Knudsen’s law

Knox-Out2FM Knudsen cells Knudsen diffusion Knudsen s law Koavone [86115-11-9]... 

Ba.rrier Flow. An ideal separation barrier is one that permits flow only by effusion, as is the case when the diameter of the pores in the barrier is sufficiently small compared to the mean free path of the gas molecules. If the pores in the barrier are treated as a collection of straight circular capillaries, the rate of effusion through the barrier is governed by Knudsen s law (eq. 46) ... 

Before being used for flow experiments, each disk was tested with helium for acceptability. The rate of helium flow at 10-cm. pressure gradient across the disk was measured at several mean pressures between 5 and 50 cm. Unless the mass of gas passing per unit time was constant—i.e., the flow through the disk followed Knudsen s law—the disk was rejected. About one disk in ten passed this test the others showed increased flow at high pressures. 

Pure gases. A typical diffusion barrier consists of a thin sheet of material perforated by a very large number of small holes of nearly uniform diameter. If the diameter of the holes and the thickness of the sheet are smaller than the mean free path of UFg at the pressure upstream of the barrier, individual molecules of UFg will flow through the holes without colliding with other molecules in what is known as molecular flow. The rate of molecular flow through a circular capillary is given by Knudsen s law [K3] ... 

D b eff, AB varies inversely as p, and approximately directly as (see Chap. 2). If, however, the pore diameter and the gas pressure are such that the molecular mean free path is relatively large, d/ less than about 0.2, the rate of diffusion is governed by the collisions of the gas molecules with the pore walls and follows Knudsen s law. Since molecular collisions are unimportant under these conditions, each gas diffuses independently. In a straight circular pore of diameter d and length /... 

If conditions of pore diameter and pressure occur for which Knudsen flow prevails (d/ < 0,2), the flow will be described by Knudsen s law, Eqs. (4.17) to (4.20). There will be of course a range of conditions for a transition from hydrodynamic to Knudsen flow. If the gas is a mixture with different compositions and different total pressure on either side of the porous solid, the flow may be a combination of hydrodynamic, Knudsen, and diffusive. Younquist [20] reviews these problems. 

The Knudsen equation can only be applied to the viscous laminar flow regime, the molecular flow regime or the transition flow regime. For predominantly laminar flow, the calculated contribution for molecular flow becomes negligible, and similarly for predominantly molecular flow, the calculated contribution for laminar flow becomes negligible. The first part of this equation represents the viscous laminar flow component, Q, which is derived from Poiseuille s law for laminar flow, and the second part represents the molecular flow component, Q, which is derived from Knudsen s law for free molecular flow. 

The drag coefficients for disks (flat side perpendicular to the direction of motion) and for cylinders (infinite length with axis perpendicular to the direclion of motion) are given in Fig. 6-57 as a Function of Reynolds number. The effect of length-to-diameter ratio for cylinders in the Newton s law region is reported by Knudsen and Katz Fluid Mechanics and Heat Transfer, McGraw-Hill, New York, 1958). 

Thus, in an isothermal system, the mass flow rate depends on the difference in pressures of the gas across the orifice and does not depend upon the thickness of the plate. One may define an area-normalized resistance, R, for mass transfer through the orifice using a generalization of Ohm s law, i.e., Resistance = force/ flux. For Knudsen flow, the force is the pressure difference (analogous to voltage difference in Ohm s law) and the flux is the mass flow per unit area of the hole (analogous to the electrical current density in Ohm s law). Thus, we have... 

The main emphasis in this chapter is on the use of membranes for separations in liquid systems. As discussed by Koros and Chern(30) and Kesting and Fritzsche(31), gas mixtures may also be separated by membranes and both porous and non-porous membranes may be used. In the former case, Knudsen flow can result in separation, though the effect is relatively small. Much better separation is achieved with non-porous polymer membranes where the transport mechanism is based on sorption and diffusion. As for reverse osmosis and pervaporation, the transport equations for gas permeation through dense polymer membranes are based on Fick s Law, material transport being a function of the partial pressure difference across the membrane. 

In connection with multiphase diffusion another poorly understood topic should be mentioned—namely, the diffusion through porous media. This topic is of importance in connection with the drying of solids, the diffusion in catalyst pellets, and the recovery of petroleum. It is quite common to use Fick s laws to describe diffusion through porous media fJ14). However, the mass transfer is possibly taking place partly by gaseous diffusion and partially by liquid-phase diffusion along the surface of the capillary tubes if the pores are sufficiently small, Knudsen gas flow may prevail (W7, Bl). 

Diffusivity in Pores and Fick s Laws Diffusion in Gas-Filled Pores Knudsen Effect Diffusion in Liquid-Filled Pores Renkin Effect Diffusion in the Unsaturated Zone of Soils... 

Fick s law is derived only for a binary mixture and then accounts for the interaction only between two species (the solvent and the solute). When the concentration of one species is much higher than the others (dilute mixture), Fick s law can still describe the molecular diffusion if the binary diffusion coefficient is replaced with an appropriate diffusion coefficient describing the diffusion of species i in the gas mixture (ordinary and, eventually, Knudsen, see below). However, the concentration of the different species may be such that all the species in the solution interact each other. When the Maxwell-Stefan expression is used, the diffusion of... 

Knudsen diffusion can be taken into account also when Fick s law or the Maxwell-Stefan rate of mass transport are employed [34, 59] by combining the molecular diffusion, DLm, and the Knudsen diffusion as follows ... 

This result was first observed experimentally by Graham and is called Graham s law of diffusion. Knudsen diffusion membranes have been used to separate gas isotopes that are difficult to separate by other methods, for example tritium from... 

The mean free path, /, of the C02 molecules at the temperatures and pressures of the permeation experiment are by far smaller than the membrane pore size, d, that is, d, /.. Then, Knudsen flow is not possible since the determining process is gaseous laminar flow through the membrane pores [18]. It is therefore feasible to apply Darcy s law for gaseous laminar flow (Equations 10.19 through 10.23). 

Most studies have assumed equation (3) to apply, so that equation (1) takes the form of Fick s law, with the composite (effective) diffusion taking account of both bulk and Knudsen diffusion. For the stealy state operation of the Wicke-Kallenbach cell, this can often be a reasonable assumption. Smith et al (18) have also used this description of the transport processes to analyze the situation when a pulse of the trace component is applied at z=0 and the concentration is monitored at z=L. For sufficiently high flow rates of the carrier gas, the first moment of the response curve to a pulse input is ... 

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... 

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