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

2

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 three-

phase-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.

3

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

4

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.

i

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).

ii

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. 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

III. 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

IV. 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 9th

Fuel Cell Science Conference, ESFuelCell2011-54015, 2011

II. H. Paradis, M. Andersson, J. Yuan, B. Sundén, CFD Modeling Concerning Different Kinetic

Models For Internal Reforming Reactions in an Anode-Supported SOFC, ASME 8th Fuel Cell

Science Conference, FuelCell2010-33045; pp. 55-64, 2010

III. H. Paradis, M. Andersson, J. Yuan, B. Sundén, Review of Different Renewable Fuels for

Potential Utilization in SOFCs, 5th International Green Energy Conference, 2010

IV. M. Andersson, H. Paradis, J. Yuan, B. Sundén, Modeling Analysis of Different Renewable

Fuels in an Anode-Supported SOFC, ASME 8th 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, 5th International Green Energy Conference 2010

iii

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

iv

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

v

Nomenclature

a lattice direction, –

AV surface area-to-volume ration, m2/m

3

Bi Biot number, –

c lattice speed of sound, lu/ts

cT total concentration, mol/m3

cp specific heat at constant pressure, J/(kg·K)

C concentration, mol/m3

Da Darcy number, –

Dij Maxwell-Stefan binary diffusion coefficient, m2/s

D thermal diffusion coefficient, kg/(m·s)

E activation energy, kJ/mol

e lattice speed, lu/ts

fa particle density distribution function, –

f eq

equilibrium distribution function, –

F force vector, N/m3

F Faraday constant, 96485 C/mol

F gravitational force, mu·lu/ts2

h enthalpy, kJ/mol

h heat transfer coefficient, W/(m2·K)

hv volume heat transfer coefficient, W/(m3·K)

i current density, A/cm2

i0 exchange current density, A/cm2

J mole flux, mol/(m2s

1)

J* dimensionless mole flux, –

k thermal conductivity, W/(m·K)

k reaction rate constant, mol/(m3·bar

2·s)

kc mass transfer coefficient, m/s

k´´ pre-exponential factor, 1/(·m2)

Ke equilibrium constant, Pa2 or dimensionless

Mj molecular weight of species j, kg/mol

N number of grid points, –

Niter number of iterations, –

Nu Nusselt number, –

ne number of electrons transferred per reaction, –

P pressure, bar

p partial pressure, Pa

vi

Q source term (heat), W/m3

q heat flux, W/m2

r reaction rate, mol/(m3·s), mol/(m

2·s)

r mean pore radius, m

R gas constant, 8.314 J/(mol·K)

Re Reynolds number, –

Si source term for the reaction rate, kg/(m3·s)

T temperature, K

T viscous stress tensor, N/m2

t time step, –

u, v velocity, m/s

wa weight factor, –

wi 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, m2

κdv deviation from thermodynamic equilibrium, Pa·s

μ dynamic viscosity, Pa·s

ρ density, kg/m3

ζ ionic/electronic conductivity, Ω-1

m-1

η tortuosity, –

η relaxation time

υ kinematic viscosity, m2/s

χ Damköhler number for the heat transfer, –

ω Damköhler number for the mass transfer, –

ΩD 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, ca,e

eff effective

el electrolyte

vii

g gas phase

i molecule i

j molecule j

LB discrete system, lattice Boltzmann

P physical system

r steam reforming reaction

s solid phase or water-gas shift reaction

w 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

CH4 methane

CO carbon monoxide

CO2 carbon dioxide

H2 hydrogen

H2O water

Ni nickel

O2 oxygen

1

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.

2

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.

3

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].

4

(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

5

between 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

– 101

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 103 s and cathode

thermal diffusion appears in 104 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.

6

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

7

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 H2

and H2O 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

8

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].

9

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 fa 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 ea, 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 ea, 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 fa 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].

10

(2.6)

where fa is the PDF, ea 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 ea and the PDF fa 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 Ωa and the D2Q9 equilibrium distribution function faeq

are defined as [3]:

(2.9)

(2.10)

where wa 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 advection-

diffusion is simulated by another equilibrium distribution fζ,aeq

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

11

streaming [3]. The concentration ρζ 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 fai is the PDF, ea

i the velocity and a

i the collision term at any spatial location x and

time t along the direction a for species i.

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. Off-

lattice 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.

12

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

l0 and a characteristic time scale t0. The transition from the dimensionless to the discrete

system is done by implementing a discrete length step δx and a discrete time step δt [25]. To

13

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 δx and δt 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

l0, e.g., approximate size of an obstacle, and a time scale t0, 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]:

14

(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 δx and time step δt. If the reference

variables, i.e., δxD and δtD, in the dimensionless system are defined, t0D and x0D 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 Niter 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 δx and δt are attached with some constraints in LBM. The value of uLB 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 δt, it is important to note that the LB model is of second-order

accuracy. This means that the lattice error ε(δx) ~ δx2. 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(δx). This leads to a more specific relationship between the reference variables,

namely δt ~ δx2 [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

15

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 Ji is the mole flux of species i, cT the total mole concentration, xi the mole fraction of

species i and Dij 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 Ji and

Xi 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 Dij 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, CA and CB 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

16

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.

17

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 fa 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.

18

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 ionic-

electronic 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.

19

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

20

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.

21

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].

22

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

23

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-color-

conversion 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.

24

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.

25

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, Dij the Stefan-Maxwell binary diffusion coefficient, x the mole

fraction, DiT the thermal diffusion coefficient and Si the source term. Si 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, Dij is calculated as:

(3.7)

(3.8)

(3.9)

(3.10)

26

where T is the temperature, P the pressure, σAB the characteristic length and D the

dimensionless collision integral. The averaged molecular weight MAB 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 MAB [g/mol] and σAB [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, Mi 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 Tg is the gas temperature, cp,g the specific heat, kg the gas thermal conductivity and Qg

the heat transfer between the gas and solid phases. The heat transfer between the gas and solid

phases is defined as:

(3.16)

27

where hv 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, εp 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. κdv 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 εp = 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 iio and iel are charge fluxes for ions and electrons, respectively, and ϕio 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 i0 is the exchange current density, F the Faraday constant, β the transfer coefficient, ne

the number of electrons transferred per reaction, ηact,e the electrode activation polarization

28

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 CO2 can be solved separately because the sum of the mass fractions is equal

to unity. The reaction rate rr is for the catalytic steam reforming reaction and rs 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

29

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 Ts 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∙m3).

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.% ZrO2

(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

m2. 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 ∙104 - 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

30

applied an active surface area to volume ratio of 5 ∙ 105 for modeling work. Janardhanan and

Deutschmann [5] applied a slightly smaller surface area to volume ratio of 102500 m2/m

3,

whereas Klein et al. [46] applied a much larger value of 2.2 ∙ 106 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 ks is the reaction rate constant and Ke,s the equilibrium constant for the water-gas shift

reaction. The value for ks is calculated according to Haberman and Young [48]. The unit for

the water-gas shift reaction rate is mol/(s∙m3). 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 Δhreac is the enthalpy change due to the reactions and Qint,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

31

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 (x-

direction) and the radial direction refers to the direction normal to the main flow direction (y-

direction). 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 BiR is the Biot number based on the reactor diameter, ΔH the enthalpy change of

reaction, rr the reaction rate, Ro the reactor diameter, ke the thermal conductivity for the solid

porous media, Tw 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/dp > 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 Ea 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.

32

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 kc 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 rr is the reaction rate, rp the particle diameter, C the concentration, Deff 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), kc 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 Ea 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]:

33

(3.48)

where, besides the parameters mentioned above which are the same, ks 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, dp the particle diameter and ks 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, kf the thermal conductivity of the gas and Do 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 μf is the dynamic viscosity of the gas, cpf the specific heat of the gas and kf 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

34

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 Ro >> rp and ke 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 dp is the particle diameter and the molecular weight MAB 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 rp is the particle radius, kc the mass transfer coefficient, C the concentration, n the

reaction order and Deff 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)

35

for an isothermal spherical particle where rr is the reaction rate, rp particle radius, C the

concentration and Deff the effective diffusivity. The results for the control of the criteria for

the heat and mass transfer limitations are provided in the next chapter.

36

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 lu1

Height 10 lu, 50 lu1

Porosity (ε) 0.4

Inlet mole fraction

H2 0.9

H2O 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.

37

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.

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.

0 5 10 15 20 25 30 35 400

0.02

0.04

0.06

0.08

0.1

0.12

Distance [nodes]

Velo

cit

y

ux (Analytical Poiseuille)

ux (LBM)

38

Figure 4.2: Velocity contours [lu/ts] in a channel with a circular obstacle.

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.

0.02

0.04

0.06

0.08

0.1

0.12

0.5

0.505

0.51

0.515

0.52

0.525

39

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.

Figure 4.5: Mole fraction distribution of H2 in a porous media with a porosity of 0.40.

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

40

In Figure 4.5 shows the whole modeled porous domain for the mole fraction distribution of

H2. The inlet mole fraction is specified as xH2 = 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.

Figure 4.6: Mole fraction distribution of H2O in a porous media with a porosity of 0.40.

In Figure 4.6 the mole fraction distribution of H2O is shown for the whole modeled domain.

The inlet mole fraction is specified as xH2O = 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.

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

41

Table 4.2: Parameters in the SOFC analysis.

Parameter Value

Cell length 0.1 m

Anode thickness 500 μm

Cathode thickness 50 μm

Particle diameter 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 vx,A 6.9 ∙10-5

m/s

Velocity vy,A 1.5∙10-3

m/s

Velocity vx,C 1∙10-3

m/s

Velocity vy,C 7.5 ∙10-4

m/s

Enthalpy change (MSR3) 226 kJ/mol

Activation energy (MSR3) 82 kJ/mol

Activation energy (CE4) 71 kJ/mol

Activation energy (AE5) 185 kJ/mol

Reaction rate (MSR3), close to inlet 15 mol/m³/s

Reaction rate (CE4) 0.53 kmol/m³/s,

5.3 kmol/m³/s 2

Reaction rate (AE5) 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

42

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 1000 K

Pressure 1 atm

Average current density 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

43

reactions 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.

44

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.

45

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).

46

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 ∙104 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% CH4 content but the SF was varied

between 1, 3, and 5. Additionally, three cases with SF equal to 3 but CH4 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 H2, to achieve numerical

stability. Steam was added to avoid carbon deposition and also to be used in the reforming

reactions. The amount of H2O was calculated from the relationship SF = [H2O]/[CH4] 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.

47

Table 4.5: Inlet mole fractions for the different case studies.

Case models CH4 CO2 H2O

CH4 0.45, SF=3 0.191 0.235 0.574

CH4 0.60, SF=3 0.214 0.143 0.643

CH4 0.75, SF=3 0.231 0.076 0.692

CH4 0.60, SF=1 0.375 0.25 0.375

CH4 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/m2. 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% CH4 (left) and 75% CH4

(right).

In Figure 4.10, the mole fractions at the centerline of the anode along the whole cell length

are presented. The change of H2O 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 CO2 that changes. The reason for this is that CO2 is linked to the choice of CH4 and

48

H2O. Both CO and H2 are initially set to small values to enable the numerical calculations, but

CO2 and H2O 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% CH4 (left) and 75% CH4

(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 CH4

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.

49

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 H2O 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 H2O exists slightly downstream the inlet. Instead

of putting all focus on the inlet mole fraction of H2O, 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).

50

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.

51

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 anode-

supported 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.

52

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.

53

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