Engineering
Gas Mean Free Path & Knudsen Number (Kn) Vacuum Flow Regimes
Estimate an ideal hard-sphere gas mean free path, calculate Kn=lambda/L, and compare it with explicitly defined rarefaction ranges using absolute pressure and consistent length units.
An ideal-gas estimate with a declared collision diameter
For a dilute ideal gas with a hard-sphere collision model, lambda=kBT/(sqrt(2) pi d² P). T is absolute temperature and P is absolute pressure, not gauge pressure. The collision diameter is a model input; the supplied 364 pm value is illustrative, not an exact gas property for every condition.
Using d in picometres and reporting lambda in millimetres converts the SI numerator to 13806.49×T. The diameter is squared. The model estimates intermolecular collision spacing in the bulk gas; walls and surface interactions are not included in that free-path calculation.
Kn=lambda/L compares that length with a relevant geometric scale. This worksheet assigns Kn<0.01 to continuum, 0.01<=Kn<0.1 to slip, 0.1<=Kn<10 to transition and Kn>=10 to free-molecular ranges. These are approximate modeling conventions, with equality assigned explicitly for consistent labels.
The labels help select a modeling approach; they do not prove that a particular Navier–Stokes solver, slip boundary condition or molecular-flow model is valid. The relevant length may be a gap or local gradient scale, not the overall chamber size. Zero pressure is outside the calculation; it is not replaced with a finite free path.
Unrounded mean free path in millimetres, D2
Positive numeric temperature, diameter and pressure are required before division.
=IF(COUNT(B2:B4)<>3,"Enter numeric gas inputs",IF(MIN(B2:B4)<=0,"Require T, diameter and absolute pressure >0",13806.49*B2/(SQRT(2)*PI()*B3^2*B4)))Unrounded dimensionless Knudsen number, D3
B5 is in millimetres, matching D2. Do not divide a millimetre free path by a metre gap without conversion.
=IF(NOT(ISNUMBER(D2)),D2,IF(COUNT(B5)<>1,"Enter numeric characteristic length",IF(B5<=0,"Characteristic length must be positive",D2/B5)))Range label in D4
The classifier uses the unrounded number and propagates input messages instead of dividing again without guards.
=IF(NOT(ISNUMBER(D3)),D3,IFS(D3<0.01,"Continuum range",D3<0.1,"Slip range",D3<10,"Transition range",TRUE,"Free-molecular range"))Try the example
Use the original values or change them to test your understanding.