Inverse Knudsen number

Dq = characteristic inverse Knudsen number po o P 2RTq-, Tq = characteristic temperature ... 

In Fig. 2, the flow rate coefficients, Qp, calculated from these different models are plotted in the range of inverse Knudsen number D from 0.01 to 100. We can see that for... 

We can see that the velocity profiles of these models become more and more different from each other with the decrease in the inverse Knudsen number. The difference between the velocity profiles of the first-order and second-order slip models is in the magnitude of the slip velocity, but the shapes of the velocity profiles are almost same. The second-order slip model predicts a larger and larger slip velocity than the first-order slip model when the inverse Knudsen number decreases. The Fukui-Kaneko model gives a medium slip velocity and velocity profile. 

It should be pointed out that the flow rate in the case of the Couette flow is independent of the inverse Knudsen number, and is the same as the prediction of the continuum model, although the velocity profiles predicted by the different flow models are different as shown in Fig. 4. The flow velocity in the case of the plane Couette flow is given as follows (i) Continuum model ... 

Fig. 5—Change of the rates of particle-particle and particle-wall collisions with the inverse Knudsen number.

As described above, the magnitude of Knudsen number, Kn, or inverse Knudsen number, D, is of great significance for gas lubrication. From the definition of Kn in Eq (2), the local Knudsen number depends on the local mean free path of gas molecules,, and the local characteristic length, L, which is usually taken as the local gap width, h, in analysis of gas lubrication problems. From basic kinetic theory we know that the mean free path represents the average travel distance of a particle between two successive collisions, and if the gas is assumed to be consisted of hard sphere particles, the mean free path can be expressed as... 

Fig. 8—Dependence of the nanoscale effect function on the inverse Knudsen number.

Figure 8 shows the change of Np with the inverse Knudsen number D. When D > 1, the nanoscale effect is weak, and the Xff is slightly less than X./,. When D < 1, however, the difference between and X becomes significant. It is worth to... 

To evaluate the intensity of the rarefaction effect, the him thickness h itself is not enough. We have to know the Knudsen number Kn or the inverse Knudsen number D. From the dehnition of Kn, Eq (2), and the relationship between and pressure, ... 

Fig. 24—Area percentage in different inverse Knudsen number ranges, D is inverse Knudsen number A, is the area satisfied with the inequalities in the horizontal abscissa, (a) sliding speed v = 9.57557 m/s (b) sliding speed v=40 m/s.

Knudsen number (Kn) is the ratio of mean free path I of fluid molecules to a t3rpical dimension of gas flow a, i.e., Kn = Ija. Rarefaction parameter 8 is the inverse Knudsen number. Velocity distribution function is defined so that the quantity /(f, r, v) dr dv is the number of particles in the phase volume dr dv near the point (r, v) at the time t. 

The main parameter determining the gas rarefaction is the Knudsen number Kn = Ha, where I is the mean free path of fluid molecules and a is a typical dimension of gas flow. If the Knudsen number is sufficiently small, say Kn < 10 , the Navier-Stokes equations are applied to calculate gas flows. For intermediate and high values of the Knudsen number, the Navier-Stokes equations break down, and the implementation of rarefied gas dynamics methods is necessary. In practical calculations usually the rarefaction parameter defined as the inverse Knudsen number, i.e.. 

Fig. IS. The collisions of gas particles with a sphere, which are taken into account in the expansion of the force on the sphere in powers of the inverse Knudsen number, (a) Collisions that are responsible for the free molecular flow force (b, c, d) dynamical events that contribute to the K correction to this value (d) represents a process where the second gas particle does not hit the sphere, but would have, had the second collision not taken place (e) represents one of the type of events that contribute to order log K .

For large times t, the exponential factor in Eq. (231), which takes the damping of the particle trajectories into account, restricts the time that each particle spends between collisions to be on the order of t the mean free time. Upon comparing (231) with (137), the expansion of the force on a sphere in a rarefied gas in powers of the inverse Knudsen number, one can see that these two expansions have a remarkably similar structure. This similarity has its source in the fact that the coefficients in the density expansion of 17/170 and in the expansion of F/Fq are determined by dynamical events of the same basic types, as may be seen by comparing Fig. 15 with Figs. 22 and 24. The dynamical events that contribute to F/Fq differ from those which contribute to 17/170 only in that in F/Fq one of the gas particles is replaced by a macroscopic object, in this case a sphere. 

The mean free path is inversely proportional to the pressure and varies from 30-100 nm at 0.1 MPa to 0.3-1 nm at 10 MPa for normal gases. As previously described, the average size of the pores of commercial catalysts varies over a wide range (1- 200 nm). Therefore, typical values of the Knudsen number are lO-4 - 102. This rough estimate shows that different flow regimes of different complexity occur in practice. 

Figure 5 a shows the variation of the inverse slip coefficient with the outlet Knudsen number considering pressure ratio of II = 1.8 and accommodation coefficient of cr = 0.93. It is observed that the IP-based model concur exactly with the experimental data [6]. This is where the analytical formula of Aubert and Colin [7] departs from the given data for Kn > 0.25. 

From the data it is clear that the Nusselt number decreases when the Knudsen number increases for a fixed Prandlt number. The rarefaction of the gas decreases the intensity of the heat transfer. When the Prandlt number increases for a fixed JCnudsen number, the Nusselt number increases this means that the convective heat transfer is enhanced for gases with larger Prandlt numbers. When the viscous dissipation increases the Nusselt number tends to increase. In fact, the increase of the viscous dissipation tends to increase the bulk temperature of the fluid. For a T boundary condition, since the Nusselt number is linked to the inverse of the difference between the wall temperature and the bulk temperature, an increase of the bulk temperature reduces this difference and increases the Nusselt number. 

Typically, the pore volume is 40 to 60% of the total volume bounded by the external surface of the catalyst, and Sp 0.50 is a reasonably good number. The range of pore radii varies from a lower limit of approximately 10 A to an upper limit slightly above 1 tim (i.e., 10" A). As illustrated below in eqnation (21-16), smaller pores correspond to a larger internal surface area per volume of catalyst, which is advantageous for converting reactants to products. However, Knudsen diffusion is restricted when the pores are too small because the mean free path of the gas, which varies inversely with gas density, is much larger than the pore diameter. 

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