### 4.1.5 Pumping speed of pumping stages connected in series

Let us consider a vacuum pump having a pumping speed $S_0$ and a compression ratio $K_0$. The pump has backflow losses through gaps with conductivity $C_R$. Let inlet pressure be $p_{inlet}$ and discharge pressure $p_{outlet}$. An additional pump having a pumping speed $S_b$ is connected on the outlet side. This could be a Roots pumping station or a turbopumping station, for instance.

The overall pumping station with a pumping speed $S$ delivers the gas quantity

\[q_{pV}=p_{inlet} \cdot S=p_{outlet} \cdot S_b= S_0 \cdot p_{inlet}-C_R (p_{outlet}-p_{inlet})\]

**Formula 4-2:** Pump combination gas flow

For backflow conductivity $C_R$, the following applies where $C_R << S_0$

\[C_R=\frac{S_0}{K_0}\]

**Formula 4-3:** Backflow conductance

and for the actual compression ratio

\[C_R=\frac{p_{inlet}}{p_{outlet}}=\frac{S}{S_b}\]

**Formula 4-4:** Actual compression ratio

Using the above formulas, it therefore follows that the pumping speed $S$ of a two-stage pumping station will be

\[S=\frac{S_0}{1-\frac{1}{K_0}+\frac{S_0}{K_0 \cdot S_b}}\]

**Formula 4-5:** Pumping speed recursion formula

This formula can also be used as the recursion formula for multiple pumping stages that are connected in series by starting with the pumping speed $S_b$ of the last stage and inserting the $K_0$ and $S_0$ of the preceding stage.