Rarefied Gas Dynamics, High-Vacuum Plumbing & Molecular Flow
Knudsen Molecular-Flow Tube Conductance & Aperture Formula
Calculate Gas Mean-Free-Path Knudsen Number Kn, Long-Tube Knudsen Conductance (L/s, scaling with D^3!), Entrance Orifice Conductance (L/s), and Short-Tube Series Conductance (L/s).
In molecular vacuum flow (Kn > 1), halving pipe diameter slashes conductance EIGHT-FOLD (D^3)—and short tubes MUST include entrance orifice impedance!
At high vacuum (P < 0.1 Pa), gas mean free path lambda = 0.0066 / P_Pa (m, or 0.66 / P_Pa cm for 20°C N2/air) exceeds pipe diameter D_cm (Knudsen Number Kn = lambda / D > 1.0): molecules bounce ballistically off chamber walls without colliding with each other! Under Knudsen’s Long-Tube Law (20°C air), conductance scales with Diameter CUBED: C_long = (12.1 * (D_cm^3)) / L_cm L/s! However, applying C_long alone to a short manifold (L / D = 3 to 10) yields an unphysical conductance exceeding the open entrance hole itself! Combining Entrance Orifice Conductance C_aperture = 11.6 * (PI() * (D_cm^2) / 4) L/s in series with C_long (Dushman short-tube approximation 1 / C_total = 1 / C_aperture + 1 / C_long) prevents over-sizing errors!
Verify on a DN100 High-Vacuum Foreline Spool—Internal_Diameter D = 10.0 cm (B2), Spool_Length L = 50.0 cm (B3), and Operating_Vacuum P = 0.010 Pa (1e-4 mbar, B4): Mean-Free-Path Knudsen Number Kn hits (0.66 / 0.010) / 10.0 = 6.60 (> 1.0 free molecular regime!, E2), Naive Long-Tube Conductance C_long = (12.1 * 1,000) / 50 = 242.00 L/s (E3), Entrance Aperture Conductance C_ap equals 911.06 L/s (E4), and True Short-Tube Conductance C_total equals 191.21 L/s (E5)!
| Knudsen Molecular Pipe KPI (D = 10 cm, L = 50 cm, P = 0.01 Pa @ 20°C Air) | Formula | Verified Output |
|---|---|---|
| Knudsen Number Kn (lambda / D) | (0.66 / Pressure_Pa) / Diameter_cm | 6.60 (13.20 @ D = 5.0 cm) |
| Long-Tube Knudsen C_long (L/s) | (12.1 * (Diameter_cm^3)) / Length_cm | 242.00 L/s (30.25 @ D = 5.0 cm!) |
| Entrance Orifice C_ap (L/s) | 11.6 * (PI() * (Diameter_cm^2) / 4) | 911.06 L/s (227.77 @ D = 5.0 cm) |
| Dushman Short-Tube C_tot (L/s) | 1 / ((1 / C_ap) + (1 / C_long)) | 191.21 L/s (26.70 @ D = 5.0 cm!) |
Dushman Short-Tube Molecular-Flow Conductance Formula
Validate Diameter D_cm > 0, Length L_cm > 0, Pressure P_Pa > 0, and verify Knudsen Number Kn >= 0.5 (transition/molecular regime; viscous Poiseuille flow governs higher pressures)!
=LET(dm_c, B2, ln_c, B3, pr_p, B4, IF(OR(dm_c <= 0, ln_c <= 0, pr_p <= 0), "Invalid Tube Input", LET(kn_n, (0.66 / pr_p) / dm_c, IF(kn_n < 0.5, "VISCOUS CONTINUUM REGIME", LET(cl_g, (12.1 * (dm_c ^ 3)) / ln_c, ca_p, 11.6 * (PI() * (dm_c ^ 2) / 4), ROUND(1 / ((1 / ca_p) + (1 / cl_g)), 2))))))Cubic D^3 Diameter Bottleneck Check (Halve Pipe Diameter from 10.0 cm to 5.0 cm)
Step down from DN100 (B2 = 10.0 cm) to DN50 (B2 = 5.0 cm): Long-Tube Conductance E3 plunges EXACTLY 8x (2^3) from 242.00 L/s down to 30.25 L/s, and Net Short-Tube Conductance E5 collapses more than 7x from 191.21 L/s to 26.70 L/s!
=IF(ISTEXT(E5), E5, ROUND((12.1 * (B2 ^ 3)) / B3, 2))Try the example
Use the original values or change them to test your understanding.