Formula and Method for the Knudsen Number
The Knudsen number is a dimensionless quantity used in fluid mechanics and rarefied gas dynamics to determine whether a gas can be treated as a continuum or must be modeled molecule by molecule. It is defined as Kn = λ / L, where λ (lambda) is the mean free path of the gas molecules — the average distance a molecule travels between collisions — and L is a characteristic length scale of the flow, such as a pipe diameter, a gap width, or a particle size. This calculator first computes λ from kinetic theory, then divides it by your chosen L to get Kn and reports the resulting flow regime.
How the calculation works
From the kinetic theory of gases, the mean free path of a molecule in an ideal gas is λ = kB·T / (√2·π·d²·P), where kB is the Boltzmann constant (1.380649×10⁻²³ J/K), T is the absolute temperature, d is the effective collision diameter of the gas molecule, and P is the pressure. The √2 factor accounts for the relative motion of all the gas molecules, not just one molecule moving through stationary neighbors. Once λ is known, the Knudsen number follows directly from Kn = λ / L. Because λ shrinks as pressure or molecular size increases and grows as temperature increases, Kn is highly sensitive to operating conditions — the same geometry can be continuum flow at atmospheric pressure and free-molecular flow inside a vacuum chamber.
Common mistakes
- Mixing temperature scales: the mean free path formula needs an absolute temperature (Kelvin) — plugging in Celsius or Fahrenheit directly gives a wrong, sometimes negative, result. This calculator converts for you, but hand calculations must convert first.
- Wrong characteristic length: L should represent the smallest relevant flow dimension — a channel height, a pore diameter, a particle radius — not an unrelated large-scale dimension, which would make Kn artificially small and hide real rarefaction effects.
- Guessing the molecular diameter: effective collision diameters vary by gas (about 0.37 nm for air/N₂, 0.36 nm for O₂, 0.26 nm for He) — using the wrong value can shift the mean free path, and therefore Kn, by 30% or more.
Flow regimes and when they apply
- Kn < 0.01 — Continuum flow: the Navier–Stokes equations with standard no-slip boundary conditions are accurate. This covers most everyday aerodynamics and pipe flow.
- 0.01 ≤ Kn < 0.1 — Slip flow: Navier–Stokes still applies, but boundary conditions need velocity-slip and temperature-jump corrections. Common in microfluidic channels and MEMS devices.
- 0.1 ≤ Kn < 10 — Transition flow: continuum equations break down; models need the Burnett equations or particle-based methods such as direct simulation Monte Carlo (DSMC).
- Kn ≥ 10 — Free-molecular flow: molecule–surface collisions dominate over molecule–molecule collisions; this regime governs high-vacuum systems and spacecraft in low Earth orbit or the upper atmosphere.