COMPUTATIONALLY-EFFICIENT MODELING AND
SIMULATION OF TRANSPORT PHENOMENA IN FUEL CELL
STACKS
ASHWINI KUMAR SHARMA
(B.Tech., Hons., NIT Durgapur, India)
A THESIS SUBMITTED
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
DEPARTMENT OF CHEMICAL AND BIOMOLECULAR
ENGINEERING
NATIONAL UNIVERSITY OF SINGAPORE
2014
Typeset with A
M
S-L
A
T
E
X.
Doctor of Philosophy thesis for public evaluation, National University of Singapore,
4 Engineering Drive 4.
c
Ashwini Kumar Sharma 2014
Preface
The thesis addresses computationally-e¢ cient modeling of transport phenomena in fuel
cell stacks which is in parts based on the following journal and conference papers:
Journal papers
Paper 1. A. K. Sharma, and E. Birgersson. Computationally-e¢ cient simulation of
transport phenomena in fuel cell stacks via electrical and thermal decoupling of the cells.
Fuel cells, In press (2014).
Paper 2. A. K. Sharma, E. Birgersson and S. H. Khor. Computationally-e¢ cient hy-

brid strategy for mechanistic modeling of fuel cell stacks. Journal of Power Sources,
247, p.481 (2014).
Paper 3. A. K. Sharma, E. Birgersson, and M. Vynnycky. An aggregate measure for
local current density coupling in fuel cell stacks. Journal of the Electrochemical Society,
160, p.F1237 (2013).
Paper 4. A. K. Sharma, E. Birgersson, M. Vynnycky, and H. Ly. On the Inter-
changeability of Potentiostatic and Galvanostatic Boundary Conditions for Fuel Cells.
Electrochimica Acta, 109, p.617 (2013).
Paper 5. M. Vynnycky, A. K. Sharma, and E. Birgersson. A …nite element method
for the weakly compressible parabolized steady 3D Navier Stokes equations in a channel
with a permeable wall. Computers and Fluids, 81, p.152 (2013).
Conference papers
Paper 6. A. K. Sharma, E. Birgersson, and M. Vynnycky. Asymptotically reduced
three-dimensional model for a proton exchange membrane fuel cell, in European Congress
of Chemical Engineering. The Hague, The Netherlands (2013).
Paper 7. A. K. Sharma, S. H. Khor, and E. Birgersson.Veri…ed hybrid simulation
strategy for a proton exchange membrane fuel cell stack model based on scale analy-
sis, in European Congress of Chemical Engineering. The Hague, The Netherlands (2013).
i
ii
Acknowledgements
First and foremost, it is my great pleasure to extend my sincere and deepest gratitude
to my supervisor Dr. Karl Erik Birgersson for his invaluable guidance and unending
support throughout my tenure. I truly admire his continuous advice and moral sup-
port, and appreciate his trust and patience especially at time of my slow progress. As
a friend-cum-supervisor, he has taught me how to frame the research questions, how to
build the path to …nd the solutions and how to convey the research …ndings via technical
writing and presentation, the elements that are necessary for my future endeavors too.
I feel privileged to be a part of his research group.
I want to convey my special thanks to Dr. Michael Vynnycky from University of

Limerick for sharing his knowledge and helping me out on several occasions with critical
insights. I highly value the support of my colleagues Dr. Ly Cam Hung, Dr. Sherlin Ee
and Ling Chun Yu, and FYP students, namely Khor Shu Heng, Ng Jun Rong, Ee Jin
Guan, Xie Mingchuan, and Shaun Ang.
I would like to thank my external and internal examiners to accept the request to
examine my thesis. I am thankful to A/Prof. Laksh and Dr. Linga for being my exam-
ination committee and their constructive comments during my qualifying examination.
I also thank the teachers from my schooling and undergraduate studies for inculcating
strong fundamentals in various subjects.
I would like to thank Mr. Boey, Ms. Samantha Fam and Ms. Lim Kwee Mei for
their support in lab related issues. I thank Ms. Yoke, Mr. Ste¤en, and Ms. Vanessa for
helping me out in academic and administrative matters. I thank National University of
Singapore for providing me the research scholarship and the opportunity to pursue my
PhD at one of the best engineering schools across the world.
I want to express my special and heartfelt thanks to KMG, Sumit, Shivom, Vaibhav,
and Manoj for taking their extra time out in memorable conversations and energizing
chitchats during entire PhD duration. I wish to thank my friends, Sattu, Shashi, Nimit,
Shailesh, Naresh, Rajneesh, Prashant, Praveen, Naviyn, Meiyappan, Anjaiah, Gudena
Krishna, Naresh Thota, Vamsi, Akshay, and Jaya Kumar to keep myself sane and also
providing me a wonderful atmosphere outside the research life.
I am grateful and indebted to my parents (Mr. Hari Prakash, Mrs. Santosh Sharma),
and my siblings (Mr. Arjun, Ms. Sarita) whose undying love supported me and all my
academic pursuits. I further want to convey my deepest gratitude and warmest thanks
to my aunts, namely Mrs. Sukhi Devi, Mrs. Dhansukhi Devi, Mrs. Kanta and Mrs.
Munni Devi, and my cousins, Mr. Dinesh, Mr. Sanjay, Mr. Pradeep and Mr. Nandk-
ishor for their kind support during my entire education, both …nancially and morally. I
also want to thank my in-laws, all other relatives and friends for their love and cherished
moments.
To my lovely wife, Chanda and my cute son, Kunal, all I can say is it would take
another thesis to express my deep love for you both. Your patience, love and encour-

agement have upheld me during the tough times.
Lastly, but most importantly, I am deeply grateful to the almighty God. Thank you
God for giving me the courage and strength to follow my dreams.
iv
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv
List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii
1 Introduction 1
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Motivation and objectives . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Layout of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Fuel Cells 9
2.1 Fuel cells: Electrochemical engines . . . . . . . . . . . . . . . . . . . . . 9
2.2 Fuel cell performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 Overview of fuel cell technologies . . . . . . . . . . . . . . . . . . . . . . 12
2.4 Components of a cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.5 Fuel cell stack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3 Literature Review 21
3.1 Stack-manifold models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2 Manifold decoupled stack models . . . . . . . . . . . . . . . . . . . . . . 27
3.2.1 Reduced models . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2.2 Detailed models . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4 Mathematical Formulation 39
4.1 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.2 Constitutive relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.3 Phenomenological membrane model . . . . . . . . . . . . . . . . . . . . 46
4.4 Electrochemistry and agglomerate model . . . . . . . . . . . . . . . . . . 48
4.5 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.6 Base-case parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.7 Modi…cation of the discontinuous relations . . . . . . . . . . . . . . . . . 56
5 Computationally-efficient Hybrid Strategy for Modeling of Fuel Cell Stacks 59
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
v
Contents
5.2 Mathematical formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.3 Hybrid coupling methodology . . . . . . . . . . . . . . . . . . . . . . . . 64
5.4 Numerics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.5 Veri…cation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.6 Computational cost and e¢ ciency . . . . . . . . . . . . . . . . . . . . . . 71
5.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
6 An Aggregate Measure for Local Current Density Coupling in Fuel Cell Stacks 75
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
6.2 Mathematical formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 76
6.3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
6.4 Veri…cation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
7 Interchangeability of Potentiostatic and Galvanostatic Boundary Conditions for
Fuel Cells 87
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
7.2 Mathematical formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 89
7.3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
7.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
7.5 Veri…cation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
7.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

8 Computationally-efficient Simulation of Transport Phenomena in Stacks via
Electrical and Thermal Decoupling of the Cells 101
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
8.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
8.3 Numerical scheme for simulation of decoupled cells . . . . . . . . . . . . 107
8.4 Veri…cation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
8.5 Computational cost and e¢ ciency . . . . . . . . . . . . . . . . . . . . . . 113
8.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
9 A Finite Element Method for the Weakly Compressible Parabolized Steady
Three-dimensional Navier Stokes Equations 117
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
9.2 Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
9.2.1 Full equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
9.2.2 Parabolized equations . . . . . . . . . . . . . . . . . . . . . . . . 123
9.2.3 Velocity-vorticity formulation . . . . . . . . . . . . . . . . . . . . 126
9.3 Numerical implementation . . . . . . . . . . . . . . . . . . . . . . . . . . 129
9.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
9.4.1 Case (i): incompressible ‡ow, impermeable walls . . . . . . . . . 135
9.4.2 Case (ii): weakly compressible ‡ow, impermeable walls . . . . . . 135
9.4.3 Case (iii): incompressible ‡ow, one permeable wall . . . . . . . . 135
9.4.4 Case (iv): weakly compressible ‡ow, one permeable wall . . . . . 138
9.4.5 Computational cost and e¢ ciency . . . . . . . . . . . . . . . . . 142
9.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
9.6 Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
vi
Contents
10 Conclusions and Future Work 149
10.1 Summary of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
10.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
Bibliography 168

vii

Summary
In the last two decades, mathematical modeling and simulations have come to play an
important role in the research and development of fuel cells. In order to capture the
wide array of physicochemical processes that occur inside the cell, the models need to
consider transport of mass, momentum, species, energy, and charge in multiple length
scales and result in a highly coupled system of non-linear partial di¤erential equations.
As such, applying these models to stacks, comprising tens or even hundreds of single
cells, will come at a hefty computational cost, both in terms of memory usage and exe-
cution time. It is therefore of interest to derive computationally-e¢ cient strategies that
can solve for and predict the local behavior of each cell in a stack at su¢ ciently low
cost, whilst preserving all the essential physics.
To reduce the overall complexity and associated computational cost for detailed
mechanistic stack models, this thesis aims to investigate and exploit the underlying
mathematical nature of the transport equations. First, a hybrid modeling strategy is
proposed for fuel cell stacks, in which the steady-state transport equations are classi…ed
based on their regions of in‡uence: conservation of mass, momentum and species are
local to cells and their governing equations can be reduced mathematically by exploiting
the slenderness at the single cell level; whereas conservation of heat and charge are global
to the stack and thus retain the original elliptic nature. These two sets of equations are
then solved iteratively. The methodology is demonstrated for a proton exchange mem-
brane fuel cell (PEMFC) stack subjected to non-uniform operating conditions. Around
80% computational savings were achieved with the hybrid strategy and it allows for the
Summary
simulation of large stacks: e.g., it takes less than an hour to simulate a 350-cell stack.
The thesis further investigates the charge transport phenomena taking place across
the cells. In this regard, steady-state conservation of charge in a bipolar plate between
two cells is analyzed, and a dimensionless number,  is identi…ed that quanti…es the
degree of local current density coupling across the cells. The same number is found to

govern the interchangeability of potentiostatic and galvanostatic boundary conditions for
fuel cells. The dimensionless number which provides an aggregate measure comprising
the design, operating conditions and material properties of the bipolar plate, is corre-
lated with the current redistribution between cells, and an upper bound is determined.
Under certain bound on the dimensionless number, i.e.,   3, there is negligible po-
tential gradient along the separator plate placed between the cells and the cells exhibit
current density distributions as if they are b eing operated ’isolatedly’. Therefore, the
transport phenomena in the individual cells can be simulated stand-alone fashion for
such stacks. However, one needs to investigate the thermal decoupling of the cells as
well–another transport phenomenon taking place across the cells. In this regard, the
heat transport is analyzed in the coolant plates installed between cells or groups of cells
and the required condition for thermal decoupling of the cells is found. Thus, it can be
argued that the electrically and thermally decoupled units can be found in a fuel cell
stack. The decoupled units are not in‡uenced by their neighboring units and thus can
be simulated one by one repeatedly; simulation of all the units provides a solution of
complete stack model.
The thesis, thus far, demonstrates various concepts with PEMFC stack equipped
with porous ‡ow …elds that allow reduction in dimensionality as well as the linear Darcy
law instead of the nonlinear Navier Stokes equations; the latter is more challenging
to solve. For fuel cells equipped with straight rectangular ‡ow channels, one needs to
resolve the three-dimensional (3D) Navier Stokes equations which add to the required
x
Summary
computational resources. In this context, a velocity-vorticity formulation is implemented
to tackle the weakly compressible parabolized steady 3D Navier Stokes equations in a
channel with a permeable wall - a situation that occurs in fuel cells. The parabolized
equations are found to be cheaper to compute both in terms of memory usage, and con-
vergence time. It should be possible to use this approach for the modeling of cells with
dozens of straight channels at unprohibitive computational cost; even more signi…cantly,
it will then be possible to model large stacks containing such cells.

In summary, this thesis proposes and investigates computational-e¢ cient strategies
for modeling and simulation of the transport phenomena in fuel cell stacks. The scala-
bility and associated low computational cost of such strategies open up the possibilities
for wide-ranging parameteric studies and optimization of stacks.
xi

List of Tables
2.1 Description of major fuel cell technologies . . . . . . . . . . . . . . . . . . . 14
4.1 Base-case parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5.1 Computational cost for the full and hybrid sets; the numbers in the brackets
indicate the time required to automatically generate the numerical stack model
before solving it. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
7.1 Base-case parameters [46, 123, 165] . . . . . . . . . . . . . . . . . . . . . . 98
8.1 Computational cost estimates for the full and decoupled stack models. . . . . 114
9.1 Computational cost for the full and reduced sets of governing equations in terms
of degrees of freedom (DoF), number of elements (NoE), CPU time and random
access memory (RAM). * denotes the DoF in an X-Y plane. . . . . . . . . . . 143
xi

List of Figures
1.1 Objectives of the study. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1 A schematic of a fuel cell. . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 A schematic of a polarization curve and power density curve. . . . . . . . . . 11
2.3 A schematic of a cross-section in a PEMFC illustrating di¤erent functional layers
of a cell. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.4 A schematic of a two-cell stack. . . . . . . . . . . . . . . . . . . . . . . . . 19
4.1 Schematic of a PEMFC equipped with porous ‡ow …elds. Boundaries are marked
with Roman numerals (N.B. h
MEA
= 2  h

cl
+ h
m
) . . . . . . . . . . . . . 40
4.2 Relationship between water content () vs water activity (a
w
). (- - -) original,
and (— –) modi…ed, discontinuities are zo omed in. . . . . . . . . . . . . . . . 57
4.3 Relationship between di¤usion coe¤cient of water in membrane (D
H
2
O;m
) vs
water content (). (- - -) original, and (— –) modi…ed, discontinuities are zoomed
in. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.1 Schematic for a PEMFC stack comprising n cells, denoted by j (a); mathemat-
ical nature of the governing equations for the full stack model (b) and reduced
stack model (c): elliptic PDEs (), parabolic PDEs (!) and ODEs (j). . . . . 62
5.2 Flowchart for hybrid simulation strategy for fuel cell stacks. . . . . . . . . . . 65
5.3 Polarization curve for a 10-cell stack with perturbed cathode inlet velocities;
symbols for the full model and lines for the hybrid counterpart. . . . . . . . . 68
5.4 Local current density distributions for a 10-cell stack (E
stack
= 3 V) along the
x-axis at the interface between the cathode catalyst layer and the membrane
for the full set of equations in cell (N) 1, () 5, () 10; and corresponding
predictions of the hybrid set in lines. . . . . . . . . . . . . . . . . . . . . . . 68
5.5 Local temperature distributions for a 10-cell stack (E
stack
= 3 V) along the

x-axis at the interface between the cathode catalyst layer and the membrane
for the full set of equations in cell (N) 1, () 5, () 10; and corresponding
predictions of the hybrid set in lines. . . . . . . . . . . . . . . . . . . . . . . 69
5.6 Local oxygen concentrations for a 10-cell stack (E
stack
= 3 V) along the x-axis
at the interface between the cathode catalyst layer and the membrane for the
full set of equations in cell (N) 1, () 5, () 10; and corresponding predictions
of the hybrid set in lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.7 Scale-up tests for the hybrid model: () memory usage, () time for setting up
the numerical stack models, and (N) convergence time (at a typical operating
voltage of roughly 0.6 V for each cell). . . . . . . . . . . . . . . . . . . . . . 72
6.1 Schematic of a 2-cell stack. . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
xiii
List of Figures
6.2 Local current density distributions in a two-cell stack at a stack voltage of 1.2
V for di¤erent values of the dimensionless number. The two cells are operating
at di¤erent cathode stoichiometries: 1:00001 for cell #1 (solid lines) and 10 for
cell #2 (dashed lines). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
6.3 R
2
of the current density distributions in the two cells decreases as the dimen-
sionless number is decreased. . . . . . . . . . . . . . . . . . . . . . . . . . . 84
7.1 Bound on the interchangeability number to establish interchangeability of BCs
in an electric conductor plate depends on the ratio of its height to length, ":
Eq. 21 (symbols), Eq. 24 (dashed line), Eq. 26 (solid line), and Eq. 27 (dotted
line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
7.2 Bound on the interchangeability number for imparting interchangeability of BCs
in a PEMFC increases with stoichiometric ratio, 
c

. . . . . . . . . . . . . . . 96
7.3 Polarization curves: (N) from experiments [123], and corresponding potentio-
static (— ) and galvanostatic () model predictions. . . . . . . . . . . . . . . 98
7.4 Local current densities measured by Noponen et al. [123] (symbols) correspond-
ing to the points A-J in Fig. 7.3, and potentiostatic (— ) and galvanostatic ()
model predictions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
7.5 Local current densities corresponding to the point H in Fig. 7.3; potentiostatic
predictions (), and galvanostatic predictions for di¤erent values of the inter-
changeability number: 1 10
2
(––), 0:1 (–N–), 0:2 (–H–), 1(–J–), 2 (–I–),
and 10 (–F–) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
8.1 Schematic of a fuel cell stack comprising n cells. . . . . . . . . . . . . . . . . 103
8.2 Heat ‡ux in the liquid coolant plate separating two cells or group of cells; the
black arrows denote the convective ‡ux which is much larger than the conductive
‡ux denoted by red arrows. . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
8.3 Simulation strategy for the decoupled stack model. . . . . . . . . . . . . . . 106
8.4 Local current density distributions at the interface between the cathode catalyst
layer and the membrane for a cell containing current-free spot [37]. . . . . . . 110
8.5 Polarization curve for a 10-cell PEMFC stack (  0:1) having a current-free
spot in one of the constituent cells (here, …fth cell); the symbols for the full
model and lines for the decoupled counterpart. . . . . . . . . . . . . . . . . . 111
8.6 Local current density and temperature distributions for the fourth cell in the
stack (  0:1, i
app
= 1:25  10
4
A m
2
) along the x-axis at the interface

between the cathode catalyst layer and the membrane for the full model; and
corresponding predictions of the decoupled model in lines. . . . . . . . . . . . 111
8.7 Local current density and temperature distributions for the fourth cell in the
stack (  10, i
app
= 1:25  10
4
A m
2
) along the x-axis at the interface
between the cathode catalyst layer and the membrane for the full model; and
corresponding predictions of the decoupled model in lines. . . . . . . . . . . . 112
8.8 Computational cost in terms of the convergence time for decoupled stack model
for an increasing number of the cells in the stack. . . . . . . . . . . . . . . . 115
9.1 Cross-section of a proton exchange membrane fuel cell . . . . . . . . . . . . . 119
9.2 Schematic of ‡ow in a slender channel with a permeable wall (shaded) . . . . 120
9.3 Axial velocity at the centre of the channel for incompressible ‡ow and imper-
meable walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
9.4 Axial velocity at the centre of the channel for incompressible ‡ow and imperme-
able walls: symbols (full 3D), solid line (parabolized 3D), dashed line (approx.
analytical solution). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
xiv
List of Figures
9.5 Pressure drop along the channel for incompressible ‡ow and impermeable walls:
symbols (full 3D), solid line (parabolized 3D), dashed line (approx. analytical
solution). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
9.6 Axial velocity at the centre of the channel for a weakly compressible ‡ow with
impermeable walls: symbols (full 3D), solid line (parabolized 3D), dashed line
(approx. analytical solution). . . . . . . . . . . . . . . . . . . . . . . . . . . 137
9.7 Pressure drop along the channel for a weakly compressible ‡ow with imperme-

able walls: symbols (full 3D), solid line (parabolized 3D), dashed line (approx.
analytical solution). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
9.8 Dimensionless axial velocity at the centre of the channel, w
mid
, for incompress-
ible ‡ow with a permeable wall (

V
0
= 1): symbols (full 3D), solid line (parab-
olized 3D), dashed line (approx. analytical solution). . . . . . . . . . . . . . 138
9.9 Dimensionless pressure drop, P, along the channel for incompressible ‡ow
with a permeable wall (

V
0
= 1): symbols (full 3D), solid line (parabolized 3D),
dashed line (approx. analytical solution). . . . . . . . . . . . . . . . . . . . 139
9.10 Dimensionless axial velocity at the centre of the channel, w
mid
, for incompress-
ible ‡ow with a permeable wall (

V
0
= 10): symbols (full 3D), solid line (parab-
olized 3D), dashed line (approx. analytical solution). . . . . . . . . . . . . . 139
9.11 Dimensionless pressure drop, P, along the channel for incompressible ‡ow
with a permeable wall (


V
0
= 10): symbols (full 3D), solid line (parabolized
3D), dashed line (approx. analytical solution). . . . . . . . . . . . . . . . . . 140
9.12 Dimensionless axial velocity at the centre of the channel, w
mid
, for compressible
‡ow with a permeable wall (

V
0
= 1): symbols (full 3D), solid line (parabolized
3D), dashed line (approx. analytical solution). . . . . . . . . . . . . . . . . . 140
9.13 Dimensionless pressure drop, P, along the channel for compressible ‡ow with a
permeable wall (

V
0
= 1): symbols (full 3D), solid line (parabolized 3D), dashed
line (approx. analytical solution). . . . . . . . . . . . . . . . . . . . . . . . 141
9.14 Dimensionless axial velocity at the centre of the channel, w
mid
, for compressible
‡ow with a permeable wall (

V
0
= 10): symbols (full 3D), solid line (parabolized
3D), dashed line (approx. analytical solution). . . . . . . . . . . . . . . . . . 141
9.15 Dimensionless pressure drop, P, along the channel for compressible ‡ow with

a permeable wall (

V
0
= 10): symbols (full 3D), solid line (parabolized 3D),
dashed line (approx. analytical solution). . . . . . . . . . . . . . . . . . . . 142
xv

Abbreviations
1D one-dimensional
2D two-dimensional
3D three-dimensional
BC boundary condition
cc current collector
CFD computational ‡uid dynamics
c¤ coolant ‡ow …eld
cl catalyst layer
CPU central processor unit
DoF degree of freedom
DMFC direct methanol fuel cell
¤ ‡ow channel
gdl gas di¤usion layer
MEA membrane electrode assembly
MCFC molten carbonate fuel cell
NASA National Aeronautics and Space Administration
NoE number of elements
N-S equations Navier-Stokes equations
ODE ordinary di¤erential equation
PDE partial di¤erential equation
PEMFC proton exchange membrane fuel cell

RAM random access memory
SOFC solid oxide fuel cell
xvii