Don't Calculate Pressure Drops in a Vacuum

Consider flow regime boundaries and the limitations of methods.

By Andrew Sloley, Contributing Editor

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Vacuum technologists and process engineers typically define vacuum conditions differently. The rule of thumb for process engineers is that operations down to roughly 40 mm Hg pressure are "high vacuum pressures," ones down to 5–10 mm Hg are "medium vacuum pressures," and those below 5 mm Hg are "low vacuum pressures." These ranges are set by the effect of cooling water temperatures on the design of the commonly used steam jet ejector and liquid-ring pump vacuum systems.

Results of pressure drop calculations at the new operating conditions posed a problem.

In contrast, vacuum technologists often use definitions based on the flow behavior of the system. "High pressure vacuum" systems operate where flow is in the viscous region. "Medium pressure vacuum" covers pressures in a transition region. "Low pressure vacuum" involves flow in the molecular region.

For normal flow we experience, molecule/molecule collisions dominate the flow pattern. This is the viscous flow region, which includes both turbulent and laminar flow. We use Reynolds number, NRe, to characterize flow: laminar flow occurs at NRe less than 2,000, transition flow at NRe between 2,000 and 4,000, and turbulent flow at NRe greater than 4,000.

For a gas, as pressure falls, density drops. Decreasing density increases the distance between molecules. Greater spacing raises the mean free path a molecule moves before a molecule-to-molecule collision occurs. Based on ideal gas behavior and spherical molecules, the mean free path length can be determined from:

λ = kBT/2½πσ2P (1)

where λ is the mean free path length, kB is the Boltzmann constant (1.3806 × 10-23 m2 kg s-2 K-1), T is temperature, σ is the effective hard-shell diameter of the molecule, and P is the pressure. The ratio of λ to the characteristic length, Lc, is called the Knudsen number, NKn. When it approaches 1, molecule-to-surface interactions determine system pressure drop. In such cases, pressure drops significantly will exceed those predicted by conventional flow calculations.

A commonly used reference (Myerson, E. B., "Empty Your Head of Misconceptions about Vacuum Systems," p. 50, CEP, July 1996) states that alternative calculation methods are needed from around 25 torr (start of transition region) and are really important below 10 torr.

A recent project provided insights about calculating such pressure drops. It involved altering operation of a wiped-film evaporator and its de-entrainment system to 4 torr from 10 torr. Estimating the change in system capacity required checking pressure drops from the evaporator inlet to the vacuum pump suction.

For the molecular-flow regime, the correct pressure-drop calculation uses formulas derived from a conductance analogy. The conductance of a flow component is defined as the throughput (power) divided by the pressure drop:

C = Q/(P1 – P2) (2)
The effective conductance for components in series is:
1/CTotal = 1/C1 + 1/C2 + …1/CFinal (3)
and for those in parallel:

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