An idealized porous electrode with straight cylindrical pores was considered above. Next our aim is to describe and characterize porous media in a way that is broadly applicable. The empty space of the electrode available to the electrolyte (or gas phase) is the void volume fraction or porosity, ε. This parameter is the most important feature of any porous media.

(5.13)

Initially, we assume that this void space is completely filled by the electrolyte. In our general description, we treat the porous medium as a continuum; that is, we’ll take a macroscopic point of view and do not try to describe the microscopic details. The key to this approach is that the size of the electrode is large compared to the features of the pores. The solid and electrolyte phases are therefore treated as two interpenetrating continua as shown in Figure 5.2. Additional phases can also be considered. For example, a gas phase is important for many fuel cells and for some batteries. Multiple solid phases are also possible. Thus, our two continua model can be readily expanded to include three or more interpenetrating phases.

As mentioned earlier, one key objective of porous electrodes is to increase the interfacial area between the two phases. Thus, a second key characteristic of the porous media is the specific interfacial area, a.

(5.14)

This quantity, which was introduced previously for our cylindrical pore electrode, has units of [m⁻¹]. An electrode geometry that is more representative of electrodes used in practice is that formed from a packed bed of spherical particles, where the interstitial space between particles forms the void volume, ε. A characteristic length between spheres, analogous to a hydraulic diameter, can be used to describe the pore size. Although still simplified, let’s explore this construct in a bit more depth. For a single sphere of radius r, the specific interfacial area is . In general, the specific interfacial area increases as the features of the porous electrode become smaller. Without worrying about the exact details of how the spherical particles pack together, we can express the specific interfacial area with

(5.15)

We see that the specific interfacial area for this packed bed depends not only on the size of the spheres, but also on the number of spheres as reflected in the solid volume fraction (1 − ε).

The characteristic pore size, r_p, can be estimated as follows:

(5.16)

Although simplified, this analysis helps us to develop some intuition into the behavior of porous media. As the void volume gets small, the pore diameter also decreases. For larger porosity, the pore diameter will be larger.

A typical porous electrode has a distribution of pore sizes rather than the single pore size that characterizes our idealized electrode. Similarly, there is also a distribution of particle sizes in the electrode. The detailed structure of the pores controls the properties; however, in general we won’t attempt to treat the geometric minutiae. Instead, we will take the specific interfacial area as a measurable quantity of the porous media or electrode. An example distribution of the pore volume is shown in Figure 5.3, which was measured with Hg-intrusion porosimetry. Here, larger pores are easier to fill than small ones. Since the pores don’t fill uniformly, we can relate the volume of Hg and required pressure to pore diameter. Not only don’t we have a single pore size, but often there is a relatively broad distribution, perhaps centered on a couple of predominant sizes. Figure 5.3 shows a bimodal distribution, which is also common. The volume labeled primary corresponds to the volume inside the individual particles, and the volume labeled secondary refers to the volume between particles. From data for pore volume, quantities such as the mean pore size and specific interfacial area can be estimated. The size of the pores impacts transport in the electrode and, as we’ll see below, is important in establishing a stable interface with electrodes involving gases.

ILLUSTRATION 5.1

A porous carbon felt is being used for the electrode of a redox-flow battery. The felt is 2 mm thick with an apparent density of 315 kg m⁻³; the true density of the carbon is 2170 kg m⁻³. The felt is made up of 20 μm fibers. Calculate the porosity and specific interfacial area.

1. 
2. Use a cylindrical filament to model the fibers.The superficial current density is 1000 A·m⁻². If at 60 °C the exchange-current density is 95 A·m⁻², how much is the kinetic polarization reduced because of the porous electrode? Assume α_a = α_b = 0.5.
3. The apparent current density is equal to the actual current density times the thickness and specific interfacial area:The true current density is 17.2 A·m⁻², and is assumed to be constant throughout the electrode. The Butler–Volmer kinetic expression is used to evaluate the overpotential, η_s.For the apparent current density, η_s = 136 mV; for the true current density, η_s = 5 mV.