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A ONE-DIMENSIONAL MODEL OF A PROTON-EXCHANGE MEMBRANE
PHOTOELECTROLYSIS CELL
A Honors Research Project
Presented to
The Honors College Faculty of The University of Akron
In Partial Fulfillment
of the Requirements for the Degree
Honors Bachelor of Science
Robert Daniel Moser III
May, 2011
ABSTRACT
A proton-exchange membrane photoelectrolysis cell (PEMPC) is a type of solar cell
which utilizes solar energy to split water into ions which are converted to oxygen
and hydrogen gas. The model proposed in this paper will examine several aspects
within the cell and track temperature, water content, electrical potential, and proton
concentration across the length of a PEMPC operating at steady-state. The solar
cell being modeled can be broken up into three regions: anode, membrane, and
cathode. Within the anode, a combination of applied and photo-induced current
densities split water molecules into hydrogen ions and oxygen gas. The hydrogen
ions then proceed into the membrane where they are attracted to the sulfonic acid
groups placed along the channels. Finally, the protons reach the cathode where they
are united with electrons to form hydrogen gas. Results from the model show that
reducing temperature and water content slightly increases hydrogen production. The
construction of the membrane as well as the number of sulfonic acid groups are shown
to play a somewhat noticeable role on hydrogen production. A key discovery is how
the relationship that current density and light intensity must be balanced to minimize
the power supply needed. The most important finding in the model is the tremendous
effect the size of electrodes and channels have on hydrogen production.
ii
TABLE OF CONTENTS
Page
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iv
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
v
CHAPTER
I.
INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
II. FORMULATION OF THE MODEL . . . . . . . . . . . . . . . . . . . . .
14
2.1 Anode Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
2.2 Anode-Membrane Interface . . . . . . . . . . . . . . . . . . . . . . .
20
2.3 Membrane Region . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
2.4 Membrane-Cathode Interface . . . . . . . . . . . . . . . . . . . . . .
25
2.5 Cathode Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
2.6 Treatment and Values of Constants . . . . . . . . . . . . . . . . . . .
28
2.7 The Complete Homogenized Model . . . . . . . . . . . . . . . . . . .
41
III. CONCLUSION AND FUTURE EXPANSIONS . . . . . . . . . . . . . .
45
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
iii
LIST OF TABLES
Table
Page
2.1
Known Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
2.2
Anode Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
2.3
Membrane Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
2.4
Cathode Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
2.5
Cell constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
iv
LIST OF FIGURES
Figure
1.1
1.2
2.1
2.2
2.3
Page
A representation of the three regions (Polymer backbone, hydration
shell, and water region) of the Nafion channels. . . . . . . . . . . . . .
4
Schematic (not to scale) of (a) the proton exchange membrane photoelectrolysis cell; (b) the silicon-germanium layers of the photocathode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
A representation of the length breakdown within the PEMPC and
the placement of the point charges . . . . . . . . . . . . . . . . . . . .
24
A two-dimensional view of the anode channel, from one electrode
rod to the next . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
The dimensions of the PEMPC cell used in the model
40
v
. . . . . . . . .
CHAPTER I
INTRODUCTION
Photovoltaic cells, colloquially known as solar cells, have become a very popular
option for an alternative, renewable, and clean energy source. With the sun producing
an abundant supply of light, devices that can transform light energy into other energy
forms are becoming increasingly more important. There are several different forms
of solar cells, all with their own positive characteristics and drawbacks. Some of
the most familiar solar cells commonly used are silicon-based solar cells, such as the
panels in calculators. While the efficiency of these cells are higher than most other
types, the major drawback is cost. Silicon is an expensive material for the largescale aspirations of photovoltaic cells. Silicon-based cells also are not flexible and can
become quite heavy. Another very popular form is fuel cells. Fuel cells take light,
hydrogen, and oxygen and convert them into electricity and water. This technology’s
research is receiving many funds and intense investigation. The drawback for the fuel
cell is the hydrogen. Though hydrogen is the most common element in the universe,
the acquisition of pure H2 is quite often done by burning coal. Using a fossil fuel to
produce hydrogen completely defeats the purpose of the fuel cells. A solution to this
problem is a solar cell that utilizes water electrolysis to produce hydrogen.
A photovoltaic cell that produces hydrogen through water electrolysis seems almost
1
poetic. The cell could produce hydrogen using solar energy, and then that hydrogen
could be used to power a fuel cell. To keep the poetry in motion, the fuel cell produces
water that could be used by the hydrogen-producing cell. The purity of such a cycle
which once only existed in the dreams of clean-energy enthusiasts has now become
a scientific reality. Hydrogen-producing solar cells through water electrolysis have
been given many names; in this paper, hydrogen-producing cells will be referred
to as either water electrolyzer cells or photoelectrolysis cells. The point needs to
be made that hydrogen production is not an advantageous investigation solely to
benefit fuel cell technology. Hydrogen is expected to take many of the places of
fossil fuels in today’s world [1]. Perhaps the best attribute of the water electrolyzer
cell is that the technology is very similar to that of the fuel cell. Thus, all of the
technology, structures, and discoveries in fuel cell research can easily be translated
to the photoelectroysis cell. One of technologies in particular that can be used in the
water electrolyzer cell is the proton exchange membrane (PEM) also referred to as
the polymer electrolyte membrane.
A water electrolyzer cell works by by breaking up water molecules into hydrogen ions,
dioxygen molecules, and electrons in the anode region. The chemical equation for this
process is
H2 O À 2 H+ + 0.5 O2 + 2 e− .
(1.1)
The separation requires energy, such as solar. Being that the hydrogen ion is surrounded by water, it will likely bond to form hydronium (H3 O+ ), Zundel (H5 O+
2 ),
or Eigen (H9 O+
4 ) cations [2]. For simplicity sake, these ions will be referred to as
2
protons. The electrons then travel through an external conductor, through a load,
and finally combine with the protons in the cathode region to produce hydrogen gas.
The path of the proton is more complex.
The proton travels through a proton exchange membrane before arriving at the cathode. There are several different PEM’s, but the one considered in this paper is Nafion.
Nafion is a polymer containing sulfonic acid groups to aid in the movement of the ions,
in our case protons [3]. Nafion channels can be thought of as having three regions: a
hydrophobic polymer backbone, a hydration shell with the hydrophilic sulfonic acid
groups, and the water region where the proton travels [3]. The hydration shell serves
to prevent the proton from bonding with the sulfonic acid groups [3]. The purpose of
the sulfonic groups are to create a “hopping” mechanism where the positively charged
H+ will be attracted to the negatively charged sulfonic acid group SO−
3 dissolved in
water. From there, the H+ will ”hop” to the next SO−
3 and so on until reaching the
cathode [2]. A representation of the proton exchange membrane is shown in Figure
1.1. After reaching the cathode, the protons are reunited with the electrons to form
hydrogen gas
2 H+ + 2 e− À H2 .
(1.2)
In the model in this paper, the anode and cathode regions are composed of layers of
silicon and germanium as well as a catalyst to aid in the separation and recombination. Silicon and germanium are common materials in the proton-exchange membrane
photoelectrolysis cell (PEMPC) and fuels cells as well as having known diffusion properties, temperature dependence, and activation energies [4]. Figure 1.2 represents the
3
Polymer Backbone
−
SO3 Hydration Shell
−
SO3
−
SO3
−
SO3
−
SO3
Water Region
−
SO3
Figure 1.1: A representation of the three regions (Polymer backbone, hydration shell,
and water region) of the Nafion channels.
general structure and movement of various particles within the proton exchange membrane photoelectrolysis cell (PEMPC).
The objective in this paper is to examine the proton concentration throughout the
PEMPC at a steady state. The model takes into account the number of the SO−
3
groups within the PEM, the temperature, varying potentials throughout the cell, water content, light intensity, cellular dimensions, and current density. By knowing the
concentration of H+ within the cell, many question can be answered including:
• How does the number of the sulfonic acid groups in the PEM affect the H+
mobility?
• What is the potential distribution while the cell is running?
• To what extent does temperature affect the ion transport?
4
Membrane
Load
Electrocatalyst
O2
H2
e−
Silicon
hv
H
+
Germanium
Photoanode
Polymer Backbone
Photocathode
H2 O
Proton Exchange
Membrane
Current Conducting Support Scaffold
H 2O
(a)
(b)
Figure 1.2: Schematic (not to scale) of (a) the proton exchange membrane photoelectrolysis cell; (b) the silicon-germanium layers of the photocathode.
• What role does water content play in a cell’s operation?
• How does the size of each region affect hydrogen production?
• How do varying light intensities and current densities alter hydrogen production
and needed power supply?
• Is the PEMPC effective enough to produce a large amount of hydrogen gas?
Answering these questions contributes to the understanding of the effectiveness and
limitations of the PEMPC. With this knowledge, it can be determined whether the
technology is worth investment and capable of meeting the function of a large-scale,
clean energy source.
Research strictly in the PEMPC seems rather limited, but as mentioned earlier, the
research in fuel cells can easily be reformed to suit the needs of a PEMPC model.
5
The research in fuel cells is quite vast but can be broken into several groups. A major
area of research is modelling the structure of the PEM. Other areas include heat and
water transport in the PEM, proton transport within the PEM, and the effects of
water content.
A prominent, early paper in the research of PEMPC is written by Nie et al. [1]. The
paper is among the first to analyze how the voltage-current, temperature, and light
intensity affect the PEMPC. A circuit model of the cell is created to represent the
water electrolysis process through solar energy. Nie’s model takes into account the
anode, cathode, and membrane utilizing Bulter-Volmer expressions, Nernst potential, Ohm’s Law, and mass balances. Several assumptions are made in the creation
of the model. The model assumes a well-mixed anode and solutions in chambers, no
transport limitations, effective transfer coefficients, and interfacial resistance. The
conclusions found by Nie et al. are that temperature affect both power supply and
hydrogen production but in opposite ways. The power supply decreased with increased temperature, yet the hydrogen production increased by nearly 11% with an
increase of 50◦ C. The major limitation of the paper is the lack of inclusion of sulfonic
acid groups’ contributions. This limitation requires a further look into the membrane
portion of PEMPC.
The proton exchange membrane is an area of immense investigation. Akinaga et
al. [3] employ the Lattice-Boltzmann method to the problem of proton conductivity
within Nafion. The problem considers electrostatic forces, channel width, and the distance between the sulfonic acid groups. The model is two-dimensional and assumes
6
protons move only due to diffusion and electrostactic forces, sulfonic acid groups are
evenly distributed throughout the membrane, and Nafion channels have a sandwich
structure similar to that shown in Figure 1.1. A finding of Akinaga et al. is that the
mobility of protons is very dependent on channel width and SO−
3 placement, a fact
that causes problems with the Nie model. Another key finding is the cyclic pattern
the electrostatic potential has in the membrane. A key limitation of the Akinaga
model is the absence of how a proton’s position changes its mobility. Though Akinaga et al. comment that the mobility was assumed to be the same throughout the
channel, the team is well aware the mobility would differ depending on a proton’s
location, especially in relation to a sulfonic acid group but comment that an average
mobility was the purpose of their paper. While the paper leaves out many variables
such as temperature and water concentration, the model contributes much information to the general potential distribution in the membrane.
As mentioned previously, the proton exchange membrane’s (in particular Nafion)
structure is an area receiving much research by the scientific community. Elliott
and Paddison [2] review different simulations of Nafion’s structure and morphology, especially when swelling and shrinking. The researchers review and analyze
methods of percolation models, mesoscale models, multiscale ONIOM methods, and
other approaches attempting to model the variation in structure and proton transport
in Nafion with varying water content, sulfonic-acid-group clusters, flexible polymer
backbones, and freedom of movement in the side chains. Elliott and Paddison find,
through simulations best matching experimental data, the polymer backbone and
7
side chain flexibility and position in low water content significantly determine the efficiency of proton transport. Elliott and Paddison comment that the simplistic models
took only into account vehicular tranpsort such as diffusion, eletrostatic force, fluid
motion, etc. while ignoring Grotthus shuttling mechanisms, ”structural diffusion” or
the proton jumping from one ion to another ion because of hydrogen bonds, which
have been found to be a contributing force when water content in the membrane is
high. They also found vehicular and Grotthus shuttling mechanisms contributed to
proton transport but worked against each other in high water content. Elliott and
Paddison’s results match that of Kreuer [5]. Kreuer reviews two different materials
for consideration of a proton exchange membrane, Nafion and sulfonated polyetherketones. Kreuer’s findings show that Nafion’s wide channels perform better in low
water content but have a negative effect with high water concentration due to water
flow, drag, and permeation. Kreuer also develops several ways to increase conductivity and mobility by increasing acidity, decreasing distance between groups, dissolving
protons in heterocycles especially in high temperatures, and incorporating methanol.
Both papers analyze Nafion at a nearly microscopic level to find its limitation, and
both find water transport and temperature to be crucial in Nafion’s efficiency.
Heat and water transport are major factors in the proton exchange membrane’s efficiency. Chen, Chang, and Fang [6] study the water transport throughout the membrane. The team tracks the water content from one catalyst layer to the other through
numerical simulation. The results show that the water within the membrane is highly
susceptible to flooding, has sudden drops at interfaces, and should be thin to produce
8
best results. The model was one-dimensional and did not take into account the location and distribution of the sulfonic acid groups nor the structure of the membrane.
Even with the assumptions of Chen et al., the results match those found by other
researchers. Afshari and Jazayeri [7] examine water transport but taking temperature
into account as well. They find the temperature distribution is the most important aspect affecting water transport. The team also find that flooding is a problem based on
various water inputs but the temperature significantly affects the extent of saturation
in the membrane. The temperature distribution is greatly affected by the voltage,
and the highest temperatures occur at the catalyst layer of the cathode in fuel cells,
which is equivalent to the anode of the photoelectolysis cell. This two-dimensional
model was non-isothermal and was based on conservation laws and electrochemical
properties. Many others have examined how varying structures and materials affect
the efficiency of the cell and water transport of fuel cells including Kang [8] and Du
[9]. Their results show various structures and materials increased efficiency but the
membrane still had flooding problems and deficiencies with low water content.
The approach taken in this paper is done by utilising conservation laws, material
properties, and numerical methods to find a steady-state solution to the temperature,
water content, potential, and proton concentration throughout an entire running proton exchange membrane photoelectrolysis cell. The cell is viewed as being composed
of three regions: anode, membrane, and cathode. Each section will have its own
governing equations and constants. How the regions interact at interfaces will also be
discussed and used in the calculation. The PEMPC being modeled lacks gas diffusion
9
layers which are common in fuel cells and water electrolyzers. The PEMPC in this
model is being supplied with completely liquid water at the anode and cathode. Our
model does not include these water chambers located along the outer boundaries of
the cell.
As most people know, a lack of sunlight is an obvious problem with a solar cell; a
PEMPC is no exception. our model allows various amounts of sunlight to hit the
surface. Like some other cells, an initial or continuous power supply most be used
to provided to begin the electrolysis process. To keep a constant current density in
the cell, a power supply is altered to maintain this condition. The amount of power
required to maintain a constant current density is evaluated for various light intensities. Another problem with current solar cells is the size and weight. Polymers are
used to reduce the weight and increase flexibility. The sizes of a PEMPC may vary.
The PEMPC modeled is this paper has a 1cm×1cm face. The lengths of the anode
and cathode are limited to less than 100µm long [10]. Most electrodes are not that
long. For example, the size of the anode and cathode catalyst layers in Kang [8] is
12µm. The Nafion membrane can vary in size. Kang [8] uses a membrane of length
18µm while Nie [1] used a membrane of length 178µm. Typically, the goal is to make
the membrane moderately small.
Based on a constant current density within the cell, an initial voltage can be found
based on voltage drops across the cell as by Nie [1]. The initial hydrogen concentration can also be found by the hydronium concentrations found in pure water. Proton
transport mechanisms such as diffusion, electrostatic forces, and source terms deter10
mine the concentration within the anode. At the anode-membrane interface, there is
a voltage drop similar to that found in Nie [1] and a continuity in the concentration of
protons, flux of protons, and electrical flux. Diffusion and potential control the transport of protons within the membrane with varying sulfonic acid group locations and
overall charge density in the membrane. The membrane-cathode interface conditions
are the same as those in the anode-membrane interface. In the cathode, concentration
is determined by diffusion, potential, and sink terms. Finally, the outer boundary
of the cathode has fixed flux in the concentration, assuming protons are escaping
through the cathode boundary at a fixed rate, and the voltage is zero because the
voltage drop across the cell should be that of the initially prescribed voltage at the
anode boundary.
The model takes into account the temperature, water content, current density, light
intensity, cell size, and the charge density within the membrane at steady state. While
many scientists have experimented with different liquid inputs and various ions dissolved in water, our model is done for a PEMPC where pure, neutral water is pumped
into the system. The one-dimensional model assumes:
• Proper hydration within the membrane
• Nafion channels are linear and fixed in size
• Isothermal regions are assumed for calculating material properties
• The convection of water is not considered to aid in the movement of protons
• No gas diffusion layers or water chambers are present
11
The simulation done in this paper is one of the first to analyze concentrations throughout an entire photoelectrolysis cell and consider the placement of the sulfonic acid
groups. Our model also shows the potential, temperature, and water content throughout the entire cell while operating at a steady state. This paper is a unique and nearly
complete view of all the major aspects affecting a PEMPC.
The results from this paper show the importance size has on a cell’s hydrogen production. The size of each region, especially the electrodes, can alter the production
by over 100% when altering the total size of the cell by as little as 50%. Factors
such as temperature and membrane size had very minute contributions to hydrogen
production even when altered rather significantly. The membrane also showed to
have a problem with over-hydration. To increase the production of a cell, the sulfonic acid groups should be placed in a way such that at steady state the charges of
hydrogen ions and the acid groups are offset. The increase in the number of sulfonic
acid groups are also shown to increase in the aid of transport and overall hydrogen
production within the cell. Finally, the light intensity and current density play an
extremely important role on the power required to keep the cell operational. The
balance between the two determine whether the PEMPC is worth investment.
Several limitation arise in the model mainly due to the assumptions. Hydration is
often hard to properly maintain and channel size varies depending on the water content. Water flow is often a factor in proton movement. Finally, the channels of Nafion
are not linear, tube-like structures but are porous, weaving chambers that often cause
Grotthus shuttling mechanisms to become significant [2]. These limitations are areas
12
of future research to more accurately model a PEMPC.
The rest of the paper will describe the formulation, analysis, and results of our model.
Chapter II covers the formulations of the governing equations used in the experiment
as well as describing more thoroughly each domain of the model. Chapter III presents
all the findings as well as their interpretations. Finally, Chapter IV concludes the
document with a summary of significant findings and future endeavors.
13
CHAPTER II
FORMULATION OF THE MODEL
As mentioned in the introduction, the photovoltaic cell under investigation can be
broken up into three sections each with their own governing equations. These sections
are the anode, the membrane, and the cathode. To help identify which region is being
described, subscripts are used. Variables and constants with a subscript ’a’ are those
used in the anode region, ’m’ in the membrane, and ’c’ in the cathode. Constants
and variables without subscripts are consistent throughout the entire cell. For ease
of understanding, this chapter is broken into subsections highlighting and explaining
the governing equations in the anode, membrane, and cathode separately as well as
the anode-membrane interface and membrane-cathode interface. A section in this
chapter will give the values, derivation, and assumptions placed on the constants
in the model. The final section in this chapter provides the complete homogenized
model.
2.1 Anode Region
The anode is where all the photochemistry occurs. The light hitting a photocatalyst
in the anode begins a chemical reaction that triggers the splitting of water creating
14
hydrogen ions, oxygen, and electrons. The electrons travel out of the cell, through
a conductive wire, and to the cathode where they combine with hydrogen ions to
produce hydrogen gas. The movement of the electrons creates a current. Often with
solar cells, a current must be induced to start or maintain the cell. The current
travelling through the system causes there to be a voltage drop across the entire cell.
The model calculates the total voltage drop across the cell from a prescribed current
density and light intensity. The total drop in potential Φ◦ can therefore be prescribed
as the initial voltage to the cell, and at the end of the cell in the cathode, a prescribed
voltage of zero.
To determine the potential throughout the entire anode, Ohm’s law was used to
determine how the potential behaves. Applying Ohm’s law to the anode, we find
dj
= −SΦ
dx
and σ
dΦ
=j
dx
(2.1)
where SΦ is a source or sink term, σ is the ionic conductivity, and j is the current
density. To get an idea of what a channel looks like in the anode as well as the
directional components see Figure 2.2. For the anode, the source of potential is
related to the current transferred to cell per volume [9]. Du [9] refers to this source
term as a volumetric current density j V , and Kang [8] and Afshari [7] have a similar
term denoted as the transfer current density. The exact calculation and derivation for
the transfer current density will be explained in a later section. Through substitution,
the governing equation for potential in the anode is reached and can be expressed as
Φa,xx = −
15
jV
.
σa
(2.2)
As shown by Afshari [7], the temperature within an operating solar cell fluctuates
based on the location within the cell. Our model assumes the outer boundary of the
anode is held at a constant temperature T◦ . This assumption is validated because
water is being pumped in at a fixed temperature and most likely keeps the entire face
of the anode at one temperature. Second, the model does not take into account water
in a gas phase. So, the governing equation lacks terms dealing with water changing
states. Third, the model does not take into account the flow of water. Thus, there
exist no advection terms in the temperature profile equations.
Taking the equation from Afshari [7] and making some adjustments, one determines
the governing equation for temperature to be
∇(k∇T ) + ST = 0,
(2.3)
where k is the thermal conductivity, T is the temperature, and ST is the source or
sink term for heat power generated per volume. There are several sources for heat
generation in the anode region. The first source of heat is due to the generation from
the overpotential. The overpotential is defined as the resistance at the surface of the
electrode [1]. This resistance causes heat generation. In order to determine the power
generated per volume from overpotential, one multiplies the transfer current density
travelling through the system by the total overpotential of the cell, j V (Φ◦ − V◦ ) [7].
The second source of heat is from the splitting of water. The separation of water
causes an increase in entropy. The rise in entropy causes heat to be added to the
system in what Afshari refers to as entropic heating. In order to determine the
16
amount of heat generated, laws from thermodynamics and electrochemistry are used.
Using the second law of thermodynamics, we find that the heat Q generated from
a reversible process is equal to the product between temperature and the change in
entropy ∆S making the equation Q = T ∆S [11]. Entropy is related to Gibbs’ free
¡ ∂G ¢
energy G in the following way S = −
∂T
[11]. Finally, electrochemistry states ∆G =
−e− F U◦ where e− are the number of moles of electrons in the reduction process, F
is the Faraday constant, and U◦ is the thermodynamic equilibrium potential [11].
Combining the three relations together, one finds heat generation is a function of
equilibrium potential, Q = e− F T
¡ ∂U◦ ¢
∂T
. Because power generation is of interest in
this process, some conversions need to be done. Instead of multiplying by charge
(e− F ), one can multiply by the charge density per time also known as the transfer
current density. Thus, the power generated from the splitting of water is j V T
¡ ∂U◦ ¢
∂T
.
The final term in the source term for heat generation in Afshari’s equations is caused
by Joule heating [7]. Joule heating is caused by the resistance of a current to flow
through a system. The power generated from this process is given by Joule’s first law
which states the power generated is equal to the square of the current multiplied by
resistance [11]. Since resistance is equivalent to the reciprocal of the conductivity, we
get the power of Joule heating to be
j2
.
σ
Summing all equations of power generation,
we arrive at the equation for the source term in the anode. Substituting this into
equation (2.3), one gets the entire governing equation for temperature in the anode
µ
V
V
∇(ka ∇Ta ) + j (Φ◦ − V◦ ) + j Ta
17
∂U◦
∂T
¶
+
j2
= 0.
σa
(2.4)
A major factor in a solar cell’s effectiveness is water content, especially in the membrane. In our model, as well as many other articles on the subject, water content is
the ratio of moles of water to the moles of sulfonic acid groups. In order to find water
content’s effect on the cell, one must model the hydration throughout the entire system. Using the laws of conservation of species from Kang [8], one finds the governing
equation for water content λ
¶
µ
³ ρ
´
j
∇
DW ∇λ − ∇ nd
+ SW = 0,
EW
F
(2.5)
where ρ is the density of the membrane, EW is the equivalent weight of the membrane
which is a ratio of the mass of the membrane over the number of moles of sulfonic acid
groups in the membrane, DW is the diffusion coefficient of water, nd is the electroosmotic drag, and SW is a source or sink term for water content. The electro-osmotic
drag coefficient is proportional to the water content present, nd =
2.5
λ
22
[8]. The first
term in the governing equation for water content is due to the diffusion of water,
and the second term is a drift term caused by the induced current in the system.
In the anode, water is lost due to electrolysis. To determine exactly how much is
lost, we need the formula for the chemical reaction and the transfer current density.
For every mole of water split, two moles of electrons are formed. Since the transfer
current density is known and can easily be converted to molar density per time, one
can find the sink term. The transfer current density divided by the Faraday constant
gives the moles of electrons per volume per time. Thus, dividing the molar density
per time by two provides the molar density of water per time. Because the splitting
18
of water takes water out of the system the sink term in the anode should be negative
V
j
and be given as SWa = − 2F
[8]. Substituting the drag and sink term into the equation
(2.5), one arrives at the governing equation for the water content in the anode
µ
¶
³ ρ
´
2.5 j
jV
∇
DWa ∇λa − ∇
λa −
= 0.
EW
22 F
2F
(2.6)
The final aspect the model tracks is hydrogen concentration. The water being pumped
into the cell is assumed to be pure water which has a pH of 7. Because the pH
calculation is the negative common logarithm of the concentration of hydrogen ions,
converting the pH 7 into the proper units provides an initial concentration n◦ at
the entrance of the anode. Thus, defining na to be the concentration of hydrogen
throughout the anode the outer boundary condition for the anode is na (0) = n◦ .
The model does not take into account the flow of water in the system. Therefore, the
flux of protons in the cell is caused by diffusion and drift. The flux of hydrogen J is
J = D∇n + µn∇Φ,
(2.7)
where D is the diffusion coefficient, n is the concentration of hydrogen, and µ is the
mobility coefficient [12]. The hydrogen ions are also created in the anode and removed
in the cathode, so the model will have a source or sink term. The general equation
for a diffusion-drift process is given by the following equation
nt = ∇J + S,
(2.8)
where S is a source or sink term [3]. Since hydrogen is created in the anode, the
anode governing equation will have a positive source term. The concentration of
19
hydrogen produced should be based on the current flowing through the system. The
concentration produced per second is found by dividing the transfer current density
by the Faraday constant, so S =
jV
F
[3]. We seek a steady-state solution. Thus, the
governing equation to solve in the anode is
∇ (Da ∇na + µa n∇Φa ) +
jV
= 0.
F
(2.9)
The values of the coefficients and constants will be discussed in the later section
entitled Treatment and Value of Constants 2.6.
2.2 Anode-Membrane Interface
At the interface between the anode and the membrane, there is expected to be an
ohmic drop due to a change in material. Being that the total interfacial drop ηi ,
which accounts for 5% of the total ohmic drop within the cell, is the sum of both the
anode-membrane and membrane-cathode interface, the anode-membrane interface is
assumed responsible for half the total interfacial drop. Thus at the interface,
Φa (xam ) −
ηi
= Φm (xam ),
2
(2.10)
where xam is the length of the anode and represents the distance where the anode
and membrane meet. The other condition that must be met at the anode-membrane
interface is continuity of electric displacement field. The electric displacement field
is defined as the product between permittivity and the electric field [13]. Being that
the electric field can be defined as the flux in the potential, the electric displacement
20
field can be expressed in terms of the voltage in the cell. To maintain continuity, the
displacement fields of each side are set equal,
²a ∇Φa · n̂(xam ) = ²m ∇ · Φm · n̂(xam ).
(2.11)
Here ² represents the permittivity and n̂ is a normal vector in the x direction.
Being that the temperature in the water throughout the PEMPC is being tracked,
the temperature is expected to be continuous at the interface,
Ta (xam ) = Tm (xam ).
(2.12)
The flux in the temperature at the interface is expected to be the same in the anode
and the membrane. Thus, the other condition imposed is
ka ∇Ta · n̂(xam ) = km ∇Tm · n̂(xam ).
(2.13)
The amount of water flowing through the cell should be continuous. This assumption
causes there to be a boundary condition where the water content in the anode at the
interface matches that of the membrane at the interface
λa (xam ) = λm (xam ).
(2.14)
The other condition the water content is desired to satisfy at the boundary is equal
flux due to diffusion of water which produces the equation
DWa ∇λa · n̂(xam ) = DWm ∇λm · n̂(xam ).
21
(2.15)
Similar to the rest of the variables, two boundary conditions are imposed for concentration at the anode-membrane interface: equal concentration and equal flux. These
boundary conditions give
na (xam ) = nm (xam )
(2.16)
Ja · n̂(xam ) = Jm · n̂(xam )
(2.17)
for equal concentration and
for equal flux.
2.3 Membrane Region
In Nafion, sulfonic acid groups are attached to a polymer backbone. As described
in the introduction, these groups are added to create a hopping mechanism for the
protons. Due to the presence of these sulfonic acid groups, the electric field in the
membrane is not uniform. To account for this, Gauss’s law is used and written as
~ = C,
∇· (²E)
(2.18)
~ is the electric field and C is the space charge density [3]. The electric field is
where E
equivalent to the negative flux in electric potential. The space charge density depends
on the location of the sulfonic acid groups and their relation to the hydrogen ions. One
can think of every sulfonic acid group as having a charge equivalent to one electron,
and each proton having the charge of a proton. The charge of a proton is equivalent
to the absolute charge of an electron q, sometimes called the fundamental charge.
22
Thus, the hydrogen ions will have a charge of positive q, and the sulfonic acid group
will have a charge of −q since the charge of an electron is negative. To determine the
space charge density, one can divide the charge found at a point in the membrane
by the total volume of the membrane channel volm . To model the location of the
groups and ions, point charges are placed for each making the governing equation for
potential in the membrane
´ 1
q X³ H
SO3−
∇ Φm = −
δk − δk
,
²m k
volm
2
(2.19)
−
+
where δ H the Dirac delta function for the presence of hydrogen, δ SO3 is the delta
function for a sulfonic acid group, and q is the magnitude of a charge of an electron.
For Akinaga [3], the hydrogen molecules were to line up with the sulfonic acid groups
in the membrane. By superposing the molecules, an electroneutral membrane is
established. The model in this paper allows the molecules to align in other ways.
Figure 2.1 provides a picture demonstrating how the point charges are placed within
the cell.
The general equation for temperature flow as seen in equation (2.3) is the same
general governing equation for the membrane. Because a current is flowing through
the system, the presence of Joule heating still remains. Because no other chemical
processes are taking place in the membrane, the equation has no other source terms
for heat. The specific governing equation for the temperature in the membrane is
∇(km ∇Tm ) +
23
j2
= 0.
σm
(2.20)
Membrane
Anode
Cathode
H+
d *pos
x
0
xam
xmc
xc
d
_
SO3
Figure 2.1: A representation of the length breakdown within the PEMPC and the
placement of the point charges
A similar story can be told for water content. The general equation for water content
(2.5) can be used in the membrane. Water is not created or lost in the membrane,
making the source term zero. Hence, the governing equation for water content in the
membrane is
µ
¶
³ ρ
´
2.5 j
∇
DWm ∇λm − ∇
λm = 0.
EW
22 F
(2.21)
For concentration, the governing equation for the drift and diffusion process (2.8)
is employed in the membrane. The membrane simply has the hydrogen molecules
flow through it. The source or sink term is removed from the equation leaving the
governing equation for the membrane
∇ (Dm ∇nm + µm nm ∇Φm ) = 0.
24
(2.22)
2.4 Membrane-Cathode Interface
The conditions and rationale regarding the boundary condition at the anode-membrane
interface are the same as those in the membrane-cathode interface. Having xmc denote
the length within the solar cell where the membrane ends and the cathode begins,
the governing equations for the interface become
Φm (xmc ) −
ηi
2
= Φc (xmc )
Voltage drop at interface
²m ∇Φm · n̂(xmc ) = ²c ∇Φc · n̂(xmc )
Continuity of electrical displacement field
Tm (xmc ) = Tc (xmc )
Continuity of temperature
km ∇Tm · n̂(xmc ) = kc ∇Tc · n̂(xam )
Continuity of flux for temperature
λm (xmc ) = ∇λc (xmc )
Continuity of water content
DWm ∇λm · n̂(xmc ) = DWc λc · n̂(xmc )
Continuity of diffusive flux for water content
nm (xmc ) = nc (xmc )
Continuity of concentration
Jm (xmc ) · n̂ = Jc · n̂(xmc )
Continuity of flux for concentration
2.5 Cathode Region
In the cathode, the extent to which the potential changes should be dependent upon
the amount of hydrogen ions reaching the cathode. The approach for finding the
value of the sink term in the cathode is to use the modified Butler-Volmer equation
to find transfer current density in the cathode. The modified Butler-Volmer equation
25
reads
³ n F ηc ´
c
jc = j◦ ref e RT
n
(2.23)
where j◦ is the exchange current density, R is the gas constant, and nref is the
reference concentration [8]. The addition of the ratio in concentration is to allow
for failure in the membrane when no hydrogen ions make it to the cathode. The
current density in the cathode jc can be used to find a transfer current density in
the cathode. When doing the conversion, one would find that the transfer current
density is, for all practical purposes, equivalent to the ratio of the concentrations
multiplied to the transfer current density found in the anode. The sink term in
the cathode will therefore be equivalent to SΦc = −
nc V
j . Doing some simple
nref
mathematical manipulation and substitution for Ohm’s law in equation (2.1), the
governing equation for the cathode becomes
Φc,xx =
nc j V
.
nref σc
(2.24)
The drop across the entire cell is Φ◦ . Thus, the boundary conditions for the end of
the cathode xc is Φ(xc ) = 0.
As was seen in the membrane, the only source of heat generation in the cathode is from
a current travelling through the system. While there is a chemical reaction occurring
that converts hydrogen ions to hydrogen gas, the process happens so readily it will
occur without requiring or providing any heat to the solar cell. Applying the source
term to the governing equation (2.3), we find the equation that controls temperature
26
in the cathode is
∇ (kc ∇Tc ) +
j2
= 0.
σc
(2.25)
As was discussed in the anode section, the temperature on the outer boundary of the
cell can be held constant. The model assumes the temperature at the entrance of the
anode is equivalent to that at the exit of the cathode. Applying this assumption to
the temperature boundary condition at the end of the cathode, we have the condition
Tc (xc ) = T◦ .
In the cathode, water simply flows through the channels. No water is created or
separated making the source term for water content equivalent to zero. The governing
equation therefore becomes
µ
¶
³ ρ
´
2.5 j
∇
DWc ∇λc − ∇
λc = 0.
EW
22 F
(2.26)
The water content is assumed to be monitored at the entrance of the anode and exit
of the cathode. If this is the case, the water content at the edge of the cathode can
be considered constant and the equation λc (xc ) = λ◦ .
The movement of hydrogen ions in the cathode is by diffusion and drift. In the
cathode, the hydrogen ions are removed from the cell by combining with the electrons.
To account for the removal, a sink term must be considered. As was seen with
the potential, the amount removed depends on the number of ions making their
way to the cathode. The sink term will become the negative source term from the
anode multiplied by the ratio of the concentration to reference concentration Sc =
−
nc j V
. Replacing the definition for the sink term into the governing equation for
nref F
27
concentration in equation (2.8), we find the cathode’s governing equation
∇ (Dc ∇nc + µc nc ∇Φc ) −
nc j V
= 0.
nref F
(2.27)
A water chamber exists on the outer boundary of the cell. For an ideal cell, the
outer boundary of the cathode would have an impermeable outer edge, Jc · n̂(xc ) = 0,
making all the protons stay in the cell until converted to hydrogen gas. To allow for
a non-ideal cell, the flux at the cathode edge is set equal to a mass transfer coefficient
KM T multiplied by the difference the concentration of protons in bulk water and the
concentration at the edge of the cell, Jc · n̂(xc ) = KM T (n◦ − nc (xc )).
2.6 Treatment and Values of Constants
Table 2.1 displays all the known scientific constants used in the model. Other constants require derivations and other manipulation to get a value. While many of
the derivation will apply to multiple regions, the tables will be grouped according
to region. Table 2.2, Table 2.3, and Table 2.4 display the constants specific to the
anode, membrane, and cathode, respectively. Table 2.5 displays other constants that
are used in multiple regions as well as the default geometry of the solar cell in the
model. Though the science of a device that absorbs and converts sunlight into electricity and hydrogen is quite complex and extremely innovative, the process begins
with simple water molecules. Water, which covers over 70% of the Earth’s surface,
is pumped into the anode region. The anode region is composed of rods of polymer
backbones surrounded by a silicon shell enclosed in a germanium shell that is covered
28
Table 2.1: Known Constants
Symbol
Description
Value
Units
c
Speed of light in a vacuum
2.9979 × 108
m
s
D◦
Diffusivity of hydrogen in bulk water
9.3 × 10−9
m2
s
F
Faraday constant
9.64853399 × 104
C
mol
h
Planck constant
6.626 × 10−34
Js
kB
Boltzmann constant
1.3806503 × 10−23
J
K
me
Mass of an electron
9.1096 × 10−31
kg
n◦
Concentration of hydrogen in bulk water
10−4
mol
m3
NA
Avogadro’s number
q
Fundamental charge
1.602176487 × 10−19
C
R
Gas constant
8.314
J
mol K
²◦
Permittivity in a vacuum
8.854187 × 10−12
F
m
µ◦
Mobility of hydrogen in bulk water
qD◦
kB T̄
m2
Vs
6.02214179 × 1023 mol−1
by a photocatalyst, in this model platinum, to assist in absorbing sunlight. These
polymer rods encased by a metal shell jut out of the current conducting scaffold.
Our model assumes the rods form sheets as seen in Figure 2.3. The water flows in
between the rods. It is at the surface of these rods where the power and usefulness
of photochemistry begins.
The polymer rods are connected to a current-conducting apparatus. The value of the
29
current flowing through the system is chosen by the experimenter and is a combination of an applied current and one induced by the photoelectric effect [1]. When
light, made of photons, strikes the surface of a metal, electrons may be excited from
the highest occupied molecular orbital (HOMO) to the least unoccupied molecular
orbital (LUMO) [12]. By moving from the HOMO to the LUMO, the electrons are
able to move much easier. The germanium-silicon combination creates a magnetic
field that encourages the electron to move from the metal toward the conducting
apparatus [4]. From there, the electron travels through a wire to a load and returns
to the system at the cathode. When the electrons depart from the metal, they leave
a positively charged metal surface, an ideal location for the electrolysis of water.
Water particles near the surface of the photocatalyst will undergo oxidation. In doing
so, two water molecules are transformed into oxygen gas, four hydrogen ions, and four
electrons which take the place of electrons removed from the photoelectric effect. The
hydrogen will very likely bond with passing water molecules to from hydronium and
other ions which is exactly what we are tracking. An in-depth description and explanation of the aforementioned processes not only paints a more elaborate picture of
the exact mechanism in the anode but also explains the reasoning for every equation
and rationale used in the formulation of the model.
Physical chemistry states the number of electrons freed F (ν) from the surface of a
metal struck by light per unit incident light at a prescribed frequency ν is given by
the formula [1]
F (ν) =
me c2 ³
w´
1
−
.
h2 ν 2
hν
30
(2.28)
In this formula, me is the mass of an electron, c is the speed of light, h is the Planck
constant, and w is the work function of the metal, which is platinum in this model.
By multiplying the number of electrons released per energy of light by the intensity
of light Iν , one gets the number electrons released per area. The conversion to moles
of electrons released per area can be done by dividing by Avogadro’s number NA .
The final conversion to current density is found simply enough by multiplying by the
Faraday constant F which leaves coulombs(charge) per area. Thus, the equation for
current density jν induced by light is given by the formula
jν = Iν
w´
F me c2 ³
1
−
.
NA h2 ν 2
hν
(2.29)
Aforesaid, the total current density j is determined by the experimenter. Thus, in
order to find the applied current density japp , one simply subtracts the photocurrent
density from the total current density leaving japp = j − jν . The applied current
density can be used to find the overpotential at the surface of the metal in the anode
and cathode as well as the amount of power required to keep the cell operating.
The overpotential tells the voltage drop due to resistances along the surface of the
electrode. Each region of the cell has an overpotential associated with it as well as
the interfaces between the regions. By summing all the overpotentials in each region
with the minimum voltage to cause water to split, one can find the initial voltage
of the cell Φ◦ . In the field of electrochemistry, it has been discovered that applied
voltage is related to the overpotential of the electrode surface by the Butler-Volmer
31
equation
³
japp = 2j◦ e
Fη
RT
Fη
− RT
´
−e
,
(2.30)
where j◦ is the exchange current density, η is the overpotential at the surface of the
electrode, R is the gas constant, and T is the temperature [1]. Finding the inverse of
this function and applying it to the anode, one finds the overpotential of the anode
to be
RT
ηa =
sinh−1
F
µ
japp
2j◦a
¶
.
(2.31)
.
(2.32)
A similar equation will arise for the cathode
RT
ηc =
sinh−1
F
µ
japp
2j◦c
¶
The exchange current density is temperature dependent. Because T represents the
temperature on the surface of the plate, it is assumed the temperature at the surface
is equal to the temperature prescribed at the entrance of the anode T◦ . If some
reference exchange current density j◦ref is known at a temperature T ref , one can find
the exchange current density for any temperature by the following equation
j◦ = j◦ref e−
EA
R
1
( T1 − T ref
),
(2.33)
where EA is the activation energy [8]. In the membrane, where no chemical reaction
takes place, the drop in potential is caused by the resistance of a current to flow
through the membrane. Since resistivity is equal to the inverse of the conductivity,
the overpotential in the membrane is found by
ηm =
Lm
j,
σm
32
(2.34)
where Lm represents the length of the membrane [1]. There is also a voltage drop at
the interface between regions due to a change in material. The total interfacial overpotential ηi , accounting for both the anode-membrane interface and the membranecathode interface, is assumed to contribute 5% of the total voltage drop in the cell.
Being that the overpotentials in each region can be calculated as well as the required
minimum voltage for water to undergo electrolysis V◦ , which is about 1.23 V, the
interfacial overpotential can be calculated by
ηi =
ηa + ηm + ηc + V◦
.
19
(2.35)
Now knowing all the overpotentials, one can determine the initial voltage Φ◦ of the
cell by [1]
Φ◦ = ηa + ηm + ηc + ηi + V◦ .
(2.36)
The anode is a three-dimensional object. Because the cell is assumed to be uniform
across the depth of the cell, integrating the governing equations over the depth of the
cell leaves just the length and width components. The model tracks the concentration
and charge within one channel of the cell. Figure 2.2 displays a channel of the anode.
Recall Gauss’s law (eq. (2.18)) states the flux in the displacement field is equal to
a space charge density. This property was used to find the governing equation for
potential in the membrane, but it can also be applied to the anode region as well.
Gauss’s law reveals
² (Φxx + Φyy ) = C,
33
(2.37)
Table 2.2: Anode Constants
Symbol
Description
Value
Units
Source
Da
Diffusivity of hydrogen ions
Eq. (2.46)
m2
s
Adams [13]
D Wa
Diffusivity of water
Eq. (2.45)
m2
s
Kang [8]
j◦a
Exchange current density
Eq. (2.33)
A
m2
Kang [8]
j◦ref
a
Exchange current density at 353K
10−4
A
m2
Nie [1]
kTa
Thermal conductivity
0.67
W
m K
Afshari [7]
²a
Permittivity
78²◦
F
m
Akinaga [3]
ηa
Overpotential
µa
Electrical mobility
Da q
kB T̄a
m2
Vs
Akinaga [3]
σa
Ionic conductivity
Eq. (2.42)
A
Vm
Kang [8]
Eq. (2.31) V
Nie [1]
where C is the space charge density. Because the model only finds changes along the
x direction, homogenization is performed by integrating over the y direction. In doing
so, Φ simply expresses the average potential in water between the polymer rods. On
the surface of the polymer rods, Ohm’s law states
σ
dΦ
= −j,
dm̂
(2.38)
where m̂ represents the normal of the surface interface and takes values ±ŷ depending
upon the top or bottom surface [3]. Integrating equation (2.37) over y with boundary
34
_
e
Membrane
H
O
H
y
x
Figure 2.2: A two-dimensional view of the anode channel, from one electrode rod to
the next
conditions, the equation becomes
µ
¶
2j
² P Φxx +
= P C,
σ
(2.39)
where P is the pitch or length from the surface of one electrode to another. By doing
some simple algebra, we find the equation
Φxx = −
2j
PC
+
.
Pσ
²
(2.40)
Being that the water being pumped into the cell is neutral, the space charge density
is assumed to be extremely small. The
PC
²
term is considered so small that it is
negligible. Without that term, the equation resembles that of the governing equation
for the anode (2.2). The transfer current density is therefore defined as
jV =
2j
.
P
(2.41)
The conductivity, like many other constants in the model, is temperature dependent.
35
Kang [8] and Afshari [7] provide a relationship between conductivity and temperature
which is
σ = (0.5139λ − 0.326)e2168( T − 303 ) .
1
1
(2.42)
The temperature chosen for each region is the temperature of the outer boundary T◦ .
This is done so the temperature profile can easily be solved analytically as well as
a hypothesis that temperature changes very little in the water electrolysing cell. A
similar idea is applied to the water content. It is known that there should be a drop
in water content in the membrane. Expecting this drop, λ is chosen to be about 90%
of the water chosen for the anode and the cathode λ◦ . For permittivity, the value is
determined by how the permittivity compares to the permittivity in a vacuum. In
each region, the values being tracked are the average value in the water channels.
Since this is the case, the permittivity should be that of water which has a relative
permittivity of 78 [3]. Thus, for each region,
² = 78²◦ ,
(2.43)
where ²◦ is the permittivity in a vacuum. For the temperature governing equations,
thermal conductivity is a constant value for each of the regions. For the anode and
cathode, the temperature in the water should depend on the thermal conductivity
of water which is 0.67 mWK [7]. In the membrane where the flow channels are much
smaller, the thermal conductivity of the membrane will play a much larger role. Thus,
in the membrane the thermal conductivity of the membrane, 0.95 mWK , is chosen per
Afshari [7]. Afshari [7] also provides an expressions for the thermodynamic equilib36
Table 2.3: Membrane Constants
Symbol
Dm
Description
Value
Diffusivity of hydrogen ions Eq. (2.46)
Units
Source
m2
s
Adams [13]
DWm
Diffusivity of water
Eq. (2.45)
m2
s
Kang [8]
kTm
Thermal conductivity
0.95
W
m K
Afshari [7]
²m
Permittivity
78²◦
F
m
Akinaga [3]
ηm
Overpotential
µm
Electrical mobility
Dm q
kB T̄m
m2
Vs
Akinaga [3]
σm
Ionic conductivity
Eq. (2.42)
A
Vm
Kang [8]
Eq. (2.34) V
Nie [1]
rium potential which is
U◦ = 1.23 − 0.9 × 10−3 (T − 298.15).
(2.44)
The temperature in the expression is assumed to be the temperature of the outer
boundary T◦ .
Many of the constants in the governing equation for water content depend on the
material composition of the membrane. While the product under investigation is
Nafion, many constants may vary from paper to paper depending upon the number
of groups and their density. To maintain consistency, many of the constants were
chosen from Kang [8]. The density of the membrane ρ, from Kang [8], is 2000 mkg3 .
kg
The equivalent weight (ratio of mass to moles of sulfonic acid groups) EW is 1.1 mol
[8].
37
The diffusivity of water is affected by the temperature. Being that the temperature
profile can be determined and isothermal regions are assumed, the temperature used
to find diffusivity can be the average temperature in each region from the temperature
profile. The notation for average temperature will be T̄ with a subscript specifying
the region. The expression for the diffusivity of water from Kang [8] is
DW = (2.563 − 0.33λ + 0.0264λ2 − 0.000671λ3 ) × 10−10 e7.9728−
2416
T̄
.
(2.45)
As was done for the temperature terms to make a constant coefficient problem that
can be solved analytically, the water content terms are assumed to be the initial and
Table 2.4: Cathode Constants
Symbol
Description
Value
Units
Source
Dc
Diffusivity of hydrogen ions
Eq. (2.46)
m2
s
Adams [13]
DWc
Diffusivity of water
Eq. (2.45)
m2
s
Kang [8]
j◦c
Exchange current density
Eq. (2.33)
A
m2
Kang [8]
j◦ref
c
Exchange current density at 353K
30
A
m2
Nie [1]
kTc
Thermal conductivity
0.67
W
m K
Afshari [7]
²c
Permittivity
78²◦
F
m
Akinaga [3]
ηc
Overpotential
µc
Electrical mobility
Dc q
kB T̄c
m2
Vs
Akinaga [3]
σc
Ionic conductivity
Eq. (2.42)
A
Vm
Kang [8]
Eq. (2.32) V
38
Nie [1]
final water content values in the anode and cathode λ◦ and the value in the cathode
is 90% of λ◦ for the expected drop.
The effectiveness of the PEMPC is known to be affected by the water content. In
each region, diffusivity and mobility are functions of water content and thus length.
Knowing two diffusivities at two different water contents, Adams [13] fit a linear
equation to the data given by
D = (8 × 10−10 )λ − 3.1 × 10−9 .
(2.46)
By plugging in the appropriate λ for each region, the diffusivity of each region can be
found. Being that the cell is well hydrated, the diffusivity in the anode and cathode
should be relatively close to the diffusivity of protons in bulk water D◦ given in Table
2.1 [3]. To determine the drift coefficient, the Einstein relation was used which states
[3]
µ=
Dq
,
kB T̄
(2.47)
where kB is the Boltzmann constant and T̄ is the average temperature in each region.
The final piece of discussion is the geometry of the cell. The size and dimensions
of the proton-exchange membrane photoelectrolysis cell may differ from experiment
to experiment. For this model, the dimensions are those seen in Figure 2.3. The
values were chosen because they fell in the range of many papers investigating the
subject. While the size of the cell is varied, the major goal in the model is to compare
the results to a cell of the size in Figure 2.3. The flow channels through which the
protons will move in the membrane are assumed to be roughly 3nm by 3nm. The
39
15 m
30 m
15 m
Scaffold
thickness
1 cm
Membrane
Pitch
1 cm
Electrode Sheet
Anode
Cathode
Figure 2.3: The dimensions of the PEMPC cell used in the model
dimensions of the flow channels are used in the governing equation for potential
in the membrane in the vol term which is volume of the channel. The placement
of the sulfonic acid groups are also important, especially their relationship to the
placement of the hydrogen charges. Two cases will be run in the model. One,
the placement of the hydrogen and sulfonic acid groups will coincide forming an
electroneutral membrane. An approach like this was done by Akinaga [3]. Two, the
model will allow, at steady state, the placement of the hydrogen point charges offset
by some proportion pos of the distance between each sulfonic acid group, allowing a
non-electroneutral membrane to exist. The pos proportion can be as low as 0, the
electroneutral case, but less than 1. To see how pos is used to place the H+ charges,
see Figure 2.1.
40
Table 2.5: Cell constants
Symbol
Description
Value
Units
EA
Activation energy
7.3 × 104
J
mol
EW
Effective weight
1.1
kg
mol SO−
3
La
Length of the anode
15 × 10−6
m
Lm
Length of the membrane
30 × 10−6
m
Lc
Length of the cathode
15 × 10−6
m
jν
Current density induced by light
Eq. (2.29)
A
m2
jV
Transfer current density
Eq. (2.41)
A
m3
nref
Reference concentration
40.88
mol H+
U◦
Thermodynamic equilibrium
Eq. (2.44)
V
vol
Volume of the membrane channels (9 × 10−18 ) Lm m−3
w
Work function of platinum
5.12q
J
ν
Frequency of light
6.5 × 1015
s−1
ρ
Density of Nafion
2000
kg
m3
2.7 The Complete Homogenized Model
The model in this paper is one-dimensional. To meet this condition, the assumption
is made that all variables are uniform along the height and depth of the cell. Integrating over the height and depth leaves just the length component. The homogenized
41
governing equations in the anode are
V
Φa,xx = − jσa
ka Ta,xx + j V
ρ
DWa λa,xx
EW
−
¡ ∂U◦ ¢
∂T
¡ 2.5 j ¢
22 F
2
Ta = − σj a − j V (Φ◦ − V◦ )
λa,x =
(Da na,x + µa na Φa,x )x +
Electrical Potential
jV
F
jV
2F
Temperature
Water Content
= 0.
Proton Concentration
(2.48)
Though not seen in the anode equations, the electrical potential and proton concentration form a coupled system. We therefore solved both electrical potential and proton
concentration numerically in all the regions. Even though water content constants
depend on temperature, we can solve the temperature equation analytically, use the
values found to calculate the constants in the water content, and then solve the water
content equations analytically. By solving the differential equation for temperature
in the anode, we find the temperature profile in the anode is
 s ¯ ¯
 s ¯ ¯
∂U◦ ¯
◦¯
V
¯
j ∂T
j V ¯ ∂U
∂T 
 + cTa,2 exp −x
Ta (x) = cTa,1 exp x
−
ka
ka
j2
σa
+ j V (Φ◦ − V◦ )
¡ ◦¢
,
j V ∂U
∂T
(2.49)
where cTa,1 and cTa,2 are constants that will discussed later in this section. Once the
constants are known, the coefficients in the water content differential equation can
be found. The solution to the water content differential equation in the anode is
µ
λa (x) =
cW
a,1
exp
¶
j V 22
2.5 j EW
W
x + ca,2 −
x,
22 F ρDWa
2j 2.5
(2.50)
W
where, once again, cW
a,1 and ca,2 are constants that will be found later in this section.
After the homogenization of the governing equations in the membrane, the membrane
42
equations are
P ³
Φm,xx = − ²qm
k
SO3−
δkH − δk
´
1
volm
2
km Tm,xx = − σjm
ρ
DWm λm,xx
EW
−
¡ 2.5 j ¢
22 F
Electrical Potential
Temperature
λm,x = 0
Water Content
(Dm nm,x + µm na Φm,x )x = 0.
Proton Concentration
(2.51)
Just as was done in the anode, the potential and concentration are solved numerically,
while the temperature and water content are solved analytically. The temperature
equation for the membrane is
j2
Tm (x) = −
x2 + cTm,1 x + cTm,2 .
2km σm
(2.52)
The equation for water content is
µ
λm (x) =
cW
m,1
exp
¶
2.5 j EW
x + cW
m,2 .
22 F ρDWm
(2.53)
The one-dimensional governing equations for the cathode are
nc j V
nref σc
j2
= −
σc
Φc,xx =
kc Tc,xx
ρ
DWc λc,xx
EW
−
¡ 2.5 j ¢
22 F
(Dc nc,x + µm nc Φc,x )x −
λc,x = 0
nc j V
nref F
= 0.
Electrical Potential
Temperature
Water Content
Proton Concentration
(2.54)
The cathode temperature equation is
Tc (x) = −
j2 2
x + cTc,1 x + cTc,2 ,
2kc σc
43
(2.55)
and the equation for the water content is
µ
λc (x) =
cW
c,1
exp
¶
2.5 j EW
x + cW
c,2 .
22 F ρDWc
(2.56)
Since the concentration in a channel is assumed constant along the height and depth
of the channel in the homogenization process, the flux in concentration J will only
occur in the x direction and will therefore be
J = Dnx + µnΦx .
(2.57)
The boundary conditions are
x= 0
x = xam
Φa = Φ◦
Φa = Φm +
Ta = T◦
λa = λ◦
x = xmc
ηi
2
Φm = Φc +
²a Φa,x = ²m Φm,x
²m Φm,x = ²c Φc,x
Ta = Tm
Tm = Tc
ka Ta,x = km Tm,x
km Tm,x = kc Tc,x
λa = λm
λm = λc
x = xc
ηi
2
Φc = 0
Tc = T◦
λc = λ◦
DWa λa,x = DWm λm,x DWm λm,x = DWc λc,x
na = n◦
na = nm
nm = nc
Ja = Jm
Jm = Jc
Jc = −KM T (nc − n◦ ).
(2.58)
By plugging in the equations for temperature and water content into the boundary
condition, a system of six equations and six unknowns arise for each variable. Solving
the system gives the values of the cT ’s and cW ’s as well as the complete equation for
temperature and water content.
44
CHAPTER III
CONCLUSION AND FUTURE EXPANSIONS
The proton-exchange membrane photoelectrolysis cell is a solar cell that produces
hydrogen gas through the electrolysis of water. There are many aspects that affect
the hydrogen gas production as well as the power supply needed including: membrane
charge, temperature, water content, size, permeability, current density, and light
intensity. All of these factors contribute to the efficiency of the PEMPC and are
considered in our model. Many assumptions were made to arrive upon the conclusions
in the model. As a reminder, some of these assumptions include:
• Well hydrated cell
• Nafion channels are linear and fixed in size
• Isothermal regions are assumed for calculating material properties
• Evenly distributed charges in the membrane
• The convection of water is not considered to aid in proton movement
The results show that one of the most important contributions to a cell is its size.
The size of the regions, especially the electrodes, can alter the production of hydrogen
by over 100% with small change in the total size of the cell. Many of the expected
factors, such as temperature and membrane size, showed to have little effect on the
45
cell’s production unless perhaps altered by several orders of magnitude. The PEMPC
seemed over hydrated in the default condition and should be lowered to maximize
production. The sulfonic acid groups in the membrane can be used to increase the
flow of hydrogen molecules. Results from the model simulate how offset charges can
assist in ion movement. The number of the sulfonic acid groups in the membrane
increases the production to a point. The final important discovery of the study is
the effect light intensity and current density play on the power required to keep the
cell operational. The balance between applied current density and photocurrent is
extremely crucial to the required power. In summary, the finding of our model are
• Longer electrodes increase hydrogen production
• Membrane length must be significantly changed to alter hydrogen production
• Temperature change has little effect on the PEMPC
• A less hydrated cell can produce nearly an equivalent amount of hydrogen gas
• The charge of the membrane changes the production level
• Light intensity must be balanced with current density to reduce required power
Future areas of investigation are discussed next. Perhaps the most important expansion to this model would be the addition of the water chambers. By adding water
chambers, the model will have higher dependence upon temperature and water content as demonstrated by Kang [8] and Afshari [7]. With the layers, more work can
be done on optimizing the cell to find its maximum potential for being a hydrogen46
producing source. The other major expansion would be adding more dimensions to
the model to better understand the path of the proton, especially in the membrane.
The cell may be the science of the future, and any expansion upon this model will lead
to a better overall understanding and usefulness of the proton-exchange membrane
photoelectrolysis cell.
47
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