Inside a pore, a gas molecule may hit other molecules or the pore wall. The Knudsen number tells you which one controls, and that decides whether you use $D_{\mathrm{AB}}$, the Knudsen diffusivity or a combination.

Learning objectives

• Calculate the mean free path and the Knudsen number, and classify the regime as bulk, transition or Knudsen.
• Calculate the Knudsen diffusivity and explain why it does not depend on pressure or on the partner gas.
• Combine bulk and Knudsen resistances for a given flux ratio, and correct for porosity and tortuosity.
• Predict how pressure and pore size move a pore from one regime to another.

Background

Catalyst pellets, adsorbents, membranes and porous solids being dried all carry gas through pores that may be only nanometres wide. Compare the mean free path $λ$ with the pore diameter (Perry’s Eq. 5-261 and Table 5-16, p. 5-55):

Below $\mathrm{Kn} \approx 0.01$, molecule-molecule collisions control and the ordinary bulk $D_{\mathrm{AB}}$ applies (Fuller or Table 2-141), corrected for the solid by Eq. 5-249, $D_{\mathrm{eff}} = \frac{ε_{p}D_{\mathrm{AB}}}{τ}$. Above $\mathrm{Kn} \approx 10$, wall collisions control, and the Knudsen diffusivity applies (Eq. 5-250):

($D_{\mathrm{K}}$ in m²/s, $d_{\mathrm{pore}}$ in m, $T$ in K, $M$ in kg/kmol)

$D_{\mathrm{K}}$ has no pressure term, while $D_{\mathrm{AB}} \propto \frac{1}{P}$. So raising the pressure pushes a pore from Knudsen toward bulk behavior. In between, add the resistances, with $α = 1 + \frac{N_{\mathrm{B}}}{N_{\mathrm{A}}}$ (Eq. 5-252):

Perry’s writes Eq. 5-252 with the effective values $\frac{ε_{p}D_{\mathrm{AB}}}{τ}$ and $\frac{ε_{p}D_{\mathrm{K}}}{τ}$; applying $\frac{ε}{τ}$ after combining gives the same $D_{\mathrm{eff}}$. For equimolar counterdiffusion ($α = 0$) this reduces to $\frac{1}{D} = \frac{1}{D_{\mathrm{AB}}} + \frac{1}{D_{\mathrm{K}}}$ (Eq. 5-253). Table 5-16 labels that case $N_{\mathrm{A}} = N_{\mathrm{B}}$; since $α = 0$ requires $N_{\mathrm{B}} = -N_{\mathrm{A}}$, it means $N_{\mathrm{A}} = -N_{\mathrm{B}}$. Perry’s gives a normal tortuosity range of 2 to 6 for silica gel, alumina and other porous solids, and 5 to 65 for activated carbon (p. 5-55). In pure Knudsen flow with no reaction, Graham’s law fixes the flux ratio: $\frac{N_{\mathrm{B}}}{N_{\mathrm{A}}} = -\left(\frac{M_{\mathrm{A}}}{M_{\mathrm{B}}}\right)^{1/2}$.

Simulator

Open this simulator on its own page.

Try this

• Sweep the pore diameter from 2 nm to 10 µm at 1 atm. Where do the regime boundaries fall, and which diffusivity dominates in each?
• Fix the pore size and change the pressure from 0.02 atm to 20 atm. Why does the flux plot flatten at one end?
• Switch the flux condition from equimolar to Graham’s law to stagnant B at the same composition. How much does the combined $D$ change, and when would using the wrong $α$ matter?

Credits and sources

• Perry’s Chemical Engineers’ Handbook, 9th ed.: Table 5-16 (Eqs. 5-249 to 5-253), Eq. 5-261 and text, p. 5-55.
• Secondary: Geankoplis, 3rd ed., Sec. 7.6 (diffusion of gases in porous solids and capillaries).
• Simulator: Dr. Ray’s Learning Center. Drafted with AI assistance, then checked against Perry’s.

Course: Transport Phenomena. Previous: Pseudo-Steady Evaporation. Next: The Semi-Infinite Solid.