Free molecular flow

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. 

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... 

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.

The conductance for free molecular flow in an orifice of nonzero thickness may be written... 

Knudsen s result for free molecular flow in a tube is given by Eq. (73). With/ = 1, Knudsen s result and the second term in Eq. (84) differ only by a numerical factor. In Knudsen s result, the numerical factor is 2/3 in Eq. (84), the corresponding factor is n/8. Thus, except for a modest difference in the numerical factor, the slip term in Eq. (84) is the Knudsen free molecular flow term, and transition flow in a tube appears as a mixture of free molecular flow and viscous flow. That is, the total flow behaves approximately as a sum of two parallel flow mechanisms. 

Knudsen writes the conductivity coefficient as a linear combination of the viscous flow coefficient, Fv, and the free molecular flow coefficient denoted Fm,... 

Figure 22 The pressure dependence of conductance A plot of the ratio of the total conductance to the free molecular flow conductance as a function of the ratio of tube radius to mean free path.

Figure 25 Energy transfer between two surfaces via a gas at low pressure (L

Table 3 Experimental Data for Free Molecular Flow Heat Conductivity... 

As the pressure increases from low values, the pressure-dependent term in the denominator of Eq. (101) becomes significant, and the heat transfer is reduced from what is predicted from the free molecular flow heat transfer equation. Physically, this reduction in heat flow is a result of gas-gas collisions interfering with direct energy transfer between the gas molecules and the surfaces. If we use the heat conductivity parameters for water vapor and assume that the energy accommodation coefficient is unity, (aA0/X)dP — 150 I d cm- Thus, at a typical pressure for freeze drying of 0.1 torr, this term is unity at d 0.7 mm. Thus, gas-gas collisions reduce free molecular flow heat transfer by at least a factor of 2 for surfaces separated by less than 1 mm. Most heat transfer processes in freeze drying involve separation distances of at least a few tenths of a millimeter, so transition flow heat transfer is the most important mode of heat transfer through the gas. 

Some additional work was done upon thermal transfer from cylinders in order to supplement the satisfactory correlations available (Ml). The measurements of Cole and Roshko (C4) and of Brier and co-workers (B12) at normal pressures supplemented by the measurements of Bell (B7) for the free molecular flow of a nonuniform gas are of particular interest. 

In the free molecular flow regime, where the molecular mean free path X is much larger than the orifice diameter, the speed of molecules exiting the reactor can be taken as the mean molecular speed, (v) = SRT/nMW, and the flux, FM, is given by... 

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