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Model Development

This section explains in detail how the novel electrode design was modeled. Beginning from the governing equations and material characteristics. This section does not explain how the model was solved, as that is done in Chapter 4: Solution Approach.

Description of Catalyst Mat

This section explains the physical description of the mat and provides a brief description of the model. A detailed development is presented in succeeding sections.

Physical Description

The proposed electrode design uses an electro-spun carbon nanofiber mat as a substrate with catalyst deposited on the fiber and then ionomer coated over the catalyst to produce concentric conductors for electrons and ions with an interstitial catalyst layer. Highly flexible and durable carbon nanofiber mats are currently used by the Moore research group to improve battery performance [8]. Two ways that such a mat could be used to form an electrode can be seen in Figure 3-1. This figure shows each component that will be discussed throughout this work. In Figure 3-1 (a), the catalyst layer is used without a traditional GDL, and therefore it is the only thing between the flow field plate and the ionomer membrane. The “post” area is formed when the shoulders on the flow field plates compress the catalyst layer, therefore creating a semi-solid area consisting of ionomer and carbon fibers compressed together. Oxygen can flow from the flow channels down into the catalyst layer in between each post. While the H+ ions can flow from the anode, through the membrane, into the post and into a fiber. The length of the fiber is established by the spacing of the shoulders of the flow field plate. In Figure 3-1 (b) a carbon mat is placed between the catalyst layer and the flow field plate. This mat is used to compress parts of the catalyst layer in the post area. By using a mat with known fiber orientation, spacing and diameter, the size of the posts, and the distance between the posts is known. This design allows for smaller fiber lengths than would be possible with machined or stamped flow field plates. This mat could be hot pressed onto the catalyst layer when the catalyst layer is hot pressed onto the ionomer membrane to ensure continuous ionic pathways from the anode side of the membrane through the post to the top of the fibrous catalyst layer.
For the CNF catalyst layer, current is produced along the length of each fiber that runs perpendicular to the post. Only the portions of the fibers in between the posts generate current. Fibers parallel to the post (not shown in diagrams) serve only to provide spacing and structural integrity and are not modeled here.
The proposed concentric carbon/ionomer structure formed by coating the fibers is shown in Figure 3-2. By controlling the fiber diameter, the coating thickness, and the mat porosity, many of the  factors that control the performance of the electrode are adjustable. Due to the high porosity of the mat, there will be plenty of empty space surrounding each fiber. This empty space is shown in Figure 3-2 by the dashed lines surrounding the fiber.
This electrode is modeled assuming it is utilized in a traditional fuel cell assembly, where the fuel travels across the catalyst mat in groves cut into a flow field plate just like other PEM fuel cells (similar to Figure 3-1a).
Throughout this work, several geometric variables are used to describe the physical phenomena occurring within the electrode in terms of the catalyst layer design parameters. The remainder of this section explains the definitions of these variables and, where appropriate, their derivations in terms of the catalyst layer design parameters such as fiber diameter, ionomer thickness, porosity, etc. Table 3-1 provides a summary of the key geometric variables. When studying parameters with similar dimensions it is common practice to create a non-dimensional ratio of the two parameters. This model uses a non-dimensional ratio of the fiber diameter and the ionomer thickness surrounding it. Most of the geometry terms are derived as functions of the porosity and this ratio.
Electrons and ions flow through the cross-sectional area of the carbon and ionomer respectively. The cross-sectional area of the carbon fiber is found by using the equation for the area of a circle as shown below in Eq.(3.1). A similar approach is used to find the cross-sectional area of the ionomer coating surrounding the fibers as shown in Eq. (3.2).
Applying the non-dimensional coating thickness to fiber diameter ratio that was previously discussed and simplifying Eq. (3.2) yields:
The reactants and products flow through the empty space surrounding the fibers. As seen in Figure 3-2 the total space containing the fibers is modeled as a square that runs the length of the fiber. The cross-sectional area of this square is simply the area of a square as shown in Eq. (3.4).
the solid volume (fiber and coating), Eq. (3.7), an expression can be found for the dimensions of the volume associated with fiber as seen in Eq. (3.8).
The final geometric expression that is required, is the surface area of the fiber as a function of fiber length, where the axial length of the fiber is given by L. This is given by Eq. (3.9) and simplified into Eq. (3.10) after applying the non-dimensional diameter ratio.
All of the derived expressions above will be mentioned throughout the following sections of the thesis. Descriptions of all the variables are in Table 3-1.

Description of Model

This model focuses on several key aspects of fuel cell performance. With the main goal of the new electrode design being to increase the amount of catalyst material that is exposed to oxygen, correctly modeling the oxygen transport is very important. This model assumes an oxygen concentration at the outer layer of the mat that decreases as it moves towards the inner fibers that are closer to the membrane. The model assumes this is the only gradient in concentration and that the concentration is constant along the axial length of the fibers.
Another major factor affecting fuel cell performance is the ohmic resistance, which is a measure of how difficult it is for the electrons and ions to reach the reaction site. This model accounts for these resistances along the length of the fibers, as well as the ohmic losses that occur as the ions travel across the membrane and enter the fibers.
Most fuel cell models assume the catalyst layer thickness is small enough to be negligible, and therefore current is normalized based on the membrane area instead of the catalyst layer volume. Often the current is stated as a “current density” but the units for current density are amps per area, e.g., A/cm2. In this work, current is normalized by membrane area in the results section while discussing overall performance, which allows for easy comparison to other electrode designs. However, during the modeling process, current, without any normalization is often used.

Introduction to Variables

The variables used when modeling the geometry of the catalyst mat are shown in Table 3-1. A description of each variable and the symbol that is used to reference that variable throughout the model is also shown.

Butler-Volmer Equation

The  Butler-Volmer   (BV)   equation   is   a   common   model   for  the  kinetics   of  many electrochemical reactions. The BV equation provides a relationship between the exchange current density (referenced to the reaction surface area), the activation overvoltage, and the reactant concentration. The form of the BV typically used in fuel cell electrode modeling, also known as the current-overpotential equation, is shown by Eq. (3.11), and the variables used in it are describe  The activity of platinum for catalyzing the oxygen reduction reaction has been extensively studied, and published literature provides the exchange current density,   ̂, referenced to the catalyst area for many different platinum-carbon catalysts. By using the geometric relationships from Section 3.1, the BV equation referenced to platinum surface area can be modified to yield the current per unit area of fiber by defining an area ratio, rPt-C representing the platinum surface area to the carbon fiber surface area. The exchange current density referenced to the fiber area is then given by

Fiber Modeling

The fibers and the ionomer coating surrounding the fibers provide continuous pathways for electrons and ions to reach the reaction sites. However, these pathways present a resistance to the transport of the respective species. Due to this transport resistance, the potential available to drive the electrochemical reaction, and hence the current production, varies along the length of the fiber. Thus, to estimate the current, the variation in potential in each phase (carbon and ionomer) along the axial length of the fiber must be determined. The following two sections explain the process used to model the variation in potential in each phase and describe how the variations in potential are coupled with the BV equation to determine the current production along the length of an individual fiber.

Variations in potential in the carbon and ionomer phases

The variation in potential resulting from the electrical resistance of the carbon fibers is modeled using a system of first order differential conditions. The first equation arises from the application of conservation of charge to a differential element along the fiber. As illustrated in Figure 3-3, the current entering the left face of the element is ix and the current leaving the right face of the element is ix+Dx. The current produced along the length of the element is the product of the surface area, pDDx, and the current produced per unit area of the fiber surface as given by the BV equation, Eq. (3.13). For the steady case, in which there is no storage of charge, conservation of charge is given by Rearranging Eq. (3.14), dividing by Dx, and taking the limit as Dx ® 0, yields Introducing the BV equation, Eq. (3.13), for the local current density yields the final form for the derivative of current as a function of the local activation overpotential and the local oxygen concentration.
The second differential equation used to determine the potential along the carbon fiber arises from Ohm’s law expressed as Where   (  ) is the electrical current at a particular location, x; Ax-c is the cross-sectional area of the carbon fiber; rc is the resistivity of the carbon; and fc is the potential in the carbon phase. The positive sign indicates that electrons (negatively charged) flow in the direction of increasing potential. Rearranging Eq. (3.17) yields a first order differential equation that can be solved for the potential in the carbon phase Equations (3.16) and (3.18) could be combined to yield a second order equation for the potential in the carbon phase, but because we are interested in the current and because of the particular solution approach used for this problem, it is more straightforward to solve the equations as a pair of first order differential equations.
In the ionomer phase, H+ ions travel in the same direction and at a rate that is numerically equal to the flow of electrons, hence the ionic current is the same as the electrical current as given by the solution to Eq. (3.16). The variation in potential in the ionomer phase is modeled in a manner similar to that for the electrical phase and is determined by the following equation:
where i(x) is the electrical current at a particular location, x; Ax-i is the cross-sectional area of the ionomer coating; ri is the resistivity of the ionomer; and fi is the potential in the ionomer phase. In Eq. (3.19) the negative sign indicates that the H+ ions (positively charged) travel in the direction of decreasing potential.
The system of equations comprised of Eq. (3.16), (3.18), and (3.19) incorporates a source term that represents the production of current at each point along the fiber. This source term originates from the BV equation, which predicts activation voltage (h). Figure 3-4 illustrates the calculation of the activation overvoltage, per Table 3-2. At any point along the fiber, the activation overvoltage can be found as the theoretical open circuit voltage for the fuel cell reaction, ∅c•€•‚C less the potential difference between the carbon and ionomer phases at the reaction site. Thus, By solving this system of equations for a given concentration level, and subject to a set of boundary conditions, the current generated by one fiber is known.

Fiber Boundary Conditions

In order to solve the above system of equations, there must be three boundary conditions. The first of those boundary conditions arises from the symmetry about the midpoint of the fiber. This symmetry allows for the use of a boundary condition commonly used in fluid flow and heat transfer applications. Due to symmetry, the current flowing through the midpoint of the fiber must be zero, as shown in (3.21), i.e.,
The other two boundary conditions deal with the potential in the carbon and ionomer phases. Figure 3-4 illustrates these boundary conditions and the change in potential within each phase. As the electrons move along the fiber, they encounter resistance leading to an increase in potential in accordance with Eq. (3.18). Conversely, the potential in the ionomer phase decreases as H+ ions travel along the length of the fiber to the cathode reaction site in accordance with Eq. (3.19). The H+ ions travelling through the ionomer phase originate at the anode, where the potential must be high enough to overcome the resistance through the membrane and to transfer ions to the end of the fiber at x = 0. The difference in potential between the anode and the end of the fiber is labelled as ∆  ‚ €‚ in Figure 3-4 and is determined by modeling the ion transport in the membrane and collector post as discussed in section 3.5. Each fiber will have a unique ∆  ‚ €‚. The external electrical load is connected between the anode and the carbon phase of the fiber at x Thus, at x = 0, the potential difference between the ionomer phase of the fiber and the carbon phase of the fiber is given by The value of this potential difference depends on the position of the fiber in the catalyst layer, with fibers closer to the membrane experiencing a smaller differential (because DVmem is smaller near the membrane) and fibers further from the membrane experiencing a higher differential.
For convenience in solving the system of differential equations, (3.16), (3.18), and (3.19) for each fiber, the potential in the ionomer phase is assigned a potential of 0 at the left boundary so that In accordance with Eq. (3.22), the potential in the carbon phase must then be Equations (3.21), (3.23), and (3.24) establish the complete set of boundary conditions required to solve the set of differential equations expressed by equations. (3.16), (3.18), and (3.19).

Mass Transport

The preceding approach for calculating the current generation along the fiber relies on the BV equation, which is dependent on the concentration of oxygen at the reaction site. While it is assumed the concentration will be constant along the axial length of the fiber, it will vary from one fiber to another. The concentration will be the highest at the fiber adjacent to the flow channel and will decrease for fibers closer to the membrane due to the consumption of oxygen. This section describes how the variation of concentration with depth into the catalyst layer is described by applying species conservation and Fick’s law. The variables used in this section are listed in Table 3-3 along with descriptions, while Figure 3-5 shows the domain of the mass transfer model.

Oxygen Conservation

The development of a model of oxygen concentration through the catalyst layer begins with the species conservation equation.     As depicted in Figure 3-6, oxygen enters the control volume surrounding each fiber from the upper surface of the volume. Some of the oxygen entering the volume is consumed by the electrochemical reaction, and some passes through the bottom surface of the volume to the next fiber. By applying species conservation to the steady flow control volume surrounding fiber k, the following expression is obtained consumption by the chemical reaction in mole/s for fiber k. Since the oxygen flow is primarily in the y-direction, concentration gradients in the x-direction are neglected and the fluxes across the top and bottom face are considered uniform. With this assumption, the species conservation equation becomes The molar rate of oxygen consumption for a fiber is related to the current generated in the fiber by For a given flux at the top face, the flux through the bottom face is given by The molar flux at each face is related to the concentration gradient by Fick’s first law of diffusion, which relates the molar flux to the concentration gradient through the diffusivity, D, expressed in m2/s. Hence.

Analysis of Ionomer-coated Carbon Nanofibers for use in PEM Fuel Cell Catalyst Layers

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