Knudsen Number Calculator

The Knudsen number (Kn) is a dimensionless quantity defined as the ratio of the molecular mean free path (λ) to a representative physical length scale (L) of a system: Kn = λ/L. It characterises how 'rarefied' a gas is relative to the geometry it occupies. A small Kn means the gas behaves like a continuum; a large Kn means individual molecular effects dominate.

The Knudsen number formula is simply Kn = λ / L, where λ is the mean free path of the gas molecules and L is the characteristic length scale of the physical system (e.g., pipe radius, channel width, or body dimension). Both values must be expressed in the same units before dividing.

Four regimes are commonly identified: Continuum flow (Kn < 0.001), where the Navier-Stokes equations apply; Slip flow (0.001 ≤ Kn < 0.1), where Navier-Stokes with slip boundary conditions apply; Transition flow (0.1 ≤ Kn < 10), requiring DSMC or Boltzmann equation approaches; and Free molecular flow (Kn ≥ 10), where statistical mechanics and kinetic theory are necessary.

The mean free path (λ) is the average distance a gas molecule travels between successive collisions with other molecules. It depends on gas pressure, temperature, and molecular size. For air at standard atmospheric pressure and room temperature (20 °C), the mean free path is approximately 68 nanometres.

As pressure decreases (higher vacuum), the mean free path increases significantly because molecules collide less frequently. This raises the Knudsen number. In high vacuum (< 10⁻³ hPa), the mean free path can exceed tens of metres, pushing Kn >> 1 for any typical apparatus, so free-molecular flow and statistical mechanics must be used rather than continuum equations.

Yes. There is an alternative relationship: Kn ≈ (Ma / Re) × √(γπ/2), where Ma is the Mach number, Re is the Reynolds number, and γ is the ratio of specific heats. This form is useful in aerodynamics when Mach and Reynolds numbers are already known, and it gives the same dimensionless Kn value.

You should transition to statistical mechanics (e.g., the Boltzmann equation or Direct Simulation Monte Carlo methods) when Kn > 0.1. In the transition regime (0.1–10) both formulations partially apply but continuum assumptions break down. For Kn > 10, the gas is so rarefied that individual particle interactions must be modelled explicitly and continuum fluid equations are no longer valid.

The Knudsen number is widely used in microfluidics and MEMS design, spacecraft aerodynamics in near-vacuum upper atmosphere, vacuum system engineering, semiconductor manufacturing processes, and nanoscale heat transfer analysis. Any system where the physical length scale approaches the molecular mean free path requires Kn analysis to choose the correct governing equations.