Why compare molecular travel with device size
The Navier–Stokes equations and Fourier's law assume that gas molecules collide many times before reaching a body. Kn tests this by comparing the mean free path λ (average distance between collisions) with a length L such as a channel width, gap or body size. A NASA-supported survey of high-atmosphere thermometry literature notes that at very small Kn continuum concepts apply and Re and Ma alone characterise the flow; at large enough Kn they must be modified.
Kn matters in low-pressure equipment (long λ), MEMS devices (micrometre L) and high-altitude or planetary-entry vehicles (both). It is the starting point for choosing no-slip walls, slip boundary conditions or a particle method such as DSMC.
Kn = λ/L and how to get the mean free path
MIT course notes (3.185) define Kn as mean free path over process length and give the hard-sphere form Kn = kBT/(√2 π d² p L); dropping L gives the λ above (kB Boltzmann's constant, T absolute temperature, d molecular diameter, p pressure). At fixed temperature λ∝1/p: a tenth of the pressure means ten times the λ and the Kn.
Equation (10) of a NIST slip-correction paper writes kinetic-theory viscosity as μ = φρc̄λ, with c̄ the mean molecular speed, of the same order as the speed of sound. Substituting into Re = ρUL/μ and Ma = U/a shows Kn scales with Ma/Re, with a constant that depends on the λ model. NASA TM 107730 likewise speaks of "low-density (low Reynolds number) effects".
Enter λ and L in metres: nm × 10⁻⁹, μm × 10⁻⁶.
Open this calculatorHow far λ and Kn grow as pressure drops
NIST takes air λ₀ = 67.3 nm at 23 °C and 101.3 kPa and corrects it with Willeke's relation (its equation (11)), which at constant temperature reduces to λ = λ₀(p₀/p). The table entries are kinetic-theory conversions, not measurements. Regime names follow the high-atmosphere survey (below 0.01 continuum, 0.01 to 0.1 slip, 0.1 to 10 transition, above 10 free molecular).
| Pressure p | λ | Kn (L = 1 mm) | Kn (L = 25 mm) |
|---|---|---|---|
| 101.3 kPa | 67.3 nm | 6.73 × 10⁻⁵ (continuum) | 2.69 × 10⁻⁶ (continuum) |
| 10.13 kPa | 673 nm | 6.73 × 10⁻⁴ (continuum) | 2.69 × 10⁻⁵ (continuum) |
| 1.013 kPa | 6.73 μm | 0.00673 (continuum) | 2.69 × 10⁻⁴ (continuum) |
| 101.3 Pa | 67.3 μm | 0.0673 (slip) | 0.00269 (continuum) |
| 10 Pa | 0.682 mm | 0.682 (transition) | 0.0273 (slip) |
| 1 Pa | 6.82 mm | 6.82 (transition) | 0.273 (transition) |
The NIST uncertainty table also lists a reduced-pressure λ of 673 nm, matching the 10.13 kPa row. At atmospheric pressure a 1 mm channel has Kn below 10⁻⁴; below a few tens of pascals even a 25 mm pipe is in the slip range. Check this first when a vacuum-equipment CFD model disagrees with pressure measurements.
Kn for vacuum hardware, micro gaps and Mars aerobraking
Example 1. A 5 μm gap in air at atmospheric pressure
Given: λ = 67.3 nm for air at 23 °C and 101.3 kPa (NIST). The gap L = 5 μm is an assumed value representing the space between MEMS structures.
Unit conversion: λ = 67.3 nm = 67.3 × 10⁻⁹ m and L = 5 μm = 5 × 10⁻⁶ m.
Calculation: Kn = 67.3 × 10⁻⁹ ÷ (5 × 10⁻⁶) = 0.0135. The gap that gives Kn = 0.01 is 67.3 nm ÷ 0.01 = 6.73 μm, and the gap that gives Kn = 0.1 is 0.673 μm.
Interpretation: The high-atmosphere survey puts this just inside the slip range, but the Moss and Bird figure in NASA TM 107730 adds slip to Navier–Stokes only above 0.03, so 0.0135 would stay no-slip. Where limits differ by more than a factor of two, run both no-slip and slip-wall models and compare the damping force or flow rate.
Example 2. Pumping line and substrate gap in a 10 Pa vacuum chamber
Given: assumed operating conditions of 10 Pa, 23 °C, a 25 mm pumping line and a 1 mm holder-to-wall gap; λ₀ = 67.3 nm at 23 °C and 101.3 kPa (NIST).
Unit conversion: 10 Pa = 0.01 kPa, so the pressure ratio is 101.3 ÷ 0.01 = 10,130. Assuming the same temperature and applying λ∝1/p gives λ = 67.3 nm × 10,130 = 6.82 × 10⁻⁴ m (0.682 mm).
Calculation: pumping line Kn = 6.82 × 10⁻⁴ ÷ 0.025 = 0.0273; substrate gap Kn = 6.82 × 10⁻⁴ ÷ 0.001 = 0.682.
Interpretation: One chamber holds both slip (line) and transition (gap) flow, so compute Kn for each narrow passage that controls the result. In the transition range even slip-wall Navier–Stokes is hard to trust; consider DSMC.
Example 3. A power shunt on a solar panel during Mars aerobraking
Given: NASA's Mars Global Surveyor aerothermodynamics report gives a local mean free path of about 3 cm just above the power shunt, and shunt sections roughly 12 in long, 2 in wide and 1 in high.
Unit conversion: with 1 in = 0.0254 m, the length is 0.3048 m, the width 0.0508 m and the height 0.0254 m. λ = 3 cm = 0.03 m.
Calculation: length-based Kn = 0.03 ÷ 0.3048 = 0.0984; width-based Kn = 0.03 ÷ 0.0508 = 0.591; height-based Kn = 0.03 ÷ 0.0254 = 1.18.
Interpretation: The choice of L moves the answer from just below the slip limit to mid-transition. The report concluded λ is of the order of the instrument size; for local heating the smallest dimension governs. It also gives Kn of about 0.2 for the whole spacecraft at nominal periapsis density, well into transition, and notes that the local λ near the body gives a more accurate Kn than the freestream value.
Regime limits differ between references
Kn limits are modelling conventions, not physical constants; the high-atmosphere survey calls its divisions "approximate only".
| Source | Continuum and slip | Transition and free molecular |
|---|---|---|
| High-atmosphere thermometry survey (NASA-supported) | Kn < 0.01 continuum, 0.01 to 0.1 slip | 0.1 to 10 transition, Kn > 10 free molecular |
| NASA TM 107730 (quoting Moss and Bird) | Navier–Stokes applicable for Kn < 0.2, slip effects included above 0.03 | Free-molecular effects begin near 1, transition roughly 0.2 to 1.0 |
| MIT 3.185 course notes | Continuum gas models for Kn < 0.01 | Line-of-sight models for Kn > 1 |
| NASA Mars Global Surveyor report | (no separate division) | 0.1 to 10 transition, Kn > 10 free molecular |
| This site's calculator label | Kn < 0.01 continuum, 0.01 to 0.1 slip | 0.1 or more transition/free molecular |
Sensitivity is a better guide than any single limit: if Kn is within a factor of two of a boundary, apply both neighbouring models and compare flow rate, drag or heat flux. In the Mars Global Surveyor report, transitional effects lowered the drag coefficient by about 11% from its free-molecular value.
Where Kn calculations usually go wrong
- Definition of λ: Per the NIST paper, for air at 23 °C and 101.3 kPa Millikan's φ = 0.3502 gave λ = 94.17 nm, while later work used φ = 0.491 and 67.3 nm. Use the λ definition behind any correlation you borrow.
- Temperature correction: λ∝1/p holds only at fixed temperature; otherwise add the temperature and Sutherland-type terms of NIST equation (11).
- Local versus freestream λ: In shock layers and expansions, λ from the local state near the body describes the regime better than the freestream λ.
- Leading edges: NASA TM 107730 notes that with L = x from the leading edge, Kn is infinite at x = 0, so low-density effects govern the leading edge at any altitude; usually negligible, but it can matter for high-altitude hypersonic vehicles.
- Separate from compressibility: A small Kn supports only the continuum assumption; check density changes with the Mach number.
Knudsen number questions engineers ask
Which dimension of the device should be L?
Use the smallest dimension confining the flow: a plate gap, microchannel width or particle diameter. The overall device length makes Kn too small. As Example 3 shows, Kn can vary more than tenfold across one object, so record your L.
Why is the Kn in aerosol papers twice the value from this calculator?
Aerosol literature, including the NIST paper, uses Kn = 2λ/Dp (particle radius), exactly twice this calculator's λ/L with the diameter as L. Convert before applying aerosol limits or corrections.
Can I reuse the atmospheric λ of about 68 nm under other conditions?
No. It rounds NIST's 67.3 nm for air at 23 °C and 101.3 kPa. At 1/100 of the pressure λ is 100 times longer, temperature needs its own correction, and other gases have different λ.
How should CFD settings change in the slip regime?
Keep the continuum equations and add velocity-slip and temperature-jump wall conditions; NASA TM 107730 says slip must be included above Kn = 0.03. Beyond 0.1 (transition), DSMC is common, as in the Mars Global Surveyor report.
Related dimensionless numbers
Once Kn confirms the continuum assumption, these are the next numbers to compute for compressibility and viscous effects.
- Mach number (Ma): Even at small Kn, density can change strongly at high speed, so judge compressibility separately with Ma.
- Reynolds number (Re): Because Kn scales with Ma/Re, rarefaction grows as Re falls at a fixed Ma.
References
- University of Utah / NASA: Survey of Literature on Convective Heat Transfer Coefficients and Recovery Factors for High Atmosphere Thermometry. Kn regime divisions (0.01, 0.1, 10), the statement that they are approximate, and that small-Kn flow is characterised by Re and Ma
- NIST: Slip Correction Measurements of Certified PSL Nanoparticles Using a Nanometer Differential Mobility Analyzer. Air λ₀ = 67.3 nm (23 °C, 101.3 kPa), pressure and temperature correction (11), λ definitions depending on φ, Kn = 2λ/Dp
- NASA Langley: Reynolds Number Influences in Aeronautics (NASA TM 107730). Limits from the Moss and Bird figure (0.03, 0.2, 1) and the growth of Kn near leading edges
- NASA: Aerothermodynamics of the Mars Global Surveyor Spacecraft. Local λ of about 3 cm above the power shunt and the shunt dimensions, periapsis Kn of about 0.2, use of the local λ, drag coefficient difference of about 11%
- MIT OpenCourseWare (3.185): Transport Phenomena in Materials Engineering. Kn = λ/L = kBT/(√2πσ²PL) and the continuum (< 0.01) and line-of-sight (> 1) model criteria