Pore size distribution bidisperse

Of course, these shortcomings of the Wakao-Smith flux relations induced by the use of equations (8.7) and (8.8) can be removed by replacing these with the corresponding dusty gas model equations, whose validity is not restricted to isobaric systems. However, since the influence of a strongly bidisperse pore size distribution can now be accounted for more simply within the class of smooth field models proposed by Feng and Stewart [49], it is hardly worthwhile pursuing this."... 

Figure 9.25 Models of granules of monodisperse particles characteristic psds (pore size distributions) are given below (a) uniform packing (b) bidisperse packing of aggregates of particles of similar sizes (c) same as (b) but the size of aggregates vary in a wide range.

The parameters D and Dk > whether for macro (denoted by subscript m) or for micro (denoted by subscript ju) regions, are normal bulk and Knudsen diffusion coefficients, respectively, and can be estimated from kinetic theory, provided the mean radii of the diffusion channels are known. Mean radii, of course, are obtainable from pore volume and surface area measurements, as pointed out in Sect. 3.1. For a bidisperse system, two peaks (corresponding to macro and micro) would be expected in a differential pore size distribution curve and this therefore provides the necessary information. Macro and micro voidages can also be determined experimentally. 

The original limestone used has a beige color. If the cross section of a sample with small conversions is examined the color seen throughout the stone is gray. The pore size distributions of these samples give monodispersed curves. At higher conversions the cross sections of the samples have two layers. The outer layer is white while the inner core is gray. The pore size distributions of these samples show bidisperse character. 

Two Co-Mo-alumina catalysts obtained from a commercial vendor as either marketed or special research samples were used in this study. The surface area and pore-size distributions (using the mercury penetration technique) were determined by an independent commercial laboratory. The catalyst properties are given in Table II. Note that the monodispersed (MD) and bidispersed (BD) catalysts have the same metallic composition and are chemically similar. 

Figure 3.1 Pore-size distribution in catalysts (r in nm) (a) monodisperse (b) bidisperse.

The final size and shape of the catalyst particles are determined in the shape formation process, which may also affect the pore size and pore size distribution. Larger pores can be introduced into a catalyst by incorporating in the mixture 5 to 15 % wood, flour, cellulose, starch, or other materials that can subsequently be burned out. As a result, bidisperse catalyst particles are obtained. 

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 available transport models are not reliable enough for porous material with a complex pore structure and broad pore size distribution. As a result the values of the model par ameters may depend on the operating conditions. Many authors believe that the value of the effective diffusivity D, as determined in a Wicke-Kallenbach steady-state experiment, need not be equal to the value which characterizes the diffusive flux under reaction conditions. It is generally assumed that transient experiments provide more relevant data. One of the arguments is that dead-end pores, which do not influence steady state transport but which contribute under reaction conditions, are accounted for in dynamic experiments. Experimental data confirming or rejecting this opinion are scarce and contradictory [2]. Nevertheless, transient experiments provide important supplementary information and they are definitely required for bidisperse porous material where diffusion in micro- and macropores is described separately with different effective diffusivities. 

Cylindrical pellets of four industrial and laboratory prepared catalysts with mono- and bidisperse pore structure were tested. Selected pellets have different pore-size distribution with most frequent pore radii (rmax) in the range 8 - 2500 nm. Their textural properties were determined by mercury porosimetry and helium pycnometry (AutoPore III, AccuPyc 1330, Micromeritics, USA). Description, textural properties of catalysts pellets, diameters of (equivalent) spheres, 2R, (with the same volume to geometric surface ratio) and column void fractions, a, (calculated from the column volume and volume of packed pellets) are summarized in Table 1. Cylindrical brass pellets with the same height and diameter as porous catalysts were used as nonporous packing. 

In Figure 6 calculated mean transport pore radii, , from both methods are compared with pore-size distribution from mercury porosimetry. It is seen that the obtained mean transport pore radii either agree with porosimetric peak, or, for bidispersed pellets, with the porosimetric peak of wider pores, or, are positioned between porosimetric peaks closer to the peak for wider pores. It was confirmed that the gas diffusion transport takes place predominantly through wider pores and the role of narrower pores depends on their size and amount. 

Bidisperse pore-size distribution with total porosity (%) 50... 

The relevance of interphase gradients distinguishes between two different classes of problems, and this is reflected on the type of boundary condition at the pellet s surface. It is known that specifying the value of the concentration (or temperature) at the surfece (Dirichlet boundary condition) may not be realistic, and thus finite external transfer effects have to be considered (in a Robin-type boundary condition) [72]. Apart from these, a large number of additional effects have also been considered. Some examples include the nonuniformity of the porous pellet structure (distribution of pore sizes [102], bidisperse particles [103], etc.), nonuniformity of catalytic activity [104], deactivation by poisoning [105], presence of multiple reactions [106], and incorporation of additional transport mechanisms such as Soret diffusion [107] or intraparticular convection [108]. 

It appears that for porous solids with monodisperse pore-size distribution the MTPM mean-pore radii and transport-pore distributions agree with the information from standard textural analysis. For porous solids with bidisperse pore-size distribution the MTPM mean-pore radii and transport-pore distributions are close to large pore sizes fiom standard textural analysis. 

Textural properties of six porous materials with mono- and bidisperse porous structure and a range of pore radii from nanometers to microns were determined by mercury porozimetry and helium pycnometry. The obtained pore-size distributions were compared with transport characteristics obtained independently from diffusion and permeation measurements. For three chosen samples the distribution of transport-pores was obtained from LEPP. 

The pore size distribution of the caibonized membranes for polymer concentrations of 12, 10 and 8 wt% are shown in Fig. 8.41a-c, respectively. The mean pore size of the membranes was in the microfiltration range, and the pore size increases with the polymer concentration. These results are in agreement with the fiber diameter of the membranes, which were noted to increase with the polymer concentration. A monodispersed pore size distribution was observed for 10 and 12 wt% polymer concentration, whereas a bidispersed pore size distribution was observed for the 8 wt% polymer concentration. It is hypothesized that this is due to the presence of beads in the 8 wt% polymer concentration, hence there are pores that are blocked by the beads, which results in a bidispersion of the pore size distribution. 

The random pore model of Wakao and Smith (1962) for a bidisperse pore structure may also be applied in order to estimate De. It was supposed that the porous solid is composed of stacked layers of microporous particles with voids between the particles forming a macroporous network. The magnitude of the micropores and macropores becomes evident from an experimental pore size distribution analysis. If Dm and Dp are the macropore and micropore diffusivities calculated from equations (4.9) and (4.10), respectively, the random pore model gives the effective diffusivity as... 

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