Knudsen Number Regimes

It has been argued that in the higher Knudsen number regime, the Burnett equations will allow continued application of the continuum approach. In practice, many problems have been encountered in the numerical solution and physical properties of the Burnett equations. In particular, it has been demonstrated that these equations violate the second law of thermodynamics. Work on use of the Burnett equations continues, but it appears to be unlikely that this approach will extend our computational capabilities much further into the high Knudsen number regime than that offered by the Navier-Stokes equations. 

The Navier-Stokes equations are valid when A is much smaller than the characteristic flow dimension L. When this condition is violated, the flow is no longer near equilibrium and the linear relations between stress and rate of strain and the no-slip velocity condition are no longer valid. Similarly, the linear relation between heat flux and temperature gradient and the no-jump temperature condition at a solid-fluid interface are no longer accurate when A is not much smaller than L. The different Knudsen number regimes are delineated in Fig. 2. 

The local value of Knudsen number determines the degree of rarefaction and the degree of validity of the continuum model in a particular flow. The different Knudsen number regimes depicted in Fig. 2 have been determined empirically and are therefore only approximate for a particular flow geometry. The pioneering experiments in rarefied gas dynamics were conducted by Knudsen in 1909 [24]. 

In Eig. 2 we show the nondimensionalized velocity distribution along the centerline and along the wall of the channels for the entire Knudsen number regime considered here, i.e., 0.01 < Kn < 30. We included in the... 

Gas density enters the question of interaction forces indirectly by modifying the particle s trajectory in the potential field of the matter it interacts with. There are highly approximative means of dealing with this for spheres where the Knudsen number regime is well defined. In the case of other shapes such as rods which may span several ranges of Kn, the effect can become impossibly complicated when orientation dependency of the forces is included. 

The above results show close agreement between the experimental and theoretical friction factor (solid line) in the limiting case of the continuum flow regime. The Knudsen number was varied to determine the influence of rarefaction on the friction factor with ks/H and Ma kept low. The data shows that for Kn < 0.01, the measured friction factor is accurately predicted by the incompressible value. As Kn increased above 0.01, the friction factor was seen to decrease (up to a 50% X as Kn approached 0.15). The experimental friction factor showed agreement within 5% with the first-order slip velocity model. 

The applicability of the two different models of gas flow is generally judged from the gas flow regimes according to the magnitude of the local Knudsen number, Kn, defined as... 

The principle quantity determining the flow regime of gases and deviations from the standard continuum description is the Knudsen number, defined as... 

Based on the Knudsen number, four different flow regimes can be distinguished [5] ... 

In the first two cases the Navier-Stokes equation can be applied, in the second case with modified boundary conditions. The computationally most difficult case is the transition flow regime, which, however, might be encountered in micro-reactor systems. Clearly, the defined ranges of Knudsen numbers are not rigid rather they vary from case to case. However, the numbers given above are guidelines applicable to many situations encoimtered in practice. 

In Figure 2.2 DSMC results of Karniadakis and Beskok [2] and results obtained with the linearized Boltzmann equation are compared for channel flow in the transition regime. The velocity profiles at two different Knudsen numbers are shown. Apparently, the two results match very well. The fact that the velocity does not reach a zero value at the channel walls (Y = 0 and Y = 1) indicates the velocity slip due to rarefaction which increases at higher Knudsen numbers. 

The vapor-layer model developed in Section IV.A.2 is based on the continuum assumption of the vapor flow. This assumption, however, needs to be modified by considering the kinetic slip at the boundary when the Knudsen number of the vapor is larger than 0.01 (Bird, 1976). With the assumption that the thickness of the vapor layer is much smaller than the radius of the droplet, the reduced continuity and momentum equations for incompressible vapor flows in the symmetrical coordinates ( ,2) are given as Eqs. (43) and (47). When the Knudsen number of the vapor flow is between 0.01 and 0.1, the flow is in the slip regime. In this regime, the flow can still be considered as a continuum at several mean free paths distance from the boundary, but an effective slip velocity needs to be used to describe the molecular interaction between the gas molecules and the boundary. Based on the simple kinetic analysis of vapor molecules near the interface (Harvie and Fletcher, 2001c), the boundary conditions of the vapor flow at the solid surface can be given by... 

This applies to diffusion to a planar surface in the continuous regime, when the Knudsen number is small, and is the expression for gas diffusion most... 

The interpolation between these limiting flow regimes is in terms of a Knudsen number-related weighting factor ... 

CVD reactors operate at sufficiently high pressures and large characteristic dimensions (e.g., wafer spacing) such that Kn (Knudsen number) << 1, and a continuum description is appropriate. Exceptions are the recent vacuum CVD processes for Si (22, 23) and compound semiconductors (156, 157, 169) that work in the transition to the free molecular flow regime, that is, Kn > 1. Figure 7 gives an example of SiH4 trajectories in nearly free molecular flow (Kn 10) in a very low pressure CVD system for silicon epitaxy that is similar to that described by Meyerson et al. (22, 23 Meyerson and Jensen, manuscript in preparation). Wall collisions dominate, and be-... 

The relative frequency of the intermolecular collisions and collisions between molecules and the pore wall can be characterized by the ratio of the length of the mean free path A and the equivalent pore diameter dt = 2rf This ratio determines the flow regime and is called the Knudsen number Kn ... 

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 20.10 Gas flow regimes in a tube as a function of system dimensions, pressure, and the range of Knudsen numbers corresponding to gas flow regimes (summarized in Table 20.1). (From Ohring, M., The Materials Science of Thin Films, Academic Press, San Diego, 1992.)... 

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