# 7.2.1 Dimensioning a Roots pumping station

Various preliminary considerations are first required in dimensioning a Roots pumping station.

## Compression ratio

The compression ratio K0 of a Roots pump is typically between 5 and 70. To determine this ratio, we first consider the volume of gas pumped and the backflow by means of conductivity LR, as well as the return flow of gas from the discharge chamber at volume flow rate SR:

#### Formula 7-1:

where pa = intake pressure and pv = backing vacuum pressure

Selecting S as being equal to 0 yields the compression ratio:

#### Formula 7-2:

K0 of a Roots pump

The following applies in the laminar flow range: S0 >> LR >> SR and thus

#### Formula 7-3:

K0 laminar

and the following applies in the molecular flow range due to

#### Formula 7-4:

K0 molecular

At laminar flow (high pressure), the compression ratio is limited by backflow through the gap between piston and housing. Since conductivity is proportional to mean pressure, the compression ratio will decline as pressure rises.

In the molecular flow range, the return gas flow SR · pv from the discharge side predominates and limits the compression ratio toward low pressure. Because of this effect, the use of Roots pumps is restricted to pressures pa of less than 10-4 mbar.

## Volume flow rate (pumping speed)

Roots pumps are equipped with overflow valves that allow maximum pressure differentials Δpd of between 30 and 60 mbar at the pumps. If a Roots pump is combined with a backing pump, a distinction must be made between pressure ranges with the overflow valve open (S1) and closed (S2).

Since gas throughput is the same in both pumps, the following applies:

#### Formula 7-5:

S1 for Δpd << pv

As long as Δpd << pv the volume flow rate (pumping speed) of the pumping station will be only slightly higher than that of the backing pump. As backing vacuum pressure nears pressure differential Δpd , the overflow valve will close and S = will apply.

Let us now consider the special case of a Roots pump working against constant pressure (e. g. condenser operation). Formula 7-3 will apply in the range of high pressures. Using value LR in Formula 7-1 and disregarding SR against LR yields:

#### Formula 7-6:

S against high pressure

At low pressures, SR from Formula 7-4 is used and yields:

#### Formula 7-7:

S against low pressure

From Formula 2-5, it can be seen that S tends toward S0 if K0 >>

Using e.g K0 = 40 and = 10, for example, yields S = 0.816 S0.

Consequently, the following should apply for rating a pumping station ≤ 10

Because the overflow valves are set to pressure differentials of around 50 mbar, virtually only the volume flow rate of the backing pump is effective for pressures of over 50 mbar. If large vessels are to be evacuated to 100 mbar within a given period of time, for example, an appropriately large backing pump must be selected.

Let us consider the example of a pumping station that should evacuate a vessel of V = 2 m3 to 5 · 10-3 mbar in 10 minutes. To do this, we would select a backing pump that can evacuate the vessel to 50 mbar in t1 = 5 minutes. The following applies at a constant volume flow rate:

#### Formula 7-8:

Pump-down time

Formula 7-8 yields the volume flow rate Sv =

We select a Hepta 100 with a pumping speed (volume flow rate) Sv of 100 m3/h as the backing pump. Using the same formula, we estimate that the pumping speed of the Roots pump will be 61 l/s = 220 m3/h, and select an Okta 500 with a pumping speed S0 of 490 m3/h and an overflow valve pressure differential Δpd of 53 mbar for the medium vacuum range.

From Table 7.1 below, we select the backing vacuum pressures on the basis of gap pv, use the corresponding pumping speeds Sv for the Hepta 100 from Figure 2.10 and calculate the throughput Q = Sv · pv.

Table 7.1: Volume flow rate (pumping speed) of a Roots pumping station

The compression ratio KΔ = is calculated for an open overflow valve to a backing vacuum pressure of 56 mbar. K0 can be found in Figure 2.14 for backing vacuum pressures of 153 mbar or less. There are two ways to calculate the pumping speed of the Roots pump:

S1 can be obtained from Formula 7-5: S1 = Sv · KΔ or an open overflow valve, or S2 on the basis of Formula 2-5 for a closed overflow valve S2 = .

As the backing vacuum pressure nears pressure differential Δpd, S1 will be greater than S2. The lesser of the two pumping speeds will always be the correct one, which we will designate as S.

The inlet pressure is obtained on the basis of:pa = Figure 7.1 shows the pumping speed curve of this pumping station.

Figure 7.1: Volume flow rate (pumping speed) of a pumping station with Hepta 100 and Okta 500

## Pump-down times

The pump-down time for the vessel is calculated in individual steps. In ranges with a strong change in volume flow rate, the backing vacuum pressure intervals must be configured close together. Formula 7-8 is employed to determine the pump-down time during an interval, with S being used as the mean value of the two volume flow rates for the calculated pressure interval. The total pump-down time will be the sum of all times in the last column of Table 7-1.

The pump-down time will additionally be influenced by the leakage rate of the vacuum system, the conductivities of the piping and of vaporizing liquids that are present in the recipient, as well as by degassing of porous materials and contaminated walls. Some of these factors will be discussed in Sections 7.2.3.1 and 7.3. If any of the above-mentioned influences are unknown, it will be necessary to provide appropriate reserves in the pumping station.

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