The pV flow through any desired piping element, i.e. pipe or hose, valves, nozzles, openings in a wall between two vessels, etc., is indicated with
Here Δp = (p1 – p2) is the differential between the pressures at the inlet and outlet ends of the piping element. The proportionality factor C is designated as the conductance value or simply “conductance”. It is affected by the geometry of the piping element and can even be calculated for some simpler configurations.
In the high and ultrahigh vacuum ranges, C is a constant which is independent of pressure; in the rough and medium-high regimes it is, by contrast, dependent on pressure. As a consequence, the calculation of C for the piping elements must be carried out separately for the individual pressure ranges.
From the definition of the volumetric flow it is also possible to state that: The conductance value C is the flow volume through a piping element. The equation (1.11) could be thought of as “Ohm’s law for vacuum technology”, in which qpV corresponds to current, Δp the voltage and C the electrical conductance value. Analogous to Ohm’s law in the science of electricity, the resistance to flow
has been introduced as the reciprocal value to the conductance value. The equation (1.11) can then be re-written as:
The following applies directly for connection in series:
When connected in parallel, the following applies:
The effective pumping speed required to evacuate a vessel or to carry out a process inside a vacuum system will correspond to the inlet speed of a particular pump (or the pump system) only if the pump is joined directly to the vessel or system. Practically speaking, this is possible only in rare situations. It is almost always necessary to include an inter mediate piping system comprising valves, separators, cold traps and the like. All this represents a resistance to flow, the consequence of which is that the effective pumping speed Seff is always less than the pumping speed S of the pump or the pumping system alone. Thus, to ensure a certain effective pumping speed at the vacuum vessel it is necessary to select a pump with greater pumping speed. The correlation between S and Seff is indicated by the following basic equation:
Here C is the total conductance value for the pipe system, made up of the individual values for the various components which are connected in series (valves, baffles, separators, etc.):
Equation (1.24) tells us that only in the situation where C = ∞ (meaning that the flow resistance is equal to 0) will S = Seff. A number of helpful equations is available to the vacuum technologist for calculating the conductance value C for piping sections. The conductance values for valves, cold traps, separators and vapor barriers will, as a rule, have to be determined empirically.
It should be noted that in general that the conductance in a vacuum component is not a constant value which is independent of prevailing vacuum levels, but rather depends strongly on the nature of the flow (continuum or molecular flow) and thus on pressure. When using conductance indices in vacuum technology calculations, therefore, it is always necessary to pay attention to the fact that only the conductance values applicable to a certain pressure regime may be applied in that regime.
Conductance values will depend not only on the pressure and the nature of the gas which is flowing, but also on the sectional shape of the conducting element (e.g. circular or elliptical cross section). Other factors are the length and whether the element is straight or curved. The result is that various equations are required to take into account practical situations. Each of these equations is valid only for a particular pressure range. This is always to be considered in calculations.
a) Conductance for a straight pipe, which is not too short, of length l, with a circular cross section of diameter d for the laminar, Knudsen and molecular flow ranges, valid for air at 68°F or 20 °C (Knudsen equation):
d = Pipe inside diameter in cm
l = Pipe length in cm (l ≥ 10 d)
p1 = Pressure at start of pipe (along the direction of flow) in mbar
p2 = Pressure at end of pipe (along the direction of flow) in mbar
If one rewrites the second term in (1.26) in the following form
it is possible to derive the two important limits from the course of the function
Limit for laminar flow
Limit for molecular flow
In the molecular flow region the conductance value is independent of pressure!
The complete Knudsen equation (1.26) will have to be used in the transitional area
Conductance values for straight pipes of standard nominal diameters are shown in Figure 9.5 (laminar flow) and Figure 9.6 (molecular flow). Additional nomograms for conductance determination will also be found in Figures 9.8 and 9.9.
Example: What diameter d must a 1.5-m-long pipe have so that it has a conductance of about C = 1000 l / sec in the region of molecular flow? The points l = 1.5 m and C = 1000 l/sec are joined by a straight line which is extended to intersect the scale for the diameter d. The value d = 24 cm is obtained. The input conductance of the tube, which depends on the ratio d / l and must not be neglected in the case of short tubes, is taken into account by means of a correction factor α. For d / l < 0.1, α can be set equal to 1. In our example d/l = 0.16 and α = 0.83 (intersection point of the straight line with the a scale). Hence, the effective conductance of the pipeline is reduced to C · α = 1000 · 0.83 = 830 l/sec. If d is increased to 25 cm, one obtains a conductance of 1200 · 0.82 = 985 l / sec (dashed straight line).
Procedure: For a given length (l) and internal diameter (d), the conductance Cm, which is independent of pressure, must be determined in the molecular flow region. To find the conductance C* in the laminar flow or Knudsen flow region with a given mean pressure of p in the tube, the conductance value previously calculated for Cm has to be multiplied by the correction factor a determined in the nomogram: C* = Cm · α.
Example: A tube with a length of 1 m and an internal diameter of 5 cm has an (uncorrected) conductance C of around 17 l/s in the molecular flow region, as determined using the appropriate connecting lines between the “l” scale and the “d” scale. The conductance C found in this manner must be multiplied by the clausing factor γ = 0.963 (intersection of connecting line with the γ scale) to obtain the true conductance Cm in the molecular flow region: Cm · γ = 17 · 0.963 = 16.37 l/s. In a tube with a length of 1 m and an internal diameter of 5 cm a molecular flow prevails if the mean pressure p in the tube is < 2.7 · 10-3 mbar. To determine the conductance C* at higher pressures than 2.7 · 10-3 mbar, at 8 · 10-2 mbar (= 6 · 10-2 torr), for example, the corresponding point on the p scale is connected with the point d = 5 cm on the “d” scale. This connecting line intersects the “α“ scale at the point α = 5.5. The conductance C* at p = 8 · 10-2 mbar is: C* = Cm · α = 16.37 · 5.5 = 90 l/s.
b) Conductance value C for an orifice A
(A in cm2): For continuum flow (viscous flow) the following equations (after Prandtl) apply to air at 68°F (20°C) where p2/p1 = δ:
δ = 0.528 is the critical pressure situation for air
Flow is choked at δ < 0.528; gas flow is thus constant. In the case of molecular flow (high vacuum) the following will apply for air:
Given in addition in Figure 1.3 are the pumping speeds S*visc and S*mol referenced to the area A of the opening and as a function of δ = p2/p1. The equations given apply to air at 68°F (20 °C). The molar masses for the flowing gas are taken into consideration in the general equations, not shown here.
When working with other gases it will be necessary to multiply the conductance values specified for air by the factors shown in Table 1.1.
The conductance values for piping and openings through which air and other gases pass can be determined with nomographic methods. It is possible not only to determine the conductance value for piping at specified values for diameter, length and pressure, but also the size of the pipe diameter required when a pumping set is to achieve a certain effective pumping speed at a given pressure and given length of the line. It is also possible to establish the maximum permissible pipe length where the other parameters are known. The values obtained naturally do not apply to turbulent flows. In doubtful situations, the Reynolds number Re should be estimated using the relationship which is approximated below.
Here qpV = S · p is the flow output in mbar l/s, d the diameter of the pipe in cm.
Nomograms which have proved to be useful in practice can be seen in Fig 9.8 and Fig. 9.9.
Where the line contains elbows or other curves (such as in right-angle valves), these can be taken into account by assuming a greater effective length leff of the line. This can be estimated as follows:
laxial : axial length of the line (in cm)
leff : Effective length of the line (in cm)
d : Inside diameter of the line (in cm)
θ : Angle of the elbow (degrees of angle)
The technical data in the Leybold catalog states the conductance values for vapor barriers, cold traps, adsorption traps and valves for the molecular flow range. At higher pressures, e.g. in the Knudsen and laminar flow ranges, valves will have about the same conductance values as pipes of corresponding nominal diameters and axial lengths. In regard to right-angle valves the conductance calculation for an elbow must be applied.
In the case of dust filters which are used to protect gas ballast pumps and roots pumps, the percentage restriction value for the various pressure levels are listed in the catalog. Other components, namely the condensate separators and condensers, are designed so that they will not reduce pumping speed to any appreciable extent.
The following may be used as a rule of thumb for dimensioning vacuum lines: The lines should be as short and as wide as possible. They must exhibit at least the same cross-section as the intake port at the pump. If particular circumstances prevent shortening the suction line, then it is advisable, whenever this is justifiable from the engineering and economic points of view, to include a roots pump in the suction line. This then acts as a gas entrainment pump which reduces line impedance.
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