= Knudsen number
{c}
{title2=$\mathrm{Kn}=\ell_{\mathrm{mfp}}/L$}
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 convention
$$
\ell_{\mathrm{mfp}}=\frac{1}{\sqrt2\,n\pi d^2}=\frac{k_BT}{\sqrt2\,\pi d^2p},\qquad \mathrm{Kn}=\frac{k_BT}{\sqrt2\,\pi d^2pL}.
$$
Here $n$ 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 $\sqrt2$ 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. https://farside.ph.utexas.edu/teaching/355/Surveyhtml/node219.html[Fitzpatrick, Mean Free Path, equations 5.209–5.219]{ref} https://arxiv.org/abs/1309.1197[Steve T. Paik, Is the mean free path the mean of a distribution?, dilute hard-sphere equilibrium convention]{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 $\ell_{\mathrm{mfp}}\simeq68.1\,\mathrm{nm}$ at $p=10^5\,\mathrm{Pa}$. It yields $\mathrm{Kn}\simeq6.81\times10^{-6}$ for a $1\,\mathrm{cm}$ vessel but $\mathrm{Kn}\simeq0.681$ for a $100\,\mathrm{nm}$ gap. At $p=1\,\mathrm{Pa}$ the same toy gas has $\ell_{\mathrm{mfp}}\simeq6.81\,\mathrm{mm}$, 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>. https://ntrs.nasa.gov/citations/20230013741[NASA, Comparison Between DSMC and CFD for Hypersonic Planetary Entry Simulations (2023), local gradient-length breakdown analysis]{ref}
\Image[/knudsen-scale-and-flight-survival.png]
{title=Mean free path versus apparatus scale, with a fixed-rate single-flight toy model}
{description=The three chosen ratios are examples, not regime boundaries. The curve assumes a constant collision hazard per distance and plots $\exp(-L/\ell_{\mathrm{mfp}})$; it is neither measured gas data nor the full free-path distribution of an equilibrium hard-sphere gas.}
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