas temperature, c p,g the specific heat, k g the gas thermal conductivity and Q g
the heat transfer between the gas and solid phases. The heat transfer between the gas and solid
phases is defined as:
(3.16)
26

where h v is the volume heat transfer coefficient and AV the active surface area to volume
ratio.
3.2.3 Momentum transport
The approach for analysis of the momentum transport is to solve the Darcy equation for the
porous electrodes and the Navier-Stokes equations for the channels. The Darcy-Brinkman
equation is then used to solve the gas flow in the gas phase [5, 33]:
(3.17)
where μ is the dynamic viscosity, κ the permeability of the porous medium, ε p the porosity, T
the viscous stress tensor and F the volume force vector. λ is the second viscosity and for gases
it is normally set to λ = -2/3∙μ. κ dv is the deviation from the thermodynamic equilibrium and is
by default set to zero. The Darcy-Brinkman equation is converted into the Darcy equation
when the Darcy number Da 0 in the porous layers and into the Navier-Stokes equation
when κ and ε p = 1 in the fuel and air channels [5].
The velocity profile is defined at the air and fuel channel inlets as laminar flow and the outlets
are treated as pressure surfaces. The boundaries at the top and the bottom of the cell model
are defined by symmetry because the cell is considered to be surrounded by other similar cells
with the identical temperature distribution. The temperatures at the air and fuel channels inlets
are defined as constant and the outlet boundaries are defined as convective flux surfaces.
3.2.4 Electrochemical reactions
The electrochemical reactions occur at the TPB. The function of the electrolyte is on the one
hand to transport the oxygen ions to the anode and on the other hand to prevent the electrons
to cross from the anode to the cathode. The flow of electronic charge through the external
circuit balances the flow of ionic charge through the electrolyte. This transport is described in
terms of the ion transport from the conservation of charge [11, 14, 22]:
(3.18)
(3.19)
where i io and i el are charge fluxes for ions and electrons, respectively, and ϕ io is the ionic
potential in the electrolyte. The Nernst potential is calculated as the sum of the potential
differences across the anode and the cathode as [41]:
(3.20)
where E is the reversible electrochemical cell voltage and ϕ the charge potential. At the
interface between the electrode and electrolyte the Butler-Volmer equation is used to
calculate the volumetric current density [22]:
(3.21)
where i 0 is the exchange current density, F the Faraday constant, β the transfer coefficient, n e
the number of electrons transferred per reaction, η act,e the electrode activation polarization
27

over-potential, and finally R the ideal gas constant. If the transfer coefficient β is assumed to
be 0.5, the Butler-Volmer equation is reduced to [2, 5]:
(3.22)
(3.23)
(3.24)
where k´´ is the pre-exponential factor and E the activation energy. The gas species
distributions are implemented by source terms due to the electrochemical reaction as [5]:
(3.25)
(3.26)
(3.27)
where i is the current density and F the Faraday constant.
3.2.5 Internal reforming reactions
The internal reforming reaction rates are taken into account by the source terms in the Stefan-
Maxwell equation. The mass source terms due to the reforming reactions are expressed as:
(3.28)
(3.29)
(3.30)
(3.31)
The equation for CO 2 can be solved separately because the sum of the mass fractions is equal
to unity. The reaction rate r r is for the catalytic steam reforming reaction and r s is for the
water-gas shift reaction.
The reaction rates for the methane steam reforming reaction are evaluated by kinetic models
and for the water-gas shift reaction an equilibrium approach is applied. The three reaction
kinetic approaches applied are from [41-45]. It is worth noting that both Achenbach &
Riensche’s [42-43] (equation (3.32)) together with Leinfelder’s [44] (equation (3.33)) kinetics
are an Arrhenius type kinetics reaction rate, while Dreschers kinetics [45] (equation (3.34)) is
a Langmuir-Hinshelwood type. They are selected on the basis of the different order of the
partial pressure and the broad range of the activation energy. The differences in kinetics
depend on how the experimental configuration is set up and, how the material decomposition
and operating conditions are selected. From the studies of Achenbach & Riensche [42] and of
28

Achenbach [43], it was found that the reaction order of the partial pressure of methane is
unity and the partial pressure of water has no catalytic effect on the reaction [42-43]. Note
that Leinfelder [44] found a positive reaction order of water and Achenbach & Riensche [42-
43] found a reaction order of zero. The reaction rates from these three different experimental
studies are shown below:
(3.32)
(3.33)
(3.34)
where p is the partial pressure and T s the solid phase temperature. AV is the active surface area
to volume ratio. The units for all the steam reforming reaction rates are mol/(s∙m 3 ).
The reaction rate kinetic models, equations (3.32) and. (3.33), consist of three parts: partial
pressures, pre-exponential factor and activation energy. These parameters differ substantially
in the literature among different research works. The pre-exponential factor describes the
number of collisions between the molecules within the reaction. The exponential expression
including the activation energy describes the probability for the reaction to occur. As the
activation energy increases, the catalytic reaction becomes less probable. The activation
energy is based on the catalytic characteristics, such as chemical composition. Even though
the activation energy may be high, leading to a decrease in the reaction rate, the overall
reaction rate can still be high due to the pre-exponential value. The pre-exponential factors
depend strongly on both the temperature and properties of the anode material. It is possible to
change the reaction rate, either by changing the particle size of the active catalysts or the
porous structure, i.e., the active catalytic area. The large difference between the activation
energies found in the literature, [1, 41-46] suggests that additional parameters have significant
influence on the reaction rate.
According to Nagel et al. [41] a small steam-to-carbon (SC) ratio yields positive reaction
orders and a high SC ratio yields negative reaction orders. For this study, the steam to carbon
ratio is around 2, which agrees with the three kinetic models. Achenbach & Riensche [42-43]
applied a 14 mm thick nickel cermet semi-disk consisting of 20 wt.% Ni and 80 wt.% ZrO 2
(stabilized). The active surface area was 3.86 ∙ 10 -4 m 2 . The temperature was varied from 700
to 940 ºC and the system pressure from 1.1 to 2.8 bar. Leinfelder [44] applied a 50 µm thick
anode built up by two layers with 64 wt.% Ni and 36 wt.% YSZ and 89 wt.% Ni and 11 wt.%
YSZ, respectively. The active surface area for the anode was 2.5 ∙ 10 -3 m 2 . The test was
conducted for temperatures from 840 to 920 ºC and at a pressure of 1 bar. Drescher [45]
studied an anode consisting of 50 wt% Ni and 8 mol% YSZ. Achenbach & Riensche’s model
is based on work on a reformer while the other two, Leinfelder’s and Drescher’s, are based on
a unit cell.
In this study, the temperature is varied from 727 to 827 ºC (1000 to 1100 K) because this is in
the range which the experiments were carried out. The active surface area to volume ratio is
varied 10 ∙10 4 - 5 ∙10 5 m 2 /m 3 . The active surface area to volume ratio has been adopted
according to a commonly used value in the literature [5-6, 39, 45]. Several authors have
29

applied an active surface area to volume ratio of 5 ∙ 10 5 for modeling work. Janardhanan and
Deutschmann [5] applied a slightly smaller surface area to volume ratio of 102500 m 2 /m 3 ,
whereas Klein et al. [46] applied a much larger value of 2.2 ∙ 10 6 m 2 /m 3 . Note that only a
small part of the whole active surface acts as a locus for the chemical reactions. The trend for
the development during the last years is in the direction of employing smaller particles to get
larger AV.
The water-gas shift reaction is considered to be very quick and to remain in equilibrium by
active several authors in the literature [2, 46-47]. The equilibrium approach in the fuel
channel and the anode can be defined as:
(3.35)
(3.36)
where k s is the reaction rate constant and K e,s the equilibrium constant for the water-gas shift
reaction. The value for k s is calculated according to Haberman and Young [48]. The unit for
the water-gas shift reaction rate is mol/(s∙m 3 ). The heat generation and heat consumption are
defined as source terms in the governing equations. The heat generation in the fuel channel
enters in the gas phase. The heat generation and the heat consumption are assumed to occur
on the solid surface. The heat generation and heat consumption due to the reforming reactions
are implemented in equation (3.14) and are defined as:
(3.37)
where Δh reac is the enthalpy change due to the reactions and Q int,ref the heat generation.
3.3 Heat and mass transfer limitations of the kinetic model
For fuel cell research it is interesting to identify whether any transport process at any level of
scale limits the whole process. An analysis on the basis of interparticle, interphase and
intraparticle heat and mass transport is performed to provide knowledge of the limiting steps
at each level. The different domains are explained by the division below [49]:
Interparticle, also called intrareactor, is defined between the local fluid regions or
catalyst particles.
Interphase is defined between the external surfaces of the particles and fluid adjacent
to them.
Intraparticle is defined within individual catalyst particles.
The structure and equation methodology of evaluating the limiting steps for heat and mass
transfer at different scales in SOFCs consists of catalytic kinetic equations in terms of criteria
obtained from experimental work by Mears [49]. The analysis explained by Mears [49] was
performed on a reactor bed. Here, it is transferred to the anode for the steam reforming
reaction and for the electrochemical reactions at the anode and the cathode of the SOFC. The
main difference between the two reactor environments is that a reactor bed has walls along
the flow direction, compared to an anode and a cathode that are supplied with fuel and air,
respectively. This means that special consideration needs to be applied to calculate the
interparticle heat and mass transport limitations along the flow direction. For the
interphase and the intraparticle heat and mass transport limitations the SOFC anode and
30

cathode are assumed to be similar to the case of the reactor bed. The aim of this case study is
to examine whether the kinetics used for previous models [1] fulfill these criteria so no
limiting effects occur for the heat and mass transport.
The macroscale (2D) computational fluid dynamics (CFD) model of an intermediate
temperature anode-supported SOFC operating on 30% pre-reformed natural gas is the base
for the calculations performed here. First, the criteria for the different domains are described
and defined below. Then, the results of the analysis for the SOFC are presented in the next
chapter.
3.3.1 Interparticle transport
The largest scale in this analysis is for the interparticle transport which is also sometimes
called intrareactor scale because it applies to gradients within the reactor as a whole.
Transport phenomena can occur both radially and axially within the reactor and these are hard
to control and evaluate. For the SOFC the axial direction refers to the main flow direction (xdirection)
and the radial direction refers to the direction normal to the main flow direction (ydirection).
But if neglected, radial temperature gradients can force the reaction rates to be
thousandfold greater for parts of the reactor often close to the axis [49]. For the SOFC this
would occur in the anode and near the electrolyte close to the inlet of the cell and would mean
a risk for disturbing “hot spots”. This can be checked by radial dispersion [50]:
(3.38)
where Bi R is the Biot number based on the reactor diameter, ΔH the enthalpy change of
reaction, r r the reaction rate, R o the reactor diameter, k e the thermal conductivity for the solid
porous media, T w the temperature at the solid surface and γ the dimensionless activation
energy. The axial dispersion is a less frequent limitation in a severe manner. Axial
temperature gradients and axial conduction are possible to neglect if the length-to-particle
diameter ratio is large enough (L/d p > 30) which is the case for SOFCs [50]. The criterion for
the limitation for the temperature gradient across the reactor diameter is defined as [49]:
(3.39)
where the parameters are the same as for equation (3.38) except the Biot number, R the gas
constant and E a the activation energy.
Mears [49] described an approach to adopt a differential reactor and this seems to be a very
useful approach for the SOFC reactor beds. A differential reactor consists of different
amounts of catalyst throughout the reactor bed to compensate for unfavorable effects such as
extremely high reaction rates in parts of the reactor. For the SOFC one would wish to level
out the reactions and the electricity generating action over the whole bed. This can be
achieved by either increasing the amount of catalyst or to use finer particles to increase the
reaction rate. However, the SOFC has contradicting needs for the reaction rates depending on
whether the focus is on the methane steam reforming reaction or electrochemical reactions.
For the steam reforming reaction, the reaction rate is very high at the inlet and then gradually
decreases in the flow direction for the cell. But the reaction rate for the electrochemical
reactions requires a higher reaction activity right at the inlet for a limited area which would
increase if more catalyst material was provided or finer particles were present. By adjusting
the reaction rate activity for its needs, severe effects of temperature or concentration gradients
could be minimized.
31

3.3.2 Interphase transport
The limitation of heat transport at the interphase transport level is normally less severe than
the interparticle transport level and the greater part of the resistance is often in the boundary
layer around the particle rather than within it [48]. This can be expected as the thermal
conductivity of the solid is often much larger than that for the gas. For a low Reynolds
number Re, Bizzi et al. [51] described the equation for the mass transfer coefficient k c as:
(3.40)
(3.41)
(3.42)
where ρ is the fluid density, v the fluid velocity, κ the permeability, μ the dynamic viscosity
and ψ the shape factor. For spherical particles it is assumed that the shape factor is ψ = 1 [51].
Then equation (3.41) reduces to:
(3.43)
The definition of the criterion for the concentration gradient across the gas boundary film
along the particle is [49]:
(3.44)
where r r is the reaction rate, r p the particle diameter, C the concentration, D eff the effective
diffusivity and n reaction order. The Damköhler number for interphase mass transport is
defined as [49]:
(3.45)
where, besides those mentioned in equation (3.44), k c is the mass transfer coefficient. The
Damköhler number for interphase heat transport is defined as [49]:
(3.46)
where ΔH is the enthalpy change of reaction, h heat transfer coefficient and T the bulk
temperature. Further, the dimensionless activation energy is defined as [49]:
(3.47)
where E a is the activation energy, R the gas constant and T the bulk temperature. The
dimensionless maximum temperature rise is used in the criterion for the concentration
gradient across the gas film and the criterion is defined as [49]:
32

(3.48)
where, besides the parameters mentioned above which are the same, k s is the thermal
conductivity of the particle. Mass transport cannot be a significant limitation unless the
effectiveness factor is low for the intraparticle range. A criterion for the heat transport can be
formulated as [49]:
(3.49)
where all the parameters are the same as mentioned above.
To detect the limitation for the boundary layer it is necessary to find out if transport
limitations exist in the particle or not. Heat transport limitations can be expected when the
reaction rates are high and the flow rates are low (small h) [49]. The criterion for the
interphase transport for the heat transfer is defined as [49]:
(3.50)
where h is the heat transfer coefficient, d p the particle diameter and k s the thermal
conductivity of the particle. It can be safely assumed a uniform temperature distribution in the
solid part if the Bi is less than 0.1 for an SOFC. But if Bi is larger than 10, then the
conduction resistance dominates which will generate temperature gradients in the solid
particle. The convective gas particle heat transfer coefficient h is defined as [52]:
(3.51)
where Nu is the Nusselt number, k f the thermal conductivity of the gas and D o the hydraulic
diameter of the whole reactor bed, e.g., anode or cathode. The Nu in this case is calculated as
[52]:
(3.52)
where Re is the Reynolds number, and Pr the Prandtl number which are defined as in [52]:
(3.53)
where μ f is the dynamic viscosity of the gas, c pf the specific heat of the gas and k f the thermal
conductivity of the gas. Mears [49] pointed out that the heat transfer over the boundary layer
causes larger deviation from the criterion long before the mass transfer limitations.
3.3.3 Intraparticle transport
The smallest scale for the analysis in this work is for the intraparticle transport and it has been
more widely studied for reactor beds than the two previous scales. When diffusion might have
a strong influence, the objective is often to calculate the effectiveness factor which is shown
to be inversely proportional to the characteristic dimensions of the particle. Though, this
33

approach often requires detailed kinetic behavior which is close in representation to the
realistic kinetic catalysis.
The heat transport limitation is evaluated first by the larger scale interparticle transport. If the
criterion is fulfilled for the heat transport in the range for the interparticle scale, then there is
no risk for too high temperature gradient across the reactor y-axis for the intraparticle
transport, since R o >> r p and k e approaches the value of λ at low Reynolds numbers. The
interparticle transport criterion for heat transport is considered much stricter than that for
intraparticle transport [49].
The mass transport at the intraparticle scale range is analyzed for the internal diffusion within
the SOFC anode and the Knudsen diffusion is taken under consideration in the calculations.
The effective diffusivity by Knudsen diffusion is defined as [51]:
(3.54)
where d p is the particle diameter and the molecular weight M AB of substance A and B is
defined as in equation (3.11). The effective diffusivity which is based on the ordinary
diffusion is defined here as in equation (3.7). Both of the effective diffusivities are then
averaged as below [51]:
(3.55)
The averaged effective diffusivity is needed in the Thiele Modulus and is here defined as in
[53]:
(3.56)
where r p is the particle radius, k c the mass transfer coefficient, C the concentration, n the
reaction order and D eff the effective diffusivity.
The effectiveness factor is defined in words as [49, 51]:
It is calculated as:
(3.57)
where Φ is the Thiele Modulus. Also, to ensure η ≥ 0.95 it is required that [49]:
(3.58)
34

for an isothermal spherical particle where r r is the reaction rate, r p particle radius, C the
concentration and D eff the effective diffusivity. The results for the control of the criteria for
the heat and mass transfer limitations are provided in the next chapter.
35

4 Results
and discussion
This section presents the results from the LB model and the criteria for the kinetic parameters
are checked so that no critical effects occur on the transport processes. Both the LB model
and the validation of the kinetic effects are viewed from a microscale perspective. Also the
results of the macroscale model are provided and divided into two parts; change of internal
reforming reaction rate model and change of amount of methane content and steam-to-fuel
ratio.
4.1 Microscale model by LBM
The LB model stops when the maximum deviations of the mean velocity differ with less than
10 -10 over the last iteration. Reynolds number is calculated based on the velocity and is
relatively low Re typically in the order of 0.1 to 1. The physical geometry of the LB model
and material data for the anode is presented in Table 4.1. In the LB model discrete units are
used for the length and time. The lattice unit lu represents the fundamental measure of length
and time step ts the measure of time.
Table 4.1: Anode geometry and relevant parameters [1].
Anode
Size
Length 40 lu, 200 lu 1
Height 10 lu, 50 lu 1
Porosity (ε) 0.4
Inlet mole fraction
H 2 0.9
H 2 O 0.1
The study is conducted in an order of increasing complex geometries to validate the method
for future modeling of all the physical processes in an SOFC. First of all, a small test is
1 Two different values for this parameter are tested.
36

Velocity
carried out to check whether the LBM shows comparable results with an analytical solution.
For this case, the velocity profile for a channel is modeled by LBM and compared to the
analytical solution of a Poiseuille flow. The test can be seen as comparable to a fuel cell
channel. The results can be seen in Figure 4.1 and the agreement between the two velocity
profiles is good.
0.12
0.1
0.08
0.06
0.04
u x
(Analytical Poiseuille)
u x
(LBM)
0.02
0
0 5 10 15 20 25 30 35 40
Distance [nodes]
Figure 4.1: Velocity profile [lu/ts] for a cross section in the middle of a channel domain compared to
the analytical solution of a Poiseuille flow.
Secondly, a circular obstacle is placed in the channel to increase the complexity of geometry.
In Figure 4.2 shows the flow field past a circular obstacle in a channel. The case was able to
be simulated by LBM with good results in MATLAB. Both the wake and the no-slip at the
walls are obtained efficiently.
37

0.12
0.1
0.08
0.06
0.04
0.02
Figure 4.2: Velocity contours [lu/ts] in a channel with a circular obstacle.
0.525
0.52
0.515
0.51
0.505
0.5
Figure 4.3: Mole fraction in a channel with a circular obstacle.
In Figure 4.3 a simple mass diffusion case is provided for the channel with a circular obstacle
where the convection-diffusion approach is applied. The component enters with almost its
maximum of 0.525 and diffuses continuously through the channel. Around the obstacle, there
is a slight bounce-back before and a wake behind. This case is just a test case with fictitious
numbers and provides useful information of LBMs functionality.
38

Figure 4.4: Velocity field [lu/ts] in part of the porous media domain with a porosity of 0.40.
Thirdly, a porous geometry is tested for simulation of an SOFC anode. In Figure 4.4, a porous
domain is provided with a porosity of 0.4. Here the velocity field is given to illustrate that
LBM can easily handle a porous domain. Note that only part of the porous domain is shown
from Figure 3.2 to see the velocity arrows better and the following Figure 4.5 and 4.6 show
the whole modeled domain as can be seen in Figure 3.2. The velocity arrows provide an
understanding of the bounce-back theory and provide intuitive feeling for the flow process in
the porous media.
0.8
0.75
0.7
0.65
0.6
0.55
0.5
0.45
0.4
Figure 4.5: Mole fraction distribution of H 2 in a porous media with a porosity of 0.40.
39

In Figure 4.5 shows the whole modeled porous domain for the mole fraction distribution of
H 2 . The inlet mole fraction is specified as x H2 = 0.9 and the mole flux is specified at the
outlet. Note that no reaction effects are included in the model. The mass diffusion of
hydrogen predicts, in a similar manner as the simpler channel case, the continuous reduction
of hydrogen along the flow direction and the contours of the mole fraction around the
obstacles normal to the surface indicates that mass diffusion occurs parallel to the surface.
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
Figure 4.6: Mole fraction distribution of H 2 O in a porous media with a porosity of 0.40.
In Figure 4.6 the mole fraction distribution of H 2 O is shown for the whole modeled domain.
The inlet mole fraction is specified as x H2O = 0.1 and the mole flux is specified at the outlet in
a negative direction, i.e., normal to the boundary interface in opposite flow direction. Note
that no reaction effects are included. This is a heavier species where the diffusion is not as
fast as for the hydrogen case. Only the interaction between the two species is illustrated in this
case. The local spots with higher mole fraction are because these are a closed pores and the
velocity effect is very low there. For future studies, the production and consumption of
species will be included and the effect of the chemical reactions on the mass density
distribution.
4.2 Evaluation of kinetics at smaller scales
As mentioned before, the analysis is carried out for the steam reforming reaction in the anode
and the electrochemical reactions in the anode and cathode. The most significant parameters
concerning the cell structure and catalytic activity, which are used for the calculations, are
presented in Table 4.2, while the inlet and operating conditions are presented in Table 4.3.
Note that the x-direction is the main flow direction and the direction set at the inlet. The y-
direction is normal to the main flow direction.
40

Table 4.2: Parameters in the SOFC analysis.
Parameter
Value
Cell length
Anode thickness
Cathode thickness
Particle diameter
0.1 m
500 μm
50 μm
0.34 μm
Anode TPB thickness (y-dir) 10 μm, 1 μm 2
Cathode TPB thickness (y-dir) 10 μm, 1 μm 2
Tortuosity 3
Porosity 0.3
Velocity v x,A
Velocity v y,A
Velocity v x,C
Velocity v y,C
Enthalpy change (MSR 3 )
Activation energy (MSR 3 )
Activation energy (CE 4 )
Activation energy (AE 5 )
Reaction rate (MSR 3 ), close to inlet
Reaction rate (CE 4 )
Reaction rate (AE 5 )
6.9 ∙10 -5 m/s
1.5∙10 -3 m/s
1∙10 -3 m/s
7.5 ∙10 -4 m/s
226 kJ/mol
82 kJ/mol
71 kJ/mol
185 kJ/mol
15 mol/m³/s
0.53 kmol/m³/s,
5.3 kmol/m³/s 2
1.06 kmol/m³/s,
10.1 kmol/m³/s 2
It should be noted that for the thickness of the anode and cathode TPB as well as the reaction
rate for the anode and the cathode have been tested for two different values of each
parameter.
2 Two different values for this parameter are tested.
3 MSR stands for methane steam reforming reaction
4 CE for the electrochemical reactions in the cathode
5 AE for the electrochemical reactions in the anode
41

Table 4.3: Inlet and operating conditions.
Parameter
Value
Fuel utilization 80 %
Oxygen utilization 20 %
Inlet mole fraction of methane 0.171
Inlet mole fraction of hydrogen 0.2626
Inlet mole fraction of water 0.4934
Inlet mole fraction of carbon monoxide 0.0294
Inlet mole fraction of carbon dioxide 0.0436
Inlet temperature
Pressure
Average current density
1000 K
1 atm
3000 A/m²
The results are presented in Table 4.4 and commented below for all levels of the analysis of
the kinetic criteria.
Table 4.4: Results from the criteria analysis.
Transport domain Heat transport Mass transport
Interparticle
No limitation
–
Equation (3.40): ~10 -5 –10 -9 < ~10 -2
Interphase
No limitation
Bi = 2 – 4∙10 -2 < 10
No limitation
Equation (3.45):
~10 -7 –10 -8 < ~1–10 -2
Intraparticle No limitation No limitation
Equation (3.61):
η ≈ 0.999 > 0.95
The results are homogeneous with no limitations for any of the involved scales and reactions.
As the criterion is fulfilled for the heat transport at the interparticle transport level then no
limitation occurs for the heat transport at interphase or intraparticle level. The main difference
between the steam reforming reaction and the electrochemical reactions is that the latter ones
have a somewhat higher reaction rate. These reactions occur only in a small part of the
anode/cathode domain and the reaction rate is highest at the boundary between the anode and
electrolyte or between cathode and electrolyte. But the criteria for the electrochemical
42

eactions are safely fulfilled. This is a good verification of the chosen parameters for the
computational model examined in previous studies.
However, it is interesting to reflect over which parameters can form a potential risk for
limiting the transport processes. For the heat transport at all the domains, a larger enthalpy
change or an increased reaction rate can increase the risk. Also, a lowering of the temperature
can cause an increased risk but this is less significant. For the mass transport, an increased
reaction rate or a decreased concentration of reactant can cause a higher risk. As mentioned
before the effectiveness factor is affected by the characteristic dimensions of the particle
which can also be confirmed here. If a change in particle diameter causes a change in the
reaction rate, this may have a strong influence on the transport processes [6].
To summarize, the elimination of transport gradients which limit the reaction and catalytic
kinetics is complex to study. This case study, by tools to assess the transport limiting issues,
seeks to locate the limiting sources and improve these for the desired outcome. The reaction
rate is the most direct risk for limitation on the transport processes. If the reaction rate is
increased it will affect every criterion in the analysis and can cause severe gradients which
will create transport limitations. The anode and cathode structure and catalytic characteristics
have an impact on the reaction rates, especially on the steam reforming reaction, which will in
turn affect the cell performance.
4.3 Macroscale model by CFD
A two-dimensional model for an anode-supported SOFC has been developed and
implemented in the commercial software COMSOL Multiphysics (version 3.5a). Equations
for momentum, mass and heat transport are solved simultaneously. The cell geometry and
SOFC operating parameters are defined in Table 4.3. It should be mentioned that this
macroscale model is 2D only, and the connection between the electrodes and interconnect
cannot be explicitly observed in this case.
4.3.1 Case study: Internal reforming reaction rates
The flow direction is set to be from left to right for air and fuel channels as well as the anode
and the cathode. It is also possible with counter flow but this is not included in this study. It
should be explicitly mentioned that the length of the cell is 100 times longer than the height of
the air or the fuel channel.
43

Figure 4.7: Temperature distribution for Leinfelder’s kinetics.
The predicted gas phase temperature in the cell is plotted in Figure 4.7 for Leinfelder’s
kinetics. Achenbach & Riensche’s and Drescher’s kinetics are not shown here due to
similarity in the plots. In the fuel and air channels, there is a decrease in temperature after a
short distance from the inlet. In the fuel channels it is due to the steam reforming reaction,
which consumes the heat when the methane is reformed to hydrogen and carbon monoxide.
The temperature on the air side is lower due to a higher air flow rate which affects the
convective heat transfer. The decrease in temperature close to the inlet is 50 K for both
Achenbach & Riensche’s and Leinfelder’s kinetics. The temperature distribution for
Drescher’s kinetics does not drop initially as much as the other two. The area of the
temperature drop is larger for Achenbach & Riensche’s kinetics than those from Leinfelder’s
and Drescher’s kinetics. But the recovery to a higher temperature occurs faster for both
Leinfelder’s and Drescher’s kinetics than for Achenbach & Riensche’s kinetics. This might be
due to the fact that the latter is affected by the fast conversion of methane to hydrogen and
carbon monoxide.
44

Figure 4.8: Mole fraction distribution in the middle of the anode along the flow direction for
Achenbach & Riensche’s (left) and Drescher’s kinetics (right).
The effect on mole fraction distribution for the different gas species is similar for both
Achenbach & Riensche’s and Leinfelder’s kinetics and therefore only the mole fraction
distribution for Achenbach & Riensche’s (left) along with Drescher’s kinetics are presented in
Figure 4.8. Drescher’s kinetics obtained the maximum mole fraction of hydrogen faster and a
higher maximum mole fraction than Achenbach & Riensche’s and Leinfelder’s kinetics. The
initial consumption of water and the initial generation of hydrogen for Drescher’s kinetics
result in larger gradients of the mole fractions. All three kinetics are fast although Drescher’s
kinetics, expressed by a Langmuir-Hinshelwood type, differs slightly more from the others. It
deserves to be pointed out that Drescher’s kinetics includes both positive and negative orders
of the partial pressure of methane and water, as well as two different activation energies for
the denominator and the numerator, which can have some effect on the results.
Figure 4.9: Reaction rate distribution in the middle of the anode along the flow direction for Achenbach
& Riensche’s (left) and Drescher’s model (right).
45

The reaction rates for both the steam reforming reaction and the water-gas shift reaction are
plotted in Figure 4.9 for Achenbach & Riensche’s (left) and Drescher’s kinetics (right). It
should be clearly noted that the reaction rates are only plotted for the entrance region, through
0.01 m. Close to the inlet where the concentration of methane is high the reaction rate for the
steam reforming reaction is high. The reaction rate for the steam reforming and the water-gas
shift is much higher for Drescher’s kinetics than for both Achenbach & Riensche’s and
Leinfelder’s kinetics. Leinfelder’s kinetics is higher than Achenbach & Riensche’s. Close to
the inlet in the anode where the carbon monoxide generation is high, the reaction rate for the
water-gas shift reaction is at the highest. The high generation of carbon monoxide is due to
the steam reforming reaction. Furthermore, more hydrogen is produced when steam is
generated due to the fact that the water-gas shift equation is in equilibrium through the
process. As hydrogen is consumed, steam is generated thanks to the electrochemical reaction
at the TPB. The reaction rate for the water-gas shift reaction reaches a higher value due to the
faster reaction rate for the steam reforming reaction of the Drescher’s and Leinfelder’s
kinetics compared to Achenbach & Riensche’s. The comparison between the different kinetic
models needs to be evaluated on a more detailed level as it cannot be correctly explained by
just a few empirical parameters, such as the activation energy and the pre-exponential value.
To fully understand the effect and dependence of the parameters, microscale modeling is
needed. What can be concluded from this study is that the configuration and geometrical
properties of the anode and the chemical composition and catalytic characteristics are
important. To draw firm conclusions from the modeling work, it is important to reveal the
difference of the kinetic models from experimental work carried out on SOFC and a reformer
based on the same properties.
A parameter study was carried out for Leinfelder’s kinetics by increasing the inlet
temperature by 50 K. The other parameters were kept the same as in the base case. The
temperature distributions for both cases result in similar effects but obviously resulted in a
higher temperature range. The mole fraction distribution of the fuel gas species was
maintained in the same range and trend as the base case at 1000 K. The reaction rates are
slightly higher for a higher inlet temperature. Due to the increased inlet temperature the
maximum reaction rates are almost doubled compared to the base case at the inlet.
Another parameter study was conducted for the active surface area-to-volume ratio (AV) from
10 ∙10 4 to 5 ∙10 5 m 2 /m 3 , which is a frequently used interval in the literature. All the other
parameters were kept the same as the base case. The temperature profile and each mole
fraction profile were distributed in a similar overall trend as for the base case. The mole
fractions reach approximately the same maximum value for different AV but occur at different
distances from the inlet. A higher ratio moves in the maximum of all the mole fraction species
closer to the inlet. The characteristics of reaction rates for Leinfelder’s kinetics with an
increased AV are distributed similarly to the base case, but the maximum value is more or less
doubled for an increased active surface area to volume ratio.
4.3.2 Case study: Methane content and steam-to-fuel ratio
A case study was performed to simulate biogas as a fuel by varying the amount of methane
and SF. Three cases were performed, all with 60% CH 4 content but the SF was varied
between 1, 3, and 5. Additionally, three cases with SF equal to 3 but CH 4 varied as 45%, 60%
and 75%. For all the cases, a small fraction of hydrogen is added to enable electrochemical
reactions close to the inlet as well. CO is also added, similarly as H 2 , to achieve numerical
stability. Steam was added to avoid carbon deposition and also to be used in the reforming
reactions. The amount of H 2 O was calculated from the relationship SF = [H 2 O]/[CH 4 ] as SF
is specified for the different cases. In Table 4.5 the inlet mole fractions are presented for the
different cases with varying amount of methane content and SF.
46

Table 4.5: Inlet mole fractions for the different case studies.
Case models CH 4 CO 2 H 2 O
CH 4 0.45, SF=3 0.191 0.235 0.574
CH 4 0.60, SF=3 0.214 0.143 0.643
CH 4 0.75, SF=3 0.231 0.076 0.692
CH 4 0.60, SF=1 0.375 0.25 0.375
CH 4 0.60, SF=5 0.15 0.1 0.75
Similar to the previous section, the fuel flow rates for the different cases were calculated to
keep the fuel utilization at 80 percent. Note that each molecule of methane corresponds to a
generation of four molecules of hydrogen by both the steam reforming and water-gas shift
reaction and each molecule of carbon monoxide corresponds to one molecule generation of
hydrogen. The flow rate of air was kept constant for all cases and the oxygen utilization is set
to 20 percent. The inlet temperature was set to 1100 K to ensure functional conversion of the
fuel and the current density was kept constant at 3000 A/m 2 . The Knudsen diffusion was
neglected in this model to reduce computational cost. For both the steam reforming reaction
and the water-gas shift reaction, the equilibrium model is chosen for the kinetic model.
Figure 4.10: Mole fraction distribution in the middle of the anode for 45% CH 4 (left) and 75% CH 4
(right).
In Figure 4.10, the mole fractions at the centerline of the anode along the whole cell length
are presented. The change of H 2 O between the different cases is directly connected to the
methane content, in this case, to ensure significant steam at the inlet of the cell. Depending on
the inlet fraction a decrease in the mole fraction of water can be observed close to the inlet.
The biogas contains no hydrogen in the collected data but for numerical calculations the
hydrogen is initially set to have a small inlet value to enable the electrochemical reactions at
the inlet of the cell. The variations of the mole fractions are however quite small and it is
mostly CO 2 that changes. The reason for this is that CO 2 is linked to the choice of CH 4 and
47

H 2 O. Both CO and H 2 are initially set to small values to enable the numerical calculations, but
CO 2 and H 2 O are decided in accordance with the chosen methane content and SF.
Figure 4.11: Reaction rate distribution in the middle of the anode for 45% CH 4 (left) and 75% CH 4
(right).
The reaction rates for 45% and 60% methane contents are rather similarly distributed, but in
the case of 75 % methane the steam reforming reaction rate has initially higher values and
much lower values at the outlet of the cell. Also, the water-gas-shift reaction rate for high CH 4
content increases to the maximum value closer to the inlet of the cell than it does for the other
two cases. For the situations 45% and 75% content of methane shown in Figure 4.11, the
steam reforming reaction rate (within the anode) is high as long as a high concentration of
methane is available. The reaction rate increases as the temperature and concentration of
steam increase, and decreases as the concentration of methane decreases. Note that there is a
difference in scale between the x- and y-axes. It is possible to change the reaction rate, either
by changing the particle size of the active catalyst, catalytic material composition or the
porous structure, i.e., the active catalytic area. The limitation to be considered is that the
probability of carbon deposition increases where there is almost no hydrogen present. A
higher risk for carbon deposition occurs when there is a high temperature gradient close to the
cell inlet. This is not the case here as the gradients are not so high.
48

Figure 4.12: Mole fraction distribution in the middle of the anode along the flow direction for SF = 1
(left) and SF = 5 (right).
Only the two extreme cases are shown here to visualize the effect of a change in SF. In Figure
4.12 the mole fraction for SF=1 can be viewed to the left and SF=5 to the right. It is
important to verify that there is a sufficient amount of H 2 O throughout the cell or else no
efficient reaction will occur and there will be a risk for carbon deposition. In the figure for the
mole fractions, it can be seen that a drop of H 2 O exists slightly downstream the inlet. Instead
of putting all focus on the inlet mole fraction of H 2 O, one should also consider whether there
is a sufficient amount to handle this drop and adjust the inlet mole fraction subsequently. This
mole fraction drop increases when a faster reaction rate is applied which was shown in the
previous section.
Figure 4.13: Reaction rate distribution in the middle of the anode for SF = 1 (left) and SF = 5 (right).
49

The reaction rates for SF=1 (left) and SF=5 (right) are presented in Figure 4.13. It should be
mentioned that the temperature distribution was overall quite similar for the cases. When
SF=1 the temperature was overall lower for the larger part of the cell. For the case with
SF=5, the lowest temperature was much closer to the inlet than for SF=1 and the three cases
with varying methane content. The temperature distribution has an effect on the reaction rates.
The profiles of the reaction rates are quite different. For SF=1, the maximum rate value of the
water-gas-shift reaction is not reached until just upstream from the outlet but, on the other
hand, for SF=5, it is reached close to the inlet. The steam reforming reaction shows the same
tendency but it is slightly higher for SF=1 to begin with. The water-gas-shift reaction rate was
low because the mole fraction of CO is initially so small. Furthermore, the water-gas shift
reaction is connected to the steam reforming reaction, which affects the profile throughout the
cell.
50

5 Conclusions
The physics and the transport processes in SOFCs can be described at different length and
time scales. This constitutes a challenge for the development of multiscale models for fuel
cell simulations. In this study, a LBM microscale model was developed for the D2Q9 case
(two-dimensional nine speed case). The kinetic model was examined so that no severe
limiting effects on heat and mass transport occurred. Also, a FEM based model for an anodesupported
SOFC was developed to better understand the internal reforming reactions of
methane and the effects on the transport processes. The model was implemented in COMSOL
Multiphysics for the analysis of three different kinetic models found in the literature. An
equilibrium equation was employed for the water-gas shift reforming reaction rate. Parameter
studies were also conducted for the methane content and SF.
Five conclusions can be made in this study. First, LBM was found to be a functional method
to microscale modeling predicting the velocity profile and mass diffusion well. LBM could
handle both a simple geometry as a channel to a more complex geometry such as a porous
media. For the velocity field, the LBM was able to illustrate the flow correctly around the
obstacles. The mass diffusion for hydrogen was reduced from the inlet to the outlet as
expected and contours were seen around the obstacles where mass diffusion of hydrogen
occurred parallel to the surface. The detailed information from LBM at microscale regarding
the transport processes and chemical reactions can improve the macroscale model by
including this information for the TPB areas.
Second, it was shown that the reaction rates were very fast and differed slightly across the
three models due to the great differences of the pre-exponential value and the activation
energy. The model was found to be sensitive to variation of the steam reforming reaction rate.
Both the inlet temperature and active surface area to volume ratio showed an effect on the
reaction rates in terms of the maximum value.
Third, it was found that a fuel containing a high percentage of methane in combination with a
high inlet temperature produced a steep temperature gradient close to the cell inlet. Fourth, a
higher steam-to-fuel ratio showed a decreased risk of carbon deposition at the anode catalytic
active area.
Finally, there was no direct significant risk for heat and mass transport limitations for the
SOFC model with the kinetic parameters in this study. Care should be taken if the reaction
rate is increased since this will affect almost every criterion in the analysis. It transpired not to
be sufficient only to describe the reaction rates with a few empirical parameters. It was
necessary to develop a suitable microscale model for the SOFC. However, the global kinetic
models have still predicted valuable behaviors. The reason why the kinetics models differed
to a large extent is that they were sensitive to how the experiment was designed.
51

6 Future
work
Future work will involve a study of an SOFC at multiscale which will offer promising
knowledge to understand in detail the effect of design. To approach a successful electricity
producing device with improved durability and life time, the understanding of multiscale
transport and reaction phenomena within the cell is crucial. The next step is to model the
involvement of microscale thermal diffusion through LBM connected to a macroscale CFD
model. In the extended model the Knudsen diffusion, which describes collisions between the
gas molecules and the porous structure (inside the porous electrodes), is also taken into
account. Also the electrochemical reactions are prospected to contribute to capture valuable
microstructural effects. These reactions occur at a limited part of the cell, the TPB, which can
only be captured if modeled at microscale or smaller. Here the Monte Carlo method could
offer advantages to improve the multiscale development.
Another extension of the model is to include catalytic chemical surface reactions (instead of
global kinetics expressions). These surface reactions can provide knowledge of the interaction
between the transport processes and the reactions which will be valuable for fuel cell model
development. The reforming reaction rate is dependent on temperature, concentrations, type
and catalyst available. If the chemical reactions can be simulated on a microscale level, it
would open up to involve all the detailed multistep chemical reactions. More knowledge and
understanding of the effect behind the activation energy is important to enable reduction of
the operating temperature. Physical and material properties are calculated from data found in
literature and therefore experimental work is desired for validation of the model.
52

7 References
[1] Andersson M., Yuan J., Sundén B., Review on Modeling Development for
Multiscale Chemical Reactions coupled Transport Phenomena in Solid Oxide Fuel
Cells, Applied Energy, 87, pp. 1461-1476, 2010
[2] Lehnert W., Meusinger J., Thom F., Modelling of Gas Transport Phenomena in
SOFC Anodes, J. Power Sources, 87, pp. 57-63, 2000
[3] Sukop M.C., Thorne D.J. Jr., Lattice Boltzmann Modeling – An Introduction for
Geoscientists and Engineers, Springer, NY, U.S., 2007
[4] Sanchez D., Chacartegui R., Munoz A., Sanchez T., On the Effect of Methane
Internal Reforming in Solid Oxide Fuel Cells, Int. J. Hydrogen Energy, 33, pp. 1834-
1844, 2008
[5] Janardhanan V., Deutschmann O., CFD Analysis of a Solid Oxide Fuel Cell with
Internal Reforming, J. Power Sources, 162, pp. 1192-1202, 2006
[6] Hecht E., Gupta G., Zhu H., Dean A., Kee R., Maier L., Deutschmann O., Methane
Reforming Kinetics within a Ni-YSZ SOFC Anode Support, Applied Catalysis A:
General, 295, pp. 40-51, 2005
[7] Yang Y., Du X., Yang L., Huang Y., Xian H., Investigation of Methane Steam
Reforming in Planar Porous Support of Solid Oxide Fuel Cell, Applied Thermal
Engineering, 29, pp. 1106-1113, 2009
[8] Yuan J., Huang Y., Sundén B., Wang W.G., Analysis of Parameter Effects on
Chemical Coupled Transport Phenomena in SOFC Anodes, Heat Mass Transfer, 45,
pp. 471-484, 2009
[9] Yuan J., Sundén B., Analysis of Chemically Reacting Transport Phenomena in an
Anode Duct of Intermediate Temperature SOFCs, ASME J. Fuel Cell Sci. Tech., 3,
pp. 687-701, 2006
[10] Nam J. H., Jeon D. H., A Comprehensive Micro-scale Model for Transport and
Reaction in Intermediate Temperature Solid Oxide Fuel Cells, Electrochim. Acta, 51
pp. 3446-3460, 2006
53

[11] Kakac S., Pramuanjaroenkij A., Zhou X., A Review of Numerical Modeling of Solid
Oxide Fuel Cells, Int. J. Hydrogen Energy, 32, pp. 761-786, 2007
[12] Andersson M., Paradis H., Yuan J., Sundén B., Review of Catalyst Materials and
Catalytic Steam Reforming Reactions in SOFC Anodes, Int. J. Energy Research,
2011 (in press)
[13] Li. P. W., Schaefer L., Chyu M. K., Multiple Transport Processes in Solid Oxide
Fuel Cells, Chapter 1 in Sundén B., Faghri M. (eds.), Transport Phenomena in Fuel
Cells, WIT Press, UK, 2005
[14] Grew K.N., Joshi A.S., Peracchio A.A., Chiu W.K.S., Pore-Scale Investigation of
Mass Transport and Electrochemistry in a Solid Oxide Fuel Cell Anode, J. Power
Sources, 195, pp. 2331-2345, 2010
[15] Joshi A.S., Peracchio A.A., Grew K.N.,Chiu W.K.S., Lattice Boltzmann Method for
Multi-Component, Non-Continuum Mass Diffusion, J. Phys. D: Appl. Phys., 40, pp.
7593-7600, 2007
[16] Joshi A.S., Grew K.N., Peracchio A.A., Chiu W.K.S., Lattice Boltzmann Modeling
of a 2D Gas Transport in a Solid Oxide Fuel Cell Anode, J. Power Sources., 164, pp.
631-638, 2007
[17] Wang X., Lau K.C., Turner C.H., Dunlap B.I., Kinetic Monte Carlo Simulation of
the Elementary Electrochemistry in a Hydrogen-powered Solid Oxide Fuel Cell, J.
Power Sources, 195, pp. 4177-4184, 2010
[18] Voter A.F., Introduction to the Kinetic Monte Carlo Method, Theoretical Division,
Los Alamos National Laboratory, Report for U.S. DOE, 2005
[19] Lau K.C., Turner C.H., Dunlap B.I., Kinetic Monte Carlo Simulation of the Yttria
Stabilized Zirconia (YSZ) Fuel Cell Cathode, Solid State Ionics, 179, pp. 1912-1920,
2008
[20] Tarancón A., Morata A., Peiró F., Dezanneau G., A Molecular Dynamics Study on
the Oxygen Diffusion in Doped Fluorites: The Effect of the Dopant Distribution,
Fuel Cells, 11, pp. 26-37, 2011
[21] Chiu W.K.S., Joshi A.S., Grew K.N., Lattice Boltzmann Model for Multi-
Component Mass Transfer in a Solid Oxide Fuel Cell Anode with Heterogeneous
Internal Reformation and Chemistry, Eur. Phys. J. Special Topics, 171, pp. 159-165,
2009
[22] Sohn S., Nam J.H., Jeon D.H., Kim C., A Micro/Macroscale Model for Intermediate
Temperature Solid Oxide Fuel Cells with Prescribed Fully-Developed Axial Velocity
Profiles in Gas Channels, Int. J. Hydrogen Energy, 35, pp. 11890-11907, 2010
[23] Joshi A.S., Peracchio A.A., Grew K.N., Chiu W.K.S., Lattice Boltzmann Method for
Multi-Component Mass Diffusion in Complex 2D Geometries, J. Phys. D: Appl.
Phys., 40, pp. 2961-2971, 2007
[24] Joshi A.S., Grew K.N., Izzo J.R. Jr., Peracchio A.A., Chiu W.K.S., Lattice
Boltzmann Modeling of Three-Dimensional, Multicomponent Mass Diffusion in a
Solid Oxide Fuel Cell Anode, ASME J. Fuel Cell Sci. Tech., 7, pp. 1-8, 2010
54

[25] Latt J., Hydrodynamic Limit of the Lattice Boltzmann Equations, PhD Thesis,
University of Geneva, 2007
[26] Haslam I.W., Crouch R.S., Seaïd M., Coupled Finite Element-Lattice Boltzmann
Analysis, Comput. Methods Appl. Mech. Eng.., 197, pp. 4505-4511, 2008
[27] Suzue Y., Shikazono N., Kasagi N., Micro Modeling of Solid Oxide Fuel Cell Anode
Based on Stochastic Reconstruction, J. Power Sources, 184, pp. 52-59, 2008
[28] Bernsdorf J., Simulation of Complex Flows and Multi-Physics with the Lattice
Boltzmann Method, PhD Thesis, University of Amsterdam, 2008
[29] Kang Q. J., Lichtner P. C., Janecky D.R., Lattice Boltzmann Method for Reacting
Flows in Porous Media, Adv. Appl. Math. Mech., 5, pp. 545-563, 2010
[30] Servan Camas B., Lattice Boltzmann Modeling for Mass Transport Equations in
Porous Media, PhD Thesis, Dept. Civil & Environmental Engineering, Louisiana
State University, USA, URN etd-11072008-084923, 2008
[31] Kim S.D., Lee J.J., Moon H., Hyun S.H., Moon J., Kim J., Lee H.W., Effects of
Anode and Electrolyte Microstructures on Performance of Solid Oxide Fuel Cells, J.
Power Sources, 169, pp. 265-270, 2007
[32] Asinari P., Calí Quaglia M., von Spakovsky M.R., Kasula B.V., Direct Numerical
Calculation of the Kinematic Tortuosity of Reactive Mixture Flow in the Anode
Layer of Solid Oxide Fuel Cells by the Lattice Boltzmann Method, J. Power Sources,
170, pp. 359-375, 2007
[33] Nam J. H., Jeon D. H., A Comprehensive Micro-scale Model for Transport and
Reaction in Intermediate Temperature Solid Oxide Fuel Cells, Electrochim. Acta, 51
pp. 3446-3460, 2006
[34] Janardhanan V. M., Deutschmann O., Numerical Study of Mass and Heat Transport
in Solid-Oxide Fuel Cells Running on Humidified Methane, Chem. Eng. Sci., 62, pp.
5473-5486, 2007
[35] Gorte R.J., Vohs J.M., Nanostructured Anodes for Solid Oxide Fuel Cells, Current
Opinion in Colloid & Interface Science, 14, pp. 236–244, 2009
[36] Sadykov V.A., Mezentseva N.V., Bunina R.V., Alikina G.M., Lukashevich A.I.,
Kharlamova T.S, Rogov V.A, Zaikovskii V.I.,. Ishchenko A.V, Krieger T.A.,
Bobrenok O.F., Smirnova A., Irvine J., Vasylyev O.D., Effect of Complex Oxide
Promoters and Pd on Activity and Stability of Ni/YSZ (ScSZ) Cermets as Anode
Materials for IT SOFC, Catalysis Today, 131, pp. 226-237, 2008
[37] Yuan J., Faghri M., Sundén B., On Heat and Mass Transfer Phenomena in PEMFC
and SOFC and Modeling Approaches, Chapter 4 in Sundén B., Faghri M.(eds.),
Transport Phenomena in Fuel Cells, WIT Press, UK, 2005
[38] Yakabe H., Hishinuma M., Uratani M., Matsuzaki Y., Yasuda I., Evaluation and
Modeling of Performance of Anode-Supported Solid Oxide Fuel Cell, J. Power
Sources, 86, pp. 423-431, 2000
55

[39] Hussain M. M., Li. X., Dincer I., Mathematical Modeling of Transport Phenomena in
Porous SOFC Anodes, Int. J. Thermal Sciences, 46, pp. 48-86, 2007
[40] Reid R.C., Prausnitz J.M., Poling B.E., The Properties of Gases and Liquids, 4th ed.,
McGraw Hill, USA ISBN 0-07-051799-1, 1987
[41] Nagel F.P., Schildhauer T.J., Biollaz S.M.A., Stucki S., Charge, Mass and Heat
Transfer Interactions in Solid Oxide Fuel Cells Operated with Different Fuel Gases-
A Sensitivity Analysis, J. Power Sources, 184, pp. 129-142, 2008
[42] Achenbach E., Riensche E., Methane/Steam Reforming Kinetics for Solid Oxide
Fuel Cells, J. Power Sources, 52, pp. 283-288, 1994
[43] Achenbach E., Three-Dimensional and Time-Dependent Simulation of a Planar Solid
Oxide Fuel Cell Stack, J. Power Sources, 49, pp. 333-348, 1994
[44] Leinfelder R., Reaktionskinetische Untersuchungen zur Methan-Dampf-
Reformierung und Shift-Reaktion an Anoden Oxidkeramischer Brennstoffzellen”,
Doctoral Thesis, Der Technischen Fakultät, Universität Erlangen-Nürnberg,
Erlangen-Nürnberg, Deutschland, 2004 (in German)
[45] Drescher I., Kinetik der Methan-Dampf-Refomierung, Institut für Werkstoffe und
Verfahren der Energietechnik, Forschungszentrums Jülich, Jülich, Deutschland,
ISSN 0944-2952, 1999 (in German)
[46] Klein J.M., Bultel Y., Georges S., Pons M., Modeling of a SOFC Fuelled by
Methane: From Direct Reforming to Gradual Internal Reforming, Chem. Eng. Sci.,
62, pp. 1636-1649, 2007
[47] Aguiar P., Adjiman C.S., Brandon N.P., Anode-Supported Intermediate Temperature
Direct Internal Reforming Solid Oxide Fuel Cell I: Model-Based Steady-State
Performance, J. Power Sources, 132, pp. 113-126, 2004
[48] Haberman B.A., Young J.B., Three-Dimensional Simulation of Chemically Reacting
Gas Integrated-Planar Solid Oxide Fuel Cell”, Int. J. Heat and Mass Transfer, 47, pp.
3617-3629, 2004
[49] Mears D.E., Tests for Transport Limitations in Experimental Catalytic Reactors, Int.
Eng. Chem. Process Des. Develop., 10, pp. 541-547, 1971
[50] Mears D.E., On Criteria for Axial Dispersion in Nonisothermal Packed-Bed Catalytic
Reactors, Ind. Eng. Chem., Fundam., 15, pp. 20-23, 1976
[51] Nsofor E.C., Adebiyi G.A., Measurement of the Gas-Particle Heat Transfer
Coefficient in a Packed Bed for High-Temperature Energy Storage, Exp. Thermal
and Fluid Sci., 24, pp. 1-9, 2001
[52] Bizzi M., Basini L., Saracco G., Specchia V. Short Contact Time Catalytic Partial
Oxidation of Methane: Analysis of Transport Phenomena Effects, Chem. Eng. J., 90,
pp. 97-106, 2002
56