There are situations where a stable three-phase structure is required. For example, the cathode of a low-temperature fuel cell involves three phases (gas, liquid, solid) in intimate contact. Let’s examine the reduction of oxygen in acid:
The protons are supplied from the electrolyte, electrons supplied from a solid phase, and oxygen is in the gas phase. The product water may be either in the gas or liquid phase depending on temperature and pressure. The interface of two phases that we have seen before is a surface, but now with three phases the intersection is reduced to the line at which all three phases meet (Figure 5.10b). Thus, the region of intimate contact will be small and represents an obstacle that must be overcome in the development of practical electrodes. This challenge was recognized almost immediately by investigators such as Sir William Grove, the inventor of the fuel cell (1839), who described it as a “notable surface of action.” One approach to address the small contact region is with what is known as a flooded-agglomerate electrode.
This approach is illustrated in Figure 5.12, which shows the electrode at three different length scales to clarify its critical features. The porous electrode consists of agglomerate particles that are packed together to form a porous bed. Each agglomerate consists of a porous, solid, electronic conductor that supports catalyst and is completely filled with electrolyte. These agglomerates are organized to provide both a continuous electronic and ionic path across the electrode. Electrons are transferred through the electrode via contacts between the electrically conducting solid phase (carbon) of the particles. Ionic conduction occurs through liquid-phase connections between the particles. However, in contrast to the electrodes considered earlier, a large fraction of the volume between particles is occupied by a gas phase, which permits rapid gas transport and allows ready access of the small agglomerate particles to the needed oxygen. In analyzing these electrodes using the methodology described previously, we would use the respective volume fractions of gas and liquid in the electrode rather than the electrode porosity since we have added an extra phase (gas).
There is another important difference between this model and the porous electrode treatment that we have considered to the point. In the treatment above, the reaction took place at the interface between the particles and the electrolyte, and there were no physically important processes taking place inside the particles other than electrical conduction of the current. In contrast, the reaction of interest in the flooded-agglomerate model takes place throughout the agglomerate particle as oxygen diffuses into the particle. Therefore, we need to model the processes inside the particle and couple that model with the balance equations previously developed for porous electrodes. Development of the fully coupled model is beyond the scope of this chapter. We will, however, develop the model for the single agglomerate particle in order to illustrate this aspect of the overall approach.
If we now look more closely at a single agglomerate, depicted for a spherical geometry in Figure 5.12, we see that the agglomerate of catalysts dispersed on a support and is completely filled with a liquid electrolyte. Oxygen from the gas phase dissolves into the electrolyte where it simultaneously diffuses and reacts inside the agglomerate. If the dimension of the agglomerate is small enough, then oxygen can access all of the catalyst.
Starting with Equation 5.11,
(5.46)
which is applied to oxygen at steady state. We then will use Fick’s law to describe the transport of oxygen and Tafel kinetics for the reduction of oxygen, which is assumed to be first order in oxygen concentration.
(5.47)
(5.48)
For this analysis, we assume that the overpotential inside the particle is constant. This makes sense since the particle is so small that potential losses are not likely to be significant. The analysis is performed for the local partial pressure of oxygen at the surface of the particle at that location in the bed. The particle is assumed to be spherical with a radius of rp. The boundary conditions are
(5.49)
Substitution into the differential material balance and solving for concentration gives
where
(5.51)
Equation 5.50 provides the flux and, through Faraday’s law, the current needed for the equations that model the transport and reaction through the porous electrode (as opposed to the particle). In turn, the local potential difference and partial pressure of oxygen are provided by the porous electrode model to the particle model. Hence, the additional physics that take place in the particle can be accounted for while retaining the advantages of the macro-homogeneous porous electrode model.
Exploring the particle behavior further, the concentration is given by
(5.52)
The dimensionless concentration is shown in Figure 5.13. For low values of K, diffusion is rapid compared to the rate of reaction. Here the concentration is nearly constant. In contrast, for large values of K, the oxygen is consumed near the surface of the agglomerate, and for all intents and purposes no reaction occurs in the middle of the agglomerate. In this case, the effectiveness of the agglomerate is low; that is, most of the volume is not being used. We can define an effectiveness factor, similar to what is done for reaction and diffusion problems in reaction engineering. For a spherical agglomerate,
Although analytic solutions are not possible for the current distribution in this electrode, the general rules learned in the previous section are applicable. In this particular instance, the kinetics for oxygen reduction is slow and therefore the reaction tends to be spread out across the thickness of the electrode. As the current density increases, the distribution becomes more nonuniform. Because of the slow kinetics, more concern is applied to the structure of the agglomerate rather than the thickness of the electrode.
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