2.2 Calculations
2.2.1 Dimensioning a Roots pumping station
Various preliminary considerations are first required in dimensioning a Roots pumping station.
Compression ratio
The compression ratio $K_0$ eof 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 $C_R$, as well as the return flow of gas from the discharge chamber at pumping speed $S_R$:
\[p_a \cdot S = p_a \cdot S_0 - C_R\left(p_v-p_a\right)-S_R \cdot p_v\]
Formula 2-1: Roots pump gas load
| $S$ | Volume flow rate (pumping speed) |
| $S_0$ | Theoretical pumping speed on the intake side |
| $S_R$ | Pumping speed of return gas flow |
| $C_R$ | Conductivity |
| $p_a$ | Inlet pressure |
| $p_v$ | Backing vacuum pressure |
Selecting $S$ as being equal to 0 we obtain the compression ratio
\[\frac{p_a}{p_v} =K_0= \frac{S_0+C_R}{C_R+S_R}\]
Formula 2-2: Compression ratio of Roots pump
| $K_0$ | Compression ratio |
In the case of laminar flow the conductance is significantly greater than the pumping speed of the backflow. This simplifies Formula 2-2 to
\[K_0= \frac{S_0}{C_R}\]
Formula 2-3: Compression ratio of Roots pump for laminar flow
In the molecular flow range, the pumping speed is still greatest on the intake side, but the pumping speed of the backflow is now considerably greater than the conductance. The compression ratio is therefore:
\[K_0= \frac{S_0}{S_R}\]
Formula 2-4: Compression ratio of Roots pump for molecular flow
At laminar flow (high pressure), the compression ratio is limited by backflow through the gap between the roots lobes and the housing. Since conductance is proportional to mean pressure, the compression ratio will decrease as pressure rises.
In the molecular flow range, the return gas flow $S_R \cdot p_v$ 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 $p_a$ of more than 10-4 hPa.
Pumping speed
Roots pumps are equipped with overflow valves that allow maximum pressure differentials $\Delta p_d$ of between 30 and 60 hPa 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 ($S_1$) and closed ($S_2$).
Since gas throughput is the same in both pumps (Roots pump and backing pump), the following applies:
\[S_1=\frac{S_V \cdot p_v}{p_v \cdot \Delta p_d}\]
Formula 2-5: Pumping speed of Roots pumping station with overflow valve open and at high fore-vacuum pressure
| $S_1$ | Pumping speed with overflow valve open |
| $S_V$ | Pumping speed of backing pump |
| $p_v$ | Fore-vacuum pressure |
| $\Delta p_d$ | maximum pressure differential between the pressure and intake side of the Roots pump |
As long as the pressure differential is significantly smaller than the fore-vacuum pressure, the pumping speed of the pumping station will be only slightly higher than that of the backing pump. As backing vacuum pressure nears pressure differential, the overflow valve will close and will apply
\[S_1=\frac{S_0}{1-\frac{1}{K_0}+\frac{S_0}{K_0 \cdot S_V}}\]
Formula 2-6: Pumping speed of Roots pumping station with overflow valve closed and fore-vacuum pressure close to differential pressure
Let us now consider the special case of a Roots pump working against constant pressure (e. g. condenser mode). Formula 2-3 will apply in the high pressure range. Using the value $C_R$ in Formula 1 and disregarding the backflow $S_R$ against the conductance value $C_R$ we obtain:
\[S=S_0 \cdot \left[1-\frac{1}{K_0}\left(\frac{p_v}{p_a}-1 \right) \right] \]
Formula 2-7: Pumping speed of Roots pumping station at high intake pressure
At low pressures, $S_R$ from Formula 2-4 is used and we obtain
\[S=S_0 \cdot \left(1-\frac{p_v}{K_0 \cdot p_a} \right) \]
Formula 2-8: Pumping speed of Roots pumping station at low intake pressure
From Formula 2-6 , it can be seen that $S$ tends toward $S_0$ if the compression ratio $K_0$ is significantly greater than the ratio between the theoretical pumping speed of the Roots pump $S_0$ and the fore-vacuum pumping speed $S_V$.
Selecting the compression ratio, for example, as equal to 40 and the pumping speed of the Roots pump as 10 times greater than that of the backing pump, then we obtain $S$ = 0.816 $\cdot S_0$
For the purposes of adjustment for use in a pumping station the theoretical pumping speed of the Roots pump should therefore not be more than ten times greater than the pumping speed of the backing pump.
Since the overflow valves are set to pressure differentials of around 50 hPa, virtually only the volume flow rate of the backing pump is effective for pressures of over 50 hPa. If large vessels are to be evacuated to 100 hPa 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 with a volume of 2 m³ to a pressure of 5 · 10-3 hPa in 10 minutes. To do this, we would select a backing pump that can evacuate the vessel to 50 hPa in 5 minutes. The following applies at a constant volume flow rate:
\[t_1=\frac{V}{S} \mbox{ln} \frac{p_0}{p_1}\]
Formula 2-9: Pump-down time
| $t_1$ | Pump-down time of backing pump |
| $V$ | Volume of vessel |
| $S$ | Pumping speed of backing pump |
| $p_0$ | Initial pressure |
| $p_1$ | Final pressure |
By rearranging Formula 2-9, we can calculate the required pumping speed:
\[S=\frac{V}{t_1} \mbox{ln} \frac{p_0}{p_1}\]
Formula 2-10: Calculating the pumping speed
Using the numerical values given above we obtain:
\[S=\frac{2,000 l}{300 s} \mbox{ln} \frac{1,000}{50}=20\frac{l}{s}=72\frac{m^3}{h}\]
We select a Hepta 100 with a pumping speed $S_V$ = 100 m³ h-1 as the backing pump. Using the same formula, we estimate that the pumping speed of the Roots pump will be 61 l s-1 = 220 m³ h-1, and select an Okta 500 with a pumping speed $S_0$ = 490 m³ h-1 and an overflow valve pressure differential of $\Delta p_d$ = 53 hPa for the medium vacuum range.
From the table below, we select the fore-vacuum pressures given in the column $p_v$, use the corresponding pumping speeds $S_V$ for the Hepta 100 from its pumping speed curve and calculate the throughput: $Q=S_V \cdot p_v$.
he compression ratio $K_\Delta = \frac{p_v+ \Delta p_d}{p_v}$
is calculated for an open overflow valve up to a fore- vacuum pressure of 56 hPa. $K_0$ for fore-vacuum pressures ≤ 153 hPa is taken from Figure 2.1. There are two ways to calculate the pumping speed of the Roots pump:
$S_1$ can be obtained from Formula 2-5 for an open overflow valve, or $S_2$ on the basis of Formula 2-6 for a closed overflow valve.
Figure 2.2: Volume flow rate (pumping speed) of a pumping station with Hepta 100 and Okta 500
As the fore-vacuum pressure nears pressure differential $\Delta p_d$,$S_1$ will be greater than $S_2$. The lesser of the two pumping speeds will always be the correct one, which we will designate as $S$. The inlet pressure is obtained with the formula:
$p_a=\frac{Q}{S}$
Figure 2.2 shows the pumping speed graph for this pumping station.
Figure 2.1: No-load compression ratio for air with Roots pumps
| Pa / hPa | Pv / hPa | Sv / (m3 / h) | Q / (hPa · m3/ h) | K$\Delta$ | K0 | S1 / (m3 / h) | S2 / (m3 / h) | t / h | t / s |
|---|---|---|---|---|---|---|---|---|---|
| Pump-down time: 344.94 s | |||||||||
| 1,000.0000 | 1,053.00 | 90.00 | 94,770.00 | 1.05 | 94.77 | 0.00490 | 17.66 | ||
| 800.0000 | 853.00 | 92.00 | 78,476.00 | 1.07 | 98.10 | 0.00612 | 22.04 | ||
| 600.0000 | 653.00 | 96.00 | 62,688.00 | 1.09 | 104.48 | 0.00827 | 29.79 | ||
| 400.0000 | 453.00 | 100.00 | 45,300.00 | 1.13 | 113.25 | 0.01359 | 48.93 | ||
| 200.0000 | 253.00 | 104.00 | 26,312.00 | 1.27 | 131.56 | 0.00652 | 23.45 | ||
| 100.0000 | 153.00 | 105.00 | 16,065.00 | 1.53 | 7.00 | 160.65 | 321.56 | 0.00394 | 14.18 |
| 50.0000 | 103.00 | 105.00 | 10,815.00 | 2.06 | 13.00 | 216.30 | 382.20 | 0.00608 | 21.87 |
| 14.9841 | 56.00 | 110.00 | 6,160.00 | 18.70 | 18.00 | 2,053.33 | 411.10 | 0.00822 | 29.58 |
| 2.5595 | 10.00 | 115.00 | 1,150.00 | 36.00 | 449.30 | 0.01064 | 38.30 | ||
| 0.2300 | 1.00 | 105.00 | 105.00 | 50.00 | 456.52 | 0.00670 | 24.13 | ||
| 0.0514 | 0.30 | 75.00 | 22.50 | 46.00 | 437.39 | 0.00813 | 29.27 | ||
| 0.0099 | 0.10 | 37.00 | 3.70 | 40.00 | 375.17 | 0.00673 | 24.23 | ||
| 0.0033 | 0.06 | 15.00 | 0.90 | 39.00 | 270.42 | 0.00597 | 21.51 | ||
| 0.0018 | 0.05 | 5.00 | 0.25 | 37.00 | 135.29 | ||||
Table 2.1: Pumping speed of a Roots pumping station and pump-down times
Pump-down times
The pump-down time for the vessel is calculated in individual steps. In areas with a strong change in pumping speed, the fore-vacuum pressure intervals must be configured close together. Formula 2-9 is used to determine the pump-down time during an interval, with $S$ being used as the mean value of the two pumping speeds for the calculated pressure interval. The total pump-down time will be the sum of all times in the last column of Table 2-1.
The pump-down time will additionally be influenced by the leakage rate of the vacuum system, the conductances of the piping and of vaporizing liquids that are present in the vacuum chamber, as well as by degassing of porous materials and contaminated walls. Some of these factors will be discussed in Sections 2.2.3.1 and 2.3. If any of the above-mentioned influences are unknown, it will be necessary to provide appropriate reserves in the pumping station.
