4.1- Introduction: Flow Regimes

How does the flow of gas in vacuum systems compare with familiar examples of fluid flow in the world around us? At pressures sufficiently low that the molecular mean free path is comparable with or exceeds the size of the equipment, that is for Kn values > 1, it is totally different. But for vacua not so rarefied, in which mean free paths are such that gases still demonstrate fluid behavior (Kn < 0.01), the concepts and measures developed to describe  fluid  flow at atmospheric pressure remain appropriate.

To fix ideas we can imagine gas flow through a pipe of diameter D. This will be its characteristic dimension to be used in evaluating the Knudsen number as Kn =  λ /D  and will determine what is called the flow regime. As earlier introduced in previous chapter to describe the condition of static gas, flow regimes are de fined by

Kn < 0.01 continuum flow regime

Kn > 1 molecular flow regime

Whereas for 0.01 < Kn < 1 the flow regime is described as transitional.

These defining values are not as sharp as is implied, but their general correctness is founded in experimental results, particularly those involving viscous effects. Flow has distinct characteristics, to be discussed, in each regime. Flow in the transitional regime is difficult to analyse.

When considering flow through pipes of diameter D, it is useful to be able to determine the  flow regime directly in terms of the prevailing pressure. Because for air λ = 64 mm at  p =10-3 mbar, so that  λ p = 0.064 mm mbar, the criteria for continuum flow and molecular flow become

pD >  6.4 mbar mm.............. continuum  flow

pD < 0.064 mbar mm........... molecular  flow

Vacuum systems of typical ~ 0.5 m dimension frequently operate at pressures of 10-5 mbar and less so that conditions are molecular; even at 10-4 mbar, λ  has increased to be 0.64 m.

4.2 Measures of Flow: Throughput and Pumping Speed

Before considering details of flow in the different regimes, it is necessary to define measures of flow. Figure 4.1 shows a pipe that connects two large volumes and through which gas  flows at a steady rate and at constant temperature. In the volume at the left, the pressure is taken to be higher with a value  pU, where the subscript U signifies upstream and flow is from left to right. In the other volume, the downstream pressure is pD. In the pipe at a cross-sectional plane 1 near the entrance, the pressure is p1; at plane 2 further downstream, it is  p2.

Figure 4.1: Flow of a gas through a pipe            

Throughput in terms of Volumetric Flow Rate

The mass of gas flowing per second through plane 1 at pressure  p1 would have an associated volume V1 at that pressure. Downstream at plane 2 and the lower pressure  p2, the associated volume  V2 would be larger because, gases expand in  flowing from a higher to a lower pressure through a pipe. The magnitude of the effect (gas expansion) depends on pressure difference conditions and may be quite small.

Under conditions of steady isothermal  flow, and assuming that the gas behaves ideally,  p1V1 =p2V2. Denoting the volumes per second as volumetric flow rates at the associated pressures, this becomes    . Here, the product    of pressure at any cross-section multiplied by the volumetric flow rate is called the throughput  Q, gives a straightforward measure of the rate at which gas  flows. Thus, defining throughput,

For steady  flow, Q is continuous, i.e., it has the same value at every position along the pipe, reflecting the conservation of mass. In particular,  Qin = Qout - as much gas leaves the pipe downstream as enters it upstream.

The unit of Q depends on the base units used. In the SI system it is the Pascal meter3 per second  (Pa m3s−1). The more practical unit, widely accepted in Europe, and used henceforth in the text, is the millibar liter per second (mbar l s−1). Throughput is an easily assembled and manipulated measure of flow and is extensively used.

Throughput in terms of Mass Flow Rate

When mass flow rates need to be specified directly in units of kg per second, conversions are easily made. Let  be the mass flow rate of gas in kg s−1. The mass W of a gas may be expressed as

                                                           W = nM ×  M

the product of the number of moles and the molar mass. Now  

                                                       nM= pV/R0T 

and the flow rate in moles may be denoted as moles per second. At a particular plane of measurement where the pressure is  p,  this will become

Thus                                                    

And because     we have

Q = R0T/M ˟  

Throughput in terms of Molecular Flow Rate

It is sometimes useful to be able to relate throughput Q to ( dN/dt), the number of molecules  flowing per second, also called the particle flow rate. Dividing both numerator and denominator in the right-hand side of above equation by Avogadro’s number NA , we get

 where m is the mass of a molecule  and k is Boltzmann’s constant. But also

 And therefor                                                                

The volumetric flow rate is frequently given the symbol S and called the pumping speed. This is particularly so when it refers to the intake port of a pump or the entrance to a pipe that has a pump connected to its other end. In pumping practice, typical units used are liters per second, liters per minute, and m3 per hour; m3 per second is rare!

Remembering that S is a volumetric flow rate, the defining Equation of throughput now becomes

                                                         Q = S  × p

This is the usual form of the  first of two basic defining equations that describe gas flow in vacuum practice. It expresses the quantity of gas flowing as the product of the pressure and the volumetric flow rate at that pressure. Allied with the condition for continuity, it is an important tool for analysis.

4.3 Conductance

The other fundamental equation of flow relates throughput Q to the difference between the upstream and downstream pressures  pU and  pD in the two volumes that the pipe connects and serves to define the quantity conductance.Thus, referring again to Figure 4.1,                                                          

                                                  Q = C (pU– pD)

Evidently, C has the same dimensions as S, i.e., volume per second. A typical unit is liter per second.

It may be helpful to note that although in dc electrical circuits the connection between current (analogue: Q) and potential differences (analogue: pU − pD) is expressed in Ohm’s law by a resistance, in vacuum practice the link is made by its inverse, the conductance. Thus, conductance is a measure of ease of  flow in response to a pressure difference, and the greater the conductance for a given pressure difference, the greater the throughput.

Accordingly, and pursuing circuit analogies, one may expect that there are simple rules of combination for conducting elements in series and in parallel.

It is easily shown, and intuitively reasonable, that for conductances  C1,  C2, etc., in parallel, the effective conductance of the combination is given by

 while for elements in series

Effect of conductance on pumping speed

Consider, as in Figure 4.2, a vessel within which the pressure is p connected via a pipe of conductance C to a pump of speed  S*. The pumping speed at the vessel is  S. Let the pressure at the pipe exit and the entrance to the pump be p*. The throughput from the vessel, through the pipe and into the pump, is Q                     

 Figure 4.2: Effect of conductance on pumping speed      

This yields for  S, after a little algebra                                                                  

 Figure 4.3: Variation of pumping speed at the vessel with connecting pipe conductance

The significance of this formula is illustrated in Figure 4.3 in which S/ S*, the effective pumping speed at the vessel expressed as a fraction of the pump speed S*, is plotted against C/S*, the ratio of the connecting pipe conductance to S*. From this graph we can draw following conclusions.

1. Necessarily, therefore,  S is less than  S*.
2. Clearly, only when  C is appreciably greater than S* is the pumping speed at the vessel comparable with that of the pump.
3. It is halved if the conductance and the pump speed are equal.

We may visualize this result in terms of the volumes pumped at the vessel and at the pump. Gas expands as it moves downstream through the pipe to the pump, and the volumetric  flow rate that is fixed at the pump end of the pipe by the pump’s speed must, therefore, be smaller at the vessel.