PERFORMANCE ENHANCEMENT IN PROTON EXCHANGE MEMBRANE FUEL CELL -

PERFORMANCE ENHANCEMENT IN PROTON EXCHANGE MEMBRANE FUEL CELL -
PERFORMANCE ENHANCEMENT IN PROTON
EXCHANGE MEMBRANE FUEL CELL NUMERICAL MODELING AND OPTIMISATION
by
Surajudeen Olanrewaju Obayopo
Submitted in partial fulfilment of the requirements for the degree
PHILOSOPHIAE DOCTOR in Mechanical Engineering
in the
Faculty of Engineering, Built Environment and Information Technology
University of Pretoria
Pretoria
Supervisor: Prof. T. Bello-Ochende
Co-Supervisor: Prof. J.P. Meyer
2012
© University of Pretoria
Abstract
ABSTRACT
TITLE:
PERFORMANCE ENHANCEMENT IN PROTON
EXCHANGE MEMBRANE FUEL CELL NUMERICAL MODELING AND OPTIMISATION
AUTHOR:
S.O. Obayopo
SUPERVISOR:
Prof. T. Bello-Ochende
CO-SUPERVISOR: Prof. J.P. Meyer
DEPARTMENT:
Mechanical and Aeronautical Engineering
UNIVERSITY:
University of Pretoria
DEGREE:
Philosophiae Doctor (Mechanical Engineering)
Sustainable growth and development in a society requires energy supply that is
efficient, affordable, readily available and, in the long term, sustainable without
causing negative societal impacts, such as environmental pollution and its attendant
consequences. In this respect, proton exchange membrane (PEM) fuel cells offer a
promising alternative to existing conventional fossil fuel sources for transport and
stationary applications due to its high efficiency, low-temperature operation, high
power density, fast start-up and its portability for mobile applications. However, to
fully harness the potential of PEM fuel cells, there is a need for improvement in the
operational performance, durability and reliability during usage. There is also a need
to reduce the cost of production to achieve commercialisation and thus compete with
ii
Abstract
existing energy sources. The present study has therefore focused on developing novel
approaches aimed at improving output performance for this class of fuel cell.
In this study, an innovative combined numerical computation and optimisation
techniques, which could serve as alternative to the laborious and time-consuming
trial-and-error approach to fuel cell design, is presented. In this novel approach, the
limitation to the optimal design of a fuel cell was overcome by the search algorithm
(Dynamic-Q) which is robust at finding optimal design parameters. The methodology
involves integrating the computational fluid dynamics equations with a gradient-based
optimiser (Dynamic-Q) which uses the successive objective and constraint function
approximations to obtain the optimum design parameters. Specifically, using this
methodology, we optimised the PEM fuel cell internal structures, such as the gas
channels, gas diffusion layer (GDL) - relative thickness and porosity - and reactant
gas transport, with the aim of maximising the net power output. Thermal-cooling
modelling technique was also conducted to maximise the system performance at
elevated working temperatures.
The study started with a steady-state three-dimensional computational model to study
the performance of a single channel proton exchange membrane fuel cell under
varying operating conditions and combined effect of these operating conditions was
also investigated. From the results, temperature, gas diffusion layer porosity, cathode
gas mass flow rate and species flow orientation significantly affect the performance of
the fuel cell. The effect of the operating and design parameters on PEM fuel cell
performance is also more dominant at low operating cell voltages than at higher
operating fuel cell voltages. In addition, this study establishes the need to match the
PEM fuel cell parameters such as porosity, species reactant mass flow rates and fuel
gas channels geometry in the system design for maximum power output.
This study also presents a novel design, using pin fins, to enhance the performance of
the PEM fuel cell through optimised reactant gas transport at a reduced pumping
power requirement for the reactant gases. The results obtained indicated that the flow
iii
Abstract
Reynolds number had a significant effect on the flow field and the diffusion of the
reactant gas through the GDL medium. In addition, an enhanced fuel cell performance
was achieved using pin fins in a fuel cell gas channel, which ensured high
performance and low fuel channel pressure drop of the fuel cell system. It should be
noted that this study is the first attempt at enhancing the oxygen mass transfer through
the PEM fuel cell GDL at reduced pressure drop, using pin fin.
Finally, the impact of cooling channel geometric configuration (in combination with
stoichiometry ratio, relative humidity and coolant Reynolds number) on effective
thermal heat transfer and performance in the fuel cell system was investigated. This is
with a view to determine effective thermal management designs for this class of fuel
cell. Numerical results shows that operating parameters such as stoichiometry ratio,
relative humidity and cooling channel aspect ratio have significant effect on fuel cell
performance, primarily by determining the level of membrane dehydration of the
PEM fuel cell. The result showed the possibility of operating a PEM fuel cell beyond
the critical temperature (  80C), using the combined optimised stoichiometry ratio,
relative humidity and cooling channel geometry without the need for special
temperature resistant materials for the PEM fuel cell which are very expensive.
In summary, the results from this study demonstrate the potential of optimisation
technique in improving PEM fuel cell design. Overall, this study will add to the
knowledge base needed to produce generic design information for fuel cell systems,
which can be applied to better designs of fuel cell stacks.
Keywords: PEM fuel cell; Computational fluid dynamics; Optimisation algorithm;
Design parameters; Reactant gas transport; Pin fin; Cooling channel; Higher
temperatures; Optimal performance.
iv
Acknowledgements
ACKNOWLEDGEMENTS
First, I thank the almighty God for His mercies and protection during the planning and
execution of this research work.
I would like to express my sincere gratitude to my supervisor, Professor Tunde BelloOchende, for his guidance, encouragement and support from the beginning to the final
stage of this thesis. I thank him for teaching me how to be independent and remain
focused when facing research challenges.
Special thanks go to my co-supervisor, Professor Josua Petrus Meyer, for affording
me the opportunity to be part of their research group, and for his constant support and
encouragement during the course of my study. He is a memorable mentor to younger
researchers.
I also express my appreciation to Prof. J. A. Snyman for his encouragement and
support on the Dynamic-Q optimisation algorihm used in part of this study.
My special thanks are due to all academic staff of the Department of Mechanical and
Aeronautical Engineering, University of Pretoria. They have made my stay a
successful one. The kind gestures of Dr. Dirker, Prof. Slabbert and the rest of the staff
are highly acknowledged. Thank you all for your constant advice and guidance
through the journey.
My special thanks go to the departmental administrative staff members for their
support: Tersia Evans and Elizabe Pieterse.
My gratitude also extends to my colleagues who helped me during the research and
writing of this thesis: Bode Olakoyejo, Seun Ogunronbi, Fervent Ighalo, Oke
Adekola, Darshik, Low Cotz, Willem le Roux, Loyd, Mehdi, Aggrie, Ernest,
v
Acknowledgements
Adewunmi and other members of the thermofluid research group at the University of
Pretoria. Appreciation is further extended to my friends in South Africa and abroad
for their love, affection and support: Dr & Mrs Adebesin, Dr & Mrs Oboirien, Dr &
Mrs Adeleke, Mr & Mrs Aregbesola, Dr & Mrs Raji, Dr Odusote, Dr & Mrs Musodiq
Bello, Alhaji & Alhaja Oyekunle, Dr & Mrs Oseni, Prof. M.A. Rahaman, Prof. A.A.
Asere, Dr A. Sanusi, Alhaji N. Adegun, Akeem Jimoh, Tajudeen Ogunmola,
Sirajudeen Aderoju, Moshood Adelani, Leasu Ismail, Sulaiman Yahya and Rasheed
Dauda.
I thank my parents Mr and Mrs Obayopo, for their invaluable support, love and
prayers. I thank my brothers and sister for their constant prayers. My profound
gratitude goes to my wife, Rofiat Omolola and our kids (Abdul-Rahman, Mar’yam
and Ibrahim) for their emotional support, patience, encouragement and love through
the journey. Words cannot express my gratitude and appreciation to them during the
period of this study.
Finally, I would like to acknowledge the financial support of the Advanced
Engineering Centre of Excellence at the University of Pretoria, NRF, TESP, NAC, the
SOLAR Hub with the Stellenbosch University, EEDSM Hub and the CSIR.
vi
List of Publications
PUBLICATIONS IN JOURNALS AND CONFERENCES
Articles in refereed journals
1. S.O. Obayopo, T. Bello-Ochende, J.P. Meyer. Three-dimensional
optimisation of a fuel gas channel of a PEM fuel cell for maximum current
density. International Journal of Energy Research: DOI:10.1002/er.1935.
Accepted for publication on 4 August 2011.
2. S.O. Obayopo, T. Bello-Ochende, J.P. Meyer. Modelling and optimisation of
reactant gas transport in PEM fuel cell with transverse pin fin insert in
channel
flow.
International
Journal
of
Hydrogen
Energy:
doi:10.1016/j.ijhydene.2012.03.150. Accepted for publication on 27 March
2012.
Journal article submitted for publication
1. T. Bello-Ochende, S.O. Obayopo, J.P. Meyer. Numerical Modeling and
Optimisation of Cooling Channel Geometric Configuration for Optimal
Performance of High-temperature PEM Fuel Cell System. Submitted to
International Journal of Hydrogen Energy.
Papers in refereed conference proceedings
1. S.O. Obayopo, T. Bello-Ochende, J.P. Meyer. Numerical Optimization of a
Single PEM Fuel Cell under Variable Operating Conditions. Proceedings of
the 7th International Conference on Heat Transfer, Fluid Mechanics and
Thermodynamics, HEFAT 2010, pp. 667-672. Antalya, Turkey, 19-21 July
2010.
2. S.O. Obayopo, T. Bello-Ochende, J.P. Meyer. Thermodynamic Optimization
of PEM Fuel Cell Stack Gas Channel for Optimal Thermal Performance.
Proceedings of the 14th International Heat Transfer Conference (ASME),
vii
List of Publications
IHTC-14, paper no. IHTC14-22233, Washington DC, USA, 8-13 August
2010.
3. S.O. Obayopo, T. Bello-Ochende, J.P. Meyer. Numerical Study of effect of
design and physical parameters on a PEM fuel cell performance. Proceedings
of the 8th International Conference on Heat Transfer, Fluid Mechanics and
Thermodynamics, HEFAT 2011, pp. 567, Pointe Aux Piments, Mauritius, 1113 July 2011.
4. S.O. Obayopo, T. Bello-Ochende, J.P. Meyer. Optmising the Performance of a
PEM Fuel Cell with Transverse Fins Insert in the Channel Flow using
Mathematical Algorithm. Proceedings of the ASME 2012 6th International
Conference on Energy Sustainability & 10th Fuel Cell Science, Engineering
and Technology Conference ESFuelCell2012, San Diego, CA, USA, 23-26
July 2012.
5. S.O. Obayopo, T. Bello-Ochende, J.P. Meyer. Numerical Study and
Optimisation of Channel Geometry and Gas Diffusion Layer of a PEM Fuel
Cell. Proceedings of the ASME 2012 6th International Conference on Energy
Sustainability & 10th Fuel Cell Science, Engineering and Technology
Conference ESFuelCell2012, San Diego, CA, USA, 23-26 July 2012.
6. S.O. Obayopo, T. Bello-Ochende, J.P. Meyer. Impact of Cooling Channel
Geometry on Thermal Management and Performance of a Proton Exchange
Membrane Fuel Cell. Proceedings of the 9th International Conference on Heat
Transfer, Fluid Mechanics and Thermodynamics, HEFAT 2012, Malta, 16-18
July 2012.
Papers in non-refereed conference proceedings
1. S.O. Obayopo, T. Bello-Ochende, J.P. Meyer. Thermodynamic Optimization of
Proton Exchange Membrane Fuel Cell System. Proceedings of the First
Postgraduate Renewable Energy Symposium, NCRS2010, Paper no.
NCRS017. pp. 1-15. Stellenbosch, South Africa, 11-12 November 2010.
2. S.O. Obayopo, T. Bello-Ochende, J.P. Meyer. Performance enhancement of
PEM fuel cell through reactant gas channel and gas diffusion layer
optimization. Proceedings of the Second Postgraduate Renewable Energy
Symposium, NCRS2011, Paper no. NCRS010. pp. 1-15. Stellenbosch, South
Africa, 17-18 November 2011.
viii
Table of Content
TABLE OF CONTENTS
ABSTRACT…… ..........................................................................................................ii ACKNOWLEDGEMENTS ........................................................................................ v PUBLICATIONS IN JOURNALS AND CONFERENCES ..................................vii TABLE OF CONTENTS ........................................................................................... ix LIST OF FIGURES ................................................................................................. xiii LIST OF TABLES ...................................................................................................xvii NOMENCLATURE ............................................................................................... xviii CHAPTER 1: INTRODUCTION ........................................................................... 1 1.1 BACKGROUND AND MOTIVATION ............................................................. 1 1.2 REVIEW OF RELATED LITERATURE ........................................................... 7 1.2.1 OPTIMAL OPERATING CONDITIONS FOR PEM FUEL CELLS ............. 8 1.2.2 FUEL GAS CHANNEL OPTIMISATION FOR PEM FUEL CELLS ......... 16 1.2.3 REACTANT GAS TRANSPORT IN PEM FUEL CELLS ............................ 20 1.2.4 HEAT TRANSPORT AND COOLING IN PEM FUEL CELLS .................. 25 1.3 JUSTIFICATION FOR THIS STUDY ............................................................. 28 1.4 RESEARCH OBJECTIVES .............................................................................. 30 1.5 ORGANISATION OF THE THESIS ................................................................ 31 CHAPTER 2: FUNDAMENTALS OF PEM FUEL CELL SYSTEMS ............ 33 2.1 INTRODUCTION ............................................................................................. 33 2.2 THE BASIC STRUCTURE OF A PROTON EXCHANGE MEMBRANE
FUEL CELL................................................................................................. 33 2.2.1 PROTON EXCHANGE MEMBRANE ........................................................ 36 2.2.2 CATALYST LAYERS ................................................................................... 36 2.2.3 GAS DIFFUSION LAYERS ........................................................................ 38 2.2.4 BIPOLAR PLATES ..................................................................................... 39 2.3 PEM FUEL CELL STACK DESIGN................................................................ 40 2.3.1 HYDROGEN FUEL CELL SYSTEM COMPONENTS ............................... 42 ix
Table of Content
2.4 THEORIES OF TRANSPORT AND ELECTROCHEMICAL PROCESSES IN
PEMFC ........................................................................................................ 45 2.4.1 CONSERVATION EQUATIONS................................................................. 46 2.4.2 NUMERICAL MODELS OF INDIVIDUAL PEM FUEL CELL
COMPONENTS ............................................................................... 52 CONCLUSION……………………………………………………………………...69 CHAPTER 3: NUMERICAL MODELLING FRAMEWORK ......................... 69 3.1 INTRODUCTION ............................................................................................. 69 3.2 NUMERICAL METHOD .................................................................................. 69 3.2.1 NUMERICAL MODELLING PROCEDURES............................................ 70 3.3 NUMERICAL OPTIMISATION ...................................................................... 72 3.3.1 CONSTRAINED OPTIMISATION .............................................................. 72 3.3.2 THE DYNAMIC-Q METHOD .................................................................... 73 3.3.3 DYNAMIC-Q APPROACH: CONSTRUCTING SPHERICAL QUADRATIC
SUBPROBLEMS.............................................................................. 75 3.3.4 THE OBJECTIVE AND CONSTRAINT FUNCTIONS GRADIENT
APPROXIMATION .......................................................................... 78 3.3.5 ADVANTAGE OF DYNAMIC-Q ALGORITHM ....................................... 79 CONCLUSION……………………………………………………………………...80 CHAPTER 4: NUMERICAL OPTIMISATION OF OPERATING AND
DESIGN PARAMETERS FOR A PEM FUEL CELL .............. 81 4.1 INTRODUCTION ............................................................................................. 81 4.2 MODEL DESCRIPTION ................................................................................ 84 4.2.1 MODEL ASSUMPTIONS ........................................................................... 88 4.2.2 GOVERNING TRANSPORT EQUATIONS ................................................ 88 4.2.3 CHANNEL CROSS-SECTION .................................................................... 92 4.2.4 FLUID FLOW THROUGH GAS DIFFUSION LAYER .............................. 95 4.2.5 BOUNDARY CONDITIONS ....................................................................... 96 4.2.6 SOLUTION TECHNIQUE .......................................................................... 97 4.2.7 MODEL VALIDATION ............................................................................... 98 4.3 MODEL RESULTS AND DISCUSSION ......................................................... 99 x
Table of Content
4.3.1 PRESSURE DROP IN FLOW CHANNEL .................................................. 99 4.3.2 EFFECT OF PHYSICAL PARAMETERS ON PROTON EXCHANGE
MEMBRANE FUEL CELL PERFORMANCE .............................. 100 4.3.3 EFFECT OF DESIGN PARAMETERS ON PROTON EXCHANGE
MEMBRANE FUEL CELL PERFORMANCE .............................. 105 4.3.4 OPTIMAL CHANNEL GEOMETRY ......................................................... 109 CONCLUSION…………………………………………………………………….114 CHAPTER 5: OPTIMISING REACTANT GAS TRANSPORT IN A PROTON
EXCHANGE MEMBRANE FUEL CELL WITH A PIN FIN
INSERT IN CHANNEL FLOW ..................................................116
5.1 INTRODUCTION ........................................................................................... 115 5.2 MODEL DESCRIPTION ................................................................................ 119 5.2.1 GOVERNING EQUATIONS ..................................................................... 121 5.2.2 NUMERICAL PROCEDURE ................................................................... 126 5.3 MATHEMATICAL OPTIMISATION ALGORITHM ................................... 127 5.4 OPTIMISATION PROBLEM FORMULATION ........................................... 128 5.4.1 OPTIMISATION CONSTRAINTS ............................................................. 128 5.4.2 OPTIMISATON PROCEDURE ................................................................ 130 5.5 RESULTS AND DISCUSSION ...................................................................... 132 5.5.1 RESULTS OF FLOW FIELD .................................................................... 132 5.5.2 RESULTS OF PIN FIN GEOMETRY ....................................................... 135 5.5.3 OPTIMISATION RESULTS ...................................................................... 139 5.5.4 PERFORMANCE EVALUATION ............................................................. 143 CONCLUSION…………………………………………………………………….146 CHAPTER 6: MODELLING AND OPTIMISATION OF COOLING
CHANNEL GEOMETRIC CONFIGURATION FOR
OPTIMAL THERMAL PERFORMANCE OF A PROTON
EXCHANGE MEMBRANE FUEL CELL ............................... 147 6.1 INTRODUCTION ........................................................................................... 147 6.2 MODEL DESCRIPTION ................................................................................ 152 xi
Table of Content
6.2.1 BASIC ASSUMPTIONS ............................................................................ 155 6.2.2 GOVERNING EQUATIONS ..................................................................... 156 6.2.3 NUMERICAL PROCEDURE ................................................................... 159 6.3 MATHEMATICAL OPTIMISATION ALGORITHM ................................... 160 6.4 OPTIMISATION PROBLEM FORMULATION ........................................... 161 6.4.1 DESIGN VARIABLE CONSTRAINTS ...................................................... 162 6.4.2 OPTIMISATION PROCEDURE ............................................................... 162 6.5 RESULTS AND DISCUSSION ...................................................................... 164 6.5.1 MODEL VALIDATION ............................................................................. 164 6.5.2 PARAMETRIC STUDY RESULTS ............................................................ 165 6.5.3 OPTIMISATION RESULTS ...................................................................... 172 CONCLUSION…………………………………………………………………….179 CHAPTER 7: CONCLUSIONS AND RECOMMENDATIONS .................... 180 7.1 CONCLUSIONS.............................................................................................. 180 7.2 RECOMMENDATIONS ................................................................................. 183 REFERENCES…………………………………………………………………….186 APPENDICES……………………………………………………………………..216 xii
List of Figures
LIST OF FIGURES
Figure 1.1 Comparison between hydrogen and gasoline as energy currency on service
delivery chain [10].. ....................................................................................................... 3 Figure 2.1 Schematic diagram of a single PEM fuel cell……………………………34
Figure 2.2 The basic structure of a PEM fuel cell showing the path of the
electrochemical reaction [15]………………………………………………………...35 Figure 2.3 Fuel cell stack component [71] .................................................................. 40 Figure 2.4 A schematic of a complete hydrogen-air fuel cell system [115]. .............. 43 Figure 4.1 Schematic diagram of a PEM fuel cell showing different zones and species
transport across the zones. The net water flux is the sum of: (A1) electro-osmotic
effect, (A2) diffusion effect and (A3) the permeability effect……………………….85
Figure 4.2 The discretised three-dimensional computational domain of a single PEM
fuel cell......................................................................................................................... 86 Figure 4.3 Channel cross-sectional view…………………………………………….93 Figure 4.4 Comparison of numerical model prediction and experimental polarisation
curves at base condition. .............................................................................................. 99 Figure 4.5 Pressure drop along the model flow channel at base operating
conditions
for a channel depth of 2.0 mm and width of 1.2 mm. ................................................ 100 Figure 4.6 Effect of temperature on cell performance at base conditions. ............... 101 Figure 4.7 Effect of cathode gas flow rate on cell performance at base conditions..102 Figure 4.8 Effect of gas diffusion layer porosity on cell performance at base
conditions…………………………………………………………………………...103 Figure 4.9 Effect of operating pressure on cell performance at base conditions......104 Figure 4.10 Effect of cathode gas stoichiometry on cell performance at base
conditions…………………………………………………………………………...105 Figure 4.11 The cell current density at different channel depths at a cell potential of
0.3 V, a temperature of 70C and a mass flow rate of 5e-06 kg/s…………………..106 Figure 4.12 The cell current density at different channel widths at a cell potential of
0.3 V and a temperature of 70C. .............................................................................. 107 xiii
List of Figures
Figure 4.13 The cell current density for counterflow orientation (2.61 A/cm2) and coflow orientation (2.54 A/cm2) at base case conditions, for a channel depth of 2.0 mm
and a channel width of 1.2 mm……………………………………………………..108 Figure 4.14 Contours of mass fraction of hydrogen at the anode for (a.) counterflow
and (b.) co-flow cases at the base case operating conditions ..................................... 109 Figure 4.15 Effect of porosity and channel depth on the cell current density. ......... 110 Figure 4.16 Optimum depths as a function of flow rate and gas diffusion layer
porosity. ..................................................................................................................... 111 Figure 4.17 Effect of flow rate and gas diffusion layer porosity on the cell current
density. ....................................................................................................................... 112 Figure 4.18 Effect of porosity and channel width on the cell current density. ......... 112 Figure 4.19 Optimum widths as a function of flow rate and gas diffusion layer
porosity. ..................................................................................................................... 113 Figure 5.1 PEMFC half-cell model with two transverse pin fins along the flow
channel……………………………………………………………………………...120
Figure 5.2 The representative grid system and computational domain. ................... 127 Figure 5.3 Optimisation automation flow diagram. .................................................. 131 Figure 5.4 Effect of Reynolds number on the flow field for different flow field
configurations (s/d = 5,   0.2): (a) Re = 50, (b) Re = 150, (c) Re = 250. ................ 133 Figure 5.5 Effect of Reynolds number on the flow field for different flow field
configurations (s/d = 5,   0.6 ): (a) Re = 50, (b) Re = 150, (c) Re = 250 ................ 134 Figure 5.6 Contours of tangential velocity for different flow field configurations (s/d
= 5,   0.6 ): (a) Re = 50, (b) Re = 150, (c) Re = 250. ............................................... 134 Figure 5.7 Fuel channel friction factor as a function of the Reynolds number and
pitch at a clearance ratio,   0.3 ............................................................................... 135 Figure 5.8 Fuel channel friction factor as a function of the Reynolds number and
clearance ratio at a pitch, s/d = 5.. ............................................................................. 136 Figure 5.9 Fuel channel friction factor as a function of the Reynolds number and
GDL porosity at a pitch, s/d = 5, and a clearance ratio,   0.3 ................................. 137 Figure 5.10 Effect of optimised clearance ratio on the peak channel flow resistance…
.................................................................................................................................... 138 xiv
List of Figures
Figure 5.11 Effect of optimised pitch on the channel peak fuel channel flow
resistance…………………………………………………………………………… 139 Figure 5.12 The minimised fuel channel flow resistance as a function of Reynolds
number for a fixed GDL porosity,   0.5, and a tip clearance ratio,   0.3……...140 Figure 5.13 Optimal clearance ratio as a function of Reynolds number at a fixed
pitch, s/d = 5, and a GDL porosity,   0.5………………………………………..141 Figure 5.14 Optimal pitch as a function of Reynolds number at a fixed clearance
ratio,   0.3, and a GDL porosity,   0.5………………………………………..141 Figure 5.15 Effect of channel flow resistance on the optimised clearance ratio at a
fixed pitch, s/d = 5, and a GDL porosity,   0.5, at Reynolds number of 250……142 Figure 5.16 Effect of channel flow resistance on the optimised pitch at a fixed
clearance ratio,   0.3, and a GDL porosity,   0.5, at a Reynolds number of
250…………………………………………………………………………………..143 Figure 5.17 Fuel channel pressure drop as a function of the applied pressure drop for
a channel with pin fin (s/d = 5,   0.3 ) and one without pin fin. .............................. 144 Figure 5.18 Pumping power as a function of tip clearance ratio at a pitch, s/d = 5,
and GDL porosity,  = 0.6, at a Reynolds number of 250…………………………145 Figure 6.1 A schematic diagram of a 3-D model of PEM fuel cell system with cooling
channels embedded in the bipolar plates……………………………………………153
Figure 6.2 The discretised three-dimensional computational domain of a single PEM
fuel cell with cooling channels……………………………………………………...160 Figure 6.3 Optimisation automation flow diagram. .................................................. 163 Figure 6.4 Comparison of numerical model prediction and experimental polarisation
curves at base condition. ............................................................................................ 165 Figure 6.5 Effect of temperature on the PEM fuel cell performance at base
conditions. .................................................................................................................. 166 Figure 6.6 The cell current density as a function of temperature and the operating cell
voltage………………………………………………………………………………167 Figure 6.7 I-V curve at varying stoichiometry number. P = 3.0 bar and Re = 500... 168 Figure 6.8 Effect of stoichiometry ratio on the PEM cell temperature at cell voltage
of 0.7 V……………………………………………………………………………...169 xv
List of Figures
Figure 6.9 I-V curve at varying relative humidity (RH). P = 3.0 bar and Re = 500..170 Figure 6.10 The cell current density at different aspect ratio at a cell potential of 0.7
V and a fixed Reynolds number of 500……………………………………………..171 Figure 6.11 Current density at three cases of channel aspect ratio and Re = 500….172 Figure 6.12 Effect of optimised cooling channel aspect ratio on the peak fuel cell
current density at different temperatures……………………………………………175 Figure 6.13 Effect of Reynolds number and temperature on the optimised aspect ratio
of the cooling channel………………………………………………………………176 Figure 6.14 Effect of Reynolds number on the maximum current density at different
cell temperatures…………………………………………………………………….177 Figure 6.15 The local distribution of temperature along the membrane at different
cooling channel aspect ratios and cell operating voltage of 0.7 V and Re = 500:
(a) H W = 1.875, (b) H W = 2.500 and (c) H W = 2.813…………………………178 xvi
List of Tables
LIST OF TABLES
Table 1.1 Combustion properties of hydrogen compared with other fuels [4]. ............. 4 Table 4.1 Base case geometric parameters of the modelled fuel cell………………...86
Table 4.2 Physicochemical properties of the modelled fuel cell ................................. 86 Table 5.1 Parameters of the modelled fuel cell……………………………………..121
Table 6.1 Parameters and properties used in the present model……………………154
Table 6.2 The governing equation source terms in various regions of the fuel cell. . 157 Table 6.3 Grid independence test…………………………………………………...159 Table 6.4 Dimension of the cooling channels investigated for initial simulations. ... 161 Table 6.5 Values of optimised parameters ................................................................. 173 Table 6.6 Polarisation data at optimised conditions and varying cell operating
temperatures at Re = 500……………………………………………………………173
xvii
Nomenclature
NOMENCLATURE
A
Channel width (m)
A
Hessian matrix of the objective function
Ach
Cross-sectional area of channel (m2)
Ac
Fin cross-sectional area (m2)
B
Channel depth (m)
Bi
Hessian matrix of the inequality function
C
Constant
a, b, c
Diagonals of Hessian matrices A, B, C
Cj
Hessian matrix of the equality function
CF
Quadratic drag factor
Cp
Specific heat capacity (J kg-1)
cr
Condensation rate constant
D
Gas diffusivity (m2 s-1)
Dch
Channel diameter (m)
Df
Diameter of pin fin (m)
Deff
Effective diffusivity (m2 s-1)
Dh
Hydraulic diameter (m)
E
Electrolyte
EOCV
Open-circuit voltage (V)
e-
Electron
F
Faraday constant (96, 487 C mol-1)
xviii
Nomenclature
F
Friction factor
f x 
Objective function
~
f x 
Objective approximate function
G
Computational domain width (m)
g j x 
j-th equality constraint function
g~ j x 
j-th inequality constraint approximate function
h
Enthalpy (J kg-1)
H
Computational domain height (m)
hk x 
k-th equality constraint function
~
hk x 
k-th equality constraint approximate function
hL
Enthalpy of condensation/vaporisation of water (J kg-1)
I
Exchange current density (A m-2)
io
Local current density (Am-2)
j
Volumetric transfer current
k
Thermal conductivity (W m-1 K-1)
K
Permeability
L
Channel axial length (m)
MW
Molecular weight
M
Molar mass (g/mol)
Channel mass flow rate (kg/s)
n
Electron number
xix
Nomenclature
nd
Electro-osmotic drag coefficient
P
Pressure (Pa)
Wetted perimeter
Pc
Capillary pressure (Pa)
Po
Poiseuille constant
Ppump
Pumping power (W)
P[k]
Successive sub-problem
Q
Volume flow rate (m3/s)
rp
Mean pore radius
rw
Water condensation rate (s-1)
R
Universal gas constant (8.314 J mol-1 K-1)
Re
Reynolds number
Rf
Dimensionless flow resistance
Rohm
Resistance of proton transfer through electrolyte membrane (
RH
Relative humidity
S
Liquid saturation or source term
S
Pin spacing (m)
Sh
Sherwood number
Sh
Volumetric heat source term
sw
Water saturation
n
n-dimensional real space
xx
Nomenclature
T
Time (s)
T
Temperature (K)
U
Overall heat transfer coefficient
Uo
Average velocity at inlet (m/s)
U0
Thermodynamic equilibrium potential
u, v
Velocities in the x- and y- directions (m/s)
u
Velocity vector [ms-1]
V
Volume (m3)
V
Cell potential (V)
Vavg
Mass-averaged velocity (m/s)
Vd
Volume ratio in diffusion layer
Vs
Surface ratio in diffusion layer
x, y, z
Cartesian coordinate (m)
w
Water
w
Mean velocity (m/s)
W
Molar mass fraction of oxygen
Vw
Convective velocity
x*
Design variables
xk
Design points
j,k, m,n,r
Positive integer
Greek
xxi
Nomenclature
Difference operator

Permeability (m2)

Porosity

Viscosity of flow [kg m-1 s-1]

Fluid viscosity (kg m-1 s-1)

Penalty parameter value

an
Electrical transfer coefficient (anode)
 cat
Electrical transfer coefficient (cathode)

Membrane water content

Tip clearance ratio
V
Kinematic viscosity [m2 s-1]

Ionic conductivity [S/m]

Pitch

Solid fraction

Over-potential (V)

Phase potential function (V)

Density (kg m-3)
Tortuosity

Electrical conductivity
xxii
Nomenclature
Subscripts
a
Air
an
Anode
avg
Average
c
Capillary
cat
Cathode
ch
Channel
D
Porous diffusion layer
e
Electrolyte
eff
Effective
f
Fuel
G
Gas
H
Hydraulic
k
species
L
Liquid water
m
Mass moment source
m
Membrane
max
Maximum
min
Minimum
opt
Optimum
px, py, pz
Momentum source terms
xxiii
Nomenclature
react
Electrochemical reaction
ref
Reference value
s
Electronic conductive solid matrix
sat
Saturation
T
Energy source term
w
Liquid water source
v
Vapor phase
x,y,z
Components in the x-, y- and z- directions
AC
Alternating current
BPP
Bipolar plate
BTU
British thermal unit
CESFF
Convection-enhanced serpentine flow field
CL
Catalyst layer
CO
Carbon monoxide
CO2
Carbon dioxide
CFD
Computational fluid dynamics
CHP
Combined heat and power
DC
Direct current
EMF
Electromotive force
FEM
Finite element method
GDL
Gas diffusion layer
xxiv
Nomenclature
H2
Hydrogen gas
HOR
Hydrogen oxidation reaction
HT
Higher temperature
ICE
Internal combustion engine
LFOPC
Leapfrog optimization program for constrained problems
MEA
Membrane electrode assembly
MFPM
Multi-facilitated proton membrane
NOx
Nitrogen oxides
O2
Oxygen
ORR
Oxygen reduction reaction
PEM
Proton exchange membrane
PEMFC
Proton exchange membrane fuel cells
Pt
Platinum
SQP
Sequential quadratic programming
xxv
Chapter 1: Introduction
1
CHAPTER 1:
INTRODUCTION
1.1 BACKGROUND AND MOTIVATION
Energy is becoming an issue of serious concern in the world today. It is inevitable for
human life and a secure supply of energy is required for sustainability of human
societies [1]. The need to satisfy world energy demand, which actually determines the
living standard of the populace, is increasing. This energy is utilised to generate the
electricity we need for our homes, businesses, schools and factories. It energises our
computers, lights, refrigerators, washing machines and air conditioners, to mention
only a few. Also, the quantity of energy required in the industrial sector of the
economy for its production activities is ever-increasing. This energy is mostly
obtained from fossil fuel stock combustion processes and great deals of pollutant
gases (CO2, NOX, etc.) are emitted to the atmosphere [2, 3]. Some of these gases,
especially CO2, are a major contributor to global warming and its attendant
consequences, such as rise in global average temperatures, rise in sea levels, flooding
and deforestation. Therefore, the effects of global warming have become an issue of
major concern to goverments, policy makers and environmentalists. Hence, in recent
times, numerous researches and commissioned studies have focused on the
development of carbon-free energy sources that are environment-friendly, sustainable
and cheaply available so as to minimise the amount of pollutant gases emitted into the
atmosphere as a result of energy consumption [4].
The available energy sources in the world today are divided into two groups:
renewable and non-renewable sources. Renewable energies are those that come from
natural resources and are replenished naturally. Non-renewable energies are those that
1
Chapter 1: Introduction
are not replenished or only replenished very slowly. The available renewable energy
systems range from solar power systems, wind power systems, geothermal power
systems, fuel cells, etc. Renewable systems have different comparative advantages
which usually determine their applications. Both renewable and non-renewable
energy sources can be used to produce secondary energy sources, including electricity
and hydrogen. However, most of our energy sources today are from non-renewable
sources, which include the fossil fuels, i.e. oil, natural gas and coal [3]. Renewable
energy resources become an important option to fossil fuel as the negative
environmental consequences of fossil fuel increases and its utility cost (electricity)
climbs. The quality of renewable energy technologies, that makes it a viable substitute
to fossil fuel, includes its modular nature, lower operating cost and its flexibility and
adaptability. These energy sources are considered by many as a direct replacement of
existing fossil fuel technologies and this has made the evaluation of its benefit in
terms of cost to be rated low when compared to traditional fossil technologies. The
baseline is to view these renewable technologies as a complementary modular
addition to existing energy systems with short lead-times [1]. This will adequately
reduce the pressure on the national grids and ensure availability of energy to people in
remote areas. Moreover, it will help reduce the amount of pollutant gases released
into the atmosphere as a result of fossil fuel usage.
The world energy consumption projection by 2030 is estimated at about 700
Quadrillion British thermal unit (BTU) [5]. This figure equates to two-thirds more
energy than the present usage. Fossil fuels will remain the dominant sources of
energy, accounting for more than 90% of the projected increase in demand [5].
Problems associated with energy supply and demand are much more than global
warming threats, but environmental concerns such as ozone layer depletion, pollution,
deforestation and radioactive emission are increasing today [1]. These environmental
problems need to be addressed quickly if the world is to achieve a sustainable energy
future. The drive today is to seek for sustainable development through the utilisation
of energy sources that has little or no adverse impact on the environment [6, 7]. These
2
Chapter 1: Introduction
energy sources (i.e. solar, wind, etc.) are easily replenished once consumed, as
compared to finite fossil fuels (oil, coal and natural gas).
Hydrogen, a clean and renewable fuel source, is generally available in abundance and
is a safe energy source [8, 9]. This fuel type can be generated from different kinds of
sources, including most renewable sources and fossil fuels (natural gases and coal
gasification). Figure 1.1 illustrates a typical comparison of utilising gasoline and
hydrogen as fuel for transportation and mobile applications in the service sector [10].
The figure illustrates that hydrogen sources are diverse on the energy sector side and
that the emission characteristics are quite limited on the service sector side, making
hydrogen a key candidate for future energy currency.
Figure 1.1 Comparison between hydrogen and gasoline as energy currency on service
delivery chain [10]
Hydrogen has long been recognised as a potential fuel source for application in
engines due to some unique and desirable properties [11]. These properties include its
combustion in oxygen that produces only water as a waste, though, when combusted
3
Chapter 1: Introduction
in air, could generate some oxides of nitrogen. Table 1.1 is a comparison of
combustion properties of hydrogen with other fuels. The table shows the outstanding
properties of hydrogen in terms of performance when compared with other
conventional fuels.
Recent studies [12-14] have shown the importance of hydrogen energy to sustainable
development and in resolving the prevalent global environmental issues. The
transition to hydrogen-based economy, where the main energy carrier is hydrogen and
the main non-chemical energy form is electricity, is being made gradually and interest
in this area is growing rapidly. However, generating electricity directly from
hydrogen requires specific energy technologies such as the fuel cell. Fuel cell is a
thermodynamic system that generates power by a direct conversion of the chemical
energy in fuel into electrical power through electrochemical reaction [15].
Table 1.1 Combustion properties of hydrogen compared with other fuels [4]
Property
Flammability limits (% by volume)
Minimum ignition energy (mJ)
Laminar flame speed at NTP (m/s)
Adiabatic temperature (k)
Autoignition temperature (k)
Quenching gap at NTP (mm)
Hydrogen
4-75
0.02
1.90
2318
858
0.64
Methane
5.3-15.0
0.28
0.38
2190
813
2.03
Gasoline
1.2-6.0
0.25
0.37-0.43
~2470
~500-750
~2.0
Recent advancements in fuel cells have been driven by the demand for highly
efficient power generation devices. Current fuel cell investments are mainly made by
automotive industries to increase fuel efficiency and/or to use hydrogen as an
alternative fuel. There are also opportunities of its application in power and electronic
industries. The main reason for using fuel cells in power generation is the need for
pollution reduction, back-up power, diversification of energy supply, as well as
reduction in foreign energy dependency. Fuel cells are very useful as power sources in
remote locations, such as spacecraft, remote weather stations, large parks, rural
4
Chapter 1: Introduction
locations and in certain military applications. A fuel cell running on hydrogen can be
compact and lightweight, and have no major moving parts. A new application is
micro combined heat and power (CHP), which is cogeneration for family homes,
office buildings and factories [16].
Among the various types of fuel cells, proton exchange membrane fuel cells
(PEMFCs), also termed “proton exchange membrane fuel cell” system, has attracted
much interest as a convenient and viable alternative source of power, with promising
potential to reduce the excess consumption of fossil fuel and discharge of carbondioxide [14]. The PEMFC has a high power density and a relatively low operating
temperature (ranging from 60 to 80 degrees Celsius). The low operating temperature
means that it does not take very long for the fuel cell to warm up and start generating
electricity. Hence, PEMFC may most likely power automobiles and even residential
houses in the nearest future.
Despite the potential of fuel cells to serve as clean alternative energy sources, a lot of
issues still need to be addressed, mainly its cost of production and technical issues
relating to optimal operating performance. The costs of components required to make
fuel cells are prohibitive. For PEMFC systems, costly components such as proton
exchange membranes, precious metal catalysts (usually platinum), gas diffusion
layers and bipolar plates constitute up to 70% of the cost of a typical module [15].
Also, stationary fuel cell application typically require more than 40,000 hours of
reliable operation at temperatures of -35C to 40C, while automotive fuel cells
require a 5,000-hour lifespan (equivalent of 150,000 miles) under extreme
temperatures. Automotive engines must also be able to start reliably at -30oC and have
high power to volume ratio (typically 2.5 KW per liter). Thus, there is the need to
develop fuel cells that are durable and can operate at temperatures greater than 100C
and yet function well at sub-zero ambient temperatures. In addition, the cell
membranes also tend to degrade while the fuel cell system cycles on and off,
particularly as operating temperatures rises. Hence, it is important for the membrane
5
Chapter 1: Introduction
to remain stable under cycling conditions. Also, PEMFC membranes must be
hydrated in order to transfer hydrogen protons. This is necessary because, if water in
the system evaporates too quickly, the membranes dry up and resistance across it
increases. It will eventually crack, creating a gas “short circuit” where hydrogen and
oxygen combine directly, generating heat that will damage the fuel cell. This
condition necessitates that the fuel cell continues to operate in sub-zero temperatures,
low humidity environments and high operating temperatures. Furthermore, when a
fuel cell is in operation, the temperature must be maintained throughout the cell in
order to prevent destruction of the cell through thermal loading. This is particularly
challenging, since the reaction in the fuel cell is highly exothermic (heat releasing)
and thus, large quantities of heat is generated within the fuel cell. Maintaining a
uniform operating cell temperature in the fuel cell is thus not a trivial task.
The technical issues highlighted above, have hindered the commercialisation of
PEMFC, hence there is need for in-depth research to understand and proffer solutions
aimed at improving the performance of this class of fuel cell, so as to meet the market
competitiveness compared to fossil-fuel based energy systems. One of the main
objectives of the present fuel cell research in the industry today is the need to improve
the performance of fuel cells. This can be done by better design and enhancing its
capability so as to increase its production at low cost in order for it to compete
favourably with fossil fuel-based systems. There are two primary approaches to
achieving this, that is:
 Design, build and experimentally test approach to evaluate its performance
 Simulate by numerical modelling approach
The first approach usually yields useful and physical representative information of the
phenomenon in the real system but is costly and time-consuming [10]. This becomes
more difficult when looking into the vast number of working parts involved in a fuel
6
Chapter 1: Introduction
cell system and the limited experimental techniques available. The modelling
approach can provide the much needed insight into the phenomena that characterise
fuel cell systems at a reduced cost and time [10]. Optimal design of the system can
thus be achieved and subsequently help at achieving the goal of fuel cell
commercialisation. Performance improvement in the proton exchange membrane
(PEM) fuel cell system is still an open research. More design models are being
developed with the aim of enriching the knowledge base on generic information
needed for a better design of PEM fuel cell systems. This research is one of the efforts
channelled in that direction by introducing a more novel modelling approach coupled
with optimisation techniques to improve the performance of PEM fuel cell systems.
1.2 REVIEW OF RELATED LITERATURE
A significant amount of research studies have been devoted to the study of PEM fuel
cells, ranging from theoretical to experimental studies. With regards to the former,
several empirical and mathematical models, which are aimed at understanding and
predicting PEM fuel cell performance, have been proposed. Giner and Hunter [17]
and Cutlip et al. [18, 19] have proposed the first of such models taking into
consideration diffusion and reaction in the gas-diffusion electrodes. More attention
subsequently spring up towards experimental studies [20-22] using simple 0-D
models to analyse data on PEM fuel cells. These models normally fit the experimental
data with a single equation. Although these models demonstrate good fits and are
quick and easy to implement, they are less accurate and reliable in predicting the PEM
fuel cell behaviour. More fundamental models were developed thereafter to simulate
performance and gain deeper understanding of the underlying fundamental transport
processes. Two main works in this regard are those of Bernadi and Verbrugge [23]
(hereafter referred to as B&V) and Springer et al. [24]. Both studies included the
membrane, diffusion media and catalyst layers in their respective models. B&V’s
model assumes a fully hydrated membrane and incorporates porous-electrode
7
Chapter 1: Introduction
equations and Stefan-Maxwell diffusion in the diffusion media and catalyst layers.
The model of Springer et al. [24] does not use porous-electrode equations but
changing water content in the membrane. This changing water content allows for
variable properties in the membrane such as conductivity and the water diffusion
coefficient. Most models today can conveniently trace their roots back to B&V
studies [25].
The advances in digital computer technology have spurred the progress in the area of
fuel cell development, especially in the application of numerical methods for fuel cell
optimisation. The advancement in computational fluid dynamics (CFD) allows for
effective design and optimisation of the fuel cell systems, with reduced reliance on
hardware prototyping and reduction in development cycles. CFD provides a platform
for understanding the variety of complex multi-physics transport processes
characterised by a broad spectrum of length and time scales in the fuel cell structure.
These processes include phenomena which involve fluidic, ionic, electronic and
thermal transport in concert with electrochemical reactions. B&V’s model forms the
basis for almost all the CFD models in fuel cells today [25]. The incorporated
electrochemical effects stem from the developed equations of B&V, such as their
kinetic source terms in the catalyst layers and the use of Schlogl’s equation for water
transport in the membrane. The following sections (1.2.1-1.2.4) deal with specific
literature relevant to this thesis, in which related studies addressing the design,
optmisation and performance analysis of PEM fuel cells are discussed.
1.2.1 OPTIMAL OPERATING CONDITIONS FOR PEM FUEL CELL
Fuel cell operation involves the specification of a range of operating conditions such
as temperature, pressure, stoichiometry ratio of reactant gases, porosity of the
diffusion layers, etc. Accurate specification of this range of operating conditions will
assist in predicting the fuel cell performance under these specified conditions and
8
Chapter 1: Introduction
could be used to optimise the design of a fuel cell system. Appropriate operating
conditions are also required for a PEM fuel cell system to achieve and maintain stable
operational performance. Effects of the gas hydrodynamics on the performance of the
air cathode of a PEMFC with an interdigitated gas distributor has been studied by Yi
and Nguyen [26]. In addition, pressure drop between the inlet and outlet channels,
electrode height and shoulder width on the average current density were studied. They
discovered that, with the forced flow-through condition created by the interdigitated
gas distributor design, the diffusion layer thickness is greatly reduced. However, even
with a much thinner diffusion layer, diffusion still plays a significant role in the
transportation of oxygen to the reaction surface. In addition, the average current
density generated at an air cathode increases with higher gas flowrates, thinner
electrodes and narrower shoulder widths between the inlet and outlet channels of the
interdigitated gas distributor.
Chan and Tun [27] conducted an investigation to determine the effects of the different
parameters such as catalyst layer porosity, catalyst layer thickness and ionic
conductivity on the performance of PEM fuel cells. The model showed that catalyst
layer porosity and catalyst layer thickness has a significant effect on the limiting
current density for the fuel cell. However, the ionic conductivity has no effect on the
limiting current density. Furthermore, Jaouen et al. [28] used a one-dimensional,
steady-state agglomerate model to determine the nature of mass transport limitations
in the PEM fuel cell cathode. Effects of the active layer thickness, oxygen
concentration and relative humidity of the oxygen stream were investigated. The
result of the model shows that limitation by proton migration in the active layer, or by
oxygen diffusion in the agglomerates leads to a doubling of the Tafel slope at higher
current densities. For those two types of transport limitations, the dependence of the
reaction rate on the active-layer thickness, oxygen partial pressure and relative
humidity of the specie gas were shown. When additional limitation, due to slow gas
phase diffusion, appears, the double Tafel slope is distorted. A mathematical
9
Chapter 1: Introduction
expression for the limiting current density, due to this process, was presented for use
in correcting the polarisation curves for slow gas phase diffusion.
Studies on the effects of various operational parameters such as temperature, pressure,
stoichiometric ratio, porosity and gas diffusion layer (GDL) thickness on the fuel cell
performance was performed by Berning et al. [29]. They observed that temperature,
pressure, stoichiometry ratio, GDL thickness and porosity, all have an impact on the
limiting current density at a varying degree for the fuel cell. They also observed the
need to estimate the extent of contact resistance inside the fuel cell in order to
properly assess the impact of porosity and channel width on fuel cell performance.
Kazim et al. [30] presented a two-dimensional mathematical model in which they
investigated the effects of cathode porosity, inlet oxygen mole fraction, operating
temperature and pressure on the performance of PEM fuel cells with the interdigitated
flow field. The obtained result illustrated the positive impact of an increase in the
GDL porosity on the fuel cell performance. Furthermore, it was observed that an
increase in the mole fraction, operating pressure, or temperature of the oxygen
entering the cathode GDL leads to higher fuel cell performance.
Chu et al. [31] studied the effect of variability in the porosity size of the GDL on the
performance of PEM fuel cell. They observed that a fuel cell embedded in a GDL
with a larger averaged porosity consumes a greater amount of oxygen, such that a
higher current density is generated and a better fuel cell performance of the fuel cell is
obtained. This explains partly why fuel cell performance deteriorates significantly as
the cathode is flooded with water (i.e. to give a lower effective porosity in the GDL).
In terms of the system performance, a change in GDL porosity has virtually no
influence on the level of polarisation when the current density is medium or lower,
but exerts a significant influence when the current density is high. The investigations
of Jeng et al. [32] focused on the effects of the change in the porosity of the GDL on
the performance characteristics of a PEMFC. Their results also showed that the
existence of ribs causes the GDL to be used only partly in the mass transfer process.
10
Chapter 1: Introduction
The GDL’s effectiveness decreases with the cell current density and increases with
the width of the gas flow channels. The PEM fuel cell performance decreased with an
increase in the GDL thickness when the GDL porosity is low. However, when a highporosity GDL is used, the optimal thickness becomes an indicator which determines
the maximal PEM fuel cell performance.
Wang et al. [33] conducted a study aimed at verifying the mechanisms of parameter
effects and their interrelationship by comparing modelling results with experimental
data. They observed that, when adequate humidification is provided, the performance
of the PEM fuel cell improves with the increase in fuel cell temperature. The result
also showed that anode and cathode humidification has significant effects on the
performance of the PEM fuel cell. Lee et al. [34] conducted a numerical simulation of
the species gas in the fuel channel and the diffusion layer to investigate the effects of
GDL thickness, porosity and distribution of the pore size on the PEM fuel cell
performance. The PEM electrodes were prepared by applying different porous GDLs
onto each face of a carbon cloth support. They discovered that a GDL with a more
porous structure performed better. More importantly, it was shown that a GDL’s
geometric characteristics (thickness, porosity and distribution of pore size) greatly
affect the performance of the PEM fuel cell.
Hwang et al. [35] presented a three-dimensional numerical model to simulate the
transport phenomena on the cathode air-side of a PEMFC. They compared the
polarisation curves of the interdigitated flow field and parallel flow field for a typical
PEM fuel cell. Their study ascertained the fact that an interdigitated flow field gives a
higher average current density on the catalyst layer surface than with parallel flow
field under similar mass flow rate and cathode overpotential. Effects of electron
transport through the GDL of the PEM fuel cell was investigated by Meng and Wang
[36]. They discovered that the lateral electronic resistance of the GDL, which is
affected by the electronic conductivity, GDL thickness and gas channel width played a
critical role in determining the current distribution and cell performance. It was further
11
Chapter 1: Introduction
observed that, under fully-humidified gas feed in the anode and cathode, both oxygen
and lateral electron transport in the GDL dictated the current distribution. The lateral
electronic resistance dominated the current distribution at high cell voltages, while the
oxygen concentration played a more decisive role at low cell voltages. With reduced
GDL thickness, the effect of the lateral electronic resistance on the current distribution
and cell performance became even stronger, because the cross-sectional area of the
GDL for lateral electron transport was smaller.
Du et al. [37] proposed a theoretical model to investigate the effective protonic and
electronic conductivity of the catalyst layers in PEM fuel cells. The model showed
that effective protonic conductivity increased with an increase in the Nafion volume
fraction in the catalyst layers of the PEM fuel cells. The study also showed that
effective protonic conductivity increased almost linearly with an increase in the
operating temperature for a given water activity. Pasaogullari and Wang [38]
conducted an investigation on the two-phase flow characteristics in the cathode GDL
of a PEMFC. They revealed that an onset of flooding of the porous cathode hinders
the rate of oxygen mass transport to the cathode catalyst layer. In addition, their result
showed that the rate of cell humidification and mass flow rate of the reactant gas
species are important parameters determining PEMFC two-phase flow transport
characteristics and performance.
Lu and McGurick [39] presented a model of the PEMFC cathode with an
interdigitated gas distributor to investigate the effects of various parameters such as
electrode permeability, thickness and shoulder width on the cell performance. It was
observed that changes in permeability, ranging from 10–8 to 10–13 m2, has little impact
on the cell performance. Increasing the electrode thickness and the shoulder width
resulted in poorer performance due to greater resistance to flow. In addition, their
results showed that liquid water tends to form near the outlet of the electrode when the
current density is greater than 1.0×104 A m–2. Sun et al. [40] developed a twodimensional model to investigate the influence of the GDL property and flow-field
12
Chapter 1: Introduction
geometry, such as diffusion layer diffusivity, diffusion layer conductivity, channel
width-to-area ratio and diffusion layer thickness on the local reaction rate in the
PEMFC cathode catalyst layer. Their work showed that, when the PEMFC is operated
using reformate hydrogen, the performance drops dramatically due to carbon
monoxide (CO) poisoning as the anode gas flow rate increases. More research on the
CO poisoning effect on PEMFC performance are reported in the literature [41-43].
Mawardi and Pitchumani [44] studied the effect of parameter uncertainty on the
variability in performance of PEM fuel cells and optimisation of different operating
parameters that affects fuel cell performance. They developed a sampling-based
stochastic model to measure the performance of PEM fuel cells. The results further
provided a valuable tool for the design of fuel cells under uncertainty in material and
operating parameters.
Hsieh et al. [45] investigated the effects of the operating temperature and
backpressure on the performance of micro PEM fuel cells using different flow fields.
Their study concluded that cell performance increases with an increase in cell
operating temperature until a limiting or threshold level is reached. In addition, they
observed that the higher the flow-back pressure, the better the performance of the
single micro PEMFC at a fixed cell operating temperature. The interdigitated flow
field showed better performance, while lower pressure was obtained using mesh-type
flow field at a fixed active area of the membrane electrode assembly (MEA).
Yan et al. [46] investigated the steady and transient response on performance in both
single fuel cell and stack configuration under a variety of loading cycles and operating
conditions. They discovered that different feed gas humidity, operating temperature,
feed gas stoichiometry, air pressure, fuel cell size and gas flow pattern affect both the
steady-state and dynamic response of fuel cells. They experimentally confirmed that a
decrease in the cathode humidity has a detrimental effect on the fuel cell steady state
and dynamic performance of the fuel cell. Temperature variation also significantly
13
Chapter 1: Introduction
affects fuel cell performance through its effect on membrane conductivity and water
transport in the GDL and the catalyst layer. Amirinejad et al. [47] conducted
experiments to study the effects of operating parameters on PEM fuel cell
performance by using dry and humidified hydrogen and oxygen as reactant and
oxidant gases, respectively. The result of their experiment showed that the most
important factor affecting the PEMFC performance is the mass transport limitation.
This limitation included the transport of reactant and oxidant gases to active sites of
the catalyst, the transport of the proton from the anode side to the cathode side
through the membrane, and the transport of produced water from the cathode side to
the anode side by back-diffusion mechanism. Fuel cell operating parameters such as
temperature, pressures and humidity of reactant gases could decrease the mass
transport limitation and improve the performance of the fuel cell.
Zhou et al. [48] developed a steady-state, two-dimensional model to illustrate the inlet
humidification and pressure effects on PEM fuel cell performance. Their model
asserts the fact that humidification of both the anode and the cathode is very important
for fuel cell performance. Also, the pressure drop in the PEM fuel cell flow channels
increases the pumping power requirement and attention must be paid to this pressure
situation when designing the fuel cell. Yan et al. [49] determined the electrical
characteristics of a PEMFC stack under varying operating conditions, using AC
impedance measurement technique. They documented the fact that the air humidity
and cell temperature greatly impact on the charge transfer resistance of the PEM fuel
cell stack. Similar to Yan et al.’s work is studies by Zhang et al. [50]. They
investigated the effect of reactant gas relative humidity on fuel cell performance using
the AC impedance and cyclic voltammetry methods. This study affirmed that a
reduction in the relative humidity of a fuel cell can depress the electrode kinetics,
including electron reaction and mass diffusion rates, and the proton conductivity of
the membrane, resulting in a dramatic degradation of the fuel cell performance.
14
Chapter 1: Introduction
Hung et al. [51] developed a theoretical model to investigate the effects of operating
parameters (e.g. temperature, humidification temperature, pressure, gas stoichiometry
ratios) on cell performance. Design and modelling parameters were obtained using a
regression analysis of experimental data and validating it as these operating
parameters changes. Hwang et al. [52] presented a non-isothermal model of PEMFC
in contact with an interdigitated flow field to study the effect of flow orientation on
thermal-electrochemical transport in a PEM fuel cell. The study revealed that both the
solid-matrix and fluid-phase temperatures are increased with the increase of the total
overpotential of the fuel cell. In addition, the fluid-phase and solid-matrix temperature
distributions are significantly affected by the flow orientation of the species reactant
in the PEM fuel cell.
Yuan et al. [53] proposed a three-dimensional multi-phased model of a PEM fuel cell
to predict the effects of operating parameters on the performance of PEM fuel cells.
The study revealed that fuel cell performance is enhanced with an increase in
operating pressure, temperature and air stoichiometry ratio. The study asserts the fact
that anode humidification has more significant effects on the PEM fuel cell
performance than cathode humidification. It was also documented that best
performance occurred at low air relative humidity and high hydrogen relative
humidity.
In summary, most theoretical studies on fuel cells in the literature focus on the
numerical simulation of the transport phenomenon and parametric study of the effects
of physical variables. The main objectives of the reported models are to investigate
the performance of fuel cells under various operating conditions, with a view to find
optimal performance parameters.
15
Chapter 1: Introduction
1.2.2 FUEL GAS CHANNEL OPTIMISATION FOR PEM FUEL CELLS
One of the critical issues in proton exchange membrane fuel cell design is the efficient
design of the flow channels to ensure uniform distribution of the reactant gases in the
fuel cell stack. The flow field geometry and pattern has great influence on the reactant
gas transport, water management and the efficient utilisation of the fuel gases, since
efficient species gas transport and water removal from the fuel cell system is enabled
by proper flow field design. The flow field design for fuel cells is thus one of the
important technical challenges for PEM fuel cell design and operation and impacts on
system performance and life-span [54].
Kumar and Reddy [55] studied the effect of the dimensions and shape of the flow
channels in the flow-field of a PEMFC. The flow field used for the study was the
single-path serpentine design. They concluded from their study that optimum channel
width, land width and channel depth for optimal fuel cell performance are close to
values of 1.5, 0.5 and 1.5 mm, respectively. In addition, it was found that reducing the
channel land width, increases the hydrogen consumption at the anode section of the
fuel cell. A hydrodynamic model to study flow distribution and pressure drop in
parallel-channel configurations of a planar fuel cell was developed by Maharudrayya
et al. [56]. They considered Z-type and U-type configuration channels in their study.
The obtained result shows that for a fuel cell distributor plate, low and high flow
maldistribution could occur for both the Z-type and U-type configuration. The extent
of this maldistribution is a function of the geometric factors of the parallel-channel
configuration and these factors could be manipulated to achieve a uniform flow
distribution in the fuel cell system.
Shimpalee et al. [57] investigated the impact of channel path length on a PEMFC
system. They concluded from their work that better uniformity in local temperature,
water content and current density distribution in the serpentine flow-field design of
the PEM fuel cell system is obtained by using a shorter path length rather than a
16
Chapter 1: Introduction
longer path. Hence, reducing the PEM flow-field path length is a prospective variable
for improving the performance and efficiency of the PEMFC system. Inoue et al. [58]
conducted a study to investigate the effect of gas channel depth on current density
distribution of PEM fuel cell using computational fluid dynamics, including gas flow
through a GDL. They found that output current density of the fuel cell system
increases with the decrease in the depth of the separator channel and corresponding
increase in pressure drop and current density distribution.
A model similar to that of Inoue [58], was employed by Ahmed and Sung [59] to
investigate the effect of channel geometric configuration at high operating current
density of the fuel cell. Their result demonstrated the fact that a rectangular channel
cross-section produces higher cell voltages compared with trapezoidal and
parallelogram channel cross-sections. However, the trapezoidal cross-section proves
more effective at ensuring uniform reactant and local current density distribution over
the reactant area of the fuel cell. The results further ascertain the fact that shoulder
width impacts great on fuel cell performance when compared with other geometric
factors. Also, Cheng et al. [60] conducted a study to optimise the geometrical
parameters of the PEMFC, by using a numerical approach coupled with an optimiser.
The result of their study shows that the gas channel width fraction, the gas channel
height and the thickness of the GDL all influenced the performance of the fuel cell
system. In addition, their study shows that, using the coupled optimiser at channel
width fraction of 0.3925, gas channel height of 1.2034 mm and GDL thickness of
0.176 mm, an optimal power density of 1857 W m-2 is obtained when compared with
the original fuel cell design.
Xu and Zhao [61] developed a novel flow-field for polymer electrolyte-based fuel cell
systems by re-pattering conventional single serpentine flow fields. Their studies
confirmed the effectiveness of this new design at inducing larger pressure difference
between adjacent flow channels over the electrode surface area of the fuel cell. The
effect of such large difference is that mass transport of the reactant gases and products
17
Chapter 1: Introduction
are enhanced to and from the catalyst layer. Furthermore, water clogging within the
electrode in greatly reduced.
Li et al. [62] proposed a flow-field design procedure to effectively eliminate resident
water flooding in the PEM fuel cells. They employed a design based on specifying
appropriate pressure drop along the flow channel that will ensure evaporation or drift
force removal by the gas stream in the flow channel width, therefore, dehydrating the
fuel cell membrane. They reported that the designed flow-field procedure is effective
for water removal in the fuel cell. Their claims were validated experimentally, by
using a neutron imaging technique measurement of liquid water content in the fuel
cell system. Shimpalee and van Zee [63] numerically investigated the effect of rib and
channel dimension of the reactant flow-field on the performance of PEMFC under
automotive and stationary conditions. The obtained result revealed that, for stationary
applications, employing a narrower channel with widened rib spacing produces higher
fuel cell performance, with the reverse being a case of automotive application.
Owejan et al. [64] studied the effects of flow field and GDL properties on water
accumulation in the PEMFC. They documented that flow field channels with
hydrophobic coating retain more water in the fuel system, but the spread of a higher
number of smaller water slugs improves the fuel cell performance at high current
density. The result further demonstrated the fact that cells made by using diffusion
media with lower in-plane gas permeability shows lower water accumulation capacity
in the fuel cell system. Peng et al. [65] developed a model to optimise the flow
channel design and at the same time balance the fuel cell stack performance and
formability. Their optimisation result shows that optimum dimensional values for
channel depth, channel width, rib width and transitional radius of 0.5, 1.0, 1.6 and 0.5
mm, respectively, were obtained at highest reaction efficiency of 79% and formability
of 1.0 of the fuel cell used for their study.
18
Chapter 1: Introduction
Sinha et al. [66] presented a three-dimensional, non-isothermal PEMFC model to
investigate the effect of flow field design on the performance of the system at elevated
temperatures. They compared the fuel cell performance with serpentine and parallel
flow field design when the fuel cell is operated at 95ºC under various inlet humidity
conditions. They concluded that the parallel flow field design ensure better and
uniform distributed performance on the entire cell active area when compared to the
serpentine flow field at low inlet relative humidity and elevated temperatures. Hsieh
and Chu [67] conducted a study on channel and rib geometric scale effects of flowfield plates on the performance and transient thermal characteristics of micro-PEM
fuel cell system. They found that optimum channel-to-rib width ratio for the range
considered in their study to be 0.67, considering the net power gain of the system. In
addition, they documented the fact that channel and rib geometric effect has no
significant effect on the cell system transient temperature distribution.
Ferng et al. [68] performed a numerical and experimental investigation into the
effects of flow channel patterns on the performance of PEM fuel cell by using parallel
and serpentine flow channels with the single path of uniform depth and four paths of
step-wise depth, respectively. They documented in their study that the serpentine flow
channel is better when compared with the parallel flow channel. Their result further
shows that different depth of the flow channel significantly affect the performance of
the parallel design but have no significant effect on the serpentine channel design
performance. Wang et al. [69] studied the local transport phenomena and PEM fuel
cell performance with various serpentine flow field designs. The study considered
single, double and triple serpentine flow field designs. The predicted results was
confirmed, i.e. that the single serpentine flow field has better performance when
compared with double and triple serpentine designs and the performance of the single
serpentine flow field increases as the number of the channel bend is increased. It was
also found that the performance of the fuel cell increases slowly as fuel channel width
increases.
19
Chapter 1: Introduction
Finally, a comprehensive review on flow field design in the bipolar plates of PEM
fuel cells has been published by Li and Sabir [70]. They presented reviews on various
flow-field layouts developed by different companies and research groups.
Furthermore, they enumerated and evaluated the pros and cons in those various
designs. In addition, the review concluded that improvement on flow-field design for
fuel cells can greatly improve the goal of cost reduction and performance
enhancement for the commercialisation of PEM fuel cell. However, flow-field design
is still an open ended on-going research and more novel designs, that will be suitable
for different and specific applications, are required.
1.2.3 REACTANT GAS TRANSPORT IN PEM FUEL CELLS
Flow distribution in PEMFC impacts greatly on the performance and efficiency of the
system. The efficient distribution of species reactant to ensure homogenous spread on
the GDL at reasonable pressure drop along the flow channel distributor is crucial to
both effective utilisation of fuel gases and PEMFC performance. In addition, proper
water and heat management within the fuel cell structures are required for obtaining
optimal power density from the fuel cell. Hence, enormous efforts are being devoted
by various researchers to develop novel flow structures for PEM fuel that will
enhance the interaction between the GDL and the flow field to improve the cell
performance. A discussion on these research efforts follows.
Um et al. [71] developed a transient, multidimensional model to investigate the
electrochemical and transport processes inside a PEMFC. They reported that, in the
presence of hydrogen dilution in the PEM fuel stream, there is a large decrease in
hydrogen presence at the reaction surface which results in lower current density as a
consequence of decreased hydrogen transport to the reaction site of the fuel cell
system. He et al. [72] presented a two-phase model of the cathode of the PEMFC by
using interdigitated flow fields. The model was used to investigate the effect of
20
Chapter 1: Introduction
various electrode and flow field design parameters at the cathode of PEM fuel cell
performance. Their result shows that liquid water transport and evaporation form the
mechanism for water removal at the cathode section of the PEM fuel cell. Also,
higher differential pressure between inlet and outlet channels increase oxygen
transport and liquid water removal from the electrode section, thereby increasing the
fuel cell performance. They further suggested the need to optimise the electrode
thickness for better PEM fuel cell performance.
Chang et al. [73] studied flow distribution in the PEM fuel cell stack system,
incorporating flow diffusion effects into their model. They reported that higher
channel friction factors leads to more uniform flow distribution in the fuel system and
the U-type manifold design performs better than the Z-type design. In addition, they
observed that, at higher current densities, fuel cell performance is more sensitive to
operating conditions such as cathode stoichiometry and inlet pressure. Mazumder and
Cole [74] studied liquid water transport in PEM fuel cells using a three-dimensional
model. They concluded that, at critical current density, saturation levels could exceed
50% and are more prominently so at the cathode section of the fuel cell. In addition,
they also reported that the effect of electro-osmotic drag contributes majorly to the
determination of the local saturation level in the MEA of the fuel cell, but was found
negligible at impacting on the fuel cell performance.
Dohle et al. [75] proposed a model to evaluate the interaction between the GDL and
the flow field of PEMFC. Their model was also utilised to develop a suitable match
between serpentine flow field and the diffusion layer of the fuel cell system. They
reported that, to avoid reactant depletion in the specific region of the fuel cell, the
geometry of the serpentine channel should be chosen with regard to permeability of
the GDL. In addition, to obtain higher permeability, the serpentine structure should
have low pressure loss to ensure good flow homogeneity. In the studies of Gurau et
al. [76], a multifluid, multiphase model was proposed to evaluate the two-phase
transport in PEMFC. The model developed accounts for gas- and liquid-phase
21
Chapter 1: Introduction
momentum and species transport in the whole structure of the PEM fuel cell system.
They documented that the level of water accumulation in the GDL is predominantly
determined by the saturation equilibrium at the GDL-channel interface and the GDL
permeability. Meanwhile, the level of water accumulation in the catalyst layer is
determined by the saturation level in the GDL and the saturation equilibrium at the
GDL/catalyst layer interface.
Yan et al. [77] presented a model to investigate the effect of flow distributor geometry
and diffusion layer porosity on reactant gas transport and PEM fuel cell performance.
They reported that increase in channel width fraction, number of channels and
porosity of the GDL positively enhance the performance of the studied fuel cell
system. In addition, the results shows that better uniformity in current density along
the width of the cell can be obtained at relatively low overpotential of the fuel cell.
Wang et al. [78] proposed a novel serpentine-baffle flow field design, different from
conventional serpentine flow field, to improve the PEM fuel cell performance. The
model developed was also used to analyse the reactant and product transport and the
electrochemical reactions in the fuel cell. They concluded that, at high operating fuel
cell voltages, conventional and baffled novel serpentine design shows the same
performance. However, at lower operating cell voltages, the baffle design shows
better performance than the conventional design. Their result further shows that larger
pressure differences are induced on the electrode surface with baffled channels.
Consequently, the mass transport is improved, thereby leading to enhanced fuel cell
performance.
Jang et al. [79] developed a two-dimensional model to investigate the performance of
a PEM fuel cell system based on variability in porosity and GDL thickness. Their
result shows that the mass transfer increment resulting in high reaction rates can be
achieved by increasing the GDL porosity. This improves the fuel cell performance. In
addition, they documented that the performance of the fuel cell also increases with the
decrease in the thickness of the GDL. However, performance is enhanced in the fuel
22
Chapter 1: Introduction
cell system by using a co-flow of fuel and air rather than counterflow configuration.
Wang et al. [80] presented a three-dimensional model of PEM fuel cells with parallel
and interdigitated flow fields to investigate the effects of the cathode flow rate and
flow channel area ratio on the cell performance. The model also incorporated the
effects of liquid water formation on the reactant gas transport in the fuel cell system.
It was documented that the performance of the fuel cell system is not impacted at high
operating voltages by flow channel designs and operating parameters, but these
parameters have a significant effect on the fuel cell performance at low operating
voltages. Their result further shows higher performance of the fuel cell system when
using interdigitated flow fields, owing to the forced convection created by its baffles
to improve transport rates and liquid water removal.
Kim [81] investigated the effect of relative humidity and stoichiometry of reactants on
water saturation and local transport process PEMFCs. The result shows that the
reactant relative humidity (RH) and stoichiometry significantly affect the fuel cell
performance. Also, at a constant relative humidity of the anode, RH = 100%, a lower
cathode relative humidity maintains membrane hydration, resulting in improved fuel
cell performance. Conversely, at a constant cathode RH of 100%, a lower anode RH
increases the difference in water concentration between the anode and cathode,
resulting in better fuel cell performance. In addition, higher anodic stoichiometry
results in the reduction of cathode water saturation due to increase in back-diffusion,
thereby increasing the fuel cell performance. Jang et al. [82] investigated the effect of
humidity of reactant fuel on the cell performance of PEMFC with baffle-blocked flow
field designs. They reported that fuel cell performance is enhanced with an increase in
inlet RH of the hydrogen gas species due to an increase in the chemical reaction and
mass transfer of oxygen. There is an adverse effect in performance by increasing the
inlet RH in the cathode at lower cell voltage due to oxygen depletion in the fuel cell.
Their result further shows that cell performance is enhanced with an increase in the
number of baffles as a result of an increase in areas of forced convection and oxygen
gas diffusion to the catalyst layer. Furthermore, Nguyen and White [83] developed a
23
Chapter 1: Introduction
model to investigate the effectiveness of varying humidification designs in PEM fuel
cells. The model accounts for electro-osmotic and diffusion of water transport in the
membrane, a solid phase to gas phase heat transfer and latent heat formation, resulting
from evaporation and condensation in the flow channels. Their results show that, at
high current densities, large fraction of voltage losses in the cell is due to ohmic loss
in the membrane. They proposed that the anode gas stream must be humidified in
order to maintain adequate hydration in the cell, especially when the fuel cell is
operated at high power densities.
Ko et al. [84] investigated the effect of the channel flow pattern on internal properties
distribution of a PEM fuel cell for cathode starvation conditions, numerically and
experimentally. The fuel cell system performance was investigated by using single,
double and mixed serpentine fuel channel configurations. They documented that
mixed serpentine channels enhance flow velocity better than other configurations
(single and double) and subsequently prevent the channel from flooding. Their result
further shows that local temperature and sensitivity to cathode starvation is higher in
single serpentine systems than in other configurations and that it is more liable to
thermal degradation. Liu et al. [85] investigated the application of baffle-blocked flow
channel for enhancement of reactant transport and performance of fuel cells. Their
result shows that local transport of the reactant gases, current density output and cell
performance can be enhanced by the incorporated baffles in the fuel cell flow channel.
They further documented that baffle effects enhance gas fuel transport at high
operating cell voltages and raise the local current density in the upstream, but lower
them at the downstream of the channel.
Soong et al. [86] proposed a novel configuration of partially blocked fuel channels
with baffle plates transversely inserted in the channel. They evaluated the effects of
the blockage with various gap ratios, number of baffle plates, fuel flow Reynolds
number and GDL porosity on reactant gas transport and pressure drop across the
channel length. They documented that, reducing the gap size between the baffle and
24
Chapter 1: Introduction
the GDL and/or increasing the baffle number enhances the reactant gas transport but
with adverse penalty of high pressure drop. They further proposed that, to ensure high
performance and minimal pressure drop, a baffle gap ratio no smaller than 0.1, baffle
plates number between 3 to 5 and GDL porosity of about 0.7 will be preferable as
design values. Liu et al. [87] proposed a model to investigate the reactant gas
transport and the PEM fuel cell performance with a tapered flow channel design.
Their result shows that fuel cell performance is enhanced by using the tapered flow
field design but more prominently so at lower cell voltages. They further documented
that the liquid water effect has significant impact on transport phenomena and the
performance of PEM fuel cell.
1.2.4 HEAT TRANSPORT AND COOLING IN PEM FUEL CELLS
Thermal management in PEM fuel cells has drawn increasing attention in recent times
because technological limitations encountered in PEM fuel cells today depend largely
on these aspects [88]. Operating temperature affects the maximum theoretical voltage
at which a fuel cell can operate. Higher temperatures correspond to lower theoretical
maximum voltages and lower theoretical efficiency [89]. However, increase in
temperature at the electrodes increases the electrochemical activity, thereby increasing
the fuel cell efficiency. Higher temperature operation of the fuel cell also improves
the quality of waste heat derivable from the system. Practically, there is an optimal
temperature range within which a specific fuel cell system can perform well and
reliably. The main purpose of thermal management in fuel cell systems is to ensure
effective stack operation within the specific temperature range. In recent years, efforts
have been made to investigate and predict heat/mass transfer phenomenon in PEM
fuel cell systems. Some of these research efforts are highlighted below.
Coppo et al. [90] presented a 3-D model to study the influence of temperature on the
PEM fuel cell operation, including two-phase flow in the gas distribution channel.
25
Chapter 1: Introduction
The result obtained indicate that both liquid water transport within the GDL and
liquid water removal from the surface of the GDL play an important role in
determining variations in cell performance where temperature is involved. Yan et al.
[91] presented a 1-D non-isothermal model to analyse the effect of anode and cathode
side temperatures on the membrane water distribution. The results obtained shows
that increasing the temperature on the anode side can lead to membrane dehydration,
and operating the fuel cell at high current density leads to membrane dehydration on
the anode side, due to strong electro-osmotic water drag at high current density.
Ramousse et al. [92] developed a 1-D non-isothermal model accounting for heat and
mass transfer in a complete cell, and charge and mass transfer in the electrodes. Their
study provides for temperature, concentration and potential fields in a single cell. In
addition, their work shows that the thermal gradient in MEA could lead to thermal
stresses at high current densities. Shimpalee and Dutta [93] conducted a 3-D nonisothermal numerical analysis with a two-phase flow. The effect of heat produced by
the electrochemical reaction and phase change of water on the cell performance was
critically studied. Their study shows that inclusion of heat transfer in the fuel cell
model shows degradation in the fuel cell performance. This research work enumerated
the importance of incorporating the heat transfer aspect in fuel cell modelling.
Shan and Choe [94] presented a 1-D model, taking into account the dynamics in
temperature gradient across the fuel cell; dynamics in water concentration
redistribution in the membrane; dynamics in proton concentration in the cathode
catalyst layer; and dynamics in reactant concentration redistribution in the cathode
GDL. Their result generally shows that temperature profiles in each of the cell layers
tend to follow the current waveform, due to energy losses in these layers. Higher
temperature losses are prominent in the membrane and the catalyst layer, due to
ohmic losses as a result of membrane resistance and heat released by the chemical
reaction. Yuan and Sunden [95] performed a 3-D non-isothermal numerical analysis
of heat transfer and gas flow in PEM fuel cell ducts by using a generalised extended
26
Chapter 1: Introduction
Darcy model. Effects of the effective thermal conductivity, permeability, inertia
coefficient and porous layer thickness on gas flow and heat transfer were studied.
Their result shows that higher permeability, higher effective thermal conductivity of
porous GDL and smaller thickness of the porous layer improved heat transfer in the
modelled fuel cell system.
Ju et al. [96] presented a 3-D non-isothermal, single-phase model for all seven layers
of the PEM fuel cell that accounts for various location-specific heat-generation
mechanisms, including irreversible heating due to electrochemical reactions, heating
due to entropy, and Joule (ohmic) heating due to membrane ionic resistance. They
observed that the thermal effect on PEM fuel cells becomes more critical at higher
cell current density and/or lower GDL thermal conductivity. Their result further
shows that temperature increase in the membrane is highly dependent on the GDL
thermal conductivity and inlet humidity conditions. Perng and Wu [97] proposed a
semi-implicit finite element model to investigate the blockage effect generated by a
baffle plate or a rectangular cylinder and its effect on the heat transfer enhancement in
a PEM fuel cell with the catalyst layer kept at a constant heat flux. Their results show
that the installation of transversely placed baffle plates and a rectangular cylinder in
the flow channel effectively enhance the local heat transfer performance of the fuel
cell system. Meanwhile, the rectangular cylinder has better effective heat transfer
performance than a baffle plate, and the larger the cylinder width, the better the heat
transfer performance becomes.
Yu et al. [98] presented a two-phase model with phase change to investigate the liquid
water effect, especially how the inlet water (liquid or vapour) effects on the Ballard
PEM fuel cell performance. The results of their study shows that, for the studied
Ballard PEM fuel cell stack, the more the water supplied to the anode from its inlet,
the higher the voltage and usually the lower the anode exit temperature. Berning and
Djilali [99] developed a 3-D model to account for heat and mass transfer in a
multicomponent two-phase flow, considering all seven layers of a PEM fuel cell and
27
Chapter 1: Introduction
the cooling channels. The results of their study show that phase change occurs at both
sides of the fuel cell and these phase changes are due to an intricate balance of three
competing processes: temperature change, reactant gas depletion and pressure drop
inside the GDL. In addition, their study shows that the amount of liquid water formed
depends largely on the GDL permeability. Also, condensation as well as evaporation
takes place at the cathode GDL, whereas only condensation occurs at the anode GDL
except near the inlet.
Kang et al. [100] investigated the effect of the inlet temperature and flow
configuration on the species, hydration and temperature distribution in a PEM fuel
cell system using the quasi-three-dimensional model. The results show that, of all the
configurations studied, the configuration that has a fuel-air counter flow and an aircoolant co-flow, has the highest performance in all the ranges of current density
because the membrane remains the most hydrated. In addition, they observed that,
when the operating current density increases, the effect of temperature on membrane
hydration slightly decreases. They concluded that it is possible to lower the fuel cell
operating temperature to improve the fuel cell hydration which, in turn, improves fuel
cell performance. Also, different flow configurations were observed to have effect on
the pressure losses and local current density, membrane hydration and species mole
fraction in the studied fuel cell system.
1.3 JUSTIFICATION FOR THIS STUDY
The need for commercialisation and economically viable PEMFCs necessitates
further in-depth research into fuel cell designs. Although, there is extensive literature
on methods and techniques that are aimed at optimising PEM fuel cell performance,
critical issues remain in understanding how different parameters and modifications of
the internal structures relates to affect the performance of the fuel cell under real
operating conditions. Fuel cell structures such as the gas channels, reactant species
28
Chapter 1: Introduction
distribution and thermal conditioning greatly impact on the performance of the fuel
cell system. Consequently, they are the subject of extensive theoretical and
experimental investigations. In the research studies reported in this thesis, the focus is
on the numerical approach to fuel cell engineering design, with specific exploration of
a unique combination of computational fluid dynamics (CFD) and a robust
mathematical optimisation tool to gain deeper understanding of how different fuel cell
design parameters interact to determine the overall fuel cell performance.
Furthermore, this study identifies novel optimisation techniques that, if integrated into
development procedure, will enhance PEM fuel cell performance. The numerical
approach implemented in this work provides an exceptional optimisation approach
that can be used in determining a combination of optimum operating parameters for
fuel cells under real-life operating conditions. In addition, our approach ensures
minimum errors in optimised fuel cell design parameters.
Numerical modelling in recent times has made the development of CFD codes more
robust. The availability of CFD codes has made it possible to perform an analysis on a
series of parametric design variants until a satisfactory design criterion is obtained,
whereafter a prototype development can take place. This will greatly reduce the lead
time and cost in actual development procedures. Further improvement in the system
design process is the use of mathematical optimisation tools. These optimisation tools
can be used to overcome the problems of obtaining optimum design which was
previously largely constrained by the skill and experience of the modeler. The
combination of the computational fluid dynamics and mathematical optimisation can
produce great improvement in the design process. This will ultimately reduce the lead
time, cost and ease of obtaining generic information needed for better and efficient
design of the fuel cell system. This work seeks to develop innovative approach,
through modeling and optimisation, aimed at further enhancing PEM fuel cell
performance within the identified limiting factors such as: operating conditions,
channel geometry, reactant gas transport and thermal cooling approach, which are
very crucial to fuel cell operation.
29
Chapter 1: Introduction
1.4 RESEARCH OBJECTIVES
The main aim of this research is to investigate on new approaches towards
performance enhancement in PEM fuel cell system through numerical modelling and
optimisation. It is anticipated that this would provide new insights into new
approaches for PEM fuel cell system design technology. It is hoped that this
information will be useful in maximising the efficiency and attainment of the
commercialisation drive on this new energy technology. In order to realise the aim
mentioned above, this study will focus on the following specific research activities:
 to numerically predict the performance of PEMFCs under different operating
conditions by using a CFD code;
 to optimise the performance of PEMFCs through gas channel modification, taking
into consideration the mass flow rate and porosity nature of the GDL;
 to develop a novel design approach that can improve the reactant species
distribution on the GDL, hence improving the performance and reducing parasitic
pump power losses;
 to investigate numerically cooling channel geometry scheme in conjunction with
operating parameters (that are temperature-related) of PEM fuel cell systems that
will allow operation of low temperature PEM fuel cell beyond the critical
temperature (  80ºC) to intermediate high temperatures (100-150ºC), without the
need for special compatible high temperature resistant materials which are
relatively costly.
 to carry out the numerical model validations on the investigated fuel cell models.
30
Chapter 1: Introduction
The study emphasises that the attainment of these objectives will provide
comprehensive understanding of how different fuel cell design parameters interact to
improve the performance of PEM fuel cell systems. Some manufacturing parameters
and novel approaches are established to optimise the performance of fuel cell systems.
In summary, results from this study will lead to improved performance and design
information needed for fuel cell manufacturers, which can be applied for better
designs of fuel cell stacks.
1.5 ORGANISATION OF THE THESIS
The thesis is presented in a multiple manuscript format for better organisation and
ease of reading. Chapters 4, 5, 6 are written as individual research papers. The thesis
consists of the following chapters:
 Chapter 2 gives an in-depth view into relevant literature related to the
fundamental structures of a PEMFC and discusses the function of these features in
relation to system performance. This chapter also presents the basic transport and
electrochemical processes in PEMFC systems.
 Chapter 3 exhibits an appropriate framework pertaining to the numerical
modelling of PEMFC used in this study. Furthermore, the Dynamic-Q algorithm,
used for the mathematical optimisation part of this study, is discussed in detail.
 Chapter 4 deals with the numerical study on the effect of key operating
parameters that impact on the performance of PEM fuel cells. The parameters
investigated are both design and physical parameters. In addition, numerical
optimisation of the fuel cell gas channel is carried out with interest on mutual
interdependence of the GDL porous medium, reactant gas flow rate and gas
channel geometry on the fuel cell system performance. The GDL morphology
31
Chapter 1: Introduction
influences greatly on the species distribution from the channel to the catalyst
surface in the cell. This is expected to affect the electrochemical reaction rate that
subsequently determines the fuel cell performance.
 Chapter 5 numerically investigates the reactant gas transport in PEM fuel cells
with transverse pin fins inserts in the channel flow. This is aimed at improving the
system performance via effective distribution of the reactant gases at a reduced
pumping power requirement penalty during fuel cell operation. A numerical
optimisation tool (Dynamic-Q) was coupled with the CFD code to obtain
optimum parameters required for improving PEM system performance. In this
chapter, the steps involved in linking the optimisation method to a commercial
CFD code are also indicated.
 Chapter 6 numerically investigates the impact of cooling channel geometry on
PEM fuel cell performance, specifically when the system is operated at higher
temperatures (HT), beyond the critical temperature typical of conventional lowtemperature PEM fuel cells. Optimal cooling channel geometry was obtained
using a numerical optimisation algorithm. This will ensure thermal stability of the
PEMFC, especially at high temperature conditions.
 Chapter 7 provides conclusions drawn from this study, makes recommendations
and discusses possible future research directions.
32
Chapter 2: Fundamentals of PEM Fuel Cell System
2
CHAPTER 2:
FUNDAMENTALS OF PEM FUEL CELL
SYSTEM
2.1 INTRODUCTION
In this chapter, the basic components of a PEMFC and its functions are briefly
discussed. The main physical processes occurring in the fuel cell structure are
discussed and the corresponding governing equations used in PEMFC modelling are
also presented. Relevant equations include a basic continuity equation or conservation
of mass, a momentum equation and an energy equation as applicable to fuel cells.
Equations governing kinetics of electrochemical reactions in the fuel cell, charge (i.e.
electrons and protons) transport in the MEA, as well as gas flux based on Darcy’s
diffusion formulation are also discussed. The various assumptions in the mathematical
models in each fuel cell component are emphasised.
2.2 THE BASIC STRUCTURE OF A PROTON EXCHANGE
MEMBRANE FUEL CELL
Figure 2.1 illustrates a simplified schematic showing basic components of a single
PEMFC. The single cell (or unit cell) consists of nine different regions: the cathode
current collector, the cathode channel, the cathode diffusion layer, the cathode catalyst
layer, the PEM, the anode catalyst layer, the anode diffusion layer, the anode channel
and the anode current collector.
33
Chapter 2: Fundamentals of PEM Fuel Cell System
Figure 2.1 Schematic diagram of a single PEM fuel cell
A fuel cell works by catalysis mechanism, wherein electrons and protons are
produced by the reactant fuels, such that the electrons are forced to travel through a
circuit, thereby producing electrical power. The catalyst usually comprises platinum
group metal or alloy. A similar catalytic process takes the electrons back in,
combining them with the protons and the oxidant to form simple waste compound like
water and heat. In a typical hydrogen-oxygen PEMFC design (Fig. 2.2), a protonconducting polymer membrane, the electrolyte, separates the anode and cathode sides.
On the anode side, hydrogen diffuses to the anode catalyst where it later dissociates
into protons and electrons. These protons often react with oxidants causing them to
become what is commonly referred to as multi-facilitated proton membrane (MFPM).
The protons are conducted through the membrane to the cathode, but the electrons are
forced to travel in an external circuit (supplying power) because the membrane is
34
Chapter 2: Fundamentals of PEM Fuel Cell System
electrically insulated. On the cathode catalyst, oxygen molecules react with the
electrons (which have travelled through the external circuit) and protons to form
water, the only waste product in this type of fuel cell.
Figure 2.2 The basic structure of a PEM fuel cell showing the path of the
electrochemical reaction [15]
Fuel cells are made of four major structural units. These are the following:
1. Proton Exchange Membrane (PEM),
2. Catalyst layers (anode and cathode),
3. Gas diffusion layers (anode and cathode) (GDL),
4. Bipolar plates with flow channels for reactants and coolant in larger cell
stacks.
Each of these is discussed further in the following sections.
35
Chapter 2: Fundamentals of PEM Fuel Cell System
2.2.1 PROTON EXCHANGE MEMBRANE
The proton exchange membrane (PEM) functions primarily as conductor of ions, but
it also serves as a separator between the reactant fuel gases, and acts as an electronic
insulator. A functional PEM must also have sufficient mechanical and thermal
stability during fuel cell operation. The liquid water in the PEM is transported because
of convection, diffusion, dispersion, pressure gradients and electro-osmotic forces
being dragged by the moving protons [101]. For effective performance of the PEM,
some level of membrane hydration is necessary. However, excess water accumulation
in the fuel cell electrodes can result in electrode flooding, so adequate moisture
balance must be achieved within the cell.
The most common solid polymer electrolytes consist of a hydrophobic and inert
polymer backbone sulfonated with hydrophilic acid clusters to provide adequate
conductivity. For example, the most widely used electrolyte membranes in PEMFCs
are known by their trade names and are called Nafion®. Nafion® has similar backbone
structure as Teflon® but has added sulfonic acid groups [102]. In cases where the
amount of water in the membrane becomes low, Nafion® conductance diminishes
significantly. The membrane humidification in the fuel cell is mainly achieved
through cathode reaction. Many systems utilise reactant gas humidification to
maintain hydration. Modern perflourosulfonated ionomer electrolytes for hydrogen
gas (H2) PEMFCs are 18-25 μm thick with a practical operating temperature limit of
120ºC, although PEMFC operation is rarely greater than 90ºC due to adequate
humidity requirements and operational lifetimes [103]. There are ongoing efforts
[104, 105] to develop a high temperature membrane in order to improve the
performance and efficiency of the PEMFC system.
2.2.2 CATALYST LAYERS
High activation energy is required in PEMFCs to induce electrochemical reactions.
This is usually achieved by using a catalyst. The catalyst layers (CLs) are essentially
36
Chapter 2: Fundamentals of PEM Fuel Cell System
sandwiched between the ionomer membrane and a porous, electrically-conductive
substrate. They are the layers where the electrochemical reactions take place (reaction
zone). The most common catalyst used in PEMFCs for both oxygen reduction and
hydrogen oxidation reactions is platinum. CLs in PEMFCs are made up of a porous,
three-dimensional structure, with a thickness of 5-30 μm. In supported CLs, the 2-10
nm catalyst is physically supported on considerable larger, 45-90 nm carbon particles
[103]. In preparation of the CLs, the most important is the surface area and not the
weight, so it is important to have small platinum particles (4 nm or smaller) with a
large surface area finely dispersed on the surface of the catalyst support [106].
There are two distinct ways of preparing a catalyst layer and its attachment to the
ionomer membrane. Such a combination of membrane and catalyst layer is referred to
as the MEA (membrane electrode assembly). The first way of MEA preparation is to
deposit the catalyst layer on the porous substrate, called the GDL, typically carbon
fibre paper or carbon cloth, and thereafter hot-press it to the membrane. The second
method of MEA preparation is the application of the catalyst layer directly or
indirectly (via a decal process) to the membrane, forming the so-called 3-layer MEA
or catalysed membrane. The porous substrate may be added later, either as an
additional step in the MEA preparation or in a process of stack assembly [106].
Minimising the cell potential losses due to the rate of proton transport and reactant gas
permeation in the depth of the electrocatalyst layer requires making the layer
relatively thin. Also, the metal-active surface area should be maximised by making
the platinum (Pt) particles as small as possible. The first design requirement entails
higher Pt/C ratios (>40% by wt), however smaller Pt particles and consequently larger
metal areas are achieved with lower loading. In general, higher Pt loading results in an
increased voltage gain [107], assuming equal utilisation and reasonable thickness of
the catalyst layer. An efficient catalyst layer must have facile transport of ions,
electrons, reactants and products with a high electrochemical active surface area
where the reactants, catalyst, proton and electron conduction are all available.
37
Chapter 2: Fundamentals of PEM Fuel Cell System
2.2.3 GAS DIFFUSION LAYERS
The gas diffusion layer (GDL) consists of a carbon fibre or woven cloth macroporous
layer and possibly a highly hydrophobic microporous layer developed to enable better
electrical contact between the catalyst layer and fuel system lands. For a GDL to
function efficiently in a fuel cell system, it must have the following properties [108]:
 It should be sufficiently porous to allow the flow of both reactant gases (hydrogen
and oxygen) and product water. Depending on the design of the flow field,
through-plane and in-plane diffusion is important.
 It must be both electrically and thermally conductive, again both through-plane
and in-plane conduction are important. Interfacial or contact resistance is typically
more important than bulk conductivity.
 The catalyst layer should be made of discreet small particles; hence the pores of
the GDL facing the catalyst layer must be minimum.
 It must be sufficiently rigid to support the “flimsy” MEA. However, it must have
some flexibility to maintain good electrical contacts.
In addition, the GDL should be able to conduct heat generated at the catalyst layers
mainly by conduction to the bipolar plates and by convection in gas phases to the gas
flow channels [109]. Carbon fibre based materials such as carbon-fibre papers and
woven carbon fabrics or cloths are usually used, considering the conflicting array of
GDL requirements. The GDL (both anode and cathode) material is typically treated
with a hydrophobic material, such as Teflon, to facilitate water removal and
subsequently prevent flooding in their bulk. In addition, the interface with the
adjacent catalyst layer may be fitted with a coating or a microporous layer to ensure
better electrical contacts, as well as efficient water transport into and out of the
diffusion layer. The pores in this layer are usually between 0.1 and 0.5 μm, thus much
smaller than the pore size of the carbon fibre papers (20-50 μm) [108].
38
Chapter 2: Fundamentals of PEM Fuel Cell System
2.2.4 BIPOLAR PLATES
PEM fuel cells are usually designed by connecting multiple cells in series with bipolar
plates (BPPs). These BPPs structurally support the thin MEAs in PEM fuel cells and
actually comprise almost all of the volume of the fuel cell stack, and typically over
60% of the weight and 30% of the total cost in a fuel cell stack [70]. The BPPs collect
and conduct the current from the anode of one cell to the cathode of the next, while
evenly distributing the fuel gas over the surface of the anode, and the oxygen/air over
the surface of the cathode through the flow channel. In some designs, it performs the
task of facilitating water and heat management. These functions are possible through
the plate topologies and material composition of the BPPs. The essential requirements
for BPPs, in respect of physiochemical characteristics, are efficient and uniform
distribution of the reactant gases over the electrodes to minimise the concentration
over potential; high values of electronic conductivity for current collection; adequate
mechanical strength for stack integrity; impermeability to reactant gases for safe
operation; resistance to corrosion in severe cell environment for long lifetime; cheap
materials; and easy and automated fabrication for low cost [110].
In general, two families of materials have been used for PEM fuel cell BPPs, namely
polymer-sealed graphite-composite and metallic. The polymer sealing is used to
ensure that the normally porous graphite is impermeable to water. For high power
density, low weight and robust stack design, however, metallic plates are required
[103]. Technical difficulties with metal BPPs include difficulty in scaling and
corrosion, which results in rapid electrolyte degradation and poor electrical contact
resistance. In fuel cells, a balance exists between gas supply and current conduction.
Hence, there is need for large-scale porosity in the flow fields, which requires seeking
an optimal flow field design that will efficiently supply the required reactant gases at
lower pumping power requirement. The effective design and optimisation of the gas
flow fields and BPPs remains a pertinent explored area for reasonable cost reduction
and optimum performance attainment for PEM fuel cells [111, 112].
39
Chapter 2: Fundamentals of PEM Fuel Cell System
2.3 PEM FUEL CELL STACK DESIGN
Since fuel cells operate at less than 100% efficiency, the voltage output of one cell is
less than 1.16 volt. Most applications require much higher voltages than this, (for
example, effective commercial electric motors typically operate at 200-300 volts), the
required voltage is obtained by connecting individual single fuel cells in series to
form a fuel cell stack [113]. In stack design, weight and volume is crucial. Hence, to
decrease the overall volume and weight of the stack, instead of two current collectors,
only one plate is used with a flow field cut into each side of the plate. This type of
arrangement is referred to as bipolar plates (BPPs). The bipolar plate (BPP) separates
one cell from the next, with the single plate doing dual work of carrying the hydrogen
gas on one side and air on the other side. A typical stack configuration is illustrated in
Figure 2.3.
Figure 2.3 Fuel cell stack component [71]
The bipolar configuration is best suited for large fuel cells since the current is
conducted through relatively thin conductive plates, thus it travels short distance
through a large area. This causes minimum electroresistive losses even with the use of
a relatively poor electrical conductor such as graphite (or graphite mixtures).
However, for small cells it is possible to connect the edge of one electrode to the
40
Chapter 2: Fundamentals of PEM Fuel Cell System
opposing electrode of the adjacent cell by some kind of connector [113]. A good fuel
cell stack design should have the following properties [103]:
1. Uniform distribution of reactants to each cell
2. Uniform distribution of reactants inside each cell
3. Minimum resistive losses (choice of materials, configuration, uniform contact
pressure)
4. No leak of reactant gases (internal between the cells, or external)
5. Mechanical sturdiness (internal pressure including thermal expansion, and
external forces during handling and operation, including shocks and vibrations).
Fuel cell performance is sensitive to the flow rate of the reactant gases, hence it is
highly necessary that each stack receives approximately the same amount of reactant
gases. Uniformity in flow distribution would result in even performance of the cell.
Therefore, to achieve uniform distribution, feeding of the cell in the stack is done in
parallel through a manifold that can be either external or internal. In fuel cell design,
internal manifolds are commonly employed because of better sealing and versatility in
gas flow configuration [113]. The reactant gases entering the fuel cell stack cell must
be distributed over the entire active area. This is practically achieved through a flow
field, which is basically a form of channels covering the entire area in some pattern or
porous structure. The pattern of the flow field in the plate, as well as the width and
depth of the channels, have a large impact on the effectiveness and the even
distribution of the reactant gases across the active area of the membrane/electrode
assembly. The flow field design also affects the water supply to the membrane and
water removal from the cathode [113]. The following are the key flow field design
variables [106]:
 flow field shape
 flow field orientation
 configuration of channels
41
Chapter 2: Fundamentals of PEM Fuel Cell System
 channels shape, dimensions and spacing
 pressure drop through the flow field
Flow field design is a critical aspect in fuel cell manufacturing and is a well-kept
secret by stack manufacturers [106]. CFD modelling is a great tool for the design of
fuel cell flow fields [114, 71].
2.3.1 HYDROGEN FUEL CELL SYSTEM COMPONENTS
Fuel cell stacks need to be integrated into a complete fuel cell system for it to operate.
The fuel cell system must be of appropriate weight and volume to fit into the space
provided for operation. The operation of the entire engine must maintain the near-zero
emissions and high efficiency of the fuel cells. In addition, all these requirements
must be met with components that are both inexpensive and designed for low cost,
high volume manufacturing. Figure 2.4 is a schematic of typical hydrogen PEMFC
system.
The hydrogen fuel cell system includes the following subsystems and control
components [103]:
2.3.1.1 Reactant storage, delivery and recycling
This comprises of the pumps and blowers required to supply the fuel cell stack with
prescribed flow rates of fuel and oxidiser and to recycle unused fuel back into the
anode inlet stream. Typically, only fuel storage and recycling are needed as air is used
as the oxidant.
2.3.1.2 Humidification
This system is required for humidification of the flow of reactant gases. Portable
system designs are passively humidified, thereby eliminating this subsystem
completely at the expense of reduced performance.
42
Chapter 2: Fundamentals of PEM Fuel Cell System
2.3.1.3 Cooling
Systems larger than 1 kW power typically require active cooling of the stack to
remain within membrane material tolerances and achieve uniform system
performance. Smaller, low-portable systems can be passively cooled. The choice of
coolant is an active area of research. Distilled water can be used but will freeze at
subzero temperatures. Ethylene glycol is the coolant of choice for contemporary
automotive applications and can operate at subzero temperatures, but contact with
electrolytes can result in irreversible damage.
Figure 2.4 A schematic of a complete hydrogen-air fuel cell system [115]
2.3.1.4 Hydrogen reformation
In some fuel cell systems, hydrogen fuel is obtained from liquid hydrocarbon or
alcohol fuel. This fuel cell type requires a hydrogen generation system. For stationary
applications, a fuel reformer is often incorporated. Meanwhile, in automobile or
43
Chapter 2: Fundamentals of PEM Fuel Cell System
portable applications, on-board reformation is typically avoided due to the required
excessive complexity, cost and transient control limitations.
2.3.1.5 Power conditioning and control
Power output from a fuel cell stack is normally in the form of direct current (DC)
which must be inverted to alternating current (AC) and conditioned into a suitable
voltage range to power equipment. The fuel cell control system is responsible for all
system monitoring and maintenance of stable and safe operation through feedback
from a variety of flow, pressure, voltage, current and temperature sensors [103].
2.3.1.6 Startup Power System
External power inputs are usually incorporated to assist in the system startup. An
auxiliary high-power battery to run pumps and heaters during startup, or to provide
power to overcome voltage transients and reversals in the fuel cell stack is often
required.
2.3.1.7 System Humidification
Fuel cells have a precarious balance between a moist electrolyte needed for high ionic
conductivity and a flooded cell that degrades fuel cell performance. It is possible that
some sections of the same fuel cell or individual plates in a stack will be critically dry
and other sections in the cell or different plates in a stack will be flooded. Hence,
some humidification is typically required at the inlet of the fuel cell to ensure
adequate performance. In addition, strong humidity gradients in the electrolyte can
result in internal stresses that limit system durability. Humidification in fuel cell is
basically achieved by two main procedures, passive approach and direct approach. In
the former, the water generated by the reaction in the cell is used to maintain a proper
44
Chapter 2: Fundamentals of PEM Fuel Cell System
moisture balance and humidity of the incoming reactant flow without external power.
However, in active humidification, a separate humidifier is directly employed to
provide the humidification of the incoming flow with stored or recycled water.
2.4 THEORIES OF TRANSPORT AND ELECTROCHEMICAL
PROCESSES IN PEMFCs
Understanding the transport, electrochemical and coupled spontaneous oscillations of
hydrodynamic processes in the fuel cell system requires the understanding the physics
of the various processes within the fuel cell structures. The transport and
electrochemical processes of a typical PEMFC (Figure 2.1) are discussed below:
 The hydrogen fuel is supplied through the anode gas flow channel and is
distributed to the thin catalyst layer section of the fuel cell via the anode GDL.
 The oxidation of the hydrogen molecules occur in the anode catalyst layer to
produce protons and electrons which, in turn, are transported to the cathode
through the fuel cell membrane and an external circuit, respectively.
 In the cathode catalyst layer, the oxidant molecules (transported from the cathode
gas flow channel and GDL) combine with the protons and the electrons from the
anode section to produce water.
The basic half-cell reaction occurring in the fuel cell is given as:
anodic:
H 2  2H   2e  ,
(2.1)
cathodic:
1
O 2  2 H   2e   H 2 O , and
2
(2.2)
45
Chapter 2: Fundamentals of PEM Fuel Cell System
net-reaction: H 2  1 O 2  H 2 O .
2
(2.3)
From equation 2.3, one of the key advantages of the fuel cells is seen. That is that the
only by-product of the reaction is water, which is non-pollutant to the environment
compared to the harmful products of combustion using fossil fuels.
Modelling transport and electrochemical processes taking place in the various layers
in the fuel requires solving five types of constitutive relations simultaneously: (i) the
conservation equations, (ii) constitutive relations for various fluxes, (iii) kinetic
equations for reactions, (iv) equilibrium relationships and (v) auxiliary relations such
as variable definitions and Faraday’s Law [25]. The conservation equations are
applicable to all the layers in the fuel cell, while the other four equations are related
specifically to sections within the fuel cell: membrane, GDL and the catalyst layers.
2.4.1 CONSERVATION EQUATIONS
The fundamental description of fuel cell operation involves the five conservation
principles, namely, mass, momentum, species, electrical charge and thermal energy.
Generally, unified and individual modelling approaches are used in fuel cell
modelling. These two approaches are used in the present work and have been
elaborately discussed by Yang and Pitchumani [116]. A brief discussion on these
approaches is presented in this section. A unified-domain approach allows a valid set
of governing equations for all the fuel cell layers written in vector form as [117, 118]:
mass:
  
    u   S m ,
t
momentum:
1  u 1

   uu  p     Su ,

  t


(2.4)
(2.5)
46
Chapter 2: Fundamentals of PEM Fuel Cell System
species:
 C k 
   u C k     D k eff  C k  S k ,
t
charge:
  κ eff Φe  j  0 ,


(2.6)


(2.7)


(2.8)
  σ eff Φ s  j  0 ,
and finally,
energy:
  T     c
 c p
m
t
p uT
    k eff  T  S T .
(2.9)
The unified-domain approach eliminates the requirement of prescribing assumed or
approximate boundary conditions at the interfaces between the various layers of the
fuel cell system. The main assumptions made in the conservation equations are: (i)
the reactant gases are an ideal gas mixture and (ii) flow is incompressible and laminar
due to small flow velocities occurring in the fuel cell system. The dependent variables
u, p, Ck,  e ,  s , and T in the equations denotes the superficial fluid velocity vector,
pressure, molar concentration of species k, electrolyte (membrane) phase potential,
solid phase potential, and temperature, respectively. The solid phase relates to the
electron conducting materials in the fuel cell system, i.e., the flow channel BPPs, the
GDL, or the electrocatalyst and its carbon support in the catalyst layer section. In the
succeeding section below, other individual equations and the description of their
parameters are presented:
2.4.1.1 The mass equation (equation 2.4):
In this equation, t and  are the time and porosity, respectively, and the density of the
gas mixture  is given by:
47
Chapter 2: Fundamentals of PEM Fuel Cell System
   MWk C k ,
(2.10)
k
where MWk is the molecular weight of species k, and the summation is performed
over all the gas species involved. The source term S m assumes non-zero value at the
catalyst layers, resulting from the consumption/production of electrochemical
reactions as well as diffusion and osmotic drag of water through the membrane [117]:
S m  MWH 2
 
ja
i 
 MWw    Dw,mCw  nd e  , for anode
F 
2F
 
(2.11)
and
Sm  MWo2
 
jc
j
i 
 MWH2 c  MWw   Dw,mCw  nd e  , for cathode,
F 
4F
2F
 
(2.12)
where Dw,m is the liquid water diffusion coefficient in the membrane, and n d is the
electro-osmotic drag coefficient. The current density in the membrane (resulting from
proton flux), ie , is related to the membrane phase potential,  e , through Ohm’s law
[71]:
i e  κ eff Φ e ,
(2.13)
where κ eff is the effective proton conductivity in the catalyst layer. The current
density in the solid phase, i s , could also be obtained by using Ohm’s law as [116]:
i s    Φ s ,
(2.13b)
where  is the electrical conductivity.
48
Chapter 2: Fundamentals of PEM Fuel Cell System
2.4.1.2 The momentum equation (equation 2.5):
The effect of porous media is represented by the source term, S u   u K wherein
 and K denote viscosity and hydraulic permeability, respectively. In the porous
layers (i.e, the GDLs, hydraulic layers and membrane), the viscous term from the
divergence of the viscous stress,  τ , and the inertial terms may be small and the
momentum equation is reduced to Darcy’s law [116].
2.4.1.3 The species equation (equation 2.6):
The equation denotes the reactant species diffusion in the fuel cell system. The first
term on the right-hand side of the equation indicates that the species diffusion is
modeled by Fick’s law for a binary mixture, which is an acceptable approximation for
multicomponent diffusion in PEM fuel cells [117]. The effective diffusivity for
species k is adopted to account for the effects of porous media, and the
expression D keff represents the effective diffusion coefficient for the species. The
source term S k for hydrogen and oxygen species is due to the electrochemical
reactions, which may be written in general form as [117]:
v
k
M kz  ne  ,
(2.14)
k
where v, M k and exponent z are the stoichiometric coefficient, the chemical symbol,
and charge number for species k , respectively, and n depicts the number of electrons
transferred across the charge double layer. The consumption rate of the reactant
species, S k , is related to the volumetric transfer current, j, through Faraday’s law
[116]:
49
Chapter 2: Fundamentals of PEM Fuel Cell System
Sk  
vk j
,
nF
(2.15)
with j given by the Butler-Volmer equation [118]:
  2 F 
 2 F 
j  Aav io exp a    exp  c   ,
 RT 
  RT 
(2.16)
where Aav is the electrochemically active area per unit volume, io is the exchange
current density, and F is the Faraday’s constant. The anodic and cathodic charge
transfer coefficients,  a and  c , represent the portion of the electrical energy
harnessed in driving the electrochemical reactions, and the values are between 0 and
1, depending on the reactions and material properties involved. The activation
overpotential, , is defined as:
  s  e U0 ,
(2.17)
where U 0 is the thermodynamic equilibrium potential, which is determined by using
the Nernst equation for the cathode reaction [119]:
E  E0 
RT  a H 2  aO2
ln
2 F  a w

,


(2.18)
where E 0 is the electromotive force (emf) at the standard pressure (i.e., 1 bar), R is
the universal gas constant, T is the temperature, F is the Faraday constant, and
a H 2 aO2 , and a w are the activities of the hydrogen, oxygen, and water species,
respectively. The effect of osmotic drag is also added to the source S k for the water
species in the catalyst and membrane layers [117].
50
Chapter 2: Fundamentals of PEM Fuel Cell System
2.4.1.4. The charge transport equations (Equations 2.7 and 2.8):
These equations involve the volumetric transfer current, j, as a source term. The
general Butler-Volmer equation, Eq. (2.16), may be simplified for the anode and
cathode catalyst layer depending on the characteristic of the half-cell reactions [117,
120]:
anode:
 CH
2
j a  Aav i0ref,a 
 CH
 2 , ref
cathode:
 CO
2
jc  Aav i0ref,c 
 CO
 2 ,ref
1
 2 

c
  a
F a  and
  RT


(2.19)

 exp   c F  ,
c

 RT


(2.20)
where the sub/superscripts a, c, and ref , denote the anode, cathode and reference
state, respectively. The hydrogen oxidation reaction (HOR) in the anode catalyst layer
is fast and the overpotential  a is typically small, and
j a in Eq. (2.19) is
approximately proportional to  a . In cases where the PEM fuel cell operates on a
reformate feed at the anode section, the electrochemical reactions in the anode
catalyst layer involve the adsorption process of CO, leading to a decreased active
area, Aav , at the anode section of the fuel cell [121, 122]. The oxygen reduction
reaction (ORR) is typically slow with high c , and the expression for j c , Eq. (2.20),
may be obtained by neglecting the anodic reaction term of Eq. (2.16).
2.4.1.5 The energy equation (Equation 2.9):
In the energy equation, K eff is the thermal conductivity, and the heat capacitance in a

porous material,  c p

m
, is a volume-averaged volumetric specific heat over the
solid matrix and the fluid in the micropores:
51
Chapter 2: Fundamentals of PEM Fuel Cell System
 c 
p
m
   c p   1    c p s ,
where the subscript s refers to the solid material and
(2.21)
c  pertains to the fluid.
p
The source term S T in Eq. (2.9) consists of contributions from three mechanisms, i.e.,
irreversible heat from the electrochemical reaction, reversible or entropic heat, and
Joule heating. In solving the conservation equations discussed in this section, the
material property characterisation (i.e., transport properties of the membrane, kinetic
data for electrochemical reactions, effective parameters for porous materials, and
properties of the reactants and products) must be considered.
2.4.2 MODELS OF INDIVIDUAL PEM FUEL CELL COMPONENTS
The previous discussion focuses on the simulation of entire fuel cell systems. The
subsequent discussion will focus on numerical models specific to individual cell
components such as cell membrane, GDLs, etc.
2.4.2.1 Membrane models
The membrane model discussed here will focus primarily on the two major types of
macroscopic models of the membrane, namely, the single phase and the two-phase
models. The membrane is one of the most important components of a PEM fuel cell,
which serves to effectively separate the anode fuel from the cathode oxidant and to
conduct protons at high rates during cell operation. The discussions here focus on the
Nafion membrane, owing to the fact that the governing equations for Nafion are
generally valid for other types of membranes (only with different property values)
[116]. Most macroscopic models consider a membrane consisting of three species: the
membrane polymer, proton and water. Other types of ions are neglected in the threespecies system [123].
52
Chapter 2: Fundamentals of PEM Fuel Cell System
(i) Single-phase models
The single-phase model (or diffusive) considers the vapor-equilibrated membrane.
The system is treated as a single, homogeneous phase where the water and proton
dissolve in the polymer matrix and is transported by diffusion. The membrane matrix
is considered to be stationary in the space, and the fluxes of the dissolved species may
be obtained by the dilute solution theory [124] or concentrated solution theory [125,
126]. The dilute solution theory considers only the interactions between each
dissolved species and the solvent (i.e., the polymer matrix), and the general motion of
charged carriers is governed by the Nernst-Planck equation [116]:
Ni =  z i u i Fci  e  Di ci  ci ve .
(2.22)
In Eq. (2.22), the first term represents the migration of the charged particles i in an
electrolyte potential gradient  e , where z i is the charge number, and ui and ci are
the mobility and concentration, respectively. The diffusive and convective fluxes are
governed by the second and the third term, respectively, and the diffusion coefficient
Di is related to the mobility ui by the Nernst-Einstein equation [124]:
Di  RTu i .
(2.23)
Since the one-phase analysis considers the polymer matrix as stationary solvent, the
convective velocity ve = 0, Eq. (2.22) reduces to Ohm’s law (i.e, Eq. (2.13)) for the
case of zero proton concentration gradient, and to Fick’s law when z i  0 in the case
of water transport.
The flux of the proton across the membrane induces a flow of water in the same
direction via the electro-osmotic drag effect. The electro-osmotic flow is a result of
53
Chapter 2: Fundamentals of PEM Fuel Cell System
the proton-water interaction which cannot be modelled by the dilute solution theory.
The three-species membrane system is better modelled using the concentrated
solution theory, which accounts for the interaction among all the species. Considering
the thermodynamic driving force to be a sum of frictional interactions among different
species, the following expressions for the ionic and water fluxes are obtained [125,
126]:
ie  
κn d
  w  κ Φ e
F
N w  nd
and
ie
  wμw .
F
(2.24)
(2.25)
where  w and  w , respectively, denotes the chemical potential and transport
coefficient of water. The proton-water interaction is taken into account by the two
terms containing the electro-osmotic drag coefficients, n d , in each of Eqs. (2.24) and
(2.25).
(ii) Two-phase models
The two-phase model, also known as the hydraulic model, considers the liquid
membrane to consist of two-phases, namely, the polymer matrix and the liquid water.
A major assumption concerning two-phase models is the fully hydrated state of the
membrane, corresponding to the complete filling of the membrane micro-pores with
liquid water. This implies that concentration gradient and the diffusion transport of
the liquid water species should be zero. The proton species is assumed to be dissolved
in water and moves along with the water molecules. Bernadi and Verbrugge [23, 127]
presented one of the first two-phase models and adopted the Nernst-Planck equation,
Eq. [2.22], to describe the proton flux. The convective velocity, Vw, is non-zero and is
given by Schlogl’s equation for the water species [25, 128], that is,
54
Chapter 2: Fundamentals of PEM Fuel Cell System
K
K 
Vw =   p L     z f c f F e ,

  
(2.26)
where K and K  represent the effective hydraulic and electrokinetic permeability,
respectively, p L is the liquid pressure,  is the liquid viscosity, and z f and c f
denote the charge and concentration of fixed ionic sites, respectively. In two-phase
models, the water flux is attributed to the combined effects of a potential and a
pressure gradient. The portion of water flux driven by the pressure gradient is
primarily due to the permeation of water through the micro-pore network of a fully
hydrated membrane. When the membrane is partially hydrated, water concentration
gradients exist across the membrane thickness and a modification of the hydraulic
models becomes necessary.
2.4.2.2 Gas diffusion layer models
The GDL is the porous diffusion media between the catalyst layer and the gas
channel. It provides structural support, uniform distribution of the reactant gases, and
a pathway for electrons and liquid water to or from the catalyst layer. Due to the high
conductivity of carbon in the gas diffusion layer, the conduction of electrons is
usually ignored in most GDL models. However, the electronic conduction may
become an important factor for the current distribution due to small contact areas with
the gas channels [129] or the composition of the diffusion media [36]. Ohm’s law
accounting for porous media effect is adopted for the electrical current, is , in the
GDLs:
eff
is   Φs  
s
 0Φs   1s .5 0Φs ,
s
(2.27)
55
Chapter 2: Fundamentals of PEM Fuel Cell System
where  eff depicts the effective conductivity of the porous GDL,  0 is the intrinsic
conductivity of the GDL material, and  s and  s are the volume fraction and tutorsity
of the solid conducting phase, respectively. With the assumption of one-dimensional
steady-state flow, the fluxes of the various reactants in the GDLs are constant and are
related to the current density by the stoichiometric coefficients [130, 131]. However,
the water flux may not be constant due to evaporation or condensation in the porous
media. A rate term for the phase change, rw , may be written as [130]:

rw  hm 
sat
w

  w  Sh
Dvgeff
L


f e  wsat   w ,
(2.28)
where hm is the mass transfer coefficient,  w is the partial density of water vapor and
 wsat is the saturation partial density of water, which may be correlated for the pore
effects via the Kelvin equation [132]. The mass transfer coefficient, hm , is related to
the Sherwood number, Sh , via the correlation:
hm  Sh
Dvgeff
L
fe ,
(2.29)
where L is the characteristic length scale, Dvgeff is the effective diffusion coefficient of
water vapor, and f e is the specific area of the liquid/vapor interface. In the GDL, it is
evident that two-phase flow existed. These are gas and liquid water phase flow, which
are further discussed hereafter.
(i) Gas-phase transport in gas diffusion layers
The transport of a multicomponent gas mixture through a porous media is usually
described by the Stefan-Maxwell equations [133]:
56
Chapter 2: Fundamentals of PEM Fuel Cell System
 xi 

xi N j  x j N i
cT Dijeff
j
,
(2.30)
where xi and N i are the mole fraction and the molar flux of species i , respectively,
cT is the total concentration or molar density of all the gas species, and Dijeff is the
effective binary diffusion coefficient for species i and j , which may be related to
the binary diffusion coefficient Dij via the Bruggeman relation:
D ijeff   G1 . 5 D ij ,
(2.31)
where  G denotes the volume fraction of the gas phase, and equals to the bulk
porosity of the media when the liquid water is ignored. In the case of two-phase flow,
 G must be determined from the liquid saturation condition in the GDL. With a
decrease in pore size in the GDL, the gas molecules collide more often with the pore
wall than with each other, resulting in Knudsen diffusion from the intensified gas-wall
interaction [134]. From an order-of-magnitude analysis, it is noted that the bulk
diffusion dominates when the mean-free path of a molecule is less than 1% of the
pore radius, while Knudsen diffusion dominates when the mean-free path is more than
10 times the pore radius [134]. In accounting for the gas-wall interaction, a Knudsen
diffusion term is added to the Stefan-Maxwell equation based on a dusty-gas analysis
[135]:
 xi  
Ni
cT D eff
ki


j
xi N j  x j N i
cT D ijeff
,
(2.32)
where the effective Knudsen diffusion coefficient Dkeffi is proportional to the mean
pore radius, rp , and the mean thermal velocity of the gas molecules [136]:
57
Chapter 2: Fundamentals of PEM Fuel Cell System
Dkeffi 
2
8 RT
,
rp
3
MWi
(2.33)
where R, T, and MWi are the gas constant, temperature, and molecular weight of
species i, respectively.
Most models treat the gas transport in the GDL as a pure diffusion problem where
total gas pressure remains constant through the thickness of the porous media
whereas, in computational fluid dynamics (CFD) models the average convective
velocity, vG , is computed by adopting Darcy’s law for the gas phase [133]:
vG = 
KG
G
p G ,
(2.34)
where KG and  G are the permeability and viscosity for the gas mixture, respectively,
and most computational fluid models incorporate Eq. (2.34) as a source term into the
momentum equation.
Generally, the pressure difference through the GDLs from most simulation results is
small and the assumption of uniform pressure may be valid for typical operating
conditions [127, 137]. This observation is not totally unexpected, since the gas
mixture has convective flow in the channel direction and transportation is only
feasible through the porous by diffusion due to a no-slip condition at the pore walls.
(ii) Liquid Water Transport in Gas Diffusion Layers
Liquid water transport is critical to cell performance. Sufficient liquid water is
desirable for high membrane conductivity, while excessive liquid may block the pores
in the GDL, preventing the reactants from reaching the reaction sites. In some
simplified models, liquid water is treated as a stationary species that occupies a
58
Chapter 2: Fundamentals of PEM Fuel Cell System
certain volume fraction in the GDL pores [31, 138]. The effective binary diffusivities
are thus decreased (see Eq. 2.31) and the flooding effect of liquid is accounted for to
some extent. Most of the simplified models use the liquid volume fraction as a fitting
parameter [31]. More elaborate models treat the liquid water to be fine droplets that
flow with the gas mixture [83, 139]. Evaporation and condensation may take place,
however, a separate liquid phase is not considered. In these models, the liquid is
assumed to be a component of the gas and exerts negligible influence on the gas flow
field. The models keep track of the liquid water volume fraction at various locations
without resulting in complicated two-phase transport analyses.
The two types of models mentioned above essentially describe single-phase transport,
while more accurate treatment of liquid water flow requires two-phase flow models.
To account for liquid water flux, some simple two-phase models assume isolated gas
and liquid pores in the media [130, 127, 140]. This assumption is based on the fact
that the GDL is a mixture of hydrophobic Teflon and hydrophilic carbon solid. The
flux of liquid water follows from Darcy’s law:
N w, L  
K
pL ,
Vw 
(2.35)
where the subscript L denotes the liquid phase and Vw is the molar volume of water.
However, some models adopted a phase mixture approach where all the properties
pertain to a gas-liquid mixture [141, 142]. These models use Eq. (2.35) to perform
liquid flux computation, which is omitted by the single-phase models. An
oversimplification in this approach is that the liquid flows with the same velocity as
the gas, consequently, the interaction between the gas and the liquid is not adequately
accounted for.
Gas-liquid two-phase flow in porous media is a well-known problem in a wide range
of engineering applications, and rigorous modeling of the phenomenon has been
59
Chapter 2: Fundamentals of PEM Fuel Cell System
reported in the literature [143, 144]. Here, we restrict our discussion to gas-liquid
diffusion in fuel cells. Thus, the interaction between the gas and the liquid is
characterised by a capillary pressure, pC , defined as [145]:
pC  p L  pG  
2 cos 
r
,
(2.36)
where  is the surface tension of water,  is the contact angle of a water droplet with
a pore wall, and r is the pore radius. Depending on the wetting characteristic of the
GDL material, the contact angle has a range of  o    900 for a hydrophobic
material, and 90o    180o for a hydrophilic one. An important goal of the two-phase
models is to predict the distribution of liquid saturation, s , which is defined as the
portion of pore volume filled with liquid. Thus, the volume fraction of the gas
phase,  G , is related to the porosity of the GDL,  0 as:
 G   0 1  s  .
(2.37)
The equation implies that the increase in saturation results in a decrease in gas phase
volume fraction and effective diffusion coefficients (Eq. 2.37). In determining the
liquid saturation, s , empirical constitutive equations are adopted to relate the capillary
pressure, pC , to the saturation, s [146, 147]. Wang and Cheng [147] gave a
correlation for pC as a function of s . Thus,
 
p C   cos   0 
K
0.5
1.4171  s   2.1201  s 
2

 1.2631  s  ,
3
(2.38)
where the surface tension is taken to be 0.0625 Nm-1 for the liquid water-air system at
80C and K is the effective permeability of the GDL. The functional form for the
60
Chapter 2: Fundamentals of PEM Fuel Cell System
p C  s relationship is also determined using a bundle-of-capillary model [145]. The
capillary pressure in Eq. (2.38) at various locations in the porous media must be
known to determine the liquid saturation. In typical two-phase flow models, Darcy’s
law [Eqs. (2.34) and (2.35)] is employed to calculate the pressure fields for both
liquid and gas phases, and subsequently utilised in Eqs. (2.36) and (2.38) to obtain the
liquid saturation distribution. In some models, capillary pressure is used as the driving
force for the liquid-water flow [148]:
Nw, L  
K
K
K
pL  
(pG  pC )  
pC .
Vw
Vw
Vw
(2.39)
The rightmost expression in Eq. (2.39) assumes that the gas pressure is constant
within the GDL. The effective permeability K in Eqs. (2.38) and (2.39) is commonly
related to a relative permeability, K r , as:
K

K rK
sat
,
(2.40)
where the permeability at complete saturation, K sat , depends only on the structure of
the porous medium. Many empirical relations existed in the literature for K r as a
function of the saturation. Some adopts a linear dependence of K r on saturation [149,
150]. Most other models represent the relative permeabilities for liquid and gas
phases, namely, K rl and K rg , with the following expressions [146]:
K rl  s 3 , and
(2.41)
K rg  (1  s) 3 .
(2.42)
61
Chapter 2: Fundamentals of PEM Fuel Cell System
The constitutive relations in Eqs. [2.36-2.42] is commonly incorporated in the
multiphase model to simulate two-phase flow in the PEM fuel cells [146, 147].
2.4.2.3 Catalyst layer models
The catalyst layer is a critical component of a fuel cell. The physical processes in a
catalyst layer include the electron conduction in the solid phase; the proton transport
in the membrane phase; the gas diffusion in the gas, liquid and membrane phases; and
the electrochemical reactions on the active catalyst sites. The HOR occurs in the
anode catalyst layer and the ORR takes place in the cathode catalyst layer.
Models that are often used for the catalyst layer are either microscopic or
macroscopic. The macroscopic model is classified into four different types, namely,
(i) the interface model that treat the catalyst layer with zero thickness, (ii) the
macrohomogeneous approach, (iii) the film models, and (iv) the agglomerate models.
The catalyst models are usually for the cathode, due to the fact that the cathode
reaction is slower and contributes to the principal losses in the fuel cell. However, the
modelling approaches for the cathode are generally applicable to the anode catalyst
layer, with only different kinetic expressions and values of properties.
(i) Microscopic models
The microscopic model is further divided into gas pore models [151, 152] and the
flooded-agglomerate models [153, 154]. Since the two modelling approaches are very
similar, only one, that is, the flooded-agglomerate model will be presented here. In
this model, the catalyst layer is made up of a number of porous cylinders flooded with
the electrolyte. The species diffusion and reaction occur within the cylinders [155].
During fuel cell operation, reactant gas diffuses through the gas pore, dissolves and
diffuses in the electrolyte contained in the agglomerates, and reacts on the active sites
62
Chapter 2: Fundamentals of PEM Fuel Cell System
of the catalyst particles. The diffusion of the reactant gas is governed by Fick’s law
with a source term accounting for the simultaneous bulk reaction [155]:
Dmeff
 2 C r 
1 C r 
 Dmeff
 Sk  0 ,
2
r r
r
(2.43)
where D meff is the effective diffusion coefficient of the dissolved species and S k 
vk j
nF
is the consumption rate given by Eq. (2.15). The activation overpotential, , in the
source term S k , is a function of x, and may be obtained from Ohm’s law expressed
as:
2 nFD meff   C 
d 2

,


dx 2
K eff r0   r  r  r0
(2.44)
where K eff is the effective ionic conductivity. Equation (2.44) could be solved
numerically to obtain the radial distribution of current density at various locations.
Subsequently it can be used to evaluate the performance of the electrode as a function
of physical properties such as the intrinsic activity of the catalyst, agglomerate size,
internal porosity and active surface area.
(ii) Macroscopic models
The macroscopic model is classified into different types and each is discussed below:
 Interface models: This model treats the catalyst layer as an infinitely thin interface
between the GDL and the membrane. It is used in fuel cell simulations when the
emphasis is not on the catalyst-layer but rather on the membrane, the water
balance, or the nonisothermal effects. The catalyst is treated as a location where
63
Chapter 2: Fundamentals of PEM Fuel Cell System
the reactants are consumed and the water is produced especially in models
focusing on water management [156, 157].
Faraday’s law, Eq. (2.15), is used in the boundary conditions for the mass balance
of each species between the membrane and the GDL. The overall polarisation
behaviour, as a function of the catalyst interface, is also studied by using this
model [138, 158]. Generally, the interface model assumes that the values of the
relevant variables are constant across the thickness of the catalyst layers, based on
the fact that the layers are extremely thin. However, detailed treatment of the
catalyst layer is required when the structure parameters of the layer, such as the
catalyst loading, need to be optimised.
 Macrohomogeneous and thin film models: In this model framework, the catalyst
layer is assumed to consist of a uniformly-dispersed carbon-supported catalyst and
the ionomer electrolyte without the gas pores. The gas species dissolve and diffuse
in the membrane phase and, consequently, the diffusion rate is low. In the thin
film model, gas pores are assumed to exist and the catalyst particles are covered
by a thin film of electrolyte polymer. The macrohomogeneous models [159] and
the thin film models ignore the microstructural details and share similar governing
equations. The major distinction between the two models is the values of the
diffusion coefficients, since the reactant gas diffuses in different phases, namely,
in the gas phase for the thin film models and in the membrane phase for the
macrohomogeneous models. A one-dimensional macrohomogeneous model for a
cathode catalyst with thickness  cat is summarized in Ref. 159 and is presented
here for completion. In the model, the oxygen flux, No 2 through the catalyst
thickness is determined by Fick’s law of diffusion:
dCo
dx
2
 
i x   I o
No 2
,

eff
D o2
4 FD oeff2
(2.45)
64
Chapter 2: Fundamentals of PEM Fuel Cell System
where Doeff2 is the effective oxygen diffusion coefficient, i  x  is the local proton
current density, and I 0 is the total current density through the cell. The solid phase
is considered to be equipotential (i.e.,  s  0 ), since the ohmic losses are
negligible in the highly conductive materials. Consequently, the local electrode
potential,  x  , defined as the potential difference between the membrane and
solid phase, is only determined by the local electrolyte potential. Thus,
 x    e   s   e x  .
(2.46)
Ohm’s law for the conduction of protons in the ionomer phase yields:
d 2 x 
dx
2

d 2Φ e x 
dx
2

jc  x 
κ eff
,
(2.47)
where jc x  is the volumetric transfer current and  eff is the effective proton
conductivity. The effect of the effective oxygen diffusion coefficient D0eff2 and
effective proton conductivity  eff on the cathode performance could be obtained
through Eqs. (2.46) and (2.47). Also both D0eff2 and  eff are functions of the
membrane in the catalyst layer, hence the model could be used to determine an
optimal membrane gradient that significantly improves the cathode performance
via optimising both the oxygen diffusion and proton transport [159].
 Agglomerate models: The macrohomogeneous and thin film models deal with the
transport processes on macroscale across the layer thickness, while neglecting the
variation of physical variables in the local agglomerate-scale. To investigate the
effect in changes in the agglomerate or pore-scale on the accuracy of the modeling
of the phenomenon, the characteristic length of the pore-agglomerate must be
compared with the diffusion lengths, i.e., the distances over which the physical
65
Chapter 2: Fundamentals of PEM Fuel Cell System
variables related significantly to the transport process change. The characteristic
pore scale lengths of a PEMFC are in the range 10-7 to 10-8 m, while the diffusion
lengths for the reactants in the electrolyte phase specifically changes from 10-7 to
10-8 m [160].
A one-dimensional, steady-state, isothermal agglomerate model was developed by
Wang et al. [161]. In this model, the cathode catalyst layer is assumed to consist
of uniformly distributed spherical agglomerates with radius Ra and void space.
The ohmic losses within the solid are ignored, hence the potential within the
agglomerate is constant. The diffusion and oxygen reaction within the
agglomerate is given as [161]:
Daeff
1 d  2 dC O2
r
dr
r 2 dr 

j
 c ,

nF

(2.48)
where D aeff is the effective diffusivity of oxygen in the porous agglomerate and jc
is the volumetric transfer current. When oxygen concentration in the agglomerate
is C Os 2 , an analytical solution to Eq. (2.48) is given as [161]:
CO2  COs 2
Ra sinhr 
,
r sinhRa 
(2.49)
where the product Ra is commonly called the Thiele modulus and is defined as:
Ra 
Aav i0ref,c
eff
a
nFD C
ref
O2
 F 
Ra exp c   .
 2 RT 
(2.50)
66
Chapter 2: Fundamentals of PEM Fuel Cell System
The current produced in the agglomerate, I a , is obtained by using Faraday’s law
and Eq. (2.49):
I a  nF (4Ra2 ) N O2 (r  Ra )  4nFRa Daeff COs 2 [Ra coth(Ra )  1] ,
(2.51)
where N O2 ( r  R a ) is the oxygen flux at the agglomerate surface, and
concentration C Os 2 is related to the gas concentration in the void space, CO2 ' g , by
Henry’s law:
COs 2  H O2 CO2 ' g ,
(2.52)
where H O2 is Henry’s constant for the oxygen gas.
The current density variation in the macroscopic catalyst scale, i  x  , may be
obtained from the proton mass balance in the layer, based on the solution of the
current in the agglomerate, Eq. (2.51). Thus,
di
  a I a ,
dx
(2.53)
where  a is the density of the agglomerates in the catalyst layer and is defined by:
a 
1c
,
4 3  R a3
(2.54)
where  c is the porosity of the catalyst layer. Also, the overpotential is governed
by Ohm’s law:
67
Chapter 2: Fundamentals of PEM Fuel Cell System
d
i
 eff .
dx 
(2.55)
The oxygen concentration in the gas pores may be obtained from Eq. (2.53) and
Fick’s law:
eff
O2 'c
nFD
d 2 CO2 ' g
dx 2
  a I a ,
(2.56)
where DOeff2 'c is the effective oxygen diffusivity in the gas phase. The agglomeratetype models agree better with the physical picture and they are easily implemented
in fuel-cell simulation.
CONCLUSION
This part of the thesis provided a background to the basic components of a PEM fuel
cell system and their respective functions. General overviews of numerical models of
dynamics within PEMFC structures, highlighting governing equations, were also
discussed, as well as the applicable theoretical framework employed in the PEM fuel
cell modelling carried out in this thesis. The numerical models were implemented in
the CFD code employed in this study and the results are reported in Chapters 4 to 6 of
this thesis.
68
Chapter 3:Numerical Modelling Framework
3
CHAPTER 3:
NUMERICAL MODELLING FRAMEWORK
3.1 INTRODUCTION
In this chapter, we present the numerical and optimisation procedures employed in
this research. Numerical modelling provides key benefits to fuel cell designers. It
enables the design and building of system components in parallel without other stack
hardware being in place. Therefore, as a result of numerical approaches to the design,
the production cost of a fuel cell is relatively lower when compared to cost resulting
from trial and error design approach, which is often exacerbated by expensive fuel
cell hardware components. Hence, the role of numerical modelling in fuel cell
production cannot be overemphasised. In this chapter, a general overview is presented
of domain discretisation and optimisation techniques employed in fuel cell models
used in this study. However, further details regarding grid independence and
boundary conditions, as applied for each fuel cell modelling design problem in this
thesis, are presented in subsequent sections, i.e., in Chapters 4-6.
3.2 NUMERICAL METHOD
The numerical method involves the flow modelling, discretisation of the flow domain,
solving the flow governing equations and data processing. The numerical study was
conducted by using the finite volume method [162]. A commercial computational
fluid dynamics (CFD) code, ANSYS Fluent® [163] with Gambit® [164] as a preprocessor, is used. The CFD code has an add-on package for fuel cells. The detailed
analysis of the numerical modelling techniques will be discussed later, in subsequent
sections.
69
Chapter 3:Numerical Modelling Framework
3.2.1 NUMERICAL MODELLING PROCEDURES
The governing sets of mathematical equations that describe the flow field in the
modelled fuel cells are based on fundamental fluid dynamics principles, mass
conservation, conservation of momentum and conservation of energy. CFD involves
the numerical solving of Navier-Stokes and energy equations on a discritised domain.
This numerical process commences by first defining the domain and thereafter
creating the grid. The grid generation is basically the division of the domain into
smaller control volumes. Generally, the numerical algorithm integrates the governing
equations over the control volumes and, with the aid of the discretisation, the integral
equations are converted into algebraic equations which are then solved iteratively
[162]. Navier-Stokes and energy equations are then solved in these smaller volumes.
The CFD code employed in this research solves these equations on a discretised
domain when relevant flow boundary conditions are specified. The general form of
the equation in vector form has been previously presented in section 2.4.1 of this
thesis (Eqs. 2.4-2.9).
In modelling the fuel cell, some basic assumptions were made:
 the cell operates under steady-state conditions;
 isothermal boundary conditions were used for external walls;
 the flow in the cell is considered to be laminar;
 reactant and products are assumed to be ideal gas mixtures; and
 the electrode is assumed to be an isotropic and homogeneous porous medium.
These assumptions could be varied when required and additional assumptions may
also be specified for a specific fuel cell model as will be shown in Chapters 4-6. The
numerical analysis is divided into three stages: pre-processing, the solver (solution
technique) and post-processing. The geometry development and grid generation
within the flow domain is the pre-processing stage, while solving the flow governing
equation at various nodal points within the flow domain is regarded as the solver or
70
Chapter 3:Numerical Modelling Framework
solution technique. The results analysis which involves graphical presentation of
simulation data outputs, contour, velocity fields and floods of various parameters are
classified as post-processing. A commercial automated grid generator, Gambit®
(Geometry and Mesh Building Intelligent Toolkit) that works with a graphical user
interface in grid creation, has been used to generate the grid for the pre-processing
stage. Gambit® has an added advantage of being able to parameterise the source file
of the model domain. This advantage enables a quasi-automation of the grid
generation by using journal files which are text files that contain commands that
indicate the steps to be followed in the design of the model of interest. The use of the
journal files eliminates the need for the graphical user interface or the repetition
involved in its usage. ANSYS Fluent® commercial software was used as the solver
and part of the post processing. However, the major part of the post-processing was
carried out using KaleidaGraph 4.0 software.
Several works have been done towards development of metrics for validation and
verification of the computational code used in fluid flow modelling. American
Institute of Aeronautics and Astronautics (AIAA) and American Society of
Mechanical Engineers (ASME) have also declared policy statements and guidelines
for the verification and validation of computational fluid dynamics simulations [43,
44]. These metrics include assessment for iterative convergence, spatial grid
convergence and comparison of the CFD results to experimental data. These criteria
are used in this thesis for the validation of the solved models. Numerical models used
in solving each proposed fuel cell problem in this thesis are first constructed, after
which the model is verified by conducting grid independence tests and comparing
specific cases against other reported studies in the literature. Modelling validation was
done largely by comparison with reported modelling and experimental studies in the
literature and this is presented in subsequent sections, i.e., Chapters 4-6. The detail
grid adaptation technique for each fuel cell model and the time ranges for each
simulation will be presented in subsequent chapters in relation to each model
examined. The simulations were carried out on an Intel® Core(TM) 2Duo 3.00 GHz
PC with 3.24 GB of DDRam.
71
Chapter 3:Numerical Modelling Framework
3.3 NUMERICAL OPTIMISATION
Recent advancement in digital computer technology has spurred outstanding progress
in the area of numerical methods for optimisation. Several methods have been
developed for unconstrained and constrained optimisation [165, 166]. Engineering
applications for optimisation usually involve solving a nonlinear constrained
optimisation problem. Nonlinear constrained problems basically involve the search
for a minimum of a nonlinear objective function subject to a set of nonlinear
constraints. Numerical optimisation deals with determining the best solution to
problems which can be expressed mathematically or numerically. In other words, it
implies choosing the best element from a range of available alternatives.
3.3.1 CONSTRAINED OPTIMISATION
Consider the constrained optimisation problem of the general mathematical form:
min f (x); x  [ x1 , x2 ,.....,xi ,.....,xn ]T , x  R n
subject to constraints below:
g j ( x )  0; j  1, 2 ,....., m
h k ( x )  0; k  1, 2 ,....., p  n
(3.1)
The function f (x) is the objective function to be minimised (or maximised). The
g j (x)
and hk (x) represents the inequality and equality constraint functions,
respectively. The components x i , i  2,....., n of x are referred to as the design
variables. The optimum vector x that solves the problem denoted by Eq. (3.1) is
denoted by the vector:
72
Chapter 3: Numerical Modelling Framework
x *  [ x1* , x 2* ,....., x n* ]T ,
(3.2)
*
with the corresponding lowest function value f (x ) subject to the given inequality
and equality constraints.
There are different approaches to solving the optimisation problem described in Eq.
3.1. An approach is to use the gradient-based algorithms [166] (i.e., successive
approximation sequential quadratic programming (SQP) method), or stochastic
methods (genetic algorithm). The use of genetic algorithm methods is usually too
expensive in terms of number of function evaluations (numerical simulations) when
compared with SQP [167, 168]. In this thesis, a relatively new gradient-based and
successive approximation Dynamic-Q method of Snyman and Hay [169] is employed.
This method has been found to be of equal competitiveness to the conventional SQP
method [169], with an advantage of being able to handle problems with severe noise
and mixed integer problems [170, 171]. The Dynamic-Q method consists of applying
the dynamic trajectory, LFOPC (Leapfrog Optimisation Program for Constrained
Problems) optimisation algorithm, to successive quadratic approximations of the
actual optimisation problem [169]. The Dynamic-Q method is capable of handling
general constrained optimisation problems, and it is discussed in detail in the
subsequent section.
3.3.2 THE DYNAMIC-Q METHOD
The Dynamic-Q algorithm method [172] employed in this study uses the LFOP
algorithm [173, 174] to handle constrained problems and which includes the use of
penalty function approach that is implemented in three distinct phases to increase the
capability of obtaining optimal design in a short span of time. Considering a general
optimisation problem depicted in Eq. (3.1), the associated penalty function that
73
Chapter 3: Numerical Modelling Framework
transforms the constrained problem to an unconstrained problem form, is thus
formulated as [173]:
p( x )  f ( x ) 
where

j
m
p
j 1
j 1
  j g 2j ( x )    k hk2 ( x ),
0 ifg j ( x )  0
ifg ( x )  0
j
 {
j
(3.3)
.
.
To increase the simplicity of the algorithm, the penalty parameters  j and  k takes
the same large positive value of  and the higher the value of  , the more accurate
the obtained solution. Meanwhile, at extreme values of  , the optimisation problem
becomes ill-conditioned. This is resolved by increasing the penalty function piecewise until a favourable limit value of  is obtained and then keeping it constant at
this limit value until convergence is achieved [175]. The LFOP dynamic trajectory
method phases applied to the penalty function are highlighted below.
Phase 0:
Given an initial starting guess of the design variables x 0 , the LFOP is applied with
some overall penalty parameter  0 to P ( x,  0 ) . This gives an optimum design
variable vector x * (  0 ) at convergence. The constraints are checked at this optimum
value to ensure that no active constraints are neglected (violated). If none of the
active constraints are violated, the optimal point is taken as the actual optimal
minimum of the optimisation problem being solved and the algorithm is subsequently
terminated.
Phase 1:
74
Chapter 3: Numerical Modelling Framework
In a case where there is an active constraint being violated after checks in Phase 0,
this phase is initialised. The initialisation process involves applying LFOP with an
increasing value of  and using the obtained optimum ( x * (  0 ) ) in phase 0 as the
initial guess. Thereafter, the penalty parameter is then minimised and active
constraints are identified. If no active constraints are violated, the optimisation
algorithm is terminated and the obtained solution x * ( 1 ) is accepted as the optimal
solution of the optimisation problem.
Phase 2:
This phase uses the optimal solution from the preceding Phase 1 as the starting guess
to apply the LFOP. The algorithm will search for the optimal solution which
corresponds to the intersection of the active constraints. During the search, if the
active constraints do not intersect, the algorithm will find the best probable solution,
which is usually close enough to the actual solution with the lowest possible
constraint violation.
3.3.3 DYNAMIC-Q APPROACH: CONSTRUCTING SPHERICAL
QUADRATIC SUBPROBLEMS
Dynamic-Q offers a robust optimisation algorithm due to its capability to deal with
numerical analyses from CFD and finite element method (FEM) simulations. It does
this by handling associated noises generated due to errors created by environmental
influences, grid changes, incomplete convergence and numerical accuracy of the
computer. The numerical computational time (for an objective function not
analytically given or expensive to compute numerically) is reduced in the Dynamic-Q
approach by substituting computationally expensive functions by simpler spherically
quadratic approximate functions obtained from a few expensive function evaluations
(simulations). These approximate functions are utilised to construct successive sub75
Chapter 3: Numerical Modelling Framework
problems P[i ], i  0,1,2,.... at successive design iteration points i k . The approximated
function can be an objective function and/or the constraint function depending on the
optimisation problem being handled. In this Dynamic-Q algorithm, the classical
steepest descent (SD) algorithm method used to solve the general function of the form
f (x) is modified for better overall performance by applying the SD method
successfully to a sequence of very simple quadratic approximations of f (x) . The
identical curvature entries along the diagonal of the Hessian, means that the level
~
surfaces of the quadratic approximation f k ( x ) , are concentric hyper-spheres. Hence,
the modified classical steepest descent algorithm is aptly referred to as spherical
quadratic
approximations.
The
constructions
of
these
spherical
quadratic
approximations are described below [175, 176]:
1
~
f ( x)  f ( x i )   T f ( x i )( x  x i )  ( x  x i )T A(x  x i )
2
1
g~ j ( x)  g j ( x i )   T g j ( x i )( x  x i )  ( x  x i )T B j ( x  x i )
2
1
~
hk ( x)  hk ( x i )   T hk ( x i )(x  x i )  ( x  x i )T C k ( x  x i )
2
(3.4)
A , B j and C k are Hessian matrices of the objective, inequality and equality
functions, respectively, and often take on the simple forms:
A
= diag ( a , a ,..., a )  a I
B j = bj I
C k = ck I
(3.5)
,
where I represents the identity matrix.
76
Chapter 3: Numerical Modelling Framework
 T ,  T g j and  T hk are gradient vectors. If these vectors are not known
analytically, they are approximated from functional data by means of first-order
forward finite differences [176].
Intermediate move limits are employed in the Dynamic-Q algorithm to achieve
convergence in a controlled and stable form. The move limit  j takes on the form of
a constraint by limiting the movement of each design variable, x (ji  1 ) , by preventing
the new design point from moving too far away from the current design point. An
additional constraint of the form is:
x j  x (j i 1 )   j  0
 x (j i 1 )  x j   j  0
; j = 1, 2,…,n
.
(3.6)
The Dynamic-Q algorithm terminates when the following step size and function value
criteria are satisfied:
 Step size:
 x norm 
x k  x k 1
1 x k
 x ;
(3.7)
 Function value:
f norm 
| f k  fbest |
f ;
1 | fbest |
(3.8)
where  x and  f are the step sizes and function value tolerances, respectively.
77
Chapter 3: Numerical Modelling Framework
3.3.4 THE OBJECTIVE AND CONSTRAINT FUNCTIONS GRADIENT
APPROXIMATION
The Snyman Dynamic-Q method requires the gradients of the objective and constraint
functions. When these gradient functions are not analytically available, the
components of the gradients are calculated as follows:
f (x) f (x  x i )  f (x)
, i  1,2,..., n

xi
xi
,
(3.9)
where the differencing step size is given as:
x i  [0,0,..., xi ,...,0]T .
(3.10)
The gradients of the inequality and equality constraint function components used in
the spherical approximation are similarly approximated and depicted in the Eq. (3.10).
Thus,
g i (x) g i (x  x i )  g i (x)
, i  1,2,..., n

xi
xi
,
(3.11)
hi (x) hi (x  x i )  hi (x)
, i  1,2,..., p ,

xi
xi
also with the differencing step size being:
x i  [0,0,..., xi ,...,0]T
.
(3.12)
In practice, new CFD simulation is required to approximate each of the components
(i.e., at each optimisation iteration, n+1). This tends to increase the computational
78
Chapter 3: Numerical Modelling Framework
cost but could be reduced by assuming a constant differencing step size for each
design variable.
Simulation processes are always accompanied by noises that are introduced by the
step size x , used in the differencing scheme. Therefore, choosing an appropriate step
size that gives good results and at the same time eliminates noise generation during
simulation, remains very pertinent. This is done by using the Dynamic-Q algorithm.
The experience of the modeller to solve this associated problem then comes in handy.
To ensure an appropriate step size, different starting guesses should be used a couple
of times and, if the converged solution values are the same, then the chosen step size
is accepted as sufficient. In cases where the converged solution value differs, the step
size should be modified until the variations in the results are eliminated.
In summary, the Dynamic-Q algorithm can be stated as follows [169]:
i. Choose a starting point x1 and move limits δ j , j  1,2,..., n and set i : = 1.
ii. Evaluate f ( x i ), g j (x i ), and hk ( x i ) , as well as f (x i ), g j (x i ), and  hk (x i ). If
termination criteria are satisfied then set x *  x i and stop.
iii. Construct a local approximation, P [ i ] , to the optimisation problem at x i , using
approximations for the objective and constraint functions.
iv. Solve the approximated sub-problem, P[i ] , to give x*i , by using LFOPC [173].
i
v. Set i : = i  1, x : = x  (i- 1 ) and return to step ii.
3.3.5 ADVANTAGE OF DYNAMIC-Q ALGORITHM
The use of spherically quadratic approximation in the Dynamic-Q algorithm offers a
competitive advantage when compared with other methods in terms of the
computational and storage requirements. The O(n2) calculations and storage locations
required for the second order derivatives are not required since the second derivatives
79
Chapter 3: Numerical Modelling Framework
of the objective function and constraints are approximated by using function and
gradient data. The Dynamic-Q computational and storage resources are thus
practically reduced to O(n). At the most, 4  p  q  r  s n  vectors need be stored
(where p, q, r and s are the number of inequality and equality constraints and the
number of lower and upper limits of the variables, respectively). The storage savings
becomes highly significant when the number of variables becomes large [169].
Therefore, the particular strength of the Dynamic-Q method makes it well suited for
optimisation of engineering problems with large number of variables.
CONCLUSION
This chapter focused on the description of numerical methods and mathematical
optimisation algorithm used in this study. The DYNAMIC-Q, which builds on the
LFOPC algorithm, is discussed in detail. The strength of this mathematical
optimisation choice was also highlighted in terms of storage savings where large
numbers of variables and noise handling during simulation were being considered.
80
Chapter 4:Operating and Design Parameters in PEM Fuel Cell
4
CHAPTER 4:
NUMERICAL OPTIMISATION OF OPERATING
AND DESIGN PARAMETERS FOR A PEM FUEL
CELL
4.1 INTRODUCTION
Fuel cell technology is rapidly advancing due to the need for high energy efficiency
and low environmental impact. Fuel cells are regarded as a potential reliable future
source of energy supply due to the fact that they are one of the cleanest and most
efficient alternatives for generating power. However, the large initial capital costs of
fuel cell technology have offset the advantages it offers and slowed down its adoption
for widespread applications [177]. The PEMFC using hydrogen is one of the
emerging fuel cells with many advantages ranging from emission of water as waste,
operation at low temperatures for quick start-up, and the use of solid polymers as
electrolytes, reducing both construction and safety complications [71]. This fuel cell
type is seriously being considered as an alternative power source for stationary and
mobile applications, but there are several technical challenges which have to be
overcome before it can be adopted for use in these devices.
One of the means of reducing the cost of a PEMFC is by improving its performance
through system optimisation. This facilitates the understanding of how different
parameters affect the performance of the fuel cell in real operating conditions and
subsequently reduce the cost involved in prototype development. Fuel cell modelling
has received tremendous attention in the last two decades with the ultimate aim of
better understanding the underlying phenomenon of operating fuel cells. Much
research has been carried out on PEMFCs ranging from one-dimensional models,
showing phenomena where mass transport limitation is taken into account, and twoor three-dimensional models encompassing thermal and water management. This two81
Chapter 4:Operating and Design Parameters in PEM Fuel Cell
or three-dimensional model also combines electrochemical, thermodynamic and fluid
dynamic equations. Heat transfer equations and mass and energy balances were also
incorporated into some studies to provide detailed understanding of emerging
processes in fuel cell systems.
In the landmark works on PEMFC by Bernardi [157], Bernardi and Verbrugge [23]
and Springer et al. [24], which are based on one-dimensional models, the focus is on
humidification requirements of inlet gases and issues related to variable membrane
humidification. The work by this group [23, 24, 157] provided the required
framework for the multidimensional models that followed in subsequent years. A vast
number of previous works are also CFD-based. More recent works that are CFDbased can be found in [30, 53, 66, 80, 141, 178, 179]. Available experimental work to
date has been conducted mostly to validate highly sophisticated CFD simulations
against the cell global polarisation curves. Some of the experimental studies can be
found in [68, 180-182].
Another issue of significant importance in PEM fuel cells is the pressure drop,
especially at the cathode side of the cell. The product water generated at the cathode
channel must be removed from the cell and this requires a high pressure drop. Too
high pressure drops create excessive parasitic power requirement for the pumping of
air through cells, hence, the effective design of the fuel channel is required to ensure a
balance in pressure drop requirements at the fuel cell cathode section.
Inoue et al. [58] studied gas flow through the GDL and the internal phenomena of a
single PEMFC. The result shows an increase in flow when the differential pressure
between adjoining channels is increased, and an increase in the output density as the
depth of the separator channel become smaller. Liu et al. [183] studied the two-phase
flow and water flooding of reactants in the cathode flow channels of an operating
transparent PEMFC experimentally. The effect of the flow field type, cell
temperature, cathode flow rate and operation time on the water build-up and cell
performance formed part of this study. The results indicate the adverse effect of liquid
82
Chapter 4:Operating and Design Parameters in PEM Fuel Cell
water accumulation on mass transport and the subsequent reduction of the
performance of the fuel cell.
Rodatz et al. [184] conducted studies on the operational aspects of a PEMFC stack
under practical conditions. Their study focused particularly on the pressure drop, twophase flow and effect of bends. They observed a decrease in the pressure drop at a
reduced stack current. Maharudrayya et al. [185] studied the pressure drop and flow
distribution in the multiple parallel channel configurations that are used in PEMFC
stacks. Through their study, they developed an algorithm to calculate the flow
distribution and pressure drop in multiple U- and Z-type flow configurations of a fuel
cell. Ahmed et al. [59] used a numerical model to investigate the performance of a
PEMFC at high operating current densities for various channel cross-sectional
configurations, while maintaining the same reactant flow rates and inlet boundary
conditions. The obtained results reveal that rectangular channel cross-sections give
higher cell voltages, while the trapezoidal channel cross-section gives more uniform
distributions at the membrane-cathode GDL interface. The results further reveal the
presence of an optimum channel-shoulder ratio for optimal fuel cell performance.
Most of the existing models in the literature address the effect of fuel channel
geometric parameters on the performance of the PEM fuel cell without investigating
the mutual interdependence of the GDL porous medium, reactant gas flow rate and
gas channel geometry on the fuel cell system performance. Studies on PEM fuel cell
performances which incorporate the determination of optimal operating values for
fuel cell design parameters, taking into consideration the combined mutual effect of
channel geometry, flow rate and GDL characteristics are still very limited in the
literature. A good understanding of the interactive interdependence of these fuel cell
parameters is therefore essential for optimum fuel cell design. One crucial design
consideration in fuel cell design is the reactant flow in the flow field because of the
dominant effect of the parasitic losses caused by frictional losses, reactant
consumption, species production and blockages resulting from the two-phase flow.
83
Chapter 4:Operating and Design Parameters in PEM Fuel Cell
Therefore, the purpose of this chapter in the thesis is to investigate the effect of a
range of operating conditions such as reactant flow rates, GDL porosity, channel
geometry and flow orientation on the performance of a single PEM fuel cell, and also
to determine the optimal operating conditions for this class of fuel cell. In addition,
this chapter sought to determine the optimal fuel cell performance at different
geometric configurations for a given GDL porosity and reactant species flow rate,
which has not been given much attention in the literature. In this chapter, a threedimensional steady-state computational model for a single PEM fuel cell was
developed to predict the fuel cell performance under different operating conditions,
and subsequently add to the knowledge base needed to produce generic design
information for fuel cell systems, which can be applied to better designs of fuel cell
stacks.
4.2 MODEL DESCRIPTION
Figure 4.1 shows a schematic diagram of a typical PEM fuel cell cross-section
indicating the different zones and species transport across the zones. This consists of
seven different regions: the cathode flow channel, cathode diffusion layer, cathode
catalyst layer, PEM, anode catalyst layer, anode diffusion layer and the anode flow
channel. It was assumed that the fuel used is hydrogen at the anode side which
diffuses through the porous GDL and comes into contact with the catalyst layer. At
this layer, it forms hydrogen ions and electrons. The hydrogen ion diffuses through
the polymer electrolyte membrane at the centre, while the electrons flow through the
GDL to the current collectors and into the attached electric load.
84
Chapter 4:Operating and Design Parameters in PEM Fuel Cell
Figure 4.1 Schematic diagram of a PEM fuel cell showing different zones and species
transport across the zones. The net water flux is the sum of: (A1) electro-osmotic
effect, (A2) diffusion effect and (A3) the permeability effect
The electrochemical reactions are:
anodic:
,
cathodic:
net reaction:
(4.1)
,
.
(4.2)
(4.3)
Figure 4.2 depicts the computational domain consisting of the anode flow channel,
anode diffusion layer, MEA, cathode diffusion layer, and cathode flow channel. In
this model, the numerical domain is a full single-cell geometry domain. Pure
hydrogen and air were used as reactant gases in the model. The inlet flow velocity
was controlled by the stoichiometry numbers of 1.2 at the anode and 2.0 at the
85
Chapter 4:Operating and Design Parameters in PEM Fuel Cell
cathode. The operating pressure was 101 kPa absolute at the exit of the cell. The
details of the flow field and other physicochemical parameters used for the base case
are summarised in Table 4.1 and Table 4.2.
Figure 4.2 The discretised three-dimensional computational domain of a single PEM
fuel cell
Table 4.1 Base case geometric parameters of the modelled fuel cell
Channel length (mm)
120
Channel width (mm)
1.0
Channel depth (mm)
1.2
Membrane thickness (mm)
0.036
Catalyst layer thickness (mm)
0.012
Electrode thickness (mm)
0.21
Table 4.2 Physicochemical properties of the modelled fuel cell
Description
Value
Cell operating temperature (oC)
70
Air-side/fuel-side inlet pressure (atm)
3/3
86
Chapter 4:Operating and Design Parameters in PEM Fuel Cell
Open-circuit voltage (V)
0.95
Porosity of GDL
0.4
Permeability of GDL (m2)
1.76 x 10-11
Tortuosity of GDL
1.5
Porosity of catalyst layer
0.4
Permeability of catalyst layer (m2)
1.76 x 10-11
Tortuosity of catalyst layer
1.5
Porosity of membrane
0.28
Permeability of membrane (m2)
1.8 x 10-18
Reference diffusivity of H2
11 x 10-5 m2 s-1
Reference diffusivity of O2
3.2 x 10-5 m2 s-1
Electric conductivity of catalyst layer
Electric conductivity of GDL
( 1m 1 )
( 1m 1 )
Electric conductivity in carbon plate
( 1m 1 )
190
300
4000
O2 stoichiometry ratio
1.2
H2 stoichiometry ratio
2.0
Oxygen mole fraction
0.406
Relative humidity of inlet fuel/air
100%
Reference current density of anode (A/m2)
7500
Reference current density of cathode (A/m2)
20
Anode transfer coefficient
2.0
Cathode transfer coefficient
2.0
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Chapter 4:Operating and Design Parameters in PEM Fuel Cell
4.2.1 MODEL ASSUMPTIONS
Regarding the modelling of the fuel cell, the following assumptions were made:
 the cell operates under steady-state conditions;
 isothermal boundary conditions were used for external walls;
 the flow in the cell is considered to be laminar;
 reactant and products are assumed to be ideal gas mixtures; and
 the electrode is assumed to be an isotropic and homogeneous porous medium.
4.2.2 GOVERNING TRANSPORT EQUATIONS
The basic transport equation (conservation of mass and momentum) applies to the
transport of gas mixtures in the gas channels in the fuel cell. The corresponding
governing equations are written as follows:
Continuity equation:
 vx   v y   vz 


 Sm
x
y
z
(4.4)
where Sm is the source term, owing to electrochemical reactions corresponding to the
hydrogen depletion during reactions, which is applicable at both the anode and
cathode GDL/MEA interface and calculated by [178]:
Sm = 0,
z0  z  z2
Sm =   H 2  ,
  H 2 
z2  z  z3 ,
and
(4.5)
(4.6)
where H 2  is the concentration of hydrogen in the domain of interest, and  and 
are terms of which the values are dependent upon the rate constants for the atomic
oxidation of H2 and the platinum loading in the catalyst layer. The value of  was set
88
Chapter 4:Operating and Design Parameters in PEM Fuel Cell
equal to 1 and  assumed different values, which were subject to different values of
the concentration of Pt initially in the catalyst layer, as discussed by Hontanon et al.
[178]. The momentum conservation, also referred to as the Navier-Stokes equation, is:
Momentum (x-direction):
vx
=
  v x 
  v x 
  v x 
 vy
 vz
x
y
z

P   v x    v x
 
  
x x  x  y  y
   v x 
   
  S px
 z  z 
,
(4.7)
Momentum (y-direction):
vx
=
 v y 
x

 vy
 v y 
y
 vz
 v y 
z
P   v y    v y    v y 
  S py
  
  
 
y x  x  y  y  z  z 
and
(4.8)
.
(4.9)
Momentum (z-direction):
vx
  v z 
  v z 
 v z 
 vy
 vz
x
y
z
=

P   v z    v z
 
  
z x  x  y  y
   v z 
   
  S pz
 z  z 
The source terms account for situations where a fluid passes through a porous
medium. The term is applicable to the electrode and catalyst zones. For low velocities
encountered in fuel cells, these source terms are applicable at the GDLs and are given
by Darcy’s law:
89
Chapter 4:Operating and Design Parameters in PEM Fuel Cell
S px  
v x
,
x
S py  
S pz  
v y
y
v z
z
(4.10)
and
at
(4.11)
z1  z  z 6 ,
(4.12)
where  is the fluid viscosity in the medium and  is the permeability of the
electrode material. The permeability of the medium was assumed to be isotropic as
stated in the assumptions in this model, hence  x ,  y and  z all have the same value
stated in Table 4.2 (1.76 x 10-11 m2). Other source terms for the equations above used
in the model were taken from Dutta et al. [141]. The local current density, io , is a
measure of the electrochemical reaction rate and generally given by the ButlerVolmer equation [60]:
  nF 
   nF  
io  io , ref exp an    exp  cat   
RT

 RT

 
,
(4.13)
where  is the overpotential and defined as
   s   e   Eocv .
(4.14)
The energy conservation equation is:
  v x h    v y h    v z h    T    T
k



k

z
x  x  y  y
x
y
   T 
 
  ST .
k
 z  z 
(4.15)
For the energy equation, additional volumetric sources are present, because not all
chemical energy released in the electrochemical reaction can be converted to
90
Chapter 4:Operating and Design Parameters in PEM Fuel Cell
electrical work due to irreversibilities of the process. The total source that goes to the
thermal energy equation (i.e., enthalpy) is [163]:
S h  hreact  Ran,cat an,cat  I 2 Rohm  hL .
(4.16)
PEMFCs operate under relatively low temperature (< 100oC) and thus water vapour
may condense to liquid water, especially at high current densities. The existence of
the liquid water keeps the membrane hydrated, but it also blocks the GDL passage,
reduces the diffusion rate and the effective reacting surface area. The water formation
and transport of liquid water is modelled by using a saturation model based on [186,
187]. In this approach, the liquid water formation and transport is governed by the
conservation equation for the volume fraction of liquid water, s, or the water
saturation [163]:
,
(4.17)
where the subscript represents liquid water, and
is the condensation rate modelled
as:
,
where
(4.18)
is added to the water vapor equation as well as the pressure correction (mass
source). The condensation rate constant is hardwired to
that the liquid velocity,
. It was assumed
is equivalent to the gas velocity inside the gas channel.
Inside the highly-resistant porous zones, the use of the capillary diffusion term allows
the replacement of the convective term in Eq. (4.17):
.
(4.19)
91
Chapter 4:Operating and Design Parameters in PEM Fuel Cell
Depending on the wetting phase, the capillary pressure is computed as a function of
(the Leverett function) [163, 81]:
pc  {
 cos  c
) 0 .5

 cos
c
(
K
(
K

)
0 .5
( 1 . 417 ( 1  s )  2 . 12 ( 1  s ) 2  1 . 263 ( 1  s ) 3 ,  c  90 o C
( 1 . 417 s  2 . 12 s 2  1 . 263 s 3 ),   90 o C
(4.20)
Eq. (4.17) models various physical processes such as condensation, vaporisation,
capillary diffusion and surface tension. The clogging of the porous media and the
flooding of the reaction surface are modelled by multiplying the porosity and the
active surface area by (1- s), respectively.
4.2.3 CHANNEL CROSS-SECTION
Flow channels in fuel cells are typically rectangular in cross-section, though other
configurations such as triangular, trapezoidal, and semi-circular shapes have been
explored for fuel cell designs [178]. The manufacturing processes of the flow
channels in fuel cells are quite time-consuming and expensive since graphite, which
is hard and brittle, is typically used as the material of choice. Hence, the making of
the flow channel is a major cost in the development of a complete PEM fuel cell. In
the design of small fuel cells, where the pressure drop is in the order of 0.5-1 bar
[188], serpentine or interdigitated channels could be applicable, but in larger fuel
cells this is not possible, as the pressure drop would be in the order of a few bars.
From cost considerations and manufacturing and performance requirements, the
geometrical shape of the channel cross-section has traditionally been either
rectangular or square. The rectangular cross-section was used in the design of the
PEM fuel cell in this study and is schematically shown in Figure 4.3.
92
Chapter 4:Operating and Design Parameters in PEM Fuel Cell
Figure 4.3 Channel cross-sectional view
For internal flows such as the ones in fuel cell channels, the Reynolds number is
conventionally defined as [189]:
where
(4.21)
.
For a rectangular channel in this study,
.
(4.22)
is defined as [189]:
(4.23)
93
Chapter 4:Operating and Design Parameters in PEM Fuel Cell
For the channel under consideration in Figure 4.3, the cross-sectional area is equal to
the product of the channel width and the channel depth:
Ac = ab
(4.24)
and the wetted perimeter is:
P* = 2(a + b) .
(4.25)
The pressure drop for a flow in a channel of length, L, is usually expressed by using
the following relation [189]:
,
(4.26)
where the friction factor, f , for steady fully-developed laminar flows in a channel
with a square cross-section is given as:
.
(4.27)
Substituting the above relation Eq. (4.27) for Eq. (4.26), and taking into consideration
Eqs. (4.21) to (4.25), the pressure drop can be obtained for flow channels with square
cross-section (a = b), as:
94
Chapter 4:Operating and Design Parameters in PEM Fuel Cell
.
(4.28)
Thus, the flow channel length for flow channels with a square cross-section can be
determined as:
.
(4.29)
Similarly, the flow channel length for a rectangular cross-section can be obtained by:
,
is a function of the
where
(4.30)
for rectangular flow channels [189].
The pressure drop in the channel can be obtained using the flow rate
drop
pressure
relationship for a rectangular cross-section relation [190]:
.
(4.31)
4.2.4 FLUID FLOW THROUGH GAS DIFFUSION LAYER
In fuel cells, the fluid flow diffuses through the GDL for the reaction to take place on
the MEA. The effective diffusivity for the gas-phase flow in porous media can be
written as:
95
Chapter 4:Operating and Design Parameters in PEM Fuel Cell
.
(4.32)
The tortuosity
is a difficult parameter to estimate except through experimentation.
Hence, it is usually correlated in fuel cell studies using a Bruggeman correlation. This
correlation assumes that
is proportional to
, resulting in the simpler
expression [191]:
.
(4.33)
The porosity correlation is used to adjust for the longer effective path length through
the porous media.
4.2.5 BOUNDARY CONDITIONS
Pressure boundary conditions were specified at the outlets since the reactant gas flow
is usually separate and at different pressures. The inlets were all assigned as mass
flow inlets. The GDL and the catalyst layer were surrounded by sealed plates at the
inlet and outlet planes, so the boundary conditions at the inlet and outlet planes take
the no-slip condition for the velocity and non-permeable condition for the species
mass fraction. The membrane-electrode interface was defined as a wall, primarily to
inhibit species and electron crossover through the membrane. This also prevents
pressure problems at the interface. In the areas at which the gas diffusion electrodes
were in contact with the bipolar plates, a constant reference voltage equal to zero was
assigned as a boundary condition both at the anode and at the cathode terminals. The
electron flux was set to zero at all other walls. The anode was grounded (V = 0) and
the cathode terminal was set at a fixed potential (0.75 V), less than the open-circuit
potential (0.95 V). Both anode and cathode terminals were assigned wall boundaries.
96
Chapter 4:Operating and Design Parameters in PEM Fuel Cell
4.2.6 SOLUTION TECHNIQUE
The model equations were solved using the CFD software ANSYS Fluent® 12.0 with
Gambit® (2.4.6) as a pre-processor. The CFD code has an add-on package for fuel
cells, which has the requirements of the source terms for species transport equations,
heat sources and liquid water formations [163]. Control volume technique was used
for solving the problem. The meshes were more refined at the membrane-catalyst
assembly regions. The conservation of mass, momentum and energy equations in the
three-dimensions were solved in turn, until the iterative process met the convergence
criteria. In this study, the definition of convergence criteria indicates that the largest
relative error between two consecutive iterative residuals within the overall
computational domains is less that 10-6.
The domain was divided into hexahedral volume elements. A computational mesh of
about 257 346 volume elements was obtained with the grid. The grid independence
was verified at the preliminary test runs. Four structured grid configurations were
evaluated for the PEMFC. The number of elements in the x-, y- and z-directions was:
(a) 70 × 70 × 25, (b) 87 × 87 × 34, (c) 104 × 87 × 34 and (d) 104 × 104 × 43. The
influence of the number of elements on the local current density at an operating
voltage of 0.4 V was investigated. The local current density for grid (a) differs from
that of (b-d) with a deviation of about 4.2%. However, the local current density
distributions for grids (b), (c) and (d) do not show any significant differences. The
difference between the local current densities for (b) and (c) is about 0.36% and the
difference between (c) and (d) is 0.48%. Grid (c) was chosen for the simulations as a
trade-off between accuracy and cost of time.
The solution strategy was based on the SIMPLE algorithm [192]. Momentum
equations were solved for the velocity followed by solving the continuity equation,
which updates the pressure and the flow rate. Results were then verified for
convergence. The simulation for each operating potential converged in 45-60 minutes
97
Chapter 4:Operating and Design Parameters in PEM Fuel Cell
depending on the current density on an Intel® Core(TM) 2Duo 3.00 GHz PC with
3.24 GB of DDRam.
4.2.7 MODEL VALIDATION
The validation of physical and numerical models is very important, hence comparison
with available experimental data is highly desirable. To describe the performance of
fuel cells, polarisation curve or voltage-current curve is often used. Also, by
comparing the polirisation curve from modelling with experiments, the accuracy or
otherwise of a model could be validated [193]. The polarisation curve obtained for the
base case operating conditions have been compared with experimental measurements
of Wang et al. [33] and Cheng et al. [60] and are shown in Figure 4.4. There is a good
agreement between the experimental curves in the low load region. However, the
model current density in the high mass transport limited region (> 2.75 A/cm2) is
higher than the experimental values.
This observation is common in models where the effect of reduced oxygen transport,
due to water flooding at the cathode at higher current density, cannot be properly
accounted for [99]. Nonetheless, the prediction from the model could still be used
successfully for better understanding of the complex processes in fuel cell systems.
98
Chapter 4:Operating and Design Parameters in PEM Fuel Cell
1
Present model
Wang [33]
Cheng [60]
V [volt]
0.8
0.6
0.4
0.2
0
0
0.5
1
1.5
2
I [A/cm ]
2
2.5
3
Figure 4.4 Comparison of numerical model prediction and experimental
polarisation curves at base condition
4.3 MODEL RESULTS AND DISCUSSION
4.3.1 PRESSURE DROP IN FLOW CHANNEL
Figure 4.5 shows the calculated pressure drops for the rectangular flow channel over a
range of air mass flow rates at a channel depth and width of 1.2 mm and 1.0 mm,
respectively. The results indicate that the pressure drop increases as the mass flow rate
at the cathode is increased. This is expected since an increase in the mass flow rate
increases the reaction of the reactant species and also reduces the resident water in the
cathode channel of the fuel cell. Generally, fuel cells with high pressure drops in the
flow field exhibit a more even distribution of the reactant species flow than those with
low pressure drops in their flow fields. These even distributions of reactant species
greatly enhance the fuel cell performance [194].
99
C h ann el depth = 1 .2 m m
C h ann el w idth = 1 .0 m m
150
50
100
P [Pa]
200
250
Chapter 4:Operating and Design Parameters in PEM Fuel Cell
0.0
5.0E -5
1.0E -4
m
1.5E -4
2.0E -4
[kg/s]
Figure 4.5 Pressure drop along the model flow channel at base operating
conditions for a channel depth of 2.0 mm and width of 1.2 mm
4.3.2 EFFECT OF PHYSICAL PARAMETERS ON PROTON
EXCHANGE MEMBRANE FUEL CELL PERFORMANCE
Figure 4.6 illustrates the polarisation curves obtained from the model (cell voltage 0.3
V) at several operating temperatures from 60-90oC at stoichiometry ratios of 1.2 and
2.0, respectively, for the anode and the cathode. The curve indicates that the fuel cell
performance increases with an increase in temperature and is at the optimum at
temperatures of approximately 60-80oC. This is consistent with literature [179, 195].
The increase in fuel cell performance with the increase in temperature can be
attributed to an increase in gas diffusivity and membrane conductivity at higher
operating temperatures.
The polarisation curves are also lower at 75-80 oC compared with 60-70 oC in the
lower current density region, primarily due to the lower reaction rates resulting in low
water content in the membrane. The condensation of water easily occurs at lower
100
Chapter 4:Operating and Design Parameters in PEM Fuel Cell
temperatures resulting in the flooding and deterioration of the gas diffusivity in the
catalyst layer and the GDLs. At temperatures beyond 80oC, the cell performance
declines, since membrane conductivity decreases at high temperatures due to the onset
of reduction in relative humidity of reactant gases and water content in the membrane.
1
V [volt]
0 .8
0 .6
0
T ( C )
0
60 C
0 .4
0
70 C
0
75 C
0
0 .2
80 C
0
85 C
0
0
90 C
0
0 .5
1
1 .5
2
2 .5
3
3 .5
2
I [A /c m ]
Figure 4.6 Effect of temperature on cell performance at base conditions
Hence, the fuel cell performance is adversely affected at temperatures between 80 and
90oC. Increasing the cell temperature beyond 80oC, results in higher levels of water
loss in the cell until a critical temperature is attained where the evaporated water is
greater than the amount of water being generated in the cell, thereby resulting in a
total dry-out of the membrane. This could eventually lead to fuel cell failure. This
model ascertains the fact that these fuel cells need to be operated at temperatures
below 80oC. A humidifier may be required if operation at higher temperatures is
required but this adds to the capital and running costs of fuel cells.
Figure 4.7 shows the effect of changing the oxygen mass flow rate from 5.0E-06 to
1.6E-04 kg/s on the fuel cell performance. When the cathode gas mass flow rate is
increased, the fuel cell performance is enhanced, especially at lower operating fuel
101
Chapter 4:Operating and Design Parameters in PEM Fuel Cell
cell voltages. The reason is the increase in oxygen gas through the GDL to the
reaction sites, which increases the rate of reaction. At low operating voltages, more
liquid water is produced, due to stronger electrochemical reaction rates, which is
expected to reduce fuel cell performance.
0 .9
m
0 .8
5 .0
1 .0
3 .0
5 .0
1 .0
1 .6
0 .7
V [volt]
( k g /s )
0 .6
E -0 6
E -0 5
E -0 5
E -0 5
E -0 4
E -0 4
0 .5
0 .4
0 .3
0 .2
0
0 .5
1
1 .5
2
2 .5
3
3 .5
4
2
I (A /c m )
Figure 4.7 Effect of cathode gas flow rate on cell performance at base conditions
However, the high oxygen mass flow rates in the porous layer generate high shear
forces, which aid the transport of liquid water downstream in the flow channel along
the flow direction. The effect is minimal at high operating voltages as observed on the
curves, primarily due to low membrane humidification. Wang and Liu [196] obtained
similar results in their experimental work on PEM fuel cell performance. This is
because a low amount of water presence occurs at these voltage levels, due to slow
reaction rates coupled with an increase in the oxygen gas supply which results in
reduced cell performance.
The effect of the GDL porosity on the performance of the PEM fuel cell is shown in
Figure 4.8. The results show the fact that the effect of the GDL porosity on fuel cell
performance is significant when the GDL is in the low value region (0.1-0.4).
102
Chapter 4:Operating and Design Parameters in PEM Fuel Cell
Increasing the diffusion layer porosity size has an increasingly weaker effect on the
performance. A GDL porosity beyond 0.6 does not have a significant effect on the fuel
cell polarisation curve. This observation is in agreement with the optimisation work of
Lin et al. [195]. They reported an optimum GDL porosity of 0.5913 for the PEM fuel
cell modelled in their study. Therefore, maintaining a porosity level between 0.4 and
0.6 is a reasonable value for the fuel cell if durability issues in the fuel cell structure
are taken into consideration.
1
V [volt]
0 .8
 = in c r e a s in g
0 .6





0 .4
0 .2
0
=
=
=
=
0 .1
0 .2
0 .3
0 .4
 = 0 .5
 = 0 .6
 = 0 .7
0
0 .5
1
1 .5
2
2 .5
3
2
I ( A /c m )
Figure 4.8 Effect of gas diffusion layer porosity on cell performance at base
conditions
Fuel cell performance is also largely influenced by the operating pressure. In this
study, the fuel cell operating pressure varied from 1-5 atm at a constant operating
temperature of 70oC. The polarisation curves for different operating pressures are
shown in Figure 4.9. As the operating pressure increased from 1-5 atm, the fuel cell
performance also improved. There was a significant increase in the fuel cell
performance from 1-3 atm, however, after 3 atm the increase was minimal. Increasing
pressure improves the reactant’s interaction with the electrolyte, hence increasing fuel
cell performance. The pressure impact on the fuel cell performance is prominent at a
higher current density of operation. Generally, the polarisation curve shifts position
positively as the pressure increases.
103
Chapter 4:Operating and Design Parameters in PEM Fuel Cell
1
P [atm ]
1
2
3
4
5
V [Volts]
0 .8
0 .6
0 .4
0 .2
0
0
0 .5
1
1 .5
2
2 .5
3
I [A /c m 2 ]
Figure 4.9 Effect of operating pressure on cell performance at base conditions
Figure 4.10 shows the effect of changing the cathode gas stoichiometry ratio on the
fuel cell performance. The temperature, pressure and porosity were maintained at
constant values of 70oC, 2 atm and 0.4, respectively. Very small changes in the
overall cell performance are observed at an increased cathode gas stoichiometry,
especially at higher operating current densities. Performance at low stoichiometries
(<3.0) shows a reduced fuel cell performance and at higher levels (>3.5) the
increment becomes insignificant. This increase in performance is due to the increment
in oxygen availability and the humidity of the membrane. At low cathode gas
stoichiometry, there are limitations of oxygen availability towards the end of the flow
channel and, furthermore, the water removal rate is reduced leading to a reduction in
performance. Operating a fuel cell at lower voltages increases electrical resistance
within the cell that also hinders an increase in performance. However, performance
could be augmented by increasing the stoichiometry rate at these operating voltages.
104
Chapter 4:Operating and Design Parameters in PEM Fuel Cell
1
S t o ic h . r a tio
0 .9
2 .5
3 .0
3 .5
4 .0
V [Volt]
0 .8
0 .7
0 .6
0 .5
0 .4
0 .3
0 .2
0
0 .5
1
1 .5
2
I (A /c m )
2
2 .5
3
Figure 4.10 Effect of cathode gas stoichiometry on cell performance at base
conditions
4.3.3 EFFECT OF DESIGN PARAMETERS ON PROTON EXCHANGE
MEMBRANE FUEL CELL PERFORMANCE
Simulations were performed for different sets of channel dimensions. Two different
parameters, i.e., channel width and channel depth, were chosen for the study. Figure
4.11 illustrates the effect of channel depth on the fuel cell performance at a constant
channel length. The optimal current density for the fuel cell was obtained at a channel
depth of 2.0 mm (current density: 2.62 A/cm2). A further increase in depth showed a
decline in fuel cell performance.
105
Chapter 4:Operating and Design Parameters in PEM Fuel Cell
3
   
m = 5 e -0 6 ( k g /s )
2
I (A/cm )
2 .5
2
1 .5
1
0 .5
0
0 .5
1
1 .5
2
2 .5
3
3 .5
b (m m )
Figure 4.11 The cell current density at different channel depths at a cell potential of
0.3 V, a temperature of 70 C and a mass flow rate of 5e-06 kg/s
Figure 4.12 shows the fuel cell performance for the six cases of channel widths
considered. Performance increased gradually from case 1 (0.6 mm – current density:
1.30 A/cm2) until an optimum was obtained at case 4 (1.2 mm – current density: 2.45
A/cm2).
106
Chapter 4:Operating and Design Parameters in PEM Fuel Cell
2 .6
2 .4
   
5 e -0 6 (kg /s )
m =
2
2
I (A/cm )
2 .2
1 .8
1 .6
1 .4
1 .2
0 .4
0 .6
0 .8
1
1 .2
1 .4
1 .6
1 .8
a (m m )
Figure 4.12 The cell current density at different channel widths at a cell potential of
0.3 V and a temperature of 70 ºC
Increasing the channel width beyond 1.2 mm showed a reduction in fuel cell
performance. These results were consistent with those observed by other researchers.
Watkins et al. [197] studied optimal dimension for cathode-side channels. They
claimed that the most preferred ranges are 1.02-2.04 mm for channel depths and 1.141.4 mm for channel widths. Figures 4.11 and 4.12 suggest the existence of an optimal
channel depth and width for the PEM fuel cell that will offer the best system
performance.
The effect of species flow orientation on the performance of the fuel cell was
investigated for the base case. It was found that the direction of flow affects the
performance of the fuel cell. Co-flow and counterflow affect the fuel cell performance
at different operating cell voltages. Figure 4.13 depicts the fuel cell performance at
the base case conditions and, for a channel depth and width of 2.0 mm and 1.2 mm,
respectively, the counterflow and the co-flow orientations. Current densities of 2.61
A/cm2 and 2.54 A/cm2 were obtained for the counterflow and co-flow cases,
107
Chapter 4:Operating and Design Parameters in PEM Fuel Cell
respectively. Counterflow creates better performance for the fuel cell, especially at
higher current voltages.
Figure 4.13 The cell current density for counterflow orientation (2.61 A/cm2) and coflow orientation (2.54 A/cm2) at base case conditions, for a channel depth of 2.0 mm
and a channel width of 1.2 mm
Figure 4.14 shows the contours of mass fraction for hydrogen at the anode flow
channel for counterflow (Figure 4.14a) and co-flow cases (Figure 4.14b), respectively.
The contour shows that counterflow configuration allows more uniform distribution
of the hydrogen species at the anode flow channel, which subsequently improves the
performance of the fuel cell. The effective species distribution generally aids reaction
on the membrane sites and this leads to increased current density.
108
Chapter 4:Operating and Design Parameters in PEM Fuel Cell
(a.) Counter flow
(b.) Co-flow
Figure 4.14 Contours of mass fraction of hydrogen at the anode for (a.) counterflow
and (b.) co-flow cases at the base case operating conditions
4.3.4 OPTIMAL CHANNEL GEOMETRY
The results in Section 4.3.3 (Figures 4.11 and 4.12) depict the existence of an optimal
channel depth and width for a PEM fuel cell system. The search for an optimal
channel depth and width was carried out for the PEM fuel channel at varying GDL
porosities. The first run of the simulation was carried out by fixing the cathode gas
flow rate at 5e-06 kg/s, width of channel at 1.2 mm, cell operating voltage at 0.3 V
and GDL porosity at 0.2. The channel depth was then varied between
109
Chapter 4:Operating and Design Parameters in PEM Fuel Cell
0.5 and 3.0 mm. An optimal channel depth,
, was found for this configuration.
The procedure was repeated for other values of GDL porosities in the range of 0.2
as shown in Figure 4.15, until an optimal channel depth, which
corresponds with the maximum current density, was obtained at each value of the
GDL porosity.
3

= 0 .2
= 0 .3
= 0 .4
= 0 .5
= 0 .6
2.5
2
I (A/cm )
2
1.5
m = 5 e-0 6 (kg /s )
1
w id th = 1 .2 m m
0.5
0
0
0.5
1
1.5
2
2.5
3
3.5
b (m m )
Figure 4.15 Effect of porosity and channel depth on the cell current density
Figure 4.16 gives the optimum channel depth,
, for different cathode gas mass
flow rates for different gas diffusion layer porosities. The optimal channel depth
decreases as the mass flow rate increases.
110
Chapter 4:Operating and Design Parameters in PEM Fuel Cell
2 .6

=
=
=
=
=
bopt (mm)
2 .4
2 .2
0 .2
0 .3
0 .4
0 .5
0 .6
2
1 .8
1 .6
0 .0
4 .1 E -5
8 .3 E -5
m
1 .2 E -4
1 .7 E -4
(k g / s )
Figure 4.16 Optimum depths as a function of flow rate and gas diffusion layer
porosity
Figure 4.17 shows the behaviour of the maximum current density,
, with varying
cathode gas mass flow rates. Each point of the figure depicts the result of a full
optimisation with respect to channel depth. The graph shows that maximised current
density increases as the mass flow rate of the reactant gas increases. In each case,
there is an optimal channel depth that maximises the current density of the fuel cell.
Similarly, the search for optimal channel widths,
maximum current density,
, corresponding to the
, was carried out as conducted for the channel depths.
Figure 4.18 shows the current density value as a function of the channel widths for
different values of GDL porosities. The cathode gas mass flow rate and channel depth
were initially fixed at 5e-06 kg/s and 2.0 mm, respectively.
111
Chapter 4:Operating and Design Parameters in PEM Fuel Cell
3.8
3.6
I
max
2
( A/cm )
3.4
3.2

3
=
=
=
=
=
2.8
2.6
0.0
4.1E -5
8.3E -5
m
1.2E -4
0 .2
0 .3
0 .4
0 .5
0 .6
1.7E -4
(kg/s)
Figure 4.17 Effect of flow rate and gas diffusion layer porosity on the cell current
density
3
2
I (A/cm )
2.5
2

=
=
=
=
=
b = 2.0 m m
m = 5e-06 (kg/s)
0.2
0.3
0.4
0.5
0.6
1.5
1
0.5
0.4
0.6
0.8
1
a
1.2
1.4
1.6
1.8
(m m )
Figure 4.18 Effect of porosity and channel width on the cell current density
112
Chapter 4:Operating and Design Parameters in PEM Fuel Cell
Figure 4.19 depicts the optimal value of the channel width as a function of the cathode
. The
gas mass flow rate for each of the values of GDL porosities (0.2
optimal channel widths,
, from the figure decreases as the mass flow rate
increases. The results obtained from Figures 4.16 and 4.19 both suggest that optimal
channel depth and width decrease at increasing cathode gas mass flow rates. In
designing PEM fuel cells, it can be concluded that the matching of fuel cell operating
conditions and gas fuel channel configuration is very important for optimum
operation issues.
1.3

=
=
=
=
=
1.2
aopt (mm)
1.1
0 .2
0 .3
0 .4
0 .5
0 .6
1
0.9
0.8
0.7
0.0
4.1E -5
8.3E -5
m
1.2E -4
1.7E -4
(k g/s)
Figure 4.19 Optimum widths as a function of flow rate and gas diffusion layer
porosity
CONCLUSION
In this chapter, a steady-state three-dimensional computational model was established
to study the performance of a single-channel PEMFC under varying operating
conditions. The model prediction was validated by its good agreement with available
113
Chapter 4:Operating and Design Parameters in PEM Fuel Cell
experimental results. The numerical results provided detailed information on the
effect of varying operating parameters of a single-channel fuel cell performance. It
was proved that temperature, GDL porosity, cathode gas mass flow rate and species
flow orientation affect the performance of the fuel cell. Specifically, we show that
fuel cell performance increases with an increase in temperature from 60-80oC. Further
increases in temperature, beyond 80C, shows a decline in fuel cell performance. The
porosity of the GDL also affects the fuel cell performance. The porosity effects on
fuel cell performance are more significant at porosity levels of 0.1-0.4 than at porosity
levels of 0.5-0.7. The effect of the operating and design parameters on PEM fuel cell
performance is also more dominant at low operating cell voltages than at higher
operating fuel cell voltages. In addition, this study establishes the need to match the
PEM fuel cell parameters such as porosity, species reactant mass flow rates and fuel
gas channels geometry in the system design for maximum power output.
114
Chapter 5: Optimising Reactant Gas Transport in PEM Flow Channel
5
CHAPTER 5:
OPTIMISING REACTANT GAS TRANSPORT IN
A PROTON EXCHANGE MEMBRANE FUEL
CELL WITH A PIN FIN INSERT IN CHANNEL
FLOW
5.1 INTRODUCTION
The flow distribution in a fuel cell bipolar plate is one of the most important
enhancing factors of PEM fuel cell systems. One of the critical issues in PEM fuel cell
design is the efficient design of the flow channels to ensure uniform distribution of the
reactant gases in the fuel cell stack. The flow field geometry and pattern have great
influence on the reactant gas transport, water management and the efficient utilisation
of the fuel. The flow field design of fuel cells is one of the critical technical
challenges for PEM fuel cell designs and operation and impacts on the performance
and the life-span of the system [65, 198].
Several studies have been carried out in recent years to improve fuel cell performance
through flow-field design such as parallel, serpentine, interdigitated and many other
novel combinations of these conventional types [30, 94, 69, 185, 199]. The serpentine
channel type is the most widely used among the studied flow channels due to its
outstanding performance when compared with others under the same operating and
design conditions [197]. However, a serpentine flow field has its associated problems
and is not an ideal flow field configuration. Some of the associated problems are:
 high reactant pressure loss resulting in significant parasitic power requirement to
pressurise air, especially at the cathode section [62];
115
Chapter 5: Optimising Reactant Gas Transport in PEM Flow Channel
 loss of reactant gas concentration along the channel from the inlet towards the
outlet, and membrane dehydration near the channel inlet region;
 resultant liquid water flooding near the exit region of the channel, as a result of
excessive liquid water carried downstream by the reactant gas stream and
collected along the flow channel [117];
These serpentine flow channel characteristics proved their effectiveness in small cells
(  560  250  330 mm and power rating  1000 W) where the pressure drop is in the
order of 0.5-1 bar. However, serpentine flow channels perform poor for larger cells
(> 560  250  330 mm and power rating  1500 W) where the pressure drop is in the
order of a few bars [56]. Hence, parallel flow channels have several applications,
especially for larger cell applications, but the problems of cathode gas flow
distribution and cell water management need to be solved.
In these channels, apart from issues related to maldistribution of reactant gases, water
coalescence forms droplets of varying numbers and sizes in the channels. This
subsequently forces the reactant gas to flow preferentially through the path of least
obstruction [70]. Performance improvement for this type of channel and others have
been documented in the literature, but there is little information in the open literature
regarding the design procedure and cross-sectional dimensions that includes pressure
drops for flow in the channels [62]. Performance improvement of PEM fuel cells can
be achieved in many ways and researchers have developed varieties of flow-field
layouts for this purpose.
An interdigitated flow-field design was first proposed by Nguyen [200] with the
addition of baffles at the end of the channels. The design forces the reactants through
the GDL and the generated shear forces help blow the trapped water into the inner
layer of the electrodes resulting in better fuel cell performance. Kumar and Reddy
[201] presented a three-dimensional steady-state numerical mass-transfer single-cell
model for a PEM fuel cell, by using metal foam in the flow field of the bipolar/end
116
Chapter 5: Optimising Reactant Gas Transport in PEM Flow Channel
plates rather than using conventional rectangular channels. Their result showed a
significant effect of the metal foam on the permeability of the reactant species, which
improved the performance of the fuel cell. They proposed the use of metal foam
instead of conventional rectangular channels, especially in thinner channels where
there are manufacturing constraints.
Liu et al. [85] investigated the effect of baffle-blocked channels on the reactant
transport and cell performance using a conventional parallel flow field. Their results
showed improved cell performance due to an increasing reactant spread over the GDL
which enhances chemical reactions. Soong et al. [86] developed a novel flow channel
configuration by inserting baffles in the channel of conventional flow fields to form a
partially blocked fuel channel. They discovered that enhanced fuel cell performance
could be achieved by reducing the gap size and/or increasing the baffle number along
the channel, though with the penalty of higher pressure loss.
Liu et al. [87] studied the reactant gas transport and cell performance of a PEM fuel
cell with a tapered flow channel design. The results obtained from the study revealed
that fuel cell performance can be enhanced with a tapered fuel channel and
enhancement is more prominent at lower cell voltage. The reactant gas in the tapered
channel is accelerated and forced into the GDL, thereby enhancing the
electrochemical reaction that improves cell performance. Xu and Zhao [61] presented
a new flow-field design, termed the convection-enhanced serpentine flow field
(CESFF) for polymer electrolyte-based fuel cells. They observed that the CESFF
design induces larger pressure differences between adjacent flow channels over the
electrode surface when compared with the conventional flow field. This design
characteristic increases the mass transport rates of reactants and products to and from
the catalyst layer and reduces liquid water entrapped in the porous electrode which
subsequently, enhances fuel cell performance.
117
Chapter 5: Optimising Reactant Gas Transport in PEM Flow Channel
Wang et al. [78] similarly studied the use of baffles in a serpentine flow field to
improve cell performance. The results showed that the novel baffle serpentine flow
field, even though it induces larger pressure differences between adjacent flow
channels over the entire electrode surface than the conventional serpentine design
does; helps gas diffusion which leads to enhanced current density and improved cell
performance.
These investigations have shown that the addition of bluff bodies (baffles) in the flow
channels can increase the convection of reactants through the GDL thereby enhancing
fuel cell performance. This enhanced performance and operating stability in fuel cells
are achieved through improved reactant mass transport. Meanwhile, a proper
understanding of the phenomenon of mass transfer through the GDL, under the
influence of disturbances along the flow channels and associated pressure drop, will
facilitate a proper design of PEM fuel cells.
From the literature survey above, it is clear that issues of high penalty in terms of
pressure loss due to high flow resistance occur in most of the baffle-enhanced PEM
flow-field designs and therefore need to be addressed. In addition, to the best
knowledge of the authors, the application of pin fins for performance enhancement in
PEM fuel cells has not been examined before, especially for determining the optimal
geometry of the employed pin fins in PEM flow channels. Therefore, one of the major
objectives of this study is to investigate the effect of a pin fin insert in the flow field
of a fuel cell with the aim of improving performance as well as pressure drop along
the fuel cell flow channel. The cell overpotential at the anode side of the PEM fuel
cell is negligible in comparison with the cathode-side overpotential [32], hence the
choice of considering oxygen mass transport at the cathode side of the fuel cell
system.
Extended surfaces (fins) are frequently used in heat exchanging devices for the
purpose of increasing the heat transfer between the primary surface and the
118
Chapter 5: Optimising Reactant Gas Transport in PEM Flow Channel
surrounding fluid. Extended surfaces of various shapes have been employed for this
purpose in heat and mass transfer studies, ranging from relatively simple shapes, such
as rectangular, square, cylindrical, annular, tapered or pin fins, to a combination of
different geometries. Literature shows that pin fins are some of the most widely
employed extended surfaces considering its hydrodynamics along flow channels [202204]. A pin fin is a cylinder or other shaped element attached perpendicularly to a
wall, with the transfer fluid passing in cross-flow over the element. Pin fins with a
height to diameter ratio between 0.5 and 4 are accepted as short fins, whereas long
fins have a pin height to diameter ratio exceeding 4 [203]. The effective selection of
the pin fin geometric parameters will result in the improvement of the reactant gas
distribution in the flow channel due to the mixing of the main flow and/or the flow in
the near-wall region and, subsequently, will permit effective reactant spread over the
GDL.
In the present work, the main motive is to investigate the effect of pin fins
transversely arranged along the flow channel on the reactant gas distribution, and
pressure drop characteristics of the fuel cell reactant gas channel. Pin fins of small
hydraulic diameter, which can reduce the additional pressure drop, are employed and
the effect on PEM performance is investigated. In addition, a mathematical
optimisation tool is used to select the best pin fin geometric configuration that
improves the fuel cell performance at a reduced pumping power requirement penalty
in the PEM fuel cell flow channel. This chapter presents a novel approach at
enhancing the oxygen mass transfer through the PEM fuel cell GDL at a reduced
pressure drop.
5.2 MODEL DESCRIPTION
In this chapter, a two-dimensional half-cell model of a PEM fuel cell system for the
cathode-side fuel gas channel and the GDL is considered. Figure 5.1 shows a
119
Chapter 5: Optimising Reactant Gas Transport in PEM Flow Channel
schematic diagram of the two-dimensional half-cell model with two pin fins along the
transverse section of the flow channel.
Figure 5.1 PEMFC half-cell model with two transverse pin fins along the flow
channel
The fluid considered here is air at an inlet pressure, P0, and velocity, U0. The fin
disturbance employed in this study protrudes from a rectangular base towards the
GDL at a height to diameter ratio between 0.5 and 4. The parameters h1, h2 and h3
(Figure 5.1) depict the flow channel height, tip clearance size and GDL thickness,
respectively. The tip clearance size is characterised by defining a dimensionless
parameter named, clearance ratio,   h2 h1 , for the study. The values of   0 and 1
indicate fully blocked and block-free conditions, respectively, and the values in
between are a measure of various levels of blockage [86]. Also defined, is another
dimensionless parameter: the ratio of the distance between pin distances in the
transverse direction to the fin thickness (pitch),   s d . The effects of the tip
clearance size, the pitch, the fuel flow Reynolds number (Re) and the porosity ( ) of
120
Chapter 5: Optimising Reactant Gas Transport in PEM Flow Channel
the GDL on the reactant gas transport, and the pressure drop across the channel are
critically explored. The porosity,  , of the porous medium is defined as the fraction
of the total volume of the medium that is occupied by void space. In this study,
parameters
were
varied
in
the
following
range: 0.2    0.6 , 5.0    10 ,
50  Re  350 and 0.2    0.6 . Other parameters used for the modelled PEM fuel
cell are shown in Table 5.1. The idea proposed in this chapter is aimed at improving
the reactant species distribution over the catalyst layer in the fuel cells in order to
increase the fuel cell performance at reduced pumping power requirement.
Table 5.1 Parameters of the modelled fuel cell
Channel length (mm)
120
Channel width (mm)
1.0
Channel depth (mm)
1.2
Membrane thickness (mm)
0.036
GDL thickness (mm)
0.21
Membrane porosity
0.5
Cell operating temperature (oC)
70
Cell operating pressure (atm)
3
GDL permeability (m2)
1.76 x 10-11
Electric conductivity of GDL (   1 m  1 )
300
Relative humidity
100%
5.2.1 GOVERNING EQUATIONS
The present chapter provides a two-dimensional solution for the half-cell model of a
PEM fuel cell. The following assumptions are used in this study:
 the reactant gas (air) is an ideal gas, and the flow is incompressible steady and
laminar;
 the GDL is from an isotropic porous material and uniform;
121
Chapter 5: Optimising Reactant Gas Transport in PEM Flow Channel
 the catalyst layer is treated as an ultra-thin layer (regarded as a boundary
condition), hence the reactant gas is totally consumed in the reaction;
 the reaction is assumed to be fast, ensuring that the transport time scale is
dominant when compared with the reaction time scale- this assumption allows
treating the chemical reaction simply as a boundary condition at the catalyst layer;
and
 the fuel cell operates at a constant temperature.
Based on these assumptions, the following governing equations for the gas channel
and the GDL can be written as [147]:
In the gas channel section, the governing equations are:
u v

 0,
x y
(5.1)
  2u  2u 
1 P
u
u
u
v

 v 2  2 ,
 x
x
y
y 
 x
(5.2)
u
  2v  2v 
v
v
1 P
 v  2  2 ,
v

 x
x
y
 y
y 

  2Wi  2Wi
Wi
Wi

u
v
 Di 
2
x
y
y 2
 x
and

,


(5.3)
(5.4)
where Di and Wi depict the diffusivity and mass fraction of the species, respectively.
In the GDL section, the governing equations are:
122
Chapter 5: Optimising Reactant Gas Transport in PEM Flow Channel
u v

 0,
x y
(5.5)
 3 C F u 2
  2 u  2 u   2
 u
 P
u 
u
u  v2
   2  2  
 v   
k
 x
y 
y 
k
 x
 x
  u
 3C F v 2
  2 v  2 v   2
 v
 P
v 
v
u  v2
  u  v   
   2  2  
k
 y
y 
y 
k
 x
 x
,
(5.6)
(5.7)
and,
 Wi
Wi
v
x

y

  u
  2W  2Wi

  Di ,eff  2 i 
 x
y 2






(5.8)
CF in Eqs. (5.6) and (5.7) depicts the quadratic drag factor. The Blake-Kozeny
correlation [86] is used for the relationship between the porosity and permeability, k,
of the GDL:
 D 2   3 
k   d  
,
2 
 150   1    
(5.9)
where
Dd 
6V d
S d
.
(5.10)
The last two terms in Eqs. (5.6) and (5.7) are drag force terms, added due to the
presence of the porous wall, which might increase the pressure drop. The porous
diffusion layer quantity is represented by the subscript d and Vd' Vs' is a geometrical
parameter which depicts the volume-to-surface ratio of the GDL [86]. In fuel cells,
the fluid flow diffuses through the GDL for the reaction to take place on the MEA.
The effective diffusivity ( Di,eff ) for gas-phase flow in porous media can be written as:
123
Chapter 5: Optimising Reactant Gas Transport in PEM Flow Channel
D i ,eff  D


(5.11)
The porosity,  , is the void volume fraction in the porous media. The tortuosity,  , is
a measure of the average path length of the species flow through the porous media
compared to the linear path length in the direction of the species transport. The
quantity (tortuosity) is usually estimated through experiment. Therefore, it is
conventionally correlated in fuel cell studies using the Bruggeman correlation. This
correlation assumes 
is proportional to 
 0 .5
, resulting in the simpler expression
[191]:
D i , eff  D  1 .5 .
(5.12)
The porosity correlation is used to account for geometric constraints of the porous
media.
The Reynolds number was defined as [205]:
Re  m D ch (  Ach ).
For hydraulic performance in the channel, an apparent friction factor,
(5.13)
f , was
evaluated by using the following equation [205]:
f  (  P / L ) D ch (  w 2 / 2 ),
(5.14)
where
w  m (  A ch ).
(5.15)
The channel flow resistance, (P m ), is defined as [206]:
124
Chapter 5: Optimising Reactant Gas Transport in PEM Flow Channel
(5.16)
2
R f  2Po L Dch
Ach ,
where Po is the Poiseuille constant.
The pumping power is evaluated by using the relation:
P pump

L
0
 dp ( x )

Q ( x )  dx .
 dx


(5.17)
The water formation and transport of liquid water are modelled using a saturation
model based on [72, 187]. In this approach, the liquid water formation and transport
are governed by the conservation equation for the volume fraction of liquid water, sw,
or the water saturation [163]:

 (  l s w )
   (  l V l s )  rw ,
t
where subscript
(5.18)
represents liquid water, and rw is the condensation rate modelled
as:

P  Psat
rw  c r max   (1  s w ) wv
M
RT

w ,H 2 0


 , [  s w  l ] 


,
(5.19)
where rw is added to the water vapour equation. The condensation rate is constant
1
at c r  100 s .
The clogging of the porous media and the flooding of the reaction surface are
modelled by multiplying the porosity and the active surface area by (1 – sw),
respectively.
125
Chapter 5: Optimising Reactant Gas Transport in PEM Flow Channel
5.2.2 NUMERICAL PROCEDURE
The model equations were solved using a finite-volume computational fluid dynamics
code Fluent [163] with Gambit® (2.4.6) as a pre-processor. The CFD code has an
add-on package for fuel cells, which has the requirements for the source terms for
species transport equations, heat sources and liquid water formations. The domain
was discretised using a second-order discretisation scheme. The pressure-velocity
coupling was performed with the SIMPLE algorithm [192] for convection-diffusion
analysis. Numerical convergence was obtained at each test condition when the ratio of
the residual source (mass, momentum and species) to the maximum flux across a
control surface was less than 10-6.
Uniform isothermal free stream and fully developed fluid (air) with constant
properties were assumed at the inlet and flows were fully developed at the outlet of
the channel. At the interface between the gas channel and the GDL layer interface, the
same velocity, the same concentration and the same gradients were imposed. No-slip
no-penetration boundary conditions were enforced on the pin fins and wall surfaces.
The domain was divided into hexahedral volume elements. A grid independence test
was carried out to ensure that solutions were independent of the dimensions of the
chosen grid, with consideration of both accuracy and economics. For this purpose,
four grid systems at 37 × 27, 82 × 27, 120 × 60 and 150 × 80 were tested. For the case
of Re = 350,   0.6,   7.0 and   0.5 , the maximum relative deviation for the
skin friction between the 120 × 60 grid and the 150 × 80 grid was less than 3%. It was
considered that the system of 120 × 60 was sufficient for the study as a trade-off
between accuracy and cost of time. A typical grid network for the computational
domain is shown in Figure 5.2. The model and solution were implemented using an
Intel® Core(TM) 2Duo 3.00 GHz PC with 3.24 GB of DDRam.
126
Chapter 5: Optimising Reactant Gas Transport in PEM Flow Channel
Figure 5.2 The representative grid system and computational domain
5.3 MATHEMATICAL OPTIMISATION ALGORITHM
The Dynamic-Q optimisation algorithm [166] previously discussed in Chapter 3 was
used in this study. The algorithm is a robust multidimensional gradient-based
optimisation algorithm which does not require an explicit line search and it is ideally
robust for cases where the function evaluations are computationally expensive. The
algorithm applies the dynamic trajectory LFOPC which is adapted to handle
constrained problems through approximate penalty function formulation [166]. This
dynamic approach is applied to successive quadratic approximations of the actual
optimisation problem. The successive sub-problems are formed at successive design
points by constructing spherically quadratic approximations, which are used to
approximate the objective functions or constraints (or both) if they are not analytically
given or very expensive to compute numerically [169, 207]. The use of spherically
quadratic approximation in the Dynamic-Q algorithm offers a competitive advantage
when compared with other algorithms in terms of computational and storage
127
Chapter 5: Optimising Reactant Gas Transport in PEM Flow Channel
requirements [169]. The storage savings become highly significant when the numbers
of variables becomes large. Therefore, this particular strength of the Dynamic-Q
method makes it well suited for optimisation of engineering problems with large
numbers of variables and it has been used to successfully solve a large variety of
engineering problems [207-213].
5.4 OPTIMISATION PROBLEM FORMULATION
The optimisation problem was tailored towards finding the best pin fin geometric
parameters, which would give the maximum reactant species diffusion to the GDL of
the fuel cell for a fixed Reynolds number, GDL thickness and GDL porosity at a
reduced channel flow resistance, contributing to the increase in pressure drop along
the channel. The apparent pressure drops increase the pumping power requirement for
operating a fuel cell system. The design variables which greatly affect the
hydrodynamic performance of pin fins are the geometric parameters s, d , h2 and h1
as depicted in the half-cell model shown in Figure 5.1.
The objective function for the optimisation can be written mathematically as,
I max  f (  opt ,  opt ,  opt , Re opt )
(5.20)
where I max is the maximized current density output for the optimised design
variables.
5.4.1 OPTIMISATION CONSTRAINTS
The optimisation problem was carried out subject to the following constraints:
5.4.1.1 Total pin fin area constraint
In pin fin application, the weight and material cost of pin fins are limiting factors.
Hence, the total area of pin fins is fixed to a constant value:
128
Chapter 5: Optimising Reactant Gas Transport in PEM Flow Channel
  Ac j 
 D H
j
j
Constant
C
(5.21)
and
D
j
Hj 
C
(5.22)

where j = 1, 2 and Ac is the pin fin area.
5.4.1.2 The tip clearance size
The tip clearance size,
, is the ratio of the gap size between the pin fin tip and the
GDL to the channel height. This was varied between 0.2 and 0.6:

h 
0 . 2     2   0 .6 .
h1 

(5.23)
5.4.1.3 The pitch
The pitch is the ratio of the distance between successive pin fins to the pin fin
diameter. This was allowed to vary between 5 and 10:
5    s d   10 .
(5.24)
5.4.1.4 Manufacturing constraint
The solid area fraction,  , which is defined as the ratio of the pin fin material to the
total area of the fuel cell channel was allowed to vary between 0.5 and 4. This is
based on manufacturing and size constraints [214, 215]:
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Chapter 5: Optimising Reactant Gas Transport in PEM Flow Channel
h h 

0.5    1 2   4.
d 

(5.25)
Also, the interfin spacing is limited to 50 microns based on pin fin fabrication
techniques [216, 217]:
s  50  m .
(5.26)
5.4.2 OPTIMISATON PROCEDURE
The optimisation problem defined in Section 4.1 was solved by coupling the
Dynamic-Q optimisation algorithm with CFD code FLUENT [163] and grid
generation (GAMBIT [164]) code in a MATLAB [218] environment. Figure 5.3
depicts a flow diagram of how the automation was carried out until convergence
(either by step size or function value criteria) was attained. To ensure that the
converged solution obtained was indeed the global minimum, a multi-starting guess
approach was employed.
130
Chapter 5: Optimising Reactant Gas Transport in PEM Flow Channel
Figure 5.3 Optimisation automation flow diagram
131
Chapter 5: Optimising Reactant Gas Transport in PEM Flow Channel
5.5 RESULTS AND DISCUSSION
5.5.1 RESULTS OF FLOW FIELD
The pin fins employed in this work are expected to induce high levels of mixing of the
main flow and/or the flow in the near-wall region and, subsequently, to improve the
convection of reactant gas through the GDL. The power output in the fuel cell system
is the consequence of the electrochemical reaction. Subsequently, the consumption of
oxygen through diffusion into the catalyst membrane region is an index of the cell
performance [78]. Higher oxygen mass flow rates through the GDL to the catalyst
layer result in better fuel cell performance, since this reaction gas is more available to
participate in the electrochemical reaction per unit of time. A qualitative description
of the flow velocity pattern around the pin fin and within the GDL is presented in
Figures 5.4-5.6 to illustrate the hydrodynamic phenomenon in the computational
domain.
Figure 5.4 shows the Reynolds number influence on the flow pattern for the case of
s/d = 5, and   0.2 at a fixed GDL porosity of 0.5. The Reynolds number has a
significant effect on the flow field and the diffusion of the reactant gas through the
GDL medium. The rate of diffusion increases as the Reynolds number increases,
thereby improving the reaction rate in the fuel cell system. The wake-shedding
generated by the front pin fin interacts with the pin fin immediately behind it along
the channel, which affects the flow-field characteristics. At a low Reynolds number of
50 (Figure 5.4a), there is flow attachment between the front pin and the back pin tips.
This flow attachment also occurs at a Reynolds number of 150 (Figure 5.4b), but for a
Reynolds number of 250 (Figure 5.4c), a flow separation occurs at the tip between the
front pin and the back pin.
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Chapter 5: Optimising Reactant Gas Transport in PEM Flow Channel
This increased Reynolds number creates the phenomenon of increasing the wake
generation and the diffusion pattern into the GDL of the cell. The angle of separation
of flow depends on the Reynolds number and the level of clearance ratio. As the
clearance ratio increases, the location of the boundary layer separation moves
forward. This movement is practically due to the change in the velocity distribution
inside the boundary layer formed on the pin fins.
(a.)
(b.)
(c.)
Figure 5.4 Effect of Reynolds number on the flow field for different flow field
configurations (s/d = 5,   0.2): (a) Re = 50, (b) Re = 150, (c) Re = 250
The flow pattern for a higher tip clearance (   0.6 ) is shown in Figure 5.5, where the
effect of the Reynolds number (at fixed GDL porosity of 0.5) can also be clearly
observed. The flow pattern in Figure 5.5 depicts the significant influence of the
increase in the tip clearance between the pin fin and the GDL at the rate of reactant
diffusion through the GDL into the catalyst reaction site.
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Chapter 5: Optimising Reactant Gas Transport in PEM Flow Channel
(a)
(b)
(c)
Figure 5.5 Effect of Reynolds number on the flow field for different flow field
configurations (s/d = 5,   0.6 ): (a) Re = 50, (b) Re = 150, (c) Re = 250
The reactant gases are forced down the GDL, hence improving the rate of
electrochemical reaction for improved performance. In Figure 5.6, the contours of the
tangential velocity profiles for the same case described in Figure 5.5 are shown. High
pressure points are shown at the tips of the front pin. The rate of reactant gas diffusion
into the GDL improves from Contours (a) to (c).
(a)
(b)
(c)
Figure 5.6 Contours of tangential velocity for different flow field configurations (s/d
= 5,   0.6 ): (a) Re = 50, (b) Re = 150, (c) Re = 250
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Chapter 5: Optimising Reactant Gas Transport in PEM Flow Channel
5.5.2 RESULTS OF PIN FIN GEOMETRY
In Figure 5.7, we present the friction factor, f, as a function of the channels Reynolds
number and the pitch (ratio of distance between the pin fin and the pin fin diameter).
The friction factor decreased with the increasing Reynolds number. The data obtained
in Figure 5.7 further shows that, as the pitch increases, the friction factor decreases.
This implies lower diffusion of reactant gas, consequently reducing the performance
of the fuel cell. Hence, lower pitch value, which generates more flow disturbance
between the pin fin tip and GDL surface, will be more appropriate as this improves
the fuel transport rate and subsequently, the reaction rate at the catalyst layer is
improved. However, this should also be optimised for minimum power requirement.
1.2
s/d = 4
s/d = 6
s/d = 8
1
f
0.8
0.6
0.4
0.2
0
8x10
1
2x10
2
2x10
2
3x10
2
Re
Figure 5.7 Fuel channel friction factor as a function of the Reynolds number and
pitch at a clearance ratio,   0.3
Figure 5.8 shows the variation of the friction factor as a function of the channel
Reynolds number and the clearance ratio (  ) between the pin fin and the GDL
surface. Decreasing  , means that the height of the fin towards the GDL increases.
135
Chapter 5: Optimising Reactant Gas Transport in PEM Flow Channel
Similarly, decreasing the height of the pin fin reduces the pressure drop in the gas
channel flow and subsequently reduces the convectional flow through the fuel cell
GDL, thereby reducing cell performance. Increasing the height of the pin fin increases
the fluid flow into the reaction site of the fuel cell. This is due to the tangential flow
velocity created by the pin fin and flow-mixing effects, however, with a penalty of
increasing the pumping power requirement due to increased pressure drop along the
fuel channel. This is also supported by the flow description experienced at a higher 
in Figure 5.5. An optimised clearance ratio will reduce the associated pressure drop
due to the increase in pin length towards the GDL and pumping cost will therefore
decrease.
1 .2
   
   
   
1
f
0 .8
0 .6
0 .4
0 .2
0
1
5x10
1x 10
2
2x10
2
2x 10
2
3x10
2
3x10
2
Re
Figure 5.8 Fuel channel friction factor as a function of the Reynolds number and
clearance ratio at a pitch, s/d = 5
Figure 5.9 depicts the friction factor as a function of channel Reynolds number and
the GDL porosity. The results show a decrease in the friction factor with an increase
in the GDL porosity of the fuel cell. The increased GDL porosity improves the
convection flow through the GDL and subsequently improves fuel cell performance.
The flow resistance in the channel at the larger GDL porosity (e.g. 0.7) is much less
136
Chapter 5: Optimising Reactant Gas Transport in PEM Flow Channel
than with the smaller porosity (e.g. 0.3). The pressure drop along the flow channel
enhanced with pin fins can be reduced considerably with an appropriate higher GDL
porosity.
1.2
  
  
  
1
f
0.8
0.6
0.4
0.2
0
2
1 x 10
2x10
2
2x 1 0
2
2x 10
2
3x 10
2
Re
Figure 5.9 Fuel channel friction factor as a function of the Reynolds number and
GDL porosity at a pitch, s/d = 5, and a clearance ratio,   0.3
Figure 5.10 shows the peak channel flow resistance as a function of the clearance
ratio and the GDL porosity. There is an optimum clearance ratio at   0.39 in which
the peak flow resistance in the fuel gas channel is minimised.
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Chapter 5: Optimising Reactant Gas Transport in PEM Flow Channel


96
64
R
f,max
80
48
32
0.13
0.26
0.39
0.52
0.65

Figure 5.10 Effect of optimised clearance ratio on the peak channel flow resistance
Also, Figure 5.11 shows the peak channel flow resistance as a function of the pitch
and the GDL porosity. There is also an optimal pitch at s / d  7.8, which minimises
the fuel channel friction. These results support the fact that an optimal arrangement of
the pin fin parameters could effectively minimise the fuel channel friction and reduce
the pressure drop along the fuel channel with a corresponding increase in reaction rate
on the catalyst layer, thereby improving the fuel cell performance. Figures 5.10 and
5.11 also show that the GDL porosity has a significant effect on the peak flow
resistance along the fuel gas channel. An increase in the GDL porosity reduces the
peak flow resistance in the fuel channel. This observation is in agreement with
previous work of Soong et al. [86].
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Chapter 5: Optimising Reactant Gas Transport in PEM Flow Channel
   
   
90
60
R
f,max
75
45
30
5
6
7
8
9
10
s/d
Figure 5.11 Effect of optimised pitch on the channel peak fuel channel flow
resistance
5.5.3 OPTIMISATION RESULTS
In this section, the optimisation algorithm was applied to obtain the best geometric
configuration of the pin fin that would offer optimal flow resistance along the fuel cell
channel, thus ensuring optimum performance of the fuel cell system. From the results
in Section 5.2, it is clear that the pin fin geometric parameters (clearance ratio and
pitch) optimally exist, which minimises the channel flow resistance. This optimal
geometric parameters and the porosity of the GDL have a significant influence on fuel
cell performance through reactant gas distribution and the reaction rate on the catalyst
layer. Reducing the inherent flow resistance along the flow channel will reduce the
additional pressure drop, therefore reducing the pumping power requirement. A series
of numerical optimisations and calculations were conducted within the design
constraint ranges given in Section 4.1 and the results are presented in the succeeding
section to highlight the optimal behaviour of the fuel cell system. Figure 5.12 shows
139
Chapter 5: Optimising Reactant Gas Transport in PEM Flow Channel
the effect of the minimised flow resistance as a function of the Reynolds number for a
fixed clearance ratio of 0.3 and a GDL porosity of 0.5. Minimised flow resistance
decreases with an increase in the Reynolds number.
120
100
R
f, min
80
60
40
20
0
1
6x 10
1x10
2
1x 10
2
2x10
2
2x10
2
3x10
2
Re
Figure 5.12 The minimised fuel channel flow resistance as a function of Reynolds
number for a fixed GDL porosity,   0.5, and a tip clearance ratio,   0.3
Figure 5.13 shows that the optimal pin fin clearance ratio decreases as the Reynolds
number increases. This result affirms the fact that a unique optimal pin fin clearance
ratio exists for the fuel gas Reynolds numbers. Similarly, Figure 5.14 shows the
optimal pitch as a function of the fuel gas Reynolds number at a fixed clearance ratio
of 0.3 and a GDL porosity of 0.5. The result also shows the existence of a unique
optimal pitch for the fuel gas Reynolds number.
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Chapter 5: Optimising Reactant Gas Transport in PEM Flow Channel
0 .6
0 .5
0 .4

pt
0 .3
0 .2
0 .1
0
40
80
120
16 0
200
240
280
32 0
360
Re
Figure 5.13 Optimal clearance ratio as a function of Reynolds number at a fixed
pitch, s/d = 5, and a GDL porosity,   0.5
6
5
(s/d)
opt
4
3
2
1
0
40
80
120
160
200
240
280
320
360
Re
Figure 5.14 Optimal pitch as a function of Reynolds number at a fixed clearance
ratio,   0.3, and a GDL porosity,   0.5
141
Chapter 5: Optimising Reactant Gas Transport in PEM Flow Channel
The effect of channel flow resistance on the optimised channel clearance ratio at a
porosity of 0.5, pitch of 5 and Reynolds number of 250 was investigated in Figure
5.15. The result shows that channel flow resistance has a significant effect on the
optimised clearance ratio. As the flow resistance increases, the optimal clearance ratio
decreases.
0 .5
0 .3

pt
0 .4
0 .2
0 .1
0
10
20
30
40
R
50
60
f
Figure 5.15 Effect of channel flow resistance on the optimised clearance ratio at a
fixed pitch, s/d = 5, and a GDL porosity,   0.5, at Reynolds number of 250
Also in Figure 5.16, the effect of channel flow resistance on the optimised pin fin
pitch was investigated at a clearance ratio of 0.3, GDL porosity of 0.5 and Reynolds
number of 250. The result shows that the optimised pitch decreases with an increase
in channel flow resistance. Generally, in this model, the flow resistance decreases
when the Reynolds number increases. The optimal clearance ratio and pitch also
decrease with increasing channel flow resistance, but an optimal level of these factors
(clearance ratio and pitch) exists which minimises the flow resistance of reactant
gases in the fuel cell gas channel.
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Chapter 5: Optimising Reactant Gas Transport in PEM Flow Channel
10
9
s/d
opt
8
7
6
5
4
20
30
40
50
R
60
70
80
f
Figure 5.16 Effect of channel flow resistance on the optimised pitch at a fixed
clearance ratio,   0.3, and a GDL porosity,   0.5, at a Reynolds number of 250
5.5.4 PERFORMANCE EVALUATION
Generally, in heat transfer studies using pin fins for enhancement, performance
analysis is done by using performance evaluation criteria [203, 219]. Therefore, it is
necessary to perform a similar analysis for this study and state the performance in
terms of pressure drop for a fuel cell channel equipped with pin fins and one without
pin fins. Figure 5.17 shows the pressure drop characteristic for a fuel cell channel with
and without pin fins. As can be expected, the figure shows that higher pressure drops
occur in the fuel channel with pin fins than in the fuel channel without pin fins.
However, the difference obtained along the flow channel for all the pin fin geometry
cases considered in this study was less than 6%.
143
Chapter 5: Optimising Reactant Gas Transport in PEM Flow Channel
90
C hannel without pin fin
C hannel with pin fin
80
Pressure drop (kPa)
70
60
50
40
30
20
10
0
50
100
150
200
250
300
350
400
Re
Figure 5.17 Fuel channel pressure drop as a function of the applied pressure drop for
a channel with pin fin (s/d = 5,   0.3 ) and one without pin fin
Thus, it is evident from this study that, in terms of both high performance
enhancement and reasonable pressure drop in a fuel cell system, the pin-fin-enhanced
fuel channel is a promising approach for the optimal design of a fuel cell system.
Figure 5.18 shows the pumping power as a function of the clearance ratio at a
Reynolds number of 250 for a pitch of 5 and GDL porosity of 0.6. The pumping
power is the product of the volumetric flow rate and pressure drop. The result shows a
minimum pumping power for the friction factor of the fuel channel at a fixed
Reynolds number and a specified pitch and GDL porosity of the fuel cell system. In
general, the clearance ratio,  , which has a significant effect on the fuel gas flow, can
be optimised to improve fuel cell performance at a reduced pumping power
requirement.
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Chapter 5: Optimising Reactant Gas Transport in PEM Flow Channel
0.8
0.6
P
min
[mW]
0.7
0.5
0.4
0.3
0
0.2
0.4
0.6
0.8
1

Figure 5.18 Pumping power as a function of tip clearance ratio at a pitch, s/d = 5,
and GDL porosity,  = 0.6, at a Reynolds number of 250
CONCLUSION
Using the numerical approach, the reactant gas transport phenomenon in the gas flow
channel of a half-cell model of a PEM fuel cell with pin fin insert was investigated.
The effect of the flow and geometrical parameters of the pin fin on the flow
distribution in the GDL, as well as friction characteristics in the channel were
critically studied. Pumping power requirements at varying pin fin clearance ratios to
evaluate performance was also explored. The conclusions are summarised as:
 The flow Reynolds number had a significant effect on the reactant flow field, and
the diffusion of the reactant gas through the GDL medium increased as the
Reynolds number increased.
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Chapter 5: Optimising Reactant Gas Transport in PEM Flow Channel
 The friction factor increased with an increasing clearance ratio of the pin fin in the
channel.
 The optimal clearance ratio and pitch for the considered fuel cell channel
decreased with an increase in the fuel channel friction.
 The friction factor decreased with an increase in the GDL porosity. Hence, the
channel friction and pressure drop can be reduced significantly with increased
GDL porosity.
 An optimal pin fin clearance ratio existed which offered minimum pumping
power requirement.
 An enhanced fuel cell performance was achieved by using pin fins in a fuel cell
gas channel, which ensured high performance and low fuel channel pressure drop
of the fuel cell system.
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Chapter 6: Optimising Thermal Performance in PEM Fuel Cell
6
CHAPTER 6:
MODELLING AND OPTIMISATION OF
COOLING CHANNEL GEOMETRIC
CONFIGURATION FOR OPTIMAL THERMAL
PERFORMANCE OF A PROTON EXCHANGE
MEMBRANE FUEL CELL
6.1 INTRODUCTION
A fuel cell is an electrochemical energy device that directly converts the chemical
energy in the fuel into electrical energy. Applications include: portable power,
stationary applications, vehicle propulsion and large electrical plants. PEM fuel cell
researchers are moving ahead at a rapid pace because of the many attractive features,
like rapid start-up, high power density, high efficiency and the belief of being the
most promising among fuel cell types for transportation application, due to its fast
start-up and dynamic response to changes in the demand for power during vehicular
operations [71, 220, 221]. These features have made it one of the most promising
clean and highly efficient power generation technologies in the 21st century.
Operating temperatures of fuel cell systems affect the maximum theoretical voltage at
which a fuel cell can operate [89]. Higher operating temperatures correspond to lower
theoretical maximum voltages and lower theoretical efficiency. However, higher
temperature at fuel cell electrodes increases electrochemical activity which, in turn,
increases efficiency [89]. Most current PEM fuel cells operate at low temperatures (<
80oC) encountering several performance difficulties, especially vehicular applications
such as reduced electrochemical kinetics at electrode sites; flooding due to two-phase
flows emergence; intolerance to impurities such as CO; insufficient heat rejection
capability and relatively high cost. A recent approach is to operate this class of fuel
cell at higher temperature (> 100oC) which eliminates some of these obstacles [88,
147
Chapter 6: Optimising Thermal Performance in PEM Fuel Cell
222-224]. Operating at higher temperatures increases the reaction rates at both
electrodes and consequently increases system efficiency. The quality of waste heat
from the fuel cell stack, which could be used in other system components requiring
heat or used to run an additional thermodynamic heat for additional power, is also
enhanced at higher operating temperature. Also, there is a substantial reduction in the
incidence of water “flooding” that restricts oxygen transport by blocking the channel
path and pores of the gas diffusion electrodes when fuel cells are operated at a higher
temperature. Several approaches are also on-going, especially on developing PEM
materials (polymers, catalyst layers and MEA compositions) that will be relatively
stable for compatibility with operation at high temperatures [104, 225, 226]. The
development of these materials comes with an additional cost of operating a PEM fuel
cell.
Moderate temperature ranges exist within which a specific fuel cell type will operate
efficiently and reliably. Subsequently, the goal in fuel cell thermal management is to
ensure effective stack operation at a specified temperature range. A PEMFC operated
at 80oC with an efficiency of 40-50% produces an enormous amount of heat (~ 50%
waste heat) due to the exothermic nature of the cell reaction that must be removed if
the integrity of the cell structure is to be maintained [222]. In a typical modern vehicle
based on the internal combustion engine (ICE), the cooling system rejects < 40% of
the generated waste heat and the exhaust manifold removes the bulk of the waste heat
in the system [227]. In contrast, a typical PEM fuel cell stack operating at 80oC must
reject all the heat produced via the cooling system.
The heat rejection capability of a PEM fuel cell system operating below 100oC is very
inefficient and requires elaborate cooling systems for adequate system performance
[88]. Basically, cooling methods are predominantly determined by the size of the fuel
cell system [119]. The size of the fuel cell system also has a direct link with the
required power output from the fuel cell system. Fuel cell units below 2 kW are better
cooled using air, while systems between 2 kW and 10 kW require judicious decisionmaking to use water or air cooling [89]. Water cooling requires more complex system
148
Chapter 6: Optimising Thermal Performance in PEM Fuel Cell
design when considering the need to monitor the temperature and pressure of the
cooling water and the need for an oil-free water pump to supply cooling water. Air
cooling could be achieved through increasing the reactant air flow to the fuel cell
system but with the risk of too much air drying out the PEM [119]. This associated
problem usually necessitates the use of a separate reactant air supply and a cooling
system for the fuel cell system.
Temperature distribution in fuel cells is usually non-uniform, even when there is a
constant mass flow rate in the flow channels [228]. This occurs primarily as a result of
the heat transfer and phase changes in PEM fuel cells. It usually causes temperature
fluctuations within the fuel cell system structure and affects the fuel cell performance.
Heat transfer in PEM fuel cells occurs in the following ways [228]:
 Between the cell component layers and the flowing air and fuel streams. This way
of heat transfer is usually described in terms of heat transfer coefficients ha (for
air channel) and h f (for fuel channel) due to forced convective heat transfer with
or without natural convection.
 Between the fuel and air streams across the interconnect layer, described as
overall heat transfer coefficient, U;
 In solid structures, described as heat conduction with different thermal
conductivities, k i ( i  electrolyte, electrodes and current interconnect layers).
In order to alleviate the excessive temperature build-up in a PEM fuel cell, the heat
generated by the various processes in the fuel cell structure should be removed
properly. Thermal management has a very strong impact on fuel cell performance,
since it affects the transport of water and gaseous species as well as electrochemical
reactions in the cells. Thermal management still remains a critical issue that needs to
be resolved in order for PEM fuel cell technology to be feasible for various
commercial applications [229, 230]. A number of numerical modelling works has
been carried out in the literature to investigate heat/mass transfer in PEM fuel cells.
149
Chapter 6: Optimising Thermal Performance in PEM Fuel Cell
Yu et al. [98] investigated the performance of the Ballard PEM fuel cell in terms of
electrochemical characteristics and water management. The study shows that the more
the water supplied to the anode from its inlet, the higher the voltage and usually the
lower the anode exit temperature. Coppo et al. [90] presented a 3-D model to study
the influence of temperature on PEM fuel cell operation which includes a two-phase
flow in the gas distribution channel. The obtained result indicates that both liquid
water transport within the GDL and liquid water removal from the surface of the GDL
play an important role in determining variations in cell performance as far as
temperature is concerned.
Yan et al. [91] presented a 1-D non-isothermal model to analyse the effect of anode
and cathode side temperatures on the membrane water distribution. The obtained
results shows that a temperature increase on the anode side can lead to membrane
dehydration and fuel cell operation at high current density leads to membrane
dehydration on the anode side, due to the strong electro-osmotic water drag at high
current density. Ramousse et al. [92] developed a 1-D non-isothermal model
accounting for heat and mass transfer in a complete cell with charge and mass transfer
in the electrodes. Their study provides temperature, concentration and potential fields
in a single cell. In addition, their work shows that the thermal gradient in MEA could
lead to thermal stresses at high current densities.
Shimpalee and Dutta [93] conducted a 3-D non-isothermal numerical analysis with a
two-phase flow phenomenon incorporated in their model. The effect of the heat
produced by the electrochemical reaction and phase change of water on the cell
performance was studied critically. Their study shows that the inclusion of heat
transfer in fuel cell model shows degradation in fuel cell performance. This research
work enumerated the importance of incorporating the heat transfer aspect in fuel cell
modelling. Shan and Choe [94] presented a 1-D model taking into account the
dynamics in temperature gradient across the fuel cell; the dynamics in water
concentration redistribution in the membrane; the dynamics in proton concentration in
the cathode catalyst layer; and the dynamics in reactant concentration redistribution in
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Chapter 6: Optimising Thermal Performance in PEM Fuel Cell
the cathode GDL. This study’s general result shows that temperature profiles in each
of the cell layers tend to follow the current waveform due to energy losses in these
layers. Higher temperature losses are prominent in the membrane and catalyst layer
due to ohmic losses as a result of the membrane resistance and heat released by the
chemical reaction.
Yuan and Sunden [95] performed a 3-D non-isothermal numerical analysis of heat
transfer and gas flow in PEM fuel cell ducts using a generalised extended Darcy
model. The effects of effective thermal conductivity, permeability, inertia coefficient
and porous layer thickness on gas flow and heat transfer were studied. Their result
shows that higher permeability, higher effective thermal conductivity of porous GDL
and smaller thickness of the porous layer improved heat transfer in the modelled fuel
cell system. Ju et al. [96] presented a 3-D non-isothermal single-phase model for all
seven layers of the PEM fuel cell that accounts for various location-specific heatgeneration mechanisms. These mechanisms include irreversible heating due to
electrochemical reactions and entropy, and Joule (ohmic) heating due to membrane
ionic resistance. They observed that the thermal effect on PEM fuel cells becomes
more critical at higher cell current density and/or lower GDL thermal conductivity.
Their result further shows that temperature increase in the membrane is highly
dependent on the GDL thermal conductivity and inlet humidity conditions.
A number of modelling approaches has been developed in the literature to predict the
thermal effect in PEM fuel cells as described above [90, 98, 91-96] but, although they
represent a significant contribution in fuel cell thermal modelling, there are few
reports on thermal cooling approaches to enhance thermal management in a PEM fuel
cell structure. Furthermore, most models on thermal management in PEM fuel cells
emphasise approaches to understand and improve the kinetic process for thermal
prediction aimed at improving individual fuel cell model performance, rather than a
practical approach to reduce the incident temperature generated in the fuel cell
structure. One of the enhancement techniques to reduce excessive temperature build151
Chapter 6: Optimising Thermal Performance in PEM Fuel Cell
up in a PEM fuel cell is by using air/water (depending on fuel cell size) cooling,
conveyed through cooling channels, as an integral part of the fuel cell flow structure.
To the author’s knowledge, studies on the impact of the geometric configuration of
cooling channels on effective thermal heat transfer and performance in the fuel cell
system is still limited in the literature and this phenomenon is explored in this study.
A numerical modelling study that investigates the geometrical effect of cooling
channels on the thermal performance of a PEM fuel cell is described in this chapter. A
parametric study on the effect of temperature, stoichiometry ratio, relative humidity
and the cooling channel aspect ratio on cell performance were initially conducted,
since these factors contribute to the extent of membrane hydration in a fuel cell
system. Optimal results of these parameters were subsequently combined with the
different aspect ratio of the cooling channels and the system performance was
evaluated for elevated fuel cell system temperatures (100-150C). In addition, a
mathematical optimisation tool was used to select the best geometric configuration
that would improve cooling and enhance fuel cell performance for a given cooling gas
flow Reynolds number. The results of this study will be of interest to fuel cell
engineers who are striving to improve thermal management in fuel cell systems and
enhance system performance.
6.2 MODEL DESCRIPTION
In this chapter, a numerical study and optimisation of geometric parameters of the
cooling channel of a PEM fuel cell is attempted, based on a three-dimensional full cell
model, and the impact on cell performance is explored. The single-cell PEMFC
consists of the anode flow channel, anode diffusion layer, MEA assembly, cathode
diffusion layer, cathode flow channel, as well as an array of cooling channels on the
carbon plates. Figure 6.1 shows the 3-D schematic of the model of the PEM fuel cell
system. The influential factors considered in this study that could affect the fuel cell
thermal behaviour and subsequently performances are the stoichiometry ratio, the
152
Chapter 6: Optimising Thermal Performance in PEM Fuel Cell
relative humidity, the cooling channel aspect ratio and the coolant air Reynolds
number. The construct of three (3) cooling channels transversely placed at equal
distances at each side (anode and cathode) of the bipolar plates of the PEM fuel cell
are shown in Figure 6.1. These parameters are selected as the design parameters to be
optimised in this study. The other geometric and physicochemical properties for the
fuel cell system are kept constant in this study and given in Table 6.1.
The Dynamic-Q optimisation algorithm [166] is employed herein as the optimisation
search scheme. This study is aimed at optimising these identified factors, so that the
best performance in terms of the optimal current density (the objective function) of
the PEM fuel cell system at prescribed operating conditions can be achieved. The
optimisation algorithm is expected to ensure robust optimal values for the factors
investigated in this study.
Figure 6.1 A schematic diagram of a 3-D model of PEM fuel cell system with cooling
channels embedded in the bipolar plates
153
Chapter 6: Optimising Thermal Performance in PEM Fuel Cell
1. Anode-side bipolar plate
6. Membrane
2. Cooling channel
7. Cathode catalyst layer
3. Hydrogen fuel channel
8. Cathode GDL
4. Anode GDL
9. Air gas channel
5. Anode catalyst layer
10.Cathode-side bipolar plate
Table 6.1 Parameters and properties used in the present model
Description
Value
Cell operating temperature (oC)
70
Air-side/fuel-side inlet pressure (atm)
3/3
Open-circuit voltage (V)
0.95
Porosity of GDL
0.4
Permeability of GDL (m2)
1.76 x 10-11
Tortuosity of GDL
1.5
Porosity of catalyst layer
0.4
Permeability of catalyst layer (m2)
1.76 x 10-11
Tortuosity of catalyst layer
1.5
Porosity of membrane
0.28
Permeability of membrane (m2)
1.8 x 10-18
Reference diffusivity of H2
0.915 x 10-4 m2 s-1
Reference diffusivity of O2
0.22 x 10-4 m2 s-1
Electric conductivity of catalyst layer
Electric conductivity of GDL
(
1
m
Electric conductivity in carbon plate
1
(1m1)
)
(  1 m 1 )
190
300
4000
154
Chapter 6: Optimising Thermal Performance in PEM Fuel Cell
O2 stochiometry ratio
1.2
H2 stochiometry ratio
2.0
Oxygen mole fraction
0.406
Relative humidity of inlet fuel/air
100%
Reference current density of anode
7 500
Reference current density of cathode (A/m2)
20
Anode transfer coefficient
0.5
Cathode transfer coefficient
0.5
Evaporation and condensation rate
100 s-1
6.2.1 BASIC ASSUMPTIONS
The electrochemical reactions in the fuel cell are complicated, hence the following
simplifying assumptions are made:
1. Ideal gas mixture in the flow channels and the porous electrode;
2. Incompressible and laminar flow;
3. Isotropic and homogeneous porous medium;
4. Ionic conductivity of both the membrane and catalyst layer are constant;
5. No pressure gradient between the anode and the cathode side (only gas
diffusion is considered);
6. “ultra thin” electrode layer, hence gas transport resistance through the
electrode porous layer could be neglected;
7. Identical inlet conditions for both the anode and cathode as well as the coolant
channel;
8. No gas pressure drop along the gas channels;
9. Liner temperature gradient across the layers in the fuel cell;
10. Constant thermal conductivity of the materials in the fuel cell.
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Chapter 6: Optimising Thermal Performance in PEM Fuel Cell
6.2.2 GOVERNING EQUATIONS
The employed governing equations are a single set applicable to all domains (flow
channels, GDL and catalyst layer). This approach is based on the previous work of
Um et al. [71]. The conservation equations of mass, momentum, species, proton,
electron and energy are presented below:
Continuity equation:
   u   0
(6.1)
Momentum:
1
2
  uu   p      S u
(6.2)
Species:


  uC k     Dkeff C k  S k
(6.3)
Proton:


   eff  e  S   0
(6.4)
Electron:


   seff  s  S   0
(6.5)
Energy:
156
Chapter 6: Optimising Thermal Performance in PEM Fuel Cell
  c p uT     k eff T   S T
(6.6)
The energy source term, S T , depicts the sum of the reversible heat release and the
irreversible heat generation. In the catalyst layer, the reversible and irreversible
reaction heats, as well as latent heat of water phase change, are considered. For the
membrane, the ohm heating of the current due to the large resistance of the membrane
is also considered. The detailed source terms used for the model in the equations
above are presented in Table 6.2.
Table 6.2 The governing equation source terms in various regions of the fuel cell
Gas channel
Diffusion
Catalyst layer
Membrane
layer
Mass
Sm  0
Sm  0
Sm  0
Anode:
Sm  Sh  Sw,
Cathode:
Sm  So  Sw ,
Momentum
Su  0
Su  
g
kk rg
ug
Su  
g
kk rg
ug
Su  0
Species:
O2
So  0
So  0
S o  ic 4 F M o
So  0
H2
Sh  0
Sh  0
S h  ia 2 F M h
Sh  0
Solid phase -
S ,s  0
S ,s  0
S ,s   I
S ,s  0
Membrane
S  ,m  0
S  ,m  0
S  ,m  I
S  ,m  0
Charge:
phase157
Chapter 6: Optimising Thermal Performance in PEM Fuel Cell
ST  0
Energy
ST  0
dV  I 2
I2

ST 
S T  i  T oc  
m
dT   m

The transfer current densities at the anode and the cathode are calculated using the
Butler-Volmer equation [60]:
  nF 
   nF  
io  io, ref exp  an    exp  cat   
 RT

  RT 
,
(6.7)
where  is the overpotential and defined as,
   s   e   Eocv ,
(6.8)
where F is the Faraday constant,  an and  cat represents the experimental anodic and
cathodic transfer coefficients, respectively, and R is the universal gas constant. The
effective diffusivity ( D i ,eff ) for the gas-phase flow in porous media can be written as:
D i , eff  D

.

(6.9)
The quantity (  = tortuosity) is usually estimated through experiment. Therefore, it is
conventionally correlated in fuel cell studies using the Bruggeman correlation [103].
This correlation assumes 
is proportional to 
 0 .5
, resulting in the simpler
expression [103]:
D i , eff  D  1 .5 .
(6.10)
The porosity correlation is used to account for geometric constraints of the porous
media.
The Reynolds number was defined as [205]:
158
Chapter 6: Optimising Thermal Performance in PEM Fuel Cell
Re  m D (A).
(6.11)
6.2.3 NUMERICAL PROCEDURE
The model equations were solved by using a finite-volume computational fluid
dynamics code Fluent [163] with Gambit® (2.4.6) [164] as a pre-processor. The CFD
code has an add-on package for fuel cells, which has the requirements for the source
terms for species transport equations, heat sources and liquid water formations. The
domain was discretised using a second-order discretisation scheme. The SIMPLE
algorithm [192] for convection-diffusion analysis was utilised to deal with the
pressure-velocity coupling. Numerical convergence was obtained at each test
condition when the relative error of each dependent variable between two consecutive
iterations was less than 1.0  10-7. The domain was divided into hexahedral volume
elements. A grid independence test was carried out to ensure that solutions were
independent of the dimensions of the chosen grid with consideration for both accuracy
and economics. For this purpose, five grid systems at 24 × 12 × 60, 34 × 12 × 60, 34 ×
22 × 60, 44 × 22 × 60 and 34 × 34 × 60 were tested. The obtained results of the
average current density under different grid systems, when the PEM fuel cell system
operating voltage was 0.7 V, are summarised in Table 6.3. It was considered that the
system of 34 × 22 × 60 (I = 1.7054 A/cm2) was sufficient for the present study as a
trade-off between accuracy and cost of time. A typical grid network for the
computational domain is shown in Figure 6.2. The model and solution were
implemented using an Intel® Core(TM) 2Duo 3.00 GHz PC with 3.24 GB of
DDRam.
Table 6.3 Grid independence test
Grid size
Iav [A/cm2]
24 × 12 × 60
1.7012
34 × 12 × 60
1.7048
34 × 22 × 60
1.7054
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Chapter 6: Optimising Thermal Performance in PEM Fuel Cell
44 × 22 × 60
1.7055
34 × 34 × 60
1.7057
Figure 6.2 The discretised three-dimensional computational domain of a single PEM
fuel cell with cooling channels
6.3 MATHEMATICAL OPTIMISATION ALGORITHM
The Dynamic-Q optimisation algorithm [166], previously discussed in detail in
Chapter 3 of this thesis, was used in this study. The algorithm is a robust
multidimensional gradient-based optimisation algorithm which does not require an
explicit line search and is ideally robust for cases where function evaluations are
computationally expensive. The algorithm applies the dynamic trajectory LFOPC
which is adapted to handle constrained problems through approximate penalty
function formulation [166]. This dynamic approach is applied to successive quadratic
approximations of the actual optimisation problems. The successive sub-problems are
formed at successive design points by constructing spherically quadratic
approximations, which are used to approximate the objective functions or constraints
(or both), if they are not analytically given or very expensive to compute numerically
160
Chapter 6: Optimising Thermal Performance in PEM Fuel Cell
[169, 207]. The use of spherical quadratic approximation in the Dynamic-Q algorithm
offers a competitive advantage when compared with other algorithms in terms of the
computational and storage requirements [169]. Storage savings become highly
significant when the number of variables becomes large. Therefore, this particular
strength of the Dynamic-Q method makes it well suited for optimisation of
engineering problems with a large number of variables and it has been used to
successfully solve a large variety of engineering problems [208, 210-213].
6.4 OPTIMISATION PROBLEM FORMULATION
The optimisation problem was tailored towards finding the best operating design
parameters which would give the best performance in PEM fuel cells. The design
variables which greatly affect the performance of PEM fuel cells, especially at high
operating temperatures are the air stoichiometry ratio, relative humidity (RH), the
aspect ratio of cooling channels and the coolant Reynolds number. The objective
function here is the maximised current density of the fuel cell system at optimised
operating factors (stoichiometry ratio, relative humidity, cooling channel aspect ratio
and coolant Reynolds number) at a pressure drop of less than 3 atm. Table 6.4 shows
the dimensions of the cooling channels used at base case condition for this study.
Table 6.4 Dimension of the cooling channels investigated for initial simulations
W(mm)
H(mm)
L(mm)
 ( H W )
1
0.8
1.5
120
1.875
2
1.2
3.0
120
2.500
3
1.6
4.5
120
2.813
Test Case
The objective function for the optimisation can be written mathematically as
I max  f (  opt , RH
opt
,H W
opt
, Re )
,
(6.12)
161
Chapter 6: Optimising Thermal Performance in PEM Fuel Cell
where I max is the maximised current density output for the optimised design
variables. The maximised current density approach in PEM fuel cell design has shown
to be robust and allows the determination of maximum parameteric values that are
sharp and robust enough for practical design applications [231]. In addition, when
identified accurately they pave way for increasing the cell stack net power efficiency,
approaching the actual PEMFC first-law efficiency level [231, 232]. The maximised
current densities in this study were examined at the fuel cell voltage of 0.7V.
6.4.1 DESIGN VARIABLE CONSTRAINTS
Total fixed volume. For each of the optimisation problems, the cooling channel
volume is kept constant.
The following constraints are imposed for the optimisation:
1    5
0.2  RH  1.0
1 .5  H W  3 .5
100  Re  500
(6.13)
(6.14)
(6.15)
(6.16)
6.4.2 OPTIMISATION PROCEDURE
The optimisation problem defined in Section 6.4.1 was automatically carried out in a
MATLAB [218] environment by simultaneously using GAMBIT [164] for mesh
generation and FLUENT [163] for modelling. This was made feasible by using both
GAMBIT [164] and FLUENT [163] journal files, which were executed in MATLAB
[218] by Windows executable files. Figure 6.3 depicts a flow diagram of how
162
Chapter 6: Optimising Thermal Performance in PEM Fuel Cell
automation is carried out until convergence (either by step size or function value
criteria) is attained. To ensure that the converged solution obtained is indeed the
global minimum, a multi-starting guess approach was employed.
Figure 6.3 Optimisation automation flow diagram
163
Chapter 6: Optimising Thermal Performance in PEM Fuel Cell
6.5 RESULTS AND DISCUSSION
6.5.1 MODEL VALIDATION
In computational modelling, grid independence study and comparison of the CFD
results with experimental data are the prescribed metrics for validation and
verification of the CFD modelling studies. For fuel cell performance description, the
polarisation or voltage-current (IV) curve is one of the most important final outcomes
of numerical simulation and is widely used for validation purposes [193]. The
simulation results for the base case operating conditions were verified against
experimental measurements of Wang et al. [33]. The computed polarisation curve
shown in Figure 6.4 is in good agreement with the experimental curves in the low
load region. However, the model current density in the high mass transport limited
region (> 1.5 A/cm2) is higher than the experimental values. This might be due to
possible experimental uncertainty or inadequate account of the effect of reduced
oxygen transport, as a result of water flooding at the cathode side of the fuel cell at
higher current density [150]. However, the predicted dependent variable distribution
patterns could still be used successfully for better understanding of the complicated
processes in fuel cell systems.
164
Chapter 6: Optimising Thermal Performance in PEM Fuel Cell
1
0
Present m odel
T = 70 C
W ang et al. [33]
V (volt)
0.8
0.6
0.4
0.2
0
0
0.5
1
2
I [A/cm ]
1.5
2
Figure 6.4 Comparison of numerical model prediction and experimental polarisation
curves at base condition
6.5.2 PARAMETRIC STUDY RESULTS
In this section, a series of simulations was performed on a range of PEM fuel cell
operating parameters to investigate their effect on the performance of the system.
These parameters were investigated at operating cell voltage of 0.7 V and results are
presented below.
First, the effect of temperature on the performance of a PEM fuel cell and the
investigation of an optimal temperature range for the modelled PEM fuel cell system
in this study is shown. Figure 6.5 shows the effect of an operating temperature from
50ºC to 90ºC on the performance of the PEM fuel cell system at steps of 10C. The
PEM fuel performance increase with the increase in cell temperature between 50ºC
and 75ºC, since the water removal is easier and prevents incidence of flooding. The
cell improvement at this temperature range is more noticeable at higher cell current
165
Chapter 6: Optimising Thermal Performance in PEM Fuel Cell
density. Meanwhile, an onset of decline in performance is observed as the operating
temperature is increased beyond 75ºC. Profound performance deterioration occurred
at temperatures between  75ºC and 90ºC, practically as a result of high membrane
dehydration at these elevated temperatures.
1
o
T C
o
50 C
0.8
o
60 C
o
V (Volt)
70 C
o
80 C
0.6
o
90 C
0.4
0.2
0
0
0.5
1
1.5
2
2.5
3
3.5
2
I (A /cm )
Figure 6.5 Effect of temperature on the PEM fuel cell performance at base conditions
Figure 6.6 clearly shows the optimal performance behaviour for the PEM fuel cell
model and the point of decline of performance as the cell temperature increases
beyond the optimal range level. This observed performance reduction phenomenon is
a critical factor hindering the operation of PEM fuel cells beyond a certain
temperature range to avoid fuel cell failure.
166
Chapter 6: Optimising Thermal Performance in PEM Fuel Cell
2.9
V = 0.7
2.8
2
I (A/cm )
2.7
2.6
2.5
2.4
2.3
2.2
40
50
60
70
o
T ( C)
80
90
Figure 6.6 The cell current density as a function of temperature and the operating cell
voltage
Figure 6.7 shows the polarisation curve at a varying stoichiometry number for a
constant temperature (70C) and pressure (P = 3 bar). For a low stoichiometry
number, the removal of the cathode outlet flow decreases, thereby keeping the water
concentration in the membrane layer increasing. This results in lower membrane
resistance and subsequent lower ohmic over-potential, hence the improvement in cell
performance. Meanwhile, at higher current density of the fuel cell, the low
stoichiometry number adversely affects the cathode over-potential due to excessive
resident water in the catalyst layer of the fuel cell system. Figure 6.8 depicts the PEM
fuel cell behaviour at varying stoichiometry ratios of the cathode for three different
current density loads. The figure shows that, at relatively low current density of the
fuel cell system (0.42 A/cm2), the air stoichiometry has little impact on the
temperature.
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Chapter 6: Optimising Thermal Performance in PEM Fuel Cell
0.8

0
T = 70 C
4
3
2
0.7
V (volt)
0.6
0.5
0.4
0.3
0.4
0.8
1.2
1.6
2
2
I (A /cm )
Figure 6.7 I-V curve at varying stoichiometry number. P = 3.0 bar and Re = 500
At this low current density, the rate of fuel consumption is small and the heat
generation in the fuel cell system is minimal. For an increased current density (i.e.
0.72 A/cm2) of the fuel cell system, which corresponds to higher reaction rates in the
fuel cell system and subsequently increases in heat generation, the effect of the
stoichiometry ratio becomes glaring on the cell temperature. The increased air stream
improves heat transfer in the fuel cell system. This shows the possibility of an optimal
match of temperature and stoichiometry ratio for improved fuel cell system
performance.
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Chapter 6: Optimising Thermal Performance in PEM Fuel Cell
140
120
2
0.62 A/cm
2
0.72 A/cm
2
80
o
T ( C)
100
0.42 A/cm
60
40
20
0
0
2
4
6
8
Stoichiometry ratio
10
Figure 6.8 Effect of stoichiometry ratio on the PEM cell temperature at cell voltage
of 0.7 V
Figure 6.9 shows the influence of relative humidity at the cathode inlet on the fuel cell
output voltage. At an increased relative humidity at the cathode inlet, air transport to
the catalyst is hindered. This results in an increase in the cathode over-potential,
especially at a high operating current density of the fuel cell system. There is an
increase in the generation of liquid water formation which results in reduced PEM
fuel cell performance. This result shows that the relative humidity of the cathode inlet
has a significant effect on liquid water formation and the extent of heat removal
within the fuel cell system. For optimal system performance, this effect could be
optimised in relation to other operating parameters.
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Chapter 6: Optimising Thermal Performance in PEM Fuel Cell
0.8
0
T = 70 C
RH
0.75
0.7
0.50
0.25
V (volt)
0.6
0.5
0.4
0.3
0.5
1
1.5
2
2
I (A/cm )
Figure 6.9 I-V curve at varying relative humidity (RH). P = 3.0 bar and Re = 500
Figure 6.10 shows the fuel cell performance at different aspect ratios of the cooling
channels for a Reynolds number of 500. The result shows that fuel cell performance
increases as the aspect ratio of the cooling channels increases to an operating
temperature of 70ºC, until it reaches an optimal aspect ratio of  3.0 mm. Beyond the
aspect ratio of about 3.0, cell performance starts to deplete. This result shows the
existence of an optimal channel aspect ratio that optimises fuel cell performance in
terms of current density. Figure 6.11 shows the IV curve for the cooling channel
aspect ratios (Table 6.4) investigated at the base operating condition of the fuel cell. It
is observed that fuel cell performance increases with the increase of the cooling
channel aspect ratio at a cell operating temperature of 70C. This increase in
performance is likely due to an improvement in the cooling within the PEM fuel cell
system, thereby increasing the cell membrane hydration and subsequently positively
aiding cell performance.
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Chapter 6: Optimising Thermal Performance in PEM Fuel Cell
1 .6
0
T = 70 C
2
I (A/cm )
1 .4
1 .2
1
0 .8
0 .6
0 .4
1.5
2
2.5
3
3 .5
H /W
Figure 6.10 The cell current density at different aspect ratio at a cell potential of 0.7
V and a fixed Reynolds number of 500
171
Chapter 6: Optimising Thermal Performance in PEM Fuel Cell
1
     
     
     
V (Volt)
0.8
0.6
0.4
0.2
0
0
0.5
1
1.5
2
2.5
3
3.5
2
I (A /cm )
Figure 6.11 Current density at three cases of channel aspect ratio and Re = 500
6.5.3 OPTIMISATION RESULTS
The results obtained previously on the effect of the stoichiometry ratio, relative
humidity and cooling channel aspect ratio on performance of the PEM fuel cells
suggest the possibility of an optimal combination of these parameters for improved
performance of PEM fuel cells when temperature is increased beyond the critical
operating temperature which is typical of low temperature PEM fuel cells. Moreover,
these factors are mutually dependent, especially in determining the rate of membrane
hydration which, in turn, determines the reaction and transport characteristics in the
fuel cell system. These factors are combined with varying cell operating temperatures
to predict cell performance, especially when operation at higher temperatures (HT) is
desired. It is well know that operating PEM fuel cells at higher operating temperatures
eliminates some of the complications hindering improved performance. An
intermediate HT-PEM (100-150C) operating situation was investigated in this study.
A series of numerical optimisations and calculations within the specified design
constraint ranges in Section 6.4.1 were conducted to highlight the optimal
172
Chapter 6: Optimising Thermal Performance in PEM Fuel Cell
performance of the PEM fuel cell model studied in this work. Table 6.5 presents the
obtained optimal values for the optimised parameters when using the Dynamic-Q
algorithm.
Table 6.5 Values of optimised parameters
Model parameters
Optimised values (0.7V)

4.161
RH
0.782
H W
3.182
The maximised fuel cell performance was investigated at varying cell operating
voltage for the combination of the optimal parameters in Table 6.5 at higher cell
operating temperatures of the fuel cell. Table 6.6 shows the polarisation data based on
the optimal design parameters for the different operating fuel cell voltages and
temperatures.
Table 6.6 Polarisation data at optimised conditions and varying cell operating
temperatures at Re = 500
Cell voltage
I (A/cm2)
I (A/cm2)
I (A/cm2)
(V)
(T = 120 C)
(T = 130 C)
(T = 150 C)
0.7
3.1421
3.6213
3.8228
0.6
4.0627
4.7341
5.1431
0.5
4.6814
5.4326
5.6314
0.4
5.3343
5.9531
6.3281
The results presented in Table 6.6 above shows that there is improvement in cell
performance at different cell voltages with increasing cell operating temperatures.
Higher performance was obtained al low cell operating voltages compared to higher
cell voltages at different temperature ranges. The increase in cell current density
difference was more prominent between the temperatures of 120C and 130C, but as
173
Chapter 6: Optimising Thermal Performance in PEM Fuel Cell
temperature increased towards 150C, cell performance started to deplete negatively.
When the operation was conducted beyond the 150C level, the cell performance
reduction became highly noticeable. This is most likely due to high level membrane
dehydration beyond this temperature (150C) level. The cooling to sustain the thermal
build-up in the cell structure was no longer effective at this higher temperature level.
Moreover, thermal stresses in PEM fuel cells are shown to rise as cell current density
increases.
Figure 6.12 shows the peak current density as a function of cooling channel aspect
ratio and cell temperature. An optimum cooling channel aspect ratio exists for the
examined temperature ranges in which the peak fuel cell current density is
maximised. This obtained result suggests that optimal arrangements of the channel
geometry (aspect ratio), that could effectively maximise the cell current density of the
fuel cell system, are feasible. In Figure 6.13, the optimal aspect ratio is shown as a
function of the coolant Reynolds number and temperatures. The figure depicts an
increase in the aspect ratio of the cooling channels as the Reynolds number increases
from 100 to  300, but the rate of increment starts declining as the Reynolds number
increases from 300.
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Chapter 6: Optimising Thermal Performance in PEM Fuel Cell
0.7
o
150 C
2
I (A/cm )
0.6
0.5
o
130 C
0.4
o
120 C
0.3
0.2
0
0.5
1
1.5
2
H /W
2.5
3
3.5
4
Figure 6.12 Effect of optimised cooling channel aspect ratio on the peak fuel cell
current density at different temperatures and cell potential of 0.7V
The result further shows that, for an increased temperature to operate the fuel cell
system, the required optimal aspect ratio increases, but at a relatively lower rate due to
reduced system temperature as the coolant Reynolds number is increased. The result
presented in Figure 6.13 can be correlated at temperature T = 110ºC by
H
W opt  2.22 Re 0.058
.
(6.17)
175
Chapter 6: Optimising Thermal Performance in PEM Fuel Cell
3.6
o
T = 15 0 C
3.4
o
(H/W)
opt
T = 13 0 C
o
3.2
T = 11 0 C
3
2.8
100
200
300
Re
400
500
600
Figure 6.13 Effect of Reynolds number and temperature on the optimised aspect ratio
of the cooling channel and cell potential of 0.7V
Figure 6.14 shows the effect of the maximised fuel cell current density as a function
of the cooling gas Reynolds number for fixed cell temperatures and a fixed cooling
channel aspect ratio of 2.50 mm. The maximised current density increases with an
increase in the cooling channel Reynolds number. The result presented in Figure 6.14
can be correlated at temperature T = 130ºC by
I max  1.93 Re 0.092
.
(6.18)
176
Chapter 6: Optimising Thermal Performance in PEM Fuel Cell
3.6
o
H /W = 2.50
Tem p ( C)
o
120 C
3.4
o
130 C
I
max
2
(A/cm )
o
150 C
3.2
3
2.8
0
100
200
300
Re
400
500
600
Figure 6.14 Effect of Reynolds number on the maximum current density at different
cell temperatures and cell potential of 0.7V
In Figure 6.15, temperature contours on the membrane section of the fuel cell system
are presented for varying cooling channel aspect ratios for a fuel cell voltage of 0.7
and a Reynolds number of 500. The result shows an improved temperature profile on
the membrane section as the aspect ratio of the cooling channel increases for the
examined cooling air Reynolds number.
177
Chapter 6: Optimising Thermal Performance in PEM Fuel Cell
Figure 6.15 The local distribution of temperature along the membrane at different
cooling channel aspect ratios and cell operating voltage of 0.7 V and Re = 500:
(a) H W = 1.875, (b) H W = 2.500 and (c) H W = 2.813
CONCLUSION
This chapter of the thesis is aimed at developing a numerical approach to improving
PEM fuel cell performance at elevated operating temperatures through the
combination of operating parameters with cooling channel aspect ratios. Numerical
results indicate that operating parameters such as the stoichiometry ratio, relative
178
Chapter 6: Optimising Thermal Performance in PEM Fuel Cell
humidity and the cooling channel aspect ratio have a significant effect on fuel cell
performance, primarily in determining the level of membrane dehydration of PEM
fuel cells. Optimal values of the stoichiometry ratio, relative humidity and cooling
channel aspect ratios were obtained by integrating a direct problem solver with an
optimiser (Dynamic-Q). For the particular PEM fuel cell model operating conditions
considered in this work, fuel cell performance is considerably enhanced when
combining the studied parameters. Performance is more outstanding at temperatures
between 120C and 130C. The performance increment then declines gradually from
130C to 150C. It should be noted that beyond 150C, there is no significant increase
in cell performance. The result of this work further shows that maximised current
density also exist for varying cooling channel aspect ratios of the fuel cell system. The
result also shows the possibility of operating low temperature PEM fuel cells beyond
the typical critical temperatures (  75-80ºC), by using the combined optimal of the
stoichiometry ratio, relative humidity and cooling channel geometry without the need
for special temperature resistant materials for the PEM fuel cell. This study can easily
be extended to varying cooling channel geometries and scaled for application in PEM
stack systems for enhanced PEM fuel cell performance.
179
Chapter 7: Conclusions and Recommendations
7
CHAPTER 7:
CONCLUSIONS AND RECOMMENDATIONS
Although fossil fuel energy systems are affordable and widely available, they are
finite and often accompanied by environmental pollution which has a negative impact
on agriculture, health, social and the economic condition of the populace. Thus, the
search for an alternative pollution-free affordable and widely-available energy source
to replace the conventional fossil fuel has been receiving increased attention in the
last decade. In this regard, the proton exchange membrane (PEM) fuel cell system has
been touted to be one of the most promising clean and highly efficient power
generation technologies of the future. In this thesis, factors that can enhance the
performance of PEM fuel cell systems have been explored by using a combined
numerical modelling and optimisation approach. The methodology developed in this
work ensures an effective and accurate prediction of PEM fuel cell performance under
different operating conditions. Novel approaches to performance enhancement were
also introduced, especially in areas of reactant gas and thermal cooling optimisation
for PEM fuel cells. The enhancement methodologies form the basis for new
component geometry development that can be utilised to improve the advancement in
system performance and manufacturing.
7.1 CONCLUSIONS
The major achievements and conclusions drawn from this study are summarised in the
following.
180
Chapter 7: Conclusions and Recommendations
(1) Development of a finite-volume model to predict the performance of a PEM fuel
cell system under different operating and design parameters:
 This model highlights that temperature, GDL porosity, cathode gas mass flow
rate and species flow orientation has significant impact on the performance of
a PEM fuel cell.
 The model further shows that the impact of operating parameters on the
performance of a PEM fuel cell is more significant at low operating cell
voltages than at higher operating fuel cell voltages
 The results from the model underscore the interactive mutual interdependence
of these fuel cell parameters during fuel cell operation and the need for an
optimal match for these parameters for optimum fuel cell design.
(2) Development of a finite-volume approach, combined with an optimisation
algorithm to model reactant gas transport in a PEM fuel cell with a pin fin insert in
the channel flow:
 This model shows that performance in PEM fuel cells could be improved
significantly by incorporating a pin fin in the channel flow. The Reynolds
number had a significant effect on the diffusion of the reactant gas through the
GDL medium.
 The fuel channel friction factor also increased with an increase in the
clearance ratio of the pin fin while it decreased with an increasing GDL
porosity. Hence, the channel friction and pressure drop can be reduced
significantly with the increasing GDL porosity, though at an optimal value.
 The optimal clearance ratio and pitch for the considered fuel cell channel
181
Chapter 7: Conclusions and Recommendations
decreased with an increase in the fuel channel friction factor. Optimal pin fin
clearance ratio exists which offered a minimum pumping power requirement.
(3) Development of a finite-volume approach, combined with an optimisation
algorithm to model the impact of cooling channel geometry on the thermal
management and performance of a PEM fuel cell system:
 The results from this model show that fuel cell performance is considerably
enhanced when PEM fuel cells operate at combined optimised design
parameters. Performance is more outstanding at temperatures between 120C
and 130C. However, the performance increment rate declines gradually from
130C to 150C.
 The result of this study shows the possibility of operating a PEM fuel cell
beyond the critical temperature range (  80C) by using the combined
optimised stoichiometry ratio, relative humidity and cooling channel
geometry, without the need for special temperature resistant materials for the
PEM fuel cell which constitute additional cost for PEM fuel cell development.
 It should also be noted that this study can easily be extended to different
cooling channels (apart from the rectangular channels used in this study) in
order to enhance the performance of PEM fuel.
In summary, this research work has shown the feasibility of designing fuel cells with
enhanced performance by using only the CFD approach or this approach in
combination with an optimiser. The use of the CFD approach alone provides
improvement in the lead time reduction for PEM fuel cell system development when
compared with development based solely on an experimental method (design and test
approach). However, the obtained results are not usually a distinct optimum for
system design. An optimiser such as the Dynamic-Q combined with the CFD codes
182
Chapter 7: Conclusions and Recommendations
show a more robust and distinct optimal result that is more accurate and suitable for
implementation during system design. Furthermore, it should be stressed that there is
a need for proper understanding of the problem formulation and implementation of
the modelling approach for a successful outcome, especially when combining the
CFD codes with the optimisation algorithm. This hinges more on the modelling skill,
expertise and experience of the modeller. The modelling methodology implemented
in this thesis can be applied to other fuel cell system designs (such as Solid oxide fuel
cell structure), provided a proper problem formulation is implemented. Also, scaling
of the obtained results in this research work is practicable for large scale PEM fuel
cell stack system design.
7.2 RECOMMENDATIONS
The numerical methodologies and models presented in this research work could be
further improved and/or extended in the following directions:
 In the models presented in this thesis, assumptions of isotropic and homogeneous
properties of the porous medium were applied. There is a need to evaluate PEM
fuel cell system performance under different material properties of the porous
medium, including the catalyst.
 In real PEM fuel cell system operation, the incidences of two-phase flow of liquid
in the channel structure are inevitable and have significant impact on fuel cell
performance. The physics governing the model could be improved in order to
increase the applicability of this model by extension to two-phase capability.
 In modelling the phase change of liquid water, the assumptions of evaporation and
condensation rate constants are commonly used. More detailed models on system
performance should incorporate a functional dependence of evaporation and
condensation rates rather than a constant value.
183
Chapter 7: Conclusions and Recommendations
 In terms of computational complexity, large scale simulation using parallel
computing will reduce the computational time, especially in such models as
presented in this thesis, where multi-parameter evaluation, which combines CFD
with an optimisation algorithm, is involved.
 In computational fuel cell models, such as the ones presented in this research,
there is a need to validate the results against adequate experimental data. Future
work should involve the design of modelled systems studied on standard fuel cell
test stations. This will ensure adequate validity and implementation of model
results in the PEM development process.
184
Appendix
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Polymer Physics, vol. 43, pp. 786-795, 2005.
226. Bai, Z., Putthanarat, S., Rodrigues, S. J., and Dang, T. D., Properties and
performance of composite electrolytes membranes based on sulfonated poly
(arylenethioethersulfone) and sulfonated polybenzimidazole, Polymer, vol. 52,
pp. 3381-3388, 2011.
227. Frank, G., Proceeding of the second European PEFC Forum P749, Lucerne,
Switzerland, 2003.
228. Yuan, J., Faghiri, M., and Sunden, B., On heat and mass transfer phenomena in
PEMFC and SOFC and modelling approaches, In: Sunden, B., Faghiri, M.,
(Eds.) Transport phenomena in fuel cells, WIT Press, pp. 133-174, 2005.
229. Zhang, Y. J., Ouyang, M. G., Luo, J. X., Zhang, Z., and Wang, Y. J.,
Mathematical modeling of vehicle fuel cell power system thermal management,
SAE Paper, vol. 1, pp. 11-46, 2003.
230. Andrew, R., and Li, X. G., Mathematical modeling of proton exchange
membrane fuel cells, Journal of Power Sources, vol. 102, pp. 82-96, 2001.
213
Appendix
231. Vargas, J.V.C., Ordonez, J.C., and Bejan, A., Constructal PEM fuel cell stack
design, International Journal Heat and Mass Transfer, vol. 48: pp. 4410-4427,
2005.
232. Vargas, J.V.C., Ordonez, J.C., and Bejan, A., Constructal flow structure for a
PEM fuel cell, International Journal Heat and Mass Transfer, vol. 47: pp. 41774193, 2004.
214
Appendix
APPENDICES
APPENDIX A:
SAMPLE GAMBIT JOURNAL FILE (GRID GENERATION AND
MESHING): SINGLE CHANNEL PEM FUEL CELL.
/
$htot = 0.8
$wtot = 3
$offr1x = ($wtot/2)
$offr1y = ($htot/2)
$ys = 0.6
$xs = 0.5
$offsy = $htot - $ys
$offsx = ($wtot/2) - ($xs/2)
$offr2x = ($xs/2)
$offr2y = ($ys/2)
face create width $wtot height $htot offset $offr1x $offr1y 0 xyplane rectangle
face create translate "edge.3" vector 0 0.21 0
face create translate "edge.7" vector 0 0.012 0
face create translate "edge.10" vector 0 0.036 0
face create translate "edge.13" vector 0 0.012 0
face create translate "edge.16" vector 0 0.21 0
face create translate "edge.19" vector 0 $htot 0
face create width $xs height $ys offset $offr2x $offr2y 0 xyplane rectangle
face move "face.8" offset $offsx $offsy 0
face cmove "face.8" multiple 1 offset 0 1.08 0
face split "face.1" connected faces "face.8"
face split "face.7" connected faces "face.9"
undo begingroup
215
Appendix
edge modify "edge.4" "edge.20" "edge.21" backward
edge picklink "edge.4" "edge.20" "edge.21" "edge.2"
edge mesh "edge.2" "edge.4" "edge.20" "edge.21" successive ratio1 1 intervals \
20
undo endgroup
undo begingroup
edge modify "edge.22" backward
edge picklink "edge.22" "edge.1"
edge mesh "edge.1" "edge.22" successive ratio1 1 intervals 30
undo endgroup
undo begingroup
edge modify "edge.33" "edge.40" "edge.38" backward
edge picklink "edge.33" "edge.40" "edge.38" "edge.36" "edge.39" "edge.19" \
"edge.3" "edge.31" "edge.32" "edge.34" "edge.37" "edge.35"
edge mesh "edge.35" "edge.37" "edge.34" "edge.32" "edge.31" "edge.33" \
"edge.3" "edge.19" "edge.39" "edge.40" "edge.38" "edge.36" successive \
ratio1 1 intervals 10
undo endgroup
undo begingroup
edge picklink "edge.7" "edge.10" "edge.13" "edge.16"
edge mesh "edge.16" "edge.13" "edge.10" "edge.7" successive ratio1 1 \
intervals 30
undo endgroup
undo begingroup
edge modify "edge.14" "edge.15" backward
edge picklink "edge.14" "edge.15" "edge.9" "edge.12" "edge.11"
edge mesh "edge.14" "edge.11" "edge.15" "edge.12" "edge.9" successive ratio1 \
1 intervals 4
undo endgroup
undo
/Undone to: undo begingroup
216
Appendix
undo begingroup
edge modify "edge.14" "edge.15" backward
edge picklink "edge.14" "edge.15" "edge.9" "edge.12" "edge.8" "edge.11"
edge mesh "edge.14" "edge.11" "edge.8" "edge.15" "edge.12" "edge.9" \
successive ratio1 1 intervals 4
undo endgroup
undo begingroup
edge modify "edge.6" backward
edge picklink "edge.6" "edge.18" "edge.17" "edge.5"
edge mesh "edge.5" "edge.17" "edge.6" "edge.18" successive ratio1 1.15 \
intervals 10
undo endgroup
face mesh "face.1" "face.2" "face.3" "face.4" "face.5" "face.6" "face.9" \
"face.8" "face.7" submap size 1
undo
/Undone to: face mesh "face.1" "face.2" "face.3" "face.4" "face.5" "face.6" "face
undo
/Undone to: undo begingroup
undo begingroup
edge modify "edge.5" "edge.6" backward
edge picklink "edge.5" "edge.6" "edge.18" "edge.17"
edge mesh "edge.17" "edge.18" "edge.5" "edge.6" successive ratio1 1 intervals \
4
undo endgroup
face mesh "face.1" "face.2" "face.3" "face.4" "face.5" "face.6" "face.9" \
"face.8" "face.7" submap size 1
edge create translate "vertex.16" vector 0 0 125
undo begingroup
edge picklink "edge.41"
edge mesh "edge.41" successive ratio1 1.1 ratio2 1.1 intervals 60
undo endgroup
217
Appendix
volume create translate "face.1" "face.2" "face.3" "face.4" "face.5" "face.6" \
"face.9" "face.8" "face.7" onedge "edge.41" withmesh
window modify invisible mesh
window modify visible mesh
window modify invisible mesh
physics create "inlet-a" btype "MASS_FLOW_INLET" face "face.54"
physics create "inlet-c" btype "MASS_FLOW_INLET" face "face.8"
physics create "outlet-a" btype "PRESSURE_OUTLET" face "face.9"
physics create "outlet-c" btype "PRESSURE_OUTLET" face "face.59"
physics create "wall-terminal-a" btype "WALL" face "face.67"
physics create "wall-terminal-c" btype "WALL" face "face.12"
physics create "wall-ch-a" btype "WALL" face "face.51" "face.53" "face.52"
physics create "wall-ch-c" btype "WALL" face "face.16" "face.14" "face.17"
physics create "wall-ends" btype "WALL" face "face.1" "face.2" "face.3" \
"face.4" "face.5" "face.6" "face.7" "face.20" "face.27" "face.32" "face.37" \
"face.42" "face.49" "face.68"
physics create "wall-gdl-a" btype "WALL" face "face.48" "face.46"
physics create "wall-gdl-c" btype "WALL" face "face.18" "face.19"
physics create "wall-sides" btype "WALL" face "face.13" "face.15" "face.24" \
"face.25" "face.29" "face.30" "face.34" "face.35" "face.39" "face.40" \
"face.44" "face.45" "face.64" "face.66"
physics create "catalyst-a" ctype "FLUID" volume "volume.5"
physics create "catalyst-c" ctype "FLUID" volume "volume.3"
physics create "channel-a" ctype "FLUID" volume "volume.7"
physics create "channel-c" ctype "FLUID" volume "volume.8"
physics create "gdl-a" ctype "FLUID" volume "volume.6"
physics create "gdl-c" ctype "FLUID" volume "volume.2"
physics create "membrane" ctype "FLUID" volume "volume.4"
physics create "current-a" ctype "SOLID" volume "volume.9"
physics create "current-c" ctype "SOLID" volume "volume.1"
window modify visible mesh
218
Appendix
export fluent5 "pem-single-channel1011.msh"
save name "C:\\pem-single101110\\pem-single-channelnew.dbs"
save
export fluent5 "C:\\pem-single101110\\pem-single-channelnew.msh"
219
Appendix
APPENDIX B:
THE DYNAMIC-Q OPTIMISATION ALGORITHM IN MATLAB
B-1 DYNQ.M
function [X,F]=dynq(x0,varargin);
tic
%
%
DYNAMIC-Q ALGORITHM FOR CONSTRAINED OPTIMISATION
%
GENERAL MATHEMATICAL PROGRAMMING CODE
%
-------------------------------------
%
% This code is based on the Dynamic-Q method of Snyman documented
% in the paper "THE DYNAMIC-Q OPTIMISATION METHOD: AN
ALTERNATIVE TO SQP?" by J.A. Snyman and A.M. Hay. Technical Report, Dept
Mech. Eng., UP.
%
%
%
%
MATLAB implementation by A.M. HAY
Multidisciplinary Design Optimisation Group (MDOG)
Department of Mechanical Engineering, University of Pretoria
%
August 2002
%
%
UPDATED : 23 August 2002
%
%
BRIEF DESCRIPTION
%
-----------------
% Dynamic-Q solves inequality and equality constrained optimisation
% problems of the form:
220
Appendix
%
%
minimise F(X) , X={X(1),X(2),...,X(N)}
% such that
%
Cp(X) <= 0
p=1,2,...,NP
Hq(X) = 0
q=1,2,...,NQ
% and
%
% with lower bounds
%
CLi(X) = V_LOWER(i)-X(NLV(i)) <= 0 i=1,2,...,NL
% and upper bounds
%
CUj(X) = X(NUV(j))-V_UPPER(j) <= 0 j=1,2,...,NU
%
% This is a completely general code - the objective function and the
% constraints may be linear or non-linear. The code therefore solves
% LP, QP and NLP problems.
%
%
-----------------
%
% User specified functions:
%
% The objective function F and constraint functions C and H must be
% specified by the user in function FCH. Expressions for the respective
% gradient vectors must be specified in function GRADFCH.
%
% {The user may compute gradients by finite differences if necessary
% - see example code in GradFCH}
%
% Side constraints should not be included as inequality constraints
% in the above subroutines, but passed to the dynq function as
% input arguments LO and UP. (Described below)
%
221
Appendix
% In addition to FCH and GRADFCH the following functions are called
% by DYNQ and should not be altered:
%
DQLFOPC,DQFUN,DQCONIN,DQCONEQ,DQGRADF,DQGRADC,DQGRADH
%
% In addition the script HISTPLOT.m plots various optimisation
% histories. To suppress automatic plotting set PRNCONST=0 below.
%
%
-----------------
%
% synopsis:
%
%
[X,F] = dynq(x0,lo,up,dml,xtol,ftol,clim,np,nq,kloop);
%
% outputs:
%
X = optimal solution (1xN)
%
F = optimal function value
%
% inputs:
%
x0 = starting point (1xN)
%
lo = NLx2 matrix associated with lower limits on the variables
%
containing variable index NLV(i) in the first column and
%
associated value V_LOWER of that limit in the second column
%
(optional, otherwise assumed no lower side constraints)
%
up = NUx2 matrix associated with lower limits on the variables
%
containing variable index NUV(i) in the first column and
%
associated value V_UPPER of that limit in the second column
%
(optional, otherwise assumed no upper side constraints)
%
%
dml = the move limit which should be approximately the same order
of magnitude as the "radius of the region of interest"
222
Appendix
%
= sqrt(n)*max-variable-range (optional, default =1)
%
xtol = convergence tolerance on the step size (optional, default =1e-5)
%
ftol = convergence tolerance on the function value (optional, default =1e-8)
%
clim = tolerance for determining whether constraints are violated
%
(optional, default =ftol*1e2)
%
np = number of inequality constraints (optional)
%
nq = number of equality constraints (optional)
%
Note: Both np and nq are optional and determined automatically
%
if not specified, but at the cost of an extra function evalution.
%
kloop = maximum number of iterations (optional, default = 100)
%
%
NOTE: use [] to activate default inputs, for example
%
% [X,F]=dynq(x0,[],[],2); uses dml=2 but default values for all other inputs.
%
%
See FCH and GRADFCH for an example problem.
%
% ---- This program is for educational purposes only ----
%*****PLOT OPTIMISATION HISTORIES AT END OF
PROGRAM?*******************
%
YES: 1
OR
NO: 0
%
PRNCONST=1;
%*******************************************************************
***
clc;
223
Appendix
N=length(x0); % Determine number of variables
X=x0;
[dum,D]=size(varargin);
vars=cell(1,9);
vars(1:D)=varargin;
LO=vars{1};
UP=vars{2};
DML=vars{3};
XTOL=vars{4};
FTOL=vars{5};
CLIM=vars{6};
NP=vars{7};
NQ=vars{8};
KLOOPMAX=vars{9};
% default values
[NL,dum]=size(LO);
if NL>0
NLV=LO(:,1)';
V_LOWER=LO(:,2)';
else
NLV=[];
V_LOWER=[];
end
[NU,dum]=size(UP);
if NU>0
NUV=UP(:,1)';
V_UPPER=UP(:,2)';
224
Appendix
else
NUV=[];
V_UPPER=[];
end
if isempty(DML)
DML=1; end
if isempty(XTOL)
XTOL=1e-5; end
if isempty(FTOL)
FTOL=1e-8; end
if isempty(CLIM)
CLIM=FTOL*1e2; end
if isempty(NP)|isempty(NQ)
[F,C,H]=fch(X);
NP=length(C);
if isempty(C)
NP=0;
end
NQ=length(H);
if isempty(H)
NQ=0;
end
end
if isempty(KLOOPMAX)
KLOOPMAX=100; end
%###################################################################
###C
%*******************************************************************
***C
225
Appendix
%
MAIN PROGRAM FOLLOWS: Do not alter!!!!
%*******************************************************************
***C
%###################################################################
###C
%*****OPEN OUPUT
FILES*************************************************C
%
fidA=fopen('Approx.out','wt+');
fidD=fopen('DynamicQ.out','wt+');
fidH=fopen('History.out','wt+');
%
%*****SPECIFY INITIAL APPROXIMATION
CURVATURES*************************C
%
ACURV=0.D0;
BCURV=zeros(1,NP);
if NP==0
BCURV=[];
end
CCURV=zeros(1,NQ);
if NQ==0
CCURV=[];
end
%
%
%
%*****INITIALIZE
OUTPUT************************************************C
226
Appendix
FEASIBLE=0;
fprintf(fidA,' DYNAMICQ OUTPUT FILE \n');
fprintf(fidA,' -------------------- \n');
fprintf(fidA,' Number of variables [N]= %i \n',N);
fprintf(fidA,' Number of inequality constraints [NP]= %i \n',NP);
fprintf(fidA,' Number of equality constraints [NQ]= %i \n',NQ);
fprintf(fidA,' Move limit= %12.8e \n',DML);
fprintf(1,'\n DYNAMICQ OPTIMISATION ALGORITHM \n');
fprintf(1,' ------------------------------- \n');
% (MAXX=Maximum number of X-values to be displayed on screen)
MAXX=4;
if N<=MAXX
fprintf(1,' Iter Function value ? XNORM
fprintf(1,'X(%i)
RFD
');
',1:N);
fprintf(1,'\n ------------------------------------------');
for I=1:N
fprintf(1,'------------',1:N);
end
fprintf(1,'\n');
else
fprintf(1,' Iter Function value ? XNORM
RFD ');
fprintf(1,'\n --------------------------------------------\n');
end
fprintf(fidD,' DYNAMICQ OPTIMISATION ALGORITHM\n');
fprintf(fidD,' -------------------------------\n');
fprintf(fidD,' Iter Function value
fprintf(fidD,'X(%i)
? XNORM
RFD
');
',1:N);
227
Appendix
fprintf(fidD,'\n');
fprintf(fidD,' --------------------------------------------------');
for i=1:N
fprintf(fidD,'---------------');
end
fprintf(fidD,'\n');
% Initialize outer loop counter
KLOOP=0;
% Arbitrary large values to prevent premature termination
F_LOW=1.D6;
RFD=1.D6;
RELXNORM=1.D6;
C_A=zeros(1,NP+NL+NU+1);
%*****START OF OUTER OPTIMISATION
LOOP*********************************C
while KLOOP<=KLOOPMAX
%*****APPROXIMATE
FUNCTIONS********************************************C
% Determine function values
[F,C,H]=fch(X);
% Calculate relative step size
228
Appendix
if KLOOP>0
DELXNORM=sqrt((X_H(KLOOP,:)-X)*(X_H(KLOOP,:)-X)');
XNORM=sqrt(X*X');
RELXNORM=DELXNORM/(1+XNORM);
end
% Determine lowest feasible function value so far
if KLOOP>0
FEASIBLE=1;
check=find(C<CLIM);
if isempty(check)&NP>0;
FEASIBLE=0;
end
check=find(abs(H)<CLIM);
if isempty(check)&NQ>0;
FEASIBLE=0;
end
for I=1:NL
if C_A(I+NP)>CLIM
FEASIBLE=0;
end
end
for I=1:NU
if C_A(I+NP+NL)>CLIM
FEASIBLE=0;
end
end
end
% Calculate relative function difference
229
Appendix
if F_LOW~=1.D6&FEASIBLE==1
RFD=abs(F-F_LOW)/(1+abs(F));
end
if FEASIBLE==1&F<F_LOW
F_LOW=F;
end
% Store function values
X_H(KLOOP+1,:)=X; % Need to adjust from Fortran version since
F_H(KLOOP+1)=F;
% Matlab does not accept 0 as a matrix index
if NP>0
C_H(KLOOP+1,1:NP)=C;
end
if NL>0
C_H(KLOOP+1,NP+1:NP+NL)=C_A(NP+1:NP+NL);
end
if NU>0
C_H(KLOOP+1,NP+NL+1:NP+NL+NU)=C_A(NP+NL+1:NP+NL+NU);
end
C_H(KLOOP+1,NP+NL+NU+1)=C_A(NP+NL+NU+1);
if NQ>0
H_H(KLOOP+1,:)=H;
end
% Determine gradients
[GF,GC,GH]=gradfch(X);
% Calculate curvatures
if KLOOP>0
230
Appendix
DELX=X_H(KLOOP,:)-X_H(KLOOP+1,:);
DELXNORM=DELX*DELX';
% Calculate curvature ACURV
DP=GF*DELX';
ACURV=2.*(F_H(KLOOP)-F_H(KLOOP+1)-GF*DELX')/DELXNORM;
for J=1:NP
DP=GC(J,:)*DELX';
% Calculate corresponding curvature BCURV(J)
BCURV(J)=2.*(C_H(KLOOP,J)-C_H(KLOOP+1,J)GC(J,:)*DELX')/DELXNORM;
end
for J=1:NQ
DP=GH(J,:)*DELX';
% Calculate corresponding curvature CCURV(J)
CCURV(J)=2.*(H_H(KLOOP,J)-H_H(KLOOP+1,J)GH(J,:)*DELX')/DELXNORM;
end
end
%*****RECORD PARAMETERS FOR THE
ITERATION******************************C
% Write approximation constants to Approx.out
fprintf(fidA,' Iteration %i \n',KLOOP);
fprintf(fidA,' --------------\n');
fprintf(fidA,' X=\n');
for I=1:N
231
Appendix
fprintf(fidA,' %12.8f ',X(I));
end
fprintf(fidA,'\n F= %15.8e\n',F);
for I=1:NP
fprintf(fidA,' C(%i)=%15.8e',I,C(I));
end
for I=1:NQ
fprintf(fidA,' H(%i)=%15.8e',I,H(I));
end
fprintf(fidA,' Acurv=%15.8e',ACURV);
for I=1:NP
fprintf(fidA,' Bcurv(%i)=%15.8e',I,BCURV(I));
end
for I=1:NQ
fprintf(fidA,' Ccurv(%i)=%15.8e',I,CCURV(I));
end
% Write solution to file
if KLOOP==0
fprintf(fidD,' %4i %+19.12e %i
',KLOOP,F,FEASIBLE);
else
if RFD~=1.D6
fprintf(fidD,' %4i %+19.12e %i %9.3e
%9.3e',KLOOP,F,FEASIBLE,RELXNORM,RFD);
else
fprintf(fidD,' %4i %+19.12e %i %9.3e
',KLOOP,F,FEASIBLE,RELXNORM);
end
end
232
Appendix
fprintf(fidD,' %+13.6e',X);
fprintf(fidD,'\n');
% Write solution to screen
if KLOOP==0
if N<=MAXX
fprintf(1,' %4i %+14.7e %i
',KLOOP,F,FEASIBLE);
fprintf(1,' %+9.2e',X);
fprintf(1,'\n');
else
fprintf(1,' %4i %+14.7e %i\n',KLOOP,F,FEASIBLE);
end
else
if N<=MAXX
if RFD~=1.D6&FEASIBLE==1
fprintf(1,' %4i %+14.7e %i %9.3e
%9.3e',KLOOP,F,FEASIBLE,RELXNORM,RFD);
else
fprintf(1,' %4i %+14.7e %i %9.3e
',KLOOP,F,FEASIBLE,RELXNORM);
end
fprintf(1,' %+9.2e',X);
fprintf(1,'\n');
else
if RFD~=1.D6&FEASIBLE==1
fprintf(1,' %4i %+14.7e %i %9.3e
%9.3e\n',KLOOP,F,FEASIBLE,RELXNORM,RFD);
else
fprintf(1,' %4i %+14.7e %i %9.3e\n',KLOOP,F,FEASIBLE,RELXNORM);
end
233
Appendix
end
end
% Exit do loop here on final iteration
if KLOOP==KLOOPMAX|RFD<FTOL|RELXNORM<XTOL
if KLOOP==KLOOPMAX
fprintf(1,' Terminated on max number of steps\n');
fprintf(fidD,' Terminated on max number of steps\n');
end
if RFD<FTOL
fprintf(1,' Terminated on function value\n');
fprintf(fidD,' Terminated on function value\n');
end
if RELXNORM<XTOL
fprintf(1,' Terminated on step size\n');
fprintf(fidD,' Terminated on step size\n');
end
fprintf(1,'\n');
fprintf(fidD,'\n');
break;
end
%*****SOLVE THE APPROXIMATED
SUBPROBLEM********************************C
[X,F_A,C_A,H_A]=dqlfopc(X,NP,NQ,F,C,H,GF,GC,GH,ACURV,BCURV,CCURV,
DML...
,NL,NU,NLV,NUV,V_LOWER,V_UPPER,XTOL,KLOOP);
% Record solution to approximated problem
234
Appendix
fprintf(fidA,'Solution of approximated problem:\n');
fprintf(fidA,'X=\n');
for I=1:N
fprintf(fidA,' %12.8f\n',X(I));
end
fprintf(fidA,' F_A=%15.8e\n',F_A);
for I=1:NP+NL+NU+1
fprintf(fidA,'C_A(%i)=%15.8e\n',I,C_A(I));
end
for I=1:NQ
fprintf(fidA,'H_A(%i)=%15.8e\n',I,H_A(I));
end
% Increment outer loop counter
KLOOP=KLOOP+1;
end
% Write final constraint values to file
if NP>0
fprintf(fidD,' Final inequality constraint function values:\n');
for I=1:NP
fprintf(fidD,' C(%i)=%15.8e\n',I,C(I));
end
end
if NQ>0
fprintf(fidD,' Final equality constraint function values:\n');
for I=1:NQ
fprintf(fidD,' H(%i)=%15.8e\n',I,H(I));
235
Appendix
end
end
if NL>0
fprintf(fidD,' Final side (lower) constraint function values:\n');
for I=1:NL
fprintf(fidD,' C(X(%i))=%15.8e\n',NLV(I),C_A(NP+I));
end
end
if NU>0
fprintf(fidD,' Final side (upper) constraint function values:\n');
for I=1:NU
fprintf(fidD,' C(X(%i))=%15.8e\n',NUV(I),C_A(NP+NL+I));
end
end
% Write final constraint values to screen
fprintf(1,' Constraint values follow:\n\n')
if NP>0
fprintf(1,' Final inequality constraint function values:\n');
for I=1:NP
fprintf(1,' C(%i)=%15.8e\n',I,C(I));
end
end
if NQ>0
fprintf(1,' Final equality constraint function values:\n');
for I=1:NQ
fprintf(1,' H(%i)=%15.8e\n',I,H(I));
end
end
if NL>0
236
Appendix
fprintf(1,' Final side (lower) constraint function values:\n');
for I=1:NL
fprintf(1,' C(X(%i))=%15.8e\n',NLV(I),C_A(NP+I));
end
end
if NU>0
fprintf(1,' Final side (upper) constraint function values:\n');
for I=1:NU
fprintf(1,' C(X(%i))=%15.8e\n',NUV(I),C_A(NP+NL+I));
end
end
% Write history vectors
fprintf(fidH,' %3i%3i%3i%3i%3i%3i\n', KLOOP,N,NP,NL,NU,NQ);
for I=1:KLOOP+1
fprintf(fidH,' %3i %15.8e',I-1,F_H(I));
for J=1:N
fprintf(fidH,' %15.8e',X_H(I,J));
end
fprintf(fidH,'\n');
end
if NP>0
for I=1:KLOOP+1
fprintf(fidH,' %3i',I-1);
for J=1:NP
fprintf(fidH,' %15.8e',C_H(I,J));
end
fprintf(fidH,'\n');
end
237
Appendix
end
if NL>0
for I=1:KLOOP+1
fprintf(fidH,' %3i',I-1);
for J=NP+1:NP+NL
fprintf(fidH,' %15.8e',C_H(I,J));
end
fprintf(fidH,'\n');
end
end
if NU>0
for I=1:KLOOP+1
fprintf(fidH,' %3i',I-1);
for J=NP+NL+1:NP+NL+NU
fprintf(fidH,' %15.8e',C_H(I,J));
end
fprintf(fidH,'\n');
end
end
if NQ>0
for I=1:KLOOP+1
fprintf(fidH,' %3i',I-1);
for J=1:NQ
fprintf(fidH,' %15.8e',H_H(I,J));
end
fprintf(fidH,'\n');
end
end
fclose(fidD);
238
Appendix
fclose(fidH);
fclose(fidA);
if PRNCONST
histplot;
% disp('Press a key to continue');
% pause;
% close all;
end
toc
239
Appendix
B-2 FCH.M
function [F,C,H]=fch(X);
% Objective and constraint function evaluation for DYNAMIC-Q
%
(USER SPECIFIED)
%
% synopsis:
%
%
[F,C,H]=fch(X);
%
% outputs:
%
F = objective function value
%
C = vector of inequality constraint functions (1xNP)
%
H = vector of equality constraint functions (1xNQ)
%
% inputs:
%
X = design vector (1xN)
%
%
-----------------
%
% The application of the code is illustrated here for the very simple
% but general example problem (Hock 71):
%
%
minimise F(X) = X(1)*X(4)*(X(1)+X(2)+X(3))+X(3)
% such that
%
%
%
C(X) = 25-X(1)*X(2)*X(3)*X(4) <= 0
and
H(X) = X(1)^2+X(2)^2+X(3)^2+X(4)^2-40 = 0
%
%
and side constraints
240
Appendix
%
%
1 <= X(I) <= 5 , I=1,2,3,4
%
% Starting point is (1,5,5,1)
%
% Solution of this problem is accomplished by:
%
(with FCH and GRADFCH unaltered)
%
%
x0=[1,5,5,1] % Specify starting point
%
lo=[1:4;1,1,1,1]' % Specify lower limits
%
up=[1:4;5,5,5,5]' % Specify upper limits
%
[X,F]=dynq(x0,lo,up); % Solve using Dynamic-Q
%
% NOTE: This function should return C=[]; H=[]; if these are
%
not defined.
%
% See also DYNQ and GRADFCH
%
%Objective Function
%Load Design Variables
%Get the Total Heat transfer
F = -LL4{2};
%Inequality Constraints
C(1)=(X(3)/(4*X(1)))-1;
C(2)=1-(2*X(3)/X(1));
C(3)=(X(4)/(4*X(2)))-1;
241
Appendix
C(4)=1-(2*X(4)/X(2));
Volu = 0.05;
%Equality Constraints
H(1)=(X(1)^2*X(3))+(X(2)^2*X(4))-(4*Volu/pi);
% To eliminate error messages
% Do not delete
if ~exist('C')
C=[];
end
if ~exist('H')
H=[];
end
242
Appendix
B-3 GRADFCH.M
function [GF,GC,GH]=gradfch(X);
% Objective and constraint function GRADIENT evaluation for DYNAMIC-Q
%
(USER SPECIFIED)
%
% synopsis:
%
%
[GF,GC,GH]=gradfch(X);
%
% outputs: Partial derivatives wrt variables X(I) of
%
GF = objective function (1xN)
%
GC = inequality constraint functions (NPxN)
%
GH = equality constraint functions (NQxN)
%
% inputs:
%
X = design vector (1xN)
%
%
COMPUTE THE GRADIENT VECTORS OF THE OBJECTIVE FUNCTION
F,
%
INEQUALITY CONSTRAINTS C, AND EQUALITY CONSTRAINTS H
%
W.R.T. THE VARIABLES X(I):
%
GF(I),I=1,N
%
GC(J,I), J=1,NP I=1,N
%
GH(J,I), J=1,NQ I=1,N
%
% NOTE: This function should return GC=[]; GH=[]; if these are
%
not defined.
%
% See also DYNQ, FCH
243
Appendix
%
% Determine gradients by finite difference
FDFLAG=1;
if FDFLAG
DELTX=1.D-4; % Finite difference interval
[F,C,H]=fch(X);
N=length(X);
for I=1:N
DX=X;
DX(I)=X(I)+DELTX;
[F_D,C_D,H_D]=fch(DX);
GF(I)=(F_D-F)/DELTX;
if ~isempty(C)
GC(1,1)=-X(3)/(4*X(1)^2);
GC(1,2)=0;
GC(1,3)=1/(4*X(1));
GC(1,4)=0;
GC(1,5)=0;
GC(2,1)=2*X(3)/(X(1)^2);
GC(2,2)=0;
GC(2,3)=-2/X(1);
GC(2,4)=0;
GC(2,5)=0;
GC(3,1)=0;
GC(3,2)=-X(4)/(4*X(2)^2);
GC(3,3)=0;
GC(3,4)=1/(4*X(2));
GC(3,5)=0;
244
Appendix
GC(4,1)=0;
GC(4,2)=2*X(4)/(X(2)^2);
GC(4,3)=0;
GC(4,4)=-2/X(2);
GC(4,5)=0;
end
if ~isempty(H)
GH(1,1)=2*X(1)*X(3);
GH(1,2)=2*X(2)*X(4);
GH(1,3)=X(1)^2;
GH(1,4)=X(2)^2;
GH(1,5)=0;
end
end
end
% To eliminate error messages
% Do not erase
if ~exist('GC')
GC=[];
end
if ~exist('GH')
GH=[];
end
245
Appendix
B-4 Execute_Finsim.m
%This program initiates DYNQ.M
clear all
clc
close all
x0=[+2.824638e-001 +1.513331e-001 +6.310029e-001 +5.814793e-001 +5.0000e002];
lo=[1 0.05
2 0.05
5 0.05];
up=[3 0.95
4 0.95];
dml=0.0005;
xtol=[];
ftol=[];
clim=[];
np=4;
nq=1;
kloop=[];
[X,F] = dynq(x0,lo,up,dml,xtol,ftol,clim,np,nq,kloop);
246
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