Micropore model, diffusion coefficients

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 ... 

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. 

Shelekhin et al. [92] have modeled this situation while in the transition region Xiao [86] describes the total micropore diffusion coefficient Dt as ... 

The above examples demonstrate that for a microporous material system where only a simple, single diffusion process occurs, the diffusion coefficients can be easily obtained by fitting the FR experimental data with the single diffusion process model using the least-square fitting routines. Compared with the PFG NMR and QENS techniques, the FR method is simpler and of low-cost and can follow a much wider range of diffusivities. 

In the majority of publications the effective diffusion coefficient takes into account the pore diffusion in the fluid phase (free or molecular diffusion and Knudsen diffusion). In the case of narrow windows of zeolite cages and large adsorptive molecules, micropore diffusion can be rate controlling. Some authors have extended the LDF model in the following way ... 

The FRFs for the pore-surface diffusion model were derived for the case of constant pore and surface diffusion coefficients. The first-order FRF can be derived analytically for all three particle geometries (the solution is analogous to the one obtained for the micropore diffusion model). On the other hand, the second- and higher-order FRFs can be derived analytically only for the slab particle geometry. These are the expressions for the first- and second-order FRFs, for Dp = const, Ds = const, and o- = 0 [56,58] ... 

The first- and second-order FRFs for the general model (micropore + macropore diffusion -F surface barrier -F film resistance), defining the relation between (q) and p, were derived analytically for the case of constant diffusion coefficients and slab geometry. The following expressions were obtained [61] ... 

Some simulation results of the first- and second-order FRFs for the nonisothermal micropore diffusion model with variable diffusion coefficient are given in Figure 11.16. They correspond to literature data for adsorption of CO2 on silicalite-1 [34], Ps= 10 kPa and Tg = 298 K, and to moderate heat transfer resistances [57], The functions H2,pp(co, —co), ff2,Tx(<. —co), and //2,px( , —co), which are identically equal to zero, are not shown. In Rgurc 11.16a we also give the FRFs corresponding to isothermal case (the parameter very large). Notice that for that case the Fp set of FRFs describes the system completely. 

This model assumes an isothermal mechanism of two parallel, independent diffusion processes in two different types of micropores. Each diffusion process is defined in the identical way as in Section III.B.3. If the diffusion coefficients differ enough, bimodal characteristic curves, such as those shown in Figure 11.17, are obtained. For constant diffusivities, the material balance equations for the two types of micropores are ... 

For the adsorption process governed by a single Fickian diffusion process, the time constant is defined as the ratio t = L /D, where L is the characteristic half-dimension and D the diffusion coefficient Accordingly, the time constant for the micropore diffusion model would be... 

The expressions for the FRFs for the isothermal micropore and pore-surface diffusion models were obtained for constant diffusion coefficients. If this assumption is not met, that is, if the concentration dependence of the diffusion coefficient has to be taken into account, the value estimated from the maximum of the — Imag(Fi p(diffusion coefficient corresponding to the steady-state concentration. 

B. Nonisothermal Micropore Diffusion Model 1. Estimation of the Micropore 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. 

It has to be pointed out that the model in the presented form is strictly speaking only correct for a homogeneous distribution of the catalytic activity and a constant diffusion coefficient within the particle, that is, for the ideal case of a material with one size of pore. For pellets with a pore size distribution the selectivities are not calculated accurately. Such a situation is quite common, for example, a pellet may contain macro-and micropores if it has been shaped from microparticles (with micropores) by compression (see Example 4.5.11 in Section 4.S.6.3). 

Interpretation of concentration dependence of micropore diffusion coefficient in terms of chemical potential driving force model... 

However, a pelletized or extruded catalyst prepared by compacting fine powder typically exhibits a bimodal (macro-micro) pore-size distribution, in which case the mean pore radius is an inappropriate representation of the micropores. There are several analytical approaches and models in the literature which represent pelletized catalysts, but they involve complicated diffusion equations and may require the knowledge of diffusion coefficients and void fractions for both micro- and macro-pores [31]. An easier and more pragmatic approach is to consider the dimensional properties of the fine particles constituting the pellet and use the average pore size of only the micropore system because diffusional resistances will be significantly higher in the micropores than in the macropores. This conservative approach will also tend to underestimate Detr values and provide an upper limit for the W-P criterion. 

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. 

The mass transfer coefficient can be calculated as the ratio of the estimated values of Op and Tp. 2, Isothermal Micropore and Pore-Surface Diffusion Models... 

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. 

Mass transfer through the external fluid film, and macropore, micropore and surface diffusion may all need to be accounted for within the particles in order to represent the mechanisms by which components arrive at and leave adsorption sites. In many cases identification of the rate controlling mechanism(s) allows for simplification of the model. To complicate matters, however, the external film coefficient and the intraparticle diffusivities may each depend on composition, temperature and pressure. In addition the external film coefficient is dependent on the local fluid velocity which may change with position and time in the adsorption bed. 

---