Diffusion Knudsen flow

Transfer mechanisms involved in SC CO2 permeation through such porous membranes can be convection (Poiseuille law), diffusion (Knudsen flow), and surface membrane interaction by adsorption, capillary condensation, etc. [11]. Mechanisms have been specifically investigated for nanofiltration and zeolite membranes. With a nanofilter presenting a pore diameter of about 1 nm, Sarrade [11] mentioned a Poiseuille flow associated with an irreversible CO2 adsorption on the micropore wall. Transfer... 

Open pores, which are connected with each other and also with the surface of the crystal. The form and distribution of the pores are most important for transport phenomena such as diffusion, Knudsen flow, and surface diffusion. Because of the technological importance of this field, it possesses an extensive literature, especially on procedures for determining porosity. We can only make mention of this literature here [12]. 

A Barrier Efficiency Eactor. In practice, diffusion plant barriers do not behave ideally that is, a portion of the flow through the barrier is bulk or Poiseuihe flow which is of a nonseparative nature. In addition, at finite pressure the Knudsen flow (25) is not separative to the ideal extent, that is, (M /Afg) . Instead, the degree of separation associated with the Knudsen flow is less separative by an amount that depends on the pressure of operation. To a first approximation, the barrier efficiency is equal to the Knudsen flow multiphed by a pressure-dependent term associated with its degree of separation, divided by the total flow. 

FIGt 22-48 Transport mechanisms for separation membranes a) Viscous flow, used in UF and MF. No separation achieved in RO, NF, ED, GAS, or PY (h) Knudsen flow used in some gas membranes. Pore diameter < mean free path, (c) Ultramicroporoiis membrane—precise pore diameter used in gas separation, (d) Solution-diffusion used in gas, RO, PY Molecule dissolves in the membrane and diffuses through. Not shown Electro-dialysis membranes and metallic membranes for hydrogen. 

Process Description Gas-separation membranes separate gases from other gases. Some gas filters, which remove hquids or sohds from gases, are microfiltration membranes. Gas membranes generally work because individual gases differ in their solubility and diffusivity through nonporous polymers. A few membranes operate by sieving, Knudsen flow, or chemical complexation. 

The two BCs of the TAP reactor model (1) the reactor inlet BC of the idealization of the pulse input to tiie delta function and (2) the assumption of an infinitely large pumping speed at the reactor outlet BC, are discussed. Gleaves et al. [1] first gave a TAP reactor model for extracting rate parameters, which was extended by Zou et al. [6] and Constales et al. [7]. The reactor equation used here is an equivalent form fi om Wang et al. [8] that is written to be also applicable to reactors with a variable cross-sectional area and diffusivity. The reactor model is based on Knudsen flow in a tube, and the reactor equation is the diffusion equation ... 

Molecular diffusion and/or Knudsen flow of reactants from the exterior surface of the catalyst particle into the interior pore structure. 

The symbols refer to a single component. Since molecular collisions are rare events in Knudsen flow, flow and diffusion are synonymous and each component of a mixture behaves as if it alone were present. Numerical values of the Knudsen diffusivity for molecules of ordinary weight at ordinary temperatures range from 0.01 cm2/sec for pores with a radius of 10 A up to about 10 cm2/sec for pores with 10,000 A radii. 

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. 

Of these three mechanisms, i.e. molecular diffusion, laminar flow and Knudsen diffusion, only two are important in pressure-driven separations. These are laminar flow and Knudsen diffusion. These can be qualitatively understood as follows. If the molecules "see each other much more than they see the pore wall (which means the mean free path of the molecules is much smaller than the mean pore radius), laminar ow a molecular diffusion are important. The laminar flow is much larger, howcver, nd thelhdlecular flow can be neglected (Present and de Bethune 1949). If the molecules see the pore wall much more than they see each other, only Knudsen diffusion will occur. Thus, the molecular diffusion can be neglected in all circumstances. From now on it will be assumed, that only laminar flow and Knudsen diffusion occur. 

In the region of Knudsen flow the effective diffusivity DeK for the porous solid may be computed in a similar way to the effective diffusivity in the region of molecular flow, i.e. Dk is simply multiplied by the geometric factor. 

Many heterogeneous reactions give rise to an increase or decrease in the total number of moles present in the porous solid due to the reaction stoichiometry. In such cases there will be a pressure difference between the interior and exterior of the particle and forced flow occurs. When the mean free path of the reacting molecules is large compared with the pore diameter, forced flow is indistinguishable from Knudsen flow and is not affected by pressure differentials. When, however, the mean free path is small compared with the pore diameter and a pressure difference exists across the pore, forced flow (Poiseuille flow see Volume 1, Chapter 3) resulting from this pressure difference will be superimposed on molecular flow. The diffusion coefficient Dp for forced flow depends on the square of the pore radius and on the total pressure difference AP ... 

We should add a note of caution here, however, for in the Knudsen flow region De is proportional to the pore radius. When the pores are sufficiently small for Knudsen diffusion to occur then the selectivity will also be influenced by pore size. Maximum selectivity would be obtained for small particles which contain large diameter pores. 

Diffusion in macropores occurs mainly by the combined effects of bulk molecular diffusion (as in the free fluid) and Knudsen flow, with generally smaller contributions from other mechanisms such as surface diffusion and Poiseuille flow. Knudsen flow, which has the characteristics of a diffusive process, occurs because molecules striking the pore wall are instantaneously adsorbed and re-emitted in a random direction. The relative importance of bulk and Knudsen diffusion depends on the relative frequency of molecule-molecule and molecule-wall collisions, which in turn depends on the ratio of the mean free path to pore diameter. Thus Knudsen flow becomes dominant in small pores at low pressures, while in larger pores and at higher pressures diffusion occurs mainly by the molecular mechanism. Since the mechanism of diffusion may well be different at different pressures, one must be cautious about extrapolating from experimental diffusivity data, obtained at low pressures, to the high pressures commonly employed in industrial processes. 

FIGURE 5.27 Transport mechanisms in porous media molecular or gaseous flow, (a) Knudsen flow, (b) surface diffusion, (c) multilayer diffusion, (d) capillary condensation, and (e) configurational diffusion. 

Then, if the formerly discussed conditions for Knudsen diffusion are satisfied for a mesopore, that is, a pore of diameter in the range between 20-50nm, the diffusion coefficient for the Knudsen flow in a straight cylindrical mesopore is described by Equation 5.94. 

Hydrogen, deuterium, neon, argon, and methane flow through saran charcoal by both Knudsen and surface flow. The latter is effected by the adsorbed molecules sliding from site to site across the surface. This is equivalent to a two-dimensional Knudsen flow where the adsorption site acts as the wall of the three-dimensional case, and a slide across the surface is the same as a flight across the pore. The activation energy for surface diffusion is 75 to 80% of the heat of adsorption. It is possible to calculate theoretically the relative contribution of each mechanism, while comparison with He, which does not adsorb, permits its experimental determination. The efficiency of surface flow is the ratio of the measured to the calculated value this decreases as the size of the molecule increases, being 80% for Ne and 12% for CH4. 

The other gases used were examined in the same way as helium. In all cases the flow exceeded the Knudsen flow as predicted from the helium data. Figure 8 gives typical data the flow in excess of the calculated Knudsen flow is attributed to surface diffusion of the adsorbed molecules. 

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... 

In common practice the rates of the countercurrent flows of the two gases A and B through a pellet are measured. Mixtures of different compositions are passed along the opposite sides of the pellet. For measurements of the effective diffusivity, no pressure gradient is allowed across the pellet unless all pores are sufficiently small and only Knudsen flow occurs. The flux of the component A at a uniform total pressure is determined by the equation... 

For a given pressure gradient across a porous medium, the mass balance equation can be described as follows, provided that Knudsen diffusion, viscous flow and surface diffusion are additive to the total flux. 

The Knudsen diffusion, viscous flow and surface diffiision for strongly adsorbing vapors are well described at low range of pressures in this paper. The collision-reflection flictor for Knudsen diffusion is found to be not constant but exhibit a modest increase with an increase in pressure. The dependence of the Knudsen diflusion for n-hexane on pressure is stronger than that of the other vapors. Moreover the activation energy for the surflice diffiision of ra-hexane exhibits a faster decreasing behavior in comparison with the others. Conclusively, the reason for the minimum appearance in the total permeability of ra-hexane can be attributed by the interplay between the Knudsen diflusion and surface diffusion. 

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