Knudsen regimes

Knudsen regime Poiseuille flow regime gas phase surface phase... 

Shindo, Y., T. Hakuta, H. Yoshitome and H. Inoue. 1983. Gas diffusion in microporous media in Knudsen regime. J. Chem. Eng. Japan 16(2) 120-26. 

It is useful to examine the consequences of a closed ion source on kinetics measurements. We approach this with a simple mathematical model from which it is possible to make quantitative estimates of the distortion of concentration-time curves due to the ion source residence time. The ion source pressure is normally low enough that flow through it is in the Knudsen regime where all collisions are with the walls, backmixing is complete, and the source can be treated as a continuous stirred tank reactor (CSTR). The isothermal mole balance with a first-order reaction occurring in the source can be written as... 

Effective diffusivity in Knudsen regime Effective diffusivity in molecular regime Knudsen diffusion coefficient Diffusion coefficient for forced flow Effective diffusivity based on concentration expressed as Y Dispersion coefficient in longitudinal direction based on concentration expressed as Y Radial dispersion coefficient based on concentration expressed as Y Tube diameter Particle diameter... 

Following the procedure developed by Cha and McCoy [31], the relationship between k in the Knudsen regime and k° for a polyatomic gas may be written as ... 

In the Knudsen regime, the rate at which momentum is transferred to the pore walls surpasses the transfer of momentum between diffusing molecules. The rate at which molecules collide with a unit area of the pore wall is [2]... 

In microporous materials where Knudsen diffusion prevails, De cannot be calculated by solving Fick s law. The use of a discrete particle simulation method such as dynamic MC is appropriate in such cases (Coppens and Malek, 2003 Zalc et al., 2003, 2004). In the Knudsen regime, relatively few gas molecules collide with each other compared with the number of collisions between molecules and pore walls. One of the fundamental assumptions of the Knudsen diffusion is that the direction in which a molecule rebounds from a pore wall is independent of the direction in which it approaches the wall, and is governed by the cosine law the probability d.v that a molecule leaves the surface in the solid angle dm forming an angle 0 with the normal to the surface is... 

The approach based on the effective diffusivity concept is justified in the region of Knudsen flow, where DeA = DA t + DAsurf. Equation 3.25 with a constant effective diffusivity also follows from the more general DGM equations in the limiting case of dilute mixtures with one species (B) in considerable excess and a negligible pressure gradient. In this case the other species diffuse independently, as in a Knudsen regime, but with the effective diffusion coefficient governed by the equation... 

We derive a numerical model for the effect of a spatial temperature gradient on the local equilibrium of a chemical reaction in a low--density gas (Knudsen regime). The gas consists of two constituents and the chemical reaction is assumed to take place at the walls of the container. The numerical results are compared with experimental results on the equilibrium 2Na - Na2. From the comparison it follows that the chemical accommodation coefficient for a Na2 wall collision is essentially equal to 1. 

Using the exact solution for an infinitely long tube of diameter /, in the Knudsen regime, Dj = / < ,/,> and substituting t=sl. Equation (7) obtains the dimensionless form ... 

Many reactions taking place within catalyst or absorbent pellets in industrial plants are diffusion-limited. Under the typical operating conditions for many absorbents, diffusion of gases into the porous solid occurs in the Knudsen regime. In such circumstances the rate of gas pick-up of these materials is strongly dependent on the pore structure. The pore structure for absorbent pellets that will deliver the most efficient operation of an absorbent bed requires a pervasive system of macropores which provide rapid transport of the gas flux into the centre of the pellet. A network of ramified mesopores branching off the macropores then provides extensive surface area for absorption of gas molecules. Therefore, when manufacturing an absorbent it is necessary to be able to determine the spatial distribution of the macropore network in a product to ensure that the pore structure is the most appropriate for the peirticular duty for which it is intended. 

Because of the principal absence of convective flow in VTC, it is not easy to derive the basic relations by analogy with the treatment of gas-solid chromatography processes. Help is offered by a concept of vacuum physics related to Knudsen regime - the conductance Cvc of a tube with the length lc. The quantity has dimension of the flow rate ... 

Equation (9.7) shows that the permeation of a single gas in the Knudsen regime is ... 

If the conditions are such that the mesoporous support is in the Knudsen regime, and so has some separation properties, the separation factor can be enhanced when the feed is applied from the support side. In this case the gas composition at the interface between support and separation layer is enriched already somewhat. This effect is reported by Keizer et al. [20] and could be described by the empirical relation ... 

Diffusion of A within the porous pellet takes place. If the pores arc very large this may be the normal type of molecular diffusion, but if the pore radius is smaller than the mean free path, a molecule will hit the pore wall more often than it hits its fellows, and this is the Knudsen regime of diffusion. Both types of diffusion can be described by Fick s law in which the flux is proportional to the concentration gradient, and if the diffusion coefficient is not in some sense large there may be large variations in the concentration of A within the pellet. Let r denote position within the catalyst particle then the concentration of A within the particle is a(r), a function of that position, and obeys the partial differential equation for diffusion with a(r) = as when r is a position on the exterior surface of the particle. Clearly, this is a complicated matter and we shall seek ways of simplifying it in Sec. 

Anderson et al. (1996) used Eq. [14], in conjunction with digitized images of thin sections, to investigate the influence of pore space geometry on diffusion in soil systems. Giona et al. (1996) applied renormalization analysis to study diffusion and convection on fractal media. Coppens (1997), Santra et al. (1997), and Levitz (1998) have studied the effects of geometrical confinement on diffusion in the Knudsen regime, in which particle collisions with a fractal internal surface dominate over particle-particle collisions. 

In this chapter we develop a unified flow model that predicts the velocity profiles, and mass flowrate in two-dimensional channels and ducts in the entire Knudsen regime. Our approach is divided into two main steps First, we will analyze the nondimensional velocity profile to identify the shape of the velocity distribution. Then, we will obtain the magnitude of the average velocity, and hence, obtain a prediction for the flowrate. 

From the DSMC results and solutions of the linearized Boltzmann equation, it is evident that the velocity profiles in pipes, channels and ducts remain approximately parabolic for a large range of Knudsen number. This is also consistent with the analysis of the Navier-Stokes and Burnett equations in long channels, as documented in Ref [1]. Based on this observation, we model the velocity profile as parabolic in the entire Knudsen regime, with a consistent slip condition. We write the dimensional form for velocity distribution in a channel of height h. 

We developed a unified flow model that can accurately predict the volumetric flowrate, velocity profile, and pressure distribution in the entire Knudsen regime for rectangular ducts. The new model is based on the hypothesis that the velocity distribution remains parabolic in the transition flow regime, which is supported by the asymptotic analysis of the Burnett equations [1]. The general velocity slip boundary condition and the rarefaction correction factor are the two primary components of this unified model. 

The general slip boundary condition gives the correct nondimensional velocity profile, where the normalization is obtained using the local channel averaged velocity. This eliminates the flowrate dependence in modeling the velocity profile. For channel flows, we obtain b = - m the slip flow regime. Evidence based on comparisons of the model with the DSMC and Boltzmann solutions shows that b = - is valid in the entire Knudsen regime. 

Knudsen regime, where the pore diameter is much less than the gas-phase mean free path, equation 8b. 

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