4.4 Continuum Flow

In this regime, gas behaves as a fluid, and molecule–molecule collisions with mean free path much less than the equipment size determine gas behavior.

The characteristic dynamic property of a gas is its viscosity. If a gas had zero viscosity, its steady flow through a pipe would be characterized by uniform parallel streamlines as shown in Figure 4.4(a), and the gas velocity u would be constant over any cross-section. In reality, however, viscosity causes the gas at the pipe wall to be stationary, so that the velocity profile is developed as shown in Figure 4.4(b), which has a maximum value at the center and some value u averaged over the section.           

Fig. 4.4 Velocity profiles for streamline flow in a pipe (a) Zero viscosity (b) Finite visosity

4.5  Dynamical Analysis of Continuum Flow through Long Pipe

The formula of Poiseuille and Hagen, which describes, as initially formulated, the flow of liquids in long pipes, is adaptable to the gas flow but is unfortunately of limited value. This is because the usual requirement in vacuum technology is that pipes that connect work chambers to pumps shall offer as little impedance as possible to gas flow, and so they are made as short and wide-bored as possible, consistent with spatial constraints. Typical conditions do not, therefore, involve long pipes. Nevertheless, because it is the only pipe flow problem with a simple analytical solution and because its use illustrates methods of analysis, it is presented here.

Consider, as in Figure 4.5, steady flow in a long pipe of diameter D in a section of  length dx between positions x and x + dx, in which the pressure falls from p to p to p- dp.

                                             Fig. 4.5  Poiseuilllle fluid flow

The volumetric flow rate of fluid through this section according to Poiseuille and Hagen law is

For a gas, multiplying above equation by p gives the throughput Q at the section x, which is constant along the pipe. Thus

Integrating over the length L of the pipe from x = 0 (p = p1) to x = L (p = p2) gives

The middle term on the right-hand side is the mean pressure.

Comparing above equation with fundamental equation of throughput below

we obtain

4.6 Molecular Flow

Molecular flow is characterized by Knudsen numbers Kn greater than unity, which physically means that the mean free path associated with the prevailing number density of the contained gas molecules is greater than the size of the container, with the consequence that molecule/wall collisions dominate gas behavior.

All semblance to fluid behavior is lost because there are no molecule–molecule collisions. These are the conditions in the work chambers of many vacuum systems because, as previously noted, the mean free path for nitrogen at 10-4 mbar is 0.64 m, so that for a chamber of typical size, conditions are molecular at pressures below this value.

4.6.1 Dynamical analysis of molecular flow through Long Pipe

In a pipe of length L and diameter D, consider a short section between coordinates x and x + dx, across which pressure changes from p to p - dp. There will be an associated change n to n - dn in the number density of molecules, but n may be taken as the density in the section for the purposes of the calculus. Molecular flow in the pipe may be considered to occur with a mean drift velocity, superposed on the thermal velocities, that is reduced to zero by the collisions that molecules have with the wall.

By Newton’s second law, equating the rate of change of momentum due to the loss of in wall collisions to the force across the element, gives

in which the first two terms on the left-hand side give the number of wall impacts per second in the element dx. After cancellations,                                                           

The number of molecules per second passing through the plane at x is

Substituting value of mean velocity  and multiplying by kT to convert to a throughput, and integrating over the pipe leads to

The conductance is therefore

Because this treatment is oversimplified in its assumptions about the drift velocity, the factor π/16 should be replaced by 1/6 (see Loeb, 1961) to give

This formula was first proposed by Knudsen and is correct for long pipes. The geometrical dependence on D3/L is its most important feature and again points towards making pipes as short and fat as possible to maximize conductance.

The factor  indicates the dependence on the particular gas and its temperature and is directly related to the molecular velocity. Note that, as one would expect, the conductance does not depend on pressure.

4.6.2 Dynamical analysis of Molecular Flow through an Aperture

Consider an aperture of area A in a very thin wall separating two regions maintained at different pressures p1 and p2, with p1 > p2 and the gas in both regions sufficiently rarefied that conditions are molecular, as shown in Figure 4.6. The molecular mean free path is greater than the diameter of the aperture, and there are no molecule–molecule collisions.

                                   Fig. 4.6  Molecular flow through an Aperture

From each side, molecules approach the opening from all directions within a 2π solid angle and with a range of speeds. The fluxes are represented by the arrows J. Corresponding to p1 and p2 are number densities n1 and n2 and associated fluxes J1 and J2, where . Molecules heading towards the aperture opening from both sides will pass through it, and so with J1 > J2 as indicated, there will be a net flow of molecules from left to right. The number of molecules per unit time will be

Multiplying by kT to convert to a throughput and substituting for J gives

Comparing this with the basic defining equation for conductance,                                                     

we obtain conductance of an aperture for molecular flow. Here, introducing the symbol CA for the molecular flow conductance for an aperture,

This is an important result, exploitable not only in its own right but also because the entrances into pipes and pumps can be regarded as apertures. The presence of the factor  that will also occur in other formulas for molecular flow conductance is noteworthy because it enables conductance values for other gases to be quickly computed once values for a particular gas of reference, usually nitrogen, are known.