Knudsen flow modified

In most cases, only the Knudsen flow, slip flow, and viscous flow are taken into account - especially at high temperature. The permeation flux expressions, Eqs (2.2)-(2.4), can be modified to account for the number of capillaries per unit volume (porosity e) and the complexities of the structure (tortuosity t). For example, the viscous flux may be expressed as... 

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 we presented the permeability coefficient K of oxygen as a function of the mean gas pressure experimentally obtained for a sample of porous material from acetylene black modified with 35% PTFE. The experimental linear dependence is obtained. The intercept with the abscissa corresponds to the Knudsen term DiK. The value obtained is 2,89.1 O 2 cm2/s. The slope of the straight line is small, so that the ratio K,/ Dik at mean gas pressure 1 atm. is small ( 0.1) which means that the gas flow is predominantly achieved by Knudsen diffusion and the viscous flow is quite negligible. At normal conditions (1 atm, 25°C) the mean free path of the air molecules (X a 100 nm) is greater than the mean pore radii in the hydrophobic material (r 20 nm), so that the condition (X r) for the Knudsen-diffusion mechanism of gas transport is fulfilled. 

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... 

We have developed several new measurement techniques ideally suited to such conditions. The first of these techniques is a High Pressure Sampling Mass Spectrometric method for the spatial and temporal analysis of flames containing inorganic additives (6, 7). The second method, known as Transpiration Mass Spectrometry (TMS) (8), allows for the analysis of bulk heterogeneous systems over a wide range of temperature, pressure and controlled gas composition. In addition, the now classical technique of Knudsen Effusion Mass Spectrometry (KMS) has been modified to allow external control of ambient gases in the reaction cell (9). Supplementary to these methods are the application, in our laboratory, of classical and novel optical spectroscopic methods for in situ measurement of temperature, flow and certain simple species concentration profiles (7). In combination, these measurement tools allow for a detailed fundamental examination of the vaporization and transport mechanisms of coal mineral components in a coal conversion or combustion environment. 

More reeently, [26] has eonfirmed the need to include the second order slip condition at higher Kn number values. Their work was both theoretical and experimental using nitrogen and helium in a silicon channels. They used the second order slip approximation to obtain the equation for the volumetric flow rate and related it to the ratio of inlet to outlet pressure. It was shown that when using the Navier-Stokes equation, the boundary conditions must be modified to include second order slip terms as the Knudsen number increases. They also studied in depth the accommodation coefficient Fv and verified the need for further study. It was shown that as the Knudsen number increases, the momentum accommodation value deviates further and further from unity for instance Kn -0.5 yields Fv 0.8 for helium. The values found for nitrogen were quite similar. The measurements agreed with past studies such as [11] for lower Kn. 

The membrane constant a is the permeability divided by the membrane thickness evaluated at the reference pressure. The physical significance of a is that it represents the proportionality constant between flux and pressure drop at the reference pressure. The parameter b indicates the extent to which Poiseuille flow contributes to the permeability and lies between 0 and 1. Equation 19.30 is know as Knudsen-Poiseuille model, where is the dimensionless pressure defined as Pavg/ ref. the Pref is chosen in such a way that P becomes close to unity for the range of application of the equation. If a different pressure is assumed, Equation 19.30 may be modified retaining its original form, but having different values for the parameters a and b with the reference pressure as a new P close to unity. In such a case the new parameters a and b can be written as [59]... 

It can be seen from Fig. 1 that gas flows in micron size channels are typically relevant to the slip flow regime, at any rate for usual pressure and temperature conditions. For lower sizes, i.e., for Knudsen numbers higher than 10 , the slip flow regime could remain valid, provided that classical velocity slip and temperature jump boundary conditions are modified (taking into account higher-order terms as explained below) and/or that Navier—Stokes equations are extended to more general sets of conservation equatiOTis, such as the quasi-gasodynamic (QGD), the quasi-hydrodynamic (QHD), or the Burnett equatiOTis [3]. 

We determine the leakage of the gases and via the Knudsen equation, which is the recommended approach [224], The penetration of the gas through small cavities depends on the thermodynamic properties of the gas, flow regimes and routes. The calculation of the ideal capillary diameter is used. It is possible to use the following modified Knudsen equation for the applicable leakage range (10 to 10 Pam s ) ... 

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