Superficial vs interstitial velocity

There are two definitions for gas velocity in a packed bed: superficial and interstitial. The superficial velocity is calculated assuming there is no porous material, simply:

$$u_s = \frac{Q}{A}$$

Where $Q$ is the volumetric flow rate, and $A$ is the cross-sectional area of the column.

From the point of view of the fluid flowing through a packed bed, there is solid material blocking the flow. The packed bed has a void fraction, $\epsilon_b$, and therefore, the available cross-sectional area at any point along the length of the bed is reduced by a factor of $\epsilon_b$. The same mass of fluid must flow through a smaller area, and therefore, conservation of mass implies that the velocity must increase to account for the reduced area.

This is the actual velocity that the fluid moves through the porous matrix at, and is known as interstitial velocity. Interstitial velocity is related to the superficial velocity by:

$$u_i = \frac{u_s}{\epsilon_b}$$

As $\epsilon_b < 1$, the interstitial velocity is always larger than the superficial velocity. It is important to clarify which "velocity" is being referred to, and I sometimes see the omission in papers causing confusion.