Knudsen Number Calculator

The Knudsen number (Kn) is a dimensionless number that measures how rarefied a gas flow is. It is the ratio of the molecular mean free path λ to a characteristic physical length scale L of the flow: Kn = λ/L. It is named after Danish physicist Martin Knudsen. Small Kn means the gas behaves as a continuum; large Kn means molecular, collisionless effects dominate.

The Knudsen number is Kn = λ/L, where λ is the mean free path and L is the characteristic length (for example a channel height, particle diameter, or pipe diameter). When λ is not known, kinetic theory gives λ = kB·T/(√2·π·d²·P), with the Boltzmann constant kB = 1.380649×10⁻²³ J/K (exact), T the absolute temperature in kelvin, d the molecular (kinetic) diameter in meters, and P the pressure in pascals.

The mean free path λ is the average distance a molecule travels between collisions with other molecules. From kinetic theory, λ = kB·T/(√2·π·d²·P). Using the kinetic diameter of air d = 3.7×10⁻¹⁰ m, this calculator gives λ = 61.19 nm for air at STP (273.15 K, 101325 Pa) and λ = 65.67 nm for air at 20 °C and 1 atm. Textbooks often quote about 68 nm; the small difference comes from the choice of effective molecular diameter.

Use the kinetic (collision) diameter of the gas. This calculator presets: air 0.370 nm, nitrogen (N₂) 0.364 nm, oxygen (O₂) 0.346 nm, hydrogen (H₂) 0.289 nm, helium (He) 0.260 nm, argon (Ar) 0.340 nm, carbon dioxide (CO₂) 0.394 nm. At STP these give, for example, λ = 63.23 nm for nitrogen, 69.98 nm for oxygen, and 123.9 nm for helium. You can also enter a custom diameter.

Four regimes are distinguished by Kn: Kn < 0.01 is the continuum regime (ordinary fluid mechanics, Navier–Stokes equations apply); 0.01 ≤ Kn < 0.1 is the slip-flow regime (continuum with velocity slip and temperature jump at walls); 0.1 ≤ Kn < 10 is the transition regime (continuum breaks down, use the Boltzmann equation or DSMC simulation); Kn ≥ 10 is the free-molecular regime (molecules collide with walls far more often than with each other, effectively collisionless).

The continuum assumption — treating the gas as a continuous fluid governed by the Navier–Stokes equations — is valid when Kn < 0.01, i.e. when the mean free path is less than one percent of the characteristic length. For air at atmospheric pressure (λ ≈ 65.67 nm at 20 °C), this holds for any geometry larger than about 6.6 µm.

Slip flow occurs for 0.01 ≤ Kn < 0.1. The gas still approximately obeys continuum equations, but molecules slip along solid walls instead of sticking to them, producing a velocity slip and a temperature jump at the boundary. It is modeled with the Navier–Stokes equations plus Maxwell slip and Smoluchowski temperature-jump boundary conditions. Example: air at 20 °C and 1 atm (λ = 65.67 nm) flowing in a 1 µm channel gives Kn = 0.0657, which is slip flow.

Take air at 20 °C and 1 atm. Kinetic theory gives λ = 65.67 nm. For a channel height L = 1 µm, Kn = λ/L = 0.0657, which falls in the slip-flow regime (0.01–0.1). For a 100 nm gap, Kn = 0.657, which is in the transition regime (0.1–10) and needs Boltzmann/DSMC methods. For a 1 m pipe, Kn ≈ 6.6×10⁻⁸, deep in the continuum regime.

The mean free path is inversely proportional to pressure (λ ∝ 1/P), so the Knudsen number grows as pressure drops. For air at 20 °C, lowering the pressure from 1 atm to 100 Pa increases λ from 65.67 nm to 66.54 µm — a factor of about 1013. That is why vacuum systems, high-altitude flight, and microdevices operate at much larger Knudsen numbers than everyday macroscopic flows.

Kn ≥ 10 is the free-molecular regime: molecules hit the walls far more often than they hit each other, so intermolecular collisions can be ignored and the gas is treated as collisionless (ballistic). Drag and heat transfer then scale with the molecular flux striking surfaces. This regime occurs for satellites in very low Earth orbit, in high-vacuum chambers, and in nanoscale gaps at low pressure. For comparison, air at 270 K and 0.032 Pa has λ = 0.19 m, so a 1 m satellite is at Kn = 0.19 — still transition regime, not yet free molecular.

All three are dimensionless, but they measure different physics. The Knudsen number Kn = λ/L measures rarefaction (how discrete the gas is). The Mach number measures compressibility (flow speed relative to the speed of sound). The Reynolds number measures the ratio of inertial to viscous forces. A flow can be low-Mach and low-Re yet high-Kn — for example creeping flow in a nanoscale channel.

Mean free path λ: meters (m), micrometers (µm), nanometers (nm). Characteristic length L: meters (m), millimeters (mm), micrometers (µm), nanometers (nm). Temperature T: kelvin (K) or degrees Celsius (°C). Pressure P: pascals (Pa), kilopascals (kPa), atmospheres (atm), bar, or mmHg (torr). Molecular diameter: nanometers (nm) or meters via the gas presets.