Long Knudsen diffusion

Fig. 3.1.5 Temperature dependence of the coefficient of long-range self-diffusion of ethane measured by PFG NMR in a bed of crystallites of zeolite NaX (points) and comparison with the theoretical estimate (line). The theoretical estimate is based on the sketched models of the prevailing Knudsen diffusion...

Here X>iA (r) is a Knudsen diffusivity, which we take as the limiting value for free-molecule flow in a long tube of radius r yfngstroms,... 

This equation is the usual one expressing Knudsen diffusion in a long capillary. 

KaX than in smaii-port zeolites such as NaCaA. For hydrocarbons in HaX it has been found that the coefficients of intracrystalline self-diffusion decrease with increasing concentration (concentration dependences of type I and II (6).i. By contrast, for not toe high gas phase concentrations (i.e., as long as molecular transfer in the intercrystalline space proceeds by Knudsen diffusion)... 

The geometry of the plate does not bear much relevance to real porous solid. Thus, the above equation is not of much use in diffusion of adsorption systems. To this end, we will consider the Knudsen diffusion through a long cylindrical capillary in the next section. 

It is worthwhile at this point to remind the reader that the above conclusion for Knudsen diffusion is valid as long as the pressure is low or the capillary size is very small. When the capillary size is larger or the pressure is higher, the viscous flow will become important and the flow will be resulted due to the combination of the Knudsen and viscous flow mechanisms. This will be discussed in Sections 7.5 and 7.6. 

We have addressed in the example 7.4-4 the steady state flux due to the Knudsen diffusion mechanism, but the question which is of significant interest is how long does it take for the system to response from some initial conditions to the final steady state behaviour. This is important to understand the pure diffusion time in a capillary. By pure diffusion time, we mean the diffusion time in the absence of adsorption. In the presence of adsorption, the time to approach equilibrium from some initial state is longer than the pure diffusion time due to the... 

The parameter characterising the diffusion through the medium is the Knudsen diffusivity, which could be determined from the time lag given in eq. (12.2-24) or from the short time solution (eq. 12.2-18). The long time solution for time lag is preferrable if the experimental data exhibit a linear asymptote behaviour at long time and the constant boundary conditions (12.2-4) are maintained throughout the course of the experiment. If the medium is rather impermeable and the time lag is practically too long to measure, then the application of the short time solution is the only possible choice. 

For gas-solid systems with a low gas density in which gas molecules diffuse through long narrow pores, the mean free path of the molecules is much larger than the pore diameter. In this type of diffusion, so-called Knudsen diffusion, the transport properties are essentially determined by eollisions of the gas moleeules with the pore walls rather than by collisions with other gas molecules. Based on the kinetic theory of gases, in the Knudsen approach all rebounds are assumed to be governed by the cosine law of reflection. In a straight cylindrical pore of diameter d ore, the Knudsen diffusion coefficient Dk,i for component i with molecular mass Mi is given by... 

In liquids, the mean free path is typically of the order of 10 10 m. Hence the Knudsen effect is not important (i.e., diffusing molecules collide with solvent molecules long before they typically arrive at a pore wall). However, diffusion is affected by a different mechanism, the viscous drag caused by the pore walls. This is known as the Renkin effect (Renkin, 1954). In essence, the ratio of pore diffusivity in the liquid-filled pore space and diffusivity in the free liquid, D(pore/D,free, is a function of the nondimensional parameter... 

When the pore channel dimensions are small compared to the gaseous mean-free-path, however, diffusion is of the Knudsen type. The effective diffusion coefficient is then independent of gas pressure (as long as the gas pressure does not rise to values where the mean-free-path is no longer larger than the pore dimensions) and the appropriate extrapolation formulae are... 

Diffusion rates for the H2-N2 system were measured by Rao and Smith for a cylindrical Vycor (porous-glass) pellet 0.25 in. long and 0.56 in. in diameter, at 25°C and 1 atm pressure. A constant-pressure apparatus such as that shown in Fig. 11-1 was used. The Vycor had a mean pore radius of 45 A, so that diffusion was by the Knudsen mechanism. The diffusion rates were small with respect to the flow rates of the pure gases on either side of the pellet. The average diffusion rate of hydrogen for a number of runs was 0.44 cm min (25°C, 1 atm). The porosity of the Vycor was 0.304. 

We assumed that diffusion in the gap between the two particles is the same as in the bulk. This assumption is valid as long as the gap is much larger than the mean free path for collision between gas molecules. At room temperature and normal pressure, typical free path lengths are of the order of 100 nm. In some cases, the gap width can be significantly below 100 nm. Then collisions with the walls become more likely than collision with other gas molecules. This is called Knudsen flow. It can significantly slow down the process [571, 572]. 

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