Micropore diffusion model

However, a solution in the Laplace domain has been derived by Kucera [30] and Kubin [31]. The solution cannot be transformed back into the time domain, but from that solution, these authors have derived the expressions for the first five statistical moments (see Section 6.4.1). For a linear isotherm, this model has been studied extensively in the literature. The solution of an extension of this model, using a macro-micropore diffusion model with external film mass transfer resistance, has also been discussed [32]. All these studies use the Laplace domain solution and moment analysis. 

The results obtained with the solid film linear driving force model, the pore diffusion model, and the micropore diffusion model were compared by Ruthven [14]. In contrast to linear chromatography, numerical solutions obtained with different models are different, especially in the initial time region. For moderate loadings i.e., for Req > 0-5), the differences remain small. As the loading increases, however, and Req becomes lower than 0.5, the differences between the numerical solutions derived from the various models studied increase. Accordingly, differences observed between experimental results and the profiles predicted by a kinetic model are most often due to the selection of a somewhat inappropriate model. 

FIGURE 11.8 First- and second-order FRFs for micropore diffusion model. (From Petkovska, M. and Do, D.D., Nonlinear Dyn., 21, 353 376, 2000. With permission.)... 

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] ... 

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. 

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... 

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

Multiscale models can be based only on continuum mathematical descriptions, e.g., as in the zeolite-based chemical reactor illustrated in Fig. 5, where the conservation equation describing the reactant transport along the reactor has a source term being calculated from the boundary condition of a macropore diffusion model with a source term calculated in turn from the boundary condition of a micropore diffusion model with a reaction kinetics source term [44]. 

Based upon the results shown in Figure 6 and the surface diffusion model given by equation (21), the values of ( Dos/is) which best fit the three systems were computed, both from the standpoint of assuming complete macropore control and from that of assuming the existence of micropore resistance as well, in the amount predicted from Figure 6. These values are given in Table III. 

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. 

Reverse osmosis, pervaporation and polymeric gas separation membranes have a dense polymer layer with no visible pores, in which the separation occurs. These membranes show different transport rates for molecules as small as 2-5 A in diameter. The fluxes of permeants through these membranes are also much lower than through the microporous membranes. Transport is best described by the solution-diffusion model. The spaces between the polymer chains in these membranes are less than 5 A in diameter and so are within the normal range of thermal motion of the polymer chains that make up the membrane matrix. Molecules permeate the membrane through free volume elements between the polymer chains that are transient on the timescale of the diffusion processes occurring. 

Steinberg et al. (1987) studied the persistence of 1,2-dibromoethane (EDB) in soils and found that low amounts of the organic were released with time, particularly if EDB had not been freshly added to the soil (Fig. 6.3). They suggested that the slow release rate was due to EDB being trapped in soil micropores where release is influenced by extreme tortuosity and/or steric restrictions. It was estimated that based on a radial diffusion model, 23 and 31 years would be required for a 50% equilibrium in EDB release to occur from two Connecticut soils. The previous studies point out that while sorption of pesticides is usually rapid and often reversible in the laboratory, extraction from field soils is extremely slow and often requires multiple extractions or even chemical dissolution of the soil matrix. 

Kinetics was determined by fitting the experimental uptake curves with a bidisperded model. Only molecular diffusion was considered for the transport in macropore. The micropore diffusion in CMS samples was expressed by a dual model of the following form [3,5] ... 

Sorption Kinetics. The adsorption and desorption data were analyzed in terms of a model based on the following main assumptions. Micropore diffusion within the sieve crystals is the rate-controlling process. Diffusion may be described by Fick s law for spherical particle geometry with a constant micropore diffusivity. The helium present in the system is inert and plays no direct role in the sorption or diffusion process. Sorption occurs under isothermal conditions. Sorption equilibrium is maintained at the crystal surface, which is subjected to a step change in gas composition. These assumptions lead to the following relation for the amount of ethane adsorbed or desorbed by a single particle as a function of time (Crank, 4). 

The internal mass transfer is modeled with Fick s diffusion inside the (macro) pores (Eq. 6.82) as well as surface or micropore diffusion in the solid phase (Eq. 6.83). Note that Eqs. 6.82 and 6.83 no longer include the number of particles (Eq. 6.20) and therefore represent the balance in one particle. 

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

Fundamentals of sorption and sorption kinetics by zeohtes are described and analyzed in the first Chapter which was written by D. M. Ruthven. It includes the treatment of the sorption equilibrium in microporous sohds as described by basic laws as well as the discussion of appropriate models such as the Ideal Langmuir Model for mono- and multi-component systems, the Dual-Site Langmuir Model, the Unilan and Toth Model, and the Simphfied Statistical Model. Similarly, the Gibbs Adsorption Isotherm, the Dubinin-Polanyi Theory, and the Ideal Adsorbed Solution Theory are discussed. With respect to sorption kinetics, the cases of self-diffusion and transport diffusion are discriminated, their relationship is analyzed and, in this context, the Maxwell-Stefan Model discussed. Finally, basic aspects of measurements of micropore diffusion both under equilibrium and non-equilibrium conditions are elucidated. The important role of micropore diffusion in separation and catalytic processes is illustrated. 

Whitaker S. 1983. Diffusion and reaction in a micropore-macropore model of a porous medium, Lat. Am. J. Appl. Chem. Eng., 13, 143-183. 

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