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Molecular length versus device length
Kn = λ/L compares the mean free path λ, the average distance a gas molecule travels between collisions, with the size L of a device or body. When λ is much shorter than L, molecules collide with each other many times before reaching a wall, so the gas can be described by averaged quantities such as pressure and viscosity. As λ approaches L, the continuum picture starts to fail, for example through velocity slip at walls.
In kinetic theory, λ is inversely proportional to pressure at a given temperature. Halving the pressure doubles λ, which is why Kn can be large in vacuum equipment even when the equipment itself is large. This calculator does not compute λ for you, so convert λ to the actual pressure and temperature before entering it.
Variables and SI units
| Symbol | Meaning | SI unit | Example |
|---|---|---|---|
| λ | Average distance a gas molecule travels between collisions | m | 6.8e-8 |
| L | Channel diameter, gap or particle size | m | 0.001 |
Entering λ and L
- Enter both in metres: multiply nm by 10⁻⁹ and μm by 10⁻⁶. The input boxes accept exponent notation such as 8.24e-7.
- The default example 6.8×10⁻⁸ m (68 nm) rounds the 67.3 nm that NIST reports for air at 23 °C and 101.3 kPa. Do not reuse it away from those conditions.
- If λ is unknown, estimate it with the hard-sphere expression λ = kBT/(√2 π d² p) from the MIT 3.185 course notes, or scale a reference value to your pressure and temperature.
- Choose L as the length that confines the flow: a channel diameter, the gap between two walls or a particle diameter. The same device gives different Kn for different choices of L.
Reading the regime limits
The calculator labels Kn < 0.01 continuum flow, 0.01 to 0.1 slip flow, and 0.1 or more transitional or free-molecular flow. A NASA-supported survey of high-atmosphere thermometry literature splits the upper range into transition (0.1 to 10) and free molecular (above 10) while stating that these limits are approximate. If your result sits near a boundary, say 0.005 to 0.02 or 0.05 to 0.2, compare the two neighbouring models rather than trusting the label alone. The label does not compute slip velocities at walls.
Same gas, different size
The example uses λ = 6.8×10⁻⁸ m and L = 1 mm, giving Kn = 6.8×10⁻⁵, labelled continuum. Values below 0.0001 appear in exponent form, here 6.8000e-5. The gas-model note keeps the same λ and shrinks L to 1 μm, giving Kn = 0.068. Nothing about the gas changed, yet the smaller device moves into the slip-flow range. Compressibility is a separate judgement made with the Mach number; a small Kn does not mean density changes can be ignored.
Examples that convert λ before entry
Example 1. A 20 nm nanoparticle measured at 8.27 kPa
Given: NIST researchers lowered the pressure in a nano differential mobility analyzer to 8.27 kPa to measure slip correction for PSL particles. The smallest particle diameter was 19.90 nm, and the nominal analyzer temperature was 296.15 K (23 °C). The reference mean free path from the same paper is λ₀ = 67.3 nm at 23 °C and 101.3 kPa.
Converting λ: the temperature equals the reference, so only the pressure ratio applies: 101.3 ÷ 8.27 = 12.25, and λ = 67.3 nm × 12.25 = 824 nm = 8.24 × 10⁻⁷ m.
Calculator entry: λ = 8.24e-7 and L = 1.99e-8 give Kn = 41.4, labelled transition/free molecular.
Interpretation: Aerosol literature uses Kn = 2λ/Dp, so the same condition becomes about 83. That matches the abstract's "Knudsen numbers as large as 83", and Table 7c lists Kn = 82.934 for the 8.27 kPa, 296.3 K runs. The difference of about 0.1% comes from the actual run temperature and the 101.33 kPa reference pressure differing slightly from this example's assumptions. When comparing with the literature, multiply the calculator value by two.
Example 2. A 10 μm hot wire at 11 km altitude
Given: the standard atmosphere table in Cornell's flight dynamics course notes gives T = 216.65 K and p = 22 631.70 Pa at 11 000 m. The wire diameter of 10 μm is an assumed value.
Converting λ: temperature changes too, so use NIST equation (11). Temperature ratio 216.65 ÷ 296.15 = 0.7316, pressure ratio 101 300 ÷ 22 631.70 = 4.476, Sutherland-type factor (1 + 110.4/296.15) ÷ (1 + 110.4/216.65) = 0.9094. So λ = 67.3 nm × 0.7316 × 4.476 × 0.9094 = 200 nm. Scaling with pressure alone would give 301 nm, an overestimate of about 50%.
Calculator entry: λ = 2.00e-7 and L = 1e-5 give Kn = 0.0200, labelled slip flow.
Interpretation: a wire treated as continuum at sea level is in the slip range at 11 km. Equation (11) is a correction used near room temperature, extended here to 216.65 K, so treat the result as an order-of-magnitude estimate.
Using the Knudsen number calculator
Why is there no separate free-molecular label above Kn = 10?
The calculator groups everything from 0.1 upward into one label. A limit of 10 between transition and free molecular is common, but some sources, such as the figure quoted in NASA TM 107730, place the onset of free-molecular effects near Kn = 1. A result far above 10 can be read as free molecular.
The result shows 6.8000e-5. Is that an error?
No. Results below 0.0001 or very large results are shown in exponent notation. 6.8000e-5 means 6.8 × 10⁻⁵, or 0.000068.
Can I use it for helium or argon instead of air?
Yes. Kn = λ/L does not depend on the gas, so enter λ for that gas. The default 68 nm is an air value and must not be used for other gases. Estimate λ by putting that gas's molecular diameter d into the MIT 3.185 expression, and record which λ definition you used.
Related guides and calculators
Guide
Calculation notes
- Choosing gas-flow models: check Mach and Knudsen numbers separately: Puts the Kn from this page next to a Mach check and uses isentropic relations to size the compressibility error.
Related calculators
- Mach number (Ma): Compute Ma for the same speed and temperature to see whether density changes are negligible.
- Reynolds number (Re): Once Kn is below 0.01, go on to the ratio of inertial to viscous forces.
References
- NIST: Slip Correction Measurements of Certified PSL Nanoparticles Using a Nanometer Differential Mobility Analyzer. λ₀ = 67.3 nm, equation (11), lowest pressure 8.27 kPa and Kn up to 83, 19.90 nm particles, Kn = 2λ/Dp
- Cornell University (MAE 5070): Introduction to Flight Dynamics. Temperature and pressure at 11 000 m in standard atmosphere Table 1.1
- University of Utah / NASA: Survey of Literature on Convective Heat Transfer Coefficients and Recovery Factors for High Atmosphere Thermometry. Kn regime divisions and the statement that they are approximate
- MIT OpenCourseWare (3.185): Transport Phenomena in Materials Engineering. Hard-sphere expression for λ
- NASA Langley: Reynolds Number Influences in Aeronautics (NASA TM 107730). Figure description placing the onset of free-molecular effects near Kn = 1