Flow regime continuum

The Boltzmann equation describes the behavior of gas flows in non-equilibrium due to molecular motion. It is applicable for all mean free paths X between the molecules, that is, for all Knudsen numbers Kn (= X/L, where L is a characteristic dimension in the flow). Thus it can be employed to calculate the flow fields in a variety of applications ranging from hypersonic flow past a space shuttle to flow in microdevices, where all the flow regimes - continuum (Kn 0), transitional (Kn 0(1)) and rarefied (Kn 1) - may be encountered. The classical Boltzmann equation (CBE) describes the flow of... 

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

In order to obtain a qualitative view of how the transition regime differs from the continuum flow or the slip flow regime, it is instructive to consider a system close to thermodynamic equilibrium. In such a system, small deviations from the equilibrium state, described by thermodynamic forces X, cause thermodynamic fluxes J- which are linear functions of the (see, e.g., [15]) ... 

Henderson 575 presented a set of new correlations for drag coefficient of a single sphere in continuum and rarefied flows (Table 5.1). These correlations simplify in the limit to certain equations derived from theory and offer significantly improved agreement with experimental data. The flow regimes covered include continuum, slip, transition, and molecular flows at Mach numbers up to 6 and at Reynolds numbers up to the laminar-turbulent transition. The effect on drag of temperature difference between a sphere and gas is also incorporated. 

In the continuum flow regime the mean free path X is much smaller than the orifice diameter, and the exit pressure from the reactor is orders of magnitude greater than the backing pressure in the ion source vacuum chamber. The Mach number is unity that is, the flow speed, t , through the orifice is sonic ... 

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-... 

Three general flow regimes may be anticipated for the flow over a flat plate shown in Fig. 12 12. First, the continuum flow region is encountered when the mean free path A is very small in comparison with a characteristic body dimension. This is the convection heat-transfer situation analyzed in preceding chapters. At lo wer gas pressures, when A L, the flow seems to slip along the surface and u 4= 0 at y = 0. This situation is appropriately called slip flow. At still lower densities, all momentum and energy exchange is the result of... 

Fig. 12-12 Three types of flow regimes for a flat plate (a) continuum fiow. ( >) slip flow, (c) free-molecule flow.

When Kn < 0.2, the particles are described as being in the continuum regime. A theoretical description of particle motion in this flow regime... 

In addition to the reactor scale, which is measured in meters, vapor-phase mass transport effects can also be important in CVD at a much smaller scale, one measured in micrometers. This is often referred to as the feature scale . On this scale, the gas is generally in the transition or molecular flow regimes, rather than continuum flow. Mass transport on this scale plays an important role in the CVI processes discussed in Chapter 6. These phenomena are also important in CVD involving high-aspect ratio features, which can occur unintentionally in some growth morphologies and deliberately in microelectronics applications. 

In the rarefied flow regime, all of these collisional mechanisms can also occur. Due to the reduced number of intermolecular collisions, however, all of the energy distributions may be in a state of nonequilibrium. This means that the rates of the kinetic processes cannot be described by the temperature-dependent forms usually employed in continuum models. Instead, models are required that describe these processes at the individual collision level. 

According to reference [1] four flow regimes for gases exist continuum flow (0iKn<0.001), slip flow (0.00l Kn<0.1), transition flow (0.l SKn<10), and free molecular flow (lOsKn). Continuum equations are valid for Kn- >0, while kinetic theory is applicable for Kn>8. Slip flow occurs when gases are at low pressure or in micro conduits. The gas slip at the surface, while in continuum flow at the surface it is immobilized. 

The continuum flow assumption will only be valid when Kn < 10. As Kn increases, the flow enters the slip flow regime (10 < Kn < 10 ), transition flow regime (10 < Kn < 10), and eventually the free-moleeular flow regime (Kn > 10). These four regimes are illustrated in Figure 1. 

The detailed flow regime flow map for hydrogen gas is shown in Figure 2.18. For the CVI techniques discussed in Chapter 5, the process is usually performed at a pressure of 10 to 30 kPa. The gas flow can be treated as continuum flow if the characteristic length is more than 1 mm. 

By use of the typical numerical value for the mean free path given above for a gas at temperature iOOK and pressure 101325Pa, this condition indicates that the continuum assumption is valid provided that the characteristic dimension of the apparatus is L > 0.001cm. Nevertheless, it is noted that at low gas pressures, say lOPa, the mean free path is increased and might be comparable with the characteristic dimensions of the apparatus. In particular, for low pressures and small characteristic dimensions we might enter a flow regime where the continuum assumption cannot be justifled. 

For approximately one-half of each run, observed cryosurface pumping speeds remained near the value determined by other experimenters in the free-molecular flow regime even though pressures were in the continuum flow regime. 

The anisotropic continuum approach to losses in multifilament conductors was first conceived by Carr, who developed the model assuming that the inductor is a continuum material with anisotropic resistivity. He applied this approach to the special case of losses in cylindrical conductors for applied transverse sinusoidal fields in the absence of transport current [ ]. Those losses resulting from pJ in the conductor are classified as eddy current or saturation hysteresis losses, depending upon the level of /. Eddy current losses result from J below Jc, with the implicit assumption of rapidly rising resistivity in the flux-flow regime with currents saturated at Jc. The magnetization loss for the continuum is essentially the magnetic hysteresis loss for the filaments times the fraction of the composite occupied by the filaments. 

The standard convection-diffusion equations rely on the assumption of local thermal equilibrium. For gas flow, this means that a Maxwell-Boltzmann distribution is assumed for the velocity of the particles in the frame-of-reference comoving with the fluid. When gas flow in microreactors at high temperature or low pressure is considered, this assumption may break down. The principle quantity determining the flow regime of gases and deviations from the standard continuum description is the Knudsen number, defined as... 

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