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.
Frequently Asked Questions
What is the Knudsen number used for?
The Knudsen number tells you whether a gas flow can be treated as a continuous fluid or must be modeled as individual molecules. It is essential in vacuum system design, microfluidics, MEMS devices, aerosol science, and high-altitude or rarefied aerodynamics, where the flow passage can become comparable to or smaller than the distance molecules travel between collisions.
How do I find the mean free path of a gas?
Use the kinetic theory formula λ = kB·T / (√2·π·d²·P), with temperature in Kelvin, pressure in pascals, and the molecular collision diameter in meters. For air at 20°C (293.15 K) and 1 atm (101,325 Pa) with d ≈ 0.37 nm, this gives a mean free path of about 66-68 nanometers.
What Knudsen number counts as continuum flow?
By convention, Kn below 0.01 is continuum flow, where the Navier-Stokes equations with no-slip walls hold. Between 0.01 and 0.1 is slip flow, 0.1 to 10 is transition flow, and above 10 is free-molecular flow, each needing a different modeling approach.
Does pressure or temperature matter more for the Knudsen number?
Pressure usually dominates: mean free path is inversely proportional to pressure, so dropping from 1 atm to a rough vacuum increases the mean free path (and Kn) by the same large factor. Temperature enters only to the first power, so pressure changes are almost always the bigger lever.
Common Tip Percentages
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| Category | % of Net Income | Examples |
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| Needs | 50% | Rent, groceries, utilities, insurance |
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Common Measurement Conversions for Everyday Use
| Quantity | US / Imperial | Metric Equivalent |
|---|
| 1 cup | 8 fl oz | 237 mL |
| 1 pint | 16 fl oz | 473 mL |
| 1 quart | 32 fl oz | 946 mL |
| 1 gallon | 128 fl oz | 3.785 L |
| 1 lb | 16 oz | 453.6 g |
| 1 ft | 12 in | 30.48 cm |