# 1.2.8 Conductivities

Generally speaking, vacuum chambers are connected to a vacuum pump via piping. Flow resistance occurs as a result of external friction (gas molecules / wall surface) and internal friction (gas molecules / gas molecule „viscosity”. This flow resistance manifests itself in the form of the volume flow rate, or pumping speed. In vacuum technology, it is customary to use the reciprocal, the conductivity of piping L, instead of flow resistance W. This is expressed in [l / s] or [m3 /h].

Gas flowing through piping produces a pressure differential Δp at the ends of the piping. The following equation applies:

#### Formula 1-14:

Definition of conductivity

The conductivity of a line is L = 1/W.

Analogously to Ohm's Law I = 1/R · U qpV represents flow I, L represents the reciprocal of resistance 1/R and Δp epresents voltage U. If the components are connected in parallel, the individual conductivities are added:

#### Formula 1-15:

Parallel connection conductivities

and if connected in series, the reciprocals are added:

#### Formula 1-16:

Series connection conductivities

Figure 1.7: Diagram for determining pipe conductivities
Source: Pupp/Hartmann, Vakuumtechnik, Grundlagen und Anwendungen, Hanser Verlag

The conductivities of pipes and pipe bends will differ in the various flow ranges. In the case of continuous flow, they are proportional to the mean pressure p and in the case of molecular flow they are not a function of pressure. Knudsen flow represents a transition between the two types of flow, and the conductivities vary with the Knudsen number. Since this range is passed through relatively quickly when generating a vacuum, reference is made to the applicable literature [2].

The conductivities of orifices and long round pipes for the laminar and molecular flow ranges are presented briefly below.

The following Formula 1-14 fundamentally applies for conductivity .

Orifices are frequently encountered in vacuum systems. Examples include constriction of cross sections in valves, ventilation systems or orifices in measuring domes that are used to measure volume flow rate. Similarly, orifice resistance must also be taken into consideration in connection with pipe openings in vessel walls.

## Blocked flow

Let us consider venting of a vacuum chamber. When the venting valve is opened, ambient air flows into the vessel at high velocity at a pressure of p1 It reaches maximum sonic velocity, and the volume flowing through it qpV is not a function of the vessel's interior pressure p2. The following applies for air:

Blocking

## Gas dynamic flow

If the pressure in the vessel now rises beyond the critical pressure [2], gas flow is reduced and we obtain:

#### Formula 1-18:

Gas dynamic flow

Ψ(p2 /p1) [3] is termed the outflow function and is shown in the following diagram (figure 1.8).

## Molecular flow

If an orifice connects two vessels in which molecular flow conditions exist (l >> d), the following will apply for orifice conductivity:

#### Formula 1-19:

Orifice conductivity

Accordingly, the following applies for flow:

#### Formula 1-20:

Orifice flow

Figure 1.8: Outflow function for gas dynamic flow
Source: Jousten (publisher) Wutz, Handbuch Vakuumtechnik, Vieweg Verlag

Let us now consider specific pipe conductivities. On the one hand, this would be laminar flow in a long pipe having a round cross section:

In the case of laminar flow, the conductivity of a pipe is proportional to the mean pressure p :

#### Formula 1-21:

Laminar pipe flow

On the other, there would be molecular flow in a long pipe having a round cross section: In the molecular flow range, conductivity is constant and is not a function of pressure. It can be considered to be the product of the orifice conductivity of the pipe opening LRm and passage probability PRm through a component:

#### Formula 1-22:

Molecular pipe flow

Passage probability PRm can be calculated for different pipe shapes, bends or valves using Monte Carlo computer simulation. In this connection, the trajectories of individual gas molecules through the component can be tracked on the basis of wall collisions

The following applies for long round pipes .

Multiplying this value by Orifice Conductivity Formula 1-19 yields

#### Formula 1-23:

Molecular pipe conductivity

 LRm = pipe conductivity [m3 / s] d = pipe diameter [m] l = pipe length [m] p = (p1+p2) / 2 pressure [Pa] p1 = pressure at piping inlet [Pa] p2 = pressure at piping outlet [Pa] η = viscosity of the gas [Pa · s] c = thermodynamic gas temperature [m / s]

Figure 1.7 [5] shows curves of identical conductivities L as a function of mean pressure p and piping diameter d of one meter long pipes. At lower pressures, the conductivities are constant, and at high pressures they increase proportionately with mean pressure p . The bends in the curves represent the Knudsen flow range.

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