Macropores, surface diffusion

The model chosen to represent the effects of surface diffusion along the walls of the macropores is a modification of the Sladeck-Gilliland model (1G, 11), which has the form ... 

The first hypothesis seems unlikely to be true in view of the rather wide variation in the ratio of carbon dioxide s kinetic diameter to the diameter of the intracrystalline pores (about 0.87, 0.77 and 0.39 for 4A, 5A and 13X, respectively (1J2)). The alternative hypothesis, however, (additional dif-fusional modes through the macropore spaces) could be interpreted in terms of transport along the crystal surfaces comprising the "walls" of the macropore spaces. This surface diffusion would act in an additive manner to the effective Maxwell-Knudsen diffusion coefficient, thus reducing the overall resistance to mass transfer within the macropores. 

Incidentally, these features cannot be accounted for by assuming different values of macropore radius or tortosity factor in the predictive equations. Even with the assumptions of negligible Knudsen resistance (rpore- ) and no tortuosity (rw = D, the predicted macropore resistances (excluding surface effects) would be lowered by only forty percent, which is still insufficient to account for the low LUB values, at least in the 5A and 13X systems. There appears, therefore, to be a fairly strong case for the presence of a surface diffusion effect in these systems, with the possibility of such an effect in the CO2/ air/4A system as well. 

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. 

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. 

Surface diffusivity along the walls of the particle macropores, cm /hr... 

Diffusion in macropores occurs mainly by the combined effects of bulk molecular diffusion (as in the free fluid) and Knudsen flow, with generally smaller contributions from other mechanisms such as surface diffusion and Poiseuille flow. Knudsen flow, which has the characteristics of a diffusive process, occurs because molecules striking the pore wall are instantaneously adsorbed and re-emitted in a random direction. The relative importance of bulk and Knudsen diffusion depends on the relative frequency of molecule-molecule and molecule-wall collisions, which in turn depends on the ratio of the mean free path to pore diameter. Thus Knudsen flow becomes dominant in small pores at low pressures, while in larger pores and at higher pressures diffusion occurs mainly by the molecular mechanism. Since the mechanism of diffusion may well be different at different pressures, one must be cautious about extrapolating from experimental diffusivity data, obtained at low pressures, to the high pressures commonly employed in industrial processes. 

Ma et al. [104] attributed a decrease in diffusivity with an increase in initial concentration to pore diffusion effects. Because zeolites are bi-dispersed sorbents, both surface and pore diffusions may dominate different regions. In micropores, surface diffusion may be dominant, while pore diffusion may be dominant in macropores. This, therefore, supports the use of a lumped parameter (De). To explore further the relative importance of external mass transfer vis-a-vis internal diffusion, Biot number (NBl — kf r0/De) was considered. Table 9 summarizes the NBi values for the four initial concentrations. The NBi values are significantly larger than 100 indicating that film diffusion resistance was negligible. 

Monoliths that were anodized extensively (72) had an anodization thickness of up to 25 pm with a BET surface area of 40 m /g, which is sufficient for many applications. However, because this layer contained only mesopores (pore diameters up to 20 nm) and no macropores, internal diffusion limitations can easily be a problem. An extensive report on the anodization of aluminum monoliths, with the aim of using the anodization layer as catalyst support, was provided by Burgos et al. (73). 

Moulijn et al. [46] extended the Kubin-KuCera model by incorporating surface diffusion. They derived the solution of their new model in the Laplace domain and calculated explicit expressions for the first and the second moments of the band. Haynes and Sarma [32] obtained the expressions for the first and second moments for the band solution of a general rate model including macropore... 

Van Den Broeke and Krishna [56] compared the calculated and the experimental breakthrough curves of single components and of mixtures containing methane, carbon dioxide, propane, and propene on microporous activated carbon and on carbon molecular sieves. They ignored the external mass transfer kinetics and assumed that there is local equilibrium for each component between the pore surface and the stagnant fluid phase in the macropores. They also assumed that the surface-diffusion contribution is much larger than that of pore diffusion and they neglected pore diffusion. They used in their calculations three different... 

Kaczmarski et al. used a similar model for the calculation of the band profiles of the enantiomers of 1-indanol on a chiral phase in HPLC [29,57]. These authors ignored the external mass transfer and assumed that local equilibrium takes place for each component between the pore surface and the stagnant fluid phase in the macropores (infinite fast kinetics of adsorption-desorption). They also assumed that surface diffusion contribution is much faster than pore diffusion and neglected pore diffusion entirely. Instead of the single file Maxwell-Stefan diffusion, these authors used the generalized Maxwell-Stefan diffusion (see Chapter 5).The calculation (see below) requires first the selection of equations to calculate the surface molecular flux [29,57,58],... 

Some typical results with capillary condensation and surface diffusion in meso- and macroporous... 

The magnitudes of gas diffusivities by different mechanisms follow the order meso-macropore gas diffusivity (Dp) > surface diffusivity (D ) in those pores > micropore diffusivity (Dm). For example. Dp... 

More generally, adsorption is controlled by a combination of transport mechanisms in macropores or micropores, depending on the pore size distribution, the sorbate concentration, the isotherm, and other conditions. The combining bulk diffusion with surface diffusion gives the effective macropore diffusivity ... 

The major contributor to the MTZ, espacially in vapor-phase ndsorplion, is diffusion resistance.15 30 Adsorptive diffusion is the sum of several different mechanisms. There is belk diffusion through the film around the adsorbent panicle. Then the molecules must ovsecoine the macropore diffusion resistance that is usually characterized as Maxwell diffusion, in those pores that have dimensions on Ihe order of molecular dimensions, the transport is by Knud sen diffusion. In these micropures. (he diffusion cna also occur by surface diffusion. Typically, the entire process is characterized by a single overall diffusion coefficient. 

In macropores which are wide in eomparison to a molecule diameter the adsorptive can be transported by convection, molecular diffusion in the fluid phase, and surface diffusion in the adsorbate. When local differences of the total pressure are present in a pore a flow is initiated A Poiseuille flow for < 0.01, a slip velocity for 0.01 molecular flow for Kn > 1. With the exception of rapid loading or deloading processes in a fixed bed (often encountered in pressure swing adsorption PSA) the mass transport by differences of the total pressure (Poiseuille flow and convection) can be neglected. Then the adsorptive is transported by the following mechanisms ... 

Thus, the temperature dependencies of surface diffusion and simple molecular diffusion should be different. In macropores, diffusion may occur by several different mechanisms. 

Apart from diffusion in continuum phase, the transport of surface-adsorbed molecules and capillary condensate takes place in meso- and macroporous media. In order to model transport of adsorbable vapor at elevated pressure, it is necessary to consider the type of adsorption occurring monolayer adsorption, multilayer adsorption, or capUlary condensation [38]. Models for surface diffusion have been proposed... 

The macropore diffusion case is basically dealt with in Chapter 9 where we dealt with parallel diffusion in homogeneous solids. Since surface diffusion on the exterior surface of the zeolite microparticle is almost negligible due to the very low capacity on those surfaces, we can ignore the surface diffusion contribution in the analysis of the last chapter when we apply such analysis to a zeolite-type particle. 

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. 

Note that there is no surface diffusion in the macropore as it is assumed that the adsorption on the exterior surface of the zeolite crystal is negligible compared to the adsorptive capacity within the zeolite crystal. 

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) ... 

Pore diffusion may occur by several different mechanisms depending on the pore size, the sorbate concentration, and other conditions. In fine micro-pores such as the intracrystalline pores of zeolites, the diffusing molecule never escapes from the force field of the adsorbent surface and transport occurs by an activated process involving jumps between adsorption sites. Such a process is often called surface diffusion, but the implication of a two-dimensional surface is unnecessarily restrictive since the micropore structure in a zeolite crystal is often three-dimensional. The more general terms micropore or intracrystalline diffusion are therefore used here to describe transport in such systems, while diffusion in larger pores such that the diffusing molecule escapes from the surface field is referred to as macropore diffusion. This distinction between micropore and macropore diffusion is useful since, in a zeolitic adsorbent, the diameter of the intracrystalline... 

Macropore diffusion has been widely studied in connection with its influence on the overall kinetics of heterogeneous catalytic reactions. Four distinct mechanisms of transport may be identified molecular diffusion, Knudsen diffusion, Poiseuille flow, and surface diffusion. The effective macropore diffusivity is thus a complex quantity which often includes contributions from more than one mechanism. Although the individual mechanisms are reasonably well understood, it is not always easy to make an accurate a priori prediction of the effective diffusivity since this is strongly dependent on the details of the pore structure. 

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