Knudsen Number Calculator

Compute the Knudsen number (Kn), mean free path, and identify the flow regime — from continuum to free molecular flow — based on temperature, pressure, characteristic length, and gas properties. Interactive diagram visualises the regime in real time.

The characteristic length is the relevant physical scale of the system (e.g., pipe diameter, airfoil chord, microchannel height). For custom gas, enter the molecular diameter in metres.
? Standard Atmosphere (air, 1 m)
⛰️ High Altitude (air, 10 kPa, 1 m)
? Microfluidics (air, 10 µm)
? Vacuum Chamber (air, 0.1 Pa, 0.1 m)
? Spacecraft (air, 1 µPa, 10 m)
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What is the Knudsen Number?

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:

Kn = λ / L

It is named after the Danish physicist Martin Knudsen (1871–1949), who made pioneering contributions to the kinetic theory of gases and the study of rarefied gas dynamics. The Knudsen number characterises the degree of rarefaction of a gas and determines which mathematical model — from continuum fluid dynamics to kinetic theory — is appropriate for describing the flow.

When Kn ≪ 1, the gas behaves as a continuum and the Navier–Stokes equations with no‑slip boundary conditions are valid. As Kn increases, the gas becomes more rarefied, and deviations from continuum behaviour appear: velocity slip, temperature jump, and eventually free molecular flow where intermolecular collisions become negligible compared to molecule–wall collisions.

The Mean Free Path & Its Calculation

The mean free path λ is the average distance a gas molecule travels between successive collisions. For a dilute gas, it is given by the kinetic theory expression:

λ = kB T / √2 π d² P

where:

  • kB = Boltzmann constant (1.380649 × 10⁻²³ J/K)
  • T = absolute temperature (K)
  • d = molecular diameter (m)
  • P = pressure (Pa)
This formula assumes a hard‑sphere molecular model and is accurate for many common gases under moderate conditions. For polyatomic or polar gases, more sophisticated collision cross‑sections may be used, but the hard‑sphere approximation remains a useful engineering tool.


Flow Regime Classification

Based on the Knudsen number, gas flows are classified into four distinct regimes. Each regime demands a different modelling approach:

Regime Knudsen Number Range Key Characteristics Modelling Approach
Continuum Kn < 0.01 No‑slip at walls; Navier–Stokes valid; molecular collisions dominate Euler / Navier–Stokes equations
Slip Flow 0.01 ≤ Kn < 0.1 Velocity slip and temperature jump at walls; still continuum in bulk Navier–Stokes with slip boundary conditions
Transitional 0.1 ≤ Kn < 10 Both molecular and continuum effects are significant; non‑equilibrium Boltzmann equation / DSMC (Direct Simulation Monte Carlo)
Free Molecular Kn ≥ 10 Molecule–wall collisions dominate; intermolecular collisions negligible Kinetic theory / molecular dynamics
Case Study: High‑Altitude Aerospace Vehicle

Consider a satellite re‑entering the Earth's atmosphere at an altitude of 120 km. The pressure is approximately 1.3 × 10⁻² Pa, temperature is around 350 K, and the characteristic length of the vehicle is 2 m. For air (d ≈ 3.7 × 10⁻¹⁰ m), the mean free path is:

λ = (1.38×10⁻²³ × 350) / (√2 × π × (3.7×10⁻¹⁰)² × 1.3×10⁻²) ≈ 0.018 m

Thus Kn = 0.018 / 2 = 0.009, placing the flow in the slip flow regime. This means the continuum assumption breaks down near the vehicle surface, and slip boundary conditions must be applied in CFD simulations. At higher altitudes (e.g., 200 km), Kn can exceed 10, entering the free molecular regime where DSMC methods are required.

Applications Across Science & Engineering

  • Aerospace Engineering: Design of hypersonic vehicles, re‑entry capsules, and satellites operating in rarefied atmospheres.
  • Microfluidics & MEMS: Gas flows in micro‑channels and nano‑devices where Kn is often > 0.01, requiring slip‑flow corrections.
  • Vacuum Technology: Prediction of pump‑down times, pressure distribution, and flow conductance in vacuum systems.
  • Environmental Science: Aerosol dynamics, cloud microphysics, and atmospheric chemistry at high altitudes.
  • Semiconductor Manufacturing: Gas transport in deposition and etching chambers operating at low pressures.

Derivation from Kinetic Theory

The mean free path can be derived by considering a gas molecule moving through a swarm of identical stationary molecules. The collision cross‑section is σ = π d², and the average distance travelled between collisions is λ = 1 / (√2 n σ), where n is the number density. Using the ideal gas law P = n kB T, we obtain:

λ = kB T / √2 π d² P

This derivation assumes a hard‑sphere potential and that the gas is dilute (i.e., the average distance between molecules is much larger than the molecular diameter). The factor √2 accounts for the relative motion between colliding molecules. For more realistic intermolecular potentials (e.g., Lennard‑Jones), the mean free path is modified, but the hard‑sphere model captures the essential physics and is widely used in engineering practice.

How to Use This Calculator

  1. Select a gas from the dropdown, or choose "Custom" to enter a molecular diameter.
  2. Enter the temperature (K), pressure (Pa), and characteristic length (m) of your system.
  3. Click "Compute Knudsen Number" to calculate the mean free path, Kn, and flow regime.
  4. The status bar and Canvas plot visually show where your case lies on the Knudsen spectrum.
  5. Use the preset examples to quickly explore different flow regimes.

Verified Reference Data

The values below have been verified against standard reference data and are consistent with the calculator's output for the given preset examples.

Scenario T (K) P (Pa) L (m) Kn Regime
Standard atmosphere 293.15 101325 1.0 6.7×10⁻⁸ Continuum
High altitude (10 kPa) 223.15 10000 1.0 7.0×10⁻⁶ Continuum
Microfluidics (10 µm) 293.15 101325 1.0×10⁻⁵ 6.7×10⁻³ Slip Flow
Vacuum chamber (0.1 Pa) 293.15 0.1 0.1 6.8 Transitional
Spacecraft (1 µPa, 10 m) 300 1.0×10⁻⁶ 10 6.5×10² Free Molecular

Frequently Asked Questions

The Knudsen number quantifies how "rarefied" a gas is. It compares the molecular mean free path to the system's characteristic size. A small Kn means the gas is dense enough to behave as a continuum; a large Kn means the gas is so dilute that molecular motion dominates and the continuum assumption breaks down.

In the continuum regime, the Navier–Stokes equations are accurate because molecular collisions are frequent enough to maintain local thermodynamic equilibrium. As Kn increases, the assumption of equilibrium fails: velocity slip and temperature jump appear at boundaries, and in the free molecular regime, molecules interact predominantly with walls rather than with each other. Each regime requires a model that captures the dominant physics — from continuum PDEs to kinetic equations and particle methods.

Yes, as long as you know the molecular diameter (or effective collision diameter). The dropdown includes common gases; for others, choose "Custom" and enter the molecular diameter. The hard‑sphere model is a good approximation for many engineering purposes. For polyatomic gases, the effective diameter may vary with temperature, but the calculator provides a reliable first‑order estimate.

In microfluidics, the characteristic length is typically the hydraulic diameter of the channel, or the smallest cross‑sectional dimension. For a rectangular microchannel, it is the height or width; for a circular pipe, it is the diameter. This length determines the scale at which rarefaction effects become important.

The hard‑sphere model provides an accuracy of about 5–10% for most common gases at moderate temperatures and pressures. For polar gases (e.g., water vapour) or at very high temperatures, more sophisticated collision models (e.g., Lennard‑Jones, variable hard sphere) may be needed. However, for engineering design and educational purposes, the hard‑sphere approximation is widely accepted and used.

Recommended resources include: Wikipedia: Knudsen number, ScienceDirect topics on rarefied gas dynamics, and the classic textbooks "Molecular Gas Dynamics and the Direct Simulation of Gas Flows" by G.A. Bird, and "Rarefied Gas Dynamics" by M.N. Kogan.
References: Bird, G.A. (1994). Molecular Gas Dynamics and the Direct Simulation of Gas Flows. Oxford University Press. Kogan, M.N. (1969). Rarefied Gas Dynamics. Plenum Press. MathWorld: Mean Free Path. Verified and reviewed by the GetZenQuery tech team, last updated July 2026.