Macropore-micropore diffusion control

Fig. 6. Concentration profiles through an idealized biporous adsorbent particle showing some of the possible regimes. (1) + (a) rapid mass transfer, equihbrium throughout particle (1) + (b) micropore diffusion control with no significant macropore or external resistance (1) + (c) controlling resistance at the surface of the microparticles (2) + (a) macropore diffusion control with some external resistance and no resistance within the microparticle (2) + (b) all three resistances (micropore, macropore, and film) significant (2) + (c) diffusional resistance within the macroparticle and resistance at the surface of the...

Macropore-Micropore diffusion This is the case often called the bimodal diffusion model in the literature. In this case the two diffusion processes both control the uptake. This is expected when the particle size is intermediate. 

The half time can be calculated from eq.( 10.4-11a) by setting the fractional uptake to one half. Unlike the parallel pore and surface diffusion model discussed in Chapter 9 where the half time is proportional to the square of the particle radius, the half time of the bimodal diffusion model is proportional to R , where a is equal to 2 when macropore diffusion dominates the transport and a is equal to zero when micropore diffusion controls the uptake. An approximate expression for the half time for a bimodal diffusion model is given by Do (1990) ... 

Figure 10.4-3 shows the plot of y versus T and we see that the demarcation temperature is about 300K. For temperature greater than 400K, micropore diffusions controls the uptake while for temperature less than 200K, macropore diffusion controls. For temperatures between 200 and 400 K, both diffusion mechanisms control the uptake. 

We see that the half-time is proportional to the square of the pellet radius for this case of macropore diffusion control while it is independent of the pellet radius in the case of micropore diffusion control (eq. 10.4-31). 

The effect of crystal radius is shown in Figure 10.4-5 with the crystal radii being 1, 5 and 20 micron. The values of the parameter y corresponding to these radii are 25, 1.02 and 0.0635. These values indicate that macropore diffusion control in zeolite pellet with crystal size of 1 micron, micropore diffusion control in zeolite pellet with crystal size of 20 micron, and both diffusions control in zeolite pellet with crystal size of 5 micron. 

In bidisperse porous adsorbents such as zeolite pellets there are two diffusion mechanisms the macropore diffusion with time constant Rp /Dp and the micropore diffusion with time constant rc /Dc. Bidisperse porous models for ZLC desorption curves have been recently developed by Brandani [28] and Silva and Rodrigues [29]. In bidisperse porous adsorbents, it is important to carry out experiments in pellets with different sizes but with the same crystal size (different Rp, same rc) or pellets with the same size but with different crystals (same Rp, different rc). If macropore diffusion is controlling, time constants for diffusion should depend directly on pellet size and should be insensitive to crystal size changes. If micropore diffusion controls the reverse is true. The influence of temperature is also important when macropore diffusion is dominant the apparent time constant of diffusion defined by Rp2(H-K)/Dp is temperature dependent in the same order of K (directly related to the heat of adsorption) which is determined independently from the isotherm. The type of purge gas is... 

The rate of n-paraffin desorption generally controls the overall production rate (18, 19). The diffusion of n-paraffins in commercial 5A molecular sieves is reported to be controlled by either micropore diffusion or macropore diffusion, or both, depending on the molecular sieve crytal size and macropore size distribution of the adsorbent (20). A 5A molecular sieve adsorbent with smaller crystal size and optimum macropore size distribution would have a faster adsorption-desorption rate and, therefore, a higher effective capacity. 

Macropore and film diffusion are relatively well understood and are therefore not discussed in any detail in this voliune. In contrast, despite intensive study over the last 30 years, our imderstanding of micropore diffusion is still far from complete. At the micropore scale diffusive transport is largely controlled by steric interactions which are dominated by repulsive forces. The relative diameter of the micropore and the diffusing molecifie is clearly a critical variable. For small spherical or spheroidal molecules there is a clear... 

Chapter 10 deals with zeolite type particle, where the particle is usually in bidisperse form, that is small pores (channels inside zeolite crystal) are grouped together within a crystal, and the intercrystal void would form a network of larger pores. In other words, there are two diffusion processes in the particle, namely micropore diffusion and macropore diffusion. In the micropore network, only one phase is possible the adsorbed phase. Depending on the relative time scales between these two diffusion processes, a system can be either controlled by the macropore diffusion, or by micropore diffusion, or by a combination of both. Isothermal as well as nonisothermal conditions will be addressed in this chapter. 

If the time scale of diffusion in the micropore is very short compared to that in the macropore, we will have a macropore diffusion model with the characteristic length being the particle dimension. This case is called the macropore diffusion control. The model equations of this macropore diffusion case are similar to those obtained in Chapter 9 for homogeneous-type solids. The only difference is that in the case of macropore diffusion control for zeolite particles, there is no contribution of the surface diffusion. 

The solution given in eq. (10.4-1 la) reduces to simple solutions when either the macropore or micropore diffusion is the controlling mechanism, that is when y is less than unity, that is the time scale for the diffusion in the macropore is much smaller than that in the micropore, we would then expect the micropore diffusion would control the overall adsorption kinetics. In this case, we have the following half time... 

As we have mentioned above that the parameter which demarcates the micropore diffusion and the macropore diffusion is the parameter y. Let us now investigate its dependence on temperature to see how temperature would influence the controlling mechanism. The temperature dependence of relevant parameters in the parameter y is given below ... 

In the case of irreversible isotherm, the higher is the temperature, the larger is the value of y while for the case of linear isotherm as discussed earlier, the higher is the temperature the smaller is the value of y. Thus, macropore diffusion controls at high temperature for irreversible isotherm, while the micropore diffusion will control the uptake at high temperature in the case of linear isotherm. 

FIGURE 6.10. Theoretical uptake curves for a biporous adsorbent calculated according to Eqs. (6.39)-(6.42) showing the transition from micropore to macropore diffusion control and the difference in the shape of the uptake curve for intermediate values of (From ref. 16 with permission.)... 

Figure 4.6 Transition from micropore to macropore diffusion control figures on curves represent the parameter p. Discontinuous curves are for A = 0 and continuous curves are for A = 7 (see text) (source Lee 1978).

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. 

Generally, however, the aim is to avoid conditions leading to film diffusion control. This means that the focus is shifted towards transport processes that occur at the intermediate level (that is, in the mesopores and macropores within the macroparticle or pellet itself) and those which occur at the smallest dimensional level (viz., in the very micropores of the molecular sieve) [45, 89]. Within the mesopores and macropores between the primary zeolite crystallites transport will be dominated by molecular and ionic intercrystalline diffusion possibly coupled to surface diffusion processes, while, in the zeolite micropores themselves, intracrystalline diffusion occurs, also possibly coupled... 

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... 

For the detailed study of reaction-transport interactions in the porous catalytic layer, the spatially 3D model computer-reconstructed washcoat section can be employed (Koci et al., 2006, 2007a). The structure of porous catalyst support is controlled in the course of washcoat preparation on two levels (i) the level of macropores, influenced by mixing of wet supporting material particles with different sizes followed by specific thermal treatment and (ii) the level of meso-/ micropores, determined by the internal nanostructure of the used materials (e.g. alumina, zeolites) and sizes of noble metal crystallites. Information about the porous structure (pore size distribution, typical sizes of particles, etc.) on the micro- and nanoscale levels can be obtained from scanning electron microscopy (SEM), transmission electron microscopy ( ), or other high-resolution imaging techniques in combination with mercury porosimetry and BET adsorption isotherm data. This information can be used in computer reconstruction of porous catalytic medium. In the reconstructed catalyst, transport (diffusion, permeation, heat conduction) and combined reaction-transport processes can be simulated on detailed level (Kosek et al., 2005). 

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. 

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 transport properties across an MIP membrane are controlled by both a sieving effect due to the membrane pore structure and a selective absorption effect due to the imprinted cavities [199, 200]. Therefore, different selective transport mechanisms across MIP membranes could be distinguished according to the porous structure of the polymeric material. Meso- and microporous imprinted membranes facilitate template transport through the membrane, in that preferential absorption of the template promotes its diffusion, whereas macroporous membranes act rather as membrane absorbers, in which selective template binding causes a diffusion delay. As a consequence, the separation performance depends not only on the efficiency of molecular recognition but also on the membrane morphology, especially on the barrier pore size and the thickness of the membrane. 

Alternatively one can in principle derive both micropore and macropore diffusivities from measurements of the transient uptake rate for a particle (or assemblage of crystals) subjected to a step change in ambient sorbate pressure or concentration. The main problem with this approach is that the overall uptake rate may be controlled by several different processes, including both heat and extraparticle mass transfer as well as intraparticle or intracrystalline diffusion. The intrusion of such rate processes is not always obvious from a cursory examination of the experimental data, and the literature of the subject is replete with incorrect diffusivities (usually erroneously low values) obtained as a result of intrusion of such extraneous effects. Nevertheless, provided that intraparticle diffusion is sufficiently slow, the method offers a useful practical alternative to the Wicke-Kallen bach method. 

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