How the calculation works
A chamber of volume \(V\) pumped with effective speed \(S_\text{eff}\) against a gas load \(Q\) obeys
With constant \(S_\text{eff}\) and no gas load this reduces to the familiar \(t = (V/S_\text{eff})\ln(p_0/p_1)\). The calculator integrates the full expression numerically, because the line conductance changes with pressure by orders of magnitude. The pump and the line act in series, like resistors in parallel:
Conductance of a round tube
Molecular flow (mean free path larger than the tube). For a long tube and for a thin orifice of the same diameter,
For air at 20 °C (\(\bar v\) = 463 m/s) these become the well-known rules of thumb \(C \approx 12.1\,d^3/L\) L/s (with d and L in cm) and 11.6 L/s per cm² of orifice area. Real tubes have an entrance as well as a length. The calculator combines the two in series (Dushman's approximation), \(C_\text{mol} = C_\text{orifice}/(1 + 3L/4d)\). This is exact for very short and very long tubes and overestimates the exact Clausing value by up to about 12 % when L is one to two diameters.
Viscous flow (mean free path much smaller than the tube). Laminar Poiseuille flow gives
where \(\bar p\) is the mean pressure in the line. For air at 20 °C this is about \(136\,d^4\bar p/L\) L/s with d and L in cm and \(\bar p\) in mbar. Conductance in this regime is proportional to pressure, so a line that is wide open at 100 mbar can choke the pump at 0.1 mbar. The calculator finds \(\bar p\) self-consistently from the chamber pressure and the pump-inlet pressure \(p\,S_\text{eff}/S\).
Transitional flow. Between the two limits the calculator uses \(C_\text{line} = C_\text{visc} + C_\text{mol}\). This simple interpolation has the correct limits on both sides. For precise work in the transition range, use Knudsen's semi-empirical formula or measured data.
Worked example
A 50 L chamber pumped by a 35 m³/h (9.72 L/s) rotary-vane pump through 1 m of 25 mm bore line (KF25), down to 0.1 mbar with air at 20 °C:
- Without the line, \(t = (50/9.72)\ln(1013/0.1) = 47\) s.
- With the line, the conductance is over 500 L/s near atmosphere but only 5.4 L/s at 0.1 mbar. There \(S_\text{eff}\) falls to 3.5 L/s, 36 % of the pump's speed, and the pump-down takes about 57 s.
- In molecular flow the same line conducts only 1.8 L/s. A turbo pump behind this line would lose most of its speed. High-vacuum pumps are mounted directly on the chamber for this reason.
- An unbaked chamber with 5000 cm² of internal surface outgassing at 10⁻⁸ mbar·L/(s·cm²) adds Q = 5 × 10⁻⁵ mbar·L/s. Through this line, that limits the base pressure to about 3 × 10⁻⁵ mbar regardless of pump size.
Assumptions and limits
- Constant pump speed. Real pump curves fall off at both ends of their range. Use the speed near the pressure that dominates the pump-down time, or run the calculation in segments.
- Outgassing is time-dependent. Desorption of water from metal surfaces typically falls roughly as \(1/t\). A constant \(q\) is a snapshot that sets a floor on pressure, but it does not model the slow tail. Below about 10⁻² to 10⁻³ mbar, outgassing, not volume, usually sets the pump-down time. Use measured or vendor data for your materials and surface condition.
- Isothermal, quasi-static flow. Fast roughing from atmosphere cools the gas and can condense water. Chokes and sonic flow in very short, high-throughput lines are not modeled.
- Straight round tubes only. Bends, flexible bellows, and rectangular or annular ducts conduct less. Use vendor data or a Monte Carlo model for them.
References
- J. F. O'Hanlon, A User's Guide to Vacuum Technology, 3rd ed., Wiley (2003), ch. 3 (gas flow and conductance) and ch. 4 (gas release from solids).
- K. Jousten (ed.), Handbook of Vacuum Technology, 2nd ed., Wiley-VCH (2016).
- S. Dushman, Scientific Foundations of Vacuum Technique, 2nd ed. (rev. J. M. Lafferty), Wiley (1962).