Dispersion in packed beds

Building upon my recent essay on dispersion, this article explores the physical meaning of dispersion in packed beds.

As a quick recap, dispersion is a macroscopic phenomenon that evolves from variable flow patterns. In the case of a packed bed, the random packing and orientation of media cause each fluid element to take random paths through the bed:

Each fluid element will move at the same average velocity and experience random fluctuations in the flow path. Some paths are short, and others are longer and more tortuous. Therefore, the random scattering results in a statistical spreading of the tracer's concentration profile.

According to the above reasoning, Fick’s law applies, with the constant of proportionality representing the dispersion coefficient, $D_{ax}$. The differential equation describing the flow of a dilute tracer through a packed bed is:

$$\frac{\partial C}{\partial t} + U\frac{\partial C}{\partial z} = D_{ax}(\frac{\partial^2 C}{\partial z^2})$$

For a lot of practical applications, axial dispersion can be neglected (plug flow). However, it is very important to account for dispersion in certain flow regimes. For example, when modelling breakthrough curves in adsorbent packed beds, dispersion can influence the shape of the breakthrough curve significantly.

There are, in fact, many other fluid flow phenomena that can affect the shape of the breakthrough curve – but that will have to be a topic for another day.