Diffusion coefficient micropore diffusivity

Here L is the thickness of the porous septum and jS the length of each dead-end micropore, the effective binary bulk diffusion coefficient... 

Knudsen diffusion coefficient for the test gas in a micropore. represents the total void fraction and c that part of of the void fraction... 

The first thing to notice about these results is that the influence of the micropores reduces the effective diffusion coefficient below the value of the bulk diffusion coefficient for the macropore system. This is also clear in general from the forms of equations (10.44) and (10.48). As increases from zero, corresponding to the introduction of micropores, the variance of the response pulse Increases, and this corresponds to a reduction in the effective diffusion coefficient. The second important point is that the influence of the micropores on the results is quite small-Indeed it seems unlikely that measurements of this type will be able to realize their promise to provide information about diffusion in dead-end pores. 

Understanding the adsorption, diffusivities and transport limitations of hydrocarbons inside zeolites is important for tailoring zeolites for desired applications. Knowledge about diffusion coefficients of hydrocarbons inside the micropores of zeolites is important in discriminating whether the transport process is micropore or macropore controlled. For example, if the diffusion rate is slow inside zeolite micropores, one can modify the post-synthesis treatment of zeolites such as calcination, steaming or acid leaching to create mesopores to enhance intracrystalline diffusion rates [223]. The connectivity of micro- and mesopores then becomes an... 

Pore diffusion can be increased by choosing a catalyst with the proper geometry, in particular the pellet size and pore structure. Catalyst size is obvious (r if pore diffusion limited for the same total surface area). The diameter of pores can have a marked influence on r) because the diffusion coefficient of the reactant Da witl be a function of dp if molecular flow in the pore dominates. Porous catalysts are frequently designed to have different distributions of pore diameters, sometimes with macropores to promote diffusion into the core of the catalyst and micropores to provide a high total area. 

The concept of transport resistances localized in the outermost regions of NS crystals was introduced in order to explain the differences between intracrystalline self-diffusion coefficients obtained by n.m.r methods and diffusion coefficients derived from non-equilibrium experiments based on the assumption that Intracrystalline transport is rate-limiting. This concept has been discussed during the past decade, cf. the pioneering work [79-81] and the reviews [2,7,8,23,32,82]. Nowadays, one can state that surface barriers do not occur necessarily in sorption uptake by NS crystals, but they may occur if the cross-sections of the sorbing molecular species and the micropore openings become comparable. For indication of their significance, careful analysis of... 

The development of mixture sorption kinetics becomes increasingly Important since a number of purification and separation processes involves sorption at the condition of thermodynamic non-equilibrium. For their optimization, the kinetics of multicomponent sorption are to be modelled and the rate parameters have to be identified. Especially, in microporous sorbents, due to the high density of the sorption phase and, therefore, the mutual Influences of sorbing species, a knowledge of the matrix of diffusion coefficients is needed [6]. The complexity of the phenomena demands combined experimental and theoretical research. Actual directions of the development in this field are as follows ... 

The case of transport through microporous membranes is different from that of macroporous membranes in that the pore size approaches the size of the diffusing solute. Various theories have been proposed to account for this effect. As reviewed by Peppas and Meadows [141], the earliest treatment of transport in microporous membranes was given by Faxen in 1923. In this analysis, Faxen related a normalized diffusion coefficient to a parameter, X, which was the ratio of the solute radius to the pore radius... 

Figure 19.17 Spherical macroparticle with radius ra consisting of an aggregate of microparticles separated by micropores filled with water. A chemical with constant concentration C° diffuses into the pore volume of the macroparticle. The local dissolved pore concentration Cw is at instantaneous equilibrium with the local sorbed phase C ( K d is microscopic equilibrium coefficient). Note that the macroscopic distribution coefficient Kd is time dependent (see Eq. 19-78.)...

Xenon has been considered as the diffusing species in simulations of microporous frameworks other than faujasite (10-12, 21). Pickett et al. (10) considered the silicalite framework, the all-silica polymorph of ZSM-5. Once again, the framework was assumed to be rigid and a 6-12 Lennard-Jones potential was used to describe the interactions between Xe and zeolite oxygen atoms and interactions between Xe atoms. The potential parameters were slightly different from those used by Yashonath for migration of Xe in NaY zeolite (13). In total, 32 Xe atoms were distributed randomly over 8 unit cells of silicalite at the beginning of the simulations and calculations were made for a run time of 300 ps at temperatures from 77 to 450 K. At 298 K, the diffusion coefficient was calculated to be 1.86 X 10 9 m2/s. This... 

The only published work on the diffusion of gas in coals of different rank appears to be that of Bolt and Innes (2) who studied the diffusion of carbon dioxide from eleven samples of coal at 38°C. They found the diffusion coefficient to range from 3.5 to 9.2 x 10 8 sq. cm./sec., with no apparent correlation with coal rank. Diffusion data on coals of different rank at temperatures higher than 38°C. have only been reported by the present authors (6). It has been shown (7) that the diffusion of inert or noble gases from coal above room temperatures can be rigorously analyzed by using simple diffusion theory, and that true diffusion parameters of the micropore systems can be obtained. In this paper our measurements on the unsteady state release of argon from coals of various rank, over a temperature range, are reported. 

The experimental diffusion parameters, D /r., at 30°C. are presented in Table II for all the coals. Clearly, no correlation exists between diffusion parameter and rank. If r<> is taken as the average particle radius for the 200 X 325 mesh samples, an upper limit to the values of diffusion coefficient, D, is obtained. The diffusion coefficient ranges from 1.92 X 10 9 sq. cm./sec. for Kelley coal to 1.41 X 10"8 sk. cm./sec. for the Dorrance anthracite. Our previous studies on the change of D /n with particle size suggested that n is not necessarily the particle radius (7) but is a smaller distance related to the average length of the micropores in the particles. That is, the calculated... 

Diffusion coefficients of molecules in the lattice of H3PW12O40 are ca. 103 times less than those of molecules in the micropores of zeolites (235). Niiyama et al. reported that the effective diffusion coefficient is in the order of... 

Using the computer programs discussed above, it is possible to extract from these breakthrough curves the effective local mass transfer coefficients as a function of CO2 concentration within the stable portion of the wave. These mass transfer coefficients are shown in Figure 15, along with the predicted values with and without the inclusion of the surface diffusion model. It is seen that without the surface diffusion model, very little change in the local mass transfer coefficient is predicted, whereas with surface diffusion effects included, a more than six-fold increase in diffusion rates is predicted over the concentrations measured and the predictions correspond very closely to those actually encountered in the breakthrough runs. Further, the experimentally derived results indicate that, for these runs, the assumption that micropore (intracrystalline) resistances are small relative to overall mass transfer resistance is justified, since the effective mass transfer coefficients for the two (1/8" and 1/4" pellets) runs scale approximately to the inverse of the square of the particle diameter, as would be expected when diffusive resistances in the particle macropores predominate. 

Although the systems investigated here exhibited predominantly macropore control (at least those with pellet diameters exceeding 1/8" or 0.32 cm), there is no reason to believe that surface diffusion effects would not be exhibited in systems in which micropore (intracrystalline) resistances are important as well. In fact, this apparent surface diffusion effect may be responsible for the differences in zeolitic diffusion coefficients obtained by different methods of analysis (13). However, due to the complex interaction of various factors in the anlaysis of mass transport in zeolitic media, including instabilities due to heat effects, the presence of multimodal pore size distribution in pelleted media, and the uncertainties involved in the measurement of diffusion coefficients in multi-component systems, further research is necessary to effect a resolution of these discrepancies. 

Figure 2.37 Permeability coefficients as a function of the gas kinetic diameter in micro-porous silica hollow fine fibers [58]. Reprinted from J. Membr. Sci. 75, A.B. Shelekhin, A.G. Dixon and Y.H. Ma, Adsorption, Permeation, and Diffusion of Gases in Microporous Membranes, 233, Copyright 1992, with permission from Elsevier...

An experimental facility was described in Section 4.5.9 (see Figure 4.22) that was used to carry out the characterization of the groups present on the surface of a porous material or the channels and/or cavities of a microporous material applying the FTIR methodology. With this methodology, it is also possible to measure different diffusion coefficients in microporous materials with the help of the FTIR method [87-92], Here, a laboratory-assembled facility similar to that reported in Section 4.5.9 that has two manifolds (Figure 5.31) instead of one, for the introduction of the diffusing molecules, and thus has the capability to deliver two different hydrocarbons to the IR cell, is described [90],... 

With the help of Equation 5.107, as was previously done with Equation 5.86, we obtain a transport or chemical diffusion coefficient that is a result of Fick s laws. We now interpret the meaning of this coefficient if we consider diffusion in a microporous solid, as a special case of binary diffusion, where A is the mobile species and the diffusivity of the microporous framework atoms is zero, then, the frame of reference are the fixed coordinates of the porous solid consequently, we have a particular case of interdiffusion where the diffusion coefficient is simply the diffusivity of the mobile species [12,20],... 

For pore sizes ranging from 50 to 200 A, which are comparable to the sizes of the diffusing solute molecules and are called microporous, the diffusion of solutes may be substantially restricted by polymer materials. A diffusing molecule may be hindered from entering the pores and be chafed against the pores walls. Equation (6.30) incorporates these factors into the effective diffusion coefficient as ... 

The described treatment of mass transport presumes a simple, relatively uniform (monomodal) pore size distribution. As previously mentioned, many catalyst particles are formed by tableting or extruding finely powdered microporous materials and have a bidisperse porous structure. Mass transport in such catalysts is usually described in terms of two coefficients, a effective macropore diffusivity and an effective micropore diffusivity. 

A single effective diffusion coefficient cannot adequately characterize the mass transfer within a bidisperse-structured catalyst when the influence of the two individual systems is equally important. In a realistic model the separate identity of the macropore and micropore structures must be maintained, and the diffusion must be described in... 

The model fits adequately the experimental data allowing the extraction of the adsorption parameters and the diffusion coefficients for the transport inside the micropores. Table 3 reports these parameters for both samples. Similar diffusion parameters were found for both samples. Despite the difference in volume of the micropores as calculated by the HK method (Table 1), a big difference between Sbet and Sdr for each sample is found indicating problems of accessibility for nitrogen molecules at the low temperature at which the adsorption process is carried out (77 K). The micropore volume according to the DR method gives similar values for both samples (Table 1) The values of the diffusion parruneters found by modelling of the transient responses for both samples are very close. 

---