# 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 `K _{0}` 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

`L`, as well as the return flow of gas from the discharge chamber at volume flow rate

_{R}`S`:

_{R}#### Formula 7-1:

Roots pump gas load

where `p _{a}` = intake pressure and

`p`= backing vacuum pressure

_{v}Selecting `S` as being equal to 0 yields the compression ratio:

#### Formula 7-2:

`K _{0}` of a Roots pump

The following applies in the laminar flow range: `S _{0}` >>

`L`>>

_{R}`S`and thus

_{R}#### Formula 7-3:

`K _{0}` laminar

and the following applies in the molecular flow range due to

#### Formula 7-4:

`K _{0}` 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 `S _{R}`
·

`p`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

_{v}`p`of less than 10

_{a}^{-4}mbar.

## Volume flow rate (pumping speed)

Roots pumps are equipped with overflow valves that allow maximum pressure differentials
`Δp _{d}` 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:

`S _{1}` for

`Δp`

_{d}<< p_{v}As long as `Δp _{d} << p_{v}` 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

`Δp`, the overflow valve will close and

_{d}`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
`L _{R}` in Formula 7-1 and disregarding

`S`against

_{R}`L`yields:

_{R}#### Formula 7-6:

`S` against high pressure

At low pressures, `S _{R}` 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 `S _{0}`
if

`K`>>

_{0}Using e.g `K _{0}` = 40 and = 10, for example, yields

`S`= 0.816

`S`.

_{0}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 m^{3} 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 `t _{1}` = 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 `S _{v}` =

We select a Hepta 100 with a pumping speed (volume flow rate) `S _{v}` of 100 m

^{3}/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 m

^{3}/h, and select an Okta 500 with a pumping speed

`S`of 490 m

_{0}^{3}/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 `p _{v}`, use
the corresponding pumping speeds

`S`for the Hepta 100 from Figure 2.10 and calculate the throughput

_{v}`Q = S`.

_{v}· p_{v}

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

The compression ratio `K _{Δ}` = is calculated for an open overflow valve to a backing
vacuum pressure of 56 mbar.

`K`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:

_{0}`S _{1}` can be obtained from Formula 7-5:

`S`or an open overflow valve, or

_{1}= S_{v}· K_{Δ}`S`on the basis of Formula 2-5 for a closed overflow valve

_{2}`S`= .

_{2}As the backing vacuum pressure nears pressure differential `Δp _{d}`,

`S`will be greater than

_{1}`S`. The lesser of the two pumping speeds will always be the correct one, which we will designate as

_{2}`S`.

The inlet pressure is obtained on the basis of:`p _{a}` = 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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