Clearly, the porous structure has an effect on transport through the media. The way to treat this influence is to define effective transport properties that can be related to the bulk transport properties discussed in Chapter 4. The need for the effective transport properties can be seen with the cartoon illustrated in Figure 5.4. There are two phenomena that we need to consider. First, with a porous medium, the cross-sectional area available for transport is reduced by the presence of the solid phase. The effective area can be approximated as the superficial area multiplied by the porosity. Second, the path length for transport is increased. A molecule diffusing through the electrode must go around the solid obstacles following a tortuous path. This characteristic is known as the tortuosity, τ. The tortuosity can have values as high as 6–20, although values of 2–3 are typical for many applications. Effective transport properties are related to those in the bulk by the combination of these two effects. This correction is represented below for conductivity, but applies equally to diffusion:

(5.17)

Thus, a higher tortuosity leads to a lower effective conductivity as would be expected. The effective transport properties can be measured directly, although a common estimate that accounts for porosity and tortuosity is

(5.18)

which is known as the Bruggeman relationship. A similar expression can be used for other transport properties such as the diffusivity and mobility. With these effective transport properties, we can now write the flux equations in terms of the appropriate driving forces. For example, using the definition for conductivity (Equation 4.7), Equation 4.5 can be written for the solution-phase current density:

(5.19)

All that is needed to apply this relationship to a porous electrode is to replace the transport properties, conductivity, and diffusivities, with effective values that account for the porosity and tortuosity.

(5.20)

ILLUSTRATION 5.2

The effective conductivity of a porous separator is measured to be 0.035 S·m⁻¹. The porosity is 0.5. Estimate the tortuosity if the accepted bulk value for conductivity is 0.2 S·m⁻¹.