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Chapter 1: Introduction
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
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
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
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]
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)
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
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
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
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.
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
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.
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
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
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.
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
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
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
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.
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
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.
Chapter 1: Introduction
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
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
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.
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
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.
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
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
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
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
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
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
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.
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.
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
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
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.
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
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.
Chapter 1: Introduction
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.
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.
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
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.
Chapter 2: Fundamentals of PEM Fuel Cell System
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.
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.
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
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
Each of these is discussed further in the following sections.
Chapter 2: Fundamentals of PEM Fuel Cell System
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.
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
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.
Chapter 2: Fundamentals of PEM Fuel Cell System
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].
Chapter 2: Fundamentals of PEM Fuel Cell System
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].
Chapter 2: Fundamentals of PEM Fuel Cell System
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
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
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
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].
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
The hydrogen fuel cell system includes the following subsystems and control
components [103]: 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. 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.
Chapter 2: Fundamentals of PEM Fuel Cell System 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] 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
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. 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]. 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. 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
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.
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:
H 2  2H   2e  ,
O 2  2 H   2e   H 2 O , and
Chapter 2: Fundamentals of PEM Fuel Cell System
net-reaction: H 2  1 O 2  H 2 O .
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.
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]:
  
    u   S m ,
1  u 1
   uu  p     Su ,
  t
Chapter 2: Fundamentals of PEM Fuel Cell System
 C k 
   u C k     D k eff  C k  S k ,
  κ eff Φe  j  0 ,
  σ eff Φ s  j  0 ,
and finally,
  T     c
 c p
p uT
    k eff  T  S T .
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: 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:
Chapter 2: Fundamentals of PEM Fuel Cell System
   MWk C 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
 
i 
 MWw    Dw,mCw  nd e  , for anode
F 
 
Sm  MWo2
 
i 
 MWH2 c  MWw   Dw,mCw  nd e  , for cathode,
F 
 
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
i e  κ eff Φ e ,
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 ,
where  is the electrical conductivity.
Chapter 2: Fundamentals of PEM Fuel Cell System 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]. 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]:
M kz  ne  ,
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
Chapter 2: Fundamentals of PEM Fuel Cell System
Sk  
vk j
with j given by the Butler-Volmer equation [118]:
  2 F 
 2 F 
j  Aav io exp a    exp  c   ,
 RT 
  RT 
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 ,
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
2 F  a w
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].
Chapter 2: Fundamentals of PEM Fuel Cell System 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,
 CH
j a  Aav i0ref,a 
 CH
 2 , ref
 CO
jc  Aav i0ref,c 
 CO
 2 ,ref
 2 
  a
F a  and
  RT
 exp   c F  ,
 RT
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). 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
, is a volume-averaged volumetric specific heat over the
solid matrix and the fluid in the micropores:
Chapter 2: Fundamentals of PEM Fuel Cell System
 c 
   c p   1    c p s ,
where the subscript s refers to the solid material and
c  pertains to the fluid.
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.
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. 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].
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 .
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 .
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
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,
ie  
κn d
  w  κ Φ e
N w  nd
  wμw .
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
(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,
Chapter 2: Fundamentals of PEM Fuel Cell System
K 
Vw =   p L     z f c f F e ,
  
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. 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
is   Φs  
 0Φs   1s .5 0Φs ,
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 
  w  Sh
f e  wsat   w ,
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
fe ,
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]:
Chapter 2: Fundamentals of PEM Fuel Cell System
 xi 
xi N j  x j N i
cT Dijeff
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 ,
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
 xi  
cT D eff
xi N j  x j N i
cT D ijeff
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]:
Chapter 2: Fundamentals of PEM Fuel Cell System
Dkeffi 
8 RT
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 = 
p G ,
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
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  
pL ,
Vw 
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
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 
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  .
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 
1.4171  s   2.1201  s 
 1.2631  s  ,
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
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  
pL  
(pG  pC )  
pC .
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 rK
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
K rg  (1  s) 3 .
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]. 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
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]:
 2 C r 
1 C r 
 Dmeff
 Sk  0 ,
r r
where D meff is the effective diffusion coefficient of the dissolved species and S k 
vk j
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
2 nFD meff   C 
d 2
dx 2
K eff r0   r  r  r0
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
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:
 
i x   I o
No 2
D o2
4 FD oeff2
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  .
Ohm’s law for the conduction of protons in the ionomer phase yields:
d 2 x 
d 2Φ e x 
jc  x 
κ eff
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
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]:
1 d  2 dC O2
r 2 dr 
 c ,
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 
where the product Ra is commonly called the Thiele modulus and is defined as:
Ra 
Aav i0ref,c
 F 
Ra exp c   .
 2 RT 
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] ,
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 ,
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,
  a I a ,
where  a is the density of the agglomerates in the catalyst layer and is defined by:
a 
4 3  R a3
where  c is the porosity of the catalyst layer. Also, the overpotential is governed
by Ohm’s law:
Chapter 2: Fundamentals of PEM Fuel Cell System
 eff .
dx 
The oxygen concentration in the gas pores may be obtained from Eq. (2.53) and
Fick’s law:
O2 'c
d 2 CO2 ' g
dx 2
  a I a ,
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.
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.
Chapter 3:Numerical Modelling Framework
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.
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
Chapter 3:Numerical Modelling Framework
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
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.
Chapter 3:Numerical Modelling Framework
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.
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
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:
Chapter 3: Numerical Modelling Framework
x *  [ x1* , x 2* ,....., x n* ]T ,
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.
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
Chapter 3: Numerical Modelling Framework
transforms the constrained problem to an unconstrained problem form, is thus
formulated as [173]:
p( x )  f ( x ) 
j 1
j 1
  j g 2j ( x )    k hk2 ( x ),
0 ifg j ( x )  0
ifg ( x )  0
 {
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
Phase 1:
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.
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
approximations are described below [175, 176]:
f ( x)  f ( x i )   T f ( x i )( x  x i )  ( x  x i )T A(x  x i )
g~ j ( x)  g j ( x i )   T g j ( x i )( x  x i )  ( x  x i )T B j ( x  x i )
hk ( x)  hk ( x i )   T hk ( x i )(x  x i )  ( x  x i )T C k ( x  x i )
A , B j and C k are Hessian matrices of the objective, inequality and equality
functions, respectively, and often take on the simple forms:
= diag ( a , a ,..., a )  a I
B j = bj I
C k = ck I
where I represents the identity matrix.
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
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 ;
 Function value:
f norm 
| f k  fbest |
f ;
1 | fbest |
where  x and  f are the step sizes and function value tolerances, respectively.
Chapter 3: Numerical Modelling Framework
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
where the differencing step size is given as:
x i  [0,0,..., xi ,...,0]T .
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).
g i (x) g i (x  x i )  g i (x)
, i  1,2,..., n
hi (x) hi (x  x i )  hi (x)
, i  1,2,..., p ,
also with the differencing step size being:
x i  [0,0,..., xi ,...,0]T
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
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].
v. Set i : = i  1, x : = x  (i- 1 ) and return to step ii.
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
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.
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.
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