Knudsen number

The Knudsen number is the dimensionless ratio of an intermolecular mean free path ${\ell }_{\mathrm{mfp}}$ to a specified characteristic length $L$ of the apparatus or flow. The mean free path measures the distance travelled between gas–gas collisions; $L$ measures the distance over which boundaries or macroscopic conditions matter. Wall encounters must not be counted as gas–gas collisions in this comparison: doing so would hide the very wall-dominated behavior we want to detect. The same gas can therefore have a small Knudsen number in a large vessel and a large Knudsen number in a tiny opening.

For a dilute hard-sphere gas of identical molecular diameter $d$ in thermal equilibrium, use the conventionHere $n$ is the number density, $p$ the pressure, $T$ the absolute temperature, and $k_B$ the Boltzmann constant; the second equality uses the ideal gas law. The collision cross-section is $\pi d^2$, and the $\sqrt{2}$ accounts for both partners moving with independent equilibrium velocities. Mixtures and velocity-dependent collision cross-sections require a stated molecular model and mean free path convention; the displayed expression is not an exact formula for every real gas. ^{[ref] [ref]}

As a numerical toy example, choose $T=300\mathrm{K}$ and $d=0.37\mathrm{nm}$, without claiming that this is a measured molecular diameter for a particular gas. The formula gives ${\ell }_{\mathrm{mfp}}\simeq 68.1\mathrm{nm}$ at $p=10^5\mathrm{Pa}$. It yields $\mathrm{Kn}\simeq 6.81\times 10^{-6}$ for a $1\mathrm{cm}$ vessel but $\mathrm{Kn}\simeq 0.681$ for a $100\mathrm{nm}$ gap. At $p=1\mathrm{Pa}$ the same toy gas has ${\ell }_{\mathrm{mfp}}\simeq 6.81\mathrm{mm}$, comparable to the centimetre vessel. Reducing either pressure or apparatus size can thus expose molecular transport.

Choose $L$ for the question being asked: a tube diameter compares the mean free path with wall separation, while tube length describes a different geometric scale. For a positive macroscopic field $q$, a local gradient scale can instead be $L_q=|q|/|\nabla q|$, where its gradient is nonzero. A large vessel can contain a thin region with a large local Knudsen number. Small geometric Knudsen number alone does not guarantee a slowly varying near-equilibrium flow. The time, velocity and space scalings must also be stated when the Knudsen number appears as a coefficient in a Boltzmann equation. ^[ref]