Conductance Lab BETA

How this works

What conductance is

A conductance \(C\) relates the gas throughput \(Q\) of a component to the pressure difference across it, \(Q = C\,(p_1 - p_2)\), so it has the units of a pumping speed. In molecular flow, where molecules hit the walls far more often than each other, a component passes the fraction \(\alpha\) (its transmission probability) of the molecules that enter it:

\(A\bar v/4\) is the conductance of a thin orifice of area \(A\). For air at 20 °C, \(\bar v\) = 463 m/s, so an orifice passes 11.6 L/s for every square centimeter of its area. Every tube, duct or elbow conducts less than an orifice of its own entrance area.

Textbooks combine conductances like electrical resistors: \(1/C = \sum 1/C_i\) in series and \(C = \sum C_i\) in parallel. In molecular flow the series rule is only approximate. Each \(C_i\) is measured between two large volumes, so it includes an entrance loss, and inside a line the molecules entering the next element are not a still gas. Simply adding resistances counts the entrances more than once and underestimates the line, by up to tens of percent for short elements.

Molecular flow, element by element

Each element type has its own transmission probability, referred to its inlet:

• Round tubes use Berman's closed form, which agrees with Matuda's integral-equation values to 0.07 % for L/D from 1 to 500. Dushman's popular approximation \(\alpha \approx 1/(1 + 3L/4d)\) is up to 12 % high.
• Slits and rectangular ducts use a table of our own integral-equation solutions for the infinitely wide slot and test-particle Monte Carlo (TPMC) results for finite ducts, interpolated monotonically. A slit is a duct whose width is much larger than its gap, so its end walls are included.
• Annuli (a tube with a rod) and cones use TPMC tables of 2.3 to 30 million particles per point. A converging cone follows from the diverging one by reciprocity, \(A_1\alpha_{12} = A_2\alpha_{21}\), so the conductance does not depend on the direction.
• 90° mitered elbows combine the two legs as tubes (Oatley's rule) and multiply by a correction \(\kappa\) from TPMC: \(\alpha = \kappa(A/R, B/R)\,\alpha_\mathrm{O}\). Without \(\kappa\) the result would be off by −4.8 % to +1.0 %.
• Black boxes (a valve or trap with a datasheet conductance) are scaled to the gas and temperature and replaced by a round tube of the flange bore with the same molecular conductance. Their molecular value is exact by construction; outside molecular flow it is an estimate.

Outside the tables' ranges the values are extrapolated and the element is flagged; the accuracy is then about ±1 %.

Joining elements

The line is solved as a network: every element is an entrance edge plus an interior (core) edge between nodes, and chambers and parallel junctions are ideal mixing volumes. In molecular flow this network reduces exactly to Haefer's addition theorem:

Haefer's theorem still underestimates a line slightly, because molecules leaving one element are beamed forward. Our TPMC found the true conductance 1.4 % to 3.0 % higher for one junction between dissimilar elements and 5 % to 6.4 % higher for two. So consecutive pieces with the same cross-section (a tube and an elbow of the same bore, for example) are first merged into one element, and every result carries an error band: up to 3.5 % per remaining junction, plus the elements' own accuracy, capped at 25 %. The Monte Carlo check is the arbiter when the band is wide.

Haefer's theorem assumes the gas is fully mixed at every junction, and across a short element it is not. Behind a short diverging cone, or a narrow opening a short distance upstream, much of the gas flies straight onto the end face of a rod and bounces back: our chain simulations found the exact value up to 29 % below the formula for a cone into an annulus, and up to 68 % below for a small orifice just in front of a rod. The opposite happens where two narrow openings face each other across a short gap, such as the gaps of two annuli on one rod or two plates with aligned holes: gas flies straight from one into the next, and the exact value was up to 42 % and 59 % above the formula. For each short tube, cone or duct the tool therefore works out how the gas spreads from the opening before it to the opening after it, with a view-factor (radiosity) calculation, and widens the band by one and a half times the difference from Haefer's mixed assumption. Every junction at a rod also widens the band by up to 10 % below and 5 % above the formula. Where an opening only partly overlaps the next one, part of the gas meets a wall face that the formula does not see; the line is flagged, its band is wide, and the Monte Carlo check should decide. Openings that do not overlap at all are refused: no gas can pass.

A pump's speed \(S\) is measured on a test dome, so it already includes the entrance into its inlet. When the pump sits at the end of a line of the same bore, that entrance is counted twice unless it is removed:

This Oatley/Ho correction reproduces Livesey's example: a 300 L/s pump on a 100 mm × 200 mm tube gives \(S_\text{eff}\) = 188 L/s in air, where the simple series rule gives 156 L/s. When the pump inlet is wider than the line, the formula assumes that the gas spreads over the whole inlet, as on the test dome; the real speed then depends on the adapter and the pump throat, so add them as elements (the Monte Carlo check asks for them).

From molecular to viscous flow

The flow regime is set by the rarefaction parameter \(\delta = r\,p/(\eta\,v_m)\), with \(v_m = \sqrt{2RT/M}\), or equivalently by the Knudsen number \(\mathrm{Kn} = \lambda/D_h\), with the mean free path \(\lambda = (\sqrt{\pi}/2)\,\eta\,v_m/p\) taken from the viscosity. Flow is molecular for Kn > 0.5, transitional for 0.01 < Kn ≤ 0.5 and viscous below.

The interior of each element follows Sharipov's kinetic-theory solution for a long tube, \(G_P(\delta)\), written as a Poiseuille term plus a molecular term weighted by a rarefaction function \(Z\):

For a long round tube this is exactly Sharipov's result at any pressure ratio; in the viscous limit it becomes Poiseuille flow with the slip correction, \(G_P \to \delta/4 + 1.018\). The entrance edge adds the right end correction: a short tube in viscous flow behaves as if it were \(3\pi R/8\) longer, as creeping-flow (Stokes) theory predicts, Sampson's orifice resistance in series with Poiseuille's (Dagan, Weinbaum and Pfeffer 1982). Ducts, annuli and cones use their own laminar coefficients with \(Z\) evaluated at their hydraulic size. In a slit, a flat duct or a narrow annulus, molecules fly long distances along the gap in molecular flow; gas collisions cut those flights off, which the round-tube \(Z\) does not know. There the molecular term is limited by Sharipov's plane-channel solution, \(Q \le \frac{w h^2 v_m}{2L}\left(G_P^\text{ch}(\delta) - \frac{\delta}{6}\right)\Delta p\) with \(\delta\) on the gap \(h\), which a 0.1 mm × 100 mm slit 100 mm deep reproduces to 0.4 % (without this limit it was up to twice too high). Other transitional results for non-circular elements are approximate (about ±10 %).

Entrances and orifices pass the smaller of a linear (rarefied) law and the isentropic nozzle flow \(Q = C_d\,A\,p\,\psi(r,\gamma)\sqrt{R_sT}\). Below the critical pressure ratio \(r^* = \left(2/(\gamma+1)\right)^{\gamma/(\gamma-1)}\) the orifice chokes, and the discharge coefficient \(C_d\) in the conditions bar sets the flow. Inside ducts the flow chokes at the outlet when it reaches the isothermal speed of sound. Reynolds numbers above 2300 and Mach numbers above 0.3 are flagged: turbulence and compressibility are not modeled.

The simple sum \(C_\text{visc} + C_\text{mol}\), a common shortcut, is up to 19 % high in the transitional range compared with Sharipov's solution; Knudsen's semi-empirical formula, which this site's pump-down tool uses, is at most about 6 % high for long tubes.

The Monte Carlo method

Test-particle Monte Carlo follows molecules one at a time: each enters with a cosine (Lambertian) direction, travels in a straight line, and leaves every wall it hits in a new cosine direction, until it leaves through an opening. The fraction that reaches the far opening estimates \(\alpha\), with the statistical uncertainty \(\sigma = \sqrt{\alpha(1-\alpha)/N}\) for \(N\) particles; the results show 95 % intervals. "Verify by Monte Carlo" traces the line's exact analytic geometry in parallel Web Workers, so it checks the formula's combination rules, not just its tables.

The CAD tab traces particles through a triangle mesh. The mesh is welded and checked for holes, its open edges become the openings, and a reciprocity test (\(A_1\alpha_{12} = A_2\alpha_{21}\)) catches leaky or inconsistent meshes. Faceted circles lose area: a 16-sided tube conducts about 3.5 % less than a round one, so export at least 64 segments per circle. The wall pressure comes from the density of wall hits of the particles launched from the selected inlet, \(p_t/p_\text{in} = (h_t/A_t)\big/(N_\text{in}/A_\text{in})\): the inlet at \(p_\text{in}\) and every other opening pumping. With several inlets the map shows the gas that enters through the selected one.

Assumptions and limits

• Isothermal gas and walls; diffuse (cosine) reflection everywhere.
• No thermal transpiration, no turbulence model and no pump curves: the pumping speed is a constant for the selected gas.
• The Monte Carlo methods are free-molecular only; they verify the molecular limit, not transitional or viscous flow.
• Transitional results for short tubes and non-circular elements are good to about ±10 % (long slits and flat ducts follow the plane-channel solution), and viscous results for elements shorter than their diameter to about ±20 %.

References

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