Continuum flow

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. 

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]) ... 

A in cm ) For continuum flow (viscous flow) the follow/ing equations (after Prandtl) apply to air at 20 °C where pj/p = 5 ... 

In the creeping flow range, C is equal to the ratio of the terminal velocity to the terminal velocity in continuum flow. The value of C is sensitive to the nature of molecular reflections from the surface of the particle (E5). The accommodation coefficient, o-r, may be interpreted as the fraction of molecules undergoing diffuse reflection, the balance being specularly reflected. Typical values for lie between 0.8 and unity. For near-continuum flow. Basset (B9) showed that... 

Analogous to the slip velocity between gas and particle at Kn above the continuum flow range discussed in Section A above, a temperature discontinuity exists close to the surface at high Kn. Such a discontinuity represents an additional resistance to transfer. Hence, transfer rates are generally lowered by compressibility and noncontinuum effects. The temperature jump occurs over a distance 1.996kl 2 — a )/Fva k + 1) (K2, Sll) where is the thermal accommodation coefficient, interpreted as the extent to which the thermal energy of reflected molecules has adjusted to the surface temperature. 

The characterization of the flow in existing DPF materials has been assessed by experiments and macroscopic continuum flow in porous media approaches. However, when it comes to material design it is essential to employ flow simulation techniques in geometrically realistic representations of DPF porous media. Some first applications were introduced in Konstandopoulos (2003) and Muntean et al. (2003) and this line of research is especially important for the development of new filter materials, the optimization of catalyst deposition inside the porous wall and for the design of gradient-functional filter microstructures where multiple functionalities in terms of particle separation and catalyst distribution (for combined gas and particle emission control) can be exploited. 

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 ... 

Gas flow is generally divided into three types (see Table 1.4) defined by the Knudsen number. In continuum flow, whether laminar or turbulent conditions prevail depends on the Reynolds number. 

This value is close to that for viscous flow given in Table 1.4. Consequently, supersonic continuum flow will occur through the orifice. As the pressure in chamber 2 is below the critical pressure (4.9 mbar, see Example 6.17), a maximum gas throughput will result ... 

The interface used today between the atmospheric-pressure plasma and the low-pressure mass spectrometer is based on a differentially pumped two-stage interface similar to those used for molecular beam techniques [89-91]. The key to successful development of ICP-MS instruments was the use of a relatively large ( l-mm-diameter) sampling orifice so that continuum flow was attained with an unrestricted expansion of the plasma to form a free jet. When small orifices were used, a cold boundary layer formed in front of the orifice, resulting in substantial cooling of the plasma, including extensive ion-electron recombination and molecular oxide formation. The smaller orifices were also susceptible to clogging. 

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... 

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. 

Under hypersonic continuum flow conditions, a strong shock wave forms in front of the nose of the reentry vehicle. A typical reentry velocity at high altitude is 7 km/s (about Mach 25). The post-shock temperature for this Mach number, based on theoretical gas dynamics, is about 15 000 K and the peak temperature inside the shock is even higher (about 25 000 K). 

For the small values (Kn< 10 ), the flow is considered to be a continuum flow, while for large values (Kn> 10), the flow is considered to be a free-molecular flow. The range 10 [Pg.3]

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 validity of continuum flow assumption is unquestionable in solving many macroscopic heat transfer computations. However, when the flow is passed through microchannels, the continuum flow... 

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 aforementioned eonfinuum flow solutions are very reliable and widely aeknowledged by worldwide researehers. Continuum flow solutions should be used fo eompare any analyfieal slip flow resulfs. 

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