Molecular flow

Molecular flow occurs under conditions where Kn 0.5 - the mean free path of the particles exceeds the smallest dimension of the flow channel. Under such conditions, with thin-walled orifices, for example, gas particles will pass through almost without collision. With pipes and ducts, however, this is not the case. Particularly for low Kn values (1-10) of the particles that enter the duct, some may reach the exit whilst the remainder return to the entrance after a number of collisions with the duct walls. What is important about such collisions is that, on collision with a wall, the particles are regarded as being immobilised for a very short time before emerging in any direction with equal probability (according to the cosine law). This describes diffuse or random scattering where no particular direction is favoured. To describe this process, the concept of transmission probability (Pr) was introduced by Clausing. 

In the case of an orifice, Pr = 1. For other types of pipes and ducts Pr 1. A significant part of assessing molecular flow in ducts involves the estimation of Pr. An initial assumption is that molecules arrive at the entrance plane of a duct with an isotropic velocity distribution. Conductance under molecular flow conditions is independent of the pressure but obviously the throughput is proportional to the Ap as stated by the definition of C (e.g Equation (2.6)). 

All equations presented up to here are only valid if the mean free path lengths of the molecules are very short in comparison to the diameter d of pipes or pores. This condition is not fulfilled for gases flowing in pipes with a diameter in the millimeter range if the pressure is below 1 Pa. The pores in adsoibents or in porous materials to be dried have a width of some nanometers in most cases or even smaller than 1 nm with the result that the mean free path length is approximately the same or more than the pore diameter for gas pressures of 10 Pa. The ratio of the molecular mean free path A based on the tube or pore diameter d is the Knud-sen number 

In Fig. 3.1-9 on the left side a tube with diameter d and length L is shown where impinging molecules are reflected by the wall due to elastic impacts. On the right side a diffuse reflexion is illustrated as a result of rough walls and partially inelastic 

The number of molecules N per unit time entering the tube is 

The mass flow in the opposite direction can be described by an analogous equation. Let us consider a mean density p which is equal to the mean value between the entrance and the exit. The net mass flow rate M = M -M is 

With the equation for the mean molecule velocity presented earlier the mass flow rate M is 

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Effusion separator (or effusion enricher). An interface in which carrier gas is preferentially removed from the gas entering the mass spectrometer by effusive flow (e.g., through a porous tube or through a slit). This flow is usually molecular flow, such that the mean free path is much greater than the largest dimension of a traverse section of the channel. The flow characteristics are determined by collisions of the gas molecules with surfaces flow effects from molecular collisions are insignificant. 

If a particularly parallel beam is required in the chamber into which it is flowing the beam may be skimmed in the region of hydrodynamic flow. A skimmer is a collimator which is specially constructed in order to avoid shockwaves travelling back into the gas and increasing 7). The gas that has been skimmed away may be pumped off in a separate vacuum chamber. Further collimation may be carried out in the region of molecular flow and a so-called supersonic beam results. When a skimmer is not used, a supersonic jet results this may or may not be collimated. 

Electronic transitions in molecules in supersonic jets may be investigated by intersecting the jet with a tunable dye laser in the region of molecular flow and observing the total fluorescence intensity. As the laser is tuned across the absorption band system a fluorescence excitation spectrum results which strongly resembles the absorption spectrum. The spectrum... 

In free molecular flow, if gaseous conductance were not independent of the flow direction, a perpetual-motion machine could be constmcted by connecting two large volumes by a pair of identical ducts having a turbine in front of one of the ducts. A duct that has asymmetricaUy shaped grooves on its waU surface could alter the probabUity of molecular passage in such a way that for a tube of equal entrance and exit areas, the probabUity of passage would be made directional. 

The Knudsen number Kn is the ratio of the mean free path to the channel dimension. For pipe flow, Kn = X/D. Molecular flow is characterized by Kn > 1.0 continuum viscous (laminar or turbulent) flow is characterized by Kn < 0.01. Transition or slip flow applies over the range 0.01 < Kn < 1.0. 

Molecular Flow Under molecular flow conditions, conductance is independent of pressure. It is proportional to with the pro-... 

Slip Flow In the transition region between molecular flow and continuum viscous flow, the conductance for fully developed pipe flow is most easily obtained by the method of Brown, et al. (J. Appl. Phys., 17, 802-813 [1946]), which uses the parameter... 

For gas flow through porous media with small pore diameters, the slip flow and molecular flow equations previously given (see the Vacuum Flow subsec tion) may be applied when the pore is of the same or smaller order as the mean free path, as described by Monet and Vermeulen (Chem. E/ig. Pi og., 55, Symp. Sei , 25 [1959]). 

The pressure of gas in the source chamber was determined from the pressure in the gas reservoir (measured with a McLeod gage) by using Dushman s (11) relations for viscous and molecular flow in the higher and lower ranges of pressure, respectively. 

Kn = 0.1-10 Transition flow between slip flow and free molecular flow, treated statistically, e.g., by the Boltzmann equation... 

Kn> 10 Free molecular flow motion of individual molecules, that must be modeled and then treated statistically... 

The beauty of the reptation model is that it is able to make predictions about molecular flow both in solution and at fracture by assuming that the molecules undergo the same kind of motions in each case. For both self-diffusion in concentrated solutions and at fracture, the force to overcome in pulling the polymer molecule through the tube is assumed to be frictional. 

Figure 14 Free molecular flow through an orifice Knudsen effusion.

Knudsen effusion will be involved later as we discuss free molecular flow in channels and tubes. Knudsen effusion also finds application in the measurement of the vapor pressure of materials of low vapor pressure, typically in the... 

As the pressure is lowered, slip occurs, and the flow mechanism is referred to as transition flow. At pressures so low that collisions between gas molecules are rare compared to the collisions between the gas and the tube wall, the flow is said to be Knudsen flow or free molecular flow. Free molecular flow prevails when Lla > 1. For air at 25°C, this condition means that we have free molecular flow when aPm on < 5. We now consider an intuitive derivation of the result for Fc in the free molecular flow region. 

Figure 20 Mass transfer and momentum exchange during free molecular flow in a long tube.

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