Thermal and Fluid Dynamics Modeling of a Proton
Exchange Membrane Fuel Cell
Master Thesis
by
Abdelrahman Khaled Zohbi, 51230227
Mahmoud Radwan Fallah, 51230253
Submitted to the School of Engineering of the
Lebanese International University
Tripoli, Lebanon
in partial fulfillment of the requirements for the degree of
MASTERS OF SCIENCE IN MECHANICAL ENGINEERING
Spring 2017
Approved By:
Dr. Omar Melhem
Supervisor Name
Date
Signature
Dr. Bakri Abdulhay
Dr. Kifah Sarraf
Committee Member Name
Date
Signature
DEDICATION
Every challenging work requires personal commitment as well as the support of professionals
and loved ones who supply the fuel required for success and achievement.
Our profound thanks and appreciations go to our friends that were our motivation throughout
this project, and especially to our families who stood next to us in every step of the way.
Moreover, the hard work of our instructors cannot be overlooked, as they were there and
ready to help us in any problem we encountered. We could not have gotten this far without their
ongoing motivation, guidance, feedback and assistance.
ACKNOWLEDGEMENTS
This project consumed a large amount of work, research and dedication. Still, implementation
would not have been possible without the support of many individuals, who we would like to extend
our sincere gratitude to.
First of all, we would like to express our great appreciation to Dr. Omar Melhem for providing
us with guidance, support, and constructive recommendations for this project.
We would like to express our gratitude to Dr. Bakri Abdulhay and Dr. Kifah Sarraf for
analyzing the first part of our project, and for their proposed updates in our project.
Finally, we would like extend our greatest thanks and respect to every individual who stood
by us every step of the way.
ii
ABSTRACT
When modeling the functionality of a Proton Exchange Membrane Fuel Cell (PEMFC), an
assumption of a uniform temperature is taken into consideration to simplify the model. This
assumption has a lack in precision as the temperature in a PEMFC is not always uniform, even when
there is a constant mass flow rate through the channels. This variation in temperature appears due to
water phase change, coolant temperature, natural and forced convection, the trapping of water, the heat
produced by the chemical reactions, the tolerance in flow rate, and the heat produced by the catalyst
layer. This research will be focus around modeling a certain type of fuel cells, which is the Polymer
Exchange Membrane Fuel Cell (PEMFC).
iii
TABLE OF CONTENTS
Dedication .............................................................................................................................................. i
Acknowledgements .............................................................................................................................. ii
Abstract ................................................................................................................................................ iii
Table of Contents ................................................................................................................................ iv
List of Figures ..................................................................................................................................... vii
List of Tables ....................................................................................................................................... ix
List of Symbols and Abbreviations .................................................................................................... x
Chapter 1. Introduction ...................................................................................................................... 1
1.1. Overview.............................................................................................................................. 1
1.2. Operation ............................................................................................................................. 1
1.3. Advantages........................................................................................................................... 2
1.4. Types of fuel cells ................................................................................................................ 3
Chapter 2. Litterature Review ............................................................................................................ 5
2.1. History [3] ............................................................................................................................ 5
2.1.1. Gaseous Voltaic Battery: ............................................................................................. 5
2.1.2. Electricity Direct from Coal ......................................................................................... 6
2.1.3. The Solid Oxide Fuel Cell ........................................................................................... 7
2.1.4. Molten Carbonate Fuel Cell: ........................................................................................ 8
2.1.5. Alkaline Fuel Cell:....................................................................................................... 9
2.1.6. The Direct Methanol Fuel Cell:.................................................................................. 10
2.1.7. The Phosphoric Acid Fuel Cell: ................................................................................. 10
2.1.8. The Solid Polymer Fuel Cell: ..................................................................................... 11
2.2. Polymer exchange membrane and its reaction..................................................................... 12
2.3. PEMFC components........................................................................................................... 13
iv
2.4. PEMFC Previous studies .................................................................................................... 15
Chapter 3. Mathematical Model ....................................................................................................... 19
3.1. Fuel cell energy balances .................................................................................................... 19
3.2. General energy balance for fuel cell [15] ............................................................................ 20
3.3. Layers energy balance ........................................................................................................ 21
3.3.1. Transient conduction through End plate [16]:............................................................. 24
3.3.2. End plate, Contact and GDL layers energy balance: ................................................... 26
3.3.3. Bipolar Flow Field plate energy balance [17]: ............................................................ 28
3.3.4. Anode/Cathode Diffusion layer Energy balance: ........................................................ 30
3.3.5. Anode/Cathode Catalyst layer: ................................................................................... 31
3.3.6. Membrane Layer Energy Balance: ............................................................................. 32
3.4. Temperature in the layers ................................................................................................... 33
3.5. Modeling the proton exchange membrane .......................................................................... 36
3.5.1. Membrane Layer Energy Balance: ............................................................................. 36
3.5.2. Momentum equation: ................................................................................................. 38
3.5.3. Conservation of Energy equation: .............................................................................. 39
3.5.4. Ion Transport and other required activities: ................................................................ 40
3.5.5. Results for the Membrane: ......................................................................................... 41
3.6. Modeling the Gas Diffusion Layers .................................................................................... 43
3.6.1. Gas Diffusion Layer Model [21]: ............................................................................... 43
3.6.2. Results for the GDL: .................................................................................................. 50
3.7. Modeling the catalyst layers ............................................................................................... 51
3.7.1. Physical Description and Area of Catalyst Layer: ...................................................... 52
3.7.2. Interface and Agglomerate Models: ........................................................................... 52
3.7.3. Results and Current Density: ..................................................................................... 56
v
3.8. Modeling the flow field plates ............................................................................................ 60
3.8.1. Material and types of flow [24]: ................................................................................. 60
3.8.2. Pressure Drop in Flow Channels ................................................................................ 61
3.8.3. Mass Flow rates in the fuel cell layers ........................................................................ 63
3.8.4. Results for Pressure Drop and Flow rates ................................................................... 64
Chapter 4. Numerical simulation ..................................................................................................... 67
4.1. Introduction [25] ................................................................................................................ 67
4.2. Geometry and flow channels [25] ....................................................................................... 67
4.3. Mesh generation [25].......................................................................................................... 69
4.4. Solution and setup [25] ....................................................................................................... 71
4.5. Plots and results ................................................................................................................. 72
Chapter 5. Conclusion ....................................................................................................................... 77
Appendix A. MATLAB CODES....................................................................................................... 79
A.1. MATLAB FOR SECTION 3.3.1 ....................................................................................... 79
A.2. MATLAB FOR SECTION 3.4 .......................................................................................... 80
A.3. MATLAB FOR SECTION 3.5.5 ....................................................................................... 82
A.4. MATLAB FOR SECTION 3.6.2 ....................................................................................... 86
A.5. MATLAB FOR SECTION 3.7.3 ....................................................................................... 87
A.6. MATLAB FOR SECTION 3.8.4 ....................................................................................... 90
A.6.1. MATLAB for Pressure Drop ..................................................................................... 90
A.6.2. MATLAB for Flow Rates ......................................................................................... 91
Appendix B. ansys mathematical models and conventions ............................................................ 94
B.1. ANSYS MATHEMATICAL CONVENTIONS ................................................................. 94
B.2. ANSYS MATHEMATICAL MODEL ............................................................................... 95
References ......................................................................................................................................... 105
vi
LIST OF FIGURES
Figure 1-1: Polymer Exchange Membrane Fuel Cell [1] .................................................................... 2
Figure 2-1: Diagram of final form of the solid oxide fuel cell [3] ...................................................... 8
Figure 2-2: The development of different types of fuel cells [3] ...................................................... 12
Figure 2-3: Exploded View of a PEMFC Function[4] ..................................................................... 13
Figure 2-4: Schematic diagram of the main components of PEMFC [5] .......................................... 14
Figure 3-1: Two-dimensional node network .................................................................................... 21
Figure 3-2: Distributed nodes throughout computational domain .................................................... 22
Figure 3-3: Transient heat conduction in a polymer end plate at t=10 seconds ................................. 24
Figure 3-4: Transient heat conduction in a polymer end plate at t=120 seconds ............................... 25
Figure 3-5: Transient heat conduction through Aluminum end plate at t=60 and 120 seconds ......... 25
Figure 3-6: Energy balance around End plate, contact and GDL layers. .......................................... 26
Figure 3-7: Bipolar flow field plate energy balance ......................................................................... 28
Figure 3-8: GDL energy balance ..................................................................................................... 30
Figure 3-9: Catalyst Layer energy balance ...................................................................................... 31
Figure 3-10: Membrane Layer Energy balance ................................................................................ 32
Figure 3-11: Temperature plots for t=60, 300 and 1000 second for 1 slice per layer ........................ 34
Figure 3-12: Temperature plots for t=60, 300 and 1000 second for 10 slices per layer ..................... 35
Figure 3-13: Water concentration distribution with respect to the membrane thickness ................... 41
Figure 3-14: Temperature distribution with respect to the membrane thickness ............................... 42
Figure 3-15: Potential distribution with respect to the membrane thickness ..................................... 42
Figure 3-16: Pressure distribution with respect to the membrane thickness ...................................... 42
Figure 3-17: A 3D plot of the temperature in the interior layer ........................................................ 51
Figure 3-18: Cell Current as a function of Effectiveness Factor ....................................................... 58
vii
Figure 3-19: Cell Current Versus Butler-Volmer activation losses .................................................. 58
Figure 3-20: Cell Current as a function of voltage (polarization) ..................................................... 59
Figure 3-21: Current Density as a function of superficial flux density of hydrogen .......................... 59
Figure 3-22: A serpentine flow field design..................................................................................... 61
Figure 3-23: multiple serpentine flow channel design ...................................................................... 61
Figure 3-24: Water and hydrogen flow rates after 20 seconds of simulation time ............................ 66
Figure 3-25: Water and hydrogen flow rates after 120 seconds of time simulation .......................... 66
Figure 4-1: Layers of counter flow PEMFC .................................................................................... 68
Figure 4-2: Geometry of a counter flow PEMFC ............................................................................. 69
Figure 4-3: First Mesh, Monochrome Picture .................................................................................. 69
Figure 4-4: First Mesh Colored ....................................................................................................... 70
Figure 4-5: Second Mesh, Monochrome Picture .............................................................................. 70
Figure 4-6: Second Mesh Colored ................................................................................................... 71
Figure 4-7: Calculating the solution with 200 iterations................................................................... 72
Figure 4-8: Current flux located half-way along the length of the fuel cell using first mesh ............. 73
Figure 4-9: Current flux located half-way along the length of the fuel cell using second mesh ........ 73
Figure 4-10: Contour Plot of the Temperature distribution along the PEMFC ................................. 74
Figure 4-11: Contour Plot of Hydrogen’s Mass Fraction along the length of the PEMFC ................ 75
Figure 4-12: Contour Plot of Oxygen’s Mass Fraction along the length of the PEMFC ................... 75
viii
LIST OF TABLES
Table 1-1.1: Different types of Fuel Cells [3] .................................................................................... 4
Table 3-1- Material properties used for the heat transfer calculations .............................................. 33
Table 3-2 - Parameters for catalyst layer modeling .......................................................................... 57
ix
LIST OF SYMBOLS AND ABBREVIATIONS
A: Area
a: Anode side
a1-2: Interfacial area between membrane phases
Acell: Cell Area
AFC: Alkaline Fuel Cell
Agg: Agglomerate
C: Air heat transfer coefficient
Ci: Molar concentration
Cat: Cathode
Cp: Specific heat
Chan: Channel
Dc-H2O: Diffusion coefficient of water
D’: Diffusion coefficient at constant Temperature
D32: Volume-to-Area diameter
Dh: Hydraulic diameter
DMFC: Direct Methanol Fuel Cell
dc: Channel depth
E: Polarization
EA: Activation Energy
Ef: Effectiveness Factor
Er: Nernst Equation
F: Faraday’s constant
f: Friction factor
GDL: Gas Diffusion Layer
x
g: Gravitational Acceleration
H2Ol: Liquid Water
H2Ov: Vapor Water
H: Enthalpy
HO2-agg: Oxygen Henry’s constant in the agglomerate
HHV: Higher heating value
h: Heat transfer coefficient
I: Current density
ix: Protonic Current
Ji: Diffusive flux
K: Porous medium absolute permeability
k: Thermal conductivity
kg: Permeability of Gas Diffusion Layer to gases
𝑔
𝑘𝑟 : Relative Permeability
k𝜃: Hydraulic conductivity
L: Length
LL: Latent Heat
M: Molecular weight
MCFC: Molten Carbonate Fuel Cell
Mm: Membrane Molecular mass
m: mass flow rate
mem: Membrane
N: Number of Nodes
Ni: Nernst Planck Constant
Nx: Rate of Consumption
xi
n: Drag coefficient
ncells: Number of Cells
ne: Number of Electrons
ni: Molar flow rate
P: Pressure
Patm: Atmospheric Pressure
Pe: peclet Number of Oxygen
Pi: Inlet Pressure
PAFC: Phosphoric Acid Fuel Cell
PEMFC: Proton Exchange Membrane Fuel Cell
Pot: Potential voltage
Q: Heat rate transfer
Qc: Heat taken from the cell
Qdis: Dissipated Heat
Qgen: Generated Heat
𝑞̇ : Energy Stored
R: Gas Constant
Re: Reynold’s Number
Rm: Term of source
rO2: Oxygen content in air
S: Saturation Factor
SOFC: Solid Oxide Fuel Cell
sO2: Stoichiometric ratio of oxygen
T: Temperature
Ti: Inlet Temperature
xii
Tf: Final temperature
T’: Interior Temperature
t: Time
ti: Thickness
U: Overall Heat Transfer Coefficient
um: Mixture Velocity
𝑉: Total Voltage
𝑉𝑎𝑐𝑡: Activation Losses
𝑉𝑜ℎ𝑚𝑖𝑐: Ohmic Losses
𝑉𝑐𝑜𝑛𝑐: Concentric Losses
𝑉𝑐𝑒𝑙𝑙: Cell Voltage
𝑉𝑔𝑎𝑠: Volume of the gas
𝑉𝑣𝑜𝑖𝑑: Volume of channel space
V𝜃: Moisture Velocity
W: Work
Wel: Generated Electricity
wc: Channel width
𝑑𝑖𝑓𝑓
𝑤𝑂2 : Oxygen molar flow rate
X: Mole fraction
xi: Location of the node
𝜌: Density
𝜏𝑠𝑖𝑚 : Simulation time
𝜁: Void Fraction
∆𝑠: Change in Entropy
Ƞ: Activation over-potential
xiii
𝜆𝐻2𝑂/𝑆𝑜3 : Water Content
∅𝑚 : Membrane proton potential
𝜇: Fluid Viscosity
𝜎𝑚 : Conductivity of membrane
𝛽𝜃 : Evaporation Constant of proportionality
𝛽𝑣 : Condenstation Constant of proportionality
𝜌𝑃𝑇 : Platinum black density
𝑣𝑎𝑐𝑡 : Losses of the activation electrode
𝜕𝑎 : Anodic charge transfer coefficient
∅: Thiele Modulus
𝑣: Dynamic Viscosity
∑: Source term
xiv
CHAPTER 1. INTRODUCTION
1.1. OVERVIEW
A fuel cell is an electrochemical device that converts chemical energy from a reaction directly
into electrical energy, whenever fuel and oxidant are supplied. The basic function of a fuel cell is
reversing water electrolysis to generate electricity from hydrogen and oxygen. This functioning
principle remains unchanged till date.
Fuel cells bear similarities to batteries, as they both convert chemical energy into electrical
energy, and to continuously working engines consuming a certain type of fuel. The operation is
continuous, and takes place without combustion resulting only in the generation of heat and in the
production of water (When the fuel is only hydrogen). Accordingly, fuel cells are then called zero
emission engines.
1.2. OPERATION
In general, a fuel cell consists of two porous electrodes (anode and cathode) separated by an
electrolyte for the ions transfer between the two electrodes. The fuel is fed at the anode and the oxidant
is fed at the cathode through flow ducts. At the anode, the fuel is ionized using a catalyst resulting in
free electrons that will pass through the connected electric load, and into ions that will be transferred
through the electrolyte towards the anode. As a result of this process, heat is generated and water is
produced (See Figure 1.1).
As one fuel cell produces approximately 1 volt, fuel cells are arranged in series to attain the
needed voltage for the load to function normally.
1
Figure 1-1: Polymer Exchange Membrane Fuel Cell [1]
1.3. ADVANTAGES
Fuel cells may be a possible alternative to other currently existing energy conversion systems,
due to their advantages. As fuel cells convert chemical energy directly to electrical energy, fuel cells
operate with a good efficiency. Efficiencies of present fuel cells fall in the range of 40% to 55%, in
2
addition, their efficiency is almost independent of the connected load, which makes them suitable for
vehicle applications where the load is variable most of the time [2].
When pure hydrogen is used directly as a fuel, only water is produced and no pollutant is
rejected. However, the processing of hydrocarbon fuels into hydrogen can result in a small output of
pollutants significantly lower when compared, for example, to classical internal combustion engines.
Thus, fuel cells are considered as low emissions energy converters.
The exothermic chemical reactions in a fuel cell produces heat. This heat can be recycled and
used in another useful process, which gives the fuel cells a cogeneration capability. Fuel cells can be
sized and configured to fit a wide range of applications, ranging from few watts to Megawatts. This
scalability is a good support for using fuel cells instead of other energy conversion systems.
Natural gas, methanol and hydrocarbons are the commonly used fuels in fuel cells. Therefore,
fuels are not an issue in their operation.
The absence of mechanical systems and moving parts in fuel cells, reduces the amount of
maintenance and the noise while operating.
Despite of all its advantages, fuel cell systems have a high cost which is a major limitation.
This limitation should be improved in the future with other additional improvements in their
technologies.
1.4. TYPES OF FUEL CELLS
Fuel cells are divided into different types depending on the kind of electrolyte they employ.
Table 1 reviews some differences between some types of fuel cells. Usually, the type of fuel cell is
chosen according to the application that matches with it.
3
Table 1-1.1: Different types of Fuel Cells [3]
Fuel Cell Type Solid
Oxide Molten
(SOFC)
Carbonate
(MCFC)
Alkaline
(AFC)
Phosphoric
Direct
Acid (PAFC) Methanol
(DMFC)
𝑂2−
700-1000
𝐶𝑂3 2−
500-700
𝑂𝐻 −
100-250
𝐻+
150-250
𝐶𝐻3 𝑂−
60-130
Hydrocarbons
𝐶𝑂
2-300 Kw
Hydrocarbons
𝐶𝑂
75-250 kW
𝐻2
𝐻2
Methanol
25W250Kw
Molten
Phosphoric
Acid
55
1-50 W
Mobile Ion
Operating
Temperature
(℃)
Fuel
Power
Electrolyte
Conducting
Ceramic Oxide
Electrical
Efficiency %
(Cell)
Electrical
Efficiency %
(System)
Start-up Time
Application
60-65
Around 10
kW
Molten Alkaline Aqueous
Carbonate
Alkaline
Solution
55
60-70
55-60
45-55
62
Hours
Medium- Large
Scale
Power
Generation
Hours
Medium- Large
Scale
Power
Generation
Minutes
Aerospace
and
Underwater
applications
4
Proton
Exchange
Membrane
(PEMFC)
(𝐻2 𝑂)𝑛 𝐻 +
70-110
Polymer
Membrane
𝐻2
𝐶𝐻3 𝑂𝐻
50 W –
150 kW
Polymer
Membrane
20-30
50-70
40
10-25
30-50
Hours
MediumLarge Scale
Power
Generation
Sec-Min
Small
Mobile
Power
Sec-Min
Vehicles
CHAPTER 2. LITTERATURE REVIEW
2.1. HISTORY [3]
In the beginning of electrochemical studies, scientists faced difficulties in the improvement
of their overall efficiencies due to difficulties in the conversion of energy. Therefore, designing the
porous electrodes, and choosing their suitable electrolytes, and the selection of proper materials were
needed to be improved. At the beginning, the development of fuels cells included four types, and they
were categorized in terms of electrolyte, the “gaseous voltaic battery” that used aqueous acid, the
“direct coal” that was tested with different electrolytes, alkaline, carbonate and solid oxide electrolytes.
Additionally, the three types of fuel cells that have been used in automobile thrust systems are direct
methanol, phosphoric acid, and the polymer exchange membrane fuel cells.
2.1.1. Gaseous Voltaic Battery:
Fuel cell was previously defined as “Gaseous Voltaic Battery” that was designed by William.
R Grove in 1839, he performed his first experiment in Swansea, Wales. His experiment was done as
follows, two platinum electrodes were submerged midway in an aqueous sulfuric acid beaker, and the
tubes were inverted above the two electrodes, one of the tubes contained hydrogen and the other
contained oxygen. Lately, Grove recognized that the reaction was reliant on “Surface of action”. In
1942, Grove progressed further experiments were he used a platinized platinum electrodes, with 26
cells installed in series to increase the surface of action and electrolyze water by the products of
electrolysis- Oxygen and Hydrogen. More experiments were done by Grove were he replaced different
gases in the tubes to detect its effects, he tried several combinations of gases on both electrodes and
established that oxygen and chlorine served to the first electrode, and carbon monoxide and hydrogen
served to the second one. In 1945, Grove presented the results of his experiments to the Royal Society
5
of London about the gas voltaic battery as a testing instrument of vaporization. Finally in 1954, as a
farther application Grove presented the gas battery as an electric source from conventional fuels.
Christian Friedrich Schoebein read about the experiments of Grove and decided to do his own
experiments that was published in 1941 and 1942. Schoebein proved that it was not “mere contact”
but it was “chemical action” that lead to generation of current, the mixture of hydrogen and oxygen
that contained dissolved in water caused the current.
In 1882, Lord Rayleigh established a new form of gas battery, he increased the surface of
action between the solid electrode, the gas, and the liquid in order to increase the efficiency of the
platinum electrode. Later, coal gas was used as a fuel by Rayleigh, leading to an inferior but significant
production of current by the gas battery.
In 1889, Carl Langer and Ludwig Mond developed an updated version of gas battery that was
seen as a great progress, it was the model for the applied fuel cell.
After Alder Wright and Thompson saw the demonstration of Mond and Langer, they
reestablished a new device in 1887, this device’s electrodes were called “aeration plates”. Wright and
Thompson established that platinum black gas has the greatest voltage after testing several groupings
of materials for the aeration plate.
2.1.2. Electricity Direct from Coal
In the past, the conversion of chemical energy into mechanical energy was about 10% in
steam engine. In 1894, Ostwald recommended a new way to overcome the lack in efficiencies of
energy conversion process in the electrochemical systems. He aimed to generate electricity from coal
by applying the processes of electrochemical, but the creation of the galvanic item was unclear.
Ostwald referred to tests that was done Jablochkoff were he used Potassium nitrate to generate
electricity from coal directly.
6
In 1896, Jacques made fuel cells that generate electricity from coal, but Bruner and Haber
found that the electrochemical reaction have first been between the coal and the electrolyte and then
with the electrode.
Later in 1912, Baur and Ehrenberg attempt to test different types of electrolytes, that includes
Borate, Silicate, Carbonate and Hydroxide. In 1935, Baur and Ehrenberg used an alkali metal and
Carbonates mixture, and they noticed that Carbon Dioxide fed to the cathode enhanced the fuel cell
performance.
Finally in 1937, Baur and Brunner began investigating ceramic materials after they
determined that the solid electrolyte was more appropriate.
2.1.3. The Solid Oxide Fuel Cell
The solid Oxide Fuel Cell was established by Baur and Preis in 1937, when they recognized
the necessity of a more convenient electrolyte in comparison with the molten electrolytes. Later on,
the solid compound created by Wilhelm Nernst was used. Nernst reviewed a solid conductor at a high
temperature. Later, Nernst was demanding to design, using the solid electrolytes, an electrical lamp
and found that the mixed oxide conductivities are higher at higher temperatures.
In 1900, Nernst and Wild organized from the oxides of zirconium and thorium and another
rare elements of earth an electrolytic glow bodies, that emitted pure white light, and they noticed that
the pins began emitting light between 500 and 700 𝑜𝐶 depending on their composition.
In 1962, Weissbart and Ruka created a fuel cell that used 85% ZrO2 and 15% CaO as the
electrolyte and the electrodes was porous platinum. The area of the cell was 2.5 𝑐𝑚2 and its thickness
was 0.15 cm. The flow over the cathode was pure Oxygen and over the anode was Hydrogen or
methane.
Further experiments were done on the fuel cell by Wessbart and Ruka that lead to determine
that more steam reforming was associated with slower flow rates, and that the open circuit potential
7
increase, imitating that the reactions occurring at the anode involved the 𝐻2 and CO from the reforming
reaction rather than 𝐶𝐻4 itself.
Figure 2-1: Diagram of final form of the solid oxide fuel cell [3]
2.1.4. Molten Carbonate Fuel Cell:
Among the many mixes utilized amid the improvement of the "direct coal" fuel cell was the
Alkali metal carbonates, however they rose up out of alternate salts in light of their similarity with the
results of the oxidized fuel.
In 1943, Davtyan assigned to build up a high temperature cell that would work at 700℃ with
a solid ionic conductor as electrolyte to accomplish the target of utilizing coal gas as fuel. Davtyan
blended many mixes to build up an electrolyte that comprise of monazite sand, sodium carbonate,
tungsten trioxide, and pop glass, the upside of this mix was the enhancement of conductivity and
mechanical quality of the electrolyte.
Later in the 1960s, Broers and Ketelaar referred to the work of Davtyan and developed an
electrolyte mixture, they concluded that the electrolyte had a liquid phase and it was not completely
8
solid. Therefore, Broers and Ketelaar used carbonates as electrolyte that lead to the elimination of
decomposition of 𝐶𝑂2, furthermore, they could diminish the concentrated polarization by adding 𝐶𝑂2
to the cathode. The fuel cell configuration had the molten carbonate electrolyte held in a matrix.
At the beginning of 1960s, the Institute of Gas Technology begun to take a shot at molten
carbonate fuel cells, and understood that the porosity of the fiber nickel electrodes builds the fuel cell
execution. In the meantime, the General Electric Company utilized a porous electrode to contain the
electrolyte as opposed to utilizing a matrix, and they anticipated that a molten carbonate fuel cell with
free electrolyte and porous gas diffusion terminals would have the capacity to get higher current
densities than the matrix sort, in light of the fact that the electrode spacing could be decreased.
2.1.5. Alkaline Fuel Cell:
Alkaline were seen as an unsuitable electrolyte since the chemical reaction would lead to their
degradation. Later, it was discovered that alkaline electrolyte would be very useful if the fuel was
hydrogen, plus the use of alkaline electrolyte could decrease the risk of corrosion on the electrodes.
In 1946, Davtyan tried a low temperature fuel cell that utilized an alkaline electrolyte that
worked at atmospheric pressure and temperature, the electrolyte was an aqueous solution of potassium
hydroxide and the electrode was layered with paraffin to make it waterproof.
In 1932, Francis Tom Bacon empowered by the idea of William Grove that the electrolysis
of water could be reversible and suspected that in the event that it was reversible, the energy
transformation would be more efficient than that of the Carnot cycle, so he began to develop a cell. He
built up an electrode with layers of two diverse pore sizes.
In 1961, Justi and Winsel selected pore sizes in the DSK electrodes, then in 1965 Alford and
Niedrach attempted to increase the performance of the cell by using Teflon.
The orbiter program selected the alkaline fuel cell but was reconstructed with more active
catalysts in order to decrease the pressure and temperature. Platinum alloyed with gold was used for
anode.
9
2.1.6. The Direct Methanol Fuel Cell:
In the 1960s, the idea for a direct fuel cell raised over again, but the fuel was methanol instead
of coal, as methanol could be electrochemically oxidized at the electrodes.
Direct methanol fuel cells were created by researchers at shell (Williams et al) and ESSO
(Tarmy and Ciprios) in 1965. Williams intended to choose acid over alkaline as electrolyte since it was
not affected by the carbon dioxide formed in the methanol oxidation reaction. In 1963, a fuel cell
prototype was assembled that produced 3.15 W at 1 A. Later, they used a sulfuric acid electrolyte with
1M methanol diversified and dispersed with electrolyte. A fuel cell stack with 40 cells was made that
produced 300W at 12V and 60 ℃. Later, Murray and Grimes established a methanol fuel cell with
alkaline electrolyte.
In 1965, Binder experienced noble metals and their alloys as half-cell electrodes in alkaline
and acidic electrolytes.
In 1981, McNichol audited the methanol oxidation component on platinum in acid, and he
presumed that keeping in mind the end goal to enhance the execution of the catalyst, it ought to have
enhanced the resistance to poisoning by the dehydrogenized particle or have enhanced oxidation ability
of the atom.
At long last, in 1992, inquire about was renewed after the technical enhancements of the solid
polymer fuel cell, the researchers at the Jet Propulsion Laboratory built up an direct methanol fuel cell
utilizing a similar solid polymer electrolyte.
2.1.7. The Phosphoric Acid Fuel Cell:
The phosphoric acid fuel cell was created to utilize natural gas, however with the fuel first
chemically changed to deliver hydrogen. In order to decrease the amount of carbon monoxide
produced, the fuel cell temperature was increased.
10
In 1967, Pratt and Whitney Aircraft Division of the United Technologies Corporation built
up the phosphoric acid fuel cell. The main plan was worked on hydrogen made from natural gas,
propane and light distillate liquid fuels by the steam reforming reaction. In 1975 the development of
phosphoric fuel cells met its objective of exhibiting the innovation as electrical power frameworks for
homes provided with natural gas.
As carbon was observed to be stable in fuel cells, carbon was utilized as a part of fuel cells
because of its monetary attainability. The utilization of carbon in the fuel cell as catalyst support
diminished the measure of platinum required for the electrodes that diminished the cost of the fuel
cells to worthy levels.
2.1.8. The Solid Polymer Fuel Cell:
In 1960, General Electric (Grubb and Niedrach) built up the Solid Polymer Fuel Cell. They
designed a fuel cell with a solid exchange membrane electrolyte, where the ion exchange layer was a
polymer sheet.
The performance and permanency of the electrolyte was then improved by Grot using a new
polymer formulation in 1972.
Later in 1986, the electrode fabrication was improved by Raistrick, and it was observed as
the type that could be made useful for automobiles, but the problem was it is susceptible to poisoning
by small amounts of carbon monoxide in the fuel stream as a byproduct of the fuel cell reforming
reaction.
In 1988, Gottesfeld and Pafford stopped the poisoning difficult by producing an “oxidative
surface environment” by inserting 𝑂2 in the fuel stream.
11
Figure 2-2: The development of different types of fuel cells [3]
2.2. POLYMER EXCHANGE MEMBRANE AND ITS REACTION
Polymer Exchange Membrane Fuel Cell (PEMFC) is a type of fuel cells that uses hydrogen
gas 𝐻2 as a fuel. Hydrogen is oxidized at the negatively charged electrode (anode), resulting in free
electrons that flow through a conductive material to the load. Oxygen gas 𝑂2 is also fed at the
positively charged electrode (cathode) in order to react with the hydrogen ions and electrons. This
exothermic reaction that takes place at the cathode-side, forms water and produces heat as shown in
the following figure.
12
Figure 2-3: Exploded View of a PEMFC Function[4]
A PEMFC has the following chemical reaction:
Anode:
Cathode:
Overall:
𝐻2 (𝑔) → 2𝐻 + (𝑎𝑞) + 2𝑒 −
1⁄ 𝑂 (𝑔) + 2𝐻 + (𝑎𝑞) + 2𝑒 − → 𝐻 𝑂 (𝑙)
2
2 2
1
𝐻2 (𝑔) + ⁄2 𝑂2 (𝑔) → 𝐻2 𝑂(𝑙) + 𝑒𝑙𝑒𝑐𝑡𝑟𝑖𝑐 𝑒𝑛𝑒𝑟𝑔𝑦 + ℎ𝑒𝑎𝑡
Reactants are transported throughout the fuel cell by diffusion and convection to the cathode, where
the electrochemical reaction takes place.
2.3. PEMFC COMPONENTS
Polymer Exchange membrane fuel cell is made of five basic layers which are, proton exchange
membrane, the anode and cathode catalyst layers, and the gas diffusion layers of the anode and the
cathode. Moreover, it has some secondary components such as the end plates, contacts and gasket
materials. Each layer of the fuel cell needs a unique energy balance since they have unlike thermal
resistance, material and phase.
13
Figure 2-4: Schematic diagram of the main
components of PEMFC [5]
Proton Exchange membrane: It is also known as electrolyte membrane, it enables the positively
charged hydrogen ions to move from the anode to the cathode. The most common type of proton
exchange membrane is the Nafion membrane 112,115 and 117.
Catalyst Layers: It is made of Platinum/Carbon catalyst. Platinum is commonly used since it is
the most efficient catalyzer for the oxidation process of hydrogen and oxygen. Therefore, these layers
breaks the fuel (Hydrogen) into positively charged ions and electrons. The positive ions merge with
the oxidant to form water at the fuel cell cathode, while the electrons move to the load through an
electrical circuit to form current.
Gas Diffusion Layers: Most common used material of Gas Diffusion Layers is Carbon cloth
or Toray paper. These layers allow the fuel/oxidant to travel through the porous layer while collecting
electrons.
14
Flow Field Plates: These plates are mostly made of Graphite and Stainless Steel. They allow
the flow of hydrogen fuel to the cathode and oxygen to the anode, plus they distribute the fuel and the
oxidant to the gas diffusion layer.
Gaskets: It is made of Silicon and Teflon. Gaskets prevent fuel leakage and it helps to distribute
pressure evenly.
End Plates: These plates are made of Stainless steel, graphite, polyethylene and PVC, they are
used to hold the stack layers in place.
2.4. PEMFC PREVIOUS STUDIES
Since the first Polymer Exchange Membrane Fuel Cell was invented, studies and researches
about them started to take place till date. These studies were divided into different fields in order to
reach maximum efficiency and reliability with minimum cost. The main researches were pointed on
chemical, thermal and electrical studies. Chemical researchers focused on the type, material and
dimensions of the layers, catalysts and the exchange membrane to be used. Also, a good amount of
work was spent on Mass Transfer models throughout the fuel cell. Concerning the thermal part, studies
were conducted to model or to control the temperature variation in a fuel cell in order to achieve a
better efficiency analysis and an increase in the efficiency. The increase in efficiency was also
approached by using the heat waste from the exothermic reaction, by different studies about heat
recovery. Electrical researchers studied and modeled the power density output and focused their
researches on increasing the electrical efficiency of a single fuel and of a stack of fuel cells. Also the
flow fields of gases and their effect on efficiencies took a good amount of researches.
In 1979 R. S. Yeo and J. McBreen [6] considered electrochemical aspects of fuel cells
operation. They developed a proton diffusivity as a function of temperature by using Arrhenius
equation and performing experiments. Yeo and McBreen used hydrochloric acid as source of the
reactants and as a solvent. The results of their experiments lead to equation governing the proton
diffusivity that is being used in FLUENT fuel cell model.
15
In 1998, X. Din and E. E. Michaelis [7] were able to simulate the transport of protons and
water through the Proton Exchange Membrane Fuel Cell. Some assumptions were taken into
consideration in their research, that the membrane is a band of parallel pores which water and protons
pass. Also, they treated water and protons molecules as particles and that their motion were driven by
electrostatic interaction with pore walls.
In 1998, Lee and Lalk [8] focused on modeling a full stack of a variable number of cells after
characterizing experimentally a model of single fuel cell. The model offers an empirical polarization
curve model that simplifies the electrochemical reaction, and heat, mass and charge transport. This
simplification took place as it allows focusing on the whole stack rather than becoming lost in the
complexity of these reactions in a single cell. In this model, Lee and Lalk assumed that any chemical
energy that is not converted into electrical is in the form of heat, in order to approximate the amount
of heat rejection.
An investigation, in 2001, was conducted about the transient effects of membrane hydration
on a single cell’s performance through a mathematical model by Rowe and Li [9]. The model was a
one-dimensional model directed orthogonally through the plane of the polymer electrolyte. An
assumption of constant temperature in the entire fuel cell was taken. From this model, it was concluded
that the dynamic temperature of a fuel cell is affected by the phase change of water. It was also deduced
that the pre-humidification of the cathode air can be decreased at high loads if the anode gas
humidification is sufficiently high. However, this conclusion might be very useful in large automobile
stacks as they do typically humidify the anode gas.
In 2004, a comprehensive electrochemical model for a single cell was given by Baschuck and
Li [10]. The complete electrochemical reaction equations, proton movement equations, and detailed
mass transport equations for the gaseous products and reactants and the liquid product water are
16
negotiated by this model. The mentioned equations are practical to each section of the cell (The bipolar plates, gas flow channels, electrode backing, catalyst, and polymer electrolyte layers).
In 2008, a research was conducted by Colleen Spiegel [11] in order to develop a mathematical
model to contribute a better understanding of fuel cell mass and heat transfer phenomena. Spiegel
considered a two phase transient model, and considered that each layer is a control volume one. Each
layer was divided into number of nodes, each node has a specific transient heat and mass transfer
equations. Catalyst layers were modeled using the Butler Volume equation and the porous equations,
and the membrane was modeled using Fick’s law of diffusion. Moreover, a mathematical model was
done by Spiegel to predict the bolt torque. A simulation for this model was done to analyze the water
transport across the membrane, the pressure variation along the channel, the water phase change effect
and the energy balance. Furthermore, this simulation can be used to estimate the characteristics of the
flows inside the channel and analyze the factors that affect the performance of the fuel cell. At the end
of the research, Spiegel suggested that it is important to consider a 2-D or 3-D heat transfer to obtain
better results, and that it would be more accurate if the velocity of the mixture was calculated separately
for the gas and liquid phases.
In the same year, a research was submitted by Etim S. Udoetok [12] to Louisiana State
University, the research covered two main topics, optimal design of fuel cells and the cytometers flow,
and he used fluent computational Fluid Dynamics code in his research to simulate the results. Udoetok
designed a new fuel cell flow field that has a high mixing effect on the gas flow through the channel,
the design includes a specification about the most efficient path length. Moreover, by applying mass
conservation he designed a new model for the transport of protons in Proton Exchange Membrane Fuel
Cell. Udoetok concluded that the high pressure drop across the serpentine flow field is efficient and
that it increases the performance of the fuel cell.
In 2014, a study was published by Leeds University, Faculty of Engineering [13]. This study
conducted a simple and efficient mathematical model about air-breathing Proton Exchange Membrane
17
Fuel cell. The main objective of this study was to study the effects of joules and entropic heat sources
that is usually neglected. They concluded that the performance of fuel cell is over-predicted when one
or both of these sources is neglected, and that the performance of fuel cell is highly affected by the
state of water at the thermodynamics equilibrium. Moreover, the study lead to the deduction that at
low cell potentials, Low ambient Temperature and high humidity is more efficient to prevent
membrane dehydration.
In 2015, it was recommended to supplant the systems of straight channels by geometries
creating chaotic streams keeping in mind the end goal to enhance the performance of heat exchangers
utilized as a part of the bi-polar plate of a PEMFC. An experimental examination was led for both
chaotic geometry and a straight tube utilizing a particular test bench that was designed and intended
for testing. [14].
18
CHAPTER 3. MATHEMATICAL MODEL
3.1. FUEL CELL ENERGY BALANCES
In order to obtain an accurate model of a fuel cell, an energy balance must be done in each of
the subsystems in the Fuel cell and in the fuel cell itself in order to achieve the overall energy
requirements for the processes. Energy balance calculations determines the exit cell temperature of the
fuel cell knowing the temperatures of each layer, the reactant composition, H2 and O2 utilization,
percent of heat loss and the expected power output.
Consider a fuel enters the cell at a certain temperature T and pressure P, Oxygen enters at a
certain temperature, pressure, mass flow rate and mole fraction, they react completely to produce water
which exits at a temperature T, pressure P and at a certain mole fraction and mass flow rate. This
reaction can be described by:
1
𝐻2 + 2 𝑂2 → 𝐻2𝑂
The energy balance equation for this reaction can be written as:
𝑊
𝑚𝐻2
𝑄
+ 𝑚
𝐻2
1
= ℎ𝐻2+ 2 ℎ𝑂2 − ℎ𝐻2𝑂
(3.1)
The general heat balance on the fuel cell stack can be written as:
∑ 𝑄𝑖𝑛 − ∑ 𝑄𝑜𝑢𝑡 = 𝑊𝑒𝑙 + 𝑄𝑑𝑖𝑠 + 𝑄𝑐
(3.2)
Where 𝑄𝑖𝑛 and 𝑄𝑜𝑢𝑡 are the heats of the reactant gases and the heat produced in the product,
𝑊𝑒𝑙 is the generated electricity, 𝑄𝑑𝑖𝑠 is the dissipated heat to the surroundings and 𝑄𝑐 is the heat taken
from the cell by active cooling.
19
By equating the heat and electricity generated with the energy of the fuel reacted we can
obtain a good estimation for the fuel cell energy balance, which is:
𝐼
2𝐹
𝐻𝐻𝐻𝑉 𝑛𝑐𝑒𝑙𝑙 = 𝑄𝑔𝑒𝑛 + 𝐼𝑉𝑐𝑒𝑙𝑙 𝑛𝑐𝑒𝑙𝑙
(3.3)
Where, 𝑛𝑐𝑒𝑙𝑙 is the number of cells, 𝑉𝑐𝑒𝑙𝑙 is the voltage of the cell, and 𝑄𝑔𝑒𝑛 is the heat
generated from the stack. 𝑄𝑔𝑒𝑛 Can be obtained by:
𝑄𝑔𝑒𝑛= (1.482 − 𝑉𝑐𝑒𝑙𝑙 )𝐼𝑛𝑐𝑒𝑙𝑙
(3.4)
𝑄𝑔𝑒𝑛 = (1.254 − 𝑉𝑐𝑒𝑙𝑙 )𝐼𝑛𝑐𝑒𝑙𝑙
(3.5)
Where equation (3.4) can be used when all the produced water leaves the stack as liquid, and
equation (3.5) can be used when all the produced water leaves the stack as vapor.
3.2. GENERAL ENERGY BALANCE FOR FUEL CELL [15]
Another way to obtain the fuel cell energy equation is by summing all the energy inputs and
outputs, where the inputs are the enthalpies of the fuel, the oxidant and the water vapor present in the
reaction, and the outputs are the enthalpies of the flows out of the fuel cell, the electric power produced
and the heat leaving the fuel cell by convection, radiation or through coolant. The general energy
balance equation can be written as:
∑(ℎ𝑖)𝑖𝑛 = 𝑊𝑒𝑙 + ∑(ℎ𝑖)𝑜𝑢𝑡 + 𝑄
(3.6)
The enthalpy for each mixture can be obtained by the equation:
ℎ = 𝑚̇𝑐𝑝 𝑇
(3.7)
Where 𝐶𝑝 is the specific heat (J/ (g*K), 𝑚̇ is the mass flow rate (g/s), and T is the temperature
in℃.
For the gases having high heating value, its enthalpy can be written as:
0
ℎ = 𝑚̇(𝐶𝑝 𝑇 + ℎ𝐻𝐻𝑉
)
0
Where ℎ𝐻𝐻𝑉
is the higher heating value of the gas at 0℃.
20
(3.8)
Equation (3.8) can be used to obtain the enthalpy at 0℃, in some cases the higher heating
value is needed to be calculated at a chosen temperature.
Another parameters needed to be calculated is the enthalpy of the water vapor and liquid
water. The enthalpy of the water vapor is:
0
ℎ = 𝑚̇𝐻2𝑂(𝑔) 𝐶𝑝𝐻2𝑂(𝑔) 𝑇 + ℎ𝑓𝑔
(3.9)
The enthalpy of the liquid water is:
ℎ = 𝑚̇𝐻2𝑂(𝑙) 𝐶𝑝𝐻2𝑂(𝑙) 𝑇
(3.10)
3.3. LAYERS ENERGY BALANCE
In order to obtain an accurate heat transfer solutions, several numerical techniques can be
used such as finite-element, finite-difference, and boundary-elements and nodal network methods. In
this research we will use the nodal network method in order to determine the temperature at separated
points. This method is accomplished by diving the region or layer into small regions and a center point
must be assigned.
Figure 3-1: Two-dimensional node network
21
Figure 3-2: Distributed nodes throughout computational domain
In order to obtain the location of each node (𝑥𝑖 ), from Figure 3.1 and Figure 3.2:
(𝑖−1)
𝑥𝑖 = (𝑁−1) 𝐿
𝑓𝑜𝑟 𝑖 = 1: 𝑁
(3.11)
Where N is the number of Nodes used for solving.
The distance between adjacent nodes is:
𝐿
∆𝑥 = 𝑁−1
(3.12)
Each node is considered control volume that has a conductive heat transfer with each adjacent
node plus the energy storage, therefore we can obtain that [15]:
𝑑𝑈
𝑞̇ 𝐿𝐻𝑆 + 𝑞̇ 𝑅𝐻𝑆 = 𝑑𝑡
(3.13)
Where:
𝑞̇ 𝐿𝐻𝑆 =
𝑞̇ 𝐿𝐻𝑆 =
𝑘𝐴(𝑇𝑖−1 − 𝑇𝑖 )
(3.14)
∆𝑥
𝑘𝐴(𝑇𝑖+1 − 𝑇𝑖 )
(3.15)
∆𝑥
Where A is the area of the plate.
22
The rate of energy storage part can be calculated by multiplying the time rate of change for
the nodal temperature and the control volume thermal mass which leads to:
𝑑𝑈
𝑑𝑡
𝑑𝑇
= 𝐴∆𝑥𝜌𝑐 𝑑𝑡𝑖
(3.16)
By substituting equations (3.14), (3.15) and (3.16) in equation (3.13):
𝑑𝑇
𝐴∆𝑥𝜌𝑐 𝑑𝑡𝑖 =
𝑘𝐴(𝑇𝑖−1 − 𝑇𝑖 )
∆𝑥
+
𝑘𝐴(𝑇𝑖+1 − 𝑇𝑖 )
∆𝑥
(3.17)
The time rate of the temperature change equation can be written as:
𝑑𝑇𝑖
𝑘
= ∆𝑥 2 𝜌𝑐 (𝑇𝑖−1 + 𝑇𝑖+1 − 2𝑇𝑖 )
𝑑𝑡
𝑓𝑜𝑟 𝑖 = 2: (𝑁 − 1)
(3.18)
For the edges, the control volumes have a smaller volume and there occurs a different energy
transfers. Therefore the energy balance for the edge node can be written as:
𝑑𝑈
𝑑𝑡
= 𝑞̇ 𝐿𝐻𝑆 + 𝑞̇ 𝑐𝑜𝑛𝑣
(3.19)
Which leads to:
𝐴∆𝑥𝜌𝑐 𝑑𝑇𝑁
2
𝑑𝑡
=
𝑘𝐴(𝑇𝑁−1 − 𝑇𝑁 )
∆𝑥
+ ℎ𝐴(𝑇𝑓 − 𝑇𝑁 )
(3.20)
The time rate or temperature for this case can be written as:
𝑑𝑇𝑁
2𝑘
2ℎ
= 𝜌𝑐∆𝑥 2 (𝑇𝑁−1 − 𝑇𝑁 ) + ∆𝑥𝜌𝑐 (𝑇𝑓 − 𝑇𝑁 )
𝑑𝑡
(3.21)
Therefore, we can see that the temperature in each node is function of position (x) and time
(t). The factor i is the index that specifies the position of the node, where i=1 for the adiabatic plate
and i=N corresponds to the surface of the plate, and j is another index that is added in order to indicate
the temperature ((𝑇𝑖,𝑗 ), where j=1 related to the beginning of the simulation and j=M to the end of the
simulation. Where the total simulation time is:
𝜏
𝑠𝑖𝑚
∆𝑡 = (𝑀−1)
(3.22)
And the time associated 𝑡𝑗 is:
𝑡𝑗 = (𝑗 − 1)∆𝑡
𝑓𝑜𝑟 𝑗 = 1: 𝑀
23
(3.23)
3.3.1. Transient conduction through End plate [16]:
To obtain the temperature in the end plate, the equations of section 3.3 will be used.
Some assumptions is needed to obtain the temperature changes in the plate.
Assumptions taken:
Polymer type end plate
L = 0.01,
Plate thickness in m
K=0.2,
Conductivity (W/m-K)
𝜌= 2000,
Density (kg/m^3)
c=17,
Air heat transfer Coefficient (W/m^2-K)
𝐶𝑝 =200,
Specific heat capacity (J/kg-K)
𝑇𝑖𝑛 =343.15,
Initial Temperature (K)
𝑇𝑓 =298,
Gas Temperature (K)
h=17,
Heat transfer Coefficient (W/m^2-K)
N=10
Number of Nodes
A simulation will be done using MATLAB (refer to Appendix A.1) applying the equations
of section 3.3 at two different times, t1= 10 seconds and t2= 2 minutes.
At t=10seconds:
Figure 3-3: Transient heat conduction in a polymer end plate at t=10 seconds
24
At t=120 seconds:
Figure 3-4: Transient heat conduction in a polymer end plate at t=120 seconds
From figures 3.3 and 3.4 we can see that a small amount of heat has been transferred to the
plate for the polymer membrane.
Another simulation will be done on Aluminum End plate with a conductivity of 220 W/m-K,
a density of 2700 kg/m^3 and specific heat capacity of 900 J/kg-K. Simulation will be done on the
same time assumed before.
Figure 3-5: Transient heat conduction through Aluminum end plate at t=60 and 120 seconds
25
From Figure 3.5 it can be concluded that the temperature distribution for the aluminum end
plate is exactly the same at 60 and 120 seconds.
We can conclude from the three previous figures that the heat diffuses rapidly in the
aluminum end plate comparing it with the polymer end plate.
3.3.2. End plate, Contact and GDL layers energy balance:
Since there is no gas or liquid flows in this plates, conduction is the only heat transfer
phenomena occurs in these layers, were one side of any of these layers is exposed to the environment
or to an insulating material and the other side to a conductive current collector plate or insulating
material.
Figure 3-6: Energy balance around End plate, contact and GDL layers.
The general energy equation for these layers using figure 3.6 can be written as :
(𝜌𝐿𝑎𝑦𝑒𝑟2 𝐴𝐿𝑎𝑦𝑒𝑟2 𝑡𝐿𝑎𝑦𝑒𝑟2 𝑐𝑝𝐿𝑎𝑦𝑒𝑟2)
𝑑𝑇𝐿𝑎𝑦𝑒𝑟2
𝑑𝑡
= 𝑄𝐿𝑎𝑦𝑒𝑟1 + 𝑄𝐿𝑎𝑦𝑒𝑟3
(3.24)
Where 𝑄𝐿𝑎𝑦𝑒𝑟1 and 𝑄𝐿𝑎𝑦𝑒𝑟3 are the heat flows from layer1 and layer3.
The heat flow from Layer1 to Layer2 is:
𝑄𝐿𝑎𝑦𝑒𝑟1 = 𝑈𝐿𝑎𝑦𝑒𝑟1 𝐴(𝑇𝐿𝑎𝑦𝑒𝑟1 − 𝑇𝐿𝑎𝑦𝑒𝑟2 )
26
(3.25)
Where 𝑈𝐿𝑎𝑦𝑒𝑟1 is the overall heat transfer coefficient of Layer1, A is the layer area and T is
the layer temperature.
The heat flow from layer3 to layer2 is:
𝑄𝐿𝑎𝑦𝑒𝑟3 = 𝑈𝐿𝑎𝑦𝑒𝑟3 𝐴(𝑇𝐿𝑎𝑦𝑒𝑟3 − 𝑇𝐿𝑎𝑦𝑒𝑟2 )
(3.26)
The overall heat transfer coefficient coming from the surroundings is:
1
𝑈𝑠𝑢𝑟𝑟 = 𝑡𝐿𝑎𝑦𝑒𝑟2
(3.27)
1
+
𝑘𝐿𝑎𝑦𝑒𝑟2 ℎ𝑠𝑢𝑟𝑟
Where 𝑡𝐿𝑎𝑦𝑒𝑟2, 𝑘𝐿𝑎𝑦𝑒𝑟2 are the thickness and thermal conductivity of Layer2 and ℎ𝑠𝑢𝑟𝑟 is the
convective loss from the cell to surroundings.
The overall heat coefficient of the coming heat from Layer1 and Layer3 are:
1
𝑈𝐿𝑎𝑦𝑒𝑟1 = 𝑡𝐿𝑎𝑦𝑒𝑟2 𝑡𝐿𝑎𝑦𝑒𝑟1
(3.28)
+
𝑘𝐿𝑎𝑦𝑒𝑟2 𝑘𝐿𝑎𝑦𝑒𝑟1
1
𝑈𝐿𝑎𝑦𝑒𝑟3 = 𝑡𝐿𝑎𝑦𝑒𝑟3 𝑡𝐿𝑎𝑦𝑒𝑟2
(3.29)
+
𝑘𝐿𝑎𝑦𝑒𝑟3 𝑘𝐿𝑎𝑦𝑒𝑟2
Moreover, if the layer conducts electricity then these is additional heat generation such as
Layer2 were there is an electrical resistance, this generation can be calculated as:
𝑄𝑟𝑒𝑠_𝐿𝑎𝑦𝑒𝑟2 = (𝑖𝐴)2
𝜌𝑟𝑒𝑠_𝑙𝑎𝑦𝑒𝑟2 𝑡𝐿𝑎𝑦𝑒𝑟2
𝐴
(3.30)
Where i and A are the current density and area of the layer, 𝜌𝐿𝑎𝑦𝑒𝑟2 and 𝑡𝐿𝑎𝑦𝑒𝑟2 are the
specific resistance of the material and the thickness of the layer.
Naturally in the end plate, contact and gasket layers these is no heat generated, although in
some fuel cell designs, the end plate may be heated, so an additional heat generated must be added for
the model formulation.
27
3.3.3. Bipolar Flow Field plate energy balance [17]:
Due to the gas channels in the bipolar plate, therefore it has both conductive and convective
heat transfer.
Figure 3-7: Bipolar flow field plate energy balance
The heat transfer equation for the bipolar plate from Figure 3.7 is:
(𝑐𝑝𝑎𝑣𝑔 (𝑛𝑔𝑎𝑠𝑒𝑠 + 𝑛𝑙𝑖𝑞 ) + 𝜌𝐿𝑎𝑦𝑒𝑟2 (𝐴𝐿𝑎𝑦𝑒𝑟2 𝑡𝐿𝑎𝑦𝑒𝑟2 )𝑐𝑝𝐿𝑎𝑦𝑒𝑟2 )
𝑑𝑇𝐿𝑎𝑦𝑒𝑟2
𝑑𝑡
= 𝑄𝐿𝑎𝑦𝑒𝑟1 +
𝑄𝐿𝑎𝑦𝑒𝑟1𝑅 + 𝑄𝐿𝑎𝑦𝑒𝑟3 + 𝑄𝐿𝑎𝑦𝑒𝑟3𝑅 + 𝑄𝑟𝑒𝑠_𝐿𝑎𝑦𝑒𝑟2 𝐻𝐻2_𝑖𝑛 + 𝐻𝐻2𝑂𝑣_𝑖𝑛 + 𝐻𝐻2𝑂𝑙_𝑖𝑛 − 𝐻𝐻2𝐿𝑎𝑦𝑒𝑟3 −
𝐻𝐻2𝑂𝑣𝐿𝑎𝑦𝑒𝑟3 − 𝐻𝐻2𝑜𝑢𝑡 − 𝐻𝐻2𝑂𝑣𝑜𝑢𝑡 − 𝐻𝐻2𝑂𝑙_𝑜𝑢𝑡
(3.31)
Where 𝑄𝐿𝑎𝑦𝑒𝑟1𝑅 and 𝑄𝐿𝑎𝑦𝑒𝑟3𝑅 are heat flows from Layers 1 and 2 to the solid material and H
is the enthalpy of components.
The heat flowing from the Layer1 and Layer3 to the channels is:
𝑄𝐿𝑎𝑦𝑒𝑟1 = 𝑈𝐿𝑎𝑦𝑒𝑟1 𝐴𝑣𝑜𝑖𝑑 (𝑇𝐿𝑎𝑦𝑒𝑟1 − 𝑇𝐿𝑎𝑦𝑒𝑟2 )
(3.32)
𝑄𝐿𝑎𝑦𝑒𝑟3 = 𝑈𝐿𝑎𝑦𝑒𝑟3 𝐴𝑣𝑜𝑖𝑑 (𝑇𝐿𝑎𝑦𝑒𝑟3 − 𝑇𝐿𝑎𝑦𝑒𝑟2 )
(3.33)
Where 𝐴𝑣𝑜𝑖𝑑 is the channels area.
The heat flowing from Layer1 and Layer3 (GDL) to the solid material is:
𝑄𝐿𝑎𝑦𝑒𝑟1𝑅 = 𝑈𝐿𝑎𝑦𝑒𝑟1𝑅 𝐴1𝑅 (𝑇𝐿𝑎𝑦𝑒𝑟1 − 𝑇𝐿𝑎𝑦𝑒𝑟2 )
(3.33)
𝑄𝐿𝑎𝑦𝑒𝑟3𝑅 = 𝑈𝐿𝑎𝑦𝑒𝑟3𝑅 𝐴1𝑅 (𝑇𝐿𝑎𝑦𝑒𝑟3 − 𝑇𝐿𝑎𝑦𝑒𝑟2 )
(3.34)
28
Where 𝐴1𝑅 is the area of the solid.
The heat generation in Layer2 due to electrical resistance is:
𝑄𝑟𝑒𝑠_𝐿𝑎𝑦𝑒𝑟2 = (𝑖𝐴)2
𝜌𝑟𝑒𝑠_𝐿𝑎𝑦𝑒𝑟2 𝑡𝐿𝑎𝑦𝑒𝑟2
𝐴1𝑅
(3.35)
Where i and A are the current density and area of the layer, 𝜌𝑟𝑒𝑠_𝐿𝑎𝑦𝑒𝑟2 and 𝑡𝐿𝑎𝑦𝑒𝑟2
Are specific resistance of the material and the layer thickness.
For the gas or liquid flow out or into the layer, there enthalpy can be defines as:
𝐻𝐴 = 𝑛𝐴 ℎ𝐴 𝑇𝐿𝑎𝑦𝑒𝑟2
(3.36)
Where 𝐻𝐴 is the enthalpy of the stream leaving or entering the layer, 𝑛𝐴 is the molar flow rate
and ℎ𝐴 is the enthalpy at the temperature of the layer.
The overall heat transfer coefficient terms can be defined as:
1
𝑈𝐿𝑎𝑦𝑒𝑟1𝑅 = 𝑡𝐿𝑎𝑦𝑒𝑟1 𝑡𝐿𝑎𝑦𝑒𝑟2
+
(3.37)
𝑘𝐿𝑎𝑦𝑒𝑟1 𝑘𝐿𝑎𝑦𝑒𝑟2
1
𝑈𝐿𝑎𝑦𝑒𝑟1 = 𝑡𝐿𝑎𝑦𝑒𝑟1
1
+
𝑘𝐿𝑎𝑦𝑒𝑟1 ℎ1
1
𝑈𝐿𝑎𝑦𝑒𝑟3𝑅 = 𝑡𝐿𝑎𝑦𝑒𝑟3 𝑡𝐿𝑎𝑦𝑒𝑟2
+
(3.38)
(3.39)
𝑘𝐿𝑎𝑦𝑒𝑟3 𝑘𝐿𝑎𝑦𝑒𝑟2
1
𝑈𝐿𝑎𝑦𝑒𝑟3 = 𝑡𝐿𝑎𝑦𝑒𝑟3
1
𝑘𝐿𝑎𝑦𝑒𝑟3 ℎ1
+
(3.40)
For the calculation of the thermal mass of the gas/liquid mixture:
𝑡ℎ𝑒𝑟𝑚𝑎𝑙𝑚𝑎𝑠𝑠 = 𝑐𝑝𝑎𝑣𝑔 (𝑛𝑔𝑎𝑠𝑒𝑠 + 𝑛𝑙𝑖𝑞 )
(3.41)
Where 𝑐𝑝𝑎𝑣𝑔 is the gases average specific heat at the temperature of Layer2.
For the calculation of the molar flow rate of the gases:
𝑃𝑉𝑔𝑎𝑠𝑒𝑠
𝑛𝑔𝑎𝑠𝑒𝑠 = 𝑅𝑇
(3.42)
𝐿𝑎𝑦𝑒𝑟2
And
𝑉𝑔𝑎𝑠𝑒𝑠 = 𝜖𝑉𝑣𝑜𝑖𝑑
(3.43)
𝑉𝑣𝑜𝑖𝑑 = 𝐴𝑣𝑜𝑖𝑑 𝑡𝐿𝑎𝑦𝑒𝑟2
(3.44)
29
Where 𝑉𝑔𝑎𝑠𝑒𝑠 is the volume of the gases in the channel and 𝑉𝑣𝑜𝑖𝑑 is the volume of the channel
space and 𝜖 is the void fraction.
3.3.4. Anode/Cathode Diffusion layer Energy balance:
The gas diffusion layer is located between the catalyst layer and the flow field plate, it allows
the liquids and gases to diffuse through it in order to reach the catalyst layer. It has a very low thermal
conductivity comparing it to the bipolar plates. Gas diffusion layer is divided into two portion, the
solid portion that has a conductive heat transfer and the gas/liquid flow that has convective heat
transfer. Heat generated in the GDL is mainly due to ohmic heating.
Figure 3-8: GDL energy balance
From Figure 3.8 we can obtain the overall energy balance equation for the anode GDL side,
that is:
(𝑐𝑝𝑎𝑣𝑔 (𝑛𝑔𝑎𝑠𝑒𝑠 + 𝑛𝑙𝑖𝑞 ) + 𝜌𝐿𝑎𝑦𝑒𝑟3 (𝐴𝐿𝑎𝑦𝑒𝑟3 𝑡𝐿𝑎𝑦𝑒𝑟3 )𝑐𝑝𝐿𝑎𝑦𝑒𝑟3 )
𝑑𝑇𝐿𝑎𝑦𝑒𝑟3
𝑑𝑡
= 𝑄𝐿𝑎𝑦𝑒𝑟4 +
𝑄𝐿𝑎𝑦𝑒𝑟2 + 𝑄𝐿𝑎𝑦𝑒𝑟2𝑅 + 𝑄𝑟𝑒𝑠_𝐿𝑎𝑦𝑒𝑟3 + 𝐻𝐻2_𝐿𝑎𝑦𝑒𝑟2 + 𝐻𝐻2𝑂𝑣_𝐿𝑎𝑦𝑒𝑟2 + 𝐻𝐻2𝑂𝑙_𝐿𝑎𝑦𝑒𝑟2 − 𝐻𝐻2𝐿𝑎𝑦𝑒𝑟3 −
𝐻𝐻2𝑂𝑣_𝐿𝑎𝑦𝑒𝑟3 − 𝐻𝐻2𝑂𝑙_𝐿𝑎𝑦𝑒𝑟3
(3.45)
30
3.3.5. Anode/Cathode Catalyst layer:
The anode and cathode catalyst layer are a porous layers made of platinum and carbon that is
located on both sides of the membrane. It is divided into two portion, solid portion that has a conductive
heat transfer and the gas/liquid portion that produce convective heat transfer.
Figure 3-9: Catalyst Layer energy balance
From Figure 3.9, the overall heat energy balance equation is:
(𝑐𝑝𝑎𝑣𝑔 (𝑛𝑔𝑎𝑠𝑒𝑠 + 𝑛𝑙𝑖𝑞 ) + (𝜌𝑎_𝑐𝑎𝑡 𝐴𝑎_𝑐𝑎𝑡 𝑡𝑎_𝑐𝑎𝑡 )𝑐𝑝𝑎_𝑐𝑎𝑡 )
𝑑𝑇𝑎_𝑐𝑎𝑡
𝑑𝑡
= 𝑄𝑎_𝑔𝑑𝑙 + 𝑄𝑚𝑒𝑚 +
𝑄𝑎_𝑐𝑎𝑡_𝑖𝑛𝑡 + 𝑄𝑟𝑒𝑠_𝑎_𝑐𝑎𝑡 + 𝐻𝐻2_3 + 𝐻𝐻2𝑂𝑣_3 + 𝐻𝐻2𝑂𝑙_3 − 𝐻𝐻2_4 − 𝐻𝐻2𝑂𝑣_4 − 𝐻𝐻2𝑂𝑙_4
(3.46)
Moreover, due to the electrochemical reaction and voltage over potential a heat generates in
the catalyst layer. This heat generation in the catalyst layer can be written as:
𝑖
𝑇𝐿𝑎𝑦𝑒𝑟4 ∆𝑆
𝐿𝑎𝑦𝑒𝑟4+η
𝑛𝐹
𝑄𝑖𝑛𝑡_𝐿𝑎𝑦𝑒𝑟4 = 𝑡
Where:
𝑇𝐿𝑎𝑦𝑒𝑟4: Local Catalyst Temperature
i: Current density
𝑡𝐿𝑎𝑦𝑒𝑟4: Layer thickness
n: Number of electrons
31
+ η
(3.47)
F: Faraday’s Constant
∆𝑆: The change in entropy (for platinum catalyst, ∆𝑆 equals to 0.104 𝑚𝑜𝑙 −1 𝐾 −1 for the
anode and -326.36 for the cathode
η: Activation over-potential
3.3.6. Membrane Layer Energy Balance:
The membrane Layer is a persulfonic acid layer that separates the anode and cathode sides of
a fuel cell and conducts protons, most commonly used type of membranes is the DuPont’s Nafion
membrane. The heat transfer in the membrane layer is mostly conductive, and the heat generation in
the membrane consist of just joule heating.
Figure 3-10: Membrane Layer Energy balance
From Figure 3.10, the overall heat energy equation in the membrane layer is:
(𝑐𝑝𝑎𝑣𝑔 (𝑛𝑔𝑎𝑠𝑒𝑠 + 𝑛𝑙𝑖𝑞 ) + (𝜌𝑚𝑒𝑚 𝐴𝑚𝑒𝑚 𝑡𝑚𝑒𝑚 )𝑐𝑝𝑚𝑒𝑚 )
𝑑𝑇𝑚𝑒𝑚
𝑑𝑡
= 𝑄𝑎_𝑐𝑎𝑡 + 𝑄𝑐_𝑐𝑎𝑡 +
𝑄𝑟𝑒𝑠_𝑚𝑒𝑚 + 𝐻𝐻+_4 + 𝐻𝐻2𝑂𝑣_4 + 𝐻𝐻2𝑂𝑙_4 − 𝐻𝐻2_5 − 𝐻𝐻2𝑂𝑣_5 − 𝐻𝐻2𝑂𝑙_5
32
(3.48)
3.4. TEMPERATURE IN THE LAYERS
In the previous section an overall Energy balance were done on each layer of the fuel in order
to obtain the temperature distribution and temperature changes with layers. To solve these equations
some parameters is needed to be taken into account. Table 3-1 summarize these parameters for the
material properties used.
Table 3-1- Material properties used for the heat transfer calculations
Fuel Cell Thickness (m)
Area (𝑚2 )
Layer
Density
Thermal
Specific Heat
𝑘𝑔
Conductivity
(J/kg-K)
(𝑚 2 )
(W/m-K)
End Plate
0.01
0.0064
1300
0.2
1200
Gasket
0.001
0.001704
1400
1.26
1000
Flow field 0.0005
0.003385
8000
65
500
0.0016
1300
26
864
0.003385
8000
65
500
plate
Membran
0.001
e
Flow field 0.0005
plate
Gasket
0.001
0.001704
1400
1.26
1000
End Plate
0.01
0.0064
1300
0.2
1200
Assume initial Temperature is 298.15 K
33
Capacity
By using MATLAB (refer to Appendix A.2) to solve the equations from section3.3 and plot
the graphs of the temperature change with time and position, solve the equations from section3.3, at 3
different times: 60, 300 and 1000 second, and at two different cases.
Case1: One node per layer
Case2: 10 nodes per layer
We obtained the following results:
Figure 3-11: Temperature plots for t=60, 300 and 1000 second for 1 slice per layer
34
Figure 3-12: Temperature plots for t=60, 300 and 1000 second for 10 slices per layer
35
Figures 3.11 and 3.12 show the temperature obtained at three different times 60, 300 and 1000
seconds for one slice per layer and ten slices per layer. We can see that the temperature distribution
starts to be more accurate when the number of slices increases, and that the temperature differs slightly
in the cell with time.
3.5. MODELING THE PROTON EXCHANGE MEMBRANE
The Proton Exchange Membrane contains the solid polymer membrane that is mostly made
of DuPont Nafion, liquid water and it may also contains a trace amounts of 𝐻2 , 𝑂2 or 𝐶𝑂2 resulted
from the purity of Hydrogen coming into the system. Modeling the Proton Exchange Membrane
includes different topics to be covered. In this section, specific topics will be covered in order to model
the membrane which are [19]:

Mass and species conservation

Ion transport phenomena

Momentum conservation

Conservation of energy
3.5.1. Membrane Layer Energy Balance:
The mass conservation for water and protons can be written as:
𝜕𝑐𝑖
𝜕𝑡
𝜕
= − 𝜕𝑡 𝑁𝑖
(3.49)
Where c and N are the molar concentration and the molar flux due to electroosmotic driving
forces and convection, and i refers to 𝐻2 𝑂 or 𝐻 + .
𝑁𝑖 is given by Nernst-Planck equation, which is:
𝑁𝑖 = 𝐽𝑖 + 𝑐𝑖 𝑢𝑚
(3.50)
𝑢𝑚 is the mixture velocity and 𝐽𝑖 is the diffusive flux.
36
In the membrane there two type of fluxes that present which are the back diffusion and
electroosmotic drag flux. These two fluxes can be obtained using the same equation which is:
𝑀
𝐽𝐻2𝑂
= −𝐷𝑐𝐻2𝑂,𝑇
𝑀
𝜕𝑐𝐻2𝑂
𝜕𝑥
𝑖
+ 𝑛𝑑𝑟𝑎𝑔 𝐹𝑥
(3.51)
Where F is the Faraday’s constant, 𝑛𝑑𝑟𝑎𝑔 is the drag coefficient, 𝑖𝑥 is the protonic current in
the x direction, 𝐷𝑐𝐻2𝑂,𝑇 is the diffusion coefficient
The drag coefficient can be described by the equation:
𝜆
𝑛𝑑𝑟𝑎𝑔 = 2.5 𝐻2𝑂/𝑆𝑂3
22
(3.52)
𝑐𝑚
𝜆𝐻2𝑂/𝑆𝑂3 = 𝜌𝑚 𝐻2𝑂
(3.53)
𝑑𝑟𝑦
𝑚
−𝑏𝑐𝐻2𝑂
𝑀𝑚
𝑚
Where 𝜆𝐻2𝑂/𝑆𝑂3 is the water content, 𝑀𝑚 is the membrane molecular mass and 𝜌𝑑𝑟𝑦
is the
dry membrane density and b is the membrane extension coefficient in the x-direction. b is determined
experimentally and its value is 0.0126.[20]
For the diffusion coefficient 𝐷𝑐𝐻2𝑂,𝑇 , it can be expressed by the following equation:
1
1
1
1
𝐷𝑐𝐻2𝑂,𝑇 = 𝐷′ [𝑒𝑥𝑝2416 (303 − 𝑇)] 𝜆𝐻2𝑂/𝑆𝑂3 𝑎 17.81−78.9𝑎+108𝑎2
(3.54)
Where 𝐷′ is the diffusion coefficient calculated at constant temperature and a is the activity
of the water. 𝐷′ at temperature equals to 30℃ is defined by the equation:
𝐷′ = 2.64227𝑒(−13) 𝜆𝐻2𝑂/𝑆𝑂3
for 𝜆𝐻2𝑂/𝑆𝑂3 ≤ 1.23
(3.55)
𝐷′ = 7.75𝑒(−11) 𝜆𝐻2𝑂 − 9.5𝑒(−11)
for 1.23 < 𝜆𝐻2𝑂/𝑆𝑂3 ≤ 6
(3.56)
for 6 < 𝜆𝐻2𝑂/𝑆𝑂3 ≤ 14
(3.57)
𝑆𝑂3
𝐷′ = 2.5625𝑒(−11)𝜆𝐻2𝑂 + 2.1625𝑒(−10)
𝑆𝑂3
Water total molar flux can be defined as:
𝑚
𝑁𝐻2𝑂 = 𝐽𝐻2𝑂 + (𝑐𝐻2𝑂
𝑢𝑚 )
Where 𝑢𝑚 is the mixture velocity, its equation will be given in the next section.
Moreover, water mass conservation can be described by:
37
(3.58)
𝑀
𝜕𝑐𝐻2𝑂
𝜕𝑡
𝜕
𝜕
𝑚
= − 𝜕𝑥 𝐽𝐻2𝑂 + − 𝜕𝑥 (𝑐𝐻2𝑂
𝑢𝑚 )
(3.59)
Due to the electroneutrality and the homogenous distribution of charge sites, the mass
conservation of protons can be simplified to:
𝜕𝐶𝐻+
𝜕𝑥
= 0,
𝜕𝐶𝐻+
𝜕𝑡
=0
(3.60)
The proton charges is equal to the fixed charges and its concentration remains constant. The
molar flux diffusivity of protons can be written as:
𝐹
𝜕∅
𝐽𝐻+ = − 𝑅𝑇 𝐷𝐻+ 𝐶𝐻+ 𝜕𝑥𝑚
(3.61)
Where 𝐷𝐻 is the diffusivity of proton and ∅𝑚 is the membrane proton potential.
Therefore, the total flux for the hydrogen protons is the combination of the diffusive flux with
the convective flux, which can be defined by:
𝑁𝐻+ = 𝐽𝐻+ + 𝑐𝐻+ 𝑢𝑚
(3.62)
3.5.2. Momentum equation:
The velocity for the mixture water and protons, it is assumed that the momentum equation
takes the form of the generalized Darcy relation, so the mixture velocity can be written as:
𝐾𝑘
𝑔
𝜕𝑝
𝑢𝑚 = − 𝜇 𝑟 [𝜕𝑥 − 𝜌𝑔 cos 𝜃]
(3.63)
Where:
𝜇=
𝑀𝐻+ 𝑐𝐻+
𝜌
𝜇𝐻+ +
𝑀𝐻2𝑂 𝑐𝐻2𝑂
𝜌
𝜇𝐻2𝑂
(3.64)
𝑔
𝑘𝑟 is the relative permeability, K is porous medium absolute permeability, 𝑢𝑚 is the mixture
velocity, g is the gravitational acceleration and 𝜃 is the angle that the x-axis makes with the direction
of gravity, 𝜌 is the mixture density and 𝜇 is the dynamic viscosity.
38
3.5.3. Conservation of Energy equation:
Due to the presence of the three phases in the membrane (polymer, liquid and gas), the energy
and heat within the membrane is transferred by conduction and convection, plus the effects of ohmic
losses in the membrane as an additional source is taken into account. Therefore, the energy balance
equation within the membrane can be written as:
𝜌𝑐𝑝
𝜕𝑇
𝜕𝑡
= 𝜆𝑚
𝜕2 𝑇
𝜕𝑥 2
− 𝑀𝑐𝑝 𝑁
𝜕𝑇
𝜕𝑥
+ 𝑅𝑚
(3.65)
Where:
𝑑𝑟𝑦
𝑑𝑟𝑦
𝑑𝑟𝑦
𝜌𝑐𝑝 = 𝜌𝑚 𝑐𝑝𝑚 + 𝜌𝐻2𝑂 𝑐𝑝,𝐻2𝑂 + 𝜌𝐻+ 𝑐𝑝𝐻+
(3.66)
𝑚
𝑚
𝜌𝐻+ = 𝑀𝐻+ 𝑐𝐻+ 𝜌𝐻2𝑂
= 𝑀𝐻2𝑂 𝑐𝐻2𝑂
(3.67)
𝑚
𝑀𝑐𝑝 𝑁 = 𝑀𝐻2𝑂 𝑐𝑝,𝐻2𝑂
𝑁𝐻2𝑂 + 𝑀𝐻+ 𝑐𝑝,𝐻+ 𝑁𝐻+
(3.68)
Substitute the values of 𝑁𝐻2𝑂 and 𝑁𝐻+ in the 𝑀𝑐𝑝 𝑁 expression, to obtain the expanded
equation which is:
𝑚
𝑚
𝑀𝑐𝑝 𝑁 = 𝑀𝐻2𝑂 𝑐𝑝,𝐻2𝑂
(𝑐𝐻2𝑂
𝑢𝑚 − 𝐷𝑐𝐻2𝑂,𝑇
𝜕𝑐𝑚
𝑝,𝐻2𝑂
𝜕𝑥
𝜆𝐻2𝑂
𝑖
𝑆𝑂3 𝑚
+ 2.5 22
) + 𝑀𝐻+𝑐𝑝,𝐻+ (𝑐𝐻+ 𝑢𝑚 −
𝐹
𝜕∅
𝐷𝐻+ 𝑐𝐻+ 𝜕𝑥𝑚)
(3.69)
𝑅𝑚 the term of source, can be defined by the equation:
𝑖2
𝑅𝑚 = 𝜎
(3.70)
𝑚
Where 𝜎𝑚 is the conductivity of the membrane, and it can be written in terms of the
temperature and the water content:
1
1
𝜎𝑚 = 𝜎𝑚303 exp[1268 (303 − 𝑇)]
(3.71)
Where 𝜎𝑚33 is the membrane conductivity at 303 K, which can be defined by the equation:
𝜎𝑚33 = 100 ∗ (0.005139𝜆𝐻2𝑂 − 0.00326)
𝑆𝑂3
39
𝑓𝑜𝑟 𝜆𝐻2𝑂 > 1
𝑆𝑂3
(3.72)
3.5.4. Ion Transport and other required activities:
For Ion transport, the proton potential which is derived from ohm’s law represents the proton
flux divided by the conductivity of the membrane. Due to the assumption of electroneutrality the total
molar flux can be directly related to the velocity mixture and the current density. The results can be
defined by the equation:
𝜕∅𝑚
𝜕𝑥
=−
𝑖
𝜎𝑚
+
𝐹
𝑐 𝑢𝑚
𝜎𝑚 𝐻+
(3.73)
Another activities to be studied is the interface and membrane water activities relation. For
the membrane interface, the water activity can be written as:
𝑎=𝑃
𝑅𝑇
𝑠𝑎𝑡 (𝑇)
𝑔
𝑐𝐻2𝑂 + 2𝑠
𝑎 ∈ [0, … ,3]
(3.74)
𝑔
Where, s is the saturation factor and 𝑐𝐻2𝑂 is the water vapor concentration. S is assumed to
be zero for activities less than 1 which means that no liquid water is present in the membrane pores
until the activity a exceeds the value of 1.
For the membrane water activity, it is given by the reciprocal of the sorption curve. The
equation for the membrane water activity in Nafion 117 at 30℃ can be written as:

1 1
1
134183
𝑎 = 2160 𝑐1 + 𝑐2𝜆𝐻2𝑂 + (216(𝑐3 − 𝑐4𝜆𝐻2𝑂 + 𝑐5𝜆𝐻2𝑂 2 )2 )3 − 2160
+ 𝑐2𝜆𝐻2𝑂 +
𝑐1
𝑆𝑂3
𝑆𝑂3
1 1
(216(𝑐3 − 𝑐4𝜆𝐻2𝑂 + 𝑐5𝜆𝐻2𝑂 2 )2 )3 +
𝑆𝑂3
𝑆𝑂3
Where:
c1=-41956e4
c2=139968e3
c3=382482e6
c4=251739e3
c5=419904e6
40
797
2160
𝑆𝑂3
𝑆𝑂3
𝑓𝑜𝑟 𝜆𝐻2𝑂 ≤ 14
𝑆𝑂3
(3.75)

𝑎 = 0.7143𝜆𝐻2𝑂 − 9.0021
𝑓𝑜𝑟 14 ≤ 𝜆𝐻2𝑂 ≤ 16.8
𝑆𝑂3

(3.76)
𝑆𝑂3
𝑎=3
𝑓𝑜𝑟 16.8 ≤ 𝜆𝐻2𝑂
(3.77)
𝑆𝑂3
3.5.5. Results for the Membrane:
After knowing the topics that affect the membrane modeling and obtaining the needed
equations, MATLAB (refer to Appendix A.3) will be used to solve these equations in order to achieve
the required results. To solve these equations, assume that the change in concentration of water and
𝑑𝑇
𝑑
temperature are zero with respect to time ( 𝑑𝑡 = 0, 𝐶_𝐻2𝑂
= 0), to solve for the steady state distribution
𝑑𝑡
of the other variables.
In order to solve these equations, some assumption is needed to be taken into account:
𝑑𝑇
𝑑𝑡
=0
𝑑𝑐_𝐻2𝑂
𝑑𝑡
=0
𝑁 = 10
(N: Number of Slices)
t= 60s
(t: Simulation time)
Outside-Pressure= 1 atm
Membrane thickness= 0.00005 m
After solving these equations by using MATLAB, the following results were obtained:
Figure 3-13: Water concentration distribution with respect to the membrane thickness
41
Figure 3-14: Temperature distribution with respect to the membrane thickness
Figure 3-15: Potential distribution with respect to the membrane thickness
Figure 3-16: Pressure distribution with respect to the membrane thickness
From the above figures, we can see that the temperature, pressure and concentration of
hydrogen increases slightly with the change of position, while the potential decreases with respect to
the position.
42
3.6. MODELING THE GAS DIFFUSION LAYERS
The gas diffusion layer is the layer that exists between the catalyst layer and the bipolar plates.
The gas diffusion layer, the catalyst layer and the membrane are inserted together to form the
membrane electrode assembly (MEA). They are responsible of the distribution of the reactants all over
the catalyst layer and of the rejection of product water out of the electrode surface, moreover they also
allow the electrical contact between the bipolar plates and the electrodes. In this section, the topics that
will be covered in order to model the gas diffusion layer will include:

Physical description of the gas diffusion layer

Basics of modeling porous media

Modes of transport in porous media

Types of models
3.6.1. Gas Diffusion Layer Model [21]:
In this section, Beuscher et Al derived models will be used for the modeling of the Gas
Diffusion Layer in a PEM fuel cell. These models are derived from multiphase flow from the
hydrogeological concepts in porous media. Of course there are some differences between modeling
the gas diffusion layer and modeling hydrogeological cases such as unsaturated soils. Some of these
differences are that gas diffusion layers are hydrophobic whereas soils are hydrophilic, the pore-size
distributions are different, and the gas diffusion layers are non-homogeneous texture of carbon fiber.
Regardless of all these dissimilarities, the hydrogeological models are still useful for modeling the gas
diffusion layer of a PEM fuel cell.
First of all, Richard’s equation which governs the moisture velocity 𝑉𝜃 of the liquid and vapor
in porous media.
𝑉𝜃 = −𝑘𝜃 (𝜃)∇𝛹
(3.78)
43
Where, 𝑘𝜃 is the hydraulic conductivity of the Gas Diffusion Layer to the liquid water, and
𝛹 is the moisture potential.
Consider the non-hysterestic case, therefore 𝜃 will be a single-valued function of 𝜃 only.
Assuming incompressibility (𝜌𝑤𝑎𝑡𝑒𝑟 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡), the conservation equation becomes:
∇. 𝑉𝜃 = ∑
(3.79)
∑: the source term introduced to include condensation and evaporation.
∇. (−𝑘𝜃 (𝜃)∇𝛹) = ∑
(3.80)
The diffusion coefficient of water can be defined by:
𝑑𝛹
𝐷𝜃 (𝜃) = 𝑘𝜃 (𝜃) 𝑑𝜃
(3.81)
According to the chain rule of differentiation:
∇. [ 𝐷𝜃 (𝜃) ∇θ] + ∑ = 0
(3.82)
Now using Arrhenius’ law to model evaporation as it is a temperature dependent process:
𝐸
𝑒𝑣𝑎𝑝𝑜𝑟𝑎𝑡𝑖𝑜𝑛 ∞ exp (− 𝑅𝑇𝐴 ) 𝜃
(3.83)
- 𝐸𝐴 : 𝐴𝑐𝑡𝑖𝑣𝑎𝑡𝑖𝑜𝑛 𝐸𝑛𝑒𝑟𝑔𝑦
-𝑅: 𝐺𝑎𝑠 𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡
Condensation depends on the concentration of water vapor and is independent of temperature.
𝑐𝑜𝑛𝑑𝑒𝑛𝑠𝑎𝑡𝑖𝑜𝑛 ∞ 𝜐̃
(3.84)
Introducing the constant of proportionality, 𝛽𝜃 for evaporation and 𝛽𝑣 for condensation:
𝐸
𝛽𝜃 exp (− 𝑅𝑇𝐴 ) 𝜃 + 𝛽𝑣 𝜐̃ = ∑
(3.85)
Therefore, this becomes:
𝐸
∇. [ 𝐷𝜃 (𝜃)∇θ] − 𝛽𝜃 exp (− 𝑅𝑇𝐴 ) 𝜃 + 𝛽𝑣 𝜐̃ = 0
(3.86)
To describe the diffusion processes for the gases, either Fickian diffusion, or the StephanMaxwell equation can be used. Considering Fick’s equation:
∇. (𝐷𝑢 (𝜃)∇𝑢̃ − 𝑢̃𝑉̃𝑔 ) = 0
44
(3.87)
Where 𝑉̃𝑔 is the velocity of the gas phase, and 𝐷𝑢 is the diffusion coefficient of oxygen. In
order to model the vapor transport, convection must be included:
𝐸
∇. (𝐷𝑦 (𝜃)∇𝑣̃ − 𝑣̃𝑉̃𝑔 ) + 𝑣1 [𝛽𝜃 exp (− 𝑅𝑇𝐴 ) 𝜃 + 𝛽𝑣 𝜐̃] = 0
(3.88)
Where 𝐷𝑦 is the water vapor diffusion coefficient and 𝑣1 the normalization factor.
In order to conduct a thermal study and model the temperature, the following must be taken
into consideration:

Fourier’s law for heat conduction

Convection

Condensation which causes a heat gain

Evaporation which causes a heat loss.
Thermal transport from gas and liquid can be neglected as the gas and liquid velocities are
respectively small.
Therefore:
𝐸
∇. [𝑘̃(𝜃)∇𝑇̃] + 𝜌𝜃 𝐿 [𝛽𝜃 exp (− 𝑅𝑇𝐴 ) 𝜃 + 𝛽𝑣 𝜐̃] = 0
(3.89)
Where, 𝑘̃ is the thermal conductivity, 𝜌𝜃 is the density of the liquid, and 𝐿 is the latent heat.
Now, taking the assumption of no liquid, all terms due to 𝜃 in the equations are neglected.
Also, all terms related to evaporation and condensation drop out of the equations. If the gas phase
convects, Darcy’s law governs the velocity.
𝑘𝑔 (𝜃)
𝑉̃𝑔 = − 𝜇 ∇𝑃̃
(3.90)
𝑘𝑔 : The permeability of the gas diffusion layer to gases. (𝑘𝑔 is dependent on 𝜃 as liquid water
is responsible of the removal of available pore spaces for the gas)
𝜇: The dynamic viscosity of the gas.
45
Solving for pressure, continuity equation can be used:
∇. 𝑉̃𝑔 = 0
𝑘𝑔 (𝜃)
∇𝑃̃) = 0
(3.92)
∇. (𝑘𝑔 (𝜃)∇𝑃̃) = 0
(3.93)
∇. (−
−1
𝜇
(3.91)
𝜇
Since all terms of 𝜃 are neglected, 𝑘𝑔 (𝜃) is constant.
𝑘𝑔 (𝜃)
𝜇
∇. ∇𝑃̃ = 0
(3.94)
Implies:
𝐷𝑢 . ∇. (∇𝑢̃) − ∇. (𝑢̃𝑉̃𝑔 ) = 0
𝑘𝑔 (𝜃) ∂𝑢
̃ ∂𝑃̃
̃ ∂𝑃̃
∂𝑢
∇. 𝑢̃ + 𝜇𝐷 [∂𝑥̃ . ∂𝑥̃ + ∂𝑦̃ . ∂𝑦̃] = 0
𝑢
(3.95)
(3.96)
Due to no liquid assumption, the condensation and evaporation terms are dropped.
𝑘𝑔 (𝜃) ∂𝑣̃ ∂𝑃̃
∂𝑣̃ ∂𝑃̃
∇. 𝑣̃ + 𝜇𝐷 [∂𝑥̃ . ∂𝑥̃ + ∂𝑦̃ . ∂𝑦̃] = 0
𝑣
(3.97)
Equations (3.97) becomes:
∇2 . 𝑇̃ = 0
(3.98)
According to the governing equations and boundary conditions, the following dimensionless
parameters are motivated:
𝑥̃
𝑥=𝑑 ;
(3.99)
𝑦̃
𝑦=𝑑 ;
(3.100)
̃
𝑢
𝑢=𝑢 ;
(3.101)
1
𝑣̃
𝑣=𝑣 ;
(3.102)
1
𝑇(𝑥, 𝑦) =
𝑇(𝑥̃,𝑦̃)−𝑇𝑚
𝑇1 −𝑇𝑚
;
46
(3.103)
2𝑃̃ (𝑥̃,𝑦̃)−(𝑃1 +𝑃𝑟 )
𝑃(𝑥, 𝑦) =
(3.104)
𝑃1 +𝑃𝑟
By substitution of these into the previous equations:
𝑃1 −𝑃𝑟 ∂2 𝑃̃
𝑃 −𝑃 ∂2 𝑃̃
+ 12ℎ2 𝑟 ∂𝑦̃ 2 = 0
2𝑑2 ∂𝑥̃ 2
∂2 𝑃̃
(3.105)
∂2 𝑃̃
𝜀 2 ∂𝑥̃ 2 + ∂𝑦̃ 2 = 0
̃
1 ∂2 𝑢
(3.106)
̃
1 ∂2 𝑢
𝑢1 (𝑑2 ∂𝑥̃ 2 + ℎ2 ∂𝑦̃ 2 ) +
𝑘𝑔 𝑢1 (𝑃1 −𝑃𝑟 )
2𝜇𝐷𝑢
̃ ∂𝑃̃
1 ∂𝑢
̃ ∂𝑃̃
1 ∂𝑢
(𝑑2 ∂𝑥̃ . ∂𝑥̃ + ℎ2 ∂𝑦̃ . ∂𝑦̃) = 0
(3.107)
̃
∂2 𝑢
̃
∂2 𝑢
̃ ∂𝑃̃
∂𝑢
̃ ∂𝑃̃
∂𝑢
𝜀 2 (∂𝑥̃ 2 + ∂𝑦̃ 2 ) + 𝑃𝑒𝑢 (𝜀 2 ∂𝑥̃ . ∂𝑥̃ + ∂𝑦̃ . ∂𝑦̃ ) = 0
𝑘𝑔 𝑢1 (𝑃1 −𝑃𝑟 )
𝑃𝑒𝑢 =
(3.108)
(3.109)
2𝜇𝐷𝑢
𝑃𝑒: Peclet Number of oxygen.
When 𝑣 is replaced by 𝑢, then:
∂2 𝑣
∂2 𝑣
∂𝑣
∂𝑃
∂𝑣
∂𝑃
𝜀 2 (∂𝑥 2 + ∂𝑦 2 ) + 𝑃𝑒𝑣 (𝜀 2 ∂𝑥 . ∂𝑥 + ∂𝑦 . ∂𝑦 ) = 0
𝑃𝑒𝑣 =
𝑘𝑔 (𝑃1 −𝑃𝑟 )
(3.110)
(3.111)
2𝜇𝐷𝑣
Therefore,
(𝑇1 −𝑇𝑚 ) ∂2 𝑟
+
∂𝑥 2
𝑑2
(𝑇1 −𝑇𝑚 ) ∂2 𝑟
ℎ 2 ∂2 𝑟
∂2 𝑇
𝑑
∂𝑦 2
( )
∂𝑦 2
=0
(3.112)
=0
(3.113)
𝜀 2 ∂𝑥 2 + ∂𝑦 2 = 0
(3.114)
∂𝑥
+
2
ℎ2
Substituting into:
∂2 𝑇
∂2 𝑇
Adding to the assumption of no liquid, and assuming a constant heat flux and no convection,
the transport will now be Fickian, and the pressure is constant. Hence:
∂2 𝑢
∂2 𝑢
𝜀 2 ∂𝑥 2 + ∂𝑦 2 = 0
47
(3.115)
∂2 𝑣
∂2 𝑣
∂2 𝑇
∂2 𝑇
𝜀 2 ∂𝑥 2 + ∂𝑦 2 = 0
(3.116)
𝜀 2 ∂𝑥 2 + ∂𝑦 2 = 0
(3.117)
After examining the boundaries with constant pressure, the positive and negative sections are
symmetric about the y-axis. Taking the section −2 ≤ 𝑥 ≤ 0, the boundary conditions are as follows:
∂𝑇 𝑚
∂𝑥
∂𝑢𝑚
∂𝑥
∂𝑣 𝑚
∂𝑥
(0, 𝑦) = 0
(3.118)
(0, 𝑦) = 0
(3.119)
(0, 𝑦) = 0
(3.120)
At the cathode catalyst layer interface, assume constant flux. Oxygen drift out of the gas
diffusion layer, therefore:
̃
∂𝑢
−𝐷𝑢 ∂𝑦 (𝑥̃, 0) = −𝑞̃𝑢
(3.121)
The temperature’s fluxes and water vapor fluxes are given:
∂𝑣̃
𝐷𝑣 ∂𝑦 (𝑥̃, 0) = −𝑞
̃𝑣
(3.122)
̃𝑐 ∂𝑇̃ (𝑥̃, 0) = −𝑞̃𝑇
𝑘
∂𝑦
(3.123)
By the substitution of equations (3.99), (3.100), (3.101), (3.102) and (3.103) into equations
(3.121), (3.122) and (3.123):
𝑢 ∂𝑢
𝐷𝑢 ℎ1 ∂𝑦 (𝑥, 0) = 𝑞̃𝑢
∂𝑢
∂𝑦
(𝑥, 0) = 𝑞𝑢 𝜀 2
(3.124)
𝑞 ℎ
𝑞𝑢 𝜀 2 = 𝐷 𝑢𝑢
;
𝑢 1
𝑣 ∂𝑣
𝐷𝑣 ℎ1 ∂𝑦 (𝑥, 0) = −𝑞
̃𝑣
∂𝑣
∂𝑦
(3.126)
𝑞 ℎ
(𝑥, 0) = −𝑞𝑣 𝜀 2 ;
𝑞𝑣 𝜀 2 = 𝐷𝑣𝑣
(3.127)
𝑣 1
𝑇 −𝑇 ∂𝑇
𝑘𝑐 1 ℎ 𝑚 ∂𝑦 (𝑥, 0) = −𝑞̃𝑇
∂𝑇
∂𝑦
(𝑥, 0) = −𝑞𝑇
(3.128)
𝑞 ℎ
𝑞𝑇 = 𝑘 (𝑇 𝑇−𝑇 )
;
𝑐
48
(3.125)
1
𝑚
(3.129)
Using 𝜀 = 0.2 as the perturbation parameter, the dependent variable can be expressed as:
𝑇(𝑥, 𝑦, 𝜀) = 𝑇0 (𝑥, 𝑦) + 𝑜(𝜀)
(3.130)
Substitute equation (3.117) in equation (3.130):
∂2 𝑇
∂2 𝑇
𝜀 2 ∂𝑥 20 + ∂𝑦 20 + 𝑜(𝜀) = 0
(3.131)
Then:
∂2 𝑇0
∂𝑦 2
=0
(3.132)
Using the following boundary conditions:
∂𝑇 𝑚
∂𝑥
(0, 𝑦) = 0;
∂𝑇
∂𝑦
(𝑥, 0) = −𝑞𝑇 ;
∂𝑇 1
∂𝑥
(−2, 𝑦) = 0
(3.133)
Integrating the boundary conditions:
∂𝑇0
∂𝑦
= −𝑞𝑇 ;
𝑇0 = −𝑞𝑇 𝑦 + 𝑓(𝑥);
𝑓 ′ (−2) = 𝑓 ′ (0) = 0
(3.134)
In this problem now, we have 2 different regions:
𝑇01 (𝑥, 𝑦) = 𝑞𝑇 (1 − 𝑦) + 1
; −2 ≤ 𝑥 ≤ −1
(3.135)
𝑇0𝑚 (𝑥, 𝑦) = 𝑞𝑇 (1 − 𝑦) + 1
; −1 ≤ 𝑥 ≤ 0
(3.136)
To consider the discontinuity, at 𝑥 = 0 an interior layer is introduced:
𝑧=
𝑥+1
𝜀
;
𝑇(𝑥, 𝑦) = 𝑞𝑇 (1 − 𝑦) + 𝑇 𝑖 (𝑧, 𝑦)
(3.137)
𝑇 𝑖 : Interior Temperature
And by substituting equation (3.137) in equations (3.117), (3.129) and (3.112):
∂2 𝑇 𝑖
∂2 𝑇 𝑖
+ ∂𝑦 2 = 0
∂𝑧 2
(3.138)
∂𝑇 𝑖
−𝑞𝑇 + ∂𝑦 (𝑧, 0) = −𝑞𝑇
∂𝑇 𝑖
∂𝑦
(3.139)
(𝑧, 0) = 0
(3.140)
𝑇 𝑖 (𝑧, 1) = 1
;𝑧 <0
49
(3.141)
𝑇 𝑖 (𝑧, 1) = 0
;𝑧 >0
(3.142)
Matching the outer solution:
𝑞𝑇 (1 − 𝑦) + 𝑇 𝑖 (−∞, 𝑦) = 𝑇 1 (−1− , 𝑦) = 𝑞𝑇 (1 − 𝑦) + 1 (3.143)
𝑇 𝑖 (−∞, 𝑦) = 1
(3.144)
𝑞𝑇 (1 − 𝑦) + 𝑇 𝑖 (∞, 𝑦) = 𝑇 𝑚 (−1+ , 𝑦) = 𝑞𝑇 (1 − 𝑦)
(3.145)
𝑇 𝑖 (∞, 𝑦) = 0
(3.146)
Using the following set of transformation:
𝑓1 = 𝑧 + 𝑖𝑦, 𝑓2 = exp(𝜋. 𝑓1 ) ,
𝑓 −1
𝑓3 = 𝑓2 +1 ,
2
1
𝑓4 = 0.5 + 𝜋 sin−1 (𝑓3 )
(3.147)
Finally, the solution can be written as:
𝑇 𝑖 = ℜ𝑓4
(3.148)
3.6.2. Results for the GDL:
A three-dimensional plot of the temperature for the interior layer will be done by using
MATLAB (refer to Appendix A.4) using the equations derived in section 3.6.2.
Assumptions will be:

Length in x direction = 1.6mm

Length in y direction = 1.0mm

Perturbation factor = 0.2
50
Figure 3-17: A 3D plot of the temperature in the interior layer
From Figure 3.17, the temperature at maximum x and z is the highest temperature, it keeps
decreasing until it reaches the lowest temperature, and therefore the GDL is able to withstand the
temperatures of the fuel cell, and it must be chemically stable and a good proton conductor.
3.7. MODELING THE CATALYST LAYERS
Catalyst Layers are where electrochemical reactions occur, in the anode hydrogen is broken
into protons and negatively charged ions. The catalyst layer must have high efficiency at breaking
molecules into protons and electrons or negatively charged ions, plus a high surface area. The catalyst
layers are frequently the thinnest layer in the fuel cell (5 to 30 microns), but due to porosity, multiple
phases and electrochemical reactions they are the most complex layer.
In order to model the catalyst layer, some specific topics will be covered:

Conservation of mass and other species

Ion transport

Momentum conservation

Energy conservation
51
3.7.1. Physical Description and Area of Catalyst Layer:
The Catalyst Layer is usually made of a porous mixture of carbon-supported platinum and
ionomer. The particles of the catalyst must have contact to both electronic and protonic conductors in
order to catalyze the reactions. The effective area of the active catalyst sites must be higher than the
geometric area of the electrode by several times to achieve acceptable reaction rates.
Therefore, the surface area of the catalyst is a very important parameter of the catalyst layer.
In order to designate the size distribution of the particle and the surface area of the platinum particle,
we assume all the platinum particles are spherical, so the area per unit mass will be [15]:
𝐴𝑠 =
∫ 𝑓(𝐷)𝜋𝐷 2 𝑑𝐷
𝜋𝐷3
∫ 𝑓(𝐷)𝜌𝑃𝑡 ( 6 )𝑑𝐷
=𝜌
6
𝑃𝑡 𝐷32
(3.149)
Where:
𝜌𝑃𝑡 : Platinum black density
𝐷32 : Volume-to-surface area mean diameter of all the particles
A typical value of the active area per unit mass is 28𝑚2 /𝑔𝑃𝑡, that can be estimated from the
mean diameter 𝐷32 .
The thinness of the catalyst later is to minimize the cell potential losses due to the reactant
gas permeation in the depth of the electrocatalyst layer and the rate of the proton transport.
3.7.2. Interface and Agglomerate Models:
In modeling the catalyst layers, we will implement a combination of interface and
agglomerate models. Interface models assume that the catalyst layers locate at the interface of the Gas
Diffusion layer and the Membrane, and that the interface layer of the catalyst is where water is
produced and oxygen/hydrogen are consumed. Agglomerate models assume that the rate of reaction
distribution is uniform, and it represent the catalyst layers actual structure.
52
The assumption that the catalyst layer locate at the GDL/membrane interface means that the
catalyst layers are infinitely thin, so the structure and geometry can be ignored. In order to accomplish
this in a model, treat the catalyst layer as a location where water is produced and hydrogen and oxygen
are consumed. Faraday’s law is the rate where water is generated and hydrogen/oxygen are consumed,
this can be shown in the following equation:
1
𝑁𝐻2 = 2𝐹
(3.150)
1
𝑁𝑂2 = 4𝐹
(3.151)
1
𝑁𝐻2𝑂 = 2𝐹
(3.152)
Where I is the current in Amperes, N is the rate of consumption in mole/s and F is Faraday’s
constant in C/mol.
The general equation for the polarization of the fuel cell curve is:
𝑅𝑇
𝐸 = 𝐸𝑟 − 𝛼𝐹 ln (
𝑖+𝑖𝑙𝑜𝑠𝑠
𝑖0
𝑅𝑇
𝑖
𝐿
) − 𝑛𝐹 ln (𝑖 −𝑖
) − 𝑖𝑅𝑖
𝐿
(3.153)
Where 𝐸𝑟 is the Nernst equation which is used to determine the reaction theoretical electrical
potential. 𝐸𝑟 can be calculated by the equation:
𝑃
∆𝐺
.𝑃
1/2
𝑂2
𝐸𝑟 = 2𝐹 + 𝑅𝑇𝑙𝑛 𝐻2
𝑃
𝐻2𝑂
(3.154)
The rate of species generation and consumption and the activation over-potential can be
determined by the current density and the kinetics of the electrochemical. The mass flow rates and the
rate of reaction can be related to the electric current by Faraday’s law:
1
𝑚𝐻2 = 2𝐹 𝑀𝐻2
(3.155)
1
𝑚𝑂2 = 4𝐹 𝑀𝑂2
(3.156)
1
𝑚𝐻2𝑂 = 2𝐹 𝑀𝐻2𝑂
(3.157)
Where M is the molecular weight.
53
The relation between the activation losses and the current density for the anode can be
obtained by using Butler-Volmer equation, so the equation for the anode can be written as:
𝑖𝑎 = 𝑖0,𝑎 exp (−𝛼𝑎
𝐹𝜐𝑎𝑐𝑡,𝑎
𝑅𝑇
) − exp ((1 − 𝛼𝑎 )
𝐹𝜐𝑎𝑐𝑡,𝑎
𝑅𝑇
)
(3.158)
Where, 𝑣𝑎𝑐𝑡 is the losses of the activation electrode, 𝑖𝑜 and 𝑖𝑎 is the exchange and transfer
current density in A/𝑚3 and 𝛼𝑎 is the coefficient of the anodic charge transfer.
The equation for the cathode side can be written as:
𝑖𝑐 = 𝑖0,𝑐 exp (−𝛼𝑐
𝐹𝜐𝑎𝑐𝑡,𝑐
𝑅𝑇
) − exp ((1 − 𝛼𝑐 )
𝐹𝜐𝑎𝑐𝑡,𝑐
𝑅𝑇
)
(3.159)
The exchange current density depends on the local temperature and the reactants partial
pressure, when the partial pressure decreases and exchange current density will decrease also, so the
performance will decrease also. The exchange current density for the cathode and the anode can be
written as:
−𝐸
𝑃
1
1
𝐴,𝑐
0
𝛾
𝑖0,𝑐 = 𝑖0,𝑐
( 𝑃𝑂2
(𝑇 − 𝑇 0 )]
0 ) exp [ 𝑅
(3.160)
𝑂2
𝑃
𝑃
0
𝛾1
𝑖0,𝑎 = 𝑖0,𝑎
( 𝑃𝐻2
(𝑃𝐻2𝑂
)𝛾2 exp [
0
0 )
𝐻2
𝐻2𝑂
−𝐸𝐴,𝑎 1
𝑅
1
(𝑇 − 𝑇 0 )]
(3.161)
Where 𝑖0,𝑎 and 𝑖0,𝑐 are the exchange current density of the anode and the cathode, 𝛾1 and 𝛾1
are the order of the reaction and T is the temperature (303K) and EA is the activation energy.
Finally, the average current density which is the total current generated in the fuel cell can be
written as [22]:
1
1
𝑖𝑎𝑣𝑔 = 𝐴 ∫𝑉𝑎 𝐽𝑎 𝑑𝑉 = 𝐴 ∫𝑉𝑐 𝐽𝑐 𝑑𝑉
(3.162)
The porous catalyst equation can be used for the anode and cathode and it can be modified
with an effectiveness factor, so the actual rate of reaction can be written as [23]:
∇. 𝑖2 = 𝑎1,2 𝑖ℎ,1−2 𝐸
(3.163)
Where, 𝑎1−2 is the interfacial area between the membrane phase with no flooding and the
electrical conduct, 𝑖ℎ,1−2 is the transfer current for reaction between the conducting solid and the
54
membrane, ∇. 𝑖2 represents the total anodic rate of electrochemical reactions per unit volume and E is
the effectiveness factor, Where 𝐸𝑓 is:
1
𝐸𝑓 = 3∅2 (3∅ coth(3∅) − 1)
(3.164)
Where ∅ is the Thiele modulus of the system, and it can be expressed by the equation:
𝑘′
∅ = 𝜁 √ 𝑒𝑓𝑓
(3.165)
𝐷𝑂2,𝑎𝑔𝑔
Where 𝜁 is the agglomerate characteristic length (volume per area) and k’ is a constant rate
which can be defined by the equation:
𝑘′ =
𝑎1−2 𝑖0,𝑂𝑟𝑟
𝑟𝑒𝑓
4𝐹𝑐𝑂2
𝛼 𝐹
𝑐
exp (− 𝑅𝑇
(ƞ𝑂𝑅𝑅,1−2 ))
(3.166)
And the reference concentration in the agglomerate is in equilibrium with the reference
pressure, where the concentration can be defined by the equation:
𝑟𝑒𝑓
𝑟𝑒𝑓
𝑐𝑂2 = 𝑃𝑂2 𝐻𝑂2,𝑎𝑔𝑔
(3.167)
Where 𝐻𝑂2,𝑎𝑔𝑔 is the oxygen Henry’s constant in the agglomerate.
The limitation of the external mass transfer can be neglected, so the bulb concentration can
be assumed equal to the surface concentration, and it can be assumed uniform all over the catalyst
layer. The diffusion oxygen to the surface agglomerate expression can be written as:
𝑑𝑖𝑓𝑓
𝑊𝑂2
𝑠𝑢𝑟𝑓
𝑐 𝑏𝑢𝑙𝑘 −𝑐𝑂2
= 𝐴𝑎𝑔𝑔 𝐷𝑂2,𝑓𝑖𝑙𝑚 𝑂2𝛿
(3.168)
𝑓𝑖𝑙𝑚
𝑑𝑖𝑓𝑓
𝐴𝑎𝑔𝑔 is the agglomerate external surface area, 𝑊𝑂2
is the oxygen molar flow rate to the
agglomerate. The film can be water or membrane.
Finally, the equation that covers the cathode catalyst layer agglomerate models without the
limitations of the external mass transfer can be deduced from the resultant equation in the conservation
equation, it can be written as:
1
𝑏𝑢𝑙𝑘
∇. 𝑖1 = 4𝐹𝑐𝑂2
(
𝛿𝑓𝑖𝑙𝑚
1
+
𝐴𝑎𝑔𝑔 𝐷𝑂2,𝑓𝑖𝑙𝑚 𝑘′ 𝐸
55
)
(3.169)
3.7.3. Results and Current Density:
After achieving the needed equations, it is now allowable to model the catalyst layers for the
anode and the cathode. The first step for modeling the catalyst layer is to calculate the Nernst voltage,
were the partial pressures of hydrogen, oxygen and water will be used.
In order to calculate the pressure saturation of water, the following expression can be used:
log 𝑃𝐻2𝑂 = −2.1794 + 0.02953 ∗ 𝑇𝑐 − 9.1837 ∗ 10−5 ∗ 𝑇𝑐2 + 1.4454 ∗ 10−7 ∗ 𝑇𝑐3
(3.170)
And the partial pressures of hydrogen and oxygen can be calculated by the equations:
𝑝𝐻2
𝑃𝐻2 = 0.5 ∗ (
𝑖
) − 𝑃𝐻2𝑂
(3.171)
exp(1.653∗ 1.334 )
𝑇𝑘
𝑃𝑂2 = (
𝑝𝑎𝑖𝑟
𝑖
) − 𝑃𝐻2𝑂
(3.172)
exp(4.192∗ 1.334 )
𝑇𝑘
To calculate the actual total voltage, it is the Nernst voltage plus the voltage losses, Therefore
the actual voltage can be written as:
𝑉 = 𝐸𝑁𝑒𝑟𝑛𝑠𝑡 + 𝑉𝑎𝑐𝑡 + 𝑉𝑜ℎ𝑚𝑖𝑐 + 𝑉𝑐𝑜𝑛𝑐
(3.173)
Now use the equations from section 3.7.2 to obtain the model of the catalyst layer, and to be
able to plot the current density versus effectiveness factor, activation losses, voltage (polarization) and
the hydrogen flux density.
In order to solve these equations some assumptions is needed to be taken into account and
some parameters to be known that will be summarized in table, then by using MATLAB (refer to
Appendix A.5) solve the equations from section 3.7.2 to obtain the results.
56
Table 3-2 - Parameters for catalyst layer modeling
Paramter
Value
Temperature
348.15 K
Total gas pressure
1 atm
Hydrogen pressure
1 atm
Air pressure
1 atm
O2 Permeation in agglomerate
1.5e-11
H2 Permeation in agglomerate
2e-11
Agglomerate radius for anode
110e-5
Agglomerate radius for cathode
110e-5
Anode Transfer Coefficient
1
Cathode Transfer Coefficient
0.9
Saturation
0.6e-12
Limiting Current density
1.4 A/cm2
Constant Ohmic resistance
0.02 ohm-cm2
Amplification Constant
0.085
Mass transport Constant
1.1
Electrode-specific interfacial area
10000
Gibbs function in liquid form
-228 170 J/mol
Current density
1 to 1.2 A/cm2
By solving on MATLAB, the following results were obtained:
57
Figure 3-18: Cell Current as a function of Effectiveness Factor
Figure 3-19: Cell Current Versus Butler-Volmer activation losses
58
Figure 3-20: Cell Current as a function of voltage (polarization)
Figure 3-21: Current Density as a function of superficial flux density of hydrogen
59
From the previous figures, it can be seen that these factors affect the current density in the
catalyst layer, which are the effectiveness factor, activation losses, polarization and the hydrogen flux
density. From Figure 3.18, the green curve represents the effectiveness factor of hydrogen and the blue
one the oxygen, so the effect of oxygen is larger than that of hydrogen on the current density. While
from figures y and z, we can conclude that when the current density increases the activation losses will
increase too, therefore the voltage or polarization will decrease. From the last figure, when the
hydrogen flux density is zero the current density is zero, so we don’t have any current in the layer, so
we can conclude that the current cannot be formed if no hydrogen is existed which is the fuel for the
PEMFC, and that when the hydrogen density increases, we will have more current density.
3.8. MODELING THE FLOW FIELD PLATES
Flow field plate is placed in the fuel cell to distribute the oxidant and fuel within the cell,
collect the current, and carry the water away from each layer, coolant for the cell and to humidify the
gases. It is important to study the geometry of the flow field plate because of its effects on the reactant
mass transfer and flow velocity, therefore on the fuel cell performance. Moreover, the type of material
used in modeling the flow field plate is an important manner too in order to perform its functions, and
the type of flow within the plate.
3.8.1. Material and types of flow [24]:
There are many types that has been used in designing the flow field plates. The most common
used materials are stainless steel and graphite, other materials are also used such as steel, aluminum,
titanium, nickel, and polymer.
One of the most important properties to choose the type of material is the electrical
conductivity to decrease the bulk resistance losses. The bipolar plates of the flow field are exposed to
very corrosive environment which lowers the ionic conductivity and reduces the life of the fuel cell,
therefore a coating is needed to avoid corrosion while promoting conductivity.
60
The flow fields in the fuel cell must be designed to decrease the pressure drop in order to
increase the flow velocity. Serpentine type (Figure 3.22) is used for the anode where the fuel flows, it
is a continuous one path from start to the finish, and the benefit of this type is that the flow reaches all
of the active area, but it has some disadvantages which is the pressure drop is high. To avoid this
weakness a several continuous flow channels can be used and it is called the multiple serpentine flow
field (Figure 3.23).
Figure 3-22: A serpentine flow field design
Figure 3-23: multiple serpentine flow channel design
3.8.2. Pressure Drop in Flow Channels
The pressure difference between the inlet and the outlet drives the fluid, so by increasing this
difference the velocity will be increased. The flow through this plate is laminar and proportional to the
61
flow rate, so it can be treated as incompressible flow in pipes, therefore the pressure drop can be written
as:
̅̅̅
𝑣 2̅
𝐿
̅̅̅
𝑣 2̅
∆𝑝 = 𝑓 𝑐ℎ𝑎𝑛
𝜌 2 + ∑ 𝐾𝐿 𝜌 2
𝐷
ℎ
(3.174)
Where 𝐿𝑐ℎ𝑎𝑛 is the length of the channel, 𝐷ℎ is the hydraulic diameter, f is the friction factor,
𝐾𝐿 is the local resistance, 𝜌 is the fluid density and 𝑣̅ is the flow velocity in m/s.
The hydraulic diameter for a circular flow can be obtained by the equation:
4×𝐴
𝐷ℎ = 𝑃 𝑐
(3.175)
𝑐𝑠
Where A and P are the cross-sectional area and parameter respectively
While the diameter for a rectangular flow field can be obtained by the equation:
2𝑤 𝑑
𝑐 𝑐
𝐷ℎ = 𝑤 +𝑑
𝑐
(3.176)
𝑐
𝑤𝑐 and 𝑑𝑐 are the channel width and depth.
The friction factor can be obtained by the equation:
56
𝑓 = 𝑅𝑒
(3.177)
Reynold’s number 𝑅𝑒 can be calculated by the equation:
𝑅𝑒 =
𝜌𝑣𝑚 𝐷ℎ
𝜇
𝑣 𝐷
= 𝑚𝑣 ℎ
(3.178)
Where 𝜇 and 𝑣 are the fluid viscosity and dynamic viscosity
The length of the channel is:
𝐿𝑐ℎ𝑎𝑛 = 𝑁
𝐴𝑐𝑒𝑙𝑙
(𝑤
𝑐 +𝑤𝐿 )
𝑐ℎ
(3.179)
Where 𝑁𝑐ℎ is the number of channels, 𝐴𝑐𝑒𝑙𝑙 is the active area of the cell, 𝑤𝑐 and 𝑤𝐿 are the
channel width and the space between the channels respectively.
The velocity at the fuel cell entrance can be written as:
𝑣=𝑁
𝑄
(3.180)
𝑐𝑒𝑙𝑙 𝑁𝑐ℎ 𝐴𝑐ℎ
62
Where 𝐴𝑐ℎ and 𝑁𝑐ℎ are the cross-sectional area and number of channels, 𝑁𝑐𝑒𝑙𝑙 is the number
of cells for stack design, and Q is the air flow rate at the entrance of the fuel cell which can be defined
by the equation:
𝐼 𝑆
𝑅𝑇
𝑄 = 4𝐹 𝑟𝑂2 𝑃 −𝜑𝑃 𝑖𝑛
𝑂2
𝑖𝑛
𝑠𝑎𝑡(𝑇𝑖𝑛)
𝑁𝑐𝑒𝑙𝑙
(3.181)
F is the Faraday’s constant, I is the cell current, 𝑆𝑂2 and 𝑟𝑂2 are the oxygen stoichiometric
ratio and content in air, R is the universal gas constant, 𝑇𝑖𝑛 and 𝑃𝑖𝑛 are the cell inlet temperature and
pressure, 𝜑 is the relative humidity and 𝑃𝑠𝑎𝑡 is the saturation pressure.
3.8.3. Mass Flow rates in the fuel cell layers
The flow rates is optimal for modeling a fuel cell flow field layer, which are for the liquid
water, water vapor and the amount of hydrogen going in and out of the cell. Length, width and depth
of the channels in the flow field plate must be selected carefully to guarantee a proper flow rates and
mass transfer.
In order to study the mass flow for hydrogen and water (Liquid and Vapor), the volumetric
flow rate will be converted to a molar flow rate by using the ideal gas law:
𝑃𝑉
𝑛𝐻2−𝑖𝑛 = 𝑅𝑇
(3.182)
Where P and T are the inlet pressure and temperature, V is the volume and R is the universal
gas constant, and n is the molar mass.
Consider a transient model, therefore the total molar accumulation can be defined as:
𝑑𝑛𝑡𝑜𝑡
𝑑𝑡
= 𝑛𝑡𝑜𝑡_𝑖𝑛 − 𝑛𝑡𝑜𝑡_𝑜𝑢𝑡
(3.183)
Therefore, The rate of accumulation of 𝐻2 and 𝐻2 𝑂 can be written as:
𝑑
𝑑𝑡
𝑑
𝑑𝑡
(𝑥𝐻2 𝑛𝑡𝑜𝑡 ) = 𝑥𝐻2_𝑖𝑛 𝑛𝑡𝑜𝑡_𝑖𝑛 − 𝑥𝐻2_𝑜𝑢𝑡 𝑛𝑡𝑜𝑡_𝑜𝑢𝑡
(3.184)
(𝑥𝐻2𝑂 𝑛𝑡𝑜𝑡 ) = 𝑥𝐻2𝑂_𝑖𝑛 𝑛𝑡𝑜𝑡_𝑖𝑛 − 𝑥𝐻2𝑂_𝑜𝑢𝑡 𝑛𝑡𝑜𝑡_𝑜𝑢𝑡
(3.185)
Where x is the mole fraction.
63
For the calculation of the inlet molar flow rates:
The vapor pressure of the inlet water vapor equation can be written as:
𝑃𝐻2𝑂𝑣_𝑖𝑛 = 𝑄𝑖𝑛 𝑃𝑠𝑎𝑡 (𝑇𝐻2𝑂𝑖𝑛 )
(3.186)
The water vapor mole fraction can be written as:
𝑋𝐻2𝑂𝑣_𝑖𝑛 =
𝑃𝐻2𝑂𝑣_𝑖𝑛
𝑊ℎ𝑒𝑟𝑒 𝑃𝑡𝑜𝑡 = 1
𝑃𝑡𝑜𝑡
(3.187)
The liquid water mole fraction can be written as:
𝑋𝐻2𝑂𝑙_𝑖𝑛 =
𝑋𝐻2𝑂𝑣_𝑖𝑛 ∗𝑃𝑠𝑎𝑡 (𝑇𝐻2𝑂𝑖𝑛 )
𝑃𝑡𝑜𝑡
(3.188)
Therefore, the total mole fraction of water is the sum of vapor and liquid:
𝑋𝐻2𝑂_𝑖𝑛 = 𝑋𝐻2𝑂𝑣_𝑖𝑛 + 𝑋𝐻2𝑂𝑙_𝑖𝑛
(3.189)
The total mole fraction of hydrogen can be concluded from the total mole fraction of water
such that:
𝑋𝐻2_𝑖𝑛 = 1 − 𝑋𝐻2𝑂_𝑖𝑛
(3.190)
And the inlet molar flow rate of hydrogen is:
𝑛𝐻2_𝑖𝑛 = 𝑥𝐻2_𝑖𝑛 𝑛𝑡𝑜𝑡_𝑖𝑛
(3.191)
The total molar flow rate of vapor water, liquid water and total inlet water can be written as:
𝑛𝐻2𝑂𝑣_𝑖𝑛 = 𝑋𝐻2𝑂𝑣_𝑖𝑛 𝑛𝑡𝑜𝑡_𝑖𝑛
(3.192)
𝑛𝐻2𝑂𝑙_𝑖𝑛 = 𝑋𝐻2𝑂𝑙_𝑖𝑛 𝑛𝑡𝑜𝑡_𝑖𝑛
(3.193)
𝑛𝐻2𝑂_𝑖𝑛 = 𝑥𝐻2𝑂_𝑖𝑛 𝑛𝑡𝑜𝑡_𝑖𝑛
(3.194)
For the outlet calculation, same equations can be applied to obtain the outlet mole fractions
and molar flow rates.
3.8.4. Results for Pressure Drop and Flow rates
After obtaining the needed equations, some assumptions and parameters will be taken into
account to obtain the pressure drop.
64
Assume a 100-cm2 cell area, operates at 3 atm and 60℃, and that the flow field consist of 24
parallel serpentine channels with 1mm deep, 1mm wide and 1mm apart. Assume the current density
of the cell at 0.7A/cm2 and voltage of 0.65V, the kinematic viscosity 2*10e-5 kg/ms, O2 content in
the air is 0.21, stoichiometric ratio of O2 equals to 1 and relative humidity is equal to 1. [15]
Assume no Bends.
Solving the equations in section 3.8.2 by using MATLAB (refer to Appendix A.6.1), we can
obtain that the pressure drop is equal to 33.36 Pa.
For the flow rates, by using MATLAB (refer to Appendix A.6.2) for the equations obtained
in section 3.4.3 to obtain the results. The assumptions and parameters to take into account:
Number of layers = 6
Current (amp) = 0.6
Simulation time (s) = 20
Faraday’s Constant (coulomb/mole) = 96485.3385
Ideal gas constant (J/K-mol) = 8.314472
Total pressure (bar) = 1
Active area (cm^2)—only used for current (Amps) = 0.03
Phi = 1
Molecular weight of water = 18
Volumetric flow rate of wet hydrogen (m^3/s) = 1.7e-8
Volumetric flow rate of air (m^3/s) = 1e-6
Initial temperature (K) = 293.2
Initial fluid temperature (K) = 353.3
Initial air temperature (K) = 273.5
Inlet humidity of air = 1
Fraction of O2 in air = 0.21
65
Fraction of N2 in air = 0.79
Damping factor = 0.6
By Solving on MATLAB, the following results were obtained:
Figure 3-24: Water and hydrogen flow rates after 20 seconds of simulation time
Figure 3-25: Water and hydrogen flow rates after 120 seconds of time simulation
From Figure 3.24 and Figure 3.25, it is allowable to see that the amount of hydrogen and air entering
the cell reach a specific value and continue constant in order for the cell to function normally and
with maximum efficiency, while for the water it keeps increasing with time until it reaches a time
were the removal of water should be done on the cell, so the flow rate of water will be zero till it
reaches the peak again, and this process keeps going in the cell which is the water removal.
66
CHAPTER 4. NUMERICAL SIMULATION
4.1. INTRODUCTION [25]
In this chapter, a numerical simulation will be conducted in order to plot and discuss results
about the variation of some parameters throughout the Polymer Exchange Membrane fuel cell under
some conditions that will be mentioned later on in this chapter. This simulation will be done on ANSYS
as follows:

The geometry will be first created on WORKBENCH.

The mesh will be generated using the Add-In Mesh Generator of ANSYS.

The setup of boundary conditions, materials selection, and solution will be proceeded
on Fluent.
The model will give comprehensive information on the current flux all over the electrically
conductive regions, temperature distribution, and the mass fraction distribution in the anode side and
cathode side.
For further information about the mathematical models and mathematical conventions used
in ANSYS Fluent, refer to APPENDIX B.
4.2. GEOMETRY AND FLOW CHANNELS [25]
For this simulation, the fuel cell geometry will be symmetric with respect to the membrane
and a Counter-Flow of the gases across the membrane will be considered. Figure 4.1 will illustrate the
sections of the PEMFC that will be sized.
67
Figure 4-1: Layers of counter flow PEMFC
The following sizing will be considered: [24]

𝑤𝑖𝑑𝑡ℎ 𝑜𝑓 𝑓𝑢𝑒𝑙 𝑐𝑒𝑙𝑙 = 2.4 𝑚𝑚

𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑓𝑢𝑒𝑙 𝑐𝑒𝑙𝑙 = 125 𝑚𝑚

ℎ𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑐𝑜𝑙𝑙𝑒𝑐𝑡𝑜𝑟 = 1.2 𝑚𝑚

ℎ𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑔𝑎𝑠 𝑑𝑖𝑓𝑓𝑢𝑠𝑖𝑜𝑛 𝑙𝑎𝑦𝑒𝑟 = 0.21 𝑚𝑚

ℎ𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑐𝑎𝑡𝑎𝑙𝑦𝑠𝑡 𝑙𝑎𝑦𝑒𝑟 = 0.012 𝑚𝑚

ℎ𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑚𝑒𝑚𝑏𝑟𝑎𝑛𝑒 = 0.036 𝑚𝑚

𝑜𝑣𝑒𝑟𝑎𝑙𝑙 ℎ𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑓𝑢𝑒𝑙 𝑐𝑒𝑙𝑙 = 2.88 𝑚𝑚
Figure 4.2 shows the generated geometry using the previous dimensions.
68
Figure 4-2: Geometry of a counter flow PEMFC
4.3. MESH GENERATION [25]
As a first step, a primary mesh shown in Figure 4.3 and figure 4.4, was done on our geometry
in order to solve the case. The mesh element length was selected to be of a constant value of 𝟎. 𝟏 𝒎𝒎
in all directions, resulting in a cubic element of volume 𝟎. 𝟎𝟎𝟏 𝒎𝒎𝟑. In this mesh, the change of
element size on proximity and curvatures was not taken into consideration and neglected, leading to
the division of the geometry into 864400 volumetric elements.
Figure 4-3: First Mesh, Monochrome Picture
69
Figure 4-4: First Mesh Colored
Now, as a second step, in order to increase the accuracy of the results and get smoother plots,
the mesh element length was modified to be 0.08 𝑚𝑚 in the x-direction, 0.07 𝑚𝑚 in the y-direction,
and 0.1 𝑚𝑚 in z-direction. Moreover, the change of element size on proximity and curvatures was
enabled with a minimum element length of 0.01 𝑚𝑚. This change in the mesh element size almost
doubled the number of volumetric elements as the number of the volumetric elements became
1,506,033. Refer to figure 4.5 and figure 4.6, that show the new mesh.
Figure 4-5: Second Mesh, Monochrome Picture
70
Figure 4-6: Second Mesh Colored
This previous mesh will be considered as the number of cells is large enough for the solution
and analysis of the PEMFC.
4.4. SOLUTION AND SETUP [25]
In order to model and simulate the PEMFC, an assumption of constant temperature of 353 K
all over the fuel cell at the beginning is considered, and the fuel cell is operating under a pressure of
200 kPa. [15]
At the inlet of the anode side, humidified hydrogen stream enters the fuel cell with an
assumption of no liquid. The following boundary conditions were taken:
1)
𝑇𝑖,ℎ𝑦𝑑𝑟𝑜𝑔𝑒𝑛 = 298 𝐾
2)
𝑚̇ℎ𝑦𝑑𝑟𝑜𝑔𝑒𝑛 = 6.0 × 10−7 𝑘𝑔/𝑠
3)
Mass fractions of:
- 𝐻2 : 0.6;
- 𝐻2 𝑂: 0.2
At the inlet of the cathode side, humidified air stream enters, with the same assumption of no
liquid.
1. 𝑇𝑖,𝑎𝑖𝑟 = 298 𝐾
2. 𝑚̇𝑎𝑖𝑟 = 5.0 × 10−6 𝑘𝑔/𝑠
71
3. Mass fractions of:
- 𝑂2 : 0.21;
- 𝐻2 𝑂: 0.1
For the outlet of both sides, cathode and anode, a backflow temperature 𝑇𝑜𝑢𝑡 is considered to
be 298 K.
𝑇𝑜𝑢𝑡 = 298 𝐾
For this model, a 200 iterations solution was proceeded with lowering the under-relaxation
factors to achieve the convergence of the solution. (Figure 4.7)
Figure 4-7: Calculating the solution with 200 iterations
4.5. PLOTS AND RESULTS
At the beginning, using the first mesh, a vector plot shown in Figure 4.8, which describes the
current flux, in the xy-plane located half-way along the length of the fuel cell. The vectors are colored
according to their magnitude. But according to the plot, the vectors need to be smoothed as the plot
shows a bit of broken edges.
72
Figure 4-8: Current flux located half-way along the length of the fuel cell using first mesh
In order to do so, and smooth the vectors, the second pre-mentioned mesh was generated and
a new plot for the current flux in the xy-plane located half-way along the length of the fuel cell, shown
in Figure 4.9, was created.
Figure 4-9: Current flux located half-way along the length of the fuel cell using second mesh
73
After creating the second mesh, and analyzing the results, we can conclude that no further
meshing is needed as the results were almost consistent, and now our results are grid independent.
Further analysis of Figure 4.9, gives a clear deduction that the current-flux is mostly
concentrated near the gas flow channels, especially on the edges that are in contact with the gas
diffusion layer and close to the membrane, this is where all the process takes place. Where the currentflux is mostly concentrated is where the free electrons are accumulated and waiting for a load to flow
through, which means the load should be connected at these points. As we go further from the center
of the fuel cell, we can notice that the current-flux decreases until it is almost diminished at the edges.
This is due to the escape of free electrons or found a vacancy to locate in an ion.
Secondly, in order to analyze the temperature distribution in the PEMFC, and where the
temperature is generally concentrated, a contour plot of the static temperature distribution was plotted.
The static temperature contour shown in Figure 4.10, shows that the highest temperature in
the fuel cell of magnitude 353 K, is located at the current collectors. Then it decreased gradually
through the gas diffusion layers and catalyst layers to reach an average temperature of 335K, and kept
on decreasing as it goes further away towards the outside of the PEMFC. This decrease lead to an
average temperature of 315 K at the outer walls of the fuel cell.
Figure 4-10: Contour Plot of the Temperature distribution along the PEMFC
74
Lastly, a study of the mass fraction at both, anode side and cathode side, was conducted. This
study included a contour plot of the mass fraction of hydrogen at the anode side, and another plot of
the oxygen’s mass fraction at the cathode side. (Refer to Figure 4.11 and Figure 4.12)
Figure 4-11: Contour Plot of Hydrogen’s Mass Fraction along the length of the PEMFC
Figure 4-12: Contour Plot of Oxygen’s Mass Fraction along the length of the PEMFC
75
Using Figure 4.11, it is obvious that the mass fraction of hydrogen decreases as it flows from
the anode inlet throughout the fuel cell to its outlet. At the inlet, its mass fraction was 0.6 then decreased
to an average value of 0.25 at the section that is in contact with the gas diffusion layer, and to another
value of 0.4 at the section that is in contact with the flow channel walls. Similarly, at the cathode side,
the mass fraction of oxygen, showed a sharper decrease at the section that is in contact with the gas
diffusion layer. At cathode side, the oxygen’s mass fraction diminished from its initial value of 0.21
to 0.095 at the section that is in contact with the gas diffusion layer, and to 0.13 at the section that is
in contact with the flow channel walls.
This decrease is quiet logical, as the reacting process takes place initially at the gas diffusion
layers.
76
CHAPTER 5. CONCLUSION
A mathematical model is the most efficient and realistic way to conduct an analysis about the
fuel cell, as the practical measurements of mechanical and electrical parameters is difficult due to the
thinness of the layers of the fuel cell and the fuel cell itself.
In order to analyze and examine these processes, a mathematical model was generated and a
numerical simulation was created, using MATLAB and ANSYS. The model considers thermal, fluid
dynamics, and mass transport equations. After proceeding in this study, a clear deduction was obtained
about the behavior of parameters with respect to each other, and with respect to time. Later then, a
numerical simulation was conducted using design parameters taken from several real PEMFC, with
necessary constants from previous literatures.
As mentioned before, fuel cell is formed of flow fields, catalyst layers, gas diffusion layers,
an electrolyte membrane and end plates to bond the layers of the fuel cell together. This model differs
from most other models as it considers each layer by itself as a control volume.
Firstly, starting with the electrolyte membrane, after analyzing the results, it can be seen that
the temperature and pressure changes are negligible relatively. From previous literature, it is mentioned
that high water concentrations in the fuel cell may cause corrosion and an efficiency drop. Therefore,
attention should be paid on the water concentration formed in the membrane and the removal of water
should be proceeded continuously.
Secondly, highly conductive catalysts should be chosen to form the catalyst layer in order to
reduce the Ohmic losses. As known, the power lost is a function of current, and as the current increases
the power lost also increases. This increase in the lost power will cause a voltage drop across the fuel
cell. Hence, a clear relation between the current density and the voltage is obtained, that is, as the
current density increases the voltage across the fuel cell decreases.
77
Concerning the gas diffusion layer, it is seen from the thermal analysis that it experiences a
high range in temperature, as it is in contact with the relatively cold gases and with the catalyst layer
having a high relative temperature. As a result, the material that should be chosen must have its
properties not highly affected with the change in temperature.
Furthermore, when talking about the fluid transport, the pressure difference between the inlet
and the outlet of the gas flow fields should be maintained high in order to achieve a higher velocity of
the flow. If higher velocities are achieved, this will cause an increase in the efficiency of the fuel cell
as mentioned in previous studies. [15]
Finally, from the numerical simulation proceeded on ANSYS, it can be seen that the current
flux is mostly concentrated near the membrane electrode assemblies as all the process takes place in
this region. This concertation in heat flux results in a higher temperature in the same region, that is
also seen from the previously plotted temperature contour.
As a future plan, in order to proceed further in this project, a study about the changes in the
behavior of the parameters will be conducted and check its variability with respect to flow field
geometry, and with respect to different materials.
78
APPENDIX A. MATLAB CODES
A.1. MATLAB FOR SECTION 3.3.1
79
A.2. MATLAB FOR SECTION 3.4
80
81
A.3. MATLAB FOR SECTION 3.5.5
82
83
84
85
A.4. MATLAB FOR SECTION 3.6.2
86
A.5. MATLAB FOR SECTION 3.7.3
87
88
89
A.6. MATLAB FOR SECTION 3.8.4
A.6.1. MATLAB for Pressure Drop
90
A.6.2. MATLAB for Flow Rates
91
92
93
APPENDIX B. ANSYS MATHEMATICAL MODELS AND
CONVENTIONS
B.1. ANSYS MATHEMATICAL CONVENTIONS
94
B.2. ANSYS MATHEMATICAL MODEL
95
96
97
98
99
100
101
102
103
104
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