Pfeiffer Vacuum A Passion for Perfection

7.3.1 Laminar conductivity

Let us consider the pumping station for the drying system in Section 7.2.2 (Figure 7.2) and calculate the drop in pressure between condenser and backing pump. In this case, a gas throughput of Q = 4,285 · 2.9722 10-2 = 127 Pa · m3/s is specified as a result of the pressure of 4,285 Pa and the volume flow rate Sv of the backing pump of 107 m3/h = 2.9722 · 10-2 m3/s. The DN 63 piping has an inside diameter of 0.07 m and a length of 2 m. Two 90° pipe bends having an equivalent length of 0.2 m each are also taken into consideration.

From p2 = 4,285 mbar and l · p2 = 6.65 · 10-3 Pa · m as well as the pipe diameter d = 0.07 m we use the Knudsen number Kn = to determine the flow range and obtain Kn = 2.22 · 10-5. Since Kn is less than 0.01, this results in viscous flow. This can be either laminar or turbulent, although we prefer laminar flow as the conductivities here are significantly higher than for turbulent flow, which means that significantly lower volume flow losses will occur. The Reynolds number Re must be less than 2,300 for laminar flow.

To calculate the Reynolds number, we first determine the flow velocity v in the piping:
v = 8.66 m/s
and the density ρ of the air at 42.85 mbar from the air density ρ0 = 1.293 kg/m3 at atmospheric pressure.
ρ = 0.0547 kg/m3 and in accordance with Formula 1-10:
Re = = 1.822, i.e. laminar flow.

We use Formula 1-21 to calculate pressure p1:

and multiply by Δp = p1 - p2 to obtain the gas throughput
.

Since p2 = 4,285 Pa and Q = 127 Pa · m3/ s it is possible to directly determine p1 from these values:
Qdiff = 4,287.2 Pa

We have a pressure loss Δp of merely 2.2 Pa, a very low value.
The conductivity of the piping is obtained from Formula 1-14:
LR = 57.73 m3/s

The effective volume flow rate:
Seff = 0.029707 m3/s
is only slightly lower than the volume flow rate without the piping: Sv = 0.029722 m3/s.

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