Knudsen Number Calculator — Mean Free Path & Flow Regime

Formula

How this is calculated

The mean free path λ is the average distance a gas molecule travels between successive collisions. For an ideal gas it is λ = k_B T / (√2 π d² P), where k_B = 1.381 × 10⁻²³ J/K is the Boltzmann constant, T is temperature in kelvin, d is the effective collision diameter, and P is pressure. Raising T increases λ (faster molecules, less time between collisions in terms of path length); raising P decreases λ (more molecules per unit volume, more frequent collisions). At 20 °C and 1 atm, air has λ ≈ 66 nm.

The Knudsen number divides this by the characteristic length scale L of the problem — a channel width, pore diameter, MEMS gap, or aircraft chord at altitude. When Kn ≪ 0.001 the gas behaves as a continuum and classical Navier-Stokes equations apply with standard no-slip boundary conditions. In the slip-flow regime (0.001 < Kn < 0.1) the fluid equations still hold but velocity slip at walls must be included. In the transition regime (0.1 < Kn < 10) neither continuum nor free-molecular models are fully accurate; Boltzmann-equation solvers or DSMC (Direct Simulation Monte Carlo) are needed. Above Kn ≈ 10, gas behaves more like individual billiard balls than a fluid — free-molecular flow.

This calculator uses the hard-sphere ideal-gas model with a single-species effective diameter. Real gas mixtures require mixture-averaged collision integrals. The molecular diameter 3.7 × 10⁻¹⁰ m is the standard kinetic diameter for air (∼79% N₂, ∼21% O₂); for other gases, use tabulated kinetic diameters (e.g. He ≈ 2.6 × 10⁻¹⁰ m, CO₂ ≈ 4.0 × 10⁻¹⁰ m).