Knudsen Number Calculator
Enter the mean free path and a characteristic length to get the Knudsen number and its flow regime instantly. Switch to kinetic-theory mode to compute the mean free path from temperature, pressure, and molecular diameter using the Boltzmann constant formula, then pipe that straight into the Knudsen number. A gauge and reference table show where your result falls across the four regimes: continuum, slip, transition, and free-molecular flow.
What is the Knudsen number?
The Knudsen number (Kn) is a dimensionless quantity that compares the mean free path of a gas molecule to a characteristic length of the flow geometry - typically a pipe diameter, channel height, or the size of an object immersed in the gas. It was introduced by the Danish physicist Martin Knudsen in the early 20th century while studying low-pressure gas flows through tubes. When Kn is much less than one, molecules collide with each other far more often than with surfaces, and the gas behaves like a continuous fluid that classical fluid mechanics can describe well. When Kn is much larger than one, molecules travel long distances between collisions and interact mainly with surfaces; kinetic theory takes over from continuum mechanics. The transition between these extremes is what the Knudsen number quantifies.
The formula: direct and kinetic-theory routes
The basic definition is Kn = lambda / L, where lambda is the mean free path and L is the characteristic length. For a known lambda this is a single division. When you need to derive lambda from gas properties, kinetic theory gives lambda = k_B T / (sqrt(2) pi d^2 P), where k_B = 1.3806 x 10^-23 J/K is the Boltzmann constant, T is absolute temperature in Kelvin, d is the effective molecular collision diameter, and P is the absolute pressure. For air at 293 K and 101 325 Pa, this formula yields lambda ~ 68 nm. Doubling the pressure halves the mean free path; doubling the temperature increases it by a factor of two (since molecules move faster and collide less often per unit volume). The characteristic length L depends on the geometry: pipe inner diameter for tube flow, gap height for slider bearings, or device feature size for MEMS structures.
The four flow regimes and why they matter for engineering
Continuum flow (Kn below 0.001): the Navier-Stokes equations apply with standard no-slip and no-temperature-jump boundary conditions, and conventional CFD tools are reliable. Slip flow (0.001 to 0.1): Navier-Stokes equations remain a useful approximation, but molecules close to surfaces slip rather than stopping dead, so velocity-slip and temperature-jump corrections must be added to wall boundary conditions. Transition flow (0.1 to 10): neither continuum nor purely free-molecular assumptions hold; the Boltzmann equation or Direct Simulation Monte Carlo (DSMC) methods are required. Free-molecular flow (Kn above 10): molecules travel from surface to surface without inter-molecular collisions, and kinetic theory - not fluid mechanics - governs mass flow, heat transfer, and drag. This regime is encountered in ultra-high vacuum systems, nanotechnology, atmospheric entry vehicles at high altitude, and MEMS sensors operating in low-pressure environments.
Practical examples across scales
At atmospheric pressure, air has a mean free path of about 68 nm. A 1 cm diameter pipe gives Kn ~ 7 x 10^-6 (deep continuum), while a 100 nm nanochannel gives Kn ~ 0.68 (transition). Semiconductor fabrication chambers operate at pressures of 1 to 100 Pa, pushing Kn well above 0.1 even for millimetre-scale features, which is why deposition uniformity models must account for slip. At 100 km altitude, the mean free path for air exceeds 1 cm, so satellite drag cannot be computed by aerodynamics alone. MEMS gyroscopes and accelerometers sealed at near-atmospheric pressure typically operate around Kn = 0.01 to 0.1 - the slip regime - where squeeze-film damping models need slip corrections. Knowing the regime before choosing a simulation tool saves both time and avoids results that are systematically wrong.
Knudsen number flow regimes
| Regime | Kn range | Governing approach | Typical application |
|---|---|---|---|
| Continuum | Kn < 0.001 | Navier-Stokes (no-slip) | Pipes, external aerodynamics, weather |
| Slip flow | 0.001 - 0.1 | Navier-Stokes + slip/jump BC | MEMS channels, low-pressure CVD |
| Transition | 0.1 - 10 | Boltzmann equation, DSMC | Microchannels, satellite drag, vacuum pumps |
| Free-molecular | Kn > 10 | Kinetic theory, collisionless | Ultra-high vacuum, nano-scale devices |
The four standard flow regimes and the governing equation approach for each. Boundaries shift slightly depending on the author and application.
Frequently asked questions
What does a Knudsen number of 1 mean?
A Kn of 1 means the mean free path equals the characteristic length. This falls in the transition regime (0.1 to 10), where neither continuum fluid mechanics nor purely kinetic theory applies cleanly. Accurate simulation requires Boltzmann-based approaches or DSMC.
How does pressure affect the Knudsen number?
Mean free path is inversely proportional to pressure (lambda = k_B T / (sqrt(2) pi d^2 P)), so halving the pressure doubles lambda and doubles Kn. This is why rarefaction effects appear in vacuum equipment even when the physical dimensions are large: low pressure can push an otherwise continuum system into the slip or transition regime.
What molecular diameter should I use for air?
The effective collision diameter of air molecules is approximately 370 pm (picometres), which is an average weighted toward N2 (~364 pm) and O2 (~346 pm). For pure nitrogen, use 364 pm; for CO2, use 455 pm; for hydrogen, use 289 pm. This calculator defaults to 370 pm for air.
Can I use the Knudsen number for liquids?
The Knudsen number concept applies primarily to gases, where the mean free path is a well-defined quantity from kinetic theory. For liquids, the mean free path is on the order of molecular spacing and the concept loses its utility. For nano-confined liquids, other dimensionless parameters such as the molecular Knudsen number based on the ratio of molecular size to channel width are sometimes used, but interpretation differs from gas-phase Kn.
What is the difference between the Knudsen number and the Mach number?
The Mach number (Ma) describes compressibility: how fast the flow moves relative to the speed of sound. The Knudsen number describes rarefaction: how often molecules collide relative to the device scale. The two are related by Kn = Ma x sqrt(pi gamma / 2) / Re (where Re is the Reynolds number and gamma is the specific heat ratio), so a hypersonic re-entry vehicle combines high Ma and high Kn at altitude. In everyday subsonic flows, Kn is the relevant rarefaction measure and Ma can be ignored.
When does the Navier-Stokes equation break down?
The Navier-Stokes equations are a continuum model that assumes molecules collide so frequently the gas averages out into smooth flow variables. This breaks down when Kn exceeds roughly 0.001 to 0.01. In the slip regime (up to Kn ~ 0.1) the equations are still usable with modified wall boundary conditions. Above Kn ~ 0.1, higher-order methods or kinetic solvers are needed.