Transport coefficients micropore diffusion coefficient

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],... 

The predictive calculation of micropore diffusion coefficients is difBcult because the mechanisms which lead to a molecule transport are not sufficiently known. When the molecules preferably move in the fluid phase it can be assumed that the diffusion coefficient is a function of the molecule velocity and the width of the (zeolitic) micropores. Molecules will be adsorbed on the pore walls and have to be accelerated to return into the fluid phase. So far the transport mechanism is an activated process according to the relationship (Sehweighait 1994)... 

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... 

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 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... 

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. 

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. 

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. 

The dynamics of methane, propane, isobutane, neopentane and acetylene transport was studied in zeolites H-ZSM-5 and Na-X by the batch frequency response (FR) method. In the applied temperature range of 273-473 K no catalytic conversion of the hydrocarbons occurred. Texturally homogeneous zeolite samples of close to uniform particle shape and size were used. The rate of diffusion in the zeolitic micropores determined the transport rate of alkanes. In contrast, acetylene is a suitable sorptive for probing the acid sites. The diffusion coefficients and the activation energy of isobutane diffusion in H-ZSM-5 were determined. 

The diffusion coefficient (or diffiisivity) and viscosity represent transport properties which affect rates of mass transfer. In general, these properties are at least an order of magnitude higher and lower, respectively, compared with liquid solvents. This means that the diffusion of a species through an SCF medium will occur at a faster rate than that obtained in a liquid solvent, which implies that a solid will dissolve more rapidly in an SCF. In addition, an SCF will be more efficient at penetrating a microporous solid structure. However, this does not necessarily mean that mass transfer limitations will always be absent in an SCF process. For example, in the extraction of a solute from a liquid to an SCF phase, the resistance to diffusion in the liquid phase will probably control the overall rate of mass transfer. Stirring will therefore continue to be an important factor in such systems. 

The pores structure is affected by the presence of mineral additions in two different ways. At first, the addition particles reduce the voids of granular skeleton, and secondly, due to pozzolanic properties they cause C-S-H content increase in cement paste. As a consequence, mesopores i.e. capillary pores content decreases and the ratio of gel pores, which belong to micropores, is increasing. The micropores contribute marginally to the flow of liquid through the paste, but they participate in diffusion. Diffusion is concerning the transport on the level of ions and molecules. This process depends on the effective diffusion coefficient and was discussed in Chap. 5. 

The major contributor to the MTZ, especially in vapor- ffiase adsorption, is diffusion resistance. " Ad-soiptive diffusion is the sum of sevei different mechuisms. There is bulk diffusion through the film around the adsoibent particle. Then the molecules most overcome the macropoie difiiision resistance that is usually characterized as Maxwell diffusion. In fliose pores that have dimensions on the ordo of molecular dimensions, the transport is by Knudsen diffusion. In these micropores, the diffusion can also occur by sorfece diffusion. Typically, the entire process is characterized by a single overall diffusion coefficient. 

Modeling of transient data. The model takes mass transport into account at two different levels Knudsen flow in the interstitial voids of the bed and in the macropores of the matrix, lumped into one diffusion coefficient and an activated diffusion process inside the micropores of the zeolite. The reversible sorption between the gas phase and zeolite sorbate... 

An industrial DMTO fluidized bed catalyst pellet is basically composed of SAPO-34 zeofite particles and catalyst support (or matrix). The pores of zeolite particles and matrix are interconnected as a complex network. The pores inside zeofite particles are typically micropores (less than 2 nm) and the matrix normally has either mesopores (2-50 nm) or macropores (>50 nm), or both (Krishna and Wesselingh, 1997). The bulk diffusion coefficients in the meso- and macropores might be several orders of magnitude larger than surface diffusion coefficients in the micropores. Kortunov et al. (2005) found that the diffusion in macro- and mesopores also plays a crucial part in the transport in catalyst pellets. Therefore, other than a model for SAPO-34 zeofite particles, a modeling approach for diffusion and reaction in MTO catalyst pellets, which are composed of SAPO-34 zeofite particles and catalyst support, is needed. 

The data did not permit an estimation of the activation energy for the micropore diffusion significantly different from zero. Therefore, the model applied has only 3 adjustable parameters the diffusivity in the micropores, the Henry coefficient for reversible sorption on the surface and the adsorption enthalpy. The Knudsen diffiisivities for the extraparticle transport were determined from independent experiments. The other estimated parameters together with their 95% confidence interval are presented in Table 2. 

Fig.3. Diffiision coefficients D as derived from a fit of the experimental data in Fig.2 with the solution of the diffusion equation for cylindrical sample (open circles) in addition, values corrected for the amount of gas to be transported (D d>pseudo, fuU circles) are depicted. For comparison the theoretical diffusion coefficients for gas phase diffusion in cylindrical pores are also included (dashed lines) hereby the value of the macroporosity (50%) and a tortuosity factor of 3 are taken into account. The macroporosity was calculate fix)m the bulk density of the sample and the micropore volume (macroporDsity=total porosity—microporosity= 86 % - 36%).

The surface contribution clearly correlates with the isotherm resulting from the adsorption in the micropores. Comparing samples with different macropore sizes, the absolute value of the (total, transient) diffusion coefficients is found to be proportional to the macropore size of the carbons this indicates a transport that is dominated by molecular diffusion coupled with stirface transport at low relative pressures. 

In Example 3.4.5 involving microporous cellulose membranes used in hemodialysis, consider the situation where there are boundary layer resistances on two sides of the membrane. For the transport of a microsolute through the membrane from an aqueous solution, Lane and Higgle (1959) have found that, for their membrane, feed and permeate aqueous solutions, the mass-transfer coefficients on each of the feed and permeate are given by kif= kip= lOOODjjcm/min, where Da is the diffusion coefficient of solute i in water in cm /s. Obtain an estimate of the overall mass-transfer coefBcient of Na2S04 for such a membrane system with boundary layer resistances. (Ans. K = 3 x 10 cm/s.)... 

The diffusivity (D) defined in this way is not necessarily independent of concentration. It should be noted that for diffusion in a binary fluid phase the flux (/) is defined relative to the plane of no net volumetric flow and the coefficient D is called the mutual diffusivity. The same expression can be used to characterize migration within a porous (or microporous) sohd, but in that case the flux is defined relative to the fixed frame of reference provided by the pore walls. The diffusivity is then more correctly termed the transport diffusivity. Note that the existence of a gradient of concentration (or chemical potential) is implicit in this definition. 

The external fluid film resistance (the corresponding mass-transfer coefficient ki from equations (3.4.32a,b)) is in series with the intraparticle transport resistance. The flux of a species through a porous/mesoporous/microporous adsorbent particle consists, in general, of simultaneous contributions from the four transport mechanisms described earlier for gas transport in Section 3.1.3.2 (for molecular diffusion, where (Dak/T>ab) 2> 1) ... 

Pore sixe distribution data obtained from gas desorption (Barret et al. 1951) and mercury porisimetry experiments together with a knowledge of adsorbate molecular size thus enables the mode of diffusive transport to be ascertained. It should be noted that both molecular and Knudsen diffusion may occur in the same porous medium when the porous medium contains both macropores and micropores (revealed from an analysis of a bimodal pore size distribution curve). Unconstrained molecular diffusion. Dm, and Knudsen diffusion, Dk, coefficients are subsequently calculated from formulae derived from transport properties of fluids (gaseous and liquid) and the kinetic theory of gases. The molecular diffusivity for a binary gas mixture of A and B is evaluated from the Chapman-Enskog theory (Chapman and Cowling 1951) equation... 

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