1.2.8 Conductance
Generally speaking, vacuum chambers are connected to a vacuum pump via piping. Flow resistance occurs as a result of external friction between gas molecules and the wall surface and internal friction between the gas molecules themselves (viscosity). This flow resistance manifests itself in the form of pressure differences and volume flow rate, or pumping speed, losses. In vacuum technology, it is customary to use the reciprocal, the conductivity of piping $L$ or $C$ (conductance) instead of flow resistance $W$. The conductivity has the dimension of a volume flow rate and is normally expressed in [l s^{-1}] or [m^{3} h^{-1}].
Gas flowing through piping produces a pressure differential $\Delta p$ at the ends of the piping. The following equation applies:
\[C=\frac lW=\frac{q_{pV}}{\Delta p}\]
Formula 1-18: Definition of conductance
This principle is formally analogous to Ohm’s law of electrotechnology:
\[R=\frac UI\mbox{ or }\frac 1R=\frac IU\]
Formula 1-19: Ohm’s law
In a formal comparison of Formula 1-18 with Formula 1-19 $q_{pV}$ represents flow $I$, $C$ the reciprocal of resistance $1/R$ and $\Delta p$ the voltage $U$. If the components are connected in parallel, the individual conductivities are added:
\[C_\mbox{ges}=C_1+C_2+\dots+C_n\]
Formula 1-20: Parallel connection conductance
and if connected in series, the resistances, i. e. the reciprocals, are added together:
\[\frac 1{C_\mbox{ges}}=\frac 1{C_1}+\frac 1{C_2}+\dots+\frac 1{C_n}\]
Formula 1-21: Series connection conductivities
The conductance of pipes and pipe bends will differ in the various flow regimes. In viscous flow they are proportional to the mean pressure $\bar p$ and in molecular flow they are independent of pressure. Knudsen flow represents a transition between the two types of flow, and the conductivities vary with the Knudsen number.
Figure 1.8: Conductance of a smooth round pipe as a function of the mean pressure in the pipe
A simple approximation for the Knudsen range can be obtained by adding the laminar and molecular conductivities. We would refer you to special literature for exact calculations of the conductance still in the laminar flow range and already in the molecular flow range as well as conductance calculations taking into account inhomogeneities at the inlet of a pipe.
This publication is restricted to the consideration of the conductivities of orifices and long, round pipes for laminar and molecular flow ranges.
Orifices are frequently flow resistances in vacuum systems. Examples of these are constrictions in the cross-section of valves, ventilation devices or orifices in measuring domes for measuring the pumping speed. In pipe openings in vessel walls the orifice resistance of the inlet opening must also be taken into account in addition to the pipe resistance.
Choked flow
Let us consider the venting of a vacuum chamber. When the venting valve is opened, ambient air flows into the vessel at high velocity at a pressure of p. The flow velocity reaches not more than sonic velocity. If the gas has reached sonic velocity, the maximum gas throughput has also been reached at which the vessel can be vented. The throughput flowing through it $q_{pV}$ is not a function of the vessel’s interior pressure $p_i$. The following applies for air:
\[q_{pV}=15.7\cdot d^2\cdot p_a\]
Formula 1-22: Blocking of an orifice [11]
$d$ | Diameter of orifice | [cm] |
$p_a$ | External pressure on the vessel | [hPa] |
Gas dynamic flow
If the pressure in the vessel now rises beyond a critical pressure, gas flow is reduced and we can use gas dynamic laws according to Bernoulli and Poiseuille to calculate it. The immersive gas flow $q_{pV}$ and the conductance are dependent on
- Narrowest cross-section of the orifice
- External pressure on the vessel
- Internal pressure in the vessel
- Universal gas constant
- Absolute temperature
- Molar mass
- Adiabatic exponent (= ratio of specific or molar heat capacities at constant pressure $c_p$ or constant volume $c_V$) [12]
Molecular flow [13]
If an orifice connects two vessels in which molecular flow conditions exist (i.e. if the mean free path is considerably greater than the diameter of the vessel), the following will apply for the displaced gas quantity $q_{pV}$ per unit of time
\[q_{pV}=A\cdot \frac{\bar c}4\cdot(p_1-p_2)\]
Formula 1-23: Orifice flow
$A$ | Cross-section of orifice | [cm^{2}] |
$\bar c$ | Mean thermal velocity | [m^{-1}] |
According to Formula 1-23 the following applies for the orifice conductivity
\[C_\mathrm{or,\,mol}=A\cdot \frac{\bar c}4=A\cdot\sqrt{\frac{kT}{2\pi m_0}}\]
Formula 1-24: Orifice conductivity
For air with a temperature of 293 K we obtain
\[C_\mathrm{or,\,mol}=11.6\cdot A\]
Formula 1-25: Orifice conductivity for air
$A$ | Cross-section of orifice | [cm^{2}] |
$C$ | Conductivity | [l s^{-1}] |
This formula can be used to determine the maximum possible pumping speed of a vacuum pump with an inlet port A. The maximum pumping speed of a pump under molecular flow conditions is therefore determined by the inlet port.
Let us now consider specific pipe conductivities. In the case of laminar flow, the conductivity of a pipe is proportional to the mean pressure:
\[C_\mathrm{pipe,\,lam}=\frac{\pi\cdot d^4}{256\cdot\eta\cdot l}\cdot(p_1+p_2)=\frac{\pi\cdot d^4}{128\cdot\eta\cdot l}\cdot\bar p\]
Formula 1-26: Conductance of a pipe in laminar flow
For air at 20°C we obtain
\[C_\mathrm{pipe,\,lam}=1.35\cdot\frac{d^4}l\cdot\bar p\]
Formula 1-27: Conductance of a pipe in laminar flow for air
$l$ | Length of pipe | [cm] |
$d$ | Diameter of pipe | [cm] |
$\bar p$ | Pressure | [Pa] |
$C$ | Conductivity | [l s^{-1}] |
In the molecular flow regime, conductance 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 $C_\mathrm{pipe,\,mol}$ and passage probability $P_\mathrm{pipe,\,mol}$ through a component:
\[C_\mathrm{pipe,\,mol}=C_\mathrm{orifice,\,mol}\cdot P_\mathrm{pipe,\,mol}\]
Formula 1-28: Molecular pipe flow
The mean probability $P_\mathrm{pipe,\,mol}$ can be calculated with a computer program for different pipe profiles, bends or valves using a Monte Carlo 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:
\[P_\mathrm{pipe,\,mol}=\frac 43\cdot\frac dl\]
Formula 1-29: Passage probability for long round pipes
If we multiply this value by the orifice conductivity (Formula 1-24), we obtain
\[C_\mathrm{pipe,\,mol}=\frac{\bar c\cdot\pi\cdot d^3}{12\cdot l}\]
Formula 1-30: Molecular pipe conductivity
For air at 20°C we obtain
\[C_\mathrm{pipe,\,mol}=12.1\cdot \frac{d^3}l\]
Formula 1-31: Molecular pipe conductivity
$l$ | Length of pipe | [cm] |
$d$ | Diameter of pipe | [cm] |
$C$ | Conductivity | [l s^{-1}] |