1.2.6 Types of flow
The ratio of the mean free path to the flow channel diameter can be used to describe types of flow. This ratio is referred to as the Knudsen number:
\[\mathit{Kn}=\frac{\bar l}d\]
Formula 1-13: Knudsen number
| $\bar l$ | Mean free path | [m] |
| $d$ | Diameter of flow channel | [m] |
| $\mathit{Kn}$ | Knudsen number | dimensionless |
The value of the Knudsen number characterizes the type of gas flow and assigns it to a particular pressure range. Table 1.7 gives an overview of the various types of flow in vacuum technology and their significant characterization parameters.
Profiles of the various types of flow regimes are shown in Figure 1.6.
Figure 1.6: Profiles of the various types of flow regimes
Viscous flow in low vacuum
In viscous flow, also known as continuous flow, there are frequent collisions between gas molecules, but less frequently with the walls of the vessel. In this case, the mean free path of the gas molecules is significantly shorter than the dimensions of the flow channel.
In the case of viscous flow, a distinction is made between laminar and turbulent flow. In laminar, or layered, flow the gas particles remain in the same displaced layers that are constantly parallel to each other. If the flow velocity increases, these layers are broken up and the fluid particles run into each other in a completely disordered way. This is termed turbulent flow. The boundary between these two areas of viscous flow can be expressed by the Reynolds number:
\[\RE = \frac{\rho\cdot\nu\cdot l}\eta\]
Formula 1-14: Reynolds number
| $\RE$ | Reynolds number | dimensionless |
| $\rho$ | Density of the liquid | [kg m-3] |
| $\nu$ | Mean velocity of the flow | [m s-1] |
| $l$ | Characteristic length | [m] |
| $\eta$ | Dynamic viscosity | [Pa s] |
Up to values of Re < 2,300 the flow will be laminar, and where Re > 4,000 the flow will be turbulent. In the range between 2,300 < Re < 4,000, the flow is predominantly turbulent. Laminar flow is also possible, however, both types of flow being unstable in this range.
Turbulent flow in a vacuum only occurs during pump-down operations from atmospheric pressure or when rapid venting is carried out. In vacuum systems, the pipes are dimensioned in such a manner that turbulent flow occurs only briefly at relatively high pressures, as the high flow resistance that occurs in this process necessitates that the pumps used produce higher volume flow rates.
Knudsen flow in medium vacuum
If the Knudsen number is between 0.01 and 0.5, this is termed Knudsen flow. Since many process pressures are in the medium vacuum range, this type of flow occurs with corresponding frequency.
Molecular flow in high vacuum and ultra-high vacuum
At Knudsen numbers of $\mathit{Kn}$ > 0.5 molecular interaction virtually no longer occurs. What prevails is molecular flow. In this case, the mean free path is significantly greater than the diameter of the flow channel. In molecular flow, the product of pressure and component diameter is approximately ≤ 1.3 · 10-2 hPa cm.
A graph showing an overview of flow ranges as a function of the product of pressure and component diameter is displayed in Figure 1.7.
Figure 1.7: Flow ranges in vacuum as a function of p · d
This graph clearly shows that the classification, also found in Table 1.7, into vacuum ranges purely according to pressure is an inadmissible simplification. Since this classification is still in common usage, however, it is cited here.
| Viscous flow | Knudsen flow | Molecular flow | |
|---|---|---|---|
| Low vacuum | Medium vacuum | High / Ultra-high vacuum | |
| Pressure range [hPa] | 103…1 | 1…10-3 | < 10-3 bzw. < 10-7 |
| Pressure range [Pa] | 105…10 2 | 102…10 -1 | < 10-1 bzw. < 10-5 |
| Knudsen number | Kn < 0.01 | 0.01 < Kn < 0.5 | Kn > 0.5 |
| Reynolds number | Re < 2,300: laminar Re > 4,000: turbulent |
||
| p · d [hPa cm] | p · d > 0.6 | 0.6 > p · d > 0.01 | p · d < 0.01 |
Table 1.7: Overview of types of flow regimes
