A ONE-DIMENSIONAL MODEL OF A PROTON-EXCHANGE MEMBRANE PHOTOELECTROLYSIS CELL. A Honors Research Project. Presented to

Transcription

1 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

2 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

3 TABLE OF CONTENTS Page LIST OF TABLES LIST OF FIGURES iv v CHAPTER I. INTRODUCTION II. FORMULATION OF THE MODEL Anode Region Anode-Membrane Interface Membrane Region Membrane-Cathode Interface Cathode Region Treatment and Values of Constants The Complete Homogenized Model III. CONCLUSION AND FUTURE EXPANSIONS BIBLIOGRAPHY iii

4 LIST OF TABLES Table Page 2.1 Known Constants Anode Constants Membrane Constants Cathode Constants Cell constants iv

5 LIST OF FIGURES Figure Page 1.1 A representation of the three regions (Polymer backbone, hydration shell, and water region) of the Nafion channels Schematic (not to scale) of (a) the proton exchange membrane photoelectrolysis cell; (b) the silicon-germanium layers of the photocathode A representation of the length breakdown within the PEMPC and the placement of the point charges A two-dimensional view of the anode channel, from one electrode rod to the next The dimensions of the PEMPC cell used in the model v

6 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 H 2 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

7 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 H 2 O 2 H O 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 (H 3 O + ), Zundel (H 5 O + 2 ), or Eigen (H 9 O + 4 ) cations [2]. For simplicity sake, these ions will be referred to as 2

8 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 e H 2. (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

9 Polymer Backbone SO Hydration Shell 3 SO 3 SO3 Water Region SO 3 SO 3 SO 3 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

10 Load O2 H2 e Membrane Electrocatalyst Silicon h v H + Germanium Photoanode Photocathode Polymer Backbone Proton Exchange H 2O Membrane H 2O Current Conducting Support Scaffold (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

11 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

12 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

13 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

14 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

15 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 deter- 10

16 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

17 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

18 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

19 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

20 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 dx = S Φ and σ dφ dx = j (2.1) where S Φ 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 Φ a,xx = jv σ a. (2.2) 15

21 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 ) + S T = 0, (2.3) where k is the thermal conductivity, T is the temperature, and S T 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

22 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 energy G in the following way S = ( G 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. Summing all equations of power generation, σ 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 ( ) (k a T a ) + j V (Φ V ) + j V U T a + j2 = 0. (2.4) T σ a 17

23 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 ( ρ ) ( ) EW D j W λ n d + S W = 0, (2.5) F 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, D W is the diffusion coefficient of water, n d is the electroosmotic drag, and S W is a source or sink term for water content. The electro-osmotic drag coefficient is proportional to the water content present, n d = 2.5 λ [8]. The first 22 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

24 of water takes water out of the system the sink term in the anode should be negative and be given as S Wa = jv 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 ( ρ ) ( 2.5 EW D W a λ a 22 j F ) λ a jv 2F = 0. (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 n a to be the concentration of hydrogen throughout the anode the outer boundary condition for the anode is n a (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 n t = 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

25 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 (D a n a + µ a n Φ a ) + jv F = 0. (2.9) The values of the coefficients and constants will be discussed in the later section entitled Treatment and Value of Constants 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 η i, 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, Φ a (x am ) η i 2 = Φ m(x am ), (2.10) where x am 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

26 field can be expressed in terms of the voltage in the cell. To maintain continuity, the displacement fields of each side are set equal, ɛ a Φ a ˆn(x am ) = ɛ m Φ m ˆn(x am ). (2.11) Here ɛ represents the permittivity and ˆn 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, T a (x am ) = T m (x am ). (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 k a T a ˆn(x am ) = k m T m ˆn(x am ). (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 λ a (x am ) = λ m (x am ). (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 D Wa λ a ˆn(x am ) = D Wm λ m ˆn(x am ). (2.15) 21

27 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 n a (x am ) = n m (x am ) (2.16) for equal concentration and J a ˆn(x am ) = J m ˆn(x am ) (2.17) 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 (ɛ E) = C, (2.18) where E is the electric field and C is the space charge density [3]. The electric field is 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

28 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 vol m. To model the location of the groups and ions, point charges are placed for each making the governing equation for potential in the membrane 2 Φ m = q ɛ m k ( ) δk H δ SO 3 1 k, (2.19) vol m where δ H+ the Dirac delta function for the presence of hydrogen, δ SO 3 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 (k m T m ) + j2 σ m = 0. (2.20) 23

29 Anode Membrane Cathode H + d * pos x x x 0 am mc c x 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 ( ρ ) ( 2.5 EW D W m λ m 22 j F ) λ m = 0. (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 (D m n m + µ m n m Φ m ) = 0. (2.22) 24

30 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 x mc denote the length within the solar cell where the membrane ends and the cathode begins, the governing equations for the interface become Φ m (x mc ) η i 2 = Φ c(x mc ) Voltage drop at interface ɛ m Φ m ˆn(x mc ) = ɛ c Φ c ˆn(x mc ) T m (x mc ) = T c (x mc ) k m T m ˆn(x mc ) = k c T c ˆn(x am ) λ m (x mc ) = λ c (x mc ) D Wm λ m ˆn(x mc ) = D Wc λ c ˆn(x mc ) n m (x mc ) = n c (x mc ) J m (x mc ) ˆn = J c ˆn(x mc ) Continuity of electrical displacement field Continuity of temperature Continuity of flux for temperature Continuity of water content Continuity of diffusive flux for water content Continuity of concentration 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

31 reads ( nc ) j c = j n e F ηc ref RT (2.23) where j is the exchange current density, R is the gas constant, and n ref 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 j c 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 S Φc = n c n ref jv. Doing some simple mathematical manipulation and substitution for Ohm s law in equation (2.1), the governing equation for the cathode becomes Φ c,xx = n c n ref j V σ c. (2.24) The drop across the entire cell is Φ. Thus, the boundary conditions for the end of the cathode x c is Φ(x c ) = 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

32 in the cathode is (k c T c ) + j2 σ c = 0. (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 T c (x c ) = 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 ( ρ ) ( 2.5 EW D W c λ c 22 j F ) λ c = 0. (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 λ c (x c ) = 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 S c = n c n ref j V F. Replacing the definition for the sink term into the governing equation for 27

33 concentration in equation (2.8), we find the cathode s governing equation (D c n c + µ c n c Φ c ) n c n ref j V F = 0. (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, J c ˆn(x c ) = 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 K MT multiplied by the difference the concentration of protons in bulk water and the concentration at the edge of the cell, J c ˆn(x c ) = K MT (n n c (x c )). 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

34 Table 2.1: Known Constants Symbol Description Value Units c Speed of light in a vacuum m s 9 m2 D Diffusivity of hydrogen in bulk water s F Faraday constant C mol h Planck constant J s k B Boltzmann constant J K m e Mass of an electron kg 4 mol n Concentration of hydrogen in bulk water 10 m 3 N A Avogadro s number mol 1 q Fundamental charge C R Gas constant J mol K ɛ Permittivity in a vacuum F m µ Mobility of hydrogen in bulk water qd k B T m 2 Vs 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

35 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) = m ec 2 (1 w ). (2.28) h 2 ν 2 hν 30

36 In this formula, m e 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 I ν, 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 N A. 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 j ν induced by light is given by the formula F m e c 2 j ν = I ν (1 w ). (2.29) N A h 2 ν 2 hν Aforesaid, the total current density j is determined by the experimenter. Thus, in order to find the applied current density j app, one simply subtracts the photocurrent density from the total current density leaving j app = j j ν. 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

37 equation ( j app = 2j e F η RT ) e F η RT, (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 η a = RT F sinh 1 ( japp 2j a ). (2.31) A similar equation will arise for the cathode η c = RT F sinh 1 ( japp 2j c ). (2.32) 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 E A R ( 1 T 1 T ref ), (2.33) where E A 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 η m = L m σ m j, (2.34) 32

38 where L m 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 η i, 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 η i = η a + η m + η c + V. (2.35) 19 Now knowing all the overpotentials, one can determine the initial voltage Φ of the cell by [1] Φ = η a + η m + η c + η i + 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 ɛ (Φ xx + Φ yy ) = C, (2.37) 33