4.8 The Pumping Process, Pump-Down Time, and Ultimate Pressure

Having introduced the quantities throughput Q and pumping speed S earlier in this chapter, and the subject of vapor release and outgassing from surfaces in Chapter 3, we can now set up the basic equation of the pumping process.

Figure 4.8 is a schematic representation of a pumping system. A vessel of volume  V is connected via pipe of conductance C to a pump of speed  S*. The pumping speed at the vessel will be S.                                               

                              Fig. 4.8  Schematic representation of a pumping system

Various sources may contribute to the gas load that has to be pumped. In addition to gas originally in the volume, outgassing from the interior surfaces will commence as soon as the pressure is reduced, as discussed in Chapter 3. Its magnitude may be represented by a throughput QG. There may be gas entry into the volume with throughput QL by unintended leaks or, in some applications, the intentional steady inflow of a specific gas. At some stage, gas may be produced internally as a result of an operating process for which the system has been designed, and when activated, it will contribute a throughput, say, QP , to the load. These contributions are represented schematically in the figure 4.8.

Let QT be the total of all such contributions and any others, such as vaporization, that cause the entry of gas into the volume, then 

In some cases, for example, for a system with no leaks in which there are no gas generating processes, QT will simply be due to outgassing.

The pumping equation assumes the isothermal conditions normally encountered and expresses the fact that the change in the quantity of gas in the volume V, which is associated with a change dp in the pressure p in a small time-interval dt , must be the difference in the quantities entering the volume and leaving it. Thus in pressure–volume units:                                                   

In the context of pumping to evacuate the vessel, the rate of exit of gas exceeds that of entry, and pressure will be falling so that dp and therefore dp/dt are negative. The equation may be written to express the positive rate of reduction of gas in the volume, which is  − V(dp/dt), as                                                     

This differential equation is the fundamental pumping equation, expressing the fact that the rate of change of the amount of gas in the volume at any instant is the difference between the rate of its removal  S × p by the pump and the in flux rate QT . Although this is an exact equation true at all times  t, integrating it to get realistic information about how pressure falls with time is often complicated for a number of reasons.

In many applications, pumping speed S at the vessel depends on pressure. This may be due to either the pressure dependence of the speed S* of the pump itself or, unless flow is in the molecular regime, of the conductance of the connection, or both. Secondly, the gas in flux rate due to outgassing varies significantly with time, and will depend on the previous conditions of use of a system, slowly diminishing as pumping proceeds to a small and sensibly constant value, but only being dramatically reduced in normal experimental times if special procedures such as baking are adopted.

There are, nevertheless, two results of prime importance that may be obtained from above pumping equation. They relate to the lowest pressure achievable and the pump-down time when pumping speed can be considered constant. It is evident from pumping equation that when pressure eventually ceases to fall so that dp/dt becomes zero, the steady pressure achieved in the vessel, called the system’s  ultimate pressure or its base pressure, and denoted  pu is given by                                                           

This confirms common-sense thinking that, for a given pumping speed, low pressures will be achieved for small gas loads. Equally, for a given gas load, the best vacuum is obtained for the largest pumping speed. When steady state is eventually attained, the gas load and the pump’s gas handling capacity are in balance.

We may note that in the early stages of pumping, starting at atmospheric pressure, because the system will be free of large leaks and the contribution of outgassing negligibly small, the term  S × p will be very much larger than the term QT, which can be ignored. Therefore, with rearrangement.

Furthermore, many of the primary pumps used in these early stages of pumping, and particularly the rotary pump, have pumping speeds that are sensibly constant over several decades of pressure, from 1000 down to 10-1 mbar or less. Therefore, with S constant, the above equation may be straightforwardly integrated to give                                                    

where p is the pressure at time  t and  p0  its value at  t = 0 when pumping starts. Under these conditions, therefore, pressure falls exponentially with time,  p = p0 exp(− t/τ ), and with a time constant  τ  = V/S.

Above equation may be restated as                                                 

So that the time taken for the pressure to fall from  p0 to  p may be determined

Above equation may also be used to determine the pumping speed necessary to pump down a volume to a given pressure in a specified time. Thus