Vacuum Technology

At the entrance of a pump, the relationship between the pressure and the volumetric flow rate is

\begin{equation} Q = p \cdot S \end{equation}

where S is the pumping speed. Pumping speed depends on pressure, although it can be treated as a constant for simplicity.

Pumping down process

attachment:_20240531_193442screenshot.png

$S$	pumping speed at the vessel of volume $V$
$p$	pressure in the vessel
$S^*$	pumping speed of the pump
$p*$	pressure at the pump entrance

We can model all the gas loads entering the vessel as $Q_{tot}$, which is hopefully as small as possible. Unfortunately, air tends to get into a vacuum, so it's necessary to pump it out.

Continuity equation

The difference between the quantity of gas entering the volume and the one leaving it in a small interval of time $dt$ is equal to the net change in the quantity of gas $d(pV) = V \cdot dp$ in the volume $V$.

\begin{equation} V \cdot dp = Q_{tot} \cdot dt - S \cdot p \cdot dt \end{equation}

$Q_{tot}$	Amount of gas getting into the free volume in a time $dt$ at pressure $p$ and speed $S$
$S \cdot p \cdot dt$	Gas pumped away in a time $dt$ at pressure $p$ and speed $S$
$V \cdot dp$	Change of amount of free gas in the volume $V$

This can rewritten as:

\begin{equation} -V (\frac{dP}{dt}) = S \cdot p - Q_{tot} \end{equation}

Initially, the pumpdown proces is dominated by evacuation of the free gas in the volume. Therefore $Q_{tot} = 0$ and the above equation can be simplified to

\begin{equation} V (\frac{dP}{dt}) = - S \cdot p \end{equation}

This is a first order differential equation, which can be solved as:

$$ ln(p_{final}) = ln(p_{initial}) - \frac{S}{V} t $$ $$ p_{final} = p_{initial} e^{ -\frac{S}{V}t } $$

Given that we usually know our initial and final pressure and we care about the time taken to get there, a useful shorthand is:

$$ t = \frac{V}{S} ln \frac{p_{initial}}{p_{final}} $$

Steady state behaviour

When the pressure ceases to fall and becomes constant on the time scale of observation, we've reached steady state behaviour

\begin{align} -V (\frac{dP}{dt}) &= S \cdot p - Q_{tot} 0 &= S \cdot p_{ss} - Q_{ss}

\end{align}

Conductance

The quantity of gas which is flowing across a given pressure difference depends on the ease of flow, which is called conductance.

\begin{equation} Q = C (p_u - p_D) \end{equation}

Conductances in parallel can be added together, while conductances in series follow sum of reciprocals

We can apply this to the case above, by the following equation

\begin{align} Q &= C (p^* - p) \\ Q &= S^* \cdot p^* = S \cdot p \\ \frac{1}{S^*} &= \frac{1}{S} + \frac{1}{C} \end{align}

$C$	Conductance
$Q$	Quantity of gas passing through the pipe and into the pump
$S$	pumping speed at the vessel of volume $V$
$p$	pressure in the vessel
$S^*$	pumping speed of the pump
$p*$	pressure at the pump entrance

Estimating conductance in laminar regimes

In the laminar regime, gas flows smoothly in stream lines, where the viscous forces are greater than inertia forces. In most cases treated in vacuum technology, laminar conditions can be assumed.

The